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[ [ "Bounds on the recurrence probability in periodically-driven quantum\n systems" ], [ "Abstract Periodically-driven systems are ubiquitous in science and technology.", "In quantum dynamics, even a small number of periodically-driven spins leads to complicated dynamics.", "Hence, it is of interest to understand what constraints such dynamics must satisfy.", "We derive a set of constraints for each number of cycles.", "For pure initial states, the observable being constrained is the recurrence probability.", "We use our constraints for detecting undesired coupling to unaccounted environments and drifts in the driving parameters.", "To illustrate the relevance of these results for modern quantum systems we demonstrate our findings experimentally on a trapped-ion quantum computer, and on various IBM quantum computers.", "Specifically, we provide two experimental examples where these constraints surpass fundamental bounds associated with known one-cycle constraints.", "This scheme can potentially be used to detect the effect of the environment in quantum circuits that cannot be classically simulated.", "Finally, we show that, in practice, testing an $n$-cycle constraint requires executing only $O(\\sqrt{n})$ cycles, which makes the evaluation of constraints associated with hundreds of cycles realistic." ], [ "Introduction", "Recent developments in stochastic and quantum thermodynamics of microscopic systems provide a plethora of “second-law-like” constraints: from various fluctuation theorems [1], [2], [3], to thermodynamic uncertainty relations [4], [5], [6], [7], resource theory [8], [9], global passivity [10], and passivity deformation [11].", "Presently, the utility of these additional constraints is to a large extent unclear.", "Among other things, these theories differ in the level of difficulty in measuring the quantities they constrain.", "Resource theory requires full tomography of the energy populations in the beginning and at the end of the evolution.", "Fluctuation theorems require trajectory information which amounts to process tomography at the population level.", "That is, the output distribution is recorded for each possible input in the computational basis, which means the method is not scalable with system size.", "Fluctuation theorems provide equalities rather than inequalities, but they are difficult to measure and not scalable as the system size grows.", "Global passivity inequalities [10] involve expectation values of higher order moments of the energy.", "Since theses expectation values are local, the number of experiments needed to obtain the desired statistical uncertainty grows only polynomially in the system size (see [12], [13] and in particular the appendix of [14] in the context of global passivity).", "Unfortunately, these expectation values presently do not have a clear operational meaning in terms of energy flows.", "Passivity deformation [11] addresses a wider range of observables and produces tighter bounds compared to global passivity.", "In particular, it can also handle the case where some of the objects are initially at zero temperature - a scenario where the second law in microscopic systems and other constraints provide trivial information that cannot be exploited.", "What is common to all these methods is that they provide no information when all the elements in the setup are at zero temperature.", "That is, when the setup is initially in a pure state.", "Several studies provide the aforementioned constraints with some operational meaning by applying them to presently available quantum computers.", "In [14] and [15] the goal was to diagnose the operation of the device and in [16] to understand which type of heat machine describes the interaction of the quantum computer with the environment.", "If the quantum computer is well isolated from the environment and the initial condition fits the studied thermodynamic constraints (e.g.", "the qubits are initially in some thermal state), the constraints mentioned above hold.", "Indeed, all these constraints require some form of isolation.", "While some permit only interactions with an implicit bath at a well-known temperature, others require unital evolution, which can be thought of as the operation of a random unitary on the setup.", "For example, decoherence is a unital map that can be described using a statistical mixture of unitaries.", "Since fluctuation theorems, global passivity, and passivity deformation require unital maps to hold, a deviation from this type of map will indicate the presence of a coupling to the environment which is more severe than decoherence interaction, e.g.", "an actual heat leak that reduces the entropy of the setup, like spontaneous emission.", "Thus, an interesting application for modern microscopic theory is the detection of coupling to the environment via violation of various bounds.", "Notably, no information on the circuit is needed, and therefore these thermodynamic methods produce “black box” tests that do not depend on the complexity of the circuit.", "Presently the most common method for diagnosing quantum computers is randomized benchmarking [17], [18], [19], [20].", "In this method, a random unitary and its inverse are sequentially implemented.", "In a perfect device, this construction yields the identity operator and the survival probability, i.e.", "the probability to remain in the initial state, is one.", "Unlike the aforementioned thermodynamic methods, this methods requires information on the circuits since the inverse transformation has to be constructed as well.", "Crucially, randomized benchmarking methods cannot distinguish between coherent errors (or unitary errors) and errors created by coupling to an environment (see Appendix I).", "A coherent error means that a unitary has been successfully implemented but it is different from the target unitary.", "These errors can be fixed by tuning, calibration, or by adding additional unitary operations.", "In contrast, errors generated by the environment are much more difficult to address and require a different set of solutions that includes changes in the fabrication process of the device.", "Thus, it is appealing to have complementary methods that can indicate the contribution of the environment in the error observed in the randomized benchmarking method.", "Our constraints differ from thermodynamic (or thermodynamically inspired) constraints in that they do not require a mixed state.", "Such states are often difficult to create in a system that is rather well isolated from the environment.", "Additionally, most quantum algorithms require pure initial state.", "Hence, the default input state in quantum computers is pure.", "To use the thermodynamic methods mentioned above the mixed state has to be artificially created by pre-processing circuits (see [21] for two different methods of creating a mixed state).", "A purification-based approach for creating Gibbs states is described in [22]).", "The creation of the initial state ensemble consumes a lot of resources and scales exponentially with the number of qubits.", "Furthermore, the use of mixed states can degrade the capability to detect hot environments.", "In [11] a bound on the maximal detectable temperature using passivity-based methods was obtained.", "For these two reasons, a method that works with pure states can be quite useful.", "On top of the potential practical applications, finding nontrivial constraints on the dynamics of pure states is interesting from a foundational point of view.", "Finally, note that these bounds are expressed in terms of observables that do not depend on the complexity of the studied circuit or unitary, making it a scalable and widely applicable black box test.", "Finding non-trivial and general constraints on observables undergoing unitary evolution without knowing anything about the dynamics is very difficult.", "Our method circumvents this difficulty by exploiting periodicity.", "We run the circuit of interest multiple times and study the survival probability as a function of the number of cycles.", "The information obtained from survival probabilities at different time points can be used to derive bounds that are tighter and hence more predictive than other constraints that are based on just two time points: the beginning and the end of the evolution.", "In particular, we show that this approach circumvents the zero-temperature problem and the previously mentioned bound on the detectability of hot environments.", "After a short review of the global passivity principle we proceed to derive our main finding: the periodicity inequalities.", "For each number of cycles there is a continuous set of constraints.", "A recipe is presented for obtaining the optimal bound for the detection of non unital errors in a given number of cycles.", "Next, we study an interesting features of a family of bounds which are economical in terms of resources.", "In the next part we present experimental results obtained from various IBM quantum processors and a trapped ion quantum computer (TIQC).", "Each example shows other aspects of these bounds.", "The first experiment shows that it is possible to detect incoherent errors with pure states.", "In this specific experiment, the environment is small and artificially-made (a qubit).", "The added value of this artificial environment for the purpose of our demonstration is that it can be coupled or decoupled from the system.", "Interestingly, in this specific example we find that a quantum initial condition has an advantage over classical inputs.", "In the second experiment we aim to detect intrinsic heat leaks of the processors.", "Now the environment is realistic, but not controllable.", "In some of the processors, we observe directly a violation of the bounds.", "In other cases, we observe violations by employing extrapolation based on the resource efficiency of our inequalities.", "The third experiment deals with thermal initial conditions and a hot environment.", "We demonstrate that since the periodicity inequalities use more than two points they can circumvent the bounds imposed by the passivity deformation framework on the detectability of a hot environment.", "The experiment is three-cycle long and shows that the optimized 4-point bound is essential for detecting the environment.", "Finally, in our forth example we compare the performance of a TIQC to the IBM superconducting quantum computer.", "The test is done on a specific class of circuits where the $S_{n}$ inequalities take a very simple form.", "Our periodicity inequalities follow from the global passivity inequalities that were first introduced in [23] and rediscovered and further explored in [10].", "For an initial density matrix $\\rho _{0}$ and a final density matrix $\\rho _{f}$ that is unitarily related to the initial state ($\\rho _{f}=U\\rho _{0}U^{\\dagger }$ ), the global passivity bound is $tr[F(\\rho _{0})(\\rho _{0}-\\rho _{f})]\\ge 0,$ for any unitary map $U$ , and any function $F(x)$ which satisfies $\\frac{dF(x)}{dx}\\ge 0$ for $0\\le x\\le 1$ .", "It is instructive to re-derive the following simpler version: $tr[\\rho _{0}(\\rho _{0}-\\rho _{f})]\\ge 0.$ This specific form can be proven without a passivity argument by using the $L2$ norm and the unitality of the dynamics (unitary dynamics maps the identity matrix to itself).", "Starting with $tr[(\\rho _{0}-\\rho _{f})^{2}] & \\ge & 0,$ $tr[\\rho _{0}^{2}]+tr[\\rho _{f}^{2}]-2tr[\\rho _{0}\\rho _{f}]\\ge 0,$ from unitality $tr[\\rho _{0}^{2}]\\ge tr[\\rho _{f}^{2}]$ , and therefore we obtain Eq.", "(REF ).", "$tr[\\rho _{0}^{2}]\\ge tr[\\rho _{f}^{2}]$ follows from the fact that for unital maps denoted by $\\rho _{f}=\\mathcal {M}(\\rho _{0})$ , $\\rho _{0}$ majorizes $\\rho _{f}$ (Schur concavity) [24].", "Next we show that $tr[\\rho _{0}^{2}]\\ge tr[\\rho _{f}^{2}]$ holds also for the traceless version of $\\rho $ .", "Writing $\\rho _{0}=a_{0}I+\\sum a_{i}Z_{i}$ where $Z_{i}$ is some traceless orthogonal basis $tr(Z_{i}Z_{j\\ne i})=0$ and applying unitality we get $\\mathcal {M}(\\rho _{0})=a_{0}I+\\sum a^{\\prime }_{i}Z_{i}$ (i.e.", "$a^{\\prime }_{0}=a_{0}$ ).", "Using this in $tr[\\rho _{0}^{2}]\\ge tr[\\rho _{f}^{2}]$ we get $a_{0}^{2}+\\sum _{i=1}\\left|a_{i}\\right|^{2}\\ge a_{0}^{2}+\\sum _{i=1}\\left|a_{i}^{\\prime }\\right|^{2}$ and therefore: $\\sum _{i=1}\\left|a_{i}\\right|^{2}\\ge \\sum _{i=1}\\left|a_{i}^{\\prime }\\right|^{2}.$ This implies that $tr[\\rho _{0}^{2}]\\ge tr[\\rho _{f}^{2}]$ also holds for traceless (and Hermitian) matrices.", "Consequently, (REF ) holds for any trace.", "That is, if $r$ is some Hermitian operator (potentially traceless) and $r_{f}=\\mathcal {M}(r_{0})$ then $tr[r_{0}(r_{0}-r_{f})]\\ge 0.$" ], [ "Periodicity inequalities", "Next we consider a periodically driven system where the density matrix after each cycle satisfies $\\rho _{n+1}=\\mathcal {M}(\\rho _{n})$ from linearity the object $r_{0}=\\sum _{i=0}^{N}\\alpha _{i}\\rho _{i},\\alpha _{i}\\in \\mathbb {R}$ also satisfies $\\mathcal {M}(r_{n})=r_{n+1}=\\sum \\alpha _{i}\\rho _{i+1}$ .", "Note that $r_{n}$ is Hermitian but its trace can take any value.", "In the following, we will use the term 'stencil' for $r$ .", "Since $r_{0}$ is Hermitian and evolves by a unital map it holds that $tr[r_{0}(r_{0}-r_{M})]\\ge 0.$ Consider the simple case where $r_{0}=\\rho _{1}-\\rho _{0}$ , and $r_{M}=r_{1}=\\rho _{2}-\\rho _{1}$ , eq.", "(REF ) leads to $0 & \\le & tr[(\\rho _{1}-\\rho _{0})(-\\rho _{0}+2\\rho _{1}-\\rho _{2})]\\nonumber \\\\& = & tr[\\rho _{0}^{2}]+2tr[\\rho _{1}^{2}]-3tr[\\rho _{0}\\rho _{1}]\\nonumber \\\\& & -tr[\\rho _{1}\\rho _{2}]+tr[\\rho _{0}\\rho _{2}].$ We note that the quantities $R_{n}=tr[\\rho _{0}\\rho _{n}]$ can be interpreted as the expectation of the observable $\\rho _{0}$ after $n$ cycles.", "Furthermore, using the $tr[\\rho _{0}^{2}]\\ge tr[\\rho _{n}^{2}]$ (unitality) it holds that $3R_{0}-3R_{1}-tr[\\rho _{1}\\rho _{2}]+R_{2}\\ge 0.$ However, $-tr[\\rho _{1}\\rho _{2}]$ is presently in an inconvenient form as both $\\rho _{1}$ and $\\rho _{2}$ are unknown.", "To overcome this we restrict ourselves to periodic unitary operations for which it holds that $tr[\\rho _{1}\\rho _{2}]=tr[\\rho _{1}U\\rho _{1}U^{\\dagger }]=tr[U^{\\dagger }\\rho _{1}U\\rho _{1}]=tr[\\rho _{0}\\rho _{1}]=R_{1}$ .", "More generally for periodic unitary operations $tr[\\rho _{n}\\rho _{n+m}] & = & tr[\\rho _{n}U^{n}\\rho _{m}U^{\\dagger n}]\\nonumber \\\\& = & tr[U^{\\dagger n}\\rho _{n}U^{n}\\rho _{m}]=R_{m}$ The same holds for the following unital maps: $\\rho ^{\\prime }_{n}=tr_{b}U^{n}(\\rho _{0}\\otimes \\frac{I_{b}}{N_{b}})U^{n\\dagger },$ where $I_{b}$ is the identity operator of some ancilla and $N_{b}$ is the dimension of that ancilla's Hilbert space.", "For readers that are familiar with “noisy operation” [25] we point out here we do not take the partial trace at the end of the operation.", "One can verify that: $tr[\\rho ^{\\prime }_{n}\\rho ^{\\prime }_{n+m}]=tr[\\rho {}_{0}\\rho {}_{m}]=R_{m}.$ Applying eq.", "(REF ) to eq.", "(REF ) we get $R_{2}-4R_{1}+3R_{0}\\ge 0,$ which is the simplest inequality in our approach since it contains only three time points.", "Note that the coefficients sum up to zero ($1-4+3$ ).", "This is compatible with yielding $0\\ge 0$ when the evolution is the identity operator and $R_{2}=R_{1}=R_{0}$ .", "Operationally, this inequality is evaluated in the following way: in one set of measurements $R_{0}$ is measured (no evolution, the system is in the initial state).", "In a different set of measurements the initial state is propagated for one cycle and then $R_{1}$ is measured (the survival probability for pure sate).", "Finally, in a third set of measurements $R_{2}$ is measured.", "In contrast to the two-point measurement protocol ([26] and references therein), here there is no evolution whatsoever after the measurement is taken.", "Thus, there is no issue with wavefunction collapse and in fact, the measurements could be fully destructive (e.g.", "ionization) and inequality (REF ) would still be valid.", "We have used the periodicity of the driving and nothing was assumed on the evolution of the density matrix.", "A random initial state will populate several Floquet modes so although each mode is periodic, their superposition is in general not periodic due to the quasi-energy phase accumulation and the interference of the modes.", "Before we continue to more general bounds and to experimental demonstrations, it is worth taking a closer look at this three-point inequality." ], [ "Comparison to the analog of the second law of thermodynamics:", "As a basis for comparison, the usual two-point global passivity inequality, eq.", "(REF ), yields $R_{0}\\ge R_{1}$ and $R_{0}\\ge R_{2}$ which for pure state is trivial since $R_{0}=1$ and $R_{0},R_{1}\\le 1$ .", "The inequality (REF ) can be assigned with different interpretations via different rearrangements: $R_{2}\\ge 4R_{1}-3R_{0},\\\\\\frac{1}{4}(R_{2}+3R_{0})\\ge R_{1},\\\\(R_{0}-R_{1})\\ge \\frac{1}{4}(R_{0}-R_{2})\\ge 0.$ Inequality (REF ) provides a lower bound on $R_{2}$ (the two-point bound yields an upper bound $R_{2}\\le R_{0}$ ), while () offers a refined upper bound on $R_{1}$ .", "Since $R_{0}\\ge \\frac{1}{4}(R_{2}+3R_{0})$ this bound is always tighter than the two-point bound $R_{0}\\ge R_{1}$ .", "Interestingly this bounds is using information on $R_{2}$ , so the two endpoints are used for bounding the midpoint.", "Inequality () compares the change in the survival probability in the first half to the cumulative change $R_{0}-R_{2}$ .", "It suggests that the change cannot occur just in the second half of the evolution; at least a quarter must take place in the first half.", "Similarly one can write $3(R_{0}-R_{1})\\ge R_{1}-R_{2}$ and directly compare the two halves.", "Note however that in this form the right hand side might become negative.", "Another added value of the form () is that it makes it easier to compare with the two-point inequalities $R_{0}\\ge R_{1}$ and $R_{0}\\ge R_{2}$ .", "The inequality () shows that $R_{0}-R_{1}$ is not just non-negative but also larger than another non-negative number, $\\frac{1}{4}(R_{0}-R_{2})$ .", "Thus, it is tighter than the two-point prediction $R_{0}-R_{1}\\ge 0$ .", "We emphasize again that pure states in two-point schemes lead to trivial and not useful results.", "Here, we obtain non-trivial bounds that involve pure states.", "This is an intrinsic feature of our framework, which plays important role when studying quantum processors." ], [ "Multi-time-point inequalities", "The following recipe can be used to obtain more general periodicity bounds: Choose a stencil of the form $r_{0}=\\sum _{i=0}^{N}\\alpha _{i}\\rho _{i}$ ($\\alpha _{i}$ are real numbers) Choose the value of $M$ in $r_{M}$ Expand the parenthesis in $tr[r_{0}(r_{0}-r_{M})]\\ge 0$ Replace $tr[\\rho _{n}^{2}]\\rightarrow R_{0}=tr[\\rho _{0}^{2}]$ Replace $tr[\\rho _{n}\\rho _{n+m}]\\rightarrow R_{m}=tr[\\rho _{0}\\rho _{m}]$ This will generate an inequality with $M+N$ time points (including the initial state).", "In the following we make the specific choice $r_{0}=(\\frac{1}{2})^{N}D_{N}$ , where $D_{N}$ is the shifted discrete derivative $D_{1}=\\rho _{1}-\\rho _{0},D_{2}=\\rho _{2}-2\\rho _{1}+\\rho _{0},...$ The shift of the center point is chosen such that the derivative will not include negative indices.", "For $r_{M}$ we choose $r_{1}$ .", "Here are the first few inequalities, $S_{3}-S_{5}$ : $\\frac{1}{8}(R_{2}-4R_{1}+3R_{0})\\ge 0,\\\\\\frac{1}{32}(-R_{3}+6R_{2}-15R_{1}+10R_{0}\\ge 0,\\\\\\frac{1}{128}(R_{4}-8R_{3}+28R_{2}-56R_{1}+35R_{0})\\ge 0,\\\\...\\nonumber $ We denote these inequalities by $S_{n}$ where $n$ is the number of time points used (cycle number $+1$ ).", "The $S_{n}$ inequalities can be written as $S_{n}=\\sum _{k=0}^{n}w_{k}^{(n)}R_{k}\\ge 0$ where the coefficients $w_{k}^{(n)}$ alternate sign as $k$ increases, and they satisfy $\\sum _{k=0}^{n}w_{k}^{(n)}=0$ and $\\sum _{k=0}^{n}\\text{$\\left|w_{k}^{(n)}\\right|$}=1$ .", "As shown later, when $n$ is sufficiently large the coefficients take the form: $w_{k}^{(n)}\\rightarrow \\frac{(-1)^{k}}{\\sqrt{\\pi n}}e^{-\\frac{k^{2}}{n}}R_{k}$ .", "The $S_{n}$ is just one family in a continuum of inequalities.", "While the $S_{n}$ inequalities have simple form and appealing analytical properties that will not be discussed in this paper, presently we make no claim of optimality according to some metric.", "In the following, we explore the continuum of four-point inequalities.", "Consider the stencil with $M=1$ , $r_{0}=\\rho _{0}+a\\rho _{1}+b\\rho _{2}.$ A slightly more general form would be $r_{0}=\\pm \\rho _{0}+a\\rho _{1}+b\\rho _{2}$ however both options lead to the same final result.", "Using the recipe above we get the following inequality: $-bR_{3}+(-a+2b+ab)R_{2} & + & (-1+2a-b-(a-b)^{2})R_{1}\\nonumber \\\\& + & (1-a+a^{2}+b^{2}+ab)R_{0}\\nonumber \\\\& & \\ge 0$ We notice that the left hand side, which we denote by $L(a,b)$ , is a second order polynomial in $a$ and $b$ , where the coefficients of $a,b,ab,a^{2},b^{2},1$ depend on $\\lbrace R_{n}\\rbrace _{n=0}^{N+1}$ ." ], [ "Optimization for incoherent error detection", "In the absence of a heat leak $L(a,b)\\ge 0$ , but in the presence of a heat leak $L(a,b)$ might be positive for some choices of $a$ and $b$ and negative for others.", "It is also possible that $L(a,b)$ is not negative for any choice of $a$ and $b$ .", "In that case, the heat leak is not detectable with four-point periodicity inequalities.", "One option to find the best values of $a$ and $b$ for heat leak detection, is to scan their values and look for negative values of $L(a,b)$ .", "Fortunately, this can be avoided by studying the explicit expression of $L(a,b)$ .", "In a given physical scenario (initial condition+driving protocol) $\\lbrace R_{n}\\rbrace _{n=0}^{N+1}$ are just constant positive numbers and $L(a,b)$ is a second order polynomial with known and fixed coefficients.", "Next we check which type of paraboloid $L(a,b)$ is.", "In the limits $a\\rightarrow \\pm \\infty $ and $b\\rightarrow \\pm \\infty $ we get $L(a,b)\\rightarrow (a^{2}+b^{2})(R_{0}-R_{1})$ .", "Since for pure states $R_{0}=1,R_{1}\\le 1$ (even in the presence of a heat leak) it follows that the paraboloid has a minimum rather than a maximum or a saddle point.", "For mixed-state initial condition, it can be shown that $R_{0}-R_{1}<0$ but this already indicates a heat leak and there is no need to study (REF ), which involves 3 cycles.", "We conclude that the most negative value of $L(a,b)$ occurs when $\\partial _{a}L(a,b)=0$ and $\\partial _{b}L(a,b)=0$ , we denote by $a_{min}$ and $b_{min}$ the solution to these two equations.", "Finding $a_{min},b_{min}$ we get that $L(a_{min},b_{min})\\ge 0$ is equal to $\\frac{(2R_{0}-R_{1}-2R_{2}+R_{3})}{(R_{0}-R_{2})(3R_{0}-4R_{1}+R_{2})}\\times \\nonumber \\\\\\left(R_{0}^{2}-R_{0}(R_{1}+R_{3})-(R_{1}-R_{2})^{2}+R_{1}R_{3}\\right) & \\ge & 0\\nonumber \\\\$ This bound is nonlinear in the expectation value $R_{n}$ since the expectation values are being used to choose the optimal stencil for detection.", "If the inequalities that involves $R_{0},R_{1},R_{2}$ , i.e.", "$R_{0}-R_{2}\\ge 0$ and $3R_{0}-4R_{1}+R_{2}\\ge 0$ ($S_{3}$ and $S_{2}$ ), hold, then the nonlinear bound (REF ) simplifies to $R_{0}^{2}-R_{0}(R_{1}+R_{3})-(R_{1}-R_{2})^{2}+R_{1}R_{3}\\ge 0.$ Just for clarity we point out that the nonlinearity in this inequality is entirely different from the density matrix nonlinearity usually considered in quantum information and quantum thermodynamics (e.g.", "the von Neumann entropy $S_{vn}=-tr[\\rho \\ln \\rho ]$ ).", "Here the nonlinearity appears outside the trace." ], [ "Low cost evaluation of the $S_{n}$ \nperiodicity inequalities", "We introduce the notation $A_{\\pm }\\rho _{n}=\\rho _{n\\pm 1}$ , and use $M=1$ and $r_{0}=(\\frac{1}{2})^{N}D_{N}$ in the recipe in Sec.", "REF to get: $S_{n+1}=\\frac{1}{2^{n}}tr[\\rho _{0}(1-A_{-})^{n}[(1-A_{+})^{n}-A_{+}(1-A_{+})^{n}\\rho _{0}]\\ge 0$ where after the expansion we substitute $A_{\\pm }^{j}\\rho _{n}=\\rho _{\\left|n\\pm j\\right|}$ .", "$S_{n+1} & = & \\frac{1}{2^{n}}tr[\\rho _{0}(1-A_{-})^{n}(1-A_{+})^{n}(1-A_{+})\\rho _{0}].\\nonumber \\\\$ Next, we use $A_{+}\\Longleftrightarrow A_{-}$ to obtain another expression for $S_{n}$ $S_{n+1} & = & \\frac{1}{2^{n}}tr[\\rho _{0}(1-A_{+})^{n}(1-A_{-})^{n}(1-A_{-})\\rho _{0}].\\nonumber \\\\$ Taking the mean of the two expressions and using $(1-A_{-})^{n}(1-A_{+})^{n}=(2-A_{-}-A_{+})^{n}$ yields $S_{n}=tr[\\rho _{0}(\\frac{1}{2}-\\frac{1}{4}A_{-}-\\frac{1}{4}A_{+})^{n-1}\\rho _{0}]$ The expression $(\\frac{1}{2}-\\frac{1}{4}A_{-}-\\frac{1}{4}A_{+})^{n-1}$ describes an $n-1$ random walk steps with probability of $1/4$ to move one step forward, $1/4$ to step backward and $1/2$ to stay in place.", "From the central limit theorem the probability for moving moving $k$ steps forward is: $\\frac{1}{\\sqrt{2\\pi \\sigma _{0}^{2}(n-1)}}e^{-\\frac{k^{2}}{2\\sigma _{0}^{2}(n-1)}}=\\frac{1}{\\sqrt{\\pi (n-1)}}e^{-\\frac{k^{2}}{(n-1)}},$ where we used: $\\sigma _{0}^{2}=\\frac{1}{2}\\cdot 0+2\\cdot \\frac{1}{4}\\cdot 1=\\frac{1}{2}$ .", "Applying this to eq.", "(REF ) we find $S_{n} & = & \\sum _{k=-n+1}^{n-1}(-1)^{k}\\frac{1}{\\sqrt{\\pi (n-1)}}e^{-\\frac{k^{2}}{n-1}}R_{\\left|k\\right|}\\nonumber \\\\& = & \\frac{1}{\\sqrt{\\pi (n-1)}}R_{0}+2\\sum _{k=1}^{(n-1)}(-1)^{k}\\frac{1}{\\sqrt{\\pi (n-1)}}e^{-\\frac{k^{2}}{n-1}}R_{k}.\\nonumber \\\\$ Next, we set a bound on the sum from $L$ to $n-1$ .", "$\\sum _{k=L}^{n-1}\\frac{(-1)^{k}}{\\sqrt{\\pi (n-1)}}e^{-\\frac{k^{2}}{(n-1)}}R_{k} & \\le & \\sum _{k=L}^{n}\\frac{1}{\\sqrt{\\pi (n-1)}}e^{-\\frac{k^{2}}{(n-1)}}\\nonumber \\\\& \\le & \\sum _{k=L}^{\\infty }\\frac{1}{\\sqrt{\\pi (n-1)}}e^{-\\frac{k^{2}}{n-1}}\\nonumber \\\\& \\simeq & \\frac{1}{2}\\text{erfc}(\\frac{L}{\\sqrt{n-1}})\\nonumber \\\\& + & \\frac{1}{2\\sqrt{\\pi (n-1)}}e^{-\\frac{L^{2}}{(n-1)}}.$ For convenience we set $L=\\xi \\sqrt{n-1}$ and get $2\\sum _{k=L}^{n-1}\\frac{1}{\\sqrt{\\pi (n-1)}}e^{-\\frac{k^{2}}{n-1}}\\simeq \\text{erfc}(\\xi )+\\frac{1}{\\sqrt{\\pi (n-1)}}e^{-\\xi ^{2}}.$ From (REF ) and (REF ) we finally obtain $S_{n}^{(L=\\xi \\sqrt{n-1})} & = & \\sum _{k=0}^{L-1}w_{k}R_{k}\\nonumber \\\\& \\ge & -2\\sum _{k=L}^{n-1}(-1)^{k}\\frac{1}{\\sqrt{\\pi (n-1)}}e^{-\\frac{k^{2}}{n-1}}R_{k}\\nonumber \\\\& \\ge & -\\text{erfc}(\\xi )-\\frac{1}{\\sqrt{\\pi (n-1)}}e^{-\\xi ^{2}}.$ For example, to get an error of $5\\times 10^{-3}$ in the calculation of $S_{1000}$ (1000 cycles), only 64 cycles need to be measured ($\\xi =2$ ).", "For an accuracy of $10^{-5}$ , 100 cycles are needed ($\\xi =\\pi $ ), and 126 measured cycles already yield an accuracy of $2\\times 10^{-8}$ .", "We emphasize that in practice the accuracy is limited by statistical noise.", "For example in the experiment in section REF the truncation error is $1/65$ of the $3\\sigma $ -width when taking 24 points ($\\xi =2.1$ ).", "For thirty points ($\\xi =2.63$ ), the truncation error is already $1/1000$ of the $3\\sigma $ width." ], [ "Reduction of the measured subspace by using mixed states", "In this section we point out that a combination of pure states and mixed states can be useful.", "If only a subsystem of the whole system is of interest, it is possible to use the following initial state $\\tilde{\\rho }_{0}=\\left|\\psi _{\\text{subsys}}\\right\\rangle \\left\\langle \\psi _{\\text{subsys}}\\right|\\otimes I_{\\text{rest}}/N_{\\text{rest}},$ where $\\left|\\psi _{\\text{subsys}}\\right\\rangle \\left\\langle \\psi _{\\text{subsys}}\\right|$ is a pure state in the space of the subsystem, and $I_{\\text{rest}}/N_{\\text{rest}}$ is a fully mixed state and $N_{\\text{rest}}=\\text{tr}(I_{\\text{rest}})$ .", "The observable $N_{\\text{rest}}\\tilde{\\rho }_{0}$ leads to the expectation value $R_{n}^{\\text{subsys}}=N_{\\text{rest}}tr[\\tilde{\\rho }_{0}\\rho _{n}]=tr[\\left|\\psi _{\\text{subsys}}\\right\\rangle \\left\\langle \\psi _{\\text{subsys}}\\right|\\rho _{n}^{\\text{subsys}}],$ which is the survival probability in the measurable subsystem.", "This can be used in systems where some degrees of freedom, e.g.", "translational or internal, cannot be easily measured." ], [ "Experimental demonstrations", "We present experimental results carried out on the IBM superconducting quantum processors and a TIQC.", "Each example illustrates different features of the new bounds." ], [ "Heat leak detection with a pure initial condition", "In some setups such as digital quantum computers, pure state initial condition are more natural to use compared to mixed states which require some dedicated preparation protocol (e.g.", "see [14] and [22]).", "It is natural, then, to ask if pure states can be used to detect heat leaks.", "One of the key assumptions behind the second law in quantum microscopic systems [27] is that the von Neumann entropy of the overall system (including baths) does not decrease with respect to its initial value, $S_{f}^{tot}-S_{0}^{tot}\\ge 0$ .", "The equality is obtained for unitary maps while the inequality is associated with more general unital maps, e.g., dynamics that include decoherence.", "A heat leak can be detected by the second law only if it decreases the total entropy, e.g.", "due to unaccounted spontaneous emission.", "If the initial state is pure, the initial entropy is zero $S_{0}^{tot}=0$ and it is not possible to decrease it any further.", "Hence pure states can not be used to detect heat leaks using the second law.", "A similar problem appears in global passivity and in resource theory.", "As an example, in the present approach $S_{2}$ yields $R_{0}\\ge R_{1}$ .", "Noting that pure states satisfy $R_{0}=\\text{tr}\\rho _{0}^{2}=1$ , and $R_{1}\\le 1$ , it follows that the two-point inequality $S_{2}$ cannot detect heat leaks when the initial state is pure.", "Surprisingly, the periodicity inequalities studied here are free from this limitation which paves the way to pure state heat leak detection.", "We present a proof-of-principle experiment for heat leak detection using pure states.", "Our setup, shown in Figure REF (a), is composed of a two-qubit system (the visible system) coupled to one environment qubit.", "The initial state is created by the single qubit gates which are not shown in the Figure.", "The single cnot gate between the upper two qubits constitutes the periodic unitary, i.e.", "the “cycle”.", "The experiment is composed of three sub-experiments for measuring $R_{0},R_{1}$ and $R_{2}$ .", "Since the detector measures in the computational basis, the inverse of the preparation circuit is applied before the detector.", "As a result the probability of measuring the state $\\left|00\\right\\rangle $ in the detector corresponds to the survival probability.", "We tested several different input states: $\\left|00\\right\\rangle ,\\left|01\\right\\rangle ,\\left|10\\right\\rangle ,\\left|11\\right\\rangle $ , and $\\left|1+\\right\\rangle $ .", "Our goal is to observe a violation of $S_{3}$ when the environment is connected, and no violation when it is disconnected.", "Figure REF (b) shows that states in the computational basis cannot detect the violation but the superposition state $\\left|1+\\right\\rangle $ can.", "The uncertainty values show that the average value is greater than the 3$\\sigma $ statistical uncertainty.", "Interestingly, the circuit is completely classical for “classical” binary input, i.e.", "diagonal states in the computational basis.", "The positive values for the computation basis input [Fig.", "REF (b)] imply that in this circuit no classical binary input (including stochastic inputs) can lead to heat leak detection.", "Yet quantum superposition of two “classical” binary states can detect the coupling to the environment.", "This shows a quantum aspect of heat leak detection.", "While this example does not involve a strong quantum feature such as entanglement it is still interesting and it naturally raises the question if entangled states can lead to detection of heat leaks that cannot be detected using classically correlated states.", "Figure: (a) a circuit for demonstrating heat leakdetection with pure initial state.", "The initial condition is ψ 0 ψ 0 \\protect \\left|\\psi _{0}\\right\\rangle \\protect \\left\\langle \\psi _{0}\\right|in the upper two qubits which constitute the system and the bottomenvironment qubit is initially in 00\\protect \\left|0\\right\\rangle \\protect \\left\\langle 0\\right|.The survival probability of the two upper spins is measured afterone cycle in one experiment, and after two cycles in a different experiment.", "(b) The experimental values of S 3 S_{3} for various input states carriedout on the IBM London processor.", "The negative value for ψ 0 =+ a 0 b \\protect \\left|\\psi _{0}\\right\\rangle =\\protect \\left|+_{a}0_{b}\\right\\rangle indicates the detection of the environment.", "This illustrates detectionof environment using pure state which is impossible in alternativethermodynamically inspired frameworks.", "Interestingly, only quantuminitial condition leads to detection in this setup (superpositionis required)." ], [ "Detection of realistic heat leaks", "The question that naturally arises is whether this method can detect the real physical environment of the quantum processors.", "The following experiment shows that the real environment is indeed detectable when exploring larger numbers of cycles.", "The one-cycle circuit we use is shown in the inset of Fig.", "REF where $\\theta =1$ .", "Qubits 1 and 2 of the IBM Santiago processor are initialized in the state $\\left|00\\right\\rangle $ .", "Figure REF (a) shows the values of $S_{n}$ as a function of the number of points (cycles +1).", "The width of the line corresponds $\\pm 3\\sigma $ uncertainty.", "Hence, starting from 17 points, one can see a violation beyond the 3$\\sigma $ uncertainty.", "Figure REF shown the results of a similar experiment carried out on the IBM Casablanca processor, for 65 cycles (66 time points) and $\\theta =0.1$ .", "The value of $\\theta $ was modified from the previous experiment as we want to demonstrate a different phenomenon.", "Crucially, as explained in Sec.", "REF the number of physical cycles for evaluating $S_{n}$ can be $M=\\xi \\sqrt{n}$ , where $\\xi $ depends on the needed accuracy.", "Setting $\\xi =\\pi $ and $M=66$ , we extrapolate up to $\\left\\lfloor (66/\\pi )^{2}\\right\\rfloor =440$ with an error of $10^{-5}$ , which is not observable on the scale of the Figure.", "The bound given in eq.", "(REF ) on the tail of $S_{n}$ is rather loose since it assumes that the signs of the coefficients $w_{k}^{(n)}$ do not alternate as a function of $k$ , and that all survival probabilities are one.", "In practice the contribution of the tail to the sum can, therefore, be much smaller.", "The red band shows extrapolation based on only 28 points.", "According to eq.", "(REF ) at $n=150$ the error should be smaller than $1.5\\times 10^{-3}$ , which is too big to confirm a violation.", "Yet, in practice, at $n=150$ the extrapolation based on 28 points is indistinguishable from the one exploiting 65 points.", "In general, it is possible to start with a small number of cycles, and then add more cycles until convergence is achieved at the extrapolated point.", "Up to about 170 cycles the right hand side of eq.", "(REF ) is negligible, which means that a violation corresponds to values below zero.", "Thus, we observe that with only 28 cycles, and the same number of shots (i.e.", "without using more data to reduce the uncertainty) the IBM heat leak is detectable with this simple circuit.", "Note, that other circuits can detect the leakage much faster, but the goal of this example is to illustrate the advantage of the $\\sqrt{n}$ scaling.", "Once can also choose the value of $\\xi $ according to the desired accuracy.", "Figure: Inset: the circuit used for evaluatingthe S n S_{n} inequalities under the intrinsic noise of the IBM processors.The initial condition is the ground state.", "R y R_{y} is a single qubitrotation around the yy axis.", "(a) For qubits 2 & 3 of theSantiago processor and θ=1.0\\theta =1.0, the environment leads to a violationafter 17 cycles.", "(b) Running the same experiment on qubits 5& 6 of the Casablanca processor and θ=0.1\\theta =0.1, no violationis observed for the number of cycle we were able to run in this experiment.", "(c) Using the extrapolation described in Sec.", "based on the favorable n\\sqrt{n} scaling, a violation is observedafter 130 cycles.", "The extrapolation can be used to detect a heatleak using the first 65 cycles (blue) or even just the first 28 cycles(red)." ], [ "Heat leak detection beyond the hot-environment limit", "Heat leaks created by hot environments deserve special attention.", "In the extreme case where the environment is fully mixed, the observed system experiences unital dynamics and therefore cannot be detected using second-law-like inequalities or global passivity inequalities.", "One can expect that if the environment is slightly less hot than a fully mixed state it will still be hard to detect.", "In [11] a bound was derived on the environment temperature $T_{env}$ for the detection of heat leaks using observables in the energy basis: $T_{\\text{env}}\\le T_{\\text{undet}}=\\frac{\\text{max}(E_{\\text{env}})-\\text{min}(E_{\\text{env}})}{\\text{min}(\\mathcal {B}_{n}^{\\text{vis}}-\\mathcal {B}_{n-1}^{\\text{vis}})},$ where $\\mathcal {B}_{n}^{\\text{vis}}$ is the $n$ -th eigenvalue of $-\\ln \\rho _{\\text{sys}}$ (sorted in an increasing order of their size).", "Hence, if the temperature of the environment exceeds $T_{\\text{undet}}$ , then no passivity-based inequality can detect the presence of the environment.", "However, this bound is based on the “one-cycle scheme”, in contrast to the multi-cycle approach presented in this paper that combines information taken from different numbers of cycles.", "Thus, the periodicity inequalities have the potential to detect this type of environments.", "In the following experiment we used the circuits shown in Figure REF (a).", "To generate thermal qubits for the environment and the system, an ensemble of pure states is used (see [14]).", "The inverse temperatures $\\beta _{h}=0.6$ , $\\beta _{c}=3.5$ and $\\beta _{e}=0.5$ were chosen so that the condition give in eq.", "(REF ) guarantees that the environment is too hot to be detected using one-cycle inequalities on observables.", "As indicated by the negative value of the 3rd bar in Figure REF (b), by evaluating the optimized four-point inequality (REF ) the heat leak becomes detectable.", "Figure: An experiment that confirms the capability ofthe periodicity inequalities to detect hot environments which cannotbe detected using two-point passivity-based bounds [see eq.", "()].Running the circuit (a) for two and three cycles, we observe in (b)that the S 3 S_{3} and S 4 S_{4} inequalities are not violated but theoptimized four-point inequality is violated and therefore confirmsthe coupling to the hot qubit 'e'.", "The experiment was done on theOurense IBM processor.", "See text for details." ], [ "A three-qubit gate in trapped ions and superconducting circuits ", "Next, we show a specific yet important scenario where in the absence of heat leaks, the $S_{n}$ inequalities have simple outputs.", "Consider a gate $U$ such as cnot, Toffoli (ccnot), Fredkin (cswap), etc.", "that satisfies $U\\left|\\psi _{A}\\right\\rangle =\\left|\\psi _{B}\\right\\rangle ,U\\left|\\psi _{B}\\right\\rangle =\\left|\\psi _{A}\\right\\rangle $ i.e.", "$U$ is a two-state permutation, i.e.", "$U^{2}$ is the identity operator.", "If the initial condition is $\\rho _{0}=\\left|\\psi _{A}\\right\\rangle \\left\\langle \\psi _{A}\\right|$ or $\\rho _{0}=\\left|\\psi _{B}\\right\\rangle \\left\\langle \\psi _{B}\\right|$ then the ideal output is $R_{n}=\\lbrace 1,0,1,0,1,...\\rbrace $ .", "As a result $S_{n}=\\sum _{k=0}^{\\left\\lfloor n/2\\right\\rfloor }w_{2k}=1/2$ .", "When the predicted $R_{n}$ is so simple, there are many ways to quantify the deviation of the device from its expected behavior.", "For example, one can use$\\text{$\\left|R_{n}^{exp}-R_{n}\\right|$}$ .", "However, this quantity is always positive and a large value of $\\left|R_{n}^{exp}-R_{n}\\right|$ can appear due to coherent error and is, therefore, not directly associated with a heat leak.", "In contrast, negative values of $S_{n}$ are a clear indication of a heat leak.", "In our last experiment we run sequentially the Toffoli gate with the initial state $\\left|1_{A}1_{B}0_{C}\\right\\rangle $ where $A$ and $B$ are the controlling qubits.", "The experiment was carried out both on the IBMQ superconducting processors and on a TIQC.", "The TIQC results were obtained on the University of Maryland Trapped Ion (UMDTI) quantum computer which is described in [28].", "This experiment was performed using a linear chain of five $^{171}\\text{Yb}^{+}$ ions in a room-temperature Paul trap under ultrahigh vacuum with the qubit encoded in two hyperfine ground states of $\\text{Yb}^{+}$ .", "We re-analyze raw data from results reported in [29], where three of the five ions treated as qubits while the others remained idle.", "For more information on the UMDTI hardware, see Appendix II.", "Figure REF (a) shows the $S_{n}$ plots for various IBM processors and the UMDTI as a function of the number of cycles $n$ .", "The $\\left|R_{n}^{exp}-R_{n}\\right|$ deviation with respect to the ideal output is depicted in Figure REF (b).", "While the $\\left|R_{n}^{exp}-R_{n}\\right|$ fluctuates, the $S_{n}$ curves are monotonically decreasing.", "The deviation from $1/2$ can arise from a coherent error in the implementation.", "It appears that the TIQC data is the closest to the predicted $1/2$ value.", "Using tools which are beyond the present paper one can proveManuscript in preparation.", "that the $S_{n}$ series must monotonically decrease if the evolution is unitary and periodic.", "Thus, an increase in the $S_{n}$ plot is an indication for a heat leak, which is not observed here.", "In our next test, we grouped pairs of Toffoli gates as a single cycle.", "The survival probability for this two-Toffoli cycle is $R_{n}^{\\prime }=R_{2n}=\\lbrace 1,1,1...\\rbrace $ .", "Since $\\sum _{k=0}^{n}w_{k}^{(n)}=0$ it follows that for an ideal evolution $S_{n}^{\\prime }=\\sum _{k=0}^{n}w_{k}^{(n)}R_{2k}=\\sum _{k=0}^{n}w_{k}^{(n)}=0$ for all n. Figure REF (c), shows the experimental values of $S_{n}^{\\prime }$ for the various quantum processors.", "Figure: When a circuit changes the initial state to an orthogonalstate, and in the next cycle returns to the initial state it holdsthat S n =1/2S_{n}=1/2.", "(a) S n S_{n} values for various quantum processorswhere the circuit contains a single Toffoli gate.", "The plot shows thatthe TIQC exceeds the performance of various IBM processors which arefurther away from 1/21/2.", "(b) The comparison of the theoretical andexperimental values of the survival probabilities, R n exp -R n \\left|R_{n}^{exp}-R_{n}\\right|,seems like a reasonable choice for comparing different processors.Yet, this quantity strongly fluctuates and in this case there is noprocessor that is consistently better at all time points.", "In contrast,the S n S_{n} shows a clear and consistent difference between the variousprocessors.", "(c) Same as (a), but this time the cycle contains twoconsecutive Toffoli gate.", "In this case, the ideal evolution yieldsS n =0S_{n}=0.", "Here as well, the TIQC performs better than the testedIBM superconducting quantum processors." ], [ "Concluding remarks", "We have presented a set of multi-cycle inequalities valid for periodically driven quantum systems.", "The measured quantity is the survival probability and the initial state can be pure.", "These two features are very appealing for isolated quantum systems such as quantum computers and simulators.", "Our first experiment demonstrates heat leak detection when the visible system starts in a pure state, and the environment was simulated by a qubit that can be either coupled or decoupled.", "To the best of our knowledge, heat leak detection with pure states is presently unique to the presented periodicity bounds.", "Furthermore, we have demonstrated a certain quantum advantage when the input state is quantum.", "While this advantage is not claimed to be a generic effect, it motivates further study.", "Specifically, it is interesting to check if entangled states can assist in heat leak detection.", "In the second experiment we put our bounds to the challenge of detecting the intrinsic heat leak of the IBM machine.", "We were able to detect the heat leak and demonstrate a scaling law that makes our bounds efficient: when the number of cycles exceed a few dozen, then n-cycle inequalities require only $O(\\sqrt{n})$ measured data points.", "For example, for the inequality $S_{1000}$ that formally needs measurements of 1000 data points, it is enough to measure the first $65-126$ points and the error will be smaller than $10^{-5}$ .", "Our third experiment demonstrates that our bounds can circumvent the fundamental limitation in detecting very hot environments using one-cycle bounds based on observables only, i.e.", "without resorting to non observable information measures such as entropy.", "Our final experiment exploits our inequalities and a circuit based on the Toffoli gate to quantify the performance difference between a TIQC and superconducting processors.", "Presently, we do not claim or suggest that these bounds will necessarily mature into a practical method for diagnosing quantum circuits.", "First, a clear meaning of the amount of violation is still missing.", "Second, it is yet unclear if any type of heat leak or malfunction can be detected using these bounds or some variants of it.", "In particular, it is unclear whether a system that passes all possible periodicity tests is indeed flawless.", "That being said, this method has three appealing features that make it worth exploring.", "The first is the operational advantage: it uses pure states and only the survival probability is measured.", "The second is that it also treats the circuits as black box and requires no information on the circuit.", "The third is the $\\sqrt{n}$ scaling law which states that only the first $\\propto \\sqrt{n}$ cycles contribute to the $S_{n}$ inequality.", "More fundamentally, these bounds can circumvent inherent limitations that exist in various other methods, and therefore can detect heat leaks that other methods cannot.", "In that respect, it is a framework that should appear in any benchmark comparison of a new heat leak detection technique.", "In parallel to the theoretical work on periodicity inequalities, it would be interesting to systematically test these bounds in other experimental platforms and identify violations, which are associated with a specific platform or a specific mechanism.", "R.U.", "is grateful for support from Israel Science Foundation (Grant No.", "2556/20).", "A.M.G.", "is supported by a JQI Postdoctoral Fellowship.", "N.M.L.", "acknowledges financial support from NSF grant no.", "PHY-1430094 to the PFC@JQI, and the Maryland-ARL Quantum Partnership, grant no.", "W911NF1920181.", "In this Appendix, we provide an example that illustrates why in random benchmarking (RB) coherent errors manifest in the same way incoherent errors do, and therefore these two errors cannot be distinguished using this approximation.", "The basic building block of RB is the construction of an inverse circuit that is supposed to undo the operation of a known circuit.", "Denoting the unitary of this circuit by $U$ , the quantum computer is instructed to implement $U^{-1}=U^{\\dagger }$ right after executing $U$ .", "In an ideal computer $U^{-1}U$ is the identity operator, and therefore the system will return to its initial state i.e.", "$\\left|\\psi _{f}\\right\\rangle =\\left|\\psi _{i}\\right\\rangle $ .", "As a result, the survival probability $\\left|\\left\\langle \\psi _{i}\\left|\\psi _{f}\\right.\\right\\rangle \\right|^{2}=1$ for an ideal device.", "In a realistic device, the implemented unitary $V$ will be different from $U$ .", "Similarly, the implemented inverse circuit denoted by $V^{\\prime }$ will be different from $U^{-1}$ .", "When the errors are incoherent, .i.e., originate in some coupling to the environment, it is expected that the error in both circuits will not cancel each other and therefore $\\left|\\left\\langle \\psi _{i}\\left|\\psi _{f}\\right.\\right\\rangle \\right|^{2}<1$ .", "Potentially, coherent errors in both circuits could compensate each other.", "The following example shows that this does not happen in general.", "In Figure REF the given circuit is a simple cnot (dashed box).", "The implemented unitary $V$ (solid red line) contains a coherent error in the form of a single qubit rotation on the upper qubit.", "Not knowing about the coherent error, the inverse of a cnot is a cnot and the computer is instructed to implement another cnot as the inverse.", "Thus since $V^{2}\\ne I$ the survival probability is smaller than one $\\left|\\left\\langle \\psi _{i}\\left|\\psi _{f}\\right.\\right\\rangle \\right|^{2}=\\left|\\left\\langle \\psi _{i}\\left|V^{\\prime }V\\right|\\psi _{i}\\right\\rangle \\right|^{2}=\\left|\\left\\langle \\psi _{i}\\left|V^{2}\\right|\\psi _{i}\\right\\rangle \\right|^{2}<1$ which exemplifies that coherent error can reduce the survival probability in the same way incoherent error can.", "Figure: A circuit for showing that in randomized benchmarkingcoherent errors manifest in the reduction of the survival probabilityas happens for incoherent errors.The UMDTI quantum computer is described in [28].", "Briefly, two-photon Raman transitions are used to control the qubit state, encoded in two magnetic-field-insensitive hyperfine ground states of $^{171}\\text{Yb}^{+}$ ions held in a linear chain in a Paul trap.", "Individual manipulation of each qubit is performed by splitting one of the Raman laser beams into several beams, each controlled by an independent acousto-optic modulator channel and focused onto a single ion in the chain.", "Single-qubit gate operations are executed by creating laser pulses of controlled phase and duration while two-qubit gates are compiled from single-qubit gates and a laser-driven entangling Ising gate (XX or $e^{i\\chi \\sigma _{x}\\sigma _{x}}$ ) following the MÞlmer-SÞrensen gate scheme [31], [32], [33], which creates entanglement between pairs of qubits via the shared harmonic oscillator modes of the ion chain in the trap.", "These modes act as an information bus with which the qubits are temporarily entangled.", "Modulation of the Raman beam amplitude is used to leave the qubits disentangled from these motional degrees of freedom at the end of the gate operation [34], [35]." ] ]
2105.11685
[ [ "Effect of large light-heavy neutrino mixing and natural type-II seesaw\n dominance to lepton flavor violation and neutrinoless double beta decay" ], [ "Abstract We derive the lower bound on the absolute scale of lightest neutrino mass for normal hierarchy and inverted hierarchy pattern of light neutrinos by studying the new physics contributions to charged lepton flavour violations in the framework of a TeV scale left-right symmetric model.", "In the model, the fermion sector comprises the usual quarks and leptons plus a fermion singlet per generation and the scalar sector consists of isospin doublets, triplets and a bidoublet.", "The framework allows large light-heavy neutrino mixing where the light neutrino mass formula is governed by a natural type-II seesaw mechanism, unlike the generic type-II seesaw dominance which assumes suppressed light-heavy neutrino mixing.", "We demonstrate how sizeable loop-induced contribution to light neutrino mass is kept under control such that the light neutrino mass formula is dominantly explained by the type-II seesaw mechanism.", "We examine the heavy neutrino contributions with large light-heavy neutrino mixing to charged lepton flavour violating processes like $\\mu \\to e \\gamma$, $\\mu \\to 3 e$ and $\\mu \\to e$ conversion inside a nucleus.", "We present a complementary study between neutrinoless double beta decay and charged lepton flavour violation taking into account single beta decay bound, double beta decay bound and cosmology bounds on the sum of light neutrino masses." ], [ "Introduction", "Neutrino oscillation data clearly indicate that neutrinos have small but non-zero masses [1].", "A simple theoretical paradigm for the origin of neutrino mass is the seesaw mechanism, which predicts the Majorana nature of neutrinos; for a review, see e.g.", "Ref. [2].", "The existence of right-handed (RH) neutrinos, as required for the type-I seesaw mechanism [3], [4], [5], [6], or the triplet scalars, as required for the type-II seesaw mechanism [7], [8], [9], [10], [11], can both be naturally motivated in ultraviolet-complete models of neutrino mass.", "One such example is the left-right symmetric model (LRSM) [12], [13], [14].", "In particular, TeV-scale models of left-right symmetry breaking have a number of testable consequences for collider signals in the gauge [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45] as well as Higgs sector [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], neutrinoless double beta decay ($0\\nu \\beta \\beta $ ) [11], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], low-energy charged lepton flavor violation (cLFV) [81], [82], [83], [84], [85], [86], [64], [21], [69], [87], [88], [89], [75], [77], [79], [90], [91] and electric dipole moment (EDM) [92], [93], [94], [66], [72], [95], all of which could together shed light on some of the unresolved issues in neutrino physics, such as the Dirac vs. Majorana nature, mass hierarchy and absolute mass of the left-handed (LH) neutrinos, and the leptonic $CP$ violation.", "These results will have far-reaching implications for beyond the Standard Model (SM) physics in general.", "In the conventional LRSM, where symmetry breaking is implemented with scalar bidoublets and triplets, the light neutrino mass is governed by both type-I [3], [4], [5], [6] and type-II [7], [8], [9], [10], [11] seesaw contributions: $M_\\nu \\ = \\ -M_DM_R^{-1}M_D^T + M_L \\ \\equiv \\ M_\\nu ^{\\rm I}+M_\\nu ^{\\rm II} \\, .$ Here $M_D$ is the Dirac neutrino mass induced by the bidoublet vacuum expectation value (VEV), while $M_R$ and $M_L$ are the Majorana masses of the right and left-handed neutrinos respectively, induced by the triplet VEVs.", "For phenomenological purposes, it is usually assumed that only one of the contributions is dominant for the low-scale LRSM, with observable ramifications for different experiments.", "For instance, in the type-I seesaw dominance, we assume $M_L\\rightarrow 0$ and the light neutrino mass crucially depends on the Dirac mass matrix $M_D$ , or the light-heavy neutrino mixing.", "In fact, for exact left-right symmetry, $M_D$ can be expressed in terms of $M_\\nu $ and $M_R$  [66], and it turns out that all the light-heavy neutrino mixing effects are suppressed for TeV-scale parity restoration.", "On the other hand, the type-II seesaw dominance can be realized with either $M_D\\rightarrow 0$ or with very high scale of parity restoration.", "The advantage here is that the light and heavy neutrino mass matrices are directly proportional to each other, thus leading to a more predictive scenario.", "In Ref.", "[96], the authors have considered these two types of dominance separately to constrain lightest neutrino mass scale and heavy neutrino masses from neutrinoless double beta decay and LFV in a TeV scale LRSM.", "But this comes with the cost of losing all the light-heavy neutrino mixing effects on the lepton number and/or flavor violating observables.", "In this paper, we explore an extended scenario for low-scale LRSM [97], whose particle content is such that the light neutrino mass generation is governed by a natural type-II seesaw mechanism, while still allowing for observable light-heavy neutrino mixing effects.", "This has important and non-trivial phenomenological consequences.", "The beautiful aspect of this model is that the dominant new physics contributions to cLFV, $0\\nu \\beta \\beta $ and electron EDM processes can be expressed in terms of the observed light neutrino oscillation parameters and lightest neutrino mass.", "As a result of this, we can derive constraints on the lightest neutrino mass from the non-observation of these rare processes.", "We also make a complementary study between the low-energy cLFV and $0\\nu \\beta \\beta $ processes within this scenario.", "However, in order to highlight the contributions of right-handed heavy neutrino and sterile neutrino to LFV decays and $0\\nu \\beta \\beta $ decay, we have focussed only on the diagrams mediated by them and ignored other possible channels.", "The paper is organized as follows; In Sec-, we recapitulate the basic model framework of left-right symmetric theory followed by a discussion on type-II seesaw dominance with large light-heavy neutrino mixing and the condition to achieve it.", "In Sec-, we discuss new physics contributions to relevant LFV processes due to this mixing and in Sec-, we demonstrate how these contributions to LFV constrain light neutrino masses.", "In Sec-, we study how the light neutrino masses are constrained by new physics contributions to $0\\nu \\beta \\beta $ decay and in Sec- we do a complementarity study between LNV and LFV decays.", "We also show the variation of dipole moment of electron with lightest neutrino mass and PMNS phase in that section.", "In Sec- we comment on the recent muon (g-2) anomaly results and summarize our results in Sec-." ], [ "Natural type-II seesaw dominance and large light-heavy neutrino mixing", "The left-right symmetric theory is based on the gauge group $\\mathcal {G}_{LR} \\equiv SU(3)_c\\times SU(2)_L \\times SU(2)_R \\times U(1)_{B-L}$  [13], [12], [14].", "The fermion sector comprises of quarks and leptons as follows, $& &q_{L}=\\begin{pmatrix}u_{L}\\\\d_{L}\\end{pmatrix}\\equiv [3,2,1,{\\frac{1}{3}}] \\quad , \\quad q_{R}=\\begin{pmatrix}u_{R}\\\\d_{R}\\end{pmatrix}\\equiv [3,1,2,{\\frac{1}{3}}]\\,,\\nonumber \\\\& &\\ell _{L}=\\begin{pmatrix}\\nu _{L}\\\\e_{L}\\end{pmatrix}\\equiv [1,2,1,-1] \\quad , \\quad \\ell _{R}=\\begin{pmatrix}\\nu _{R}\\\\e_{R}\\end{pmatrix}\\equiv [1,1,2,-1] \\,$ The electric charge of individual components are related to the third component of $SU(2)_{L,R}$ gauge groups and the difference between baryon and lepton number.", "$Q=T_{3L}+T_{3R}+\\frac{B-L}{2}$ The fermion mass generation including light neutrino masses crucially depends on how the left-right symmetry breaking happens.", "The left-right symmetry can be spontaneously broken down to SM gauge group $SU(3)_c\\times SU(2)_L\\times U(1)_Y$ either by assigning VEV to a scalar doublet $H_R$ or scalar triplet $\\Delta _R$ , or with the help of both.", "In case the spontaneous symmetry breaking is done with the help of doublet $H_R$ which is the minimal scenario, Majorana masses can't be generated for neutrinos and thus it becomes less interesting from a phenomenology point of view.", "However left-right symmetry breaking through $\\Delta _R$ generates Majorana masses for both light and heavy neutrinos thereby allowing lepton number violation which can be probed by same-sign dilepton signatures at colliders as well as $0\\nu \\beta \\beta $ decay at low-energy experiments.", "In this case the neutrino mass generation is governed by type-I plus type-II seesaw mechanism but it gives negligible contribution to left-right mixing.", "The final step of symmetry breaking occurs with the help of bidoublet $\\Phi $ which breaks the electroweak gauge group $SU(2)_L\\times U(1)_Y$ to $U(1)_{\\rm em}$ theory.", "We briefly discuss below how the addition of a sterile neutrino $S_L(1,1,1,0)$ per generation to this type-I plus type-II seesaw scheme results in large left-right mixing and makes the scenario more interesting phenomenologically.", "The neutral lepton sector of generic left-right symmetric theories contains three active left-handed neutrinos ($\\nu _L$ ) and three right-handed neutrinos ($N_R$ ) which are their $SU(2)_{R}$ counterparts.", "We extend the theory only by adding three sterile neutrinos ($S_L$ ), for the purpose of generating light neutrino mass through natural type-II seesaw term, as the type-I seesaw term gets exactly cancelled out in the presence of $S_L$ .", "More importantly this scenario may lead to new non-standard contributions to neutrinoless double beta decay, lepton flavour violation and the $T$ and $CP$ -violating electric dipole moment (EDM) of charged leptons because of the light-heavy neutrino mixing.", "Even though type-II seesaw dominance is assumed in many left-right models  [64], [65], [66], [69], [70], [73] in the context of low energy phenomenology, the light-heavy neutrino mixing is very much suppressed in such cases.", "Our model differs from these frameworks by naturally getting type-II seesaw term instead of assuming it, along with large light-heavy neutrino mixing.", "The detailed derivation of neutrino masses and mixing within this natural type-II seesaw and the implications of large light-heavy neutrino mixing to $0\\nu \\beta \\beta $ decay can be found in ref [97].", "The structure of mass matrix in the basis $(\\nu _L, N^c_R, S_L)$ leading to natural type-II seesaw dominance is given as, $ &&\\mathcal {M} =\\begin{split}\\left[\\begin{array}{c | c}\\begin{array}{c c}{\\color {blue} M_L } & M_D \\\\M^T_D & {\\color {blue} M_R }\\end{array} &\\begin{array}{c}\\leavevmode {\\color {red}\\bf 0} \\\\ M\\end{array} \\\\\\hline \\begin{array}{c c}\\leavevmode {\\color {red}\\bf 0} &\\quad M^T\\end{array} &\\begin{array}{c}{\\color {blue} \\bf 0 }\\end{array} \\\\\\end{array}\\right]\\end{split}\\mathop {\\xrightarrow{}}^{{\\color {blue} M_R \\gg M > M_D \\gg M_L }}_{}\\left\\lbrace \\begin{array}{c}m_\\nu = M_L \\quad \\mbox{(type-II seesaw)}\\\\[0.2cm]\\mbox{light-heavy mixing} \\simeq M_D/M\\\\[0.2cm]m_{S} \\simeq M M^{-1}_R M^T\\,,\\quad m_N = M_R\\end{array} \\right.", "\\nonumber $ where $M_D$ is the Dirac neutrino mass matrix connecting $\\nu $ and $N_R$ , $M$ is the mixing matrix in the $\\nu -S_L$ sector, $M_L$ $(M_R)$ is the Majorana mass matrix for left-handed (right-handed) neutrinos.", "Table: Particle content of left-right theories with type-II seesaw dominance.The symmetry breaking steps in our model and the subsequent mass generation for fermions and bosons can be summed up as follows.", "We use both scalar doublets and scalar triplets for left-right symmetry breaking in order to obtain natural type-II seesaw mechanism even with non-negligible $M_D$ .", "The first step of symmetry breaking happens by assigning VEVs to both Higgs doublet $H_R$ and Higgs triplet $\\Delta _R$ .", "As an immediate result of this symmetry breaking the new gauge bosons $W^\\pm _R$ , $Z_R$ and right-handed Majorana neutrinos get mass.", "The next step of symmetry breaking is done with the help of SM Higgs doublet contained in the bidoublet $\\Phi $ , at the scale $M_Z$ .", "The SM fermions and gauge bosons $W_L$ and $Z$ get their mass at this stage of symmetry breaking.", "The complete particle spectrum of the model is given in Table REF .", "For fermions we have suppressed the family index, which runs from 1 to 3.", "With these fermions and scalars, the interaction lagrangian for leptons can be written as, $-\\mathcal {L}_{Yuk} &=& \\,\\overline{\\ell _{L}} \\left[Y_3 \\Phi + Y_4 \\widetilde{\\Phi } \\right] \\ell _R+ f\\, \\left[\\overline{(\\ell _{L})^c} \\ell _{L} \\Delta _L+\\overline{(\\ell _{R})^c}\\ell _{R}\\Delta _R\\right] \\, \\nonumber \\\\&&+F\\, \\overline{(\\ell _{R})} H_R S^c_L + F^\\prime \\, \\overline{(\\ell _{L})} H_L S_L + \\mu _S \\overline{S^c_L} S_L\\ + \\mbox{h.c.}\\,.$ The scalars take VEVs as follows, $&&\\langle \\Phi \\rangle = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} v_1 & 0 \\\\ 0 & v_2 \\end{pmatrix}\\, , \\quad \\langle \\Delta _{R} \\rangle = \\frac{1}{\\sqrt{2}} \\begin{pmatrix} 0 & 0 \\\\ v_R & 0 \\end{pmatrix}\\, , \\quad \\langle \\Delta _{L} \\rangle = \\frac{1}{\\sqrt{2}} \\begin{pmatrix} 0 & 0 \\\\ v_L & 0 \\end{pmatrix}\\, , \\nonumber \\\\&&\\langle H_R \\rangle = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0 \\\\ u_R \\end{pmatrix}\\, , \\quad \\langle H_L \\rangle = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix}\\, .", "\\quad $ After spontaneous symmetry breaking with the above assignment of VEVs to Higgs scalars, the resulting mass matrix for neutral leptons in the basis $\\left(\\nu _L, N^c_R, S_L\\right)$ can be written as, $\\mathcal {M}_\\nu = \\left( \\begin{array}{ccc}M_L & M_D & 0 \\\\M^T_D & M_R & M^T \\\\0 & M & 0\\end{array} \\right) \\, ,$ where $M_D$ is the Dirac neutrino mass matrix connecting $\\nu _L$ and $N_R$ , $M$ is the mixing matrix in the $N_R-S_L$ sector, $M_L$ $(M_R)$ is the Majorana mass matrix for left-handed (right-handed) active neutrinos generated dynamically by non-zero VEV of scalar triplet $\\Delta _L$ $(\\Delta _R)$ ." ], [ "Relation between neutrino masses and mixings", "One of the elegant features of this framework is that it connects heavy neutrinos with light neutrinos by expressing heavy neutrino masses in terms of oscillation parameters.", "The light and heavy neutrino masses, $M_L= f_L \\langle \\Delta _L \\rangle = f v_L $ $\\left( M_R= f_R \\langle \\Delta _R \\rangle = f v_R \\right)$ are related as, $m_\\nu =M_L\\propto M_R \\,.$ Also in this set up the right-handed neutrino mixing is fully determined by its left-handed counterpart and thus the right-handed and left-handed PMNS matrices are of the same form, $V^{PMNS}_R=V^{PMNS}_L\\,.$ Our prime goal is to derive a bound on the lightest neutrino mass from new physics contributions to lepton flavour violating prcoesses and the $T$ and $CP$ -violating electric dipole moment (EDM) of charged leptons.", "Before estimating different new physics contributions to lepton flavor violation and lepton number violation like neutrinoless double beta decay, we fix here the involved input model parameters.", "Masses and mixing of light neutrinos:- We consider absolute value of lightest neutrino mass as a free parameter and express other light neutrino masses in terms of lightest neutrino mass.", "For normal hierarchy of light neutrinos ($ m_1 \\sim m_2 << m_3$ ), different light neutrino masses are related as, $&&m_1 = \\mbox{lightest neutrino mass} \\;\\qquad \\nonumber \\\\&&m_2 = \\sqrt{m_1^2 +\\Delta m_{\\rm sol}^2}\\;\\qquad \\nonumber \\\\&&m_3 = \\sqrt{m_1^2 +\\Delta m_{\\rm atm}^2 + \\Delta m_{\\rm sol}^2}\\;.", "$ Similarly for inverted hierarchy ($m_3 << m_1 \\sim m_2$ ), different light neutrino masses are related as, $&&m_3 = \\mbox{lightest neutrino mass} \\;\\qquad \\nonumber \\\\&&m_1 = \\sqrt{m_3^2 +\\Delta m_{\\rm atm}^2}\\;\\qquad \\nonumber \\\\&&m_2 = \\sqrt{m_1^2 +\\Delta m_{\\rm sol}^2 +\\Delta M_{\\rm atm}^2 }\\;.", "$ The leptonic PMNS mixing matrix is parametrized in terms of neutrino mixing angles and phases as, $U_{\\rm {PMNS}}&=& \\begin{pmatrix} c_{13}c_{12}&c_{13}s_{12}&s_{13}e^{-i\\delta }\\\\-c_{23}s_{12}-c_{12}s_{13}s_{23}e^{i\\delta }&c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\\delta }&s_{23}c_{13}\\\\s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\\delta }&-c_{12}s_{23}-s_{12}s_{13}c_{23}e^{i\\delta }&c_{13}c_{23}\\end{pmatrix}\\cdot \\mbox{P}, $ where the mixing angles are denoted by $s_{ij}=\\sin \\theta _{ij}$ , $c_{ij}=\\cos \\theta _{ij}$ and the diagonal phase matrix carrying Majorana phases $\\alpha $ and $\\beta $ is denoted by $\\mbox{P}=\\mbox{diag}\\left(1, e^{i\\alpha }, e^{i \\beta } \\right)$ .", "We vary the Majorana phases from $0\\rightarrow \\pi $ .", "The experimental values of different oscillation parameters for both NH and IH patterns of light neutrinos are presented in Table.REF .", "The light neutrino masses are in general diagonalised in terms of unitary mixing matrix $U\\equiv U_{\\rm PMNS}$ in a basis where charged lepton are already diagonal.", "$m^{\\rm diag}_\\nu = U^\\dagger _{\\rm PMNS} m_\\nu U^*_{\\rm PMNS} = \\mbox{diag}\\left(m_1, m_2, m_3 \\right)\\,,$ and the physical masses are related to the mass matrix in flavour basis as, $m_\\nu = U_{\\rm PMNS} m^{\\rm diag}_\\nu U^T_{\\rm PMNS}\\,.$ Table: Neutrino oscillation parameters in 3σ\\sigma range.", "Masses of heavy right-handed neutrinos:- Under the type-II seesaw dominance scheme the light and heavy neutrino masses can be written as $m_\\nu = f \\langle \\Delta _L\\rangle $ and $M_N = f \\langle \\Delta _R\\rangle = (v_R/v_L) m_\\nu $ with $f_L =f_R =f$ .", "Since $v_L$ and $v_R$ are constants, the light left-handed and heavy right-handed neutrino masses are diagonalized by the same unitary mixing matrix, i.e $U_{\\rm PMNS}$ .", "Thus the physical masses for right-handed neutrinos $M_i$ are related to light neutrino mass eigenvalues $m_i$ as $M_i \\propto m_i$ .", "This relation implies that if the light neutrinos are normal hierarchical then the heavy right-handed neutrinos would also be hierarchical in the same manner, i.e.", "if $m_{1} < m_{2} << m_{3}$ then $M_{N_1} < M_{N_2} << M_{N_3}$ .", "Thus, if we fix the largest mass eigenvalue of heavy right-handed neutrino as $M_N=M_{N_3}$ then the other two mass eigenvalues of right-handed neutrinos can be expressed in terms of normal hierarchy (NH) pattern of light neutrino masses as, $M_{N_1} &= \\frac{m_1}{m_{3}} M_{N},\\text{ NH}, \\\\M_{N_2} &= \\frac{m_2}{m_{3}} M_{N},\\text{ NH}.$ and when the largest mass eigenvalue of right-handed neutrino is fixed as $M_N=M_{N_2}$ then the other physical masses for heavy right-handed neutrinos would be related to inverted hierarchy (IH) pattern of light neutrino masses as $M_{N_1} &= \\frac{m_1}{m_{2}} M_{N},\\text{ IH}, \\\\M_{N_3} &= \\frac{m_3}{m_{2}} M_{N},\\text{ IH}.$ Masses of sterile neutrinos:- The approximate seesaw block diagonalization scheme for type-II seesaw dominance gives mass formulas for sterile neutrinos as $M_{S} = -M M^{-1}_R M^T$ .", "Assuming the matrix $M$ proportional to identity matrix $M=m_S \\mbox{diag}\\lbrace 1,1,1\\rbrace $ , the physical masses are inversely proportional to heavy right-handed neutrinos and therefore inversely proportional to light neutrino masses, i.e, $M_{S_i} \\propto 1/M_{N_i} \\propto 1/m_i$ .", "As a result of this relation, when the light neutrinos are normal hierarchical, then the sterile neutrinos would be hierarchical in the inverse way, i.e.", "when $m_{1} < m_{2} << m_{3}$ then $M_{S_3} < M_{S_2} << M_{S_1}$ .", "Either we can fix the value of $m_S$ or take the largest sterile neutrino mass as constant value and express other sterile neutrino masses in terms of light neutrino mass eigenvalues.", "We fix the largest sterile neutrino mass eigen value as $M_S=M_{S_1}$ and the other sterile neutrino mass eigenvalues can be expressed in terms of normal hierarchy (NH) pattern of light neutrino masses as, $M_{S_2} &= \\frac{m_1}{m_{2}} M_{S},\\text{ NH}, \\\\M_{S_3} &= \\frac{m_1}{m_{3}} M_{S},\\text{ NH},$ Similarly by fixing the largest sterile neutrino mass as $M_S=M_{S_3}$ the physical masses of other sterile neutrinos can be expressed in terms of inverted hierarchy (IH) pattern of light neutrino masses as, $M_{S_1} &= \\frac{m_3}{m_{1}} M_{S},\\text{ IH}, \\\\M_{S_2} &= \\frac{m_3}{m_{2}} M_{S},\\text{ IH}.$ Neutrino Mixings:- The flavor states $\\nu _L$ , $N_R$ and $S_L$ are related to their mass eigenstates in the following way $\\begin{pmatrix}\\nu _{L} \\\\ S_{L} \\\\ N^c_{R}\\end{pmatrix}_\\alpha &=&\\begin{pmatrix}{\\mbox{V}}^{\\nu \\nu } & {\\mbox{V}}^{\\nu {S}} & {\\mbox{V}}^{\\nu {N}} \\\\{\\mbox{V}}^{S\\nu } & {\\mbox{V}}^{SS} & {\\mbox{V}}^{SN} \\\\{\\mbox{V}}^{N\\nu } & {\\mbox{V}}^{NS} & {\\mbox{V}}^{NN}\\end{pmatrix}_{\\alpha i}\\begin{pmatrix}\\nu _i \\\\ S_i \\\\ N_i\\end{pmatrix} \\nonumber \\\\&=& \\begin{pmatrix}U_{\\rm PMNS} & \\frac{1}{m_S} M_D U^*_{\\rm PMNS} & \\frac{v_L}{v_R} M_D U^{-1}_{\\rm PMNS} {m^{\\rm diag.", "}_{\\nu }}^{-1} \\\\\\frac{1}{m_S} M^\\dagger _D U_{\\rm PMNS} & U^*_{\\rm PMNS} & \\frac{v_L}{v_R} m_S U^{-1}_{\\rm PMNS} {m^{\\rm diag.", "}_{\\nu }}^{-1} \\\\\\mathcal {O} & \\frac{v_L}{v_R} m_S U^{-1}_{\\rm PMNS} {m^{\\rm diag.", "}_{\\nu }}^{-1} & U_{\\rm PMNS}\\end{pmatrix}_{\\alpha i}\\begin{pmatrix}\\nu _i \\\\ S_i \\\\ N_i\\end{pmatrix}.", "\\nonumber \\\\$ We express below only those input model parameters in terms of neutrino oscillation parameters that are required for estimating branching ratios for LFV decays.", "The individual mixing matrices are expressed in terms of Dirac neutrino mass matrix $M_D$ , mixing term $M$ and right-handed Majorana mass term $M_R$ as, $&&{\\mbox{V}}^{\\nu \\nu } = U_{\\rm PMNS} \\,, \\quad {\\mbox{V}}^{\\nu {S}} = \\frac{1}{m_S} M_D U^*_{\\rm PMNS}\\, , \\quad {\\mbox{V}}^{\\nu {N}} = \\frac{v_L}{v_R} M_D U^{-1}_{\\rm PMNS} {\\widehat{m}_{\\nu }}^{-1}\\,, \\nonumber \\\\&&{\\mbox{V}}^{S\\nu } = \\frac{1}{m_S} M^\\dagger _D U_{\\rm PMNS}\\,, \\quad {\\mbox{V}}^{SS} = U^*_{\\rm PMNS} \\, ,\\quad {\\mbox{V}}^{SN} = \\frac{v_L}{v_R} m_S U^{-1}_{\\rm PMNS} {\\widehat{m}_{\\nu }}^{-1} \\,, \\nonumber \\\\&&{\\mbox{V}}^{N\\nu } = \\mathcal {O}\\,, \\quad {\\mbox{V}}^{NS}=\\frac{v_L}{v_R} m_S U^{-1}_{\\rm PMNS} {\\widehat{m}_{\\nu }}^{-1} \\,, \\quad {\\mbox{V}}^{NN} =U_{\\rm PMNS}\\,.$ For simplification we have considered $M$ to be diagonal and degenerate.", "In general the Dirac neutrino mass matrix $M_D$ is either of up-type quark mass matrix or charged lepton mass matrix.", "However we have considered an $SO(10)$ GUT motivated structure for $M_D$ including RGE effects as, $M_D&=&\\left(\\begin{array}{ccc}0.0111 & 0.0384-0.0103\\,i & 0.038- 0.4433\\,i \\\\0.0384 +0.0103\\,i & 0.29281 & 0.8623+ 0.0002\\,i \\\\0.038+ 0.4433\\,i &0.8623-0.0002\\,i & 77.7573\\end{array}\\right)\\text{GeV}\\,.", "\\nonumber $ Other model parameters:- The spontaneous symmetry breaking of left-right symmetric model to SM is done by assigning a non-zero VEV to scalar triplet $\\Delta _R$ denoted by $v_R$ .", "The value of $v_R$ decides the masses of right-handed charged gauge bosons $W_R^{\\pm }$ , doubly charged scalar triplet $\\Delta ^{++}$ , right-handed neutrinos and others.", "Considering the bound on $v_R > 6$  TeV [100], [101], [102], [103], [54], we present below the other input model parameters.", "$&&v_R \\ge \\mbox{15\\,TeV}\\,, \\quad M_{W_R} \\ge \\mbox{10\\,TeV}\\,,\\quad M_{\\Delta ^{++}} \\simeq \\mbox{10\\,TeV}\\,, \\quad M_{N} \\simeq \\mbox{1\\,TeV}\\,.$ The parameters are chosen in such a way that they not only provide the plot for natural type-II seesaw domiance but also ensure that the contributions from charged scalar triplets and right-handed charged gauge boson $W_R$ are negligible." ], [ "Lepton flavour violation in left-right symmetric model", "A flavour violating process involving charged leptons has not been observed yet.", "Many new physics models that discuss lepton flavour violation (LFV) are constrained by muon decay experiments since the current limits on $\\tau $ observables are less stringent.", "For the decay $l_{\\alpha }\\rightarrow 3l_{\\beta }$ , the SINDRUM experiment has set a limit of $\\text{BR}(l_{\\alpha }\\rightarrow 3l_{\\beta })< 10^{-12}$  [104] since a long time, which is expected to improve significantly by the future Mu3e experiment.", "Similarly for the decay $l_{\\alpha }\\rightarrow l_{\\beta }\\gamma $ , an impressive bound on its branching ratio $\\text{BR}(l_{\\alpha }\\rightarrow l_{\\beta }\\gamma )< 4.2\\times 10^{-13}$ is provided by the MEG collaboration [105] which will be improved by the upgraded MEG-II.", "Experiments like Mu2e, COMET, PRIME focus on $\\mu \\rightarrow e$ conversion that will have a sensitivity ranging from $10^{-14}$ to $10^{-18}$ .", "The present bound and future sensitivity of these lepton flavour violating processes are given in Table.REF .", "Table: Branching ratios for different LFV processes and their present experimental bound and future sensitivity values taken from refs , , , , , , .Within canonical seesaw models LFV can be induced via light neutrino mixing.", "However in such models the branching ratios are found to be very much suppressed which are well below the present and planned experimental sensitivity.", "In left-right symmetric models dominant new contributions to LFV arise when symmetry breaking occurs at few TeV scale.", "A detailed discussion on LFV within manifest left-right symmetric model can be found in [69], [84].", "The relevant interaction terms which can mediate the processes in a LRSM are given below.", "Charged-current interactions in the lepton sector: The charged-current interactions in the lepton sector within the present model where neutral leptons comprising of $\\nu _L, N_R, S_L$ are given by $\\mathcal {L}^{\\rm \\ell }_{\\rm CC} &=& \\sum _{\\alpha =e, \\mu , \\tau }\\bigg [\\frac{g_L}{\\sqrt{2}}\\, \\overline{\\ell }_{\\alpha \\,L}\\, \\gamma _\\mu {\\nu }_{\\alpha \\,L}\\, W^{\\mu }_L+\\frac{g_R}{\\sqrt{2}}\\, \\overline{\\ell }_{\\alpha \\,R}\\, \\gamma _\\mu {N}_{\\alpha \\,R}\\, W^{\\mu }_R \\bigg ] + \\text{h.c.}\\nonumber $ Using masses and mixing relation for neutral leptons, $&&\\nu _{eL}= \\mbox{V}^{\\nu \\nu }_{e\\, i}\\, \\nu _i + \\mbox{V}^{\\nu \\, S}_{e\\, i}\\, S_i + \\mbox{V}^{\\nu \\, N}_{e\\, i}\\, N_i, \\nonumber \\\\&&N_{eR} = 0\\times \\,\\nu _i +\\mbox{V}^{N\\, S}_{e\\, i}\\,S_i + \\mbox{V}^{NN}_{e\\, i}\\, N_i$ the above CC interaction Lagrangian modifies as $\\mathcal {L}^{\\rm \\ell ,m}_{\\rm CC}&=& \\frac{g_L}{\\sqrt{2}}\\,\\bigg [ \\overline{e}_{\\,L}\\, \\gamma _\\mu \\lbrace \\mbox{V}^{\\nu \\nu }_{e\\, i}\\, \\nu _i + \\mbox{V}^{\\nu \\, S}_{e\\, i}\\, S_i +\\mbox{V}^{\\nu \\, N}_{e\\, i}\\, N_i \\rbrace \\,W^{\\mu }_L \\bigg ] +\\mbox{h.c.} \\nonumber \\\\& & + \\frac{g_R}{\\sqrt{2}}\\,\\bigg [ \\overline{e}_{\\,R}\\, \\gamma _\\mu \\lbrace \\mbox{V}^{N\\, S}_{e\\, i}\\,S_i +\\mbox{V}^{NN}_{e\\, i}\\, N_i \\rbrace \\,W^{\\mu }_R \\bigg ] + \\mbox{h.c.}$ Charged-current interactions in the quark sector: The relevant charge-current interaction for quarks are $\\mathcal {L}^{\\rm q}_{\\rm CC} &=& \\bigg [\\frac{g_L}{\\sqrt{2}}\\, \\overline{u}_{\\,L}\\, \\gamma _\\mu {d}_{\\,L}\\, W^{\\mu }_L+\\frac{g_R}{\\sqrt{2}}\\, \\overline{u}_{R}\\, \\gamma _\\mu d_{R}\\, W^{\\mu }_R \\bigg ] + \\text{h.c.} \\nonumber \\\\$ Scalar triplet interactions: The other relevant terms involving scalar triplets are given by ${\\cal L}_{\\Delta ^{\\pm }_{L}} &=& \\frac{\\Delta _L^+}{\\sqrt{2}}\\left[\\overline{{\\nu _L}^c} f \\ell _{L}+\\overline{{\\ell _L}^c} f \\nu _L\\right] +{\\rm h.c.}\\,,\\\\{\\cal L}_{\\Delta ^{\\pm \\pm }_{L,R}} &=& \\Delta _{L,R}^{++}\\overline{{\\ell }^c} f P_{L,R}\\ell + \\Delta _{L,R}^{--}\\overline{\\ell } f^\\dagger P_{R,L}{\\ell }^c\\, .", "$ In our model, LFV decays can be mediated by heavy right-handed neutrinos $N_R$ , extra sterile neutrinos $S_L$ , charged scalar triplets $\\Delta ^{\\pm \\pm }_{L,R}, \\Delta ^{\\pm }_{L,R}$ and gauge bosons $W_{L,R}$ which we classify as follows, due to purely left-handed currents (LL) arising from i) exchange of light neutrinos, ii) exchange of heavy right-handed neutrinos and extra sterile neutrinos with the involvement of light-heavy neutrino mixing, iii) exchange of scalar triplets; due to purely right-handed currents (RR) via exchange of right-handed charged gauge boson, right-handed scalar triplet and heavy neutrinos; due to involvement of left-handed as well as right-handed currents (LR).", "However we focus here only on those contributions which involve large active-sterile neutrino mixing, i.e.", "due to the neutrinos $N_R$ and $S_L$ in order to constrain light neutrino masses from LFV decays.", "We ignore the other possible contributions by imposing the following limiting conditions; The mass of right-handed gauge boson $W_R$ is assumed to be much heavier than the SM gauge boson $W_L$ , i.e, $M_{W_R} \\gg M_{W_L}$ .", "The mass eigenvalues for charged gauge bosons are given by $&&M_{W_1}^2 \\approx \\frac{1}{4} g_L^2 v^2\\,, \\quad M_{W_2}^2 \\approx \\frac{1}{2} g_R^2 v^2_R\\,.$ With $v_R\\simeq 15$  TeV and $g_L=g_R\\simeq 0.623$ , the $W_L-W_R$ mixing is found to be $|\\tan \\, 2\\xi | \\simeq \\frac{v_1 v_2}{v^2_R} \\propto \\frac{M^2_{W_L}}{M^2_{W_R}} \\le 10^{-4}\\,.$ The mass of $W_R$ is found to be around $7~$ TeV leading to negligible contributions to LFV processes via right-handed currents and mixed currents.", "The masses of scalar triplets and other scalars contained in the bidoublet are assumed to be larger than heavy neutrinos, i.e, $M_{\\Delta _{L,R}}, M_{H_1} \\gg M_{N,S}$ .", "Figure: Feynman diagram for lepton flavor violating process μ→eγ\\mu \\rightarrow e\\gamma due to the exchange of mass eigenstates of heavy neutrinos N i N_i and S i S_i." ], [ "Constraints on light neutrino masses from LFV", "In this section we derive constraints on absolute scale of lightest neutrino mass inluding both normal hierarchy (NH) and inverted hierarchy (IH) patterns from charged lepton flavour violating processes like $\\mu \\rightarrow e \\gamma $ , $\\mu \\rightarrow 3e$ and $\\mu \\rightarrow e$ conversion inside a nucleus.", "We do so by considering the new contributions to these decays due to large light-heavy neutrino mixing in our model.", "The general expression for such a decay can be written as, $&&\\Gamma ^{(0)}_{\\mu }\\equiv \\Gamma _\\nu (\\mu ^+ \\rightarrow e^+ \\nu _e \\overline{\\nu _\\mu }) \\\\&&\\Gamma ^{\\rm Z}_{\\rm capt.", "}\\equiv \\Gamma \\left(\\mu ^{-} + A(Z,N) \\rightarrow \\nu _\\mu + A(Z-1,N+1)\\right)$ and the expressions for branching ratios as, $&&\\text{Br}_{\\mu \\rightarrow e\\gamma } \\equiv \\frac{\\Gamma (\\mu \\rightarrow e \\gamma )}{\\Gamma ^{(0)}_{\\mu }} \\\\&&\\text{R}_{\\mu \\rightarrow e}^A \\equiv \\frac{\\Gamma \\left(\\mu + A(N,Z)\\rightarrow e + A(N,Z)\\right)}{\\Gamma ^{\\rm Z}_{\\rm capt.}}", "\\\\&&\\text{Br}_{\\mu \\rightarrow 3e}\\equiv \\frac{\\Gamma (\\mu \\rightarrow 3e)}{\\Gamma ^{(0)}_{\\mu }} \\, .$" ], [ "Bound on light neutrino mass from $\\mu \\rightarrow e\\gamma $", "The effective Lagrangian relevant for the lepton flavor violating process $\\mu \\rightarrow e \\gamma $ in our present work can be expressed as [69], [84], $\\begin{split}{\\cal L}_{\\mu \\rightarrow e} = &-\\frac{e g^2}{4(4\\pi )^2m_{W_L}^2}m_\\mu \\overline{e}\\sigma _{\\mu \\nu }(\\mathcal {G}^\\gamma _LP_L+\\mathcal {G}^\\gamma _RP_R)\\mu F^{\\mu \\nu } \\\\&-\\frac{\\alpha _W^2}{2m_{W_L}^2}\\sum _q \\left\\lbrace \\overline{e}\\gamma _\\mu \\left[\\mathcal {W}^q_LP_L+\\mathcal {W}^q_RP_R\\right]\\mu \\; \\overline{q}\\gamma ^\\mu q\\right\\rbrace + {\\rm h.c.},\\end{split}$ where $\\sigma _{\\mu \\nu } \\equiv \\frac{i}{2}[\\gamma _\\mu ,\\gamma _\\nu ]$ and the form factors are $\\mathcal {G}^\\gamma _{L,R}$ and $\\mathcal {W}^{u,d}_{L,R}$ .", "The relevant contributions to $\\mu \\rightarrow e\\gamma $ is given by $\\begin{split}i{\\cal M}(\\mu \\rightarrow e\\gamma ) &= \\frac{e\\alpha _W}{8\\pi m_{W_L}^2}\\epsilon _\\gamma ^\\mu \\overline{e}\\left[\\left(q^2\\gamma _\\mu - q_\\mu {q}\\right)\\left(\\mathcal {F}^\\gamma _L P_L+\\mathcal {F}^\\gamma _R P_R\\right) \\right.", "\\\\& \\left.", "-im_\\mu \\sigma _{\\mu \\nu }q^\\nu \\left(\\mathcal {G}^\\gamma _L P_L+\\mathcal {G}^\\gamma _R P_R\\right)\\right]\\mu ,\\end{split}$ where the anapole and dipole form factors $\\mathcal {F}^{\\gamma }_{L,R}$ and $\\mathcal {G}^\\gamma _{L,R}$ will be defined in subsequent paragraph.", "Figure: Branching ratio of the lepton flavor violating process, μ→eγ\\mu \\rightarrow e \\gamma as a function of lightest neutrinomass m 1 m_1 for NH and m 3 m_3 for IH.", "The blue (NH) and red (IH) coloured regions display the allow due to the exchange of heavy right handed neutrino (N R N_R) (left-panel) and heavy sterile neutrino (S L S_L) (right-panel).The analytic expression for the branching ratio for the lepton flavor violating process $\\mu \\rightarrow e\\gamma $ shown in Feynman diagram in Fig.REF due to mediation of heavy neutrinos We have neglected the other contributions to LFV processes due to right-handed gauge boson $W_R$ and $W_L-W_R$ mixing so that we can quantify the effect of light-heavy neutrino mixing on $\\text{Br}_{\\mu \\rightarrow e\\gamma }$ .", "Even though there are sizable contributions from scalar triplets [84], [111], [112], [85] we have not included them in our analysis by keeping very large mass scalar triplets.", "($N$ and $S$ ) is given by $\\text{Br}_{\\mu \\rightarrow e\\gamma } = \\frac{\\alpha _W^3s_W^2}{256\\pi ^2}\\frac{m_\\mu ^4}{M_{W_L}^4}\\frac{m_\\mu }{\\Gamma _\\mu }|\\mathcal {G}_\\gamma ^{\\mu e}|^2\\,, $ where $\\alpha _W=1/29.0$ as weak fine struture constant, $m_\\mu =105$  MeV being the muon mass, $M_{W_L}$ is the SM $W$ -boson mass, $s_W\\equiv \\sin \\theta _W$ in which $\\theta _W$ is the weak mixing angle and $\\Gamma _\\mu =2.996\\times 10^{-19}$ GeV [1] is the total decay width of the muon.", "The important parameter,$G_\\gamma ^{\\mu e}$ , for deriving constraints on light neutrino masses, is of the following form, $G_\\gamma ^{\\mu e} &=& \\bigg | \\sum _{i=1}^3 \\bigg \\lbrace {\\mbox{V}^{\\nu N}_{\\mu i}}^* {\\mbox{V}^{\\nu N}_{e i}} \\mathcal {G}_{\\gamma } \\left(x_{N_i}\\right)+ {\\mbox{V}^{\\nu S}_{\\mu i}}^* {\\mbox{V}^{\\nu S}_{e i}} \\mathcal {G}_{\\gamma } \\left(x_{S_i}\\right) \\bigg \\rbrace \\bigg |^2,$ where $x_{N_i}=m_{N_i}^2/M_{W_L}^2$ , $x_{S_i}=m_{S_i}^2/M_{W_L}^2$ and the form of loop function is given by $G_\\gamma (x) &=& -\\frac{x(2x^2+5x-1)}{4(1-x)^3}-\\frac{3x^3}{2(1-x)^4}\\ln x \\, .$ The form of loop function $\\mathcal {G}_\\gamma (x)$ and its dependance with change of lightest neutrino mass $m_1$ (for NH) and $m_3$ (for IH) is presented in appendix- .", "The other parameters $\\mbox{V}^{\\nu N}$ and $\\mbox{V}^{\\nu S}$ are mixing matrices representing mixing between light-active neutrinos with $N_R$ and $S_L$ , respectively.", "The variation of $\\text{Br}_{\\mu \\rightarrow e\\gamma }$ as a function of lightest neutrino mass is displayed in Fig.REF with contributions coming from purely $N_R$ is presented in left-panel and for $S_L$ contributions is shown right-panel while the combine contributing is presented in Fig.REF .", "The blue color (NH) and red color (IH) regions are model prediction on $\\mu \\rightarrow e \\gamma $ within $3-\\sigma $ allowed range of neutrino mixing angles as well as for mass squared differences.", "In x-axis, $m_1$ for NH ($m_3$ for IH) represents absolute value of light neutrino mass.", "As given in Table.REF , the current experimental limit and future sensitivity by MEG ($\\text{Br}_{\\mu \\rightarrow e\\gamma } \\le 4.2\\times 10^{-13}$ ), MEG upgrade ($\\text{Br}_{\\mu \\rightarrow e\\gamma } \\le 5.0\\times 10^{-14}$ ) and PRISM/PRIME ($\\text{Br}_{\\mu \\rightarrow e\\gamma } \\le 1.0\\times 10^{-16}$ ) on branching ratio for $\\mu \\rightarrow e \\gamma $ are presented in dashed horizontal lines.", "The vertical shaded regions are PLANCK bound on lightest neutrino mass with 95$\\%$ C.L..", "The estimated values of $\\text{Br}_{\\mu \\rightarrow e\\gamma }$ is mostly depend on the sum of the heavy-light neutrino mixing parameters like ${\\mbox{V}^{\\nu S}_{\\mu i}}^* {\\mbox{V}^{\\nu S}_{e i}}$ for $S_L$ mediated contribution and ${\\mbox{V}^{\\nu N}_{\\mu i}}^* {\\mbox{V}^{\\nu N}_{e i}}$ for $N_R$ mediated contribution.", "The band strucure of NH and IH scenarios are coming since we have taken $3\\sigma $ range of oscillation parameters presented in Table.REF as well as vary the phases; $\\delta $ between [$0,2\\pi $ ] and $\\alpha , \\beta $ between [$0,\\pi $ ].", "As presented in Table.REF , the current experimental limit on $\\text{Br}_{\\mu \\rightarrow e\\gamma }$ from MEG [105] ($<4.2 \\times 10^{-13}$ ) and the future experimental sensitivity from MEG Upgrade [106] ($< 5.0 \\times 10^{-14}\\,$ ) $\\&$ PRISM/PRIME [113] ($<1.0 \\times 10^{-16}\\,$ ) are satisfied by the predicted branching ratio and one can derive bound on lightest neutrino masses for NH as well as IH case.", "Saturating the current experimental MEG bound ($\\text{Br}_{\\mu \\rightarrow e\\gamma } \\le 4.2\\times 10^{-13}$ ), one can derive the limit on lightest neutrino mass in the range of meV scale in IH case while most of the parameters are ruled out for NH case.", "The projected experimental sensitivity from MEG Upgrade [106] ($< 5.0 \\times 10^{-14}\\,$ ) limits lightest neutrino mass less than few meV for NH while 4 meV for IH case.", "However, most of the paremeter spaces satisfying PRISM/PRIME [113] ($<1.0 \\times 10^{-16}\\,$ ) bound are lying in the quasi-degenerate (QD) pattern of light neutrino masses which already ruled out by PLANCK data.", "The predicted branching ratio is allowed by the future sensitivity for lightest neutrino mass $m_1 \\le 10^{-3}$  eV in case of $N_R$ exchange (left-panel Fig.REF ) and for interference term (Fig.REF ).", "But in case of $S_L$ exchange (right-panel Fig.REF ) small range of lightest neutrino mass ($m_1,m_3$ ) is allowed by future limit.", "Figure: Branching ratio of the process μ→eγ\\mu \\rightarrow e \\gamma vs lightest neutrino mass, m 1 m_1 (m 3 m_3) for NH (IH) due tothe combined effect of heavy right handed and sterile neutrino." ], [ "Bound on light neutrino mass from $\\mu \\rightarrow 3e$ and {{formula:f0f350ca-ba1d-46c0-977b-1a90ffbc9787}} conversion", "Under the assumptions that $M_{W_R} \\gg M_{W_L}$ , $\\tan \\xi \\rightarrow 0$ and for heavy scalar masses, the only relevant terms contributing to $\\mu \\rightarrow 3e$ as a result of light-heavy neutrino mixing is given by $\\text{Br}^{}_{\\mu \\rightarrow 3e}&=&\\frac{\\alpha _W^4 m_\\mu ^5}{24576\\pi ^3m_{W_L}^4\\Gamma _\\mu }\\bigg \\lbrace 2\\bigg [\\bigg | \\frac{1}{2} \\mathcal {B}^{\\mu 3e}_{LL} +\\mathcal {F}^{Z_1}_L - 2s_W^2(\\mathcal {F}^{Z_1}_L-\\mathcal {F}^\\gamma _L) \\bigg |^2 +\\bigg |\\frac{1}{2} \\mathcal {B}^{\\mu 3e}_{RR} \\bigg |^2 \\bigg ]\\nonumber \\\\&&\\hspace*{56.9055pt}+ \\bigg |2s_W^2(\\mathcal {F}^{Z_1}_L - \\mathcal {F}^\\gamma _L) \\bigg |^2+8 s_W^2 \\mbox{Re}\\left(2 \\mathcal {F}^{Z_1}_L+\\mathcal {B}^{\\mu 3e}_{LL} \\right) {\\mathcal {G}^\\gamma _R}^* \\nonumber \\\\&&\\hspace*{56.9055pt} -48 s_W^2 \\mbox{Re} \\left(\\mathcal {F}^{Z_1}_L-\\mathcal {F}^\\gamma _L \\right) {\\mathcal {G}^\\gamma _R}^*+32 s_W^4 |\\mathcal {G}^\\gamma _R|^2 \\left[\\ln \\frac{m^2_\\mu }{m^2_e} -\\frac{11}{4} \\right] \\bigg \\rbrace \\,$ where $m_e (m_\\mu )$ is mass of electron (muon), $s_W^2=\\sin ^2\\theta _W$ and other loop factors are presented in appendix-.", "The estimated branching ratio for the process $\\mu \\rightarrow 3e$ is satisfied by the experimental limit SINDRUM [104] ($<10^{-12}$ ) for a broad range of lightest neutrino mass $m_1,m_3 \\le 0.03$  eV.", "Figure: Branching ratio of the process μ→3e\\mu \\rightarrow 3e as a function of lightest neutrino mass, m 1 m_1 (m 3 m_3) for NH (IH) pattern.Fig.REF shows the variation of branching ratio of the process $\\mu \\rightarrow 3e$ with respect to the lightest neutrino mass where NH and IH pattern of light neutrno mass are represented by blue and red bands respectively.", "The bounds on the branching ratio of this process are given by SINDRUM ($10^{-12}$ ) and Mu3e ($10^{-16}$ ) and the plot shows that both the bounds are satisfied by the model's predictions on this decay (due to light-heavy mixing) for both patterns of light neutrino mass.", "Similarly, the bound on light neutrino mass from $\\mu \\rightarrow e$ conversion rate with the mediation of heavy neutrino N and sterile neutrino S can be found as follows.", "${\\rm R}^{A(N,Z)}_{\\mu \\rightarrow e} = \\frac{\\alpha _{\\rm em}^3\\alpha _W^4m_\\mu ^5}{16\\pi ^2m_{W_L}^4\\Gamma _{\\rm capt}}\\frac{Z_{\\rm eff}^4}{Z}\\left|\\mathcal {F}(-m_\\mu ^2)\\right|^2\\left(|\\mathcal {Q}_L^W|^2+|\\mathcal {Q}_R^W|^2\\right), $ where the relevant parameters are given in appendix .", "In Fig.REF we have plotted the variation of $\\mu \\rightarrow e$ conversion against the lightest neutrino mass where the red curve represents the IH pattern of light neutrino mass and blue band represents the NH pattern of light neutrino mass.", "The experimental sensitivity of $\\mu \\rightarrow e$ conversion is represented by the horizontal dashed line.", "The plot shows that the model's predictions on decay rate of this conversion (due to contributions from heavy neutrino $N_R$ and sterile neutrino $S_L$ ) are not sensitive to the experimental bounds, for both NH and IH pattern of light neutrino mass.", "Figure: The rate for μ→e\\mu \\rightarrow e conversion in Au nucleus as a function of the lightest neutrino mass, m 1 m_1 (m 3 m_3) for NH (IH) due to light-heavy mixing." ], [ "Constraints from neutrinoless double beta decay", "We discuss here how the light-heavy neutrino mixing in left-right theories with type-II seesaw dominance leads to sizable new contributions to neutrinoless double beta decay.", "We give emphasis on left-handed current effects due to the exchange of heavy neutrinos $N_R$ and $S_L$ .", "To ignore the effects of right-handed currents and the contributions of doubly charged scalar triplets to $0\\nu \\beta \\beta $ transition we have assumed $M_{W_R} \\gg M_{W_L}$ and large masses for scalar triplets.", "For a detailed discussion on various new physics contributions to neutrinoless double beta decay in TeV scale left-right symmetric model with large light-heavy neutrino mixing through type-II seesaw dominance one may refer  [97].", "Figure: Feynman diagrams for 0νββ0\\, \\nu \\, \\beta \\beta transition due to purely left-handedcurrents with the exchange of virtual Majorana neutrinos ν i \\nu _i, N j N_j and S k S_k." ], [ "Half-life and effective Majorana mass", "The relevant contributions arising from purely left-handed currents with the exchange of light active neutrinos (standard mechanism) and heavy neutrinos $N$ and $S$ are shown in Fig.REF .", "The inverse half-life of $0\\nu \\beta \\beta $ transition for a given isotope is, $\\left[T_{1/2}^{0\\nu }\\right]^{-1} &=& G^{0\\nu }_{01}\\bigg [ |{\\cal M}^{0\\nu }_\\nu \\cdot \\eta _\\nu |^2 +|{\\cal M}^{0\\nu }_N \\cdot \\left(\\eta _{N}+ \\eta _{S} \\right) \\big |^2 \\bigg ]$ where the dimensionless lepton number violating parameters are given as, $& &|\\mathcal {\\eta }_{\\nu }| = \\sum _{i=1,2,3} \\frac{{\\mbox{V}^{\\nu \\nu }_{ei}}^2\\, m_{\\nu _i}}{m_e} \\,,|\\mathcal {\\eta }_{N}| = m_p \\sum _{i=1,2,3} \\frac{{\\mbox{V}^{\\nu N}_{ei}}^2}{M_{N_i}} \\,,|\\mathcal {\\eta }_{S}| = m_p \\sum _{i=1,2,3} \\frac{{\\mbox{V}^{\\nu S}_{ei}}^2}{M_{S_i}} \\,.$ In order to derive the bound on lightest neutrino mass by satuaring the current experimental bounds on half-life of a given isotope, one has to rewrite the inverse half-life in terms of a particle physics paremeter called Effective Majorana mass that contains lepton number violating information in it.", "$\\left[T_{1/2}^{0\\nu }\\right]^{-1} &=& G^{0\\nu }_{01} \\bigg | \\frac{{\\cal M}^{0\\nu }_\\nu }{m_e} \\bigg |^2 \\bigg [\\big |m^{\\nu }_{ee} \\big |^2+ \\big |m^{N}_{ee} \\big |^2+ \\big |m^{S}_{ee} \\big |^2 \\bigg ] \\nonumber \\\\&=&G^{0\\nu }_{01} \\bigg ( \\frac{{\\cal M}^{0\\nu }_\\nu }{m_e}\\bigg )^2 \\cdot |m^{\\rm eff}_{\\beta \\beta }|^2\\,.$ Here $|m^{\\rm eff}_{\\beta \\beta }|^2$ is the sum of contributions from light active neutrinos $\\nu _L$ , heavy right-handed neutrinos $N$ and sterile neutrinos $S$ .", "Thus the half-life of neutrinoless double beta decay process is estimated by three kinds of contributions; Phase-space factor $G^{0\\nu }$ which is responsible for detailed kinematics of the neutrinoless double beta decay process and is highly energy dependent, Nuclear Matrix Elements (NMEs), $\\mathcal {M}^{0\\nu }_{\\nu }$ and $\\mathcal {M}^{0\\nu }_{N}$ for light and heavy neutrinos that take care of the transition of the nucleus into daughter nuclei, Particle physics parameter called Effective Majorana mass $m^{\\rm eff}_{\\beta \\beta }$ of the transition $2 d \\rightarrow 2 u + 2 e^-$ inside the involved nucleons.", "Table: G 01 0ν G^{0\\nu }_{01} and NMEs The values of $G^{0\\nu }$ and NMEs are different for different isotopes as presented in Table.REF and the effective Majorana mass parameter $m^{\\rm eff}_{\\beta \\beta }$ is expressed in terms half-life of a given isotope as $&&\\left[T_{1/2}^{0\\nu }\\right]^{-1} = G^{0\\nu }_{01} |\\mathcal {M}^{0\\nu } (A) |^2 \\cdot \\bigg (\\frac{m^{\\rm eff}_{\\beta \\beta }}{m_e} \\bigg )^2 \\, \\nonumber \\\\&&m^{\\rm eff}_{\\beta \\beta } = \\frac{m_e}{\\sqrt{G^{0\\nu }_{01} T^{0\\nu }_{1/2}}}\\,.$ Using Table.REF and Table.REF , one can derive theoretical limits on effective Majorana mass which we use in our numerical estimations to derive limits on absolute value of light neutrino masees.", "Table: Limits on the half-life of 0νββ0\\nu \\beta \\beta ." ], [ "Bound on lightest neutrino mass from $0\\nu \\beta \\beta $", "Standard mechanism: $m^{\\nu }_{ee}$ from light neutrinos $\\nu $ :- The standard contribution to effective Majorana mass due to exchange of light neutrinos is given by, $m^{\\nu }_{ee} = \\sum _{i=1,2,3} \\left(\\mbox{V}^{\\nu \\nu }\\right)^2_{ei}\\, m_{i}\\, m_{i} \\simeq \\sum _{i = 1}^3 (U_\\text{PMNS})_{e i}^2 \\, m_i\\,.$ where $V^{\\nu \\nu }$ is the mixing matrix containing non-unitarity effects which is approximated to be $U_{\\rm PMNS}$ .", "Figure: Left panel: Effective Majorana mass parameter (|m ee ||m_{ee}|) due to standard mechanism as a function of lightest neutrino mass m 1 m_1 (m 3 m_3) for NH (IH).", "Right panel: SM contribution to the half-life of 0νββ0\\nu \\beta \\beta transition as a function of lightest neutrino mass.", "The blue horizontal bands show the limits on effective Majorana mass and half-life from GERDA and EXO+KamLAND-Zen experiments.", "The vertical purple bands are for the constraints on sum of light neutrino masses from cosmology data (PLANCK 2018) and KATRIN detector.New physics contribution: $m^{N,S}_{ee}$ from heavy neutrinos $N_R$ and $S_L$ :- Figure: Left panel: New physics contribution to the plot of effective Majorana mass as a function of lightest neutrino mass, m 1 m_1 (m 3 m_3) for NH (IH) via W L -W L W_L-W_L channel with the exchange of heavy neutrino N R N_R and sterile neutrino S L S_L.", "Right panel: Contributions of N R N_R and S L S_L to the plot of half-life vs lightest neutrino mass.The effective Majorana masses due to the contributions of heavy neutrinos $N_R$ and $S_L$ to $0\\nu \\beta \\beta $ decay (represented by the second and third Feynman diagrams in Fig.REF ) are as follows, $&&|m^{N}_{ee} | = \\langle p^2 \\rangle \\sum _{i=1,2,3} \\frac{\\left(\\mbox{V}^{\\nu N}\\right)^2_{ei}}{M_{N_i}} \\,, \\nonumber \\\\&&|m^{S}_{ee} | = \\langle p^2 \\rangle \\sum _{i=1,2,3} \\frac{\\left(\\mbox{V}^{\\nu S}\\right)^2_{ei}}{M_{S_i}} \\,.$ Here $M_{N_i} (M_{S_i})$ are the mass eigenvalues of heavy right-handed (sterile) neutrinos.", "It is to be noted here that the type-II seesaw dominance scheme in our model establishes a relationship between these mass eigenvalues and the light neutrino masses.", "$\\mbox{V}^{\\nu N}$ and $\\mbox{V}^{\\nu S}$ are the mixing matrices which represent the mixing between light-heavy neutrinos ( $\\nu $ and $N_R$ ) and active-sterile neutrinos ($\\nu $ and $S_L$ ) respectively.", "The neutrino virtual momentum $\\langle p^2 \\rangle $ plays an important role as the half-life formula for $0\\nu \\beta \\beta $ transition differs for $m_i(M_i) \\ll \\langle p^2 \\rangle $ or $m_i(M_i) \\gg \\langle p^2 \\rangle $ .", "The typical expression for neutrino virtual momentum $\\langle p^2 \\rangle $ is written as, $\\langle p^2 \\rangle = - m_e\\,m_p\\, \\frac{{\\cal M}^{0\\nu }_N}{{\\cal M}^{0\\nu }_\\nu } \\simeq (\\mbox{200\\, MeV})^2\\,.$ Fig.REF shows the effective Majorana mass parameter in the left panel and half-life in the right panel due to the standard mechanism as a function of lightest neutrino mass $m_1$ (NH) and $m_3$ (IH).", "Both the plots show that the quasi-degenerate (QD) pattern of light neutrinos, i.e.", "$m_1\\simeq m_2\\simeq m_3$ is disfavoured by cosmology data on sum of light neutrino masses whereas the NH(green band) and IH(red band) patterns may not be probed even by the next generation experiments.", "This propels the idea of exploring possible new physics contributions to neutrinoless double beta decay which might give a hint on mass hierarchy and lightest neutrino mass.", "Such possibility occurs when the contributions of heavy neutrino $N_R$ and sterile neutrino $S_L$ are considered.", "The same is shown in Fig.REF .", "The contributions of $N_R$ and $S_L$ to effective Majorana mass parameter and half-life predictions makes both NH(blue band) and IH(orange band) patterns of light neutrino masses sensitive to the current experimental bounds.", "In Fig.REF the contributions of all three types of neutrinos ($\\nu , N_R, S_L$ ) are summed up and plotted against lightest neutrino mass.", "In this case also the both NH(green band) and IH(red band) patterns saturate the current experimental bounds on effective Majorana mass and half-life.", "In all the three figures the horizontal blue bands stand for the improved limits on $0\\nu \\beta \\beta $ decay on effective Majorana mass and half-life with the combined results of GERDA and KamLAND-Zen experiments.", "The vertial magenta region is disfavoured by Planck-2018 [115] and KATRIN data [116] on sum of light neutrino masses.", "In these plots the largest value of right-handed neutrino mass is fixed at $M_{N_R}=1$ TeV while keeping $M_{W_R}, M_{\\Delta ^{++}} >> M_{N_R}$ .", "The Majorana phases and Dirac CP-phase are varied between $0 \\rightarrow \\pi $ for all the plots while other neutrino oscillation parameters are taken in their allowed $3\\sigma $ range.", "The predictions of our model on lightest neutrino mass due to heavy and sterile neutrino contributions to neutrinoless double beta decay are as follows, When the contributions of only $N_R$ and $S_L$ (excluding standard contribution) are considered, the allowed values of lightest neutrino mass are found to be in the range of $10-25$  meV for $m_1$ (NH) and $25-40$  meV for $m_3$ (IH).", "This is done by saturating the effective Majorana mass $m^{N+S}_{ee}$ with the current GERDA and KamLAND-Zen bounds.", "It remains the same for half-life.", "When the contributions of all three types of neutrinos ($\\nu , N_R, S_L$ ) are considered the predicted values of lightest neutrino mass lies in the range of $2-15$  meV for $m_1$ (NH) and $15-25$  meV for $m_3$ (IH) by sturating $m^{\\nu +N+S}_{ee}$ with the current experimental limits.", "Using these predicted values of lightest neutrino mass other light neutrino mass eigenvalues and heavy neutrino masses can be derived.", "Figure: Effective Majorana mass (left panel) and half-life (right panel) as a function of lightest neutrino mass m 1 m_1 (m 3 m_3) for NH (IH) due to the combined contributions of standard mechanism, N R N_R and S L S_L mediated diagrams." ], [ "Correlation between $m_{\\beta }$ and sum of light neutrino masses {{formula:daab3bda-ceea-46d4-84fd-d89b0a1b1250}}", "In this subsection we examine how the combined constraints from single beta decay and cosmology can limit absolute scale of light neutrino mass.", "The important parameter for single $\\beta $ decay sensitive to the electron neutrino mass is defined as follows.", "$m_{\\beta } = \\sqrt{\\sum _{i} \\mid U^2_{ei} m^2_i\\mid }$ In the present model $ U^2_{ei}$ is replaced by $\\mathcal {V}^{\\nu \\nu ^2}_{ei}$ due to sizeable non-unitarity effects.", "The current limit on $m_{\\beta }$ from KATRIN experiment is $m_{\\beta } < 1.1\\,$ eV while an improved bound of $0.2$  eV [116] is expected in the future.", "The sum of light neutrino masses is tightly constrained by Planck 2018 data [115] which is $\\sum m_i=m_1+m_2+m_3 < 0.12\\,$  eV.", "Fig.REF shows the variation of $\\sum m_i$ and $m_{\\beta }$ with lightest neutrino mass.", "Figure: Plot showing the variation of ∑m i \\sum m_i and m β m_\\beta with lightest neutrino mass (m 1 m_1 for NH, m 3 m_3 for IH)." ], [ "Correlation between $m_{ee}$ and sum of light neutrino masses {{formula:a128db61-de7d-47cd-a3aa-582467c45810}}", "Here we discuss the model predictions on effective Majorana mass by changing the sum of light neutrino masses $\\Sigma m_i$ .", "We use the following limits on sum of light neutrino masses [117], [118], [119] for displaying the allowed region of $m_{ee}$ in Fig.REF .", "$&&m_\\Sigma < \\mbox{84\\,meV}\\quad \\quad ~(1\\sigma ~\\mbox{C.L.})", "\\nonumber \\\\&&m_\\Sigma < \\mbox{146\\,meV}\\quad \\quad (2\\sigma ~\\mbox{C.L.})", "\\nonumber \\\\&&m_\\Sigma < \\mbox{208\\,meV}\\quad \\quad (3\\sigma ~\\mbox{C.L.", "})$ In both the plots of Fig.REF the horizontal bands represents the experimental bounds on effective Majorana mass $m_{ee}$ by GERDA and EXO+KamLAND-Zen while the vertical dashed line shows the bound on sum of light neutrino masses from cosmology.", "The plot in the left-panel shows the variation of effective Majorana mass parameter due to standard mechanism with sum of light neutrino masses.", "It shows that both NH (green band) and IH (red band) patterns of light neutrino masses are not sensitive to the current experimetal bounds on $m_{ee}$ and also disfavoured by cosmology when only light neutrino ($\\nu $ ) contribution is considered.", "The plot in the right-panel shows the variation of effective Majorana mass parameter due to new physics contributions (contributions of $N_R$ and $S_L$ ) with sum of light neutrino masses.", "It shows that NH (purple band) pattern of light neutrino masses saturates the experimental bound on $m_{ee}$ as well as the cosmology bound on sum of light neutrino masses whereas IH (blue band) pattern is disfavoured by cosmology.", "Thus the model with type-II seesaw dominance gives an important result on the hierarchy of light neutrino masses when the contributions of heavy and sterile neutrinos are considered.", "Figure: Allowed region of effective Majorana mass parameter (|m ee ||m_{ee}|) as a function of sum of light neutrino masses (Σm i \\Sigma m_i) for standard mechanism (left-panel) and new physics contributions from N R N_R and S L S_L mediated diagrams (right-panel).We also do a comparative analysis between neutrinoless double beta decay and LFV processes to examine how light-heavy neutrino mixing is constrained from a combined study of LNV and LFV.", "Figure: Correlation plot between LFV processes μ→eγ\\mu \\rightarrow e \\gamma and LNV 0νββ0\\nu \\beta \\beta decay for 76 ^{76}Ge isotope." ], [ "Electric dipole moment (EDM) of charged leptons", "The dipole moment of electron is an important process whose experimental observation will reveal violation of Parity $P$ and Time reversal $T$ symmetry (or violation of CP invariance) in nature.", "The electric dipole moment formula derived for charged leptons $\\ell _\\alpha (\\alpha =e, \\mu , \\tau )$ is given by [93], [66], [120], [69] $d_\\alpha = \\frac{e \\alpha _{\\rm \\small W}}{8 \\pi M^2_{W_L}}\\mbox{\\Large Im}\\bigg [\\sum ^{3}_{i=1} {\\mbox{V}}^{\\nu N}_{\\alpha i} {\\mbox{V}}^{N N}_{i \\alpha }\\, \\xi \\,\\mathcal {G}^{\\gamma }_{2}(x_{N_i}) M_{N_i}\\bigg ]\\,,$ where $\\alpha _W \\simeq 1/30$ is weak fine structure constant, $M_{W_L}=80.3$  GeV, $\\xi \\le 10^{-4}$ is the $W_L-W_R$ mixing and $M_N$ is the mass matrix for heavy neutrinos.", "The mixing matrices are $\\mbox{V}^{\\nu N}=(v_L/v_R) M_D U_{\\rm PMNS} m^{-1}_\\nu $ , $\\mbox{V}^{N N} =U_{\\rm PMNS}$ .", "The resulting dipole moment for electron can be expressed as a function of lightest neutrino mass and one can derive similar bound on absolute scale lightest neutrino mass by saturating the experimental bound.", "In the present work, we can go to a basis where both Dirac neutrino mass matrix $M_D$ connecting $\\nu _L-N_R$ and heavy Dirac mass $M$ connecting $N_R-S_L$ can be diagonal simultaneously.", "Taking $M_D \\simeq 80 \\mbox{diag}(m_e, m_\\mu , m_\\tau )$  GeV and $M_D \\simeq 500 \\mbox{diag}(1, 1, 1)$  GeV, we found that the dipole moment only depends upon PMNS phases contributing to the imaginary part.", "Considering vanishing Majorana phases, the variation of electron dipole moment with Dirac CP-phase $\\delta _{\\rm CP}$ is shown in Fig.REF The horizontal line represents the current bound on electron EDM set by ACME [121].", "Figure: Electric dipole moment, d e d_e as a function of Dirac CP-phase, δ CP \\delta _{\\rm CP} for NH and IH patterns of light neutrino masses considering diagonal structures for Dirac neutrino mass matrix M D M_D and N R -S L N_R - S_L mixing matrix MM." ], [ "Comments on muon (g-2) anomaly", "An intensive activity is going on to explain the recent results declared by Fermi National Accelerator Laboratory (FNAL) on the measurement of muon magnetic dipole moment, $a_{\\mu }= \\frac{g_{\\mu }-2}{2}$ that just confirmed the existing anomaly in the muon sector.", "The theoretical value of $a_{\\mu }$ estimated by Standard Model [122], [123], [124], [125], the measured value by Brookhaven National Laboratoty (BNL) [124] and the recent result by FNAL [125] are as follows.", "$a_{\\mu }^{\\text{SM}} = (11659181.0 \\pm 4.3) \\times 10^{-10} \\\\a_{\\mu }^{\\text{BNL}} = (11659208.0 \\pm 6.3)\\times 10^{-10} \\\\a_{\\mu }^{\\text{FNAL}} = (11659204.0 \\pm 5.4)\\times 10^{-10}$ Combining both the results of BNL and FNAL, the new world average now stands at, $a_{\\mu }^{\\text{2021}} = (11659206.1 \\pm 4.1)\\times 10^{-10}\\, .", "\\\\$ Previously there existed a 3.7 $\\sigma $ deviation in the SM's estimated value and the measured value by BNL which has now sharpened to 4.2 $\\sigma $ considering the FNAL result with a precise deviation of, $\\Delta a_{\\mu }^{\\text{2021}}=(a_{\\mu }^{\\text{BNL}}+a_{\\mu }^{\\text{FNAL}})-a_{\\mu }^{\\text{SM}}= (25.1 \\pm 5.9)\\times 10^{-10} .\\\\$ Many new physics scenarios have been proposed so far to explain the anomaly, for an incomplete list of which one may refer [126], [90], [127], [128] (and references therein) .", "In this framework new contributions to $a_{\\mu }$ arise from the interactions of sterile neutrino $S_L$ due to large mixing with muons.", "This contribution depends on sterile neutrino mass and its mixing with muons, which are related to light neutrino masses and oscillation parameters in type-II seesaw dominance scheme as discussed earlier.", "Thus, this scenario opens the possibility of constraning light neutrino masses and mass hierarchy from FNAL results on $a_{\\mu }$ .", "Other sizeable contributions can arise from $W_R$ mediated channels and scalar exchange if one considers left-right symmetry breaking at TeV scale.", "Such cases are studied in one of our recent works [127], where we have extensively calculated the contributions of neutral fermions, gauge bosons and scalars to $a_{\\mu }$ in a TeV scale extended Left-right model." ], [ "Conclusion", "We considered a new mechanism of natural type-II seesaw dominance that allows large light-heavy neutrino mixing within a framework of left-right symmetric model.", "The fermion sector of the model contains usual quarks and leptons plus an extra sterile neutrino per generation while its scalar sector consists of Higgs doublets, triplets and bidoublet.", "The extra particles help in generating large light-heavy neutrino mixing that gives new contributions to various LFV processes like $\\mu \\rightarrow e \\gamma $ , $\\mu \\rightarrow 3 e$ and $\\mu \\rightarrow e$ conversion inside nuclei.", "As a result of these new contributions the new branching ratios can be accessible to present as well as planned experiments.", "We have demonstrated how the model parameters are suitably adjusted to make the contribution of inverse seesaw and loop induced light neutrino masses sub-dominant.", "Rather we have generated the light neutrino masses through type-II seesaw mechanism.", "All the physical masses and mixing of neutral leptons are expressed in terms of light neutrino masses and PMNS mixing matrix.", "Thus the new physics contributions to various LFV processes arising from heavy and sterile neutrinos depend upon light neutrino mass.", "As a result of this, the bound on absolute scale of light neutrino mass can be derived by saturating the experimental limits on LFV processes, which we find to be in the meV range.", "We have plotted branching ratios for LFV processes such as $\\mu \\rightarrow e \\gamma $ , $\\mu \\rightarrow 3 e$ and $\\mu \\rightarrow e$ conversion inside nuclei as a function of the lightest neutrino mass for NH as well as IH pattern of light neutrinos in order to derive the bound on light neutrino mass.", "We have also plotted dipole moment of electron as a function of light neutrino mass and found that the result depends only upon PMNS phase.", "Thus in our model the dipole moment of electron can uniquely say about the relative phases of PMNS mixing matrix.", "We have also studied the new contributions to neutrinoless double beta decay arising from purely left-handed current due to exchange of heavy Majorana neutrinos in our model.", "It is found that these new contributions can saturate the experimental bound for $m_{1} > 0.001$ eV for NH.", "We have shown the correlation between LFV and LNV by focusing on $\\text{Br}_{\\mu \\rightarrow e\\gamma } \\mbox{vs.} T_{1/2}^{0\\nu } [^{76}\\mbox{Ge}]$ in Fig.", "REF ." ], [ "Acknowledgements", "The authors would like to thank P.S.", "Bhupal Dev for the useful discussions during the early stage of this work." ], [ "ELRSM mass matrix and form of unitary mixing matrix", "We discuss here the implementation of extended seesaw mechanism for neutrino masses and mixing and derivation of type-II seesaw dominance within left-right symmetric models.", "Apart from usual quarks and leptons, the fermion sector is extended with an extra sterile neutrino $\\nu _{S_L}$ (or $\\nu _S$ in short-hand notation) while scalar sector consists of usual scalar bidoublet $\\Phi $ , doublets ($H_R, H_L$ ) and triplets ($\\Delta _R, \\Delta _L$ ).", "The left-right symmetry is broken down to SM theory through spontaneous symmetry breaking when the scalars $H_R$ or $\\Delta _R, \\Delta _L$ take VEV.", "The scalar $H_L$ doesn't play any role here, rather it is present due to the left-right symmetry.", "The SM theory further breaks down to low energy theory with the help of bidoblet $\\Phi $ .", "We call this scheme the extended left-right seesaw model (ELRSM).", "The neutral fermions neeeded for ELRSM are active left-handed neutrinos, $\\nu _L$ , active right-handed neutrinos, $\\nu _R$ and sterile neutrinos, $\\nu _S$ .", "The relevant mass terms within ELRSM are given by $\\mathcal {L}_{\\rm ELRSM} &=& \\mathcal {L}_{M_D}+\\mathcal {L}_{M} + \\mathcal {L}_{M_{L}} + \\mathcal {L}_{M_R} + \\mathcal {L}_{\\mu _S} \\nonumber \\\\\\mathcal {L}_{M_D} &=& - \\sum _{\\alpha , \\beta } \\overline{\\nu _{\\alpha L}} [M_D]_{\\alpha \\beta } \\nu _{\\beta R} \\mbox{+ h.c.}\\nonumber \\\\\\mathcal {L}_{M_{}} &=& - \\sum _{\\alpha , \\beta } \\overline{\\nu _{\\alpha S}} [M_{}]_{\\alpha \\beta } \\nu _{\\beta R} \\mbox{+ h.c.}\\nonumber \\\\\\mathcal {L}_{M_{L}} &=& - \\frac{1}{2} \\sum _{\\alpha , \\beta } \\overline{\\nu ^c_{\\alpha L}} [M_{L}]_{\\alpha \\beta } \\nu _{\\beta L} \\mbox{+ h.c.}\\nonumber \\\\\\mathcal {L}_{M_{R}} &=& - \\frac{1}{2} \\sum _{\\alpha , \\beta } \\overline{\\nu ^c_{\\alpha R}} [M_{R}]_{\\alpha \\beta } \\nu _{\\beta R} \\mbox{+ h.c.}\\nonumber \\\\\\mathcal {L}_{\\mu _{S}} &=& - \\frac{1}{2} \\sum _{\\alpha , \\beta } \\overline{\\nu ^c_{\\alpha S}} [\\mu _{S}]_{\\alpha \\beta } \\nu _{\\beta S} \\mbox{+ h.c.}$ The flavour states for active left-handed neutrinos $\\nu _{\\alpha L}$ , sterile neutrinos $\\nu _{\\beta S}$ and right-handed neutrinos $\\nu _{\\gamma R}$ are defined as follows $\\nu _{\\alpha L} = \\begin{pmatrix}\\nu _{eL} \\\\ \\nu _{\\mu L} \\\\ \\nu _{\\tau L}\\end{pmatrix}\\, , \\quad \\nu _{\\beta S} = \\begin{pmatrix}\\nu _{S_{1 L}} \\\\ \\nu _{S_{2 L}} \\\\ \\nu _{S_{3 L}}\\end{pmatrix}\\, , \\quad \\nu _{\\gamma R} = \\begin{pmatrix}\\nu _{N_{1 R}} \\\\ \\nu _{N_{2 R}} \\\\ \\nu _{N_{3 R}}\\end{pmatrix}\\, .$ Similarly, the mass states for neutral fermions are given by $\\nu _{i L} = \\begin{pmatrix}\\nu _{1L} \\\\ \\nu _{2 L} \\\\ \\nu _{3L}\\end{pmatrix}\\, , \\quad S_{j L} = \\begin{pmatrix}S_{1 L} \\\\ S_{2 L} \\\\ S_{3 L}\\end{pmatrix}\\, , \\quad N^c_{k R} = \\begin{pmatrix}N^c_{1 R} \\\\ N^c_{2 R} \\\\ N^c_{3 R}\\end{pmatrix}\\, .$ In the flavour basis $\\left( \\nu _L, \\nu _S, \\nu ^c_R\\right)$ the resulting $9\\times 9$ neutral lepton mass matrix within ELRSM is given by, $\\mathcal {M}_{\\rm ELRSM}= \\begin{pmatrix}M_L & 0 & M_D \\\\0 & \\mu _S & M_{} \\\\M^T_D & M^T_{} & M_R\\end{pmatrix}$ Here $M_D$ is the Dirac mass term connecting $\\nu _L-\\nu _R$ , $M_{}$ is the mixing term between $\\nu _R-\\nu _S$ while $M_L$ , $M_R$ , $\\mu _S$ are Majorana mass terms for $\\nu _L$ , $\\nu _R$ and $\\nu _S$ , respectively.", "The mass hierarchy is given by $M_R \\gg M_{} > M_D > \\mu _S \\gg M_L\\,.$ The diagonalisation of $\\mathcal {M_{\\rm ELRSM}}$ after changing it from flavour basis to mass basis is done by a generalized unitary transformation as, $&&\\mid \\Psi \\rangle _{\\rm flavor} = \\mathcal {V}\\mid \\Psi \\rangle _{\\rm mass}\\\\&\\mbox{or,}& \\begin{pmatrix}\\nu _{\\alpha L}\\\\ \\nu _{\\beta S}\\\\ \\nu ^c_{\\gamma R}\\end{pmatrix}=\\begin{pmatrix}\\mathcal {V}_{\\alpha i}^{\\nu \\nu } & \\mathcal {V}_{\\alpha j}^{\\nu S} & \\mathcal {V}_{\\alpha k}^{\\nu N}\\\\\\mathcal {V}_{\\beta i}^{ S \\nu } & \\mathcal {V}_{\\beta j}^{S S} & \\mathcal {V}_{\\beta k}^{S N}\\\\\\mathcal {V}_{\\gamma i}^{N \\nu } & \\mathcal {V}_{\\gamma j}^{N S} & \\mathcal {V}_{\\gamma k}^{N N}\\end{pmatrix}\\begin{pmatrix}\\nu _{i}\\\\ S_{j}\\\\ N^c_{k}\\end{pmatrix} \\, \\\\&& \\mathcal {V}^{\\dagger } \\mathcal {M_{\\rm ELRSM}} \\mathcal {V}^*= \\mathcal {\\widehat{M}}_{\\rm ELRSM} \\nonumber \\\\&&\\hspace*{62.59596pt} = \\mbox{diag} \\left(m_{i},m_{S_j},m_{N_k} \\right) \\nonumber \\\\&&\\hspace*{62.59596pt} = \\mbox{diag} \\left(m_{1},m_{2},m_{3},m_{S_1},m_{S_2},m_{S_3},m_{N_1},m_{N_2},m_{N_3} \\right)$ Here the indices $\\alpha , \\beta , \\gamma $ run over three generations of light left-handed neutrinos, sterile neutrinos and heavy right-handed neutrinos in flavor basis respectively, whereas the indices $i,j,k$ run over corresponding mass states." ], [ "Seesaw block diagonalization for ELRSM neutrino mass matrix", "The block diagonaliztion is done in two steps.", "In the first step, the ELRSM neutrino mass matrix $\\mathcal {M_{\\rm ELRSM}}$ is reduced to an `Intermediate Block Diagonal' form $\\mathcal {M_{\\rm IBD}}$ and in the second step this $\\mathcal {M_{\\rm IBD}}$ is further diagonalised to $\\mathcal {M_{\\rm BD}} = \\mbox{diag}\\left(m_\\nu , m_S, m_N \\right)$ , from which we obtain the mass formulae for three types of neutrinos $\\nu _L, \\nu _S, \\nu _R$ as the three diagonal elements.", "Finally, in order to get the physical masses for all types of neutrinos we need unitary transformations for individual mass matrices as, $\\mathcal {M_{\\rm diag}} = \\mathcal {\\widehat{M}}_{\\rm ELRSM} =\\mbox{diag} \\left(m_{i},m_{S_j},m_{N_k} \\right) = \\mbox{diag} \\left(m_{1},m_{2},m_{3},m_{S_1},m_{S_2},m_{S_3},m_{N_1},m_{N_2},m_{N_3} \\right)$" ], [ "Determination of $\\mathcal {M_{\\rm IBD}}$", "Let us first rewrite the ELRSM mass matrix $\\mathcal {M}_{\\rm ELRSM}$ given in eq.", "(REF ) to generic type-I+II seesaw [129] as, $\\mathcal {M_{\\rm ELRSM}} &=& \\begin{pmatrix}\\mathcal {M}_{L} & \\mathcal {M}^T_{D} \\\\\\mathcal {M}_{D} & \\mathcal {M}_{R}\\end{pmatrix} \\, , \\nonumber \\\\\\mbox{where,} && \\mathcal {M}_{L}= \\begin{pmatrix}M_L & 0 \\\\0 & \\mu _S\\end{pmatrix}_{6 \\times 6}\\,, \\quad \\mathcal {M}_{D}= \\begin{pmatrix}M^T_D & M^T_{}\\end{pmatrix}_{3 \\times 6}\\, , \\quad \\mathcal {M}_{R} = M_R\\,.$ Using the seesaw approximations given in eq.", "(REF ), it can be shown that $|\\mathcal {M}_{R}| \\gg \\mathcal {M}_{D} \\gg \\mathcal {M}_{L}$ .", "First $\\mathcal {M_{\\rm ELRSM}}$ can be simplified to intermediate block diagonalized form $\\mathcal {M_{\\rm IBD}} $ by integrating out the heaviest right-handed neutrinos from other neutral states.", "Thus, the first block diagonalized approximate unitary mixing matrix $\\mathcal {W}_1$ gives $&&\\mathcal {W}^T_1 \\mathcal {M}^{}_{\\rm ELRSM} \\mathcal {W}_1=\\mathcal {M}^{}_{\\rm IBD} \\nonumber \\\\\\text{where},&& \\mathcal {M}^{}_{\\rm IBD} = \\begin{pmatrix}M^{\\rm light}_{\\rm Eff} & {\\bf 0}_{3 \\times 6} \\\\{\\bf 0}_{3 \\times 6} & M^{\\rm heavy}_{}\\end{pmatrix}, \\nonumber \\\\\\text{and}&&\\mathcal {W}_1=\\begin{pmatrix}\\sqrt{1-\\mathcal {B}\\mathcal {B}^{\\dagger }} & \\mathcal {B}\\\\-\\mathcal {B}^{\\dagger } & \\sqrt{1-\\mathcal {B}^{\\dagger }\\mathcal {B}}\\end{pmatrix}$ Now $\\mathcal {M_{\\rm IBD}} $ gives effective mass matrix in the modified basis of left-handed active and sterile neutrinos as, $M^{\\rm light}_{\\rm Eff} &\\equiv & M^{\\rm Eff}_{ELRSM} = \\mathcal {M}_{L}- \\mathcal {M}^T_{D} \\mathcal {M}^{-1}_{R} \\mathcal {M}_{D} \\nonumber \\\\&=&\\begin{pmatrix}M_L & 0 \\\\0 & \\mu _S\\end{pmatrix} - \\begin{pmatrix}M_D \\\\M_{}\\end{pmatrix} M^{-1}_R\\begin{pmatrix}M^T_D & M^T_{}\\end{pmatrix} \\nonumber \\\\&=& \\begin{pmatrix}M_L & 0 \\\\0 & \\mu _S\\end{pmatrix} - \\begin{pmatrix}M_{D} M^{-1}_R M^T_{D} & M_{D} M^{-1}_R M^T_{} \\\\M_{} M^{-1}_R M_{D} & M_{} M^{-1}_R M^T_{}\\end{pmatrix} \\nonumber \\\\&=& \\begin{pmatrix}M_L-M_{D} M^{-1}_R M^T_{D} & - M_{D} M^{-1}_R M^T_{} \\\\-M_{} M^{-1}_R M_{D} & \\mu _S - M_{} M^{-1}_R M^T_{}\\end{pmatrix}$ and block diagonalized mass formula for the integrated out heavy right-handed neutrinos as, $M^{\\rm heavy}_{}&\\equiv & m_N =M_R + \\cdots $ Using the standard seesaw block diagonalization methodology, one can get $\\mathcal {B}^\\dagger _1 = \\begin{pmatrix}M_D M^{-1}_R \\\\ M M^{-1}_R\\end{pmatrix}^T$ and the approximated intermediate block diagonalized mixing matrix up to the order of $\\mathcal {O}(1/M_R)$ is given by $\\mathcal {W}_1 &=& \\begin{pmatrix}1- \\frac{1}{2} Z Z^\\dagger & -\\frac{1}{2} Z Y^\\dagger & Z \\\\-\\frac{1}{2} Y Z^\\dagger & 1- \\frac{1}{2} Y Y^\\dagger & Y \\\\- Z^\\dagger & -Y^\\dagger & 1-\\frac{1}{2}\\left(Z^\\dagger Z + Y^\\dagger Y\\right)\\end{pmatrix}$ where $Z=M_D M^{-1}_R$ and $Y=M_{} M^{-1}_{R}$ .", "In this way the light active and sterile neutrinos contained in the effective block diagonalized mass matrix $M^{\\rm light}_{\\rm Eff}$ get completely decoupled from heaviest right-handed neutrinos.", "Thus the first seesaw intermediate block diagonalization brings down a $9 \\times 9$ matrix into two smaller matrices; a block diagonalized $6\\times 6$ matrix for $\\nu _L$ and $\\nu _S$ and a $3\\times 3$ matrix for $\\nu _R$ .", "Now we have to repeat the same seesaw block diagonalization procedure for $M^{\\rm light}_{\\rm Eff}$ to further block diagonalize the light neutrino states." ], [ "Determination of $\\mathcal {M_{\\rm BD}}$", "We require another approximated unitary mixing matrix $\\mathcal {S}$ (and $\\mathcal {W}_2$ for accounting the integrated out right-handed neutrinos) to further block diagonalize the effective light neutrino mass matrix $M^{\\rm light}_{\\rm Eff}$ in order to get mass matrices for $\\nu _L$ and $\\nu _S$ .", "Let us write $M^{\\rm light}_{\\rm Eff}$ as given in eq.REF in a simpler form as, $M^{\\rm light}_{\\rm Eff} &=&-\\begin{pmatrix}M_L - M_{D} M^{-1}_R M^T_{D} & - M_{D} M^{-1}_R M^T_{} \\\\-M_{} M^{-1}_R M_{D} & \\mu _S - M_{} M^{-1}_R M^T_{}\\end{pmatrix} =\\begin{pmatrix}\\mathcal {M}^\\prime _L& \\mathcal {M}^{\\prime T}_D \\\\\\mathcal {M}^\\prime _{D} & \\mathcal {M}^\\prime _{R}\\end{pmatrix} \\, , \\nonumber \\\\\\mbox{where,} && \\mathcal {M}^\\prime _{L}= M_L-M_{D} M^{-1}_R M^T_{D} \\nonumber \\\\&& \\mathcal {M}^\\prime _{D}= -M_{} M^{-1}_R M^T_{D} \\nonumber \\\\&& \\mathcal {M}^\\prime _{R}=\\mu _S -M_{} M^{-1}_R M^T_{}$ Now repeating the same procedure of type-I+II seesaw block diagonalization along with the mass hierarchy given in eq.", "(REF ), the diagonalized mass matrix and mixing matrix look as follows, $&&\\mathcal {S}^T \\mathcal {M}^{\\rm light}_{Eff} \\mathcal {S}=\\begin{pmatrix}m_\\nu & 0 \\\\0 & m_S\\end{pmatrix}\\nonumber \\\\&& \\mathcal {S}=\\begin{pmatrix}\\sqrt{1-\\mathcal {A}\\mathcal {A}^{\\dagger }} & \\mathcal {A}\\\\-\\mathcal {A}^{\\dagger } & \\sqrt{1-\\mathcal {A}^{\\dagger }\\mathcal {A}}\\end{pmatrix}=\\begin{pmatrix}1- \\frac{1}{2} X X^\\dagger & X \\\\-X^\\dagger & 1- \\frac{1}{2} X^\\dagger X\\end{pmatrix}$ where active-sterile neutrino mixing is given by the matrix $X=M_D M^{-1}$ .", "Thus, we get the block diagonalized mass formulae for light active neutrinos, sterile neutrinos and right-handed neutrinos as follows.", "$&&\\mathcal {W}^T_2 \\mathcal {M}^{IBD}_{\\rm ELRSM} \\mathcal {W}_2=\\mathcal {M}^{BD}_{\\rm ELRSM} \\nonumber \\\\&& \\mathcal {M}^{\\rm BD}_{\\rm ELRSM} = \\begin{pmatrix}m_\\nu & 0 & 0\\\\0 & m_S & 0 \\\\0 & 0 & m_N\\end{pmatrix}$ The approximated unitary mixing matrix $\\mathcal {W}_2$ is given by, $&& \\mathcal {W}_2 =\\begin{pmatrix}\\mathcal {S} & {\\bf 0}_{6 \\times 3} \\\\{\\bf 0}_{3 \\times 6} & {\\bf 1}_{3 \\times 3}\\end{pmatrix} \\nonumber \\\\&&\\hspace*{28.45274pt} =\\begin{pmatrix}\\sqrt{1-\\mathcal {A}\\mathcal {A}^{\\dagger }} & \\mathcal {A} & {\\bf 0}_{3 \\times 3} \\\\-\\mathcal {A}^{\\dagger } & \\sqrt{1-\\mathcal {A}^{\\dagger }\\mathcal {A}} & {\\bf 0}_{3 \\times 3}\\\\{\\bf 0}_{3 \\times 3} & {\\bf 0}_{3 \\times 3} & {\\bf 1}_{3 \\times 3}\\end{pmatrix}=\\begin{pmatrix}1- \\frac{1}{2} X X^\\dagger & X & {\\bf 0}_{3 \\times 3} \\\\-X^\\dagger & 1- \\frac{1}{2} X^\\dagger X & {\\bf 0}_{3 \\times 3}\\\\{\\bf 0}_{3 \\times 3} & {\\bf 0}_{3 \\times 3} & {\\bf 1}_{3 \\times 3}\\end{pmatrix}$" ], [ "Radiative contribution to light neutrino masses", "The block diagonalized mass formulas for light active neutrinos, sterile neutrinos and heavy neutrinos are given by $m_{\\nu } &=& \\mathcal {M}^\\prime _{L}- \\mathcal {M}^{\\prime T}_{D} \\mathcal {M}^{\\prime ^{-1}}_{R} \\mathcal {M}^{\\prime }_{D}\\nonumber \\\\&=&M_L - M_{D} M^{-1}_R M^T_{D} -\\big (-M_{D} M^{-1}_R M^T_{} \\big )\\cdot \\big (-M_{} M^{-1}_R M^T_{} \\big )^{-1}\\cdot \\big (\\mu _S -M_{} M^{-1}_R M^T_{D} \\big ) \\nonumber \\\\&=&M_L-M_{D} M^{-1}_R M^T_{D} + M_{D} M^{-1}_R M^T_{D}+ \\big (\\frac{M_D}{M}\\big ) \\mu _S \\big (\\frac{M_D}{M}\\big )^T \\nonumber \\\\&=& m^{\\rm II} + m^{\\rm inv} \\\\m_{S} &=& \\mu _S -M_{} M^{-1}_R M^T_{} \\\\m_{N} &=& M_R$ An interesting feature of approximated seesaw block diagonalization scheme within ELRSM is that type-I seesaw contribution gets exactly cancelled out at tree level.", "Thus at tree level light neutrinos get mass through type-II seesaw and inverse seesaw.", "There is also a sizable contribution to light neutrino masses at 1-loop level.", "$m_\\nu = m^{\\rm II} + m^{\\rm inv} + m^{rad}_\\nu \\,.$ The analytic expression for this one loop contribution to light neutrino mass mediated by SM $W$ and $Z$ -bosons is given by $m^{\\rm rad}_\\nu \\equiv \\Delta M &\\simeq & M_D \\frac{\\alpha _W}{16\\pi m_W^2} M_R\\left[\\frac{m_H^2}{M_R^2-m_H^2{\\bf 1}_3}\\ln \\left(\\frac{M_R^2}{m_H^2}\\right) +\\frac{3m_Z^2}{ M_R^2-m_Z^2{\\bf 1}_3}\\ln \\left(\\frac{M_R^2}{m_Z^2}\\right)\\right] M^T_D\\nonumber \\\\& \\simeq &M_D M_R^{-1} x_R\\, f(x_R) M_D^{T}$ where the one-loop function $f(x_R)$ is defined as $f(x_R) =\\frac{\\alpha _W}{16\\pi }\\left[\\frac{x_H}{x_R-x_H}\\ln \\left(\\frac{x_R}{x_H}\\right)+ \\frac{3x_Z}{x_R-x_Z}\\ln \\left(\\frac{x_R}{x_Z}\\right) \\right]$ with $x_R \\equiv \\hat{M}_R^2/m_W^2$ , $x_H\\equiv m_H^2/m_W^2$ , $x_Z\\equiv m_Z^2/m_W^2$ , $\\hat{M}_R$ as diagonal matrix." ], [ "Complete diagonalization and physical neutrino masses", "After block diagonalization, the mass matrix for the three types of neutrinos are further diagonalized by respective unitary mixing matrices as follows, $\\mathcal {U} = \\begin{pmatrix}U_\\nu & 0 & 0 \\\\0 & U_S & 0 \\\\0 & 0 & U_N\\end{pmatrix}$ resulting in physical masses for all the neutrinos.", "$&&U^\\dagger _\\nu m_\\nu U^*_\\nu = \\hat{m}_\\nu = \\mbox{diag}\\left(m_{\\nu _1}, m_{\\nu _2}, m_{\\nu _3} \\right) \\nonumber \\\\&&U^\\dagger _S m_S U^*_S = \\hat{m}_S = \\mbox{diag}\\left(m_{s_1}, m_{s_2}, m_{s_3} \\right) \\nonumber \\\\&&U^\\dagger _N m_N U^*_N = \\hat{m}_N = \\mbox{diag}\\left(m_{N_1}, m_{N_2}, m_{N_3} \\right)$ $\\mathcal {U}_{9 \\times 9} = \\begin{pmatrix}{\\bf U_\\nu }_{3 \\times 3} & {\\bf 0}_{3 \\times 3} & {\\bf 0}_{3 \\times 3} \\\\{\\bf 0}_{3 \\times 3} & {U_S}_{3 \\times 3} & {\\bf 0}_{3 \\times 3} \\\\{\\bf 0}_{3 \\times 3} & {\\bf 0}_{3 \\times 3} & {U_N}_{3 \\times 3}\\end{pmatrix}$ The complete $9\\times 9$ mixing matrix is derived as, $\\mathcal {V} &=& \\mathcal {W}_1 \\cdot \\mathcal {W}_2 \\cdot \\mathcal {U} \\nonumber \\\\&&\\hspace*{-14.22636pt}=\\begin{pmatrix}1 -\\frac{1}{2} Z Z^\\dagger & -\\frac{1}{2} Z Y^\\dagger & Z \\\\-\\frac{1}{2} Y Z^\\dagger & 1-\\frac{1}{2} Y Y^\\dagger & Y \\\\-Z^\\dagger & -Y^\\dagger & 1-\\frac{1}{2}\\left(Z^\\dagger Z + Y^\\dagger Y \\right)\\end{pmatrix} \\cdot \\begin{pmatrix}1 -\\frac{1}{2} X X^\\dagger & X & 0 \\\\-X^\\dagger & 1 -\\frac{1}{2} X^\\dagger X & 0 \\\\0 & 0 & 1\\end{pmatrix} \\cdot \\begin{pmatrix}U_\\nu & 0 & 0 \\\\0 & U_S & 0 \\\\0 & 0 & U_N\\end{pmatrix} \\nonumber \\\\&\\simeq & \\begin{pmatrix}U_{\\nu }\\left(1-\\frac{1}{2}X X^{\\dagger } \\right) & -U_{S}X & Z U_N\\\\-U_{\\nu }X^{\\dagger } & U_S \\left(1-\\frac{1}{2}X^{\\dagger }X \\right) & U_{N} Y \\\\\\left(Z^\\dagger X^\\dagger X\\right)U_\\nu & -U_S Y^{\\dagger } & U_N\\left(1-\\frac{1}{2}Y^{\\dagger }Y \\right)\\end{pmatrix}$ Putting $ X = M_{D} M^{-1}_{}$ , $Y = M_{} M^{-1}_{R}$ , $Z=M_D M^{-1}_R$ and fixing the typical magnitudes for $ M_D \\simeq $  10 GeV ,$ M_{} \\simeq $  100 GeV, $M_{R} \\simeq $ 1000 GeV - 10 TeV, we get $X \\simeq 0.1 $ , $Y\\simeq 0.01$ , $Z\\simeq 10^{-3}$ .", "Since $U_{\\nu }$ , $U_N$ and $U_S$ are of $\\mathcal {O}(1)$ , the matrix elements of $\\mathcal {V}$ are approximated to be $\\begin{pmatrix}\\mathcal {V}_{\\alpha i}^{\\nu \\nu } & \\mathcal {V}_{\\alpha j}^{\\nu S} & \\mathcal {V}_{\\alpha k}^{\\nu N}\\\\\\mathcal {V}_{\\beta i}^{ S \\nu } & \\mathcal {V}_{\\beta j}^{S S} & \\mathcal {V}_{\\beta k}^{S N}\\\\\\mathcal {V}_{\\gamma i}^{N \\nu } & \\mathcal {V}_{\\gamma j}^{N S} & \\mathcal {V}_{\\gamma k}^{N N}\\end{pmatrix} \\simeq \\begin{pmatrix}1 & 0.1 & 10^{-3} \\\\0.1 & 0.91 & 0.01 \\\\0 & 0.01 & 1.0\\end{pmatrix}$" ], [ "Achieving type-II seesaw dominance by including radiative contributions", "In addition to the generic Dirac neutrino mass $M_D$ and Majorana neutrino mass $M_R$ , there are two more terms; a mixing term $M$ that connects right-handed neutrinos with extra serile neutrinos and a small Majorana mass term $\\mu _S$ for only sterile neutrinos.", "With a small but non-zero $\\mu _S$ , an additional contribution to light neutrino masses arises through inverse seesaw mechanism as $m^{\\rm inv}_{\\nu } = (M_D/M) \\mu _S (M_D/M)^T$ apart from the type-II seesaw term at tree level.", "This inverse seesaw term can be avoided and type-II seesaw dominance can be achieved with $\\mu _S \\rightarrow 0$ .", "However SM radiative corrections involving the SM $Z$ and Higgs boson also contribute to light neutrino masses $m^{\\rm rad}_{\\nu }$ as pointed out originally by [130] and later by [131], [132].", "The presence of inverse seesaw and one loop radiative corrections to light neutrino mass might spoil the plot for type-II seesaw dominance.", "We briefly discuss below how one can obtain the type-II seesaw dominance.", "By considering suppressed value of Dirac neutrino mass:- The total contribution to light neutrino masses is the sum of type-II, inverse and radiative seesaw mechanism, out of which the inverse and radiative seesaw contributions mostly depend on Dirac neutrino masses.", "Thus a suppressed value of Dirac neutrino mass will give negligible contribution to inverse as well as radiative corrections to light neutrinos masses.", "In such case (i.e.", "in the limit of suppressed or vanishing Dirac neutrino masses), the only scheme for generating light neutrino masses will be type-II seesaw dominance.", "Due to effective cancellation between inverse and radiative contributions:- Alternatively type-II seesaw dominance can be achieved by allowing exact/partial cancellation between inverse seesaw and radiative contribution such that they marginally/sub-dominantly contribute to the light neutrino masses.", "This is phenomenologically more interesting because it allows large light-heavy neutrino mixing which can contribute to lepton number violation and lepton flavour violation.", "We have followed this method in our present model, i.e.", "we have allowed cancellation to make $m_\\nu ^{inv}+m_\\nu ^{\\rm rad} = 0$ ." ], [ "Loop functions involved in lepton flavour violating processes", "The relevant loop functions arising in various lepton flavour violating (LFV) processes like $\\mu \\rightarrow e \\gamma $ , $\\mu \\rightarrow 3 e$ and $\\mu \\rightarrow e$ conversion inside a nuclei used in our analysis are given by $ \\begin{split}\\mathcal {G}^{\\gamma }_1(x) &= -\\frac{2x^3+5x^2-x}{4(1-x)^3} - \\frac{3x^3}{2(1-x)^4}\\ln {x}, \\\\[1mm]\\mathcal {G}^{\\gamma }_2(x) &= \\frac{x^2-11 x+4}{2 (1-x)^2}-\\frac{3 x^2}{(1-x)^3}\\ln {x}, \\\\[1mm]\\mathcal {F}_{\\gamma }(x) &= \\frac{7x^3-x^2-12x}{12(1-x)^3}-\\frac{x^4-10x^3+12x^2}{6(1-x)^4}\\ln {x}, \\\\[1mm]\\end{split}$ $ \\begin{split}\\mathcal {F}_{\\rm box}(x,y) &= \\left(4+\\frac{xy}{4}\\right)I_2(x,y,1)-2xyI_1(x,y,1), \\\\[1mm]\\mathcal {F}_{\\rm X box}(x,y) &= -\\left(1+\\frac{xy}{4}\\right)I_2(x,y,1)-2xyI_1(x,y,1), \\\\[1mm]\\mathcal {G}_{\\rm box}(x,y,\\eta ) &= -\\sqrt{x y} \\left[(4+x y \\eta ) I_1(x,y,\\eta )-(1+\\eta )I_2(x,y,\\eta )\\right], \\\\\\mathcal {G}_Z(0,x) &= -\\frac{x\\ln {x}}{2(1-x)}\\,, \\\\[1mm]\\mathcal {F}_{\\rm box}(0,x) &= \\frac{4}{1-x}+\\frac{4x}{(1-x)^2}\\ln {x}\\, , \\\\[1mm]\\mathcal {F}_{\\rm X box}(0,x) &= -\\frac{1}{1-x}-\\frac{x\\ln {x}}{(1-x)^2}\\,, \\\\[1mm]\\mathcal {F}_Z(x) &= -\\frac{5x}{2(1-x)}-\\frac{5x^2}{2(1-x)^2}\\ln {x}\\,, \\\\[1mm]\\end{split}$ These loop functions involve heavy right-handed neutrino mass $m_N$ and sterile neutrino mass $m_S$ .", "In our model these masses $m_N$ and $m_S$ are proportional to light neutrino mass $m_\\nu $ .", "Thus the relevant loop functions needed for various LFV processes can be expressed in terms of lightest neutrino mass." ], [ "Expression for $\\mu \\longrightarrow e$ conversion with the mediation of neutrinos {{formula:743fd916-bbb6-4bc2-a999-a210b5bafd21}} and {{formula:ff3572dd-d55f-415c-9cd7-590c13763088}}", "where, the relevant terms containing only light-heavy neutrino mixing are given by $&& \\mathcal {Q}^W_{L} = (2 Z+N)\\left[\\mathcal {W}^u_{L}-\\frac{2}{3} s_W^2 G^\\gamma _{R}\\right]+ (Z+2N)\\left[W^d_{L} +\\frac{1}{3}s_W^2 G^\\gamma _{R} \\right],\\nonumber \\\\&& \\mathcal {Q}^W_{R} = (2 Z+N)\\left[\\mathcal {W}^u_{R}\\right] + (Z+2N)\\left[W^d_{R}\\right],$ and the factor used here are expressed as, $&&W^u_{L}= \\frac{2}{3} s_W^2 \\mathcal {F}^\\gamma _{L} + \\left(-\\frac{1}{4}+\\frac{2}{3} s_W^2\\right)\\mathcal {F}^{Z_1}_{L}+\\frac{1}{4}\\mathcal {B}^{\\mu e uu}_{LL}, \\nonumber \\\\&&W^u_{R}\\rightarrow 0 , \\nonumber \\\\&&W^d_{L} = -\\frac{1}{3} s^2_W \\mathcal {F}^\\gamma _{L} + \\left(\\frac{1}{4}-\\frac{1}{3} s_W^2\\right) \\mathcal {F}^{Z_1}_{L}+\\frac{1}{4}\\mathcal {B}^{\\mu e dd}_{LL}, \\nonumber \\\\&&W^d_{R} \\rightarrow 0 \\,.$ $\\frac{V^{(p)}}{\\sqrt{Z}} = \\frac{Z_{eff}^2F(-m_\\mu ^2)\\alpha _{\\rm em}^{\\frac{3}{2}}}{4\\pi }\\, ,$ and $V^{(p)}/Z \\simeq V^{(n)}/N$ .", "The key box diagram form factors are expressed as $\\begin{split}\\mathcal {B}^{\\mu e uu}_{LL} &= \\sum _{i=1}^3 \\bigg \\lbrace {\\mbox{V}^{\\nu N}_{\\mu i}}^* {\\mbox{V}^{\\nu N}_{e i}}\\left[\\mathcal {F}_{\\rm box}(0,x_i) - \\mathcal {F}_{\\rm box}(0,0)\\right], \\\\\\mathcal {B}^{\\mu e dd}_{LL} &\\simeq \\sum _{i=1}^3 {\\mbox{V}^{\\nu N}_{\\mu i}}^* {\\mbox{V}^{\\nu N}_{e i}}\\left\\lbrace \\mathcal {F}_{\\rm X box}(0,x_i) - \\mathcal {F}_{\\rm X box}(0,0) \\right.", "\\\\& \\left.", "+ |V_{td}|^2\\left[\\mathcal {F}_{\\rm X box}(x_t,x_i)-\\mathcal {F}_{\\rm X box}(0,x_i)-\\mathcal {F}_{\\rm X box}(0,x_t)+F_{\\rm X box}(0,0)\\right]\\right\\rbrace , \\\\\\mathcal {B}^{\\mu e qq}_{RR} &= \\frac{m_{W_L}^2}{m_{W_R}^2} \\mathcal {B}^{\\mu e qq}_{LL}(\\mbox{V}^{\\nu N}\\leftrightarrow {\\mbox{V}^{N N}}^*\\,;\\,x^N_i \\leftrightarrow y^N_i\\,;\\,x_t \\leftrightarrow y_t) \\rightarrow 0\\, ,\\end{split}$ where $x_t = m_t^2/m_{W_L}^2$ and $y_t = m_t^2/m_{W_R}^2$ ." ] ]
2105.11795
[ [ "Algorithmic properties of first-order modal logics of linear Kripke\n frames in restricted languages" ], [ "Abstract We study the algorithmic properties of first-order monomodal logics of frames $\\langle \\mathbb{N}, \\leq \\rangle$, $\\langle \\mathbb{N}, < \\rangle$, $\\langle \\mathbb{Q}, \\leq \\rangle$, $\\langle \\mathbb{Q}, < \\rangle$, $\\langle \\mathbb{R}, \\leq \\rangle$, $\\langle \\mathbb{R}, < \\rangle$, as well as some related logics, in languages with restrictions on the number of individual variables as well as the number and arity of predicate letters.", "We show that the logics of frames based on $\\mathbb{N}$ are $\\Pi^1_1$-hard -- thus, not recursively enumerable -- in languages with two individual variables, one monadic predicate letter and one proposition letter.", "We also show that the logics of frames based on $\\mathbb{Q}$ and $\\mathbb{R}$ are $\\Sigma^0_1$-hard in languages with the same restrictions.", "Similar results are obtained for a number of related logics." ], [ "Introduction", "How algorithmically expressive are first-order modal logics?", "More expressive, it is reasonable to assume, than the classical first-order logic $\\mathbf {QCl}$ —just as propositional modal logics are, as a rule, computationally harder than the classical propositional logic.", "(In this context, it is natural to consider logics as sets of validities, rather than as calculi: understood as calculi conservatively extending $\\mathbf {QCl}$ with a recursively enumerable set of axioms and finitary rules of inference, first-order modal logics are a priori $\\Sigma ^0_1$ -complete.The reader in need of a reminder of the basic concepts of computability theory may consult [41].)", "Numerous first-order modal logics are, however, just as algorithmically expressive as $\\mathbf {QCl}$ , i.e.", "$\\Sigma ^0_1$ -complete: some—such as $\\mathbf { QK}$ , $\\mathbf {QT}$ , $\\mathbf {QD}$ , $\\mathbf {QKB}$ , $\\mathbf {QKTB}$ , $\\mathbf {QS4}$ and $\\mathbf {QS5}$ —are recursively axiomatizable over $\\mathbf {QCl}$ ; others—such as logics of elementary classes of Kripke frames—are recursively embeddable [47], [43] into $\\mathbf {QCl}$ by the standard translation [5], [11], [21].", "This suggests a need for a more fine-grained analysis, which takes into account the algorithmic expressivity not only of full logics but also of their fragments obtained by placing restrictions on the structure of formulas.", "Such analysis allows us to distinguish $\\Sigma ^0_1$ -complete modal logics from $\\mathbf {QCl}$ by algorithmic expressivity: while the monadic fragment of $\\mathbf {QCl}$ is decidable [32], [4], the monadic fragments of most $\\Sigma ^0_1$ -complete modal logics are not [31]; while the two-variable fragment of $\\mathbf {QCl}$ is decidable [37], [25], the two-variable fragments of most $\\Sigma ^0_1$ -complete modal logics are not [30].", "This leads to the study of the algorithmic properties of the fragments of first-order modal logics.", "This study is also motivated by an underdevelopment, relative to $\\mathbf {QCl}$  [9], of the algorithmic classification problem for first-order modal logics—an effort to identify their maximal decidable and minimal undecidable fragments.", "Despite extensive literature [31], [34], [36], [38], [16], [3], [19], [61], [30], [46], [49], whose summary can be found in the Introduction to the authors' earlier article [49], much less is known about the algorithmic properties of the fragments of first-order modal, and closely related superintuitionistic, logics than about the algorithmic properties of the fragments of $\\mathbf {QCl}$ .", "The algorithmic properties of one-variable and two-variable fragments of first-order modal logics are also of interest due to close links between those fragments and, respectively, two-dimensional and three-dimensional propositional modal logics [20], [18], [52].", "In the present paper, we attempt to identify the minimal undecidable fragments of the first-order monomodal logics of frames $\\langle {N}, \\leqslant \\rangle $ , $\\langle {N}, < \\rangle $ , $\\langle {Q}, \\leqslant \\rangle $ , $\\langle {Q}, < \\rangle $ , $\\langle {R}, \\leqslant \\rangle $ and $\\langle {R}, < \\rangle $ , as well as of closely related linear orders.Preliminary results on the logics of $\\langle {N}, \\leqslant \\rangle $ and $\\langle {N}, < \\rangle $ were reported in a conference paper [48].", "The present article improves on the earlier paper in two respects.", "First, we obtain stronger results on the logics of frames based on the naturals by proving $\\Sigma ^1_1$ -hardness for weaker languages; the results reported in this article are plausibly optimal, as discussed in Section .", "Second, we report results on logics of the rationals (and hence on $\\mathbf {QS4.3}$ , $\\mathbf {QK4.3.D.X}$ and $\\mathbf {QK4.3}$ ), the reals and infinite ordinals distinct from $\\omega $ .", "The logics of these structures are of interest on at least three counts.", "First, the structures themselves are of interest, for at least two reasons.", "They have long been considered natural models of the flow of time [39], [23], [17]; therefore, their study has been stimulated by the long-standing interest in temporal reasoning.", "Even though we focus on monomodal languages, since our results are negative, they do apply to more expressive languages with modalities for the past as well as the future.", "On a more basic level perhaps, the structures based on the naturals, the rationals and the reals are so fundamental to mathematics that the properties of the corresponding logics are of intrinsic mathematical significance: the classical theories of these structures, both first-order and second-order, have been extensively studied; in particular, it has long been known that the monadic second-order theory of $\\langle {N}, < \\rangle $ is decidable [14].", "Second, the logics considered here call for techniques substantially different from those used in the previous studies [31], [19], [30], [46], [49] of the algorithmic properties of fragments of monomodal predicate logics.", "For most such logics, known undecidability proofs for fragments with a single monadic predicate letter, a restriction considered here, require a transformation of models that increases their branching factor.", "The methods the authors used earlier [46], [49] intrinsically rely on increasing the branching factor of models, a feature inherited from the propositional-level techniques [26], [12], [42], [44], [45], [50] those methods are based on.", "On the other hand, the construction by Blackburn and Spaan [8], which is propositional but, in principle, adaptable to first-order logics, does not seem to be readily applicable to logics of transitive frames since it relies on the use of a modal operator suitable for counting transitions along the accessibility relation of a frame.", "The techniques used here should, therefore, be of relevance to the study of the algorithmic properties of fragments of monomodal logics of structures with a resticted branching factor, including trees.", "Third, the logics of frames $\\langle {N}, \\leqslant \\rangle $ and $\\langle {N}, < \\rangle $ are algorithmically hard—as follows from Theorem REF below, they are $\\Pi ^1_1$ -hard.", "Most research into the algorithmic properties of monomodal, and closely related superintuitionistic, predicate logics has been focused on decidability and undecidability.", "The only study to date [49], as far as we know, of the algorithmic properties of fragments of not recursively enumerable monomodal predicate logics concerns logics of frames with finite sets of worlds (likewise, very few results [16], [51] are known on algorithmic properties of fragments of not recursively enumerable superintuitionistic predicate logics).", "While it is natural that decidability and undecidability are the main concern of the Classical Decision Problem [9], the study of the algorithmic properties of modal first-order logics should, we believe, involve identifying minimal fragments that are as hard—in pertinent classes of the arithmetical, or the analytical, hierarchy—as the full logics.", "The algorithmic properties of fragments of not recursively enumerable logics have been, however, extensively studied in the context of first-order languages more expressive than monomodal ones considered here—most recently, by Hodkinson, Wolter and Zakharyaschev [28], [61] (for a summary, see [18]; for earlier work, see [2], [57], [58], [1], [35]).", "The methods used here have been inspired by those of Wolter and Zakharyaschev [61], who encode a $\\Sigma ^1_1$ -hard tiling problem in a first-order language with two modal operators, one corresponding to a basic accessibility relation and the other to its reflexive transitive closure.", "A similar result [28] has been obtained by Hodkinson, Wolter and Zakharyaschev for the first-order temporal logic of $\\langle {N}, \\leqslant \\rangle $ in the language with two temporal operators: “next,” corresponding to the immediate successor relation on ${N}$ , and “always in the future,” corresponding to its reflexive transitive closure, the partial order $\\leqslant $ (both operators can be expressed with a binary temporal operator “until”).We touch on such languages in Section .", "Another similar result [60] has been obtained by Wolter for the first-order logics containing, alongside the individual knowledge operators, the common knowledge operator whose semantics involves the reflexive transitive closure of the union of the accessibility relations for the individual knowledge operators.", "We extend herein to monomodal logics, which do not have expressive power for capturing the reflexive transitive closures of accessibility relations, techniques developed by Hodkinson, Wolter, and Zakharyaschev [28], [60], [61].", "The paper is structured as follows.", "In Section , we introduce preliminaries on first-order modal logic.", "In Section , we prove that satisfiability for the logic of $\\langle {N}, \\leqslant \\rangle $ is $\\Sigma ^1_1$ -hard in languages with two individual variables, a single monadic predicate letter and a single proposition letter.", "In Section , the results of Section  are extended to logics of frames $\\langle {N}, R \\rangle $ , where $R$ is a binary relation between $<$ and $\\leqslant $ , and to frames based on infinite ordinals of a special form.", "In Section , we prove, by modifying the argument of Sections  and , that satisfiability for logics of $\\langle {Q}, \\leqslant \\rangle $ , $\\langle {Q}, < \\rangle $ , $\\langle {R}, \\leqslant \\rangle $ and $\\langle {R}, < \\rangle $ is $\\Pi ^0_1$ -hard in languages with the same restrictions.", "In Section , we briefly mention some corollaries of the results proven earlier.", "We conclude, in Section , by discussing first-order temporal logics with modalities “next” and “always in the future,” as well as questions for future study." ], [ "Preliminaries", "An unrestricted first-order predicate modal language contains countably many individual variables; countably many predicate letters of every arity, including 0 (nullary predicate letters are also called proposition letters); the propositional constant $\\bot $ (falsity), the binary propositional connective $\\rightarrow $ , the unary modal connective $\\Box $ and the quantifier $\\forall $ .", "Formulas as well as the symbols $\\top $ , $\\lnot $ , $\\vee $ , $\\wedge $ , $\\leftrightarrow $ , $\\exists $ and $\\Diamond $ are defined in the usual way.", "We also use the abbreviations $\\Box ^0 \\varphi = \\varphi $ , $\\Box ^{n+1} \\varphi = \\Box \\Box ^n \\varphi $ and $\\Diamond ^n \\varphi = \\lnot \\Box ^n \\lnot \\varphi $ , for every $n \\in {N}$ .", "When parentheses are omitted, unary connectives and quantifiers are assumed to bind tighter than $\\wedge $ and $\\vee $ , which are assumed to bind tighter than $\\rightarrow $ and $\\leftrightarrow $ .", "We usually write atomic formulas in prefix notation; for some predicate letters we, however, use infix.", "A normal predicate modal logic is a set of formulas containing the validities of the classical first-order predicate logic $\\mathbf {QCl}$ , as well as the formulas of the form $\\Box (\\varphi \\rightarrow \\psi ) \\rightarrow (\\Box \\varphi \\rightarrow \\Box \\psi )$ , and closed under predicate substitution, modus ponens, generalisation and necessitation.The reader wishing a reminder of the definition of these closure conditions may consult [21]; for a detailed discussion of predicate substitution, consult [21].", "In this paper, we are interested in predicate logics defined using the Kripke semantics.For Kripke semantics for predicate modal logics, see [53], [55], [29], [15], [22], [10], [24], [21].", "A Kripke frame is a tuple ${F} = \\langle W,R\\rangle $ , where $W$ is a non-empty set of possible worlds and $R$ is a binary accessibility relation on $W$ ; if $w R v$ , we say that $v$ is accessible from $w$ and that $w$ sees $v$ .", "A predicate Kripke frame with expanding domains is a tuple ${F}_D = \\langle W,R, D\\rangle $ , where $\\langle W,R\\rangle $ is a Kripke frame and $D$ is a function from $W$ into the set of non-empty subsets of some set, the domain of ${F}_D$; the function $D$ is required to satisfy the condition that $wRw^{\\prime }$ implies $D(w) \\subseteq D(w^{\\prime })$ .", "The set $D(w)$ , also denoted by $D_w$ , is the domain of $w$.", "We also consider predicate frames satisfying the stronger condition that $D(w) = D(w^{\\prime })$ , for every $w, w^{\\prime } \\in W$ ; such frames are predicate frames with a constant domain.More precisely, such predicate frames are known as predicate frames with globally constant domains.", "For connected predicate frames, the global constancy condition given above is equivalent to the local constancy condition requiring that $D(w) = D(w^{\\prime })$ whenever $w R w^{\\prime }$ .", "Since the frames we consider are rooted, and therefore connected, the distinction between global and local constancy is immaterial for the purposes of this paper.", "Predicate frame simpliciter means a predicate frame with expanding domains.", "A Kripke model is a tuple ${M} = \\langle W,R,D,I\\rangle $ , where $\\langle W,R, D\\rangle $ is a predicate Kripke frame and $I$ , the interpretation of predicate letters with respect to worlds in $W$ , is a function assigning to a world $w\\in W$ and an $n$ -ary predicate letter $P$ an $n$ -ary relation $I(w,P)$ on $D(w)$ —i.e., $I(w,P) \\subseteq D_w^n$ .", "In particular, if $p$ is a proposition letter, then $I(w,P) \\subseteq D_w^0 = \\lbrace \\langle \\rangle \\rbrace $ ; thus, we can identify truth with $\\lbrace \\langle \\rangle \\rbrace $ and falsity with $\\varnothing $ .", "We often write $P^{I, w}$ instead of $I(w,P)$ .", "We say that a model $\\langle W,R,D,I\\rangle $ is based on the frame $\\langle W,R \\rangle $ and is based on the predicate frame $\\langle W,R,D\\rangle $ .", "We use the standard notation for binary relations: the $n$ -fold, for each $n \\in {N}^+$ , composition of a binary relation $R$ is denoted by $R^n$ ; if $R$ is a binary relation on a non-empty set $W$ and $w \\in W$ , then $R(w) = \\lbrace v \\in W : w R v \\rbrace $ .", "An assignment in a model is a function $g$ associating with every individual variable $x$ an element $g(x)$ of the domain of the underlying predicate frame.", "We write $g^{\\prime } \\stackrel{x}{=} g$ if assignment $g^{\\prime }$ differs from assignment $g$ in at most the value of $x$ .", "The truth of a formula $\\varphi $ at a world $w$ of a model ${M}$ under an assignment $g$ is defined recursively: ${M},w\\models ^g P(x_1,\\ldots ,x_n)$ if $\\langle g(x_1),\\ldots ,g(x_n)\\rangle \\in P^{I, w}$ , where $P$ is an $n$ -ary predicate letter; ${M},w \\lnot \\models ^g \\bot $ ; ${M},w\\models ^g\\varphi _1 \\rightarrow \\varphi _2$ if ${M},w\\models ^g\\varphi _1$ implies ${M},w\\models ^g\\varphi _2$ ; ${M},w\\models ^g\\Box \\varphi _1$ if ${M},w^{\\prime }\\models ^g\\varphi _1$ , for every $w^{\\prime } \\in R(w)$ ; ${M},w\\models ^g\\forall x\\,\\varphi _1$ if ${M},w\\models ^{g^{\\prime }}\\varphi _1$ , for every $g^{\\prime }$ such that $g^{\\prime } \\stackrel{x}{=} g$ and $g^{\\prime }(x)\\in D_w$ .", "Observe that, if ${M} = \\langle W,R,D,I\\rangle $ is a Kripke model, $w \\in W$ and $I_w(P) = I(w,P)$ , then ${M}_w = \\langle D_w, I_w \\rangle $ is a classical model, or structure.", "We shall often use the following notation.", "Let ${M} = \\langle W,R,D,I\\rangle $ be a model, $w \\in W$ , and $a_1, \\ldots , a_n \\in D_w$ ; let also $\\varphi (x_1, \\ldots , x_n)$ be a formula whose free variables are among $x_1, \\ldots , x_n$ and $g$ an assignment with $g(x_1) = a_1, \\ldots , g(x_n) = a_n$ .", "Then, we write ${M}, w \\models \\varphi (a_1, \\ldots , a_n)$ instead of ${M}, w \\models ^g \\varphi (x_1, \\ldots , x_n)$ .", "This notation is unambiguous since the languages we consider lack constants and the truth value of $\\varphi (x_1, \\ldots , x_n)$ does not depend on the values of variables other than $x_1, \\ldots , x_n$ .", "A formula $\\varphi $ is true at a world $w$ of a model ${M}$ (in symbols, ${M},w\\models \\varphi $ , or simply $w \\models \\varphi $ if ${M}$ is clear from the context) if ${M},w\\models ^g \\varphi $ , for every $g$ assigning to free variables of $\\varphi $ elements of $D_w$ .", "A formula $\\varphi $ is true in a model ${M}$ (in symbols, ${M} \\models \\varphi $ ) if ${M},w\\models \\varphi $ , for every world $w$ of ${M}$ .", "A formula $\\varphi $ is valid on a predicate frame ${F}_D$ if $\\varphi $ is true in every model based on ${F}_D$ .", "A formula $\\varphi $ is valid on a frame ${F}$ (in symbols, ${F} \\models \\varphi $ ) if $\\varphi $ is valid on every predicate frame $\\langle {F}, D \\rangle $ .", "These notions, and the corresponding notation, can be extended to sets of formulas, in a natural way.", "We shall often rely on the following observation: if a model ${M}$ is based on a predicate frame with a constant domain, then ${M},w\\models \\varphi $ if, and only if, ${M},w\\models ^g \\varphi $ , for every assignment $g$ .", "Let ${C}$ be a class of Kripke frames.", "The set of formulas valid on every frame in ${C}$ is a predicate modal logic, which we denote by $\\mathbf {L} ({C})$ ; we write $\\mathbf {L} (W, R)$ instead of $\\mathbf {L} (\\lbrace \\langle W, R \\rangle \\rbrace )$ .", "The set of formulas valid on every predicate frame with a constant domain based on some frame in ${C}$ also is a predicate modal logic, which we denote by ${\\mathbf {L}_{c}} ({C})$ ; we write ${\\mathbf {L}_{c}} (W, R)$ instead of ${\\mathbf {L}_{c}} (\\lbrace \\langle W, R \\rangle \\rbrace )$ ." ], [ "The first-order logic of $\\langle {N}, \\leqslant \\rangle $", "In this section, we prove that satisfiability for $\\mathbf {L}( {N}, \\leqslant )$ is $\\Sigma ^1_1$ -hard—hence, $\\mathbf {L}({N}, \\leqslant )$ is $\\Pi ^1_1$ -hard, and therefore not recursively enumerable—in languages with two individual variables, one monadic predicate letter and one proposition letter." ], [ "Reduction from a tiling problem", "We do so by encoding the following $\\Sigma ^1_1$ -complete [27] ${N}\\times {N}$ recurrent tiling problem.", "We are given a set of tiles, a tile $t$ being a $1 \\times 1$ square, with a fixed orientation, whose edges are colored with $\\textit {left}(t)$ , $right(t)$ , $up(t)$ and $down(t)$ .", "A tile type is a quadruplet of edge colors.", "Each tile has a type from the set $T = \\lbrace t_0, \\ldots , t_s \\rbrace $ , tiles of each type being in an unlimited supply.", "A tiling is an arrangement of tiles on the rectangular ${N}\\times {N}$ grid so that the edge colors of the adjacent tiles match, both horizontally and vertically.", "We are to determine whether there exists a tiling of the grid in which a tile of type $t_0$ occurs infinitely often in the leftmost column, i.e., whether there exists a function $f: {N}\\times {N}\\rightarrow T$ such that, for every $n, m \\in {N}$ , ($T_1$ )  $right(f(n,m)) = \\textit {left} (f(n+1,m))$ ; ($T_2$ )  $up(f(n,m)) = \\textit {down} (f(n,m+1))$ ; ($T_3$ )  the set $\\lbrace m \\in {N}: f(0, m) = t_0 \\rbrace $ is infinite.", "The idea of the encoding we use is based on the work of Hodkinson, Wolter and Zakharyaschev [28] (also see [18]; similar constructions have been used elsewhere [56], [33], [61], [30]), but the encoding itself is more involved since our language lacks the “next” operator available to them (we touch on languages with “next” in Section ).", "To make the underlying idea clearer, we construct, in the initial encoding, a formula of two individual variables without regard for the number of predicate letters involved; subsequently, we reduce the formula thus obtained to a formula with a single monadic and a single proposition letter.", "Let $\\triangleleft $ be a binary predicate letter, $M$ and $P_t$ —for every $t \\in T$ —monadic predicate letters and $p$ a proposition letter.", "Given a formula $\\varphi $ in such a language, define $\\begin{array}{c}{}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}\\varphi = \\Diamond (p \\wedge \\Diamond ( \\lnot p \\wedge \\varphi ));\\medskip \\\\{}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}^0 \\varphi = \\varphi ; \\quad {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}^{n+1} \\varphi ={}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}{}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}^n\\varphi , \\mbox{ for every } n \\in {N}.\\end{array}$ The operator ${}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}$ forces a transition to a different world when evaluating a formula ${}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}\\varphi $ in a reflexive model, just as $\\Diamond $ does in an irreflexive one.", "To make this explicit, we define, given a model based on the frame $\\langle {N}, \\leqslant \\rangle $ , a binary relation $R_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}}$ on ${N}$ by $\\begin{array}{lcl}w R_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}} v& \\leftrightharpoons & \\mbox{$v \\lnot \\models p$ and, for some $u \\in {N}$, both $w \\leqslant u \\leqslant v$and $u \\models p$.", "}\\end{array}$ Thus, $R_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}}$ is irreflexive and transitive.", "We also define $\\begin{array}{lcl}U (x) & = & \\displaystyle \\bigwedge \\limits _{t \\in T} \\lnot P_t(x).\\end{array}$ In such a language, define (for brevity, in formulas we write $l$ , $r$ , $u$ and $d$ instead of left, right, up and down) $\\begin{array}{rcl}A_0 & = & \\exists x\\, \\Box \\, U (x); \\\\A_1 & = & \\exists x\\, (\\lnot U (x) \\wedge M(x)); \\medskip \\\\A_2 & = & \\forall x \\exists y\\, (x \\triangleleft y); \\medskip \\\\A_3 & = & \\forall x \\forall y\\, (x \\triangleleft y \\rightarrow \\Box (\\exists x\\, M(x) \\rightarrow x\\triangleleft y)); \\medskip \\\\A_4 & = & \\forall x \\forall y\\, ( x \\triangleleft y \\rightarrow \\Box (M(x) \\leftrightarrow \\lnot p \\wedge {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}M(y) \\wedge \\lnot {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}^2 M(y) ); \\medskip \\\\A_5 & = & \\displaystyle \\forall x \\forall y\\, \\Box \\bigwedge \\limits _{t \\in T}\\big ( M(x) \\wedge P_t (y) \\rightarrow \\Box (M(x) \\rightarrow P_{t} (y)) \\big ) ; \\medskip \\\\A_6 & = & \\displaystyle \\forall x\\, \\Box \\bigwedge \\limits _{t \\in T} (P_t(x) \\rightarrow \\bigwedge \\limits _{{t^{\\prime } \\ne t}} \\lnot P_{t^{\\prime }}(x)); \\medskip \\\\A_7 & = & \\displaystyle \\forall x \\forall y\\, \\Box \\bigwedge \\limits _{t \\in T}( x \\triangleleft y\\wedge P_t(x) \\rightarrow \\bigvee \\limits _{{r(t) = l(t^{\\prime })}} P_{t^{\\prime }} (y)) ; \\medskip \\\\A_8 & = & \\displaystyle \\forall x \\forall y\\, \\Box \\bigwedge \\limits _{t \\in T} \\big (M(x) \\wedge P_t (y) \\rightarrow \\Box (\\exists y\\, (x\\triangleleft y \\wedge M(y)) \\rightarrow \\bigvee \\limits _{{u(t) =d(t^{\\prime })}} P_{t^{\\prime }} (y)) \\big ); \\medskip \\\\A_9 & = & \\forall x\\, (M(x) \\rightarrow \\Box {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}P_{t_0} (x)).\\end{array}$ Let $A$ be the conjunction of $A_0$ through $A_{9}$ .", "Observe that $A$ contains only two individual variables.", "The relation ${\\triangleleft }$ can be thought of as the immediate successor relation on the domain ${D}_0$ of the world 0 where $A$ is being evaluated.", "An element $a \\in {D}_0$ such that $w \\models M(a)$ can be thought of as marking, or labelling, world $w$ ; thus, we say that $a$ is a mark of $w$ .", "Then, $A_2$ asserts that every element of $D_0$ has an immediate successor, while $A_3$ asserts that the immediate successor relation persists throughout the part of the frame where worlds are marked by elements of $D_0$ .", "Given that, $A_1$ and $A_4$ imply the existence of an infinite sequence $a_0 \\triangleleft a_1 \\triangleleft a_2 \\triangleleft \\ldots $ of elements of $D_0$ such that every world refuting $p$ is marked, as we shall see uniquely, by some element of the sequence; they also imply that the order of the marks of successive, with respect to $\\leqslant $ , worlds agrees with the relation $\\triangleleft $ .", "This, as we shall see, gives us an ${N}\\times {N}$ grid whose rows correspond to the worlds of $\\langle {N}, \\leqslant \\rangle $ and whose columns correspond to the elements of the sequence $a_0 \\triangleleft a_1 \\triangleleft a_2 \\triangleleft \\ldots \\,$ .", "Building on this, $A_5$ through $A_9$ describe a sought tiling of thus obtained grid.", "(The element of $D_0$ whose existence is asserted by $A_0$ is not part of the tiling—its presence shall be relied upon in a subsequent reduction.)", "Lemma 3.1 There exists a recurrent tiling of ${N}\\times {N}$ satisfying ($T_1$ ) through ($T_3$ ) if, and only if, $\\langle {N}, \\leqslant \\rangle \\lnot \\models \\lnot A$ .", "Proof.", "(“if”) Suppose ${M}, w_0 \\models A$ , for some model ${M} = \\langle {N}, \\leqslant , D, I \\rangle $ and some world $w_0 \\in {N}$ .", "Since truth of formulas is preserved under taking generated submodels,The notions of generated subframe and generated submodel for predicate modal logics are straightforward extensions of the respective notions [7] for propositional modal logics.", "we may assume $w_0 = 0$ .", "Since $0 \\models A_1$ , there exists $a_0 \\in D_0$ such that $0 \\lnot \\models U (a_0)$ and $0 \\models M(a_0)$ .", "Since $0 \\models A_2$ , we obtain an infinite sequence $a_0, a_1, a_2, \\ldots $ of elements of $D_0$ such that $a_0 \\triangleleft ^{I,0} a_1 \\triangleleft ^{I,0} a_2\\triangleleft ^{I,0} \\ldots \\,\\,$ .", "Since $0 \\models A_3$ , we obtain that $a_0 \\triangleleft ^{I,w} a_1 \\triangleleft ^{I,w} a_2\\triangleleft ^{I,w} \\ldots \\,$ , for every $w \\in {N}$ such that $w \\models \\exists x\\, M(x)$ .", "Since $0 \\models A_4$ , we obtain, for every $w,n \\in {N}$ , $\\begin{array}{lcl}w \\models M(a_n) & \\Longleftrightarrow &\\left\\lbrace \\begin{array}{l}w\\phantom{^{\\prime }} \\lnot \\models p; \\\\w^{\\prime } \\models M(a_{n+1}), \\mbox{ for some } w^{\\prime } \\in R_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}}(w); \\\\w^{\\prime \\prime } \\in R_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}}^2 (w) \\mbox{ implies } w^{\\prime \\prime } \\lnot \\models M(a_{n+1}).\\end{array}\\right.\\end{array}\\qquad \\mathrm {(1)}$ Thus, a mark changes from $a_n$ to $a_{n+1}$ once we pass through a world, or an unbroken non-empty sequence of worlds, satisfying $p$ to a world refuting $p$ .", "We now show that a mark remains unchanged until we have reached a world satisfying $p$ , i.e., that for every $u, u^{\\prime }, n \\in {N}$ , $\\begin{array}{l}\\mbox{if $u\\models M(a_n)$, $u \\lnot \\models p$, $u^{\\prime } \\lnot \\models p$,and no $v$ with $u \\leqslant v \\leqslant u^{\\prime }$or $u^{\\prime } \\leqslant v \\leqslant u$} \\\\\\mbox{satisfies $v \\models p$, then $u^{\\prime } \\models M(a_n)$.", "}\\end{array}\\qquad \\mathrm {(2)}$ Assume that $u \\models M(a_n)$ , $u \\lnot \\models p$ , $u^{\\prime } \\lnot \\models p$ , and that no $v$ with $u \\leqslant v \\leqslant u^{\\prime }$ or $u^{\\prime } \\leqslant v \\leqslant u$ satisfies $v \\models p$ .", "Then, by (1), there exists $w^{\\prime } \\in R_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}} (u)$ such that $w^{\\prime } \\models M(a_{n+1})$ , and $w^{\\prime \\prime } \\lnot \\models M(a_{n+1})$ , for every $w^{\\prime \\prime } \\in R^2_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}} (u)$ .", "Let us fix the said $w^{\\prime }$ .", "It follows immediately from the assumption that, for every $w, n \\in {N}$ , $\\begin{array}{lcl}w \\in R^n_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}} (u)& \\Longleftrightarrow & w \\in R^n_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}} (u^{\\prime }).\\end{array}$ Therefore, $w^{\\prime } \\in R_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}} (u^{\\prime })$ , and $w^{\\prime \\prime } \\in R^2_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}} (u^{\\prime })$ implies $w^{\\prime \\prime } \\lnot \\models M(a_{n+1})$ , for every $w^{\\prime \\prime } \\in {N}$ .", "Since by assumption $u^{\\prime } \\lnot \\models p$ , we obtain, by (1), that $u^{\\prime } \\models M(a_n)$ .", "We next show that a mark of every world is unique, i.e.", "for every $w,n \\in {N}$ and every $j \\in {N}^+$ , $\\begin{array}{lcl}w \\models M(a_n)& \\mbox{implies}& w \\lnot \\models M(a_{n+j}).\\end{array}\\qquad \\mathrm {(3)}$ Assume $w \\models M(a_n)$ .", "By (1), there exists $w^{\\prime } \\in R^{j+1}_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}} (w)$ such that $w^{\\prime } \\models M(a_{n+j+1})$ .", "Since $j \\geqslant 1$ and $R_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}}$ is transitive, $w^{\\prime } \\in R_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}}^{2} (w)$ .", "Therefore, by (1), $w \\lnot \\models M(a_{n+j})$ .", "We next show that every element $a_n$ is tiled at every world marked by some element $a_m$ , i.e.", "that for every $w, m, n \\in {N}$ , $\\begin{array}{lcl}w \\models M(a_m)& \\mbox{implies}& \\mbox{$w \\models P_t (a_n)$, for some $t \\in T$.", "}\\end{array}\\qquad \\mathrm {(4)}$ We proceed by induction on $m$ .", "As we have seen, $0 \\lnot \\models U(a_0)$ , i.e., $0 \\models P_t (a_0)$ , for some $t \\in T$ .", "Since $0 \\models A_7$ , for every $n \\in {N}$ , there exists $t \\in T$ such that $0 \\models P_t (a_n)$ .", "Since $0 \\models A_5$ , for every $w, v, m, n \\in {N}$ and every $t \\in T$ , $\\begin{array}{lcl}\\mbox{$w \\models M(a_m)$, $v \\models M(a_m)$ and $w \\models P_t(a_n)$}& \\mbox{imply}& \\mbox{$v \\models P_t(a_n)$.", "}\\end{array}\\qquad \\mathrm {(5)}$ Therefore, (4) holds for $m = 0$ .", "Assume (4) holds for $m \\geqslant 0$ and suppose $w \\models M(a_{m+1})$ .", "We claim that, then, there exists $w^{\\prime }$ such that $w^{\\prime } < w$ and $w^{\\prime } \\models M(a_m)$ .", "To prove the claim we, first, observe that there exists $w^{\\prime }$ such that $w^{\\prime } \\lnot \\models p$ and $w \\in R_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}} (w^{\\prime })$ : otherwise, by (2), $w \\models M(a_0)$ , in contradiction with (3).", "Fix the said $w^{\\prime }$ .", "We next show that $w^{\\prime \\prime } \\in R^2_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}} (w^{\\prime })$ implies $w^{\\prime \\prime } \\lnot \\models M(a_{m+1})$ .", "Assume $w^{\\prime \\prime } \\in R^2_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}} (w^{\\prime })$ .", "Then, $w^{\\prime \\prime } \\in R_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}} (w)$ .", "Since $w \\models M(a_{m+1})$ , by (1) and (2), $w^{\\prime \\prime } \\models M(a_{m+2})$ and thus, by (3), $w^{\\prime \\prime } \\lnot \\models M(a_{m+1})$ .", "Last, since $w^{\\prime } \\lnot \\models p$ , we obtain, by (1), $w^{\\prime } \\models M(a_m)$ , thereby proving the claim.", "Now, let $n \\in {N}$ be given.", "By inductive hypothesis, there exists $t$ such that $w^{\\prime } \\models P_t(a_n)$ .", "Since $0 \\models A_8$ , this implies that $w \\models P_{t^{\\prime }}(a_n)$ , for some $t^{\\prime } \\in T$ .", "Thus, (4) is proven.", "In view of (5), for every $m \\in {N}$ , we may pick an arbitrary world marked by $a_m \\in D_0$ to be part of the sought tiling.", "For definiteness, let, for every $m \\in {N}$ , $w_m = \\min \\lbrace w \\in {N}: w \\models M(a_m) \\rbrace .$ By (4), for every $n, m \\in {N}$ , there exists $t \\in T$ such that $w_m \\models P_t (a_n)$ ; it follows from $0 \\models A_6$ that such $t$ is unique.", "We can, therefore, define a function $f\\colon {N}\\times {N}\\rightarrow T$ by $\\begin{array}{lcl}f(n, m) = t& \\mbox{ whenever } & w_m \\models P_t(a_n).\\end{array}$ We next show that $f$ satisfies ($T_1$ ) through ($T_3$ ).", "Since $0 \\models A_7$ , the condition ($T_1$ ) is, evidently, satisfied.", "To see that ($T_2$ ) is satisfied, assume $f(n, m) = t$ .", "Then, $w_m \\models P_t (a_n)$ , by definition of $f$ .", "From the definition of $w_m$ we know that $w_m \\models M(a_m)$ .", "Since $0 \\models A_8$ , if $v \\geqslant w_m$ and $v \\models M(a_{m+1})$ , then $v \\models P_{t^{\\prime }} (a_n)$ , for some $t^{\\prime }$ with $up(t) = down(t^{\\prime })$ .", "We next show that $w_{m+1} \\geqslant w_m$ .", "Assume otherwise: let $w_{m+1} < w_m$ .", "From the definition of $w_{m+1}$ we know that $w_{m+1} \\models M(a_{m+1})$ .", "Therefore, by (2) and $(3)$ , $w_m \\in R_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}} (w_{m+1})$ .", "Since $w_m \\models M(a_m)$ , there exists, by (1), $w^{\\prime } \\in R_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}}^2 (w_m)$ such that $w^{\\prime } \\models M(a_{m+2})$ .", "Since $R_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}}$ is transitive, $w^{\\prime } \\in R_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}}^2 (w_{m+1})$ , in contradiction with the third clause of (1).", "Thus, we have shown that $w_{m+1} \\geqslant w_m$ .", "Hence, $w_{m+1} \\models P_{t^{\\prime }} (a_n)$ , for some $t^{\\prime }$ with $up(t) = down(t^{\\prime })$ .", "Therefore, ($T_2$ ) is satisfied.", "It remains to show that ($T_3$ ) is satisfied.", "Since $0 \\models M(a_0)$ and $0 \\models A_9$ , the set $\\lbrace w \\in {N}: w \\lnot \\models p \\mbox{ and } w \\models P_{t_0}(a_0)\\rbrace $ is infinite.", "It follows from $0 \\models M(a_0)$ , (1) and (2) that, for every $w \\in {N}$ , if $w \\lnot \\models p$ , then there exists $m \\in {N}$ such that $w \\models M(a_m)$ .", "Therefore, by (5), the set $\\lbrace w_m : m \\in {N}\\mbox{ and } w_m \\models P_{t_0} (a_0)\\rbrace $ is infinite.", "Hence, ($T_3$ ) is satisfied.", "Thus, $f$ is a required function.", "Figure: Model M 0 {M}_0(“only if”) Suppose $f$ is a function satisfying ($T_1$ ) through ($T_3$ ).", "We obtain a model based on $\\langle {N}, \\leqslant \\rangle $ satisfying $A$ .", "Let $\\mathcal {D} = {N}\\cup \\lbrace -1\\rbrace $ and $D(w) = \\mathcal {D}$ , for every $w \\in {N}$ .", "Let ${M}_0 = \\langle {N}, \\leqslant , D, I \\rangle $ be a model such that, for every $w \\in {N}$ and every $a,b\\in {\\cal D}$ , $\\begin{array}{lcl}{M}_0,w \\models a\\triangleleft b& \\leftrightharpoons &\\mbox{$w$ is even and $b=a+1$;} \\smallskip \\\\{M}_0,w \\models p& \\leftrightharpoons &\\mbox{$w$ is odd;} \\smallskip \\\\{M}_0,w \\models M(a)& \\leftrightharpoons &\\mbox{$w=2a$;} \\smallskip \\\\{M}_0,w \\models P_t(a)& \\leftrightharpoons &\\mbox{for some $m\\in {N}$, both $w=2m$ and $f(a,m) = t$.", "}\\end{array}$ It is straightforward to check that ${M}_0, 0 \\models A$ , so we leave this to the reader.", "$\\Box $ Thus, in the proof of the “if” part of Lemma REF , we obtained a grid for the tiling by treating the worlds of model ${M}$ as rows and elements $a_0, a_1, a_2, \\ldots $ of the domain $D_0$ of the world 0 satisfying $A$ as columns." ], [ "Elimination of the binary predicate letter", "We next eliminate, following ideas of Kripke's [31], the binary predicate letter $\\triangleleft $ of formula $A$ , without increasing the number of individual variables in the resultant formula.", "From now on, we assume, for ease of notation, that $A$ contains monadic predicate letters $P_0, \\ldots , P_s$ —rather than $P_t$ , for each $t \\in \\lbrace t_0, \\ldots , t_s \\rbrace $ —to refer to the tile types.", "Recall that Kripke's construction [31] transforms a model ${M}$ satisfying, at world $w$ , a formula containing a binary predicate letter, and no modal connectives, so that, for every pair of elements of the domain of $w$ , a fresh world accessible from $w$ is introduced to ${M}$ .", "This construction cannot be applied here in a straightforward manner, for two reasons.", "First, since we are working with the frame $\\langle {N}, \\leqslant \\rangle $ , we may not introduce fresh worlds to a model satisfying $A$ ; we, rather, have to use the worlds from ${N}$ to simulate $\\triangleleft $ .", "Second, since $\\triangleleft $ occurs within the scope of the modal connective in $A$ , we need to simulate the interpretation of $\\triangleleft $ not just at the world satisfying $A$ , but at every world accessible from it.", "We resolve these difficulties by working with the model ${M}_0$ defined in the “only if” part of the proof of Lemma REF , rather than with an arbitrary model satisfying $A$ , and relying on ${M}_0$ being based on a frame with a constant domain and on the interpretation of $\\triangleleft $ being identical at every world of ${M}_0$ .", "Let $P_{s+1}$ and $P_{s+2}$ be monadic predicate letters distinct from $M, P_0, \\ldots , P_s$ and from each other, and let $\\cdot ^{\\prime }$ be the function substituting $\\begin{array}{lcl}{}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}(P_{s+1} (x) \\wedge P_{s+2} (y))& \\mbox{for}& x \\triangleleft y.\\end{array}$ Lemma 3.2 There exists a recurrent tiling of ${N}\\times {N}$ satisfying ($T_1$ ) through ($T_3$ ) if, and only if, $\\langle {N}, \\leqslant \\rangle \\lnot \\models \\lnot A^{\\prime }$ .", "Proof.", "(“if”) Suppose ${M}, w_0 \\models A^{\\prime }$ , for some model ${M} = \\langle {N}, \\leqslant , D, I \\rangle $ and some world $w_0$ , which can be assumed to be 0.", "The argument is essentially the same as in the proof of the “if” part of Lemma REF .", "The only, inconsequential, difference is that ${}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}(P_{s+1} (x) \\wedge P_{s+2} (y))$ now plays the role of $x \\triangleleft y$ : for every $w \\in {N}$ , the relation $I(w, \\triangleleft ) \\subseteq D_w \\times D_w$ is replaced by the relation $\\lbrace \\langle a, b \\rangle \\in D_w \\times D_w : {M}, w \\models {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}(P_{s+1} (a) \\wedge P_{s+2} (b))\\rbrace .$ Since ${M}, 0 \\models A^{\\prime }$ , the two relations are indistinguishable, for every $w \\in {N}$ , with respect to the properties we rely on in the proof.", "(“only if”) Suppose $f$ is a function satisfying ($T_1$ ) through ($T_3$ ).", "Let ${M}_0 = \\langle {N}, \\leqslant , D, I \\rangle $ be the model defined in the “only if” part of the proof of Lemma REF .", "As we have seen, ${M}_0, 0 \\models A$ .", "We use ${M}_0$ to obtain a model satisfying $A^{\\prime }$ .", "Let $\\alpha $ be the infinite sequence $0, \\, \\, 0, 1, \\, \\, 0, 1, 2, \\, \\, 0, 1, 2, 3, \\, \\, 0, 1, 2, 3,4,\\, \\, \\ldots $ and let $\\alpha _k$ , for each $k \\in {N}$ , be the $k$ th element of $\\alpha $ .", "Let ${M}^{\\prime }_0 = \\langle {N}, \\leqslant , D, I^{\\prime } \\rangle $ be a model such that, for every $w, c \\in {N}$ , $\\begin{array}{lcl}{M}^{\\prime }_0, w \\models P_{s+1} (c)& \\leftrightharpoons &\\mbox{for some $m\\in {N}$, both $w=2m$ and $c=\\alpha _m$;}\\medskip \\\\{M}^{\\prime }_0, w \\models P_{s+2} (c)& \\leftrightharpoons &\\mbox{for some $m\\in {N}$, both $w=2m$ and $c=\\alpha _m+1$,}\\end{array}$ and for every $w \\in {N}$ and every $S \\in \\lbrace P_0, \\ldots , P_s, M, p\\rbrace $ , $\\begin{array}{lcl}I^{\\prime }(w,S) & = & I(w,S).\\end{array}$ We show that ${M}^{\\prime }_0, 0 \\models A^{\\prime }$ .", "Since ${M}_0, 0 \\models A$ , it suffices to prove that, for every $m \\in {N}$ and every $a, b \\in \\mathcal {D}$ , $\\begin{array}{lcl}{M}_0, 2m \\models a \\triangleleft b& \\Longleftrightarrow & {M}^{\\prime }_0, 2m \\models {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}(P_{s+1} (a) \\wedge P_{s+2}(b)).\\end{array}$ Assume ${M}_0, 2m \\models a \\triangleleft b$ .", "Then, $b = a + 1$ , by definition of ${M}_0$ .", "Choose $k \\in {N}$ so that $k > m$ and $\\alpha _k = a$ ; by definition of $\\alpha $ , such a number $k$ certainly exists.", "By definition of ${M}^{\\prime }_0$ , both ${M}^{\\prime }_0, 2k \\lnot \\models p$ and ${M}^{\\prime }_0, 2k \\models P_{s+1} (a) \\wedge P_{s+2} (b)$ .", "By the same definition, ${M}^{\\prime }_0, 2k - 1 \\models p$ .", "Hence, ${M}^{\\prime }_0, 2m \\models {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}(P_{s+1} (a) \\wedge P_{s+2} (b))$ .", "Conversely, assume ${M}^{\\prime }_0, 2m \\models {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}(P_{s+1} (a) \\wedge P_{s+2} (b))$ .", "Then, for some $v > 2m$ , both ${M}^{\\prime }_0, v \\lnot \\models p$ and ${M}^{\\prime }_0, v \\models P_{s+1} (a) \\wedge P_{s+2} (b)$ .", "By definition of ${M}^{\\prime }_0$ , we have ${M}_0, v \\lnot \\models p$ ; hence $v = 2k$ , for some $k > m$ .", "Also by definition of ${M}^{\\prime }_0$ , both $a = \\alpha _k$ and $b = \\alpha _k + 1$ ; hence, $b = a + 1$ .", "Therefore, ${M}_0, 2m \\models a \\triangleleft b$ , by definition of ${M}_0$ .", "$\\Box $" ], [ "Elimination of monadic predicate letters", "We lastly simulate the occurrences of letters $p, M, P_0, \\ldots , P_{s+2}$ in $A^{\\prime }$ with one monadic and one proposition letter, without increasing the number of individual variables in the resultant formula.", "Let $P$ be a monadic letter distinct from $M, P_0, \\ldots , P_{s+2}$ , and let $q$ be a proposition letter distinct from $p$ .", "For a formula $\\varphi $ in the language containing $P$ and $q$ , define $\\begin{array}{c}{}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}\\varphi = \\Diamond (\\forall x\\, P(x) \\wedge \\Diamond ( \\lnot \\forall x\\, P(x) \\wedge \\varphi ));\\medskip \\\\{}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}^0 \\varphi = \\varphi ; \\quad {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}^{n+1} \\varphi ={}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}{}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}^n\\varphi , \\mbox{ for every } n \\in {N}.\\end{array}$ Define, for every $n \\in \\lbrace 0, \\ldots , s + 2\\rbrace $ , $\\begin{array}{rcl}\\beta _n (x) & = & \\exists y\\, \\big ( {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}^{s+4} (q \\wedge P(y)) \\wedge \\lnot {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}^{s+5} (q \\wedge P(y))\\, \\wedge \\\\& & \\phantom{\\mu \\wedge P(x) \\wedge \\exists y\\, [} {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}({}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}^{n+1} (q \\wedge P(y)) \\wedge \\lnot {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}^{n+2} (q \\wedge P(y)) \\wedge P(x)) \\big );\\medskip \\\\\\beta _n (y) & = & \\exists x\\, \\big ( {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}^{s+4} (q \\wedge P(x)) \\wedge \\lnot {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}^{s+5} (q \\wedge P(x))\\, \\wedge \\\\& & \\phantom{\\mu \\wedge P(y) \\wedge \\exists x\\, [} {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}({}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}^{n+1} (q \\wedge P(x)) \\wedge \\lnot {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}^{n+2} (q \\wedge P(x)) \\wedge P(y)) \\big ).\\\\\\end{array}$ Let $\\cdot ^\\ast $ be the function replacing $P_n(x)$ with $\\beta _n (x)$ , for every $n \\in \\lbrace 0, \\ldots , s + 2\\rbrace $ ; $P_n(y)$ with $\\beta _n (y)$ , for every $n \\in \\lbrace 0, \\ldots , s + 2\\rbrace $ ; $M(x)$ with $q \\wedge P(x)$ ; $M(y)$ with $q \\wedge P(y)$ .", "Let $A_i^\\ast $ , for each $i$ with $0 \\leqslant i \\leqslant 8$ and $i \\ne 4$ , be the result of applying the function $\\cdot ^\\ast $ to $A^{\\prime }_i$ .", "Also, let $\\begin{array}{lcl}A_4^{\\ast } & = & \\forall x \\forall y\\, \\big ( {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}(\\beta _{s+1}(x) \\wedge \\beta _{s+2} (y) )\\, \\rightarrow \\smallskip \\\\& & \\phantom{\\forall x \\forall y\\, ( {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}(\\beta _{s+1}}\\Box (q \\wedge P(x) \\leftrightarrow \\lnot \\forall x\\, P(x) \\wedge {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}^{s+4} ( q \\wedge P(y)) \\wedge \\lnot {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}^{s+5} (q \\wedge P(y))) \\big ),\\end{array}$ and $\\begin{array}{lcl}A^\\ast _{9} & = & \\forall x\\, (q \\wedge P(x) \\rightarrow \\Box {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}\\beta _{0} (x)).\\end{array}$ Lastly, let $A^\\ast $ be the conjunction of $A_0^\\ast $ through $A_9^\\ast $ .", "Observe that $A^\\ast $ contains only two individual variables, a monadic letter $P$ and a proposition letter $q$ .", "We shall show that $A^\\ast $ is satisfiable if, and only if, there exists a recurrent tiling satisfying ($T_1$ ) through ($T_3$ ).", "Figure: Model M 0 * {M}_0^\\ast To obtain a model satisfying $A^\\ast $ , we “stretch out” the model ${M}^{\\prime }_0$ defined in the “only if” part of the proof of Lemma REF to include “additional” worlds whose sole purpose is to simulate the interpretation of letters $P_0, \\ldots , P_{s+2}$ at worlds of ${M}^{\\prime }_0$ .", "We “insert” $s+3$ worlds between worlds $m$ and $m + 1$ to simulate the interpretation of letters $P_0, \\ldots , P_{s+2}$ at $m$ .", "The interpretation of $P_n$ , for each $n \\in \\lbrace 0, \\ldots , s+2\\rbrace $ , at $m$ is simulated by the interpretation of letter $P$ at a newly inserted world “$n$ steps away from” $m+1$ .", "To be able to step through the newly defined model, we also “insert” extra worlds satisfying $\\forall x\\, P(x)$ ; these play the same role the worlds satisfying $p$ played in ${M}^{\\prime }_0$ .", "The proposition letter $q$ marks off the “old” worlds from ${M}^{\\prime }_0$ .", "The resultant model is depicted in Figure REF , where $\\beta _{f(a,b)}(x)$ stands for $\\beta _n (x)$ , where $n$ is such that $f(a,b)=t_n$ .", "Lemma 3.3 There exists a recurrent tiling of ${N}\\times {N}$ satisfying ($T_1$ ) through ($T_3$ ) if, and only if, $\\langle {N}, \\leqslant \\rangle \\lnot \\models \\lnot A^\\ast $ .", "Proof.", "(“if”) Suppose ${M}, w_0 \\models A^\\ast $ , for some model ${M} = \\langle {N}, \\leqslant , D, I \\rangle $ and some world $w_0$ , which can be assumed to be 0.", "The argument is essentially the same as in the proof of the “if” part of Lemma REF , the only difference being that we use $\\beta _n(x)$ instead of $P_n(x)$ ; $\\beta _n(y)$ instead of $P_n(y)$ ; $q \\wedge P(x)$ instead of $M(x)$ ; $q \\wedge P(y)$ instead of $M(y)$ .", "(“only if”) Suppose $f$ is a function satisfying ($T_1$ ) through ($T_3$ ).", "Let ${M}^{\\prime }_0 = \\langle {N}, \\leqslant , D, I^{\\prime } \\rangle $ be the model defined in the “only if” part of the proof of Lemma REF .", "As we have seen, ${M}^{\\prime }_0, 0 \\models A^{\\prime }$ .", "We use ${M}^{\\prime }_0$ to obtain a model satisfying $A^{\\ast }$ .", "We think of the worlds from ${N}$ as being labeled, in the ascending order, $\\begin{array}{l}w_0^{\\phantom{i}}, \\bar{w}_0^{\\phantom{i}}, v^{s+2}_0,\\bar{v}^{s+2}_0, \\ldots , v^0_0, \\bar{v}^0_0, \\\\w_1^{\\phantom{i}}, \\bar{w}_1^{\\phantom{i}}, v^{s+2}_1,\\bar{v}^{s+2}_1, \\ldots , v^0_1, \\bar{v}^0_1, \\\\w_2^{\\phantom{i}}, \\bar{w}_2^{\\phantom{i}}, \\ldots \\, ,\\end{array}$ i.e., we put $w_0 = 0$ , $\\bar{w}_0^{\\phantom{i}} = 1$ , $v^{s+2}_0 = 2$ , etc.", "Let ${M}_0^\\ast = \\langle {N}, \\leqslant , D, I^\\ast \\rangle $ be a model such that, for every $u \\in {N}$ $\\begin{array}{lcll}{M}_0^\\ast , u \\models q& \\leftrightharpoons & \\mbox{$u = w_m$, for some $m \\in {N}$,}\\end{array}$ and for every $u \\in {N}$ and every $a \\in \\mathcal {D}$ , the relation ${M}_0^\\ast , u \\models P(a)$ holds if, and only if, one of the following conditions is satisfied: $u = w_m$ and ${M}^{\\prime }_0, 2m \\models M(a)$ , for some $m \\in {N}$; $u = v^n_m$ and ${M}^{\\prime }_0, 2m \\models P_n(a)$ , for some $m \\in {N}$ and some $n \\in \\lbrace 0, \\ldots , s+2\\rbrace $ ; $u = \\bar{w}_m$ , for some $m \\in {N}$ ; $u = \\bar{v}^n_m$ , for some $m \\in {N}$ and some $n \\in \\lbrace 0, \\ldots , s+2\\rbrace $ .", "Thus, by definition of ${M}^\\ast _0$ , $\\begin{array}{lcl}{M}^\\ast _0, w_m \\models q \\wedge P(a) & \\Longleftrightarrow & a = m.\\end{array}\\qquad \\mathrm {(6)}$ We now prove that ${M}_0^\\ast , w_0 \\models A^{\\ast }$ .", "First, we show that $\\begin{array}{lcl}{M}^\\ast _0, u \\models \\forall x\\, P(x)& \\Longleftrightarrow &u \\in \\lbrace \\bar{w}_m : m \\in {N}\\rbrace \\cup \\lbrace \\bar{v}^n_m : m \\in {N}, 0 \\leqslant n \\leqslant s+ 2 \\rbrace .\\end{array}\\qquad \\mathrm {(7)}$ The right-to-left implication is immediate from the definition of ${M}^\\ast _0$ .", "For the converse, assume $u \\notin \\lbrace \\bar{w}_m : m \\in {N}\\rbrace \\cup \\lbrace \\bar{v}^n_m: m \\in {N}, 0 \\leqslant n \\leqslant s + 2 \\rbrace $ .", "We have four cases to consider.", "Case $u = w_m$ : The definition of ${M}^\\ast _0$ implies that ${M}^\\ast _0, w_m \\lnot \\models P(c)$ , for every $c \\in \\mathcal {D} - \\lbrace m\\rbrace $ .", "Since $\\mathcal {D} - \\lbrace m\\rbrace \\ne \\varnothing $ , we obtain ${M}^\\ast _0, w_m \\lnot \\models \\forall x\\, P(x)$ .", "Case $u = v^{s+1}_m$ : The definition of ${M}^\\ast _0$ implies that ${M}^\\ast _0, v^{s+1}_m \\models P(a)$ if, and only if, ${M}^{\\prime }_0, 2m \\models P_{s+1} (a)$ , which by definition of ${M}^{\\prime }_0$ , holds if, and only if, ${M}_0, 0 \\models a \\triangleleft b$ and $\\alpha _m = a$ .", "Therefore, ${M}^\\ast _0, v^{s+1}_m \\lnot \\models P(c)$ , for every $c \\in \\mathcal {D} -\\lbrace \\alpha _m\\rbrace $ .", "Since $\\mathcal {D} - \\lbrace \\alpha _m\\rbrace \\ne \\varnothing $ , we obtain ${M}^\\ast _0, v^{s+1}_m \\lnot \\models \\forall x\\, P(x)$ .", "Case $u = v^{s+2}_m$ : The definition of ${M}^\\ast _0$ implies that ${M}^\\ast _0, v^{s+2}_m \\models P(a)$ if, and only if, ${M}^{\\prime }_0, 2m \\models P_{s+2} (a)$ , which by definition of ${M}^{\\prime }_0$ , holds if, and only if, $a = \\alpha _m +1$ .", "Therefore, ${M}^\\ast _0, v^{s+2}_m \\lnot \\models P(c)$ , for every $c \\in \\mathcal {D} -\\lbrace \\alpha _m + 1\\rbrace $ .", "Since $\\mathcal {D} - \\lbrace \\alpha _m + 1\\rbrace \\ne \\varnothing $ , we obtain ${M}^\\ast _0, v^{s+2}_m \\lnot \\models \\forall x\\, P(x)$ .", "Case $u = v^{n}_m$ , where $n \\in \\lbrace 0, \\ldots , s\\rbrace $ : By definitions of ${M}^\\ast _0$ and ${M}^{\\prime }_0$ , $\\begin{array}{lclcl}{M}^\\ast _0, v^{n}_m \\models P(a)& \\Longleftrightarrow &{M}^{\\prime }_0, 2m \\models P_{n} (a)& \\Longleftrightarrow &{M}_0, 2m \\models P_{n} (a).\\end{array}$ The definition of ${M}_0$ implies that ${M}_0, 2m \\lnot \\models P_n(-1)$ .", "Therefore, ${M}^\\ast _0, v^{n}_m \\lnot \\models P(-1)$ ; hence, ${M}^\\ast _0, v^{n}_m \\lnot \\models \\forall x\\, P(x)$ .", "Thus, ${M}^\\ast _0, u \\lnot \\models \\forall x\\, P(x)$ , and so (7) is proven.", "Next, we show that, for every $m \\in {N}$ , every $n \\in \\lbrace 0, \\ldots , s+2\\rbrace $ and every $a \\in \\mathcal {D}$ , $\\begin{array}{lcl}{M}_0^\\ast , w_m\\models \\beta _n(a) & \\Longleftrightarrow & {M}^{\\prime }_0, 2m \\models P_n(a).\\end{array}\\qquad \\mathrm {(8)}$ First, define a binary relation ${R}_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}}$ on ${N}$ by $\\begin{array}{lcl}w R_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}} v& \\leftrightharpoons & v \\lnot \\models \\forall x\\, P(x) \\mbox{ and, for some $u \\in {N}$, both } w\\leqslant u \\leqslant v \\mbox{ and } u \\models \\forall x\\, P(x).\\end{array}$ Now, assume ${M}^{\\prime }_0, 2m \\models P_n(a)$ .", "By (6), ${M}_0^\\ast , w_{m+1} \\models q \\wedge P(m+1)$ .", "By (7) and the definition of ${M}_0^\\ast $ , $w_{m+1} \\in R^{s+4}_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}}(w_m) - R^{s+5}_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}}(w_m)$ ; $w_m < v^n_m$ ; $w_{m+1} \\in R^{n+1}_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}}(v^n_m) - R^{n+2}_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}}(v^n_m)$ ; ${M}_0^\\ast , v^n_m \\models P(a)$ .", "Therefore, ${M}_0^\\ast , w_m \\models \\beta _n (a)$ .", "Conversely, assume ${M}_0^\\ast , w_m \\models \\beta _n(a)$ .", "Then, ${M}_0^\\ast , w_m \\models \\exists y\\, ({}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}^{s+4} (q \\wedge P(y)) \\wedge \\lnot {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}^{s+5} (q \\wedge P(y))).$ Hence, there exist $u \\in {N}$ and $b \\in \\mathcal {D}$ such that ${M}_0^\\ast , u \\models q \\wedge P(b) \\mbox{ and }u \\in R^{s+4}_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}}(w_m) -R^{s+5}_{\\scriptsize {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}}(w_m).$ By definition of ${M}^\\ast _0$ and by (6), the only choices for $u$ and $b$ are, respectively, $w_{m+1}$ and $m+1$ .", "Hence, ${M}_0^\\ast , w_m \\models {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}({}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}^{n+1} (q \\wedge P(m+1)) \\wedge \\lnot {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.60004pt}\\mbox{$\\Diamond $}^{n+2} (q \\wedge P(m+1)) \\wedge P(a)).$ Thus, by definition of ${M}_0^\\ast $ , we obtain ${M}_0^\\ast , v^n_m \\models P(a)$ and, hence, ${M}^{\\prime }_0, 2m \\models P_n (a)$ .", "Thus, (8) is proven.", "From (6), (7) and (8), we obtain ${M}_0^\\ast , w_0 \\models A_i^{\\ast }$ , for each $i$ with $0 \\leqslant i \\leqslant 8$ and $i \\ne 4$ .", "Furthermore, based on (6), (7) and (8), it is straightforward to check that ${M}_0^\\ast , w_0 \\models A^\\ast _4$ and ${M}_0^\\ast , w_0 \\models A^\\ast _9$ .", "Thus, ${M}_0^\\ast , w_0 \\models A^{\\ast }$ .", "$\\Box $ From Lemma REF we immediately obtain the following result: Theorem 3.4 Satisfiability for $\\mathbf {L}({N}, \\leqslant )$ is $\\Sigma ^1_1$ -hard in languages with two individual variables, one monadic predicate letter and one proposition letter." ], [ "Logics of discrete linear orders", "We now generalise Theorem REF to logics of descrete linear orders other than $\\langle {N}, \\leqslant \\rangle $ .", "We first consider logics of frames based on ${N}$ .", "Let $R$ be a binary relation on ${N}$ between $<$ and $\\leqslant $ .", "Define $\\begin{array}{lcl}\\Box ^{+} \\varphi & = & \\varphi \\wedge \\Box \\varphi .\\end{array}$ Then, the relation $\\leqslant $ is the reflexive closure of $R$ and, hence, it is the accessibility relation associated with the operator $\\Box ^{+}$ : for every model ${M} = \\langle {N}, R, D, I \\rangle $ , every $w \\in {N}$ and every assignment $g$ , $\\begin{array}{lcl}{M},w\\models ^g \\Box ^+\\varphi & \\Longleftrightarrow & \\mbox{${M},w^{\\prime }\\models ^g \\varphi $, for every $w^{\\prime }\\in {N}$ such that $w\\leqslant w^{\\prime }$.", "}\\end{array}$ Let $A^+$ to be the formula obtained from $A^\\ast $ by replacing every occurrence of $\\Box $ with an occurrence of $\\Box ^+$ .", "The noted correspondence between $\\Box ^+$ and $\\leqslant $ , as well as their connection with, respectively, $\\Box $ and $R$ , give us the following analogue of Lemma REF : Lemma 4.1 There exists a recurrent tiling of ${N}\\times {N}$ satisfying ($T_1$ ) through ($T_3$ ) if, and only if, $\\langle {N}, R \\rangle \\lnot \\models \\lnot A^+$ .", "From Lemma REF , we obtain the analogue of Theorem REF for $\\mathbf {L}({N}, R)$ .", "Moreover, since none of the arguments made so far depend on the assumption of properly expanding domains, we obtain the following generalisation of Theorem REF : Theorem 4.2 Let $R$ be a binary relation on ${N}$ between $<$ and $\\leqslant $ , and let $L$ be a logic such that $\\mathbf {L}({N}, R) \\subseteq L \\subseteq \\mathbf {L}_{c} ({N},R)$ .", "Then, satisfiability for $L$ is $\\Sigma ^1_1$ -hard in languages with two individual variables, one monadic predicate letter and one proposition letter.", "Figure: Frames G n {G}_n and H n {H}_nAs we next observe, Theorem REF covers countably many logics, countably many pairs of which are incompatible.", "First, note that $\\mathbf {L}({N}, <)$ and $\\mathbf {L}({N}, \\leqslant )$ are incompatible.", "Let $\\begin{array}{lcl}Z & = & \\Box (\\Box p \\rightarrow p) \\rightarrow (\\Diamond \\Box p \\rightarrow \\Box p);\\\\\\mbox{\\it ref} & = & \\Box p \\rightarrow p.\\end{array}$ It is well known [23] that $\\langle {N}, < \\rangle \\models Z$ , but $\\langle {N}, \\leqslant \\rangle \\lnot \\models Z$ ; hence, $\\mathbf {L}({N}, <) \\lnot \\subseteq \\mathbf {L}({N}, \\leqslant )$ .", "It is also clear that $\\langle {N}, \\leqslant \\rangle \\models \\mbox{\\it ref}$ , but $\\langle {N}, < \\rangle \\lnot \\models \\mbox{\\it ref}$ ; hence, ${\\bf L}({N}, \\leqslant ) \\lnot \\subseteq {\\bf L}({N}, <)$ .", "Generalising this observation, we obtain countably many logics, countably many pairs of which are incompatible.", "Let $\\begin{array}{lcl}\\varphi & =& (q \\wedge \\Box (\\lnot q \\rightarrow \\varphi )) \\vee (\\lnot q \\wedge \\Box (q \\rightarrow \\varphi )).\\end{array}$ Let ${G}_n$ be the irreflexive chain $0, \\ldots , n - 1$ , followed by the infinite reflexive chain $n, n+1, \\ldots \\,$ , shown in Figure REF on the left.", "Dually, let ${H}_n$ be the reflexive chain $0, \\ldots , n - 1$ , followed by the infinite irreflexive chain $n, n+1, \\ldots \\,$ , shown in Figure REF on the right.", "We show that ${\\bf L}({G}_k) \\ne {\\bf L}({G}_m)$ and ${\\bf L}({H}_k) \\ne {\\bf L}({H}_m)$ provided $k \\ne m$ .", "Indeed, $k > m$ implies $\\mathbf {L}({G}_k) \\subseteq \\mathbf {L}({G}_m)$ : if ${G}_m, s \\lnot \\models \\varphi $ then ${G}_k, s \\lnot \\models \\varphi $ since ${G}_m$ is a generated subframe of ${G}_k$ .", "Also, ${G}_n \\models \\Box ^n \\mbox{\\it ref}$ , but ${G}_{n+1}, 0 \\lnot \\models \\Box ^n \\mbox{\\it ref}$ .", "Hence, $\\mathbf {L}({G}_k) \\ne \\mathbf {L}({G}_m)$ if $k \\ne m$ .", "A similar argument, using the formula $^n Z$ to distinguish $\\mathbf {L}({H}_{n+1})$ from $\\mathbf {L} ({H}_{n})$ , shows that $\\mathbf {L}({H}_k) \\ne \\mathbf {L}({H}_m)$ if $k \\ne m$ .", "Thus, we have infinitely many logics ${\\bf L}({G}_n)$ and infinitely many logics ${\\bf L}({H}_n)$ .", "Note that, for every $k, m \\in {N}$ , logics ${\\bf L}({G}_k)$ and ${\\bf L}({H}_m)$ are incompatible since, for every $k, m \\in {N}$ , both $\\Box ^{k+m} \\mbox{\\it ref} \\in {\\bf L}({G}_k) - {\\bf L}({H}_m)$ and $^{k+m} Z \\in {\\bf L}({H}_m) - {\\bf L}({G}_k)$ .", "From Theorem REF , we obtain the following: Corollary 4.3 Satisfiability for ${\\mathbf {L}_{c}}({N}, \\leqslant )$ , $\\mathbf {L}({N}, <)$ and ${\\mathbf {L}_{c}}({N}, <)$ is $\\Sigma ^1_1$ -hard in languages with two individual variables, one monadic predicate letter and one proposition letter.", "The frame $\\langle {N},<\\rangle $ is isomorphic to the structure $\\langle \\omega ,< \\rangle $ , where $\\omega $ is the least infinite ordinal and $<$ , as for all ordinals, is the membership relation on $\\omega $ .", "We next generalise Theorem REF to logics of frames based on infinite ordinals of a special form, which include $\\omega $ .", "Theorem 4.4 Let $\\alpha = \\omega \\cdot m + k$ , for some $m$ with $1 \\leqslant m < \\omega $ and some $k < \\omega $ , let $R$ be a binary relation on $\\alpha $ between $<$ and its reflexive closure $\\leqslant $ , and let $L = \\mathbf {L} (\\alpha ,R)$ .", "Then, satisfiability for $L$ is $\\Sigma ^1_1$ -hard in languages with two individual variables, one monadic predicate letter and one proposition letter.", "Proof.", "The proof is similar to that of Theorem REF .", "We only comment on how to obtain an analogue of Lemma REF , an encoding of the recurrent tiling problem in $\\mathbf {L}(\\alpha , \\leqslant )$ .", "(For the general case of an arbitrary relation between $<$ and $\\leqslant $ , we use $\\Box ^+$ instead of $\\Box $ .)", "Since the frame $\\langle \\alpha , \\leqslant \\rangle $ may contain a world that does not see another world, we need to define a variant of the formula $A_9$ suitable for such a situation: $\\begin{array}{lcl}A^{\\bullet }_9 & = & \\forall x\\, \\big ( (M(x) \\rightarrow \\Box (\\exists y\\,M(y) \\rightarrow {}{\\mbox{$\\Diamond $}}\\mbox{$\\Diamond $}\\hspace{0.0pt}\\hspace{1.00006pt}\\mbox{$\\Diamond $}(\\exists y\\, M(y) \\rightarrow P_{t_0} (x))) \\big ).\\end{array}$ Let $A^{\\bullet }$ be the conjunction of formulas $A_0$ through $A_8$ from Section , as well as $A^{\\bullet }_9$ .", "We claim that there exists a recurrent tiling satisfying ($T_1$ ) through ($T_3$ ) if, and only if, $\\langle \\alpha , \\leqslant \\rangle \\lnot \\models \\lnot A^{\\bullet }$ .", "Assume ${M}, u_0 \\models A^{\\bullet }$ , for some model ${M} = \\langle \\alpha , \\leqslant , D, I \\rangle $ and some world $u_0 \\in \\alpha $ .", "Then, there exists in $\\alpha $ a last copy of $\\omega $ that has the following property: it contains a world $w$ marked by an element, say $a_k$ , of the sequence $a_0 \\triangleleft ^{I,u_0} a_1 \\triangleleft ^{I,u_0} a_2\\triangleleft ^{I,u_0} \\ldots \\,$ of elements of $D(u_0)$ whose existence follows from ${M}, u_0 \\models A_2$ .", "Then, a tiling can be obtained from the said copy of $\\omega $ , similarly to the way it was done in the proof of the “if” part of Lemma REF : columns are simulated by elements $a_0,a_1,a_2,\\ldots $ of $D(u_0)$ ; rows are simulated by worlds $w_k,w_{k+1},w_{k+2},\\ldots $ such that $w_k = w$ , $w_k < w_{k+1} < w_{k+2} < \\ldots \\,$ , $w_{k+n}\\models M(a_{k+n})$ , for every $n \\in {N}$ ; and a tiling function $f\\colon {N}\\times {N}\\rightarrow T$ is defined by $\\begin{array}{lcl}f(n,m) = t & \\mbox{ whenever } & w_{k+m}\\models P_t(a_n).\\end{array}$ Thus defined $f$ clearly satisfies ($T_1$ ) and ($T_2$ ).", "Also, since ${M}, u_0 \\models A^{\\bullet }_9$ , the set $\\lbrace w_{k+m} : m \\in {N}\\mbox{ and } w_{k+m} \\models P_{t_0} (a_0)\\rbrace $ is infinite; hence, $f$ satisfies ($T_3$ ) and so is a required function.", "For the converse, we use the first copy of $\\omega $ contained in $\\alpha $ for the satisfaction of $A^{\\bullet }$ : first, we define the interpretation of all the letters occurring in $A^{\\bullet }$ on the said copy of $\\omega $ as in the proof of the “only if” part of Lemma REF ; second, we define the interpretation of letters $p$ , $M$ , $P_t$ , for each $t \\in T$ , and $\\triangleleft $ to be empty at every world not belonging to the said copy of $\\omega $ ; last, we define $U$ to be an arbitrary non-empty subset of the domain of every world not belonging to the said copy of $\\omega $ .", "Then, $A^{\\bullet }$ is true at the least, with respect to $\\leqslant $ , world of $\\alpha $ ; hence, it is satisfiable.", "$\\Box $" ], [ "Logics of dense and continuous linear orders", "It is not clear whether $\\Sigma ^1_1$ -hardness results analogous to Theorem REF can be obtained for logics of linear orders distinct from those mentioned there.", "Perhaps the most significant logics of linear orders not covered by Theorem REF are logics of the rationals and the reals with natural partial and strict orders, i.e.", "$\\mathbf {L} ({Q}, \\leqslant )$ , $\\mathbf {L} ({Q}, <)$ , $\\mathbf {L} ({R}, \\leqslant )$ and $\\mathbf {L} ({R}, <)$ .", "The proof of Lemma REF does not carry over to either $\\mathbf {L} ({Q}, \\leqslant )$ or $\\mathbf {L} ({R}, \\leqslant )$ since we cannot ensure, given a model of the formula $A$ based on either $\\langle {Q}, \\leqslant \\rangle $ or $\\langle {R}, \\leqslant \\rangle $ , that the tiling defined as in the proof of Lemma REF satisfies $(T_3)$ .", "In the case of $\\mathbf {L} ({Q}, \\leqslant )$ , no such tiling exists: $\\mathbf {L} ({Q}, \\leqslant )$ is recursively enumerable [13] and hence $\\Sigma ^0_1$ -complete.", "The case of $\\mathbf {L} ({R}, \\leqslant )$ might turn out to be similar as it is not known whether $\\mathbf {L} ({R}, \\leqslant )$ is distinct from $\\mathbf {L} ({Q}, \\leqslant )$ .", "(The superintuitionistic logics of $\\langle {Q}, \\leqslant \\rangle $ and $\\langle {R}, \\leqslant \\rangle $ coincide [54] and are $\\Sigma ^0_1$ -complete [59]; on the other hand, the superintuitionistic, and hence modal, logics of predicate frames with constant domains over $\\langle {Q}, \\leqslant \\rangle $ and $\\langle {R}, \\leqslant \\rangle $ differ [59].)", "A slight modification of the proof of Lemma REF shows, however, that satisfiability for $\\mathbf {L} ({Q}, \\leqslant )$ and $\\mathbf {L} ({R}, \\leqslant )$ is $\\Pi ^0_1$ -hard—hence, $\\mathbf {L} ({Q}, \\leqslant )$ and $\\mathbf {L} ({R}, \\leqslant )$ are $\\Sigma ^0_1$ -hard—in languages with two variables, one monadic predicate letter and one proposition letter: simply leaving out the argument for ($T_3$ ), we obtain a reduction to satisfiability for $\\mathbf {L} ({Q}, \\leqslant )$ and $\\mathbf {L} ({R}, \\leqslant )$ in appropriate languages of the $\\Pi ^0_1$ -complete [6], [9] ${N}\\times {N}$ tiling problem whose solution is required to satisfy ($T_1$ ) and ($T_2$ ), but not ($T_3$ ).", "We do, rather, establish a more general result.", "Define $B$ to be the conjunction of formulas $A_0$ through $A_8$ (i.e., leave out $A_9$ from the formula $A$ defined in Section ).", "Lemma 5.1 Let $\\langle W, \\leqslant \\rangle $ be a partial linear order containing an infinite ascending chain of pairwise distinct elements of $W$ .", "Then, there exists a tiling of ${N}\\times {N}$ satisfying $(T_1)$ and $(T_2)$ if, and only if, $\\langle W, \\leqslant \\rangle \\lnot \\models \\lnot B$ .", "Proof.", "(“if”) The proof is identical to that the “if” part of Lemma REF , except that we leave out the argument for ($T_3$ ).", "(“only if”) Suppose $f$ is a function satisfying ($T_1$ ) and ($T_2$ ).", "We obtain a model based on $\\langle W, \\leqslant \\rangle $ satisfying $B$ .", "Let $D_w = {N}\\cup \\lbrace -1\\rbrace $ , for every $w \\in W$ .", "To define the interpretation function $I$ on $\\langle W, \\leqslant , D \\rangle $ , we use elements of the infinite ascending chain $w_0 \\leqslant w_1 \\leqslant w_2 \\leqslant \\ldots $ of worlds from $W$ that exists by assumption: we define $I$ so that, for every $k \\in {N}$ and every $a, b \\in \\mathcal {D}$ , $\\begin{array}{lcl}{M},w_k \\models a\\triangleleft b& \\leftrightharpoons &\\mbox{$k$ is even and $b=a+1$;}\\smallskip \\\\{M},w_k \\models p& \\leftrightharpoons &\\mbox{$k$ is odd;}\\smallskip \\\\{M},w_k \\models M(a)& \\leftrightharpoons &\\mbox{$k=2a$;}\\smallskip \\\\{M},w_k \\models P_t(a)& \\leftrightharpoons &\\mbox{$k=2m$ and $f(a,m) = t$, for some $m\\in {N}$,}\\end{array}$ and, for every $v \\notin \\lbrace w_i : i \\in {N}\\rbrace $ and every predicate letter $S$ of $B$ , $I(v, S) = I(w_m, S), \\mbox{ where } m = \\min \\lbrace k\\in {N}: v\\leqslant w_k \\rbrace .$ It is straightforward to check that ${M}, w_0 \\models B$ , so we leave this to the reader.", "$\\Box $ Using a modification of the formula $B$ obtained by replacing every occurrence of $\\Box $ by that of $\\Box ^+$ , we can prove the following analogue of Theorem REF (the proof uses Lemma REF in the same way Theorem REF used Lemma REF ): Theorem 5.2 Let $\\langle W, < \\rangle $ be a strict linear order containing an infinite ascending chain of pairwise distinct elements of $W$ .", "Let $\\leqslant $ be the reflexive closure of $<$ and $R$ a binary relation between $<$ and $\\leqslant $ .", "Let $L$ be a logic such that $\\mathbf {L}(W, R) \\subseteq L \\subseteq \\mathbf {L}_{c} (W, R)$ .", "Then, satisfiability for $L$ is $\\Pi ^0_1$ -hard—hence, $L$ is $\\Sigma ^0_1$ -hard—in languages with two individual variables, one monadic predicate letter and one proposition letter.", "Corollary 5.3 Logics $\\mathbf {L} ({Q}, \\leqslant )$ , ${\\mathbf {L}_{c}} ({Q}, \\leqslant )$ , $\\mathbf {L} ({Q}, <)$ , ${\\mathbf {L}_{c}} ({Q}, <)$ , $\\mathbf {L} ({R}, \\leqslant )$ , ${\\mathbf {L}_{c}} ({R}, \\leqslant )$ , $\\mathbf {L} ({R}, <)$ and ${\\mathbf {L}_{c}} ({R}, <)$ are $\\Sigma ^0_1$ -hard in languages with two individual variables, one monadic predicate letter and one proposition letter.", "Well-known axiomatically defined predicate modal logics coincide with some logics mentioned in Corollary REF .", "Let $\\mathbf {K}$ be the minimal propositional normal modal logic and, for a set of formulas $\\Gamma $ and a formula $\\varphi $ , let $\\Gamma \\oplus \\varphi $ be the closure of $\\Gamma \\cup \\lbrace \\varphi \\rbrace $ under modus ponens, necessitation and propositional substitution.", "Recall the following definitions of propositional modal logics: $\\begin{array}{lcl}\\mathbf {S4.3}& =& \\mathbf {K}\\oplus \\mathit {ref}\\oplus \\Box p \\rightarrow \\Box \\Box p\\oplus \\Box (\\Box p \\rightarrow q) \\vee \\Box (\\Box q \\rightarrow p);\\smallskip \\\\\\mathbf {K4.3.D.X}& =& \\mathbf {K}\\oplus \\Box p \\rightarrow \\Box \\Box p\\oplus \\Box (\\Box ^+ p \\rightarrow q) \\vee \\Box (\\Box ^+ q \\rightarrow p)\\oplus \\Diamond \\top \\oplus \\Box \\Box p \\rightarrow \\Box p.\\end{array}$ For a propositional modal logic $L$ , denote by $\\mathbf {Q}L$ the minimal predicate modal logic containing $\\mathbf {QCl} \\cup L$ .", "It follows from the definitions of $\\mathbf {S4.3}$ and $\\mathbf {K4.3.D.X}$ given above that logics $\\mathbf {QS4.3}$ and $\\mathbf {QK4.3.D.X}$ are finitely axiomatizable and, hence, recursively enumerable, i.e., they are in $\\Sigma ^0_1$ .", "It is well known [13] that $\\mathbf {QS4.3} = \\mathbf {L} ({Q}, \\leqslant )$ and $\\mathbf {QK4.3.D.X} = \\mathbf {L} ({Q}, <)$ .", "We, therefore, obtain the following result: Corollary 5.4 The logics $\\mathbf {QS4.3}$ and $\\mathbf {QK4.3.D.X}$ are $\\Sigma ^0_1$ -complete in languages with two individual variables, one monadic predicate letter and one proposition letter.", "Thus, $\\mathbf {L} ({Q}, \\leqslant )$ and $\\mathbf {L} ({Q}, <)$ are $\\Sigma ^0_1$ -complete in languages with two individual variables, one monadic predicate letter and one proposition letter." ], [ "Some other logics", "In this section, we note some corollaries of Theorem REF other than those mentioned in Corollary REF .", "We also note that a straightforward modification of the proof of Theorem REF establishes $\\Sigma ^0_1$ -completeness of the logic $\\mathbf {QK4.3}$ in the languages we consider.", "The first corollary concerns logics of infinite ordinals: as before, for ordinals, by $<$ and $\\leqslant $ we mean, respectively, the relation $\\in $ and its reflexive closure.", "Corollary 6.1 Let $\\alpha $ be an infinite ordinal.", "Then, $\\mathbf {L} (\\alpha , <)$ , ${\\mathbf {L}_{c}} (\\alpha , <)$ , $\\mathbf {L} (\\alpha , \\leqslant )$ and ${\\mathbf {L}_{c}} (\\alpha , \\leqslant )$ are $\\Sigma ^0_1$ -hard in languages with two individual variables, one monadic predicate letter and one proposition letter.", "The second concerns logics of non-standard models of the elementary theories of some of the structures considered thus far (the elementary theory of the structure ${A}$ is denoted by ${\\sf Th}({A})$ ): Corollary 6.2 Let ${A}$ be one of the structures $\\langle {N}, {\\leqslant } \\rangle $ , $\\langle {N}, {<} \\rangle $ , $\\langle {Q}, {\\leqslant } \\rangle $ , $\\langle {Q}, {<} \\rangle $ , $\\langle {R}, {\\leqslant } \\rangle $ and $\\langle {R}, {<} \\rangle $ , and let ${F}$ be a non-standard classical first-order model of ${\\sf Th}({A})$ .", "Then, $\\mathbf {L}({F})$ and ${\\mathbf {L}_{c}}({F})$ are $\\Sigma ^0_1$ -hard in languages with two individual variables, one monadic predicate letter and one proposition letter.", "Lastly, a slight modification of the argument of Section  gives us the following result on $\\mathbf {QK4.3}$ , the logic of strict linear orders [13].", "We recall that $\\begin{array}{lcl}\\mathbf {K4.3}& =& \\mathbf {K}\\oplus \\Box p \\rightarrow \\Box \\Box p\\oplus \\Box (\\Box ^+ p \\rightarrow q) \\vee \\Box (\\Box ^+ q \\rightarrow p).\\end{array}$ Thus, $\\mathbf {QK4.3}$ is finitely axiomatizable and, hence, recursively enumerable.", "Corollary 6.3 The logic $\\mathbf {QK4.3}$ is $\\Sigma ^0_1$ -complete in languages with two individual variables, one monadic predicate letter and one proposition letter.", "Proof.", "One can show that the formula $A^+$ defined in Section  is satisfiable in a model based on a $\\mathbf {QK4.3}$ -frame if, and only if, there exists a tiling of ${N}\\times {N}$ satisfying $(T_1)$ and $(T_2)$ .", "We only notice that, in the proof of the “only if” part, we need to show that the model satisfying $A^+$ is infinite—this readily follows by $A_1$ , $A_2$ and $A^+_4$ .", "$\\Box $" ], [ "Discussion", "We now discuss some questions arising out of the present work.", "The first question is whether our main result, Theorem REF , can be strengthened to languages with two variables and a single monadic predicate letter: is the “extra” proposition letter necessary?", "For the majority of natural predicate modal—and closely related superintuitionistic—logics similar results have been obtained [46], [49], [51] for languages with a single monadic predicate letter (the number of variables—two [46] or three [49], [51]—depends on the logic).", "Those results do not, however, cover some notable logics—the known results for the predicate counterparts of propositional modal logics $\\mathbf {GL.3}$ , $\\mathbf {Grz.3}$ and $\\mathbf {S5}$ involve “extra” proposition letters [46].", "The propositional modal logics of frames $\\langle {N}, \\leqslant \\rangle $ and $\\langle {N}, < \\rangle $ —in common with $\\mathbf {GL.3}$ , $\\mathbf {Grz.3}$ and $\\mathbf {S5}$ —are NP-complete, i.e., not as computationally hard (provided PSPSACE $\\ne $ NP) as PSPACE-hard propositional logics whose first-order counterparts are known to be undecidable in languages with a few variables and a single monadic predicate letter.", "Whether this observation points to a genuine connection is unclear; the hypothesis, however, seems to be worth investigating.", "It seems at least plausible that predicate logics of $\\langle {N}, \\leqslant \\rangle $ and $\\langle {N}, < \\rangle $ are decidable in languages with two variables and a single monadic letter.", "We note that a stronger result, $\\Sigma ^1_1$ -hardness of satisfiability for languages with two variables and a single monadic letter, is relatively easily obtainable for the logics of the naturals in the more expressive language containing, alongside $\\Box $ , the unary operator $\\circledcirc $ (“next”) with the truth condition $ {M}, n \\models ^g \\circledcirc \\varphi \\mbox{ if } {M}, n + 1\\models ^g\\varphi $ .", "The resultant logic is a notational variant of the first-order quantified linear time temporal logic $\\mathbf {QLTL}$ with temporal operators $\\Box $ (interpreted as “always in the future”) and $\\circledcirc $ —even without the more expressive binary operator “until.” It is well known [28] that satisfiability for $\\mathbf {QLTL}(\\Box , \\circledcirc )$ is $\\Sigma ^1_1$ -hard in languages with two variables and only monadic predicate letters: the proof, which inspired the proofs presented above, is a reduction of the recurrent ${N}\\times {N}$ tiling problem described in Section .", "Since the number of tile types in the recurrent tiling problem is unbounded, [28] establishes $\\Sigma ^1_1$ -hardness for languages with an unlimited supply of monadic predicate letters.", "The proof of [28] can, however, be modified to establish $\\Sigma ^1_1$ -hardness for languages with a single monadic predicate letter.", "Perhasps the easiest way is to modify the formulas used in the original proof [28] so that the satisfying model has equal-seized gaps (non-empty sequences of worlds) between the worlds corresponding to the tiling; the size of the gaps is proportional to the number of tile types.", "The binary letter can then be modelled as in the proof of Lemma REF above.", "To model all the monadic letters with a single one, we use the following observation.", "If $f$ is a tiling function satisfying ($T_1$ ) through ($T_3$ ), then for every $m \\in {N}$ , there do not exist $t, t^{\\prime } \\in T$ such that $t \\ne t^{\\prime }$ and, for every $n \\in {N}$ , both $f(n, m) = t$ and $f(n, m) = t^{\\prime }$ : an entire row cannot be tiled simultaneously with tiles of two distinct types.", "Therefore, a construction similar to the one used in the proof of Lemma REF above would not produce a model where two successive worlds satisfy $\\forall x\\, P(x)$ .", "Hence, we can use $\\circledcirc \\forall x\\, P(x) \\wedge {\\circledcirc }{\\circledcirc } \\forall x\\, P(x)$ in place of the proposition letter $q$ in the final reduction.", "This gives us the following result: Theorem 7.1 Satisfiability for $\\mathbf {QLTL}(\\Box , \\circledcirc )$ is $\\Sigma ^1_1$ -hard in languages with two individual variables and a single monadic predicate letter.", "The second question arising out of the present work is whether stronger lower bounds are obtainable for the logics of the reals—provided they are distinct from the logics of the rationals.", "One approach would be to attempt to adapt techniques developed by Reynolds and Zakharyaschev [40] for proving $\\Sigma ^1_1$ -hardness of products of two propositional modal logics of linearly ordered frames.", "In particular, Reynolds and Zakharyaschev establish [40] $\\Sigma ^1_1$ -hardness of product logics satisfying two conditions: first, the product logic admits a frame with infinite ascending chains along both accessibility relations; second, the order relation associated with one of the factor logics is Dedekind complete.", "Even though this setup appears similar to predicate modal logics of the reals, it is not immediately clear how to apply the techniques of Reynolds and Zakharyaschev [40] in our circumstances since it is not obvious how an infinite linear partial or strict order, which has to be transitive, can be defined on domains of a predicate Kripke model using formulas with only two variables.", "The third question is whether our results are tight.", "We are not aware of upper-bound results for logics considered here, the exception being the logics of the rationals, which are, as mentioned in Section , $\\Sigma ^0_1$ -complete.", "Thus, a search for upper bounds appears to be an interesting topic of future study.", "The final question we mention is whether analogous results can be obtained for superintuitionistic logic of $\\langle {N}, \\leqslant \\rangle $ .", "Whether this can be done is unclear to us: the techniques used here appear unsuitable for superintuitionistic logics given the difficulty of modelling the changing values of tile types on a linear frame with a hereditary valuation." ], [ "Acknowledgements", "We are grateful to Valentin Shehtman for discussions of an earlier version of the paper.", "We are indebted to the anonymous reviewers for helping to improve the paper." ] ]
2105.11811
[ [ "Interpolating between multi-center microstate geometries" ], [ "Abstract We study interpolation between two multi-center microstate geometries in 4d/5d that represent Lunin-Mathur geometries with circular profiles.", "The interpolating solution is a Lunin-Mathur geometry with a helical profile, and is represented by a 2-center solution with a codimension-2 source.", "The interpolating 2-center solution exhibits interesting features such as some of the charges being delocalized, and some of the charges getting transferred from the codimension-2 center to the other, codimension-3 center as the interpolation proceeds.", "We also discuss the spectral flow of this entire process and speculate on the relevance of such solutions to understanding general microstates of 3-charge black holes." ], [ "Introduction and summary", "String theory contains various extended objects—branes—and allows configurations with multiple such branes bound together by their gravitational and gauge interactions.", "In supergravity, such configurations are realized by solutions with multiple centers representing branes.", "Among such multi-center solutions in supergravity, an important class of solutions is the one in four and five dimensional supergravity that is supersymmetric and characterized by harmonic functions in ${\\mathbb {R}}^3$ [1], [2], [3], [4], [5], [6], [7].", "They have various applications such as the attractor mechanism [8], [9], [10], [11], [12], [13], [14], split attractor flows and wall crossing [15], [16], [17], [18], [3], and microstate geometries [6], [19], [20].", "In this note we will call these solutions “harmonic solutions”, because their construction heavily relies on harmonic functions.", "In most literature those harmonic solutions are assumed to have sources of codimension three, but they can also have codimension-2 sources.Codimension-1 singularities are also possible, although we do not consider them in this note.", "Codimension-2 sources can be produced by the supertube transition [21] and harmonic functions will have non-trivial monodromies around a curve in ${\\mathbb {R}}^3$ .", "Some examples of codimension-2 harmonic solutions were studied in [22], [23].", "In this note, we extend the examples of codimension-2 harmonic solutions by studying the Lunin-Mathur geometries [24], [25] in the framework.", "The Lunin-Mathur geometries are smooth, horizonless solutions of type IIB supergravity in ${\\mathbb {R}}_t\\times {\\mathbb {R}}^4\\times S^1_y\\times T^4$ and represent microstates of the D1-D5 2-charge system.", "Ignoring $T^4$ directions they can be regarded as solutions of 6d supergravity.", "The geometries are parametrized by profile functions which describe the shape in ${\\mathbb {R}}^4$ of the worldvolume of a Kaluza-Klein monopole (KKM) produced by the supertube transition of D1- and D5-branes.", "For some special choices of the profile functions, via duality transformations, the Lunin-Mathur geometries can be described by harmonic solutions with codimension-3 sources.", "Here we consider more general profile functions, for which the harmonic functions have codimension-2 sources as well.The duality transformations at the supergravity level involve smearing, and therefore the profile function in ${\\mathbb {R}}^4$ gets smeared along a direction $\\psi $ along which T-duality is taken.", "This is unlike the formulation of [26] which generalizes the harmonic function to depend on $\\psi $ .", "Our purpose here is to develop techniques to construct harmonic solutions with codimension-2 sources.", "The harmonic solution involves harmonic functions commonly denoted by $(V,K^I,L_I,M)=:{\\cal H}$ where $I=1,2,3.$ The construction proceeds in layers, in that one constructs the harmonic functions in the order of $V,K^I,L_I,M$ .", "In the previous work [22], [23], solutions with trivial $V$ (i.e.", "$V=1$ ) were studied.", "In the current note, we will extend this to solutions with non-trivial $V$ .", "This is a step forward for constructing the most general codimension-2 harmonic solutions.", "In the remainder of this section, we introduce the setup and summarize our main findings.", "The Lunin-Mathur geometry is parametrized by profile functions $g_m(\\lambda )$ , $m=1,2,3,4$ in ${\\mathbb {R}}^4$ , where $g_m(\\lambda +L)=g_m(\\lambda )$ with $L$ a constant.", "One of the simplest Lunin-Mathur geometry is given by the following profile: $g_1(\\lambda )+i g_2(\\lambda )=ae^{i k\\Omega \\lambda },\\qquad g_3(\\lambda )+i g_4(\\lambda )=0,$ where $a>0$ , $k\\in {\\mathbb {Z}}_{>0}$ and $\\Omega ={2\\pi /L}$ .", "This is a circle in the $x_1$ -$x_2$ plane (and a point at the origin in the $x^3$ -$x^4$ plane).", "Using the Hopf fibration, we can project this ${\\mathbb {R}}^4$ profile onto ${\\mathbb {R}}^3$ in which the harmonic functions $(V,K^I,L_I,M)$ of the harmonic solution live.", "If we take the Hopf fiber direction $\\psi $ to be the same as the circle direction of the profile (REF ), the harmonic functions are given by ${\\begin{array}{c}V={1\\over r}, \\qquad K^1=K^2=0,\\qquad K^3={Q_5\\Omega k\\over 2}\\left({1\\over \\Sigma }-{1\\over r}\\right),\\\\L_1={Q_1\\over 4\\Sigma },\\qquad L_2={Q_5\\over 4\\Sigma },\\qquad L_3=1,\\qquad M={Q_5\\Omega k {\\widetilde{a}}\\over 4\\Sigma }.\\end{array}}$ Here the coordinates of ${\\mathbb {R}}^3$ are ${\\bf y}=(y_i)$ , $i=1,2,3$ , and we defined $r\\equiv |{\\bf y}|$ , $\\Sigma \\equiv |{\\bf y}-{\\bf {\\widetilde{a}}}|$ , ${\\bf {\\widetilde{a}}}\\equiv (0,0,-{\\widetilde{a}})$ , ${\\widetilde{a}}={a^2/4}$ .", "These harmonic functions (REF ) have codimension-3 singularities at the origin ${\\bf y}=0$ ($r=0$ ) and at a point on the negative $y_3$ axis, ${\\bf y}={\\bf {\\widetilde{a}}}$ ($\\Sigma =0$ ).", "The singularities in $L_1,L_2$ at $\\Sigma =0$ correspond to the D1 and D5-brane charges that we start with, while the singularity in $K^3$ at $\\Sigma =0$ corresponds to the KKM charge of the D1-D5 supertube along the Hopf fiber direction $\\psi $ .", "To avoid possible confusion, in the following, we refer to this charge appearing as a codimension-3 source in $K^3$ as the D4 charge, borrowing its dual type IIA interpretation.", "This solution has a $U(1)$ rotational symmetry about the $y_3$ axis.", "The need for the $1/r$ term in $K^3$ is not so obvious from the viewpoint of the harmonic function, but this is what one gets from reducing the Lunin-Mathur solution.", "As we will see in the main text, this is necessary for the gauge field in the Lunin-Mathur solution, which involves $V^{-1}K^3$ , to vanish at infinity.", "Alternatively, we can regard this $1/r$ term as coming from the “gauge transformation” [27] of harmonic solutions, under which physical fields are invariant: $\\begin{split}V&\\rightarrow V,\\qquad K^I\\rightarrow K^I+c^I V,\\end{split}$ with $c^I$ arbitrary constants (here we are only showing the $V,K^I$ part; for the full expression including $L_I$ and $M$ , see (REF )).", "It is clear that by choosing $c^3$ appropriately we can change the coefficient of the $1/r$ term in $K^3$ as we want.", "A more general profile—the object of main interest in the current note—is $g_1(\\lambda )+i g_2(\\lambda )=ae^{i k \\Omega \\lambda },\\qquad g_3(\\lambda )+i g_4(\\lambda )=be^{-i k^{\\prime }\\Omega \\lambda },$ where $a,b\\ge 0$ and $k,k^{\\prime }\\in {\\mathbb {Z}}_{>0}$ .", "This is like a helix, going in circles in both $x_1$ -$x_2$ and $x_3$ -$x_4$ directions, with pitches $k,k^{\\prime }$ .", "When $b=0$ , this reduces to (REF ).", "For $b>0$ , the $\\psi $ direction is not an isometry of the Lunin-Mathur solution, but we can still reduce it to ${\\mathbb {R}}^3$ after smearing it along the $\\psi $ direction.", "The projected profile in ${\\mathbb {R}}^3$ is like a latitude line on the globe.", "The explicit harmonic functions can be found in section .", "As we will see in the main text, the parameters $a,b$ are not arbitrary but constrained to satisfy $a^2k^2+b^2k^{\\prime 2}=\\text{const}={Q_1\\over Q_5\\Omega ^2}.$ Let's say we start with $a>0,b=0$ , which corresponds to (REF ), and increase $b$ satisfying (REF ), finally ending with $a=0,b>0$ .", "The projected profile in the ${\\bf y}$ space starts as a point at the “south pole” of an ellipsoid (at $a>0,b=0$ ) and, as we increase $a$ , becomes a latitude line.", "As we go up the ellipsoid from the south toward the north, the latitude line gets larger and then smaller, finally collapsing to a point at the “north pole” (at $a=0,b>0$ ); see the purple and blue dots in Figure REF .", "Figure: The ℝ 3 {\\mathbb {R}}^3 profile and D4 charges of the “helical”solution, as we change the parameters a,ba,b.", "(a): the south pole limit (a>0,b=0a>0,b=0).", "The profile is a point at thesouth pole of an ellipsoid (dashed ellipse).", "The purple dot represent theD4 charge at Σ=0\\Sigma =0, while the green dot the r=0r=0 center.", "The D4charge is shown next to each center in units of Q 5 Ω/2Q_5\\Omega /2.", "(b) as we make bb nonzero, the profile goes off the south pole andbecome a curve 𝒞{\\cal C} along a latitude line of the ellipsoid.", "The bluedots represent the location of 𝒞{\\cal C}.", "The magenta zigzag line betweenthem represent the branch cut coming from the multi-valuedness of K 3 K^3.Delocalized D4 charges are distributed on it.", "(c) As we increase bb and the profile 𝒞{\\cal C} goes up, the branch cut crosses the r=0r=0 center.Some of the delocalized D4 charge has been transferred to the r=0r=0center.", "(d) In the north pole limit (a=0)(a=0), the profile collapses to a point atthe north pole.", "See the main text for more detail.When $a,b>0$ , the latitude line ${\\cal C}$ is a codimension-2 singularity, around which $K^3$ has an additive a monodromy.", "This singularity represents the KKM charge of the D1-D5 supertube along ${\\cal C}$ .", "Let us refer to this charge appearing as a codimension-2 source in $K^3$ as the NS5 charge, borrowing its dual IIA interpretation.", "This is the most general Lunin-Mathur solution reduced to 4d that preserves the $U(1)$ rotational symmetry about the $y_3$ axis.", "Even for $a,b>0$ we have the D4 charge (KKM charge along $\\psi $ ), appearing as codimension-3 sources in $K^3$ .", "It is interesting to see what happens to the D4 charge as we go from the south pole to the north pole.", "Being a dipole charge, the D4 charge is not conserved from the 5d viewpoint, because the Hopf fiber that it is wrapping is contractible in ${\\mathbb {R}}^4$ .", "However, from the 4d viewpoint, the D4 charge is an ordinary (monopole) charge as any other charges and is conserved.", "This is possible because the origin of ${\\mathbb {R}}^4$ , which is a D6-brane at $r=0$ in the IIA interpretation, can also carry D4 charges; as the profile moves on the ellipsoid, some of the D4 charge can be transferred to the D6 center at $r=0$ , so that the total D4 charge is conserved.", "This is the same mechanism as in [28], where they studied how the charge of a fundamental string wrapping the special circle of a KKM background can be conserved, even though the circle is contractible.", "Here we stop to note that there are multiple notions of charges [29] and the one we are talking about is the Page charge, which is conserved, localized, and quantized, but changes under large gauge transformations.", "For harmonic solutions, the D4 Page charge is measured by [22] $-{1\\over 4\\pi } \\int _{M} *_3 dK^3,$ where $*_3$ is the Hodge star for flat ${\\mathbb {R}}^3$ and $M$ is a Gaussian surface.", "If $K^3$ is single-valued, this simply picks up the coefficients of poles in $K^3$ .", "In the present setting, charge conservation works as follows.", "Because of the non-single-valuedness of $K^3$ , in addition to the charge source appearing as a pole in the harmonic function, there also are “delocalized” charges that can be measured by the Gaussian surface surrounding the disk whose boundary is ${\\cal C}$ .This delocalized charge is an example of “Cheshire charges” that can appear in the presence of monodromies [30].$^,$The delocalized charge here corresponds to the fundamental string charge carried by the KKM background in the context of [28].", "This disk is the branch cut across which $K^3$ is discontinuous.", "As ${\\cal C}$ moves up on the ellipsoid, when the center at $r=0$ crosses the branch cut, the delocalized charge gets transferred to the $r=0$ center.", "This process is described in Figure REF .", "For $a>0,b=0$ (south pole), there is a non-vanishing D4 charge but no NS5 charge (Figure REF (a)).", "As we move off the south pole, we start to have a non-vanishing NS5 charge along a now finite ring ${\\cal C}$ .", "The D4 charge is divided into a localized part on curve ${\\cal C}$ and a delocalized part distributed over the branch cut (Figure REF (b)).", "After the branch cut has passed the $r=0$ center, the delocalized charge has changed by the amount transferred to the $r=0$ center (Figure REF (c)).", "Finally, when we reach the north pole, curve ${\\cal C}$ shrinks to a point and we are left with a different amount of D4 charge (Figure REF (d)).", "So, the trivial-looking process of continuously changing the Lunin-Mathur profile leads to a non-trivial process in the 4d harmonic solution, in which the D4 charge gets transferred to the D6 center by way of the dimension-2 NS5-brane.", "The significance of this is further discussed in section  in relation to spectral flow in the dual CFT.", "Generalizations of such processes of charge transfer are expected to realize more general topology-changing processes that presumably play an important role in understanding the microscopic physics of black holes, as will be discussed in section .", "No solutions or geometries presented in this note are essentially new; the purpose here is to look at them from a new viewpoint and re-interpret them, potentially as a basis on which to construct new solutions.", "In the rest of the note, we will give more details of the picture explained above, and also discuss related matters.", "First, in section , we explain harmonic solutions, the Lunin-Mathur geometries, and the relation between them.", "Then, in section , we derive the harmonic functions that corresponds to the Lunin-Mathur geometry for the “helical” profile (REF ), and examine in detail the process of D4-charges getting transferred to the D6 center.", "We will also discuss various other matters such as how the constraint (REF ) is derived from a no-CTC (closed timelike curve) condition.", "In section , we confirm the harmonic functions derived in section  from the 6d Lunin-Mathur geometry side.", "In section , we discuss the dual CFT perspective of the whole process.", "We will also consider the spectral flow and fractional spectral flow of our solution, in both CFT and gravity.", "Finally, in section , we discuss the implication of the result obtained in this note.", "Some details of the computation in the main text can be found in the Appendix." ], [ "Harmonic solutions", "Here we give a brief review of the harmonic solution, which represents multi-center black-hole/ring solutions in 4d/5d.", "Our purpose here is to introduce notation; for further detail, see [1], [2], [3], [4], [5], [6], [7] (for solutions with codimension-2 sources see also [22], [23]).", "The most general supersymmetric solutions of ungauged 5d ${\\cal N}=1$ supergravity with vector multiplets have been classified in [31] (see also [2], [4], [32]).", "When one applies this result to M-theory compactified on $T^6=T^2_{45}\\times T^2_{67}\\times T^2_{89}$ (the so-called STU model) and further assumes a tri-holomorphic $U(1)$ symmetry [5], the general supersymmetric solution corresponds to the following 11d fields: $\\begin{aligned}ds_{11}^2&=-Z^{-2/3}(dt+k)^2+Z^{1/3}ds_{\\mathrm {GH}}^2+Z^{1/3}\\left(Z_1^{-1}dx_{45}^2+Z_2^{-1}dx_{67}^2+Z_3^{-1}dx_{89}^2\\right)\\, ,\\\\[.5ex]{\\cal A}_3&=\\left(B^I - Z_I^{-1}(dt+k)\\right)\\wedge J_I\\, ,\\quad J_1 \\equiv dx^4\\wedge dx^5\\,,~J_2 \\equiv dx^6\\wedge dx^7\\,,~J_3 \\equiv dx^8\\wedge dx^9\\,,\\end{aligned}$ where $I=1,2,3$ ; $Z\\equiv Z_1 Z_2 Z_3$ ; and $dx_{45}^2\\equiv (dx^4)^2+(dx^5)^2$ etc.", "Supersymmetry implies that all fields in (REF ) are written in terms of 3D harmonic functions ${\\cal H}\\equiv (V,K^I,L_I,M)$ as follows.", "First, $ds_{\\mathrm {GH}}^2$ is a 4-dimensional metric of a Gibbons-Hawking space given by $ds_{\\mathrm {GH}}^2=V^{-1}(d\\psi +A)^2+V d{\\bf y}^2\\, , \\qquad \\psi \\cong \\psi +4\\pi \\,,\\qquad {\\bf y}=(y_1,y_2,y_3)\\, .$ The 1-form $A$ and the scalar $V$ depend on the coordinates ${\\bf y}$ of the $\\mathbb {R}^3$ base and satisfy $dA= *_3\\, dV\\, ,$ where $*_3$ is the Hodge dual in flat $\\mathbb {R}^3$ .", "We generally denote 3d vectors by boldface letters, such as ${\\bf y}=(y_i)$ , $i=1,2,3$ .", "The rest of the fields are: $B^I &= V^{-1} K^I (d\\psi + A)+ \\xi ^I\\, ,\\qquad \\qquad d \\xi ^I=- *_3 dK^I\\, ,\\\\Z_I&=L_I+\\frac{1}{2} C_{IJK} V^{-1} K^J K^K\\, ,\\\\k&=\\mu (d\\psi + A)+\\omega \\, ,\\\\\\mu &=M+\\frac{1}{2} V^{-1} K^I L_I+\\frac{1}{6} C_{IJK} V^{-2} K^I K^J K^K\\, ,$ where $C_{IJK}=|\\epsilon _{IJK}|$ .", "All fields are assumed to depend only on ${\\bf y}$ and not $\\psi $ or $T^6$ coordinates.", "The physical fields in the solution, such as $Z_I,\\mu $ , are invariant under the “gauge transformation” [27] $\\begin{split}V&\\rightarrow V,\\qquad K^I\\rightarrow K^I+c^I V,\\\\L_I&\\rightarrow L_I-C_{IJK}c^J K^K-{1\\over 2}C_{IJK}c^J c^K V,\\\\M&\\rightarrow M-{1\\over 2}c^I L_I+{1\\over 12}C_{IJK}(c^I c^J c^K V+3c^I c^J K^K)\\end{split}$ where $c^I$ are arbitrary constants.", "In the 5d setup that includes $\\psi $ as a coordinate, this is a gauge transformation for which the gauge transformation parameter depends on $\\psi $ (it shifts the coefficient of $d\\psi $ in $B^I$ ).", "However, in the 4d context, such $\\psi $ dependence is not allowed and the transformation (REF ) does change the charge [33].", "As already stated, supersymmetry requires that ${\\cal H}=(V,K^I,L_I,M)$ be harmonic functions in ${\\mathbb {R}}^3$ : $\\bigtriangleup V= \\bigtriangleup K^I= \\bigtriangleup L_I= \\bigtriangleup M=0,\\qquad \\bigtriangleup \\equiv \\partial _i \\partial _i.$ The 1-form $\\omega $ satisfies $*_3 d \\omega =V d M - M dV +\\frac{1}{2}( K^I dL_I - L_I d K^I )\\, .$ Applying $d\\,*_3$ on this equation we obtain the “integrability condition” [34] (see also [6]) $0 =V {\\bigtriangleup M}- M {\\bigtriangleup V}+\\frac{1}{2} (K^I \\bigtriangleup L_I - L_I \\bigtriangleup K^I )\\, .$ Although the functions $V,K^I, L_I, M$ are harmonic, they have sources and the right-hand side of (REF ) has delta-function singularities.", "The integrability condition requires all such singularities cancel.", "Reducing the 11D solution (REF ) on the $\\psi $ circle $S^1_\\psi $ , we obtain the following supersymmetric solution of type IIA supergravity: $\\begin{split}ds_{10,\\mathrm {str}}^2&=-\\frac{1}{\\sqrt{{\\cal Q}\\,}}(dt+\\omega )^2+\\sqrt{{\\cal Q}\\,} \\, d{\\bf x}^2+ \\frac{\\sqrt{{\\cal Q}\\,}}{V}\\left(Z_1^{-1}dx_{45}^2+Z_2^{-1}dx_{67}^2+Z_3^{-1}dx_{89}^2\\right)\\, ,\\\\e^{2\\Phi }&=\\frac{{\\cal Q}^{3/2}}{V^{3} Z}\\, ,\\qquad B_2=\\left({K^I\\over V}-{\\mu \\over Z_I }\\right) J_I\\, ,\\end{split}$ where $ds^2_{\\rm 10,str}$ is the string-frame metric and ${\\cal Q}\\equiv V(Z-\\mu ^2 V)$ .", "There are also RR potentials whose explicit form can be found e.g.", "in [22] and [35].", "The complexified Kähler modulus associated with $T_{89}^2$ is defined by $\\tau ^3={R_8R_9\\over {\\alpha ^{\\prime }}}\\left(B_{89}+i\\sqrt{\\det G_{ab}}\\right)={R_8R_9\\over {\\alpha ^{\\prime }}}\\left[\\left({K^3\\over V}-{\\mu \\over Z_3 }\\right)+i{\\sqrt{{\\cal Q}}\\over Z_1 V}\\right] ,$ where $a,b=8,9$ and $R_i$ are the radii of the $x^i$ directions, $i=4,\\dots ,9$ .", "Under the gauge transformation (REF ), this transforms as $\\tau ^3\\rightarrow \\tau ^3 + {R_8 R_9\\over {\\alpha ^{\\prime }}}c^3.$ The moduli $\\tau ^1$ and $\\tau ^2$ for $T^2_{45}$ and $T^2_{67}$ are defined similarly, and the transformation under the gauge transformation (REF ) is similar." ], [ "Codimension-3 sources", "The harmonic functions ${\\cal H}=(V,K^I,L_I,M)$ can have sources that represent branes, which are D-branes in the IIA setup.", "If one assumes that all sources are of codimension 3, the harmonic functions can be written as ${\\cal H}=h+\\sum _{p=1}^N {\\Gamma _p\\over |{\\bf y}-{\\bf a}_p|},$ where ${\\bf a}_p\\in \\mathbb {R}^3$ ($p=1,\\dots ,N$ ) specify the location of the codimension-3 sources where the harmonic functions become singular, and $h,\\Gamma _p$ are constants.", "The codimension-3 sources in the harmonic functions (REF ) represent branes in string/M-theory.", "For example, in the type IIA picture (REF ), the dictionary between the singularities in the harmonic functions and the D-brane sources is [3] $V\\leftrightarrow \\text{D6(456789)}\\, ,\\quad \\begin{array}{l}K^1\\leftrightarrow \\text{D4(6789)}\\\\[5pt]K^2\\leftrightarrow \\text{D4(4589)}\\\\[5pt]K^3\\leftrightarrow \\text{D4(4567)}\\end{array}\\, ,\\quad \\begin{array}{l}L_1\\leftrightarrow \\text{D2(45)}\\\\[5pt]L_2\\leftrightarrow \\text{D2(67)}\\\\[5pt]L_3\\leftrightarrow \\text{D2(89)}\\end{array}\\, ,\\quad M \\leftrightarrow \\text{D0}\\, .$ The D-branes are partially wrapped on $T^6$ as indicated here and appear in 4D as pointlike (codimension-3) objects sourcing the harmonic functions.", "When lifted to M-theory, D4 becomes M5 wrapping $S^1_\\psi $ and D2 becomes M2.", "D0 becomes momentum (P) along $S^1_\\psi $ while D6 becomes Kaluza-Klein monopole (KKM) with $S^1_\\psi $ being its special circle.", "Depending on how to choose the base space (REF ), the harmonic solution can describe 4d solutions or 5d solutions.", "For example, take $V={1\\over r}, \\qquad K^I=0,\\qquad L_I=1+{Q_I\\over 4r},\\qquad M=0,\\qquad r\\equiv |{\\bf y}|.$ In type IIA, this describes a 4d black hole made of three stacks of D2-branes and a D6-brane.", "Meanwhile, we can interpret this also as a 5d black hole because, for this $V$ , the base space (REF ) describes flat ${\\mathbb {R}}^4$ via a Hopf fibration.", "Namely, the metric for flat ${\\mathbb {R}}^4$ with coordinates $x_m$ , $m=1,2,3,4$ can be written as $\\begin{split}ds^2_{{\\mathbb {R}}^4}&=dx_m dx_m=V^{-1}(d\\psi +A)^2+V\\bigl (dr^2+r^2(d\\theta ^2+\\sin ^2\\!\\theta \\,d\\phi ^2)\\bigr ),\\\\V&={1\\over r},\\qquad A=(1+\\cos \\theta )d\\phi ,\\end{split}$ where $x_1+ix_2&=2\\sqrt{r}\\,\\sin {\\theta \\over 2}\\,e^{i{\\psi \\over 2}},\\qquad x_3+ix_4=2\\sqrt{r}\\,\\cos {\\theta \\over 2}\\,e^{i({\\psi \\over 2}+\\phi )}.$ with $\\psi \\cong \\psi +4\\pi $ , $\\phi \\cong \\phi +2\\pi $ .", "The Cartesian coordinate ${\\bf y}$ in the 3d base is $y_1+iy_2=r\\sin \\theta \\, e^{i\\phi },\\qquad y_3=r\\cos \\theta .$ Lifted to M-theory along the $\\psi $ direction, this harmonic solution becomes a 5d black hole made of three stacks of M2-branes, the D6-brane becoming the origin of the ${\\mathbb {R}}^4$ .", "This 5d black hole can be dualized (using the duality of Appendix ) into the original Strominger-Vafa black hole [36] in a type IIB frame, where the M2/D2 charges are mapped into D1, D5, and P charges.", "Harmonic functions can have other kinds of source.", "They can have a singularity along a curve in ${\\mathbb {R}}^3$ and have non-trivial monodromy around it.", "We refer to such singularities as codimension-2 sources.", "Note that this is genuinely different from the codimension-3 source discussed above; we can have a continuous distribution of codimension-3 sources along a curve, but we will still refer to them as codimension-3 sources.", "One situation for codimension-2 sources to appear is when branes undergo a supertube transition [21], gaining dimension (or losing codimension).", "For example, the following supertube transition is possible: $\\text{D2}(45) + \\text{D2}(67)\\xrightarrow{}\\text{NS5}(4567\\lambda )+\\text{P}(\\lambda ),$ where in the final configuration there is an NS5-brane along internal (4567) directions and a closed curve ${\\cal C}$ in ${\\mathbb {R}}^3$ parametrized by $\\lambda $ , and momentum along $\\lambda $ .", "By looking at the expression for $B_2$ in (REF ), we see that, if $V^{-1}K^3$ has a monodromy around ${\\cal C}$ , there will be an NS5-brane along ${\\cal C}$ (and 4567).", "(More precisely it is $V^{-1}K^3-Z_3^{-1}\\mu $ that must be monodromic, but in the current note we assume that $Z_I,\\mu $ appearing in the metric are single-valued and therefore it is the monodromy of $V^{-1}K^3$ that is relevant.)", "As it turns out, in this case $M$ also becomes monodromic.", "In general, given codimension-2 brane sources, which harmonic functions to become monodromic is a non-trivial matter that depends on the physical situation in question." ], [ "Lunin-Mathur geometries", "The Lunin-Mathur geometry [24], [25] is a solution of type IIB supergravity in ${\\mathbb {R}}_t\\times {\\mathbb {R}}^4_{1234}\\times S^1_y\\times T^4_{6789}$ and represents microstates [37], [38] of the D1-D5 2-charge system.", "The solution is parametrized by profile functions $g_m(\\lambda )$ , $m=1,2,3,4$ satisfying $g_m(\\lambda +L)=g_m(\\lambda )$ , which describe the shape inside ${\\mathbb {R}}^4$ of the KKM dipole produced by the supertube transition of D1$(y)$ and D5($y$ 6789) branes.We do not consider $g_A$ with $A\\ge 5$ [39], which describe other possible dipole charges produced by the supertube transition.", "Just as (REF ), we can describe this by the following diagram: $\\text{D1}(y) + \\text{D5}(y6789)\\xrightarrow{}\\text{KKM}(6789\\lambda ,y)+\\text{P}(\\lambda ),$ where $\\text{KKM}(6789\\lambda ,y)$ denotes the KKM dipole with $y$ being the special circle.", "The explicit form of the 10d string-frame metric of the Lunin-Mathur geometry is $d s^2_{10} & = -{2\\over \\sqrt{Z_1 Z_2}}(dv+{\\beta })\\Bigl (du+{\\omega }+{{\\cal F}\\over 2}(dv+{\\beta })\\Bigr )+\\sqrt{Z_1Z_2}\\,ds^2_{{\\mathbb {R}}^4}+ \\sqrt{\\frac{Z_1}{Z_2}}\\,ds^2_{T^4},$ where $Z_1 = \\frac{Q_5}{L} \\int _0^{L} d\\lambda \\frac{|\\partial _\\lambda \\vec{g}(\\lambda )|^2}{|\\vec{x} -\\vec{g}(\\lambda )|^2},\\qquad Z_2 = \\frac{Q_5}{L} \\int _0^{L} \\frac{d\\lambda }{|\\vec{x} -\\vec{g}(\\lambda )|^2},\\\\{\\bf A}= - \\frac{Q_5}{L} dx^j \\int _0^{L} d\\lambda \\frac{\\partial _\\lambda g_j(\\lambda )}{|\\vec{x} -\\vec{g}(\\lambda )|^2} ,\\qquad d{\\bf B}= - *_4 d{\\bf A},\\\\Q_1 = \\frac{Q_5}{L} \\int _0^{L} d\\lambda \\,|\\partial _\\lambda \\vec{g}(\\lambda )|^2,\\\\{\\beta }= \\frac{-{\\bf A}+{\\bf B}}{\\sqrt{2}},\\qquad {\\omega }= \\frac{-{\\bf A}-{\\bf B}}{\\sqrt{2}}$ and $ds^2_{{\\mathbb {R}}^4}=dx_m dx_m$ , $m=1,2,3,4$ is the flat ${\\mathbb {R}}^4$ metric with coordinates $\\vec{x}=(x_m)$ .", "For later convenience, we have written the metric (REF ) in the general form given in [40] which represents the 3-charge solution.", "In the 2-charge case we are considering, we must set ${\\cal F}=0$ .", "Also, we have taken the decoupling limit and dropped “1” from $Z_{1,2}$ .", "The coordinates $u,v$ are related to time $t$ and the coordinate $y$ of the compact circle $S^1_y$ of radius $R_y$ as $u = \\frac{1}{\\sqrt{2}}(t-y), \\qquad v = \\frac{1}{\\sqrt{2}}(t+y).$ The periodicity $L$ is related to $R_y$ as $L=2\\pi Q_5/R_y$ .", "The quantities $Q_1$ , $Q_5$ are related to the quantized D1 and D5 numbers $N_1$ , $N_5$ by $Q_1 = \\frac{N_1 g_s \\alpha ^{\\prime 3}}{v_4}\\,,\\qquad Q_5 = N_5 g_s \\alpha ^{\\prime },$ where $(2\\pi )^4 v_4$ is the coordinate volume of $T^4$ .", "Finally, $*_4$ is the Hodge dual in flat ${\\mathbb {R}}^4$ with coordinates $x_m$ .", "The simplest example of the profile, which has already been introduced in the introduction, is a circle in the $x_1$ -$x_2$ plane: $g_1+ig_2=ae^{i k \\Omega \\lambda },\\qquad g_3+ig_4=0,$ where $a>0$ , $k\\in {\\mathbb {Z}}_{>0}$ and $\\Omega \\equiv {2\\pi \\over L}={R_y\\over Q_5}.$ Substituting this into (REF ), we find $Z_1={Q_1\\over 4\\Sigma },\\qquad Z_2={Q_5\\over 4\\Sigma },\\\\Q_1=Q_5 a^2k^2\\Omega ^2,\\\\{\\bf A}=-{1\\over 2}{Q_5 k \\Omega }\\left({s^2+a^2+w^2\\over 4\\Sigma }-1\\right)d{\\widetilde{\\phi }},\\\\{\\bf B}=+{1\\over 2}{Q_5 k \\Omega }\\left({s^2-a^2+w^2\\over 4\\Sigma }-1\\right)d{\\widetilde{\\psi }},$ where $\\Sigma \\equiv {1\\over 4}\\sqrt{[(s+a)^2+w^2][(s-a)^2+w^2]}$ and we defined the polar coordinates $(s,{\\widetilde{\\phi }}),(w,{\\widetilde{\\psi }})$ via $x_1+ix_2=se^{i{\\widetilde{\\phi }}},\\qquad x_3+ix_4=we^{i{\\widetilde{\\psi }}},$ with $s,w\\ge 0$ and ${\\widetilde{\\phi }}\\cong {\\widetilde{\\phi }}+2\\pi $ , ${\\widetilde{\\psi }}\\cong {\\widetilde{\\psi }}+2\\pi $ ." ], [ "Relation to harmonic solutions", "By appropriate duality transformations (see Appendix ), we can map the harmonic solutions in the M/type-IIA frame in section REF into the type IIB D1-D5 frame in section REF .", "Here let us study the relation between the fields in the Lunin-Mathur geometries and the harmonic functions in the harmonic solutions.", "As discussed in [41], the harmonic solutions in the D1-D5 frame is $ds_{\\rm IIB}^2&=-{1\\over Z_3\\sqrt{Z_1Z_2}}(dt+\\kappa )^2+{Z_3\\over \\sqrt{Z_1Z_2}}(dz+A^{3})^2+\\sqrt{Z_1Z_2}\\,ds^2_{\\rm GH} + \\sqrt{Z_1\\over Z_2}ds^2_{T^4},\\\\\\kappa &=\\mu (d\\psi +A) +\\omega ,\\qquad A^{3}=B^3-{1\\over Z_3}(dt+\\kappa ),$ where $B^3$ is the one given in (REF ).", "On the other hand, the first part (the first term) of the Lunin-Mathur solution (REF ) can be rewritten as $ds_{10}^2&=-{(dt-{\\bf A})^2\\over Z_3\\sqrt{Z_1 Z_2}}+{Z_3\\over \\sqrt{Z_1 Z_2}}\\biggl (dy+dt+ {\\bf B}-{\\bf A}-{dt-{\\bf A}\\over Z_3}\\biggr )^2+\\cdots ,$ where $Z_3\\equiv 1-{\\cal F}/2$ (actually $Z_3=1$ in the D1-D5 case).", "By comparing (REF ) and (REF ), we read off the relation between the 1-forms ${\\bf A},{\\bf B}$ of the Lunin-Mathur geometry and the harmonic functions: $-{\\bf A}&~~\\leftrightarrow ~~ \\kappa =\\mu (d\\psi +A)+\\omega ,\\\\-{\\bf A}+{\\bf B}&~~\\leftrightarrow ~~ B^{3}=V^{-1}K^{3}(d\\psi +A)+\\xi ^{3}$ and $y+t \\leftrightarrow z$ .", "With this dictionary, we can read off harmonic functions that correspond to a Lunin-Mathur solution, and vice versa.", "Comparison of (REF ) and (REF ) also means that the flat ${\\mathbb {R}}^4$ in which the D1-D5 profile $g_m(\\lambda )$ lives should be written as the Hopf fibration (REF ).Another possibility is to compactify ${\\mathbb {R}}^4$ trivially to ${\\mathbb {R}}^3\\times S^1_\\psi $ by taking $V=1,A=0$ .", "Codimension-2 harmonic solutions obtained this way was discussed in [22].", "Generically, the profile in ${\\mathbb {R}}^4$ is going along some curve ${\\cal C}$ in the ${\\mathbb {R}}^3$ base, at the same time moving in the $\\psi $ fiber.", "Because the harmonic solutions are independent of the $\\psi $ fiber, we must smear the profile along $\\psi $ .", "The exception is when the profile is purely in the $\\psi $ direction and ${\\cal C}$ is a point in the base.", "One such exceptional example is the circular profile (REF ), for which ${\\cal C}$ corresponds to the point on the $y_3$ axis, $r=a^2/4,\\theta =\\pi $ ($y_3=-a^2/4$ ).", "We can see which harmonic functions get sources as follows.", "The $\\lambda $ direction of the profile in (REF ) can now be along $\\psi $ or ${\\cal C}\\subset {\\mathbb {R}}^3$ , so we have two kinds of dipole charges that can be produced by the supertube transition: $\\text{D1}(y) + \\text{D5}(y6789)\\xrightarrow{}{\\left\\lbrace \\begin{array}{ll}\\text{KKM}(6789\\psi ,y)+\\text{P}(\\psi )\\\\[.5ex]\\text{KKM}(6789{\\cal C},y)+\\text{P}({\\cal C})\\end{array}\\right.", "}$ After the duality transformation of Appendix , this is mapped into the following process in IIA: $\\text{D2}(45) + \\text{D2}(67)\\xrightarrow{}{\\left\\lbrace \\begin{array}{ll}\\text{D4}(4567)+\\text{D0}\\\\[.5ex]\\text{NS5}(4567{\\cal C})+\\text{P}({\\cal C})\\end{array}\\right.", "}$ The first line means that $K^3$ will have codimension-3 sources for $\\text{D4}(4567)$ .", "The codimension-3 charge is continuously distributed along ${\\cal C}$ , and its local density is proportional to the “speed” of the profile going along $\\psi $ .", "In addition, $K^3$ will have codimension-2 sources for $\\text{NS5}(4567{\\cal C})$ ; namely, $V^{-1}K^3$ will have a constant additive monodromy as we around ${\\cal C}$ and the constant is proportional to the NS5-brane charge.", "Also, $M$ will have codimension-3 sources for $\\text{D0}$ , while momentum $\\text{P}({\\cal C})$ will be encoded in the 1-form $\\omega $ .", "We can confirm that the non-trivial data of the Lunin-Mathur geometry are encoded in $K^3$ and $M$ from the relation (REF ).", "The duality equation between $A$ and $B$ (the second equation in (REF )) means that $d(-{\\bf A}+{\\bf B})=dB^{3}$ is the self-dual part of $d(-{\\bf A})=dk$ , namely $(1+*_4)dk=dB^{3}$ .", "From this, we can derive $\\mu *_3 dA+d\\omega - V*_3 d\\mu &= V^{-1}K^3 dA+d\\xi ^3,\\\\-d\\mu + \\mu V^{-1}dV+V^{-1}*_3 d\\omega &= -d(V^{-1}K^3).$ Here we used the relations between $*_4$ and $*_3$ on the Gibbons-Hawking space (REF ), ${\\begin{array}{c}*_4 \\lambda _{(2)}=V^{-1}(d\\psi +A)\\wedge *_3 \\lambda _{(2)} ,\\qquad *_4 ((d\\psi +A)\\wedge \\theta _{(1)})=V {*_3 \\theta _{(1)}},\\\\*_4 \\theta _{(1)} = (d\\psi +A)\\wedge *_3 \\theta _{(1)},\\qquad *_4(d\\psi +A)=V^2 d^3y,\\end{array}}$ where $\\lambda _{(2)}$ is a 2-form and $\\theta _{(1)}$ is a 1-form in ${\\mathbb {R}}^3$ .", "The two equations in (REF ) are $*_3$ of each other.", "By acting with $d$ on (REF ) we can derive $d*_3dM=0, \\qquad \\mu = M+{1\\over 2}V^{-1}K^3.$ Plugging (REF ) into (), we find $*_3 d\\omega = V dM-M dV-{1\\over 2}dK^3.$ Equations (REF ) and (REF ) are consistent with the relations () and (REF ) under the identification $V={1\\over r},\\qquad L_1=Z_1,\\qquad L_2=Z_2,\\qquad L_3=1,\\qquad K^1=K^2=0.$ So, the non-trivial data of the solution will be in $K^3$ and $M$ .", "Using the above dictionary (REF ), we can readily check that the Lunin-Mathur solution based on the circular profile (REF ) corresponds to a harmonic solution with the harmonic functions given in (REF ) in the introduction." ], [ "Codimension-2 Lunin-Mathur solution", "Here we explicitly construct harmonic functions that describe a Lunin-Mathur geometry with both dipole charges in (REF ) (or equivalently (REF )).", "Specifically, we consider the following profile: $g_1+ig_2=a e^{i k\\Omega \\lambda },\\qquad g_3+ig_4=b e^{-i k^{\\prime } \\Omega \\lambda }$ with $a,b\\ge 0$  We could make $a,b$ complex but that does not make a difference after smearing.", "and $k,k^{\\prime }\\in {\\mathbb {Z}}_{>0}$ .The negative sign in front of $k^{\\prime }$ in $g_3+ig_4$ is because of the holographic dictionary [42]; see section .", "We have already discussed this profile in (REF ) in the introduction.", "When smeared along and reduced on $\\psi $ , this gives a circular ring in the base ${\\mathbb {R}}^3$ : $\\textstyle \\rho \\equiv \\sqrt{y_1^2+y_2^2}=R,\\qquad y_3=c,$ where $R={ab\\over 2},\\qquad c={-a^2+b^2\\over 4}.$ This is the codimension-2 solution that we would like to construct and study.", "Of course, we can derive the harmonic functions “top-down” by starting with the Lunin-Mathur geometry in 6d with the profile (REF ), smearing and reducing it to 4d/5d, and then reading off the harmonic function.", "However, here we go “bottom-up” by directly constructing the harmonic functions based on the expected charges that they must represent.", "In the next section, we will confirm the result from the top-down viewpoint starting from the Lunin-Mathur geometry.", "The profile (REF ) is going in the angular directions $\\phi ,\\psi $ as $\\phi =-(k+k^{\\prime })\\Omega \\lambda ,\\qquad \\psi =2k\\Omega \\lambda .$ Therefore, by the argument below (REF ) we will have $-(k+k^{\\prime })$ units of NS5 charges and $k$ units of D4(4567) charges, encoded in harmonic functions.", "As mentioned in the introduction, $a$ and $b$ are constrained to satisfy (REF ) (this will be shown in (REF )).", "If we fix the charges of the system, $Q_1$ and $Q_5$ , and also the parameters $k$ and $k^{\\prime }$ , then the ring will be on an ellipsoid in ${\\mathbb {R}}^3$ for any values of $a$ and $b$ .", "More precisely, the position of the ring $\\rho =R=ab/2$ , $y_3=c=(-a^2+b^2)/4$ can be shown to satisfy $\\biggl ({y_3-y_3^{(0)}\\over A}\\biggr )^2+\\biggl ({\\rho \\over B}\\biggr )^2=1,$ where $A\\equiv {{\\cal N}\\over 8}\\left({1\\over k^2}+{1\\over k^{\\prime 2}}\\right),\\quad B\\equiv {{\\cal N}\\over 4kk^{\\prime }},\\quad y_3^{(0)}\\equiv {{\\cal N}\\over 8}\\left({1\\over k^{\\prime 2}}-{1\\over k^2}\\right),\\quad {\\cal N}\\equiv {Q_1\\over Q_5\\Omega ^2}={Q_1 Q_5\\over R_y^2}.$ The surface (REF ) is an axi-symmetric ellipsoid (spheroid), whose symmetry axis is the $y_3$ axis.", "The origin ${\\bf y}=0$ is a focal point of the ellipse on the cross section that contains the $y_3$ axis.", "Because $A\\ge B$ , the spheroid is prolate.", "We will be interested in the process of starting with $a>0,b=0$ and then increasing $b$ , finally ending with $a=0,b>0$ .", "When $a>b$ , the $y_3$ coordinate of the ring is $c<0$ .", "We call this the “southern” case, because the ring is like a latitude line on the southern hemisphere of an ellipsoid.", "When $a<b$ , the $y_3$ coordinate of the ring is $c>0$ .", "We call this the “northern” case.", "However, the word “southern”/“northern” should not be taken literally, because the center of the ellipsoid is not at $y_3=0$ .", "See Figure REF .", "As the extreme cases, if $a>0,b=0$ (the “south pole” limit), the 4d profile is a circle of radius $a$ on the $x_1$ -$x_2$ plane discussed in (REF ), which projects in ${\\mathbb {R}}^3$ onto a point on the negative $y_3$ axis ($y_3=-a^2/4$ ).", "In this case, the $\\phi $ circle shrinks and there is no NS5 charge.", "So, this limit a codimension-3 solution.", "If $a=0,b>0$ (the “north pole” limit), on the other hand, the profile is a circle of radius $b$ on the $x_3$ -$x_4$ plane, which projects onto a point on the positive $y_3$ axis ($y_3=b^2/4$ ).", "So, this limit also gives a codimension-3 solution.", "In this case, the shrinking cycle is not $\\phi $ or $\\psi $ but their linear combination because of the nontrivial Hopf fibration." ], [ "Building blocks", "To find the harmonic solution that corresponds to the profile (REF ), we start by constructing appropriately normalized harmonic functions that have codimension-2 and codimension-3 sources along the ring (REF ), as building blocks.", "Figure: Toroidal coordinates (on the y 2 =0y_2=0 plane).It is useful to introduce toroidal coordinates $(u,\\sigma ,\\phi )$ , adapted for a ring sitting at $\\sqrt{y_1^2+y_2^2}=R$ , $y_3=c$ : $y_1={\\sqrt{u^2-1}\\over u-\\cos \\sigma }R\\cos \\phi ,\\qquad y_2={\\sqrt{u^2-1}\\over u-\\cos \\sigma }R\\sin \\phi ,\\qquad y_3^{\\prime }\\equiv y_3-c={\\sin \\sigma \\over u-\\cos \\sigma }R,$ where $1\\le u<\\infty $ , $\\phi \\cong \\phi +2\\pi $ , $\\sigma \\cong \\sigma +2\\pi $ .", "The ring sits at $u=\\infty $ , while $u=1,\\sigma =0$ corresponds to $r=|{\\bf y}|\\rightarrow \\infty $ .", "$\\phi $ is the angle along the ring while $\\sigma $ is the angle going around the ring.", "See Figure REF for a graphical description.", "The flat 3d metric can be written as $d{\\bf y}^2&=dy_1^2+dy_2^2+dy_3^{\\prime }{}^2={R^2\\over (u-\\cos \\sigma )^2}\\left[{du^2\\over u^2-1}+{d\\sigma ^2}+(u^2-1)d\\phi ^2\\right].$ The inverse relations are $\\cos \\sigma ={{\\bf y}^{\\prime 2}-R^2\\over \\Lambda },\\qquad u={{\\bf y}^{\\prime 2}+R^2\\over \\Lambda },\\qquad \\Lambda ^2=({\\bf y}^{\\prime 2}-R^2)^2+4R^2 y_3^{\\prime }{}^2,$ where ${\\bf y}^{\\prime }{}^2=y_1^2+y_2^2+y_3^{\\prime }{}^2$ .", "Near the ring, $u\\rightarrow \\infty $ , ${\\textstyle \\sqrt{y_1^2+y_2^2}}\\approx R+\\varrho \\cos \\sigma ,\\quad y_3^{\\prime }=y_3-c\\approx \\varrho \\sin \\sigma ,\\qquad \\varrho \\equiv {R\\over u}.$ Namely, $(\\varrho ,\\sigma )$ are plane polar coordinates near the position of the ring.", "Let us first introduce $H$ , which is a single-valued harmonic function with codimension-3 sources uniformly distributed along the ring.", "Explicitly, it is given by $H&={1\\over R}\\sqrt{{u-\\cos \\sigma \\over 2}}\\, P_{-1/2}(u)={4\\,{\\cal I}_{00}(P,Q),\\over s^2+a^2+w^2+b^2}$ where $P_n(u)$ is the Legendre function and we defined the following quantities: $P\\equiv {2sa\\over s^2+a^2+w^2+b^2},\\qquad Q\\equiv {2wb\\over s^2+a^2+w^2+b^2},\\\\{\\cal I}_{mn}(P,Q)\\equiv \\int _0^{2\\pi }{d\\mu \\over 2\\pi }\\int _0^{2\\pi }{d\\nu \\over 2\\pi }\\,{(P\\cos \\mu )^m (Q\\cos \\nu )^n\\over 1-P\\cos \\mu - Q\\cos \\nu }.$ For other expressions of $H$ , see Appendix .", "The behavior of $H$ near the ring and near infinity is $H={\\left\\lbrace \\begin{array}{ll}\\displaystyle {\\log (8u)\\over \\pi R}+{\\cal O}\\Bigl ({1\\over u}\\Bigr ) & \\text{(near the ring, $u\\rightarrow \\infty $),}\\\\[2ex]\\displaystyle {1\\over r}+{\\cal O}\\Bigl ({1\\over r^2}\\Bigr ) & \\text{(near infinity, $r\\rightarrow \\infty $).}\\\\\\end{array}\\right.", "}$ Because $H$ is harmonic, $\\bigtriangleup H$ vanishes except on the ring.", "The first line of (REF ) implies that $\\bigtriangleup H$ has delta-function singularities uniformly smeared along the ring, and the total charge, as measured by the coefficient of $1/r$ for large $r$ , is unity.", "Next, we want a harmonic function—call it $\\gamma $ —which has the following monodromy around the ring: $\\gamma (u,\\sigma +2\\pi )\\rightarrow \\gamma (u,\\sigma )+1.$ Actually, let us be more general for a bit and consider a harmonic function $\\gamma $ with the monodromy $\\gamma \\rightarrow \\gamma +1$ around a general closed curve ${\\cal C}$ in ${\\mathbb {R}}^3$ .", "For that, it is convenient to introduce a 1-form $\\alpha = dy_i \\int _0^{2\\pi }{d\\mu \\over 4\\pi }{\\partial _\\mu f_i(\\mu )\\over |{\\bf y}-{\\bf f}(\\mu )|},$ where $f_i(\\mu )$ , $0\\le \\mu \\le 2\\pi $ , is a 3d profile function that parametrize the curve ${\\cal C}$ (this $\\alpha $ is a 3d version of ${\\bf A}$ defined in (REF )).", "The 1-form $\\alpha $ is independent of how we parametrize ${\\bf f}(\\mu )$ .", "It is not difficult to show that, if we define $\\gamma $ by the condition $d\\gamma =*_3d\\alpha ,$ then $\\gamma $ is harmonic and has the desired monodromy $\\gamma \\rightarrow \\gamma +1$ as we go around ${\\cal C}$ .", "One can also show [22] that $\\bigtriangleup \\gamma $ , which could have delta-function singularities on ${\\cal C}$ , is identically zero: $\\bigtriangleup \\gamma =0\\qquad \\text{(no delta function)}.$ This will be important when we impose the integrability condition.", "In the present case, curve ${\\cal C}$ is the circular ring (REF ).", "Being careful to the orientation of the angle variable $\\sigma $ relative to the ring (see Figure REF ), we take the 3d profile function to be $f_1+if_2=Re^{-i\\mu },\\qquad f_3=c.$ The 1-form $\\alpha $ for this profile is found to be [43] $\\alpha (u,\\sigma )&={a(u)\\over \\sqrt{u-\\cos \\sigma }}d\\phi ,\\qquad a(u)=-{R\\over 8\\sqrt{2}}{u^2-1\\over u^{3/2}}\\,\\,_2F_1\\left(\\tfrac{3}{4},\\tfrac{5}{4};2; 1-u^{-2}\\right).$ The function $\\gamma $ that satisfies (REF ) is $\\gamma (u,\\sigma )&={1\\over R}\\left[{a(u)\\over \\sqrt{u-1}}\\,{\\bf F}(\\tfrac{\\sigma }{2}|-\\tfrac{2}{u-1})-2\\sqrt{u-1}\\,a^{\\prime }(u)\\,{\\bf E}(\\tfrac{\\sigma }{2}|-\\tfrac{2}{u-1})\\right],$ where ${\\bf F}(\\phi |m)$ and ${\\bf E}(\\phi |m)$ are the elliptic integrals of the first and second kinds, respectively.We follow the Mathematica convention for the arguments of elliptic integrals.", "One can check the monodromy (REF ) from the periodicity of the elliptic integrals.", "This $\\gamma $ vanishes at infinity $r=|{\\bf y}|\\rightarrow \\infty $ , which corresponds to $\\sigma = 0,u\\rightarrow 1$ (and if we go to other branches $\\sigma = 2\\pi n,u\\rightarrow 1$ with $n\\in {\\mathbb {Z}}$ then $\\gamma \\rightarrow n$ ; a related formula is (REF )).", "The near-ring behavior of $\\gamma $ is found to be $\\gamma (u,\\sigma )={\\sigma \\over 2\\pi }+{\\cal O}\\Bigl ({1\\over u}\\Bigr ).$" ], [ "Codimension-2 source", "Using $H,\\gamma $ defined above, let us find harmonic functions that represent the Lunin-Mathur geometry with the profile (REF ).", "First, we have $V={1\\over r},\\qquad Z_1=L_1={Q_1\\over 4}H,\\quad Z_2=L_2={Q_5\\over 4}H,\\qquad L_3=1,\\qquad K^1=K^2=0.$ $V$ is always the same for the Hopf fibration, while $Z_{1,2}$ are determined so that the total charge is the same as in the south pole limit (REF ) (or (REF )) but now the charges must distributed uniformly along the ring.", "The other harmonic functions are fixed by (REF ).", "Before studying what singularities $K^3$ must have, let us discuss its normalization.", "When there are $n$ units of KKM($6789\\psi ,y$ ) charge at $r=0$ , $K^3$ has the following codimension-3 center: $K^3\\supset {R_y n\\over 2r}={Q_5 \\Omega n\\over 2r},$ where we used (REF ).", "On the other hand, when there are $n^{\\prime }$ units of KKM($6789{\\cal C},y$ ) charge lying along curve ${\\cal C}$ , then $V^{-1}K^3$ has the following monodromy as we go around ${\\cal C}$ : ${\\mathit {\\Delta }}(V^{-1}K^3)\\equiv V^{-1}K^3|_{\\sigma +2\\pi }- V^{-1}K^3|_{\\sigma }={R_y n^{\\prime }\\over R_\\psi }={Q_5 \\Omega n^{\\prime }\\over 2}$ where $R_\\psi =2$ .", "These normalizations can be derived by standard arguments (see e.g. [44]).", "From (REF ), we know that we have $-(k+k^{\\prime })$ units of KKM($6789{\\cal C},y$ ) lying along $\\phi $ .", "So, $K^3$ must have the monodromy (REF ) with $n^{\\prime }=-(k+k^{\\prime })$ .", "Clearly, this requirement is satisfied if we take $K^3$ to be: $K^3 =-{Q_5 \\Omega (k+k^{\\prime })\\over 2}\\,(\\gamma V +{\\widetilde{K}}^3),$ where ${\\widetilde{K}}^3$ is a single-valued function.", "For $K^3$ to be harmonic, ${\\widetilde{K}}^3$ must satisfy $\\bigtriangleup {\\widetilde{K}}^3= -{\\bigtriangleup (\\gamma V)}= -\\partial _i \\gamma \\, \\partial _i V= -\\epsilon _{ijk}\\partial _j \\alpha _k\\, \\partial _i V,$ where in the second equality we used that $\\gamma ,V$ are harmonic and in the last equality we used (REF ).", "A special solution of this Poisson equation is ${\\widetilde{K}}^3=-\\int _0^{2\\pi } {d\\mu \\over 4\\pi }\\,{\\epsilon _{ijk}\\, y_i f_j\\, \\partial _\\mu f_k\\over r\\,|{\\bf f}|\\,|{\\bf y}-{\\bf f}|\\,(r+|{\\bf f}|+|{\\bf y}-{\\bf f}|)},$ where $\\alpha $ and ${\\bf f}$ are given in (REF ) and (REF ).", "In the present case where $f_i$ is given by (REF ), this can be evaluated as ${\\widetilde{K}}^3&=-{R\\over r |{\\bf f}|}\\int _0^{2\\pi } {d\\mu \\over 4\\pi }\\,{c\\rho \\cos (\\mu +\\phi ) - R y_3\\over |{\\bf y}-{\\bf f}|\\,(|{\\bf f}|+|{\\bf y}-{\\bf f}|+r)}\\\\&=-{4ab\\,V\\over (a^2+b^2)(s^2+w^2+a^2+b^2)^2}\\int _0^{2\\pi } {d\\mu ^{\\prime }\\over 4\\pi }\\,{(b^2-a^2)sw\\cos \\mu ^{\\prime } + ab(s^2-w^2)\\over \\sqrt{X}(1+\\sqrt{X})}\\,,$ where $\\mu ^{\\prime }=\\mu +\\phi $ , $ X\\equiv 1-(P^2+Q^2)-2PQ\\cos \\mu ^{\\prime }$ ($P,Q$ were defined in (REF )).", "In the first equality we went to cylindrical coordinates $(\\rho ,\\phi ,y_3)$ in ${\\mathbb {R}}^3$ and in the second equality we used relations such as (REF ), (REF ), and (REF ).", "One can show (see Appendix REF ) that this can be written in terms of ${\\cal I}_{mn}(P,Q)$ as ${\\widetilde{K}}^3&=-{b^2{\\cal I}_{10}(P,Q)-a^2 {\\cal I}_{01}(P,Q)\\over a^2+b^2}V\\\\&=-{1\\over a^2+b^2}\\,V\\int _0^{2\\pi }{d\\mu \\over 2\\pi }\\int _0^{2\\pi }{d\\nu \\over 2\\pi }\\,{b^2 P\\cos \\mu - a^2Q\\cos \\nu \\over 1-P\\cos \\mu - Q \\cos \\nu }\\, .$ So, we have fixed the form of $K^3$ with the desired codimension-2 source to be $K^3 =-{Q_5 \\Omega \\over 2}(k+k^{\\prime })\\left(\\gamma -{b^2{\\cal I}_{10}-a^2 {\\cal I}_{01}\\over a^2+b^2}\\right)V+K^3_{(3)}$ where $K^{3}_{(3)}$ is a single-valued harmonic function containing only codimension-3 sources, which we turn to next.", "${\\cal I}_{10},{\\cal I}_{01}$ can be expressed in terms of elliptic integrals; see Appendix ." ], [ "Codimension-3 sources", "We want $K^3$ to also have $k$ units of codimension-3 D4-brane charge along the ring, as discussed in (REF ).", "We cannot simply set $K^3_{(3)}$ to $kH$ , because the first term of (REF ) also already contains some codimension-3 charge, which we must take into account.", "Codimension-3 sources in a harmonic function will appear as $\\delta $ -function singularities when we act with $\\bigtriangleup $ on it.", "So, let us examine the expression $\\bigtriangleup \\left[\\left(\\gamma -{b^2{\\cal I}_{10}-a^2 {\\cal I}_{01}\\over a^2+b^2}\\right)V\\right].$ From the first term, we have $\\bigtriangleup (\\gamma V)=(\\bigtriangleup \\gamma )V+2\\partial _i\\gamma \\,\\partial _iV+\\gamma (\\bigtriangleup V)$ but $(\\bigtriangleup \\gamma ) V=0$ from (REF ) while the cross term $\\partial _i\\gamma \\,\\partial _i V$ contains no $\\delta $ function.", "So, only $\\gamma (\\bigtriangleup V)$ remains.", "For the second term, $[\\bigtriangleup (b^2 {\\cal I}_{10}-a^2 {\\cal I}_{01})]V$ has no $\\delta $ function from (REF ), while $(b^2 {\\cal I}_{10}-a^2{\\cal I}_{01})\\,{\\bigtriangleup V}$ vanishes by (REF ).", "There is no $\\delta $ function from the cross term.", "So, after all, the only $\\delta $ -function singularities representing localized codimension-3 charges are $\\bigtriangleup K^3&=-{Q_5 \\Omega \\over 2}(k+k^{\\prime })\\,\\gamma (r=0)\\, {\\bigtriangleup V}+\\bigtriangleup K^3_{(3)}\\\\&={\\left\\lbrace \\begin{array}{ll}\\displaystyle -{Q_5 \\Omega \\over 2}(k+k^{\\prime }){b^2\\over a^2+b^2}\\, {\\bigtriangleup V}+\\bigtriangleup K^3_{(3)} &\\qquad \\text{($a>b$, ``southern'')}, \\\\[2ex]\\displaystyle +{Q_5 \\Omega \\over 2}(k+k^{\\prime }){a^2\\over a^2+b^2}\\, {\\bigtriangleup V}+\\bigtriangleup K^3_{(3)} &\\qquad \\text{($b>a$, ``northern'')}, \\\\\\end{array}\\right.", "}$ where in the second equality we used (REF ) for the value of $\\gamma (r=0)$ .", "So, we have $-(k+k^{\\prime }){b^2 \\over a^2+b^2}$ units of D4 charge sitting at $r=0$ in the “southern” case, and $(k+k^{\\prime }){a^2 \\over a^2+b^2}$ units for the “northern” case.", "The discontinuity as we go from the south to the north will be discussed in section REF ." ], [ "Delocalized charge", "Actually, this is not the end of the story.", "In the current situation where we have a branch cut of the multi-valued function $\\gamma $ , we can also have delocalized charges.", "Recall that the D4 charge is measured by the Gaussian integral inside ${\\mathbb {R}}^3$ as $q_{\\rm D4}^{}= -{1\\over 4\\pi }\\int _{S^2_\\infty } *_3 dK^3,$ where $S^2_\\infty $ is the sphere at infinity, with its normal pointing outward.", "We can continuously deform $S^2_\\infty $ into two disconnected pieces one of which encloses $r=0$ and the other of which surrounds the disk $D_2$ whose boundary is the ring (see Figure REF ).", "Because $K^3$ is discontinuous across the disk, there will be non-trivial contribution to the integral there.", "The only discontinuous part in $dK^3$ is $dK^3|_{\\rm discont}=-{Q_5 \\Omega \\over 2}(k+k^{\\prime })d(\\gamma V)|_{\\rm discont}=-{Q_5\\Omega \\over 2} (k+k^{\\prime })\\gamma \\, dV$ where we used the fact that $d\\gamma =*_3 d\\alpha $ is continuous.", "Let us denote the discontinuity of $X$ by $[X]=X_{\\rm out}-X_{\\rm in}$ , where “out” and “in” mean outside and inside of $D_2$ , with the orientation derived from that of $S^2_\\infty $ , as shown in Figure REF , in the “southern” case ($a>b$ ).", "Figure: The Gaussian surfaces.", "The large gray circle representsS 2 S^2 at infinity, which can be deformed into two disconnectedsurfaces: S 0 2 S^2_0, enclosing the origin, and S 1 2 S^2_1, surrounding thedisk D 2 D_2 (red zigzag line) whose boundary is the ring (bluedots).", "The difference of a quantity XX across D 2 D_2 is defined to be[X]=X out -X in [X]=X_{\\rm out}-X_{\\rm in}, where “out” and “in” are defined asshown here, for the southern case.In the “southern” case, we have $[\\gamma ]=-1$ and the contribution to (REF ) from the delocalized charge is $q_{\\rm deloc}^{}&= -{1\\over 4\\pi }\\int _{D_2} [*_3 dK^3]={1\\over 4\\pi }{Q_5\\Omega \\over 2}(k+k^{\\prime })\\int _{D_2} [\\gamma ]*_3 dV\\\\&=-{1\\over 4\\pi }{Q_5\\Omega \\over 2}(k+k^{\\prime })\\int _{D_2} *_3 dV=-{1\\over 4\\pi }{Q_5\\Omega \\over 2}(k+k^{\\prime })\\int _{D_2} \\sin \\theta \\, d\\theta \\wedge d\\phi .$ In the “southern” case ($a>b$ ), the $\\theta $ integral is from $\\theta $ such that $\\cos \\theta ={c\\over \\sqrt{c^2+R^2}}={b^2-a^2\\over a^2+b^2}$ to $\\theta =\\pi $ .", "So, evaluating the integral, we find $q_{\\rm deloc}^{}&={\\left\\lbrace \\begin{array}{ll}\\displaystyle +{Q_5\\Omega \\over 2}(k+k^{\\prime }){b^2\\over a^2+b^2}&\\qquad \\text{($a>b$, ``southern'')}\\\\[2ex]\\displaystyle -{Q_5\\Omega \\over 2}(k+k^{\\prime }){a^2\\over a^2+b^2}&\\qquad \\text{($b>a$, ``northern'')}\\\\\\end{array}\\right.", "}$ We also included the expression for the “northern” case (in which $[\\gamma ]=+1$ ).", "The discontinuity as we go from the south to the north will be discussed in section REF .", "The sum of this delocalized charge and the charge localized at $r=0$ in (REF ) is zero.", "The codimension-3 part $K^3_{(3)}$ is chosen so that the ring carries $k$ units of D4 charge (including the localized and delocalized charges) in the southern case: $K^3_{(3)}={Q_5\\Omega \\over 2}{a^2k-b^2k^{\\prime }\\over a^2+b^2}(H-V).$ Actually, here we added a term proportional to $V$ , which can be thought of as the “gauge transformation” (REF ).", "We will see that this choice agrees with the 6d result in the next section.", "For continuity, we must take the same $K^3_{(3)}$ also for the northern case.", "So, the final expression for $K^3$ is $K^3 ={Q_5 \\Omega \\over 2}\\left[-(k+k^{\\prime })\\left(\\gamma -{b^2{\\cal I}_{10}-a^2 {\\cal I}_{01}\\over a^2+b^2}\\right)V+{a^2k-b^2k^{\\prime }\\over a^2+b^2}(H-V)\\right].$" ], [ "Integrability and no-CTC conditions", "Having determined $K^3$ , we can find the remaining harmonic function $M$ .", "First, It must contain $\\gamma $ so that $\\mu $ (REF ) is single-valued.", "Furthermore, it must satisfy the integrability condition derived from (REF ), $0=V{\\bigtriangleup M}-M{\\bigtriangleup V}-{1\\over 2}{\\bigtriangleup K^3}\\qquad \\text{(no $\\delta $ function).", "}$ Imposing these conditions, it is straightforward to show that $M$ is given by: $M={Q_5 \\Omega \\over 4}\\left[(k+k^{\\prime })\\gamma + {1\\over 4}(ka^2-k^{\\prime }b^2)H \\right].$ The 1-form $\\omega $ that satisfies the equation (REF ) is $\\omega =-{Q_5\\Omega \\over 2}\\left(k{\\cal I}_{10}\\cos ^2{\\theta \\over 2}+k^{\\prime }{\\cal I}_{01}\\sin ^2{\\theta \\over 2}\\right)d\\phi .$ The modulus $\\tau ^3$ (see (REF )) is found to be $\\tau ^3=-(k+k^{\\prime })\\gamma +\\text{(single-valued)},$ which has the monodromy $\\tau ^3\\rightarrow \\tau ^3-(k+k^{\\prime })$ due to the $-(k+k^{\\prime })$ NS5($4567\\lambda $ ) branes.", "This monodromy can be represented by an $SL(2,{\\mathbb {Z}})$ transformation ${\\cal M}= \\begin{pmatrix}1&-(k+k^{\\prime }) \\\\0 & 1\\end{pmatrix}.$ At this point, the solution contains $Q_1,Q_5,a,b,k,k^{\\prime }$ as independent parameters.", "In the Lunin-Mathur geometry, they are going to be related by the relation (); for example, in the south-pole limit it is given by ().", "In the current framework of harmonic solutions in 4d/5d, such a relation can be derived by studying the no-CTC condition.", "Let us focus on the 5d part of the 11d metric (REF ).", "If we focus on the $\\phi ,\\psi $ part, we have $Z^{2/3}ds^2_5\\supset -[\\mu (d\\psi +A_\\phi d\\phi )+\\omega _\\phi d\\phi ]^2+Z[V^{-1}(d\\psi +A_\\phi d\\phi )^2+Vr^2\\sin ^2\\theta \\,d\\phi ^2]$ where we used ().", "Near the ring, $u\\sim \\infty $ , the quantities appearing here behave as (see (REF ), (REF )) ${\\begin{array}{c}\\omega _\\phi \\sim {Q_5\\Omega \\over 2}{ab\\over a^2+b^2}(k+k^{\\prime }){\\log u\\over \\pi },\\quad \\mu \\sim {Q_5\\Omega \\over 4} \\left(k{a\\over b}-k^{\\prime }{b\\over a}\\right){\\log u\\over \\pi },\\\\A_\\phi \\sim {2b^2\\over a^2+b^2},\\quad V\\sim {4\\over a^2+b^2},\\quad Z\\sim {Q_1 Q_5\\over 4\\pi ^2a^2b^2}\\log ^2u,\\quad g_{\\phi \\phi }=r^2\\sin ^2\\theta \\sim {a^2b^2\\over 4}.\\end{array}}$ By diagonalizing the metric (REF ) and requiring that there is no CTC, namely the eigenvalues are non-negative, we find the condition $Q_1\\ge \\Omega ^2(a^2k^2+b^2k^{\\prime 2})Q_5.$ In supertube configurations, genuine microstates saturate such non-CTC inequalities [45].", "Because Lunin-Mathur geometries are genuine microstates, we must require $Q_1= \\Omega ^2(a^2k^2+b^2k^{\\prime 2})Q_5.$ This is what we used in (REF )." ], [ "Comments", "Thus far, in this section, we have demonstrated the phenomenon of charge transfer, which is summarized in Figure REF .", "The D4 charge has two parts, the one localized at the location of the ring, and the other distributed over the branch cut inside the ring.", "As we move from the south pole to the north pole, the cut crosses the D6 center at $r=0$ and, at that point, some of the D4 charge gets transferred to the D6 center.", "As the result, the ring has different D4 charges in the initial and final states.", "This can be understood in terms of a “gauge transformation”.", "Let us note that, if we apply the gauge transformation (REF ) with $c^1=c^2=0$ , $c^3=-nQ_5(k^{\\prime }+k)\\Omega /2$ to the harmonic functions (REF ), (REF ), we obtain $\\begin{split}K^3 &={Q_5 \\Omega \\over 2}\\left[-(k+k^{\\prime })\\left(\\gamma +n-{b^2{\\cal I}_{10}-a^2 {\\cal I}_{01}\\over a^2+b^2}\\right)V+{a^2k-b^2k^{\\prime }\\over a^2+b^2}(H-V) \\right],\\\\M&={Q_5 \\Omega \\over 4}\\left[(k+k^{\\prime })(\\gamma +n) + {1\\over 4}(ka^2-k^{\\prime }b^2)H \\right].\\end{split}$ Let us consider the process of starting from the south pole limit with $n=0$ and moving over to the north pole.", "In the “southern” case $a>b$ , the branch cut is at $y_3=(b^2-a^2)/4<0$ and at the position of the $r=0$ center, we have $0<\\gamma <1/2$ .", "From (REF ), we see that the D4 charge of the $r=0$ center is $-{Q_5\\Omega \\over 2}(k+k^{\\prime }){b^2\\over a^2+b^2}$ .", "If we increase $b$ and go over the to the “northern” case $a<b$ , the $r=0$ pole has gone through the ring and now $\\gamma >1/2$ there.", "If we want to bring it back to the $-1/2<\\gamma <1/2$ branch which can be connected to infinity (note that $\\gamma \\rightarrow 0$ at infinity), we must do a gauge transformation with $n=-1$ so that $-1/2<\\gamma <1/2$ there.", "This is how we get ${Q_5\\Omega \\over 2}(k+k^{\\prime }){a^2\\over a^2+b^2}$ in (REF ) as the D4 charge of the $r=0$ center.", "Namely, the “jump” in the D4 charge happens because of the gauge transformation needed to bring the $r=0$ pole, which has gone to a different gauge by going through the ring, back into the original gauge.", "In the current note, we only discussed the “helical” profile (REF ), but we could consider any 4d profile and its reduction to ${\\mathbb {R}}^3$ .", "In ${\\mathbb {R}}^3$ , it will give an NS5-brane along some arbitrary curve ${\\cal C}$ , with D4 charge density which can vary along ${\\cal C}$ .", "If we define a 1-form $\\alpha $ as in (REF ) (now by an integration along ${\\cal C}$ ) and the dual scalar $\\gamma $ by $d\\gamma =*_3d\\alpha $ , the construction of $K^3$ goes just as in (REF )–(REF ), although explicit expressions will be harder to obtain." ], [ "Deriving harmonic functions from 6d", "In the previous section, we derived the harmonic functions with codimension-2 sources that represent a Lunin-Mathur geometry in 6d, based on the information about what charges must be present (bottom-up).", "Here, we confirm that these harmonic functions are correct, starting from 6d (top-down).", "The various functions of the Lunin-Mathur geometry (REF ) computed for the profile (REF ) are given by $\\begin{split}Z_1&={Q_1\\over 4} H,\\qquad Z_2={Q_2\\over 4} H,\\qquad Q_1=\\Omega ^2(k^2a^2+k^{\\prime 2} b^2)Q_5,\\\\{\\bf A}_{\\widetilde{\\phi }}&=-{Q_5\\Omega k \\over 2} {\\cal I}_{10},\\qquad {\\bf A}_{\\widetilde{\\psi }}=+{Q_5\\Omega k^{\\prime }\\over 2} {\\cal I}_{01},\\\\{\\bf B}_{\\widetilde{\\phi }}&= {Q_5\\Omega k^{\\prime }\\over 2} \\left(-{\\cal I}_{01}+{1\\over 2}b^2H-2\\gamma \\right),\\qquad {\\bf B}_{\\widetilde{\\psi }}= {Q_5\\Omega k \\over 2} \\left({\\cal I}_{10}-{1\\over 2}a^2H-2\\gamma \\right).\\end{split}$ where ${\\widetilde{\\phi }},{\\widetilde{\\psi }}$ were defined in (REF ), and other components of ${\\bf A},{\\bf B}$ vanish.", "The computation is straightforward, if complicated.", "The only thing is that we must smear over the $\\psi $ direction.", "For some detail see (REF ).", "The functions $Z_1,Z_2$ are the same as the one found from the 4d viewpoint in (REF ).", "Also, in 6d, the relation (REF ) is automatic.", "If we set $b\\rightarrow 0$ , (REF ) reduces to (REF ).", "We can find what harmonic functions these reduce to, using the results in section REF .", "We can decompose the 1-forms ${\\bf A},{\\bf B}$ as ${\\bf A}&={{\\bf A}_{\\widetilde{\\phi }}+{\\bf A}_{\\widetilde{\\psi }}\\over 2}(d\\psi +A)+\\left(-{\\bf A}_{\\widetilde{\\phi }}\\cos ^2\\!", "{\\theta \\over 2}+{\\bf A}_{\\widetilde{\\psi }}\\sin ^2\\!", "{\\theta \\over 2}\\right)d\\phi ,\\\\{\\bf B}&={{\\bf B}_{\\widetilde{\\phi }}+{\\bf B}_{\\widetilde{\\psi }}\\over 2}(d\\psi +A)+\\left(-{\\bf B}_{\\widetilde{\\phi }}\\cos ^2\\!", "{\\theta \\over 2}+{\\bf B}_{\\widetilde{\\psi }}\\sin ^2\\!", "{\\theta \\over 2}\\right)d\\phi .$ Using the identification (REF ), we find $\\begin{split}M&=-{1\\over 4}({\\bf A}_{\\widetilde{\\phi }}+{\\bf A}_{\\widetilde{\\psi }}+{\\bf B}_{\\widetilde{\\phi }}+{\\bf B}_{\\widetilde{\\psi }}),\\qquad K^3={V\\over 2}(-{\\bf A}_{\\widetilde{\\phi }}-{\\bf A}_{\\widetilde{\\psi }}+{\\bf B}_{\\widetilde{\\phi }}+{\\bf B}_{\\widetilde{\\psi }}),\\\\\\mu &=-{1\\over 2}({\\bf A}_{\\widetilde{\\phi }}+{\\bf A}_{\\widetilde{\\psi }}),\\qquad \\omega =\\left({\\bf A}_{\\widetilde{\\phi }}\\cos ^2\\!", "{\\theta \\over 2}-{\\bf A}_{\\widetilde{\\psi }}\\sin ^2\\!", "{\\theta \\over 2}\\right)d\\phi ,\\\\\\xi ^3&=\\left[({\\bf A}_{\\widetilde{\\phi }}-{\\bf B}_{\\widetilde{\\phi }})\\cos ^2{\\theta \\over 2}+(-{\\bf A}_{\\widetilde{\\psi }}+{\\bf B}_{\\widetilde{\\psi }})\\sin ^2{\\theta \\over 2}\\right]d\\phi .\\end{split}$ It is not difficult to show that these reproduce the harmonic functions (REF ), (REF ) in the previous section, using the formula (REF ).", "The reader may think that, to find harmonic functions, it is much easier to start with the Lunin-Mathur geometry in 6d and go down to 4d/5d as above (“top-down”), rather than going from 4d/5d to 6d as we did in section  (“bottom-up”).", "However, a technical point is that, even if we know ${\\bf A}$ , the relation (REF ) only gives us the combination $\\mu =M+K^3/(2V)$ , and not $M$ and $V$ separately.", "One could use the harmonicity of $K^3,M$ to disentangle them from each other, but that is far from simple.", "If we also know ${\\bf B}$ , it is easy to find $K^3,M$ from the two relations (REF ), but finding ${\\bf B}$ from the duality relation (REF ) is not simple either.", "We found the expression for ${\\bf B}$ in (REF ) by first finding $K^3,M$ and then going back up to 6d via () and (REF )." ], [ "Dual CFT states", "The CFT dual of general Lunin-Mathur geometries are known [46], [39], [47], [42].", "Here we discuss the dual for the special case of the helical profile, focusing on its symmetry.", "The holographic dual of the D1-D5 system is a 2d SCFT called the D1-D5 CFT, which is an orbifold CFT with target space ${\\rm Sym}^N(T^4)$ , $N=N_1 N_5$ .More precisely the target space is a deformation of this but we will assume that the target space is this orbifold.", "The CFT has $SU(2)_L\\times SU(2)_R$ R-symmetry with generators $J_L^i,J_R^i$ and the states in the CFT have R-charges $(J^3_L,J^3_R)=(j_L,j_R)$ .", "For more detail of the D1-D5 CFT, see [48], [49].", "Our notation here follows [50].", "The states of the orbifold CFT can be constructed multiplying “strands” of various length together so that the total length is $N$ .", "The CFT state that is dual to the circular profile (REF ) is $\\bigl [{|{++}\\rangle }_k\\bigr ]^{N/k},\\qquad N=N_1 N_5.$ Here ${|{++}\\rangle }_k$ is a strand of length $k$ , with a Ramond-Ramond ground state with R-charge $(j_L,j_R)=({\\frac{1}{2}},{\\frac{1}{2}})$ on it, denoted by “$++$ ”.", "This state is an eigenstate of $J_L^3,J_R^3$ .", "The dual bulk statement is that the supergravity solution preserves the corresponding $U(1)_L\\times U(1)_R$ symmetry.", "More precisely, $J_{\\widetilde{\\phi }}=J_L^3+ J_R^3$ and $J_{\\widetilde{\\psi }}=J_L^3-J_R^3$ generate rotations in the 1-2 and 3-4 planes under which the circular profile (REF ) is invariant.", "On the other hand, the state dual to the helical profile (REF ) is a coherent sum of RR ground states [46], [39], [47], [42], $\\mathop {\\mmlmultiscripts{\\sum {\\mmlnone }{\\prime }}}\\limits _{N_{++},N_{-+}}\\bigl [A_{++}\\,{|{++}\\rangle }_k\\bigr ]^{N_{++}}\\,\\bigl [A_{-+}\\,{|{-+}\\rangle }_{k^{\\prime }}\\bigr ]^{N_{-+}}\\,,$ where the sum is restricted to $(N_{++},N_{-+})$ with $kN_{++}+k^{\\prime }N_{-+}=N$ , and the parameters $A_{++},A_{-+}$ are related to the bulk quantities $a,b$ via $|A_{++}|^2={R_y^2N\\over Q_1 Q_5}\\,k^2 a^2,\\qquad |A_{-+}|^2={R_y^2N\\over Q_1 Q_5}\\,k^{\\prime 2} b^2.$ The sum (REF ) is dominated by the following term: $k \\overline{N}_{++}=|A_{++}|^2,\\qquad k^{\\prime } \\overline{N}_{-+}=|A_{-+}|^2.$ The state (REF ) is not an eigenstate of $J_L^3$ and $J_R^3$ separately, but it is an eigenstate of the combination $(k-k^{\\prime })J_L^3+(k+k^{\\prime })J_R^3=kJ_{\\widetilde{\\phi }}-k^{\\prime }J_{\\widetilde{\\psi }}$ .One exceptional case is $k=k^{\\prime }$ .", "In this case, making $a$ non-vanishing can be thought of merely as an $SU(2)_L$ rotation of the original state (REF ).", "Indeed, one can show that (REF ) is an eigenstate of $J_R^3$ and a certain linear combination of $J_L^i$ , $i=1,2,3$ .", "In the bulk, this is nothing but the linear combination of ${\\widetilde{\\phi }}$ and ${\\widetilde{\\psi }}$ directions under which the helical profile is invariant.", "Once we project the profile to ${\\mathbb {R}}^3$ , this structure becomes invisible because of smearing; namely, the 4d/5d configuration is symmetric under both $\\phi $ and $\\psi $ translations, although this is an artifact of smearing.", "As we keep increasing $b$ , we end up with the $a=0,b>0$ state $\\bigl [{|{-+}\\rangle }_{k^{\\prime }}\\bigr ]^{N/k^{\\prime }}$ which corresponds to the “north pole” limit.", "This is again symmetric under $U(1)_L\\times U(1)_R$ ." ], [ "Spectral flow", "Being an ${\\cal N}=(4,4)$ SCFT, the D1-D5 CFT has spectral flow symmetry [51] which maps a state with $L_0=h,J_L^3=j_L$ (we take $L_0=0$ for Ramond ground states) to a state with $L_0=h^{\\prime },J_L^3=j_L^{\\prime }$ on a strand of length $k$ as $h^{\\prime }=h+2 n j_L+k n^2,\\qquad j_L^{\\prime }=j_L+kn.$ For $n\\in {\\mathbb {Z}}$ , this maps states in the Ramond sector into states in the Ramond sector.", "Let us denote the Ramond state obtained by spectral flowing ${|{\\pm +}\\rangle }_k$ by $n$ by ${|{\\pm +}\\rangle }_{k,n} \\qquad \\text{with}\\qquad h=\\pm n+kn^2,\\quad j_L=\\pm {1\\over 2}+kn.$ By spectral flowing the state (REF ), we obtain $\\bigl [{|{\\pm +}\\rangle }_{k,n}\\bigr ]^{N/k}.$ If we spectral flow the state (REF ), we obtain the coherent sum $\\mathop {\\mmlmultiscripts{\\sum {\\mmlnone }{\\prime }}}\\limits _{N_{++},N_{-+}}\\bigl [A_{++}\\,{|{++}\\rangle }_{k,n}\\bigr ]^{N_{++}}\\,\\bigl [A_{-+}\\,{|{-+}\\rangle }_{k^{\\prime },n}\\bigr ]^{N_{-+}}\\,,$ which has charges $h &=(n+kn^2)\\overline{N}_{++}+(-n+k^{\\prime }n^2)\\overline{N}_{-+},\\\\j_L &=\\left({1\\over 2}+kn\\right)\\overline{N}_{++}+\\left(-{1\\over 2}+k^{\\prime }n\\right)\\overline{N}_{-+},\\\\j_R &={1\\over 2}(\\overline{N}_{++}+\\overline{N}_{-+}).$ In the bulk, the spectral flow transformation is realized by the following transformation of the harmonic functions[52] ${\\begin{array}{c}\\tilde{V}=V+\\gamma ^3 K^3,\\qquad \\tilde{K}^1=K^1-\\gamma ^3 L_2,\\qquad \\tilde{K}^2=K^2-\\gamma ^3 L_1,\\qquad \\tilde{K}^3=K^3,\\\\\\tilde{L}_3=L_3-2\\gamma ^3 M,\\qquad \\tilde{L}_2=L_2,\\qquad \\tilde{L}_1=L_1,\\qquad \\tilde{M}=M,\\qquad \\tilde{\\omega }=\\omega ,\\end{array}}$ under which the moduli $\\tau ^I$ defined in (REF ) transform as $\\tau ^1\\rightarrow \\tau ^1,\\qquad \\tau ^2\\rightarrow \\tau ^2,\\qquad \\tau ^3\\rightarrow {\\tau ^3\\over {Q_5\\Omega \\over 2}\\gamma ^3\\tau ^3+1}.$ Namely, it is an $SL(2,{\\mathbb {Z}})$ duality transformation on the modulus of $T^2_{89}$ .Generalization of this to a three-parameter family of transformations with parameters $\\gamma ^I$ , $I=1,2,3$ gives an $SL(2,{\\mathbb {Z}})$ duality transformation for $\\tau ^I$ [52].", "Let us study the bulk dual of the spectral flowed states above.", "First, applying (REF ) with $\\gamma ^3=-{2n\\over Q_5 \\Omega }$ to the codimension-3 harmonic functions (REF ) and doing gauge transformation (REF ) with $c^1=-{n\\over 2\\Omega },c^2=-{n Q_1\\over 2 Q_5\\Omega },c^3=0$ so that $K^I$ are ${\\cal O}(1/r^2)$ at infinity, we find the dual of (REF ): ${\\begin{array}{c}\\tilde{V}={1+kn\\over r}-{kn\\over \\Sigma },\\qquad \\tilde{K}^3={Q_5 \\Omega k\\over 2}\\left({1\\over \\Sigma }-{1\\over r}\\right),\\\\\\tilde{K}^1={n(kn+1)\\over 2\\Omega }\\left({1\\over \\Sigma }-{1\\over r}\\right),\\qquad \\tilde{K}^2={Q_1 n(kn+1)\\over 2Q_5 \\Omega }\\left({1\\over \\Sigma }-{1\\over r}\\right),\\\\\\tilde{L}_1={Q_1\\over 4}\\left({kn+1\\over \\Sigma }-{kn\\over r}\\right),\\qquad \\tilde{L}_2={Q_5\\over 4}\\left({kn+1\\over \\Sigma }-{kn\\over r}\\right),\\\\\\tilde{L}_3=1+{Q_1 n(kn+1)\\over 4Q_5 \\Omega ^2 k }\\left({kn+1\\over \\Sigma }-{kn\\over r}\\right),\\qquad \\tilde{M}={Q_1\\over 16\\Omega k}\\left({(kn+1)^2\\over \\Sigma }-{(kn)^2\\over r}\\right).\\end{array}}$ Next, if we apply the same bulk spectral flow and gauge transformations to the codimension-2 harmonic functions given by (REF ), (REF ) and (REF ), we obtain ${\\begin{array}{c}\\tilde{V}={1\\over r}-{2n\\over Q_5 \\Omega } K^3,\\qquad \\tilde{K}^1=-{n\\over 2\\Omega }{1\\over r}+{n\\over 2\\Omega }H+{n^2\\over Q_5 \\Omega ^2}K^3,\\qquad \\tilde{K}^2={Q_1\\over Q_5} \\tilde{K}^1,\\\\\\tilde{K}^3=K^3,\\qquad \\tilde{L}_1={Q_1\\over 4}H+{n Q_1\\over 2Q_5 \\Omega } K^3,\\qquad \\tilde{L}_2={Q_5\\over 4}H+{n \\over 2\\Omega } K^3,\\\\\\tilde{L}_3=1-{n^2 Q_1\\over 4Q_5 \\Omega ^2}{1\\over r}+{n^2 Q_1\\over 2Q_5 \\Omega ^2}H+{n^3 Q_1\\over 2Q_5^2 \\Omega ^3}K^3+{4n\\over Q_5 \\Omega }M,\\\\\\tilde{M}=M+{n Q_1\\over 8\\Omega } H + {n^2 Q_1\\over 8Q_5 \\Omega ^2}K^3\\end{array}}$ where $K^3,M$ are the ones given in (REF ), (REF ).", "This must be the bulk dual of the CFT state (REF ).", "From the $1/r$ fall-off of $\\tilde{L}_3$ and $\\tilde{M}$ , we can confirm the charges (REF ).", "For example, the large $r$ expansion of $\\tilde{L}_3$ is $\\tilde{L}_3&\\sim \\left[-{n^2Q_1\\over 4Q_5\\Omega ^2}+{n^2Q_1\\over 2Q_5\\Omega ^2}+{4n\\over Q_5\\Omega }{Q_5\\Omega \\over 4}{1\\over 4}(ka^2-k^{\\prime }b^2)\\right]{1\\over r}\\\\&={1\\over 4r}\\Bigl [(n+kn^2)ka^2+(-n+k^{\\prime }n^2)k^{\\prime }b^2\\Bigr ]\\equiv {Q_p\\over 4r},$ where we used (REF ).", "Using (REF ) and (REF ), and using the relation between $Q_p$ and the quantized momentum $N_p$ , $Q_p={g_s^2 {\\alpha ^{\\prime }}^4 \\over R_y^2 v_4}N_p={Q_1Q_5\\over R_y^2N}N_p,$ it is easy to show that this reproduces the CFT result (REF ).", "So, the harmonic solution (REF ) provides the supergravity description of the CFT state (REF ).", "What is interesting is that, because $\\tilde{V}$ contains the monodromic $K^3$ that can transfer charge, the solution interpolates between Gibbons-Hawking spaces with different Taub-NUT charges.", "$\\tilde{V}={1+kn\\over r}-{kn\\over \\Sigma }~~~~\\rightarrow ~~~~\\tilde{V}={1-k^{\\prime }n\\over r}+{k^{\\prime }n\\over \\Sigma ^{\\prime }},$ where $\\Sigma ^{\\prime }=|{\\bf y}-{\\bf {\\widetilde{b}}}|$ , ${\\bf {\\widetilde{b}}}=(0,0,{\\widetilde{b}})$ , ${\\widetilde{b}}=b^2/4$ .", "In particular, if $k=k^{\\prime }=1$ , $n=-m$ , this connects the two configurations: $\\tilde{V}={1-m\\over r}+{m\\over \\Sigma }~~~~\\rightarrow ~~~~\\tilde{V}={1+m\\over r}-{m\\over \\Sigma ^{\\prime }}={1-(m+1)\\over \\Sigma ^{\\prime }}+{m+1\\over r}.$ On the right hand side, by reinterpreting $\\Sigma ^{\\prime }\\rightarrow r,r\\rightarrow \\Sigma $ , we can repeat the same process now with $m\\rightarrow m+1$ .", "Doing this over again, we can reach any $m$ .", "See Figure REF for a graphical explanation of this process.", "In CFT, the process (REF ) corresponds to the interpolation $[{|{++}\\rangle }_{1,-m} ]^N~~~~\\rightarrow ~~~~[{|{-+}\\rangle }_{1,-m}]^N=[{|{++}\\rangle }_{1,-(m+1)}]^N,$ where in the last expression we used the fact that ${|{-+}\\rangle }_{1,-m}=[{|{++}\\rangle }_{1,-(m+1)}]$ .", "So, by starting from $[{|{++}\\rangle }_1]^N=[{|{++}\\rangle }_{1,0}]^N$ and repeatedly doing this interpolation, we can get to $ [{|{++}\\rangle }_{1,-m} ]^N$ with any $m$ ; see Figure REF for a description of this spectral flow on the $J_L^3$ -$L_0$ plane.", "Figure: Repeated interpolations in the bulk.Figure: Repeated interpolations in CFT to reach [|++〉 1,-m ] N [{|{++}\\rangle }_{1,-m}]^N with arbitrary mm.From (REF ) and (REF ), we see that the modulus $\\tau ^3$ now has the $SL(2,{\\mathbb {Z}})$ monodromy $\\tilde{{\\cal M}}= \\begin{pmatrix}1 & -(k+k^{\\prime })\\\\-n & 1+n(k+k^{\\prime })\\end{pmatrix}.$ This is no longer the monodromy of NS5(4567${\\cal C}$ )-branes but the “exotic brane” $5^2_2(4567{\\cal C},89)$ [43] has mixed in.", "By following the duality transformation to the D1-D5 frame, we see that this is mapped as $5^2_2(4567{\\cal C},89)\\xrightarrow{}{\\rm KKM}(6789{\\cal C},\\psi ).$ So, in addition to the KKMs in (REF ), we have new a kind of KKM dipole, but the solution is purely geometric.", "This is as it should be, because the spectral flow transformation in the D1-D5 frame is nothing but mixing the coordinates $y$ and $\\psi $ ; part of the KKM in the second line of (REF ) got transformed into ${\\rm KKM}(6789{\\cal C},\\psi )$ above." ], [ "Fractional spectral flow", "For general CFT states, the spectral flow is defined only for $n\\in {\\mathbb {Z}}$ .", "However, for the state ${|{\\pm +}\\rangle }_k$ , the fractional spectral flow by $n={s\\over k}$ , $s\\in {\\mathbb {Z}}$ , ${|{\\pm +}\\rangle }_{k,{s\\over k}}:\\qquad h={s(s\\pm 1)\\over k},\\quad j_L=\\pm {1\\over 2 }+s$ is also a valid state [53] if $h= {s(s\\pm 1)\\over k}\\in {\\mathbb {Z}},\\qquad \\text{or}\\qquad {n(nk\\pm 1)}\\in {\\mathbb {Z}}.$ Indeed, the dual bulk solution (REF ) is known to make sense [53] if this condition (REF ) is met, and represents the most general 2-center codimension-3 harmonic solution.", "The interpolating CFT state (REF ) must also be a valid state if both the states ${|{++}\\rangle }_{k,n}$ and ${|{-+}\\rangle }_{k^{\\prime },n}$ satisfy the quantization condition, namely, $n(nk+1)\\in {\\mathbb {Z}}\\qquad \\text{and}\\qquad n(nk^{\\prime }-1)\\in {\\mathbb {Z}}.$ For example, for $m,m^{\\prime }\\in {\\mathbb {Z}}_{>0}$ , the following states with $n=1/2$ : $&{|{++}\\rangle }_{4m+2,{1\\over 2}},\\quad h=m+1\\in {\\mathbb {Z}},\\quad j_L=2m+{3\\over 2},\\\\&{|{-+}\\rangle }_{4m^{\\prime }-2,{1\\over 2}},\\quad h=m^{\\prime }-1\\in {\\mathbb {Z}},\\quad j_L=2m^{\\prime }-{3\\over 2}$ satisfy the quantization condition and the interpolation will be an allowed state.", "The dual geometry is still given by (REF )." ], [ "Discussions", "In this note, we studied the Lunin-Mathur geometry interpolating the states $[{|{++}\\rangle }_{k}]^{N/k}~~~~\\text{and}~~~~[{|{-+}\\rangle }_{k^{\\prime }}]^{N/k^{\\prime }}$ in the framework of harmonic solutions in 4d/5d.", "Although the geometries dual to (REF ) are codimension-3 solutions, the interpolating solution is of codimension two, because of the puffed-out NS5-branes lying along a curve ${\\cal C}$ .", "We discussed how to construct the associated harmonic functions with a monodromy around ${\\cal C}$ , based on the data about the charges and the puffed-out dipole charge.", "The interpolating solution exhibits some interesting features, such as some of the D4-charge being delocalized, and some of the D4-charge getting transferred from the supertube center to the Taub-NUT center as the interpolation proceeds.", "We also discussed the spectral flow of this entire process, which interpolates between the states $[{|{++}\\rangle }_{k,n}]^{N/k}~~~~\\text{and}~~~~[{|{-+}\\rangle }_{k^{\\prime },n}]^{N/k^{\\prime }}.$ This solution is valid even if $n$ is not an integer, as long as a certain quantization condition is met; in that case, the interpolating solution connects the 2-center solutions found in [53].", "Figure: An example of interpolation between[|++〉 k,n ] N/k [{|{++}\\rangle }_{k,n}]^{N/k} and [|++〉 k ' ,n ' ] N/k ' [{|{++}\\rangle }_{k^{\\prime },n^{\\prime }}]^{N/k^{\\prime }} with n≠n ' n\\ne n^{\\prime }.A natural question [54] is how to interpolate between the states $[{|{++}\\rangle }_{k,n}]^{N/k}~~~~\\text{and}~~~~[{|{-+}\\rangle }_{k^{\\prime },n^{\\prime }}]^{N/k^{\\prime }}$ where $n\\ne n^{\\prime }$ .", "For simplicity, let us set $k=k^{\\prime }=1,n=0$ , and $n^{\\prime }\\rightarrow n+1$ .", "Then the interpolation in question is between $[{|{++}\\rangle }_{1}]^{N}~~~~\\text{and}~~~~[{|{-+}\\rangle }_{1,n+1}]^{N}=[{|{++}\\rangle }_{1,n}]^{N}.$ See Figure REF for an example.", "In CFT, it is easy to write down the interpolating state, as in (REF ): $\\mathop {\\mmlmultiscripts{\\sum {\\mmlnone }{\\prime }}}\\limits _{N_{++},N_{-+}}\\bigl [A_1\\,{|{++}\\rangle }_{1}\\bigr ]^{N_{++}}\\,\\bigl [A_n\\,{|{++}\\rangle }_{1,n}\\bigr ]^{N_{-+}}\\,.$ Just as (REF ), this is an eigenstate of $J_R$ and a certain linear combination of $J_L$ and $J_R$ .", "This suggests that the bulk dual has a helical structure similar to the helical Lunin-Mathur profile that we studied.", "When reduced to ${\\mathbb {R}}^3$ , it is likely to be described by a harmonic solution involving codimension-2 sources slimier to the one studied in this note.", "Constructing such solutions would be very interesting for understanding black hole microstates, because the state ${|{++}\\rangle }_{k,n}$ in the background of ${|{++}\\rangle }_{1}$ is a stringy state in AdS$_3\\times S^3$ [55], [56], [57], which is a key for understanding the fractional and higher modes responsible for the entropy.", "The technique developed in this note should be useful for such construction.", "In [20], the moduli space of the multi-center codimension-3 solutions for small numbers of D6- and D2-branes was investigated.", "They found that, for one D6- and three D2-branes, the number of solutions agrees with the prediction from quiver quantum mechanics, although for more D6-branes the number of multi-center solutions is less than the prediction.", "It would be interesting to explore the moduli space of multi-center solutions including codimension-2 sources.", "The results in this note should be useful for such exploration." ], [ "Acknowledgments", "I would like to thank CEA Saclay for their (online) hospitality in the “Black-Hole Microstructure I & II” workshops.", "This work was supported in part by MEXT KAKENHI Grant Numbers 17H06357 and 17H06359." ], [ "Duality transformation", "Here we discuss how to map the IIB configuration in (REF ) in the D1-D5 frame into the IIA configuration (REF ).", "We start with (REF ): $\\text{D1}(y) + \\text{D5}(y6789)\\xrightarrow{}{\\left\\lbrace \\begin{array}{ll}\\text{KKM}(6789\\psi ,y)+\\text{P}(\\psi )\\\\[.5ex]\\text{KKM}(6789{\\cal C},y)+\\text{P}({\\cal C})\\end{array}\\right.", "}$ After T-duality along $y67$ , this is mapped into $\\text{D2}(67) + \\text{D2}(89)\\xrightarrow{}{\\left\\lbrace \\begin{array}{ll}\\text{NS5}(6789\\psi )+\\text{P}(\\psi )\\\\[.5ex]\\text{NS5}(6789{\\cal C})+\\text{P}({\\cal C})\\end{array}\\right.", "}$ Lifting this to M-theory along $x^{10}$ , this becomes $\\text{M2}(67) + \\text{M2}(89)\\xrightarrow{}{\\left\\lbrace \\begin{array}{ll}\\text{M5}(6789\\psi )+\\text{P}(\\psi )\\\\[.5ex]\\text{M5}(6789{\\cal C})+\\text{P}({\\cal C})\\end{array}\\right.", "}$ Renaming coordinates as $(6,7,8,9,y,10)\\rightarrow (4,5,6,7,8,9)$ , we have $\\text{M2}(45) + \\text{M2}(67)\\xrightarrow{}{\\left\\lbrace \\begin{array}{ll}\\text{M5}(4567\\psi )+\\text{P}(\\psi )\\\\[.5ex]\\text{M5}(4567{\\cal C})+\\text{P}({\\cal C})\\end{array}\\right.", "}$ This is a supertube transition in the M-theory frame of (REF ).", "Compactifying M-theory to IIA on $\\psi $ , this reduces to $\\text{D2}(45) + \\text{D2}(67)\\xrightarrow{}{\\left\\lbrace \\begin{array}{ll}\\text{D4}(4567)+\\text{D0}\\\\[.5ex]\\text{NS5}(4567{\\cal C})+\\text{P}({\\cal C})\\end{array}\\right.", "}$ This is the process (REF ) in the IIA frame of (REF )." ], [ "Coordinate systems", "Here we summarize relations between the coordinate systems that we use in the main text.", "The flat ${\\mathbb {R}}^4$ coordinates $x_m$ , $m=1,2,3,4$ , are variously written as $\\begin{array}{l@{~}l@{~}l@{~}l@{~}l}x_1+ix_2&=&se^{i{\\widetilde{\\phi }}}&=&\\displaystyle 2\\sqrt{r}\\,\\sin \\tfrac{\\theta }{2}\\,e^{i{\\psi \\over 2}},\\\\[1ex]x_3+ix_4&=&we^{i{\\widetilde{\\psi }}}&=&\\displaystyle 2\\sqrt{r}\\,\\cos \\tfrac{\\theta }{2}\\,e^{i({\\psi \\over 2}+\\phi )}.\\end{array}$ The flat ${\\mathbb {R}}^4$ metric can be written as $\\begin{split}ds_4^2&=dx_m dx_m\\\\&=ds^2+ s^2 d{\\widetilde{\\phi }}^2+dw^2+ w^2 d{\\widetilde{\\psi }}^2\\\\&={1\\over r}dr^2+r\\left(d\\theta ^2+2(1-\\cos \\theta )d{\\widetilde{\\phi }}^2+2(1+\\cos \\theta )d{\\widetilde{\\psi }}^2\\right)\\\\&=V^{-1}(d\\psi +{\\cal A})^2+V d{\\bf y}^2,\\\\{\\cal A}&=(1+\\cos \\theta )d\\phi ,\\quad V={1\\over r},\\quad *_3 d{\\cal A}=dV,\\\\\\end{split}$ where $d{\\bf y}^2&=dr^2+r^2(d\\theta ^2+\\sin ^2\\!\\theta \\,d\\phi ^2),\\\\y_1+iy_2&=r\\sin \\theta \\, e^{i\\phi },\\qquad y_3=r\\cos \\theta .$ In the toroidal coordinates $y_1+iy_2={\\sqrt{u^2-1}\\over u-\\cos \\sigma }Re^{i\\phi },\\qquad y_3^{\\prime }\\equiv y_3-c={\\sin \\sigma \\over u-\\cos \\sigma }R,$ the flat ${\\mathbb {R}}^3$ metric can be written as $d{\\bf y}^2&=d{\\bf y}^{\\prime 2}=dy_1^2+dy_2^2+dy_3^{\\prime }{}^2={R^2\\over (u-\\cos \\sigma )^2}\\left[{du^2\\over u^2-1}+{d\\sigma ^2}+(u^2-1)d\\phi ^2\\right]$ where ${\\bf y}^{\\prime }=(y_1,y_2,y_3^{\\prime })$ .", "The inverse relations are $\\cos \\sigma ={{\\bf y}^{\\prime 2}-R^2\\over \\Lambda },\\qquad u={{\\bf y}^{\\prime 2}+R^2\\over \\Lambda },\\qquad \\Lambda ^2=({\\bf y}^{\\prime 2}-R^2)^2+4R^2 y_3^{\\prime }{}^2.$ Some relations between different coordinates: $r={s^2+w^2\\over 4}={1\\over 4}|\\vec{x}|^2,\\qquad \\sin \\frac{\\theta }{2}={s\\over \\sqrt{s^2+w^2}},\\qquad \\cos \\frac{\\theta }{2}={w\\over \\sqrt{s^2+w^2}},\\\\\\cos {\\theta }={-s^2+w^2\\over s^2+w^2},\\qquad \\sin {\\theta }={2sw\\over s^2+w^2}\\\\s=2\\sqrt{r}\\sin {\\theta \\over 2}=\\sqrt{2r(1-\\cos \\theta )}=\\sqrt{2(r-y_3)},\\\\ w=2\\sqrt{r}\\cos {\\theta \\over 2}=\\sqrt{2r(1+\\cos \\theta )}=\\sqrt{2(r+y_3)}\\\\\\rho =\\sqrt{y_1^2+y_2^2}=r\\sin \\theta ={sw\\over 2},\\qquad y_3=r\\cos \\theta ={-s^2+w^2\\over 4}.$ The relation between the position of the ring in ${\\mathbb {R}}^4$ (specified by $a,b$ ) and in ${\\mathbb {R}}^3$ (specified by $R,c$ ): $R={ab\\over 2},\\quad c={-a^2+b^2\\over 4},\\qquad \\sqrt{R^2+c^2}={a^2+b^2\\over 4}.$ Some more relations: $V&={1\\over r}={4\\over s^2+w^2}={4\\over |\\vec{x}|^2}\\\\\\Lambda ^2&={1\\over 256}[(s+a)^2+(w+b)^2][(s+a)^2+(w-b)^2]\\\\[-1ex]&\\hspace{42.5pt}\\times [(s-a)^2+(w+b)^2][(s-a)^2+(w-b)^2]\\\\[1ex]u&=\\tfrac{(s^2+w^2)^2+(a^2+b^2)^2-2(a^2-b^2)(s^2-w^2)}{\\sqrt{[(s+a)^2+(w+b)^2][(s+a)^2+(w-b)^2][(s-a)^2+(w+b)^2][(s-a)^2+(w-b)^2]}},\\\\[1ex]\\zeta &=\\sqrt{1-u^{-2}}=\\tfrac{8absw}{(s^2+w^2)^2+(a^2+b^2)^2-2(a^2-b^2)(s^2-w^2)},\\\\[1ex]\\Sigma &={1\\over 4}\\sqrt{[(s+a)^2+w^2][(s-a)^2+w^2]}=|{\\bf y}-{\\bf {\\widetilde{a}}}|,\\\\{\\bf {\\widetilde{a}}}&=(0,0,-{\\widetilde{a}}),\\qquad {\\widetilde{a}}={a^2\\over 4},\\qquad {\\widetilde{b}}={b^2\\over 4}$" ], [ "Functions $H$ , {{formula:e1905741-5dfd-4ea6-917e-d3c2dfb75ffc}} , {{formula:c21b28f4-4186-47f7-b912-553bd38018af}}", "Here we summarize the functions introduced in the main text and discuss their properties." ], [ "The harmonic function $H$", "The harmonic function $H$ can be written in various ways as $\\begin{split}H&={1\\over R}\\sqrt{u-\\cos \\sigma \\over 2u}\\,\\,_{2}F_{1}\\!\\left({1\\over 4},{3\\over 4};1;\\zeta ^2\\right)={2\\over \\pi R}\\sqrt{{u-\\cos \\sigma \\over 2u}}\\,{{\\bf K}({2\\zeta \\over \\zeta -1})\\over \\sqrt{1-\\zeta }}\\\\&={1\\over R}\\sqrt{{u-\\cos \\sigma \\over 2}}\\, P_{-1/2}(u)={4\\,{\\cal I}_{00}(P,Q)\\over s^2+a^2+w^2+b^2}.\\end{split}$ where $\\zeta =\\sqrt{1-u^{-2}}$ and ${\\bf K}(m)$ is the complete elliptic integral of the first kind.", "The value of $H$ at $r=0$ is easy to see from the last expression, because $r=0$ means $P=Q=0$ : $H(r=0)={4\\over a^2+b^2}$ The behavior of $H$ near the ring and near infinity is given in (REF ).", "$H$ appears in the functions $Z_{1,2}$ of the Lunin-Mathur geometry for the helical profile (REF ) after smearing.", "For example, $Z_2$ is $Z_2&={Q_5\\over L}\\int _0^{L}{d\\lambda \\over |\\vec{x}-\\vec{g}|^2}\\\\&={Q_5\\over L}\\int _0^{2\\pi }{d\\lambda \\over s^2+a^2+w^2+b^2-2sa\\cos ({\\widetilde{\\phi }}-k\\Omega \\lambda ) -2wb\\cos ({\\widetilde{\\psi }}+k^{\\prime }\\Omega \\lambda ) }.$ After smearing along $\\psi $ , this becomes $\\begin{split}Z_2&=Q_5 \\int _0^{2\\pi } {d\\mu \\over 2\\pi }\\int _0^{2\\pi } {d\\mu \\over 2\\pi }{1\\over s^2+a^2+w^2+b^2-2sa\\cos \\mu -2wb\\cos \\nu }\\\\&={Q_5\\over s^2+a^2+w^2+b^2}\\int _0^{2\\pi }{d\\mu \\over 2\\pi }\\int _0^{2\\pi }{d\\nu \\over 2\\pi }{1\\over 1-P\\cos \\mu - Q\\cos \\nu }\\\\&={Q_5\\over s^2+a^2+w^2+b^2}{\\cal I}_{00}(P,Q)={Q_5\\over 4}H.\\end{split}$ We can relate this to an integral over a 3d profile.", "Let us set $\\nu \\rightarrow \\nu +\\mu $ in the second expression of (REF ) so that ${\\begin{array}{c}P\\cos \\mu +Q\\cos \\nu \\rightarrow P\\cos \\mu +Q\\cos (\\nu +\\mu )=S\\cos (\\mu +\\mu _0),\\\\S^2=P^2+Q^2+2PQ\\cos \\nu ,\\qquad \\cos \\mu _0={P+Q\\cos \\nu \\over S},\\qquad \\sin \\mu _0={Q\\sin \\nu \\over S}.\\end{array}}$ If we use this and carry out the $\\mu $ integral, we find $\\begin{split}Z_2&={Q_5\\over s^2+a^2+w^2+b^2}\\int _0^{2\\pi }{d\\nu \\over 2\\pi }{1\\over \\sqrt{1-S^2}}\\\\&={Q_5\\over s^2+a^2+w^2+b^2}\\int _0^{2\\pi }{d\\nu \\over 2\\pi }{1\\over \\sqrt{1-(P^2+Q^2)-2PQ\\cos \\nu }}.\\end{split}$ We can show that this is equal to the integral over the 3d profile: $Q_5\\int _0^{2\\pi } {d\\nu \\over 2\\pi }{1\\over |{\\bf y}-{\\bf f}|}={Q_5\\over 4}\\int {d\\nu \\over 2\\pi }{1\\over \\sqrt{\\rho ^2+R^2-2\\rho R\\cos \\nu + (y_3-c)^2}}$ where ${\\bf f}$ is given by (REF ), because, from (REF ) and (REF ), it follows that $\\rho ^2+R^2-2\\rho R\\cos \\nu + (y_3-c)^2&={(s^2+w^2+a^2+b^2)^2\\over 16}[1-(P^2+Q^2)-2PQ\\cos \\nu ].$ This means that $H$ can be written as an integral over a 4d or 3d profile as $H={4\\over L}\\left.\\int _0^L {d\\lambda \\over |\\vec{x}-\\vec{g}|^2}\\right|_{\\rm smear}=\\int _0^{2\\pi } {d\\nu \\over 2\\pi }{1\\over |{\\bf y}-{\\bf f}|}.$" ], [ "The integrals ${\\cal I}_{mn}$", "${\\cal I}_{mn}$ is defined as ${\\cal I}_{mn}(P,Q)\\equiv \\int _0^{2\\pi }{d\\mu \\over 2\\pi }\\int _0^{2\\pi }{d\\nu \\over 2\\pi }\\,{(P\\cos \\mu )^m (Q\\cos \\nu )^n\\over 1-P\\cos \\mu - Q\\cos \\nu },\\\\P\\equiv {2sa\\over s^2+a^2+w^2+b^2},\\qquad Q\\equiv {2wb\\over s^2+a^2+w^2+b^2}.$ The value at $r=0$ is easy to find because $r=0$ means $P=Q=0$ .", "Namely, ${\\cal I}_{mn}(r=0)={\\left\\lbrace \\begin{array}{ll}1 &\\quad (m=n=0),\\\\0 &\\quad \\text{(otherwise).}\\end{array}\\right.", "}$ One can evaluate ${\\cal I}_{00}$ as follows.", "First, ${\\cal I}_{00}&=\\int _0^{2\\pi }{d\\mu \\over 2\\pi }\\int _0^{2\\pi }{d\\nu \\over 2\\pi }{1\\over 1-P\\cos \\mu -Q\\cos \\nu }=\\int _0^{2\\pi }{d\\nu \\over 2\\pi }{1\\over \\sqrt{(1-Q\\cos \\nu )^2 -P^2}}\\\\&=\\oint {dx\\over 2\\pi i}{1\\over \\sqrt{(x-(Q/2)(x^2+1))^2-P^2x^2}},\\qquad x=e^{i\\nu },$ where the contour for the $x$ integral is the unit circle on the complex $x$ plane.", "The roots of the quartic polynomial inside the square root are $x=x_a,x_b,x_c,x_d$ with $x_d=\\tfrac{1+P-\\sqrt{(1+P)^2-Q^2}}{4Q},\\quad x_c=\\tfrac{1-P-\\sqrt{(1-P)^2-Q^2}}{4Q},\\\\x_b=\\tfrac{1-P+\\sqrt{(1-P)^2-Q^2}}{4Q},\\quad x_a=\\tfrac{1+P+\\sqrt{(1+P)^2-Q^2}}{4Q},$ where $0<x_d<x_c<1<x_b<x_a$ for $P,Q,P+Q\\in (0,1)$ .", "From this it immediately follows that [58] ${\\cal I}_{00}&={4\\over \\pi }{{\\bf K}(m)\\over Q\\sqrt{(x_a-x_c)(x_b-x_d)}},$ where $m={(x_a-x_b)(x_c-x_d)\\over (x_a-x_c)(x_b-x_d)}=\\left(\\tfrac{\\sqrt{1-(P-Q)^2}-\\sqrt{1-(P+Q)^2}}{\\sqrt{1-(P-Q)^2}+\\sqrt{1-(P+Q)^2}}\\right)^2,\\\\{1\\over Q\\sqrt{(x_a-x_c)(x_b-x_d)}}={1\\over \\sqrt{1-(P+Q)^2}+\\sqrt{1-(P-Q)^2}}.$ By using the identity [59] ${\\bf K}(z)=\\tfrac{2}{1+\\sqrt{1-z}}\\,{\\bf K}\\Bigl (\\Bigl (\\tfrac{1-\\sqrt{1-z}}{1+\\sqrt{1-z}}\\Bigr )^2\\Bigr )$ one can obtain the second expression in (REF ).", "Likewise, we can write ${\\cal I}_{01},{\\cal I}_{10}$ using elliptic integrals.", "For example, ${\\cal I}_{01}&={1\\over \\pi }\\int _{x_d}^{x_c} dx{x\\over \\sqrt{(x_a-x)(x_b-x)(x_c-x)(x-x_d)}}\\\\&={4\\over \\pi }{1\\over \\sqrt{(x_a-x_c)(x_b-x_d)}}\\left[(x_d-x_a)\\,\\Pi \\bigl (\\tfrac{x_d-x_c}{x_a-x_c}\\big |m\\bigr )+x_a{\\bf K}(m)\\right]$ where $\\Pi (n|m)$ is the complete elliptic integral of the third kind.", "The 1-form ${\\bf A}$ in the Lunin-Mathur geometry can be written in the $(s,{\\widetilde{\\phi }},w,{\\widetilde{\\psi }})$ coordinates in terms of ${\\cal I}_{10},{\\cal I}_{01}$ as $\\begin{split}A_{\\widetilde{\\phi }}&=\\mathop {\\mathrm {Im}}\\nolimits [(A_1+iA_2)e^{-i{\\widetilde{\\phi }}}]s=-{Q_5 \\Omega k\\over 2} {\\cal I}_{10},\\\\A_{\\widetilde{\\psi }}&=\\mathop {\\mathrm {Im}}\\nolimits [(A_3+iA_4)e^{-i{\\widetilde{\\psi }}}]w=+{Q_5\\Omega k^{\\prime }\\over 2} {\\cal I}_{01}.\\end{split}$" ], [ "The monodromic harmonic function $\\gamma $", "The explicit expression for $\\gamma $ was given in (REF ).", "The value of $\\gamma $ on the $y_3$ axis, namely $u=1$ , can be found by using the explicit expression (REF ) as $\\gamma (u=1,\\sigma )=\\sin ^2\\!\\left({\\pi \\over 2}\\left\\lbrace {\\sigma \\over 2\\pi }\\right\\rbrace \\right)+{\\left\\lfloor {\\sigma \\over 2\\pi } \\right\\rfloor }$ where ${\\left\\lfloor {x} \\right\\rfloor }$ is the floor function and $\\lbrace x\\rbrace \\equiv x-{\\left\\lfloor {x} \\right\\rfloor }$ is the fractional part of $x$ .", "Let us find the value of $\\gamma $ at $r=0$ .", "On the $y_3$ axis, the relation between $\\sigma $ and $y_3^{\\prime }=y-c$ is $y_3^{\\prime }=R\\cot (\\sigma /2)$ , as can be derived from (REF ).", "Using this, we can show that the value of $\\gamma $ at $r=0$ ($y_3^{\\prime }=-c=(a^2-b^2)/4$ ) is $\\gamma (r=0)\\equiv {b^2\\over a^2+b^2}+n,$ where $n\\in {\\mathbb {Z}}$ must be taken appropriately for the branch in consideration.", "Let us consider the branch $-\\pi < \\sigma < \\pi ,\\qquad \\text{therefore}\\quad -{1\\over 2}<\\gamma (u=1,\\sigma )<{1\\over 2}.$ With this choice, $\\gamma $ vanishes at infinity (which corresponds to $u=1,\\sigma =0$ ).", "In this branch, $\\gamma (r=0)$ is $\\gamma (r=0)={\\left\\lbrace \\begin{array}{ll}\\displaystyle {b^2\\over a^2+b^2} & (a>b), \\\\[2ex]\\displaystyle -{a^2\\over a^2+b^2} & (b>a).\\\\\\end{array}\\right.", "}$" ], [ "Various relations", "We can always write ${\\cal I}_{10}$ and ${\\cal I}_{01}$ as follows: $x {\\cal I}_{10}+y {\\cal I}_{01}&=\\int _0^{2\\pi }{d\\mu \\over 2\\pi }\\int _0^{2\\pi }{d\\nu \\over 2\\pi }{xP\\cos \\mu + yQ\\cos \\nu \\over 1-P\\cos \\mu - Q\\cos \\nu }\\\\&=\\int _0^{2\\pi }{d\\mu \\over 2\\pi }\\int _0^{2\\pi }{d\\nu \\over 2\\pi }\\biggl [{x-y\\over a^2+b^2}\\,{b^2 P\\cos \\mu - a^2 Q\\cos \\nu \\over 1-P\\cos \\mu - Q\\cos \\nu }\\\\&\\hspace{85.0pt}+{a^2x+b^2y\\over a^2+b^2}\\,{1-(1-P\\cos \\mu - Q\\cos \\nu )\\over 1-P\\cos \\mu - Q\\cos \\nu }\\biggr ]\\\\&={x-y\\over a^2+b^2}(b^2 {\\cal I}_{10}-a^2 {\\cal I}_{01})+{a^2x+b^2y\\over a^2+b^2}\\left({H\\over V}+{a^2+b^2\\over 4}H-1\\right).$ Let us derive the near-ring behavior of ${\\cal I}_{00}$ .", "Near the ring, where $s\\rightarrow a,w\\rightarrow b$ , we have $P\\approx {a^2\\over a^2+b^2},\\quad Q\\approx {b^2\\over a^2+b^2},\\qquad P+Q\\approx 1.$ So, in the integral ${\\cal I}_{00}=\\int {d\\mu \\over 2\\pi }{1\\over \\sqrt{(1-P\\cos \\mu )^2-Q^2}},$ the dominant contribution comes from $\\mu \\approx 0$ , which corresponds to the contribution from the part of the ring near the point where we are sitting.", "In this limit, $(1-P\\cos \\mu )^2-Q^2 \\approx (1-P+Q)(D^2+x^2)$ , where $D\\equiv \\sqrt{1-P-Q}$ is the distance from us to the ring which is now approximated to be straight, and $x=\\sqrt{P/2\\,}\\,\\mu $ is the distance along the straight ring.", "So, ${\\cal I}_{00}&\\approx {1\\over 2\\pi }\\sqrt{2\\over (1-P+Q)P}\\int _{-\\Lambda }^{\\Lambda } {dx\\over \\sqrt{D^2+x^2}}\\\\&\\approx {1\\over \\pi }\\sqrt{2\\over (1-P+Q)P}\\log {\\Lambda \\over D}={1\\over \\pi }{a^2+b^2\\over ab}\\log {\\Lambda \\over D},$ where $\\Lambda $ is a cutoff which depends on global data and cannot be determined by a local analysis here.", "The relation between $D$ and $u$ works out to be $D\\approx {ab\\over \\sqrt{2}(a^2+b^2)}{1\\over u}$ .", "So, in the end ${\\cal I}_{00}&\\sim {1\\over \\pi }{a^2+b^2\\over ab}\\log u.$ Evaluation of ${\\cal I}_{10}$ is mostly identical, because $\\cos \\mu \\approx 1$ in the $\\mu \\approx 0$ region.", "The result is ${\\cal I}_{10}\\sim {a^2\\over a^2+b^2}{1\\over \\pi }{a^2+b^2\\over ab}\\log u={a\\over \\pi b}\\log u,\\qquad {\\cal I}_{01}\\sim {b\\over \\pi a}\\log u.$ This in particular means that $\\bigtriangleup (b^2 {\\cal I}_{10}- a^2 {\\cal I}_{01})=\\text{regular}\\qquad \\text{(no $\\delta $ function)}.$ Let us briefly discuss how to show that (REF ) is equal to (REF ), namely $b^2{\\cal I}_{10}-a^2 {\\cal I}_{01}&={4ab\\,\\over (s^2+w^2+a^2+b^2)^2}\\int _0^{2\\pi } {d\\mu \\over 4\\pi }\\,{(b^2-a^2)sw\\cos \\mu + ab(s^2-w^2)\\over \\sqrt{X}(1+\\sqrt{X})}\\,,$ where $X\\equiv 1-(P^2+Q^2)-2PQ\\cos \\mu $ .", "Just as we did around (REF ), by setting $\\nu \\rightarrow \\nu +\\mu $ and carrying out the $\\nu $ integral first, we find ${\\cal I}_{10}=\\int _0^{2\\pi }{d\\mu \\over 2\\pi }{P(P+Q\\cos \\mu )\\over \\sqrt{1-S^2}\\,(1+\\sqrt{1-S^2})}.$ ${\\cal I}_{01}$ is obtained by $P\\leftrightarrow Q$ .", "Using the relations (REF ), (REF ), it is straightforward to prove (REF )." ] ]
2105.11639
[ [ "A discrete de Rham method for the Reissner-Mindlin plate bending problem\n on polygonal meshes" ], [ "Abstract In this work we propose a discretisation method for the Reissner--Mindlin plate bending problem in primitive variables that supports general polygonal meshes and arbitrary order.", "The method is inspired by a two-dimensional discrete de Rham complex for which key commutation properties hold that enable the cancellation of the contribution to the error linked to the enforcement of the Kirchhoff constraint.", "Denoting by $k\\ge 0$ the polynomial degree for the discrete spaces and by $h$ the meshsize, we derive for the proposed method an error estimate in $h^{k+1}$ for general $k$, as well as a locking-free error estimate for the lowest-order case $k=0$.", "The theoretical results are validated on a complete panel of numerical tests." ], [ "Introduction", "In this work we propose a novel discretisation method for the Reissner–Mindlin plate bending problem in primitive variables that supports general polygonal meshes and arbitrary order.", "In its lowest-order version, the method can be proved to behave robustly with respect to the plate thickness $t$ .", "Its design is based on the two-dimensional discrete de Rham (DDR) complex of [20], for which key commutation properties hold that enable the cancellation of the contribution to the error linked to the enforcement of the Kirchhoff constraint.", "We consider in what follows an elastic plate of thickness $t>0$ with reference configuration $\\Omega \\times \\left(-\\frac{t}{2},\\frac{t}{2}\\right)$ , where $\\Omega \\subset \\mathbb {R}^2$ is a bounded connected polygonal domain with boundary $\\partial \\Omega $ .", "Without loss of generality, it is assumed in what follows that $\\Omega $ has diameter 1 and that $t<1$ .", "The Reissner–Mindlin model describes the deformation of the plate in terms of the rotation $\\theta :\\Omega \\rightarrow \\mathbb {R}^2$ of the fibers initially perpendicular to its midsurface and of the transverse displacement $u:\\Omega \\rightarrow \\mathbb {R}$ .", "Introducing the shear strain $\\mathsf {\\gamma }$ and denoting by $f:\\Omega \\rightarrow \\mathbb {R}$ the transverse load, the strong formulation of the model with clamped boundary conditions reads $\\mathsf {\\gamma }+\\operatorname{\\bf div}(\\mathsf {C}\\operatorname{\\mathsf {grad}_{\\rm s}}\\theta ) &= 0&\\qquad &\\text{in $\\Omega $},\\\\-\\operatorname{div}\\gamma &=f &\\qquad &\\text{in $\\Omega $},\\\\\\mathsf {\\gamma } &= \\frac{\\kappa }{t^2}(\\operatorname{\\bf grad}u-\\theta ) &\\qquad &\\text{in $\\Omega $},\\\\\\theta &= 0,\\quad u = 0 &\\qquad &\\text{on $\\partial \\Omega $}.$ Here, $\\operatorname{\\bf div}$ is the row-wise divergence of tensors, $\\operatorname{\\mathsf {grad}_{\\rm s}}$ is the symmetric part of the gradient applied to vector-valued fields over $\\Omega $ , and $\\mathsf {C}$ is the fourth-order tensor defined by $\\mathsf {C}\\mathsf {t} = \\beta _0\\mathsf {t} + \\beta _1(\\operatorname{tr}\\mathsf {t})\\mathsf {I}$ for all second-order tensor $\\mathsf {t}$ , with $\\mathsf {I}$ the identity tensor.", "The parameters of $\\mathsf {C}$ are $\\beta _0\\frac{E}{12(1+\\nu )}$ and $\\beta _1\\frac{E\\nu }{12(1-\\nu ^2)}$ , where $E>0$ and $\\nu \\in [0,\\frac{1}{2})$ are the Young modulus and Poisson ratio of the material, respectively.", "The shear modulus $\\kappa $ is given as $\\kappa \\frac{\\kappa _0E}{2(1+\\nu )}$ , with shear correction factor $\\kappa _0$ usually taken equal to $\\frac{5}{6}$ for clamped plates.", "Denoting by $H_0^1(\\Omega )$ the space of real-valued functions that are square-integrable along with their derivatives and that vanish on $\\partial \\Omega $ in the sense of traces, the standard weak formulation of () hinges on the spaces $\\Theta H_0^1(\\Omega )^2$ for the rotation and $UH_0^1(\\Omega )$ for the transverse displacement.", "Specifically, assuming that $f\\in L^2(\\Omega )$ , it reads: Find $(\\theta ,u)\\in \\Theta \\times U$ such that $A((\\theta ,u),(\\eta ,v)) = \\ell (v)\\qquad \\forall (\\eta ,v)\\in \\Theta \\times U,$ where the bilinear form $A:\\left[\\Theta \\times U\\right]^2\\rightarrow \\mathbb {R}$ and the linear form $\\ell :U\\rightarrow \\mathbb {R}$ are such that, for all $(\\tau ,w),(\\eta ,v)\\in \\Theta \\times U$ , $A((\\tau ,w),(\\eta ,v)) a(\\tau ,\\eta ) + b((\\tau ,w),(\\eta ,v)),\\qquad \\ell (v)\\int _\\Omega f v,$ with bilinear forms $a:\\Theta \\times \\Theta \\rightarrow \\mathbb {R}$ and $b:\\left[\\Theta \\times U\\right]^2\\rightarrow \\mathbb {R}$ such that $a(\\tau ,\\eta )&\\beta _0\\int _\\Omega \\operatorname{\\mathsf {grad}_{\\rm s}}\\tau :\\operatorname{\\mathsf {grad}_{\\rm s}}\\eta + \\beta _1\\int _\\Omega \\operatorname{div}\\tau ~\\operatorname{div}\\eta ,\\qquad \\\\b((\\tau ,w),(\\eta ,v))&\\frac{\\kappa }{t^2}\\int _\\Omega (\\tau - \\operatorname{\\bf grad}w)\\cdot (\\eta - \\operatorname{\\bf grad}v).$ The role of the bilinear form $b$ is to enforce the Kirchhoff constraint that, as $t\\rightarrow 0$ , the rotation of the normal fibers equals the gradient of the transverse displacement.", "Notice that the choice of considering clamped boundary conditions is made for the sole purpose of simplifying the theoretical discussion: other standard boundary conditions can be considered with straightforward modifications.", "A critical point in the numerical approximation of problem (REF ) is robustness for small $t$ .", "Methods for which error estimates uniform in $t$ can be established are commonly referred to as (shear) locking-free.", "The finite element literature for the locking-free discretisation of problem (REF ) on standard meshes dates back to the 1980s.", "In [15], the authors proposed a reformulation involving, in addition to the primitive variables $\\theta $ and $u$ , the introduction of two additional variables corresponding to the irrotational and solenoidal parts of the transverse shear strain.", "This work pointed out the relevance of establishing a discrete version of the Helmholtz decomposition to obtain error estimates uniform in $t$ .", "A method in primitive variables was later proposed in [5], based on a nonconforming (Crouzeix–Raviart) piecewise linear space for the displacement and a bubble-enriched continuous space for the rotation, and involving a projection in the discrete version of the bilinear form $b$ .", "Recent developments of these ideas, including the extension to higher orders and the use of the Taylor–Hood element pair for the underlying Stokes problem, can be found in [30], [25] The idea of using reduced integration or projections in the enforcement of the Kirchhoff constraint can be found in several other works; see, e.g., [16], [24], [29], [6], [27].", "A different approach, resorting to a mixed formulation where the shear strain appears as a separate unknown, is considered in [2].", "The key point is, in this case, the design of a suitable coupling bilinear form, for which abstract conditions are provided.", "Recent results on mixed finite element schemes can be found in [26]; see also the references therein.", "Mixed approaches inspired by fully nonconforming (discontinuous Galerkin) methods have been proposed in [4], later leading to choices of finite element spaces that do not require reduced integration [3]; see also [28], [17] for related developments.", "Discontinuous Galerkin methods in their weakly over-penalised symmetric formulation are considered in [14], [13].", "While the use of standard (e.g., simplicial conforming) meshes can be satisfactory for simple geometries and problems, it may lack flexibility in more complex situations.", "The support of general meshes can greatly simplify the meshing process in the presence of small geometric features [1] and pave the way for advanced techniques such as nonconforming adaptive mesh refinement (which does not trade mesh quality for size) and mesh coarsening [7], [8], [22], that are crucial to exploit high-order approximation in the presence of geometric singularities.", "Owing to the onset of polygonal elements and/or hanging nodes, such strategies are inaccessible to standard conforming finite elements.", "These and similar considerations have prompted, in the last few years, the development of locking-free discretisation methods for problem (REF ) supporting general polygonal meshes.", "A first example is provided by the low-order Mimetic Finite Difference method of [10], that hinges on transverse displacements defined at mesh vertices, rotations defined at mesh vertices and edges, and uses shear forces at edges as intermediate unknowns.", "The key ingredient to establish a first-order locking-free error estimate is once again a discrete Helmholtz decomposition.", "A lowest-order Virtual Element method has also been recently proposed in [11], inspired by the reformulation of problem (REF ) originally introduced in [9] in the context of Isogeometric Analysis and using the transverse displacement and shear strain as unknowns.", "The DDR method proposed in this work contains several key elements of novelty.", "First, to the best of our knowledge, it is the first scheme to support general polygonal meshes and high-order.", "Second, it does not resort to reduced integration or projections in the discrete counterpart of the bilinear form $b$ .", "Third, it admits an inexpensive lowest-order version for which locking-free estimates can be rigorously established.", "The starting point for the design of the scheme is the two-dimensional DDR complex of [20].", "This complex satisfies a crucial commutation property between the reconstructions of the discrete displacement gradient, the continuous gradient, and the interpolators on the corresponding spaces; see (REF ) below.", "When performing a convergence analysis in the spirit of the Third Strang Lemma [18], one can leverage this commutation property to cancel the error resulting from the enforcement of the Kirchhoff constraint through the discrete counterpart of the bilinear form $b$ .", "This remark suggests the use of DDR counterparts of the $H^1_0(\\Omega )$ and $H_0(\\operatorname{rot};\\Omega )$ spaces for the displacement and the rotation, respectively.", "In order to have sufficient information to reconstruct a full strain tensor, the discrete $H_0(\\operatorname{rot};\\Omega )$ space has to be enriched by the addition of normal components at edges.", "It turns out that this enriched space can be embedded into the standard Hybrid High-Order (HHO) space for elasticity originally introduced in [21] (see also [19] and [12] for an application of HHO methods to Kirchhoff–Love plates), so that the standard HHO construction can be exploited to design the discrete counterpart of the bilinear form $a$ .", "With these ingredients, we establish in Theorem REF an estimate in $h^{k+1}$ (with $h$ denoting the meshsize and $k$ the polynomial degree of the DDR sequence) for the natural (coercivity) norm of the error.", "The right-hand side of this estimate does not explicitly depend on $t$ , but involves, as is unavoidable for high-order schemes, norms of higher order derivatives of the strain; such norms are not expected to remain bounded as $t\\rightarrow 0$ .", "Through the introduction of novel liftings of the displacement and of the rotation, we show in Theorem REF that an error estimate uniform in $t$ (and thus locking-free) can be established in the lowest order case $k=0$ .", "The rest of the paper is organised as follows.", "In Section we introduce the discrete setting.", "Section contains the statement of the discrete problem preceeded by the required constructions.", "The analysis of the method is carried out in Section , the main theorems being stated in Section REF and their proofs given in Sections REF and REF .", "Finally, Section contains a complete panel of numerical results, introducing a novel analytical solution for the model and showing that the method displays, to a certain extent, a locking-free behaviour also for $k\\ge 1$ ." ], [ "Mesh", "For any measurable set $Y\\subset \\mathbb {R}^2$ , we denote by $h_Y\\sup \\lbrace |x-y|\\,:\\,x,y\\in Y\\rbrace $ its diameter and by $|Y|$ its Hausdorff measure.", "We consider meshes $\\mathcal {M}_h\\mathcal {T}_h\\cup \\mathcal {E}_h\\cup \\mathcal {V}_h$ , where: $\\mathcal {T}_h$ is a finite collection of open disjoint polygonal elements such that $\\overline{\\Omega } = \\bigcup _{T\\in \\mathcal {T}_h}\\overline{T}$ and $h=\\max _{T\\in \\mathcal {T}_h}h_T>0$ ; $\\mathcal {E}_h$ is the set collecting the open polygonal edges (line segments) of the elements; $\\mathcal {V}_h$ is the set collecting the edge endpoints.", "It is assumed, in what follows, that $(\\mathcal {T}_h,\\mathcal {E}_h)$ matches the conditions in [19].", "The sets collecting the mesh edges that lie on the boundary of a mesh element $T\\in \\mathcal {T}_h$ and on $\\partial \\Omega $ are denoted by $\\mathcal {E}_{T}$ and $\\mathcal {E}_h^{{\\rm b}}$ , respectively.", "We also denote by $\\mathcal {E}_h^{{\\rm i}}=\\mathcal {E}_h\\setminus \\mathcal {E}_h^{{\\rm b}}$ the set of internal edges.", "The coordinates vector of $V\\in \\mathcal {V}_h$ is denoted by $x_V$ .", "Each $E\\in \\mathcal {E}_h$ is endowed with an orientation determined by a fixed unit tangent vector $t_E$ ; we then choose the unit normal $n_E$ such that $(t_E,n_E)$ forms a right-hand system of coordinates.", "For $T\\in \\mathcal {T}_h$ and $E\\in \\mathcal {E}_{T}$ , we set $\\omega _{TE}=1$ if $t_E$ points in the clockwise direction of $\\partial T$ , and $\\omega _{TE}=-1$ otherwise.", "It can be checked that $n_{TE}\\omega _{TE}n_E$ is the outer unit normal to $T$ on $E$ ." ], [ "Polynomial spaces", "For any $Y\\in \\mathcal {T}_h\\cup \\mathcal {E}_h$ , we denote by $\\mathcal {P}_{}^{\\ell }(Y)$ the space spanned by the restriction to $Y$ of two-variate polynomials of total degree $\\le \\ell $ , with the convention that $\\mathcal {P}_{}^{-1}(Y)=\\lbrace 0\\rbrace $ .", "We additionally denote by $\\pi _{\\mathcal {P},Y}^{\\ell }$ the corresponding $L^2$ -orthogonal projector.", "For all $E\\in \\mathcal {E}_h$ , the space $\\mathcal {P}_{}^{\\ell }(E)$ is isomorphic to univariate polynomials of total degree $\\le \\ell $ (see [19]).", "In what follows, with a little abuse of notation, both spaces are denoted by $\\mathcal {P}_{}^{\\ell }(E)$ .", "For $Y\\in \\mathcal {T}_h\\cup \\mathcal {E}_h$ , the vector and tensor versions of $\\mathcal {P}_{}^{\\ell }(Y)$ are respectively denoted by $\\mathcal {P}_{}^{\\ell }(Y)\\mathcal {P}_{}^{\\ell }(Y)^2$ and $\\mathsf {P}_{}^{\\ell }(Y)\\mathcal {P}_{}^{\\ell }(Y)^{2\\times 2}$ , and the corresponding $L^2$ -orthogonal projectors $\\pi _{\\mathcal {P},Y}^{\\ell }$ and $\\pi _{\\mathsf {P},Y}^{\\ell }$ are obtained applying $\\pi _{\\mathcal {P},Y}^{\\ell }$ component-wise.", "We additionally denote by $\\mathsf {P}_{\\rm s}^{\\ell }(Y)$ the subspace of symmetric-valued functions in $\\mathsf {P}_{}^{\\ell }(Y)$ .", "For all $T\\in \\mathcal {T}_h$ , let $x_T\\in T$ be such that $T$ contains a ball centered at $x_T$ of radius $\\rho h_T$ , where $\\rho $ is the mesh regularity parameter in [19].", "For any integer $\\ell \\ge 0$ , we define the following relevant subspaces of $\\mathcal {P}_{}^{\\ell }(T)$ : $\\mathcal {R}^{\\ell }(T)\\operatorname{\\bf rot}\\mathcal {P}_{}^{\\ell +1}(T),\\qquad \\mathcal {R}^{{\\rm c},\\ell }(T)(x-x_T)\\mathcal {P}_{}^{\\ell -1}(T),$ where, for a vector $y\\in \\mathbb {R}^2$ , $y^\\perp $ denotes the vector obtained rotating $y$ by $-\\frac{\\pi }{2}$ .", "We have $\\mathcal {P}_{}^{\\ell }(T) = \\mathcal {R}^{\\ell }(T) \\oplus \\mathcal {R}^{{\\rm c},\\ell }(T).$ Notice that the direct sums in the above expression are not $L^2$ -orthogonal in general.", "The $L^2$ -orthogonal projectors on the spaces (REF ) are, with obvious notation, $\\pi _{\\mathcal {R},T}^{\\ell }$ , and $\\pi _{\\mathcal {R},T}^{{\\rm c},\\ell }$ ." ], [ "DDR scheme", "The scheme for (REF ) is designed using spaces of unknowns from the DDR method [20] together with an enrichment inspired by HHO methods [19]." ], [ "Spaces and interpolators", "Let a polynomial degree $k\\ge 0$ be fixed and set $\\begin{aligned}\\underline{\\Theta }_h^k&\\Big \\lbrace \\underline{\\eta }_h=\\big ((\\eta _{\\mathcal {R},T},\\eta _{\\mathcal {R},T}^{\\rm c})_{T\\in \\mathcal {T}_h},(\\eta _E)_{E\\in \\mathcal {E}_h}\\big )\\,:\\,\\begin{aligned}[t]&\\text{$(\\eta _{\\mathcal {R},T},\\eta _{\\mathcal {R},T}^{\\rm c})\\in \\mathcal {R}^{k-1}(T)\\times \\mathcal {R}^{{\\rm c},k}(T)$ for all $T\\in \\mathcal {T}_h$,}\\\\&\\text{and $\\eta _E\\in \\mathcal {P}_{}^{k}(E)$ for all $E\\in \\mathcal {E}_h$}\\Big \\rbrace ,\\end{aligned}\\\\\\underline{U}_h^k&\\Big \\lbrace \\underline{v}_h=\\big ((v_T)_{T\\in \\mathcal {T}_h}, v_{\\mathcal {E}_h}\\big )\\,:\\,\\text{$v_T\\in \\mathcal {P}_{}^{k-1}(T)$ for all $T\\in \\mathcal {T}_h$ and $v_{\\mathcal {E}_h}\\in \\mathcal {P}_{\\rm c}^{k+1}(\\mathcal {E}_h)$}\\Big \\rbrace ,\\end{aligned}$ where $\\mathcal {P}_{\\rm c}^{k+1}(\\mathcal {E}_h)$ is spanned by the functions over the mesh edge skeleton whose restriction to each edge $E\\in \\mathcal {E}_h$ is a polynomial of total degree $\\le k+1$ and that are continuous at the edges endpoints.", "The space $\\underline{\\Theta }_h^k$ is an enrichment of the two-dimensional DDR space $\\underline{X}^k_{\\operatorname{\\bf curl},h}$ with edge unknowns representing a full vector-valued field as opposed to its tangent component only; the space $\\underline{U}_h^k$ coincides with the two-dimensional DDR space $\\underline{X}^k_{\\operatorname{\\bf grad},h}$ .", "Smooth functions are interpolated as follows: For all $\\eta \\in H^1(\\Omega )^2$ $\\underline{I}_{\\Theta ,h}^k\\eta \\big ((\\pi _{\\mathcal {R},T}^{k-1}\\eta _{|T},\\pi _{\\mathcal {R},T}^{{\\rm c},k}\\eta _{|T})_{T\\in \\mathcal {T}_h},(\\pi _{\\mathcal {P},E}^{k}\\eta _{|E})_{E\\in \\mathcal {E}_h}\\big )\\in \\underline{\\Theta }_h^k,$ while, for all $v\\in C^0(\\overline{\\Omega })$ , ${\\begin{array}{c}\\underline{I}_{U,h}^k v \\big ((\\pi _{\\mathcal {P},T}^{k-1} v_{|T})_{T\\in \\mathcal {T}_h}, v_{\\mathcal {E}_h}\\big )\\in \\underline{U}_h^k,\\\\\\text{with $\\pi _{\\mathcal {P},E}^{k-1}(v_{\\mathcal {E}_h})_{|E} = \\pi _{\\mathcal {P},E}^{k-1} v_{|E}$ for all $E\\in \\mathcal {E}_h$and $v_{\\mathcal {E}_h}(x_V) = v(x_V)$ for all $V\\in \\mathcal {V}_h$.", "}\\end{array}}$ For all $T\\in \\mathcal {T}_h$ , we denote by $\\underline{\\Theta }_T^k$ and $\\underline{U}_T^k$ , respectively, the restrictions of $\\underline{\\Theta }_h^k$ and $\\underline{U}_h^k$ to $T$ , collecting the polynomial components that lie inside $T$ and on its boundary.", "A similar convention is adopted for the elements of these spaces and for the interpolators." ], [ "Discrete differential operators and potentials", "We introduce discrete versions of the differential operators and of the rotation field reconstructed from the unknowns in the discrete spaces." ], [ "Discrete gradient and transverse displacement reconstruction on $\\underline{U}_T^k$", "We follow here standard constructions from the DDR method.", "For all $T\\in \\mathcal {T}_h$ , the polynomial transverse displacement gradient $G_T^{k}:\\underline{U}_T^k\\rightarrow \\mathcal {P}_{}^{k}(T)$ is such that, for all $\\underline{v}_T\\in \\underline{U}_T^k$ , $\\int _TG_T^{k}\\underline{v}_T\\cdot \\eta = -\\int _Tv_T\\operatorname{div}\\eta + \\sum _{E\\in \\mathcal {E}_{T}}\\omega _{TE}\\int _E v_{\\mathcal {E}_{T}}(\\eta \\cdot n_E)\\qquad \\forall \\eta \\in \\mathcal {P}_{}^{k}(T).$ We additionally define the transverse displacement reconstruction $P_{U,T}^{k+1}:\\underline{U}_T^k\\rightarrow \\mathcal {P}_{}^{k+1}(T)$ such that, for all $\\underline{v}_T\\in \\underline{U}_T^k$ , $\\int _TP_{U,T}^{k+1}\\underline{v}_T\\operatorname{div}\\eta = -\\int _TG_T^{k}\\underline{v}_T\\cdot \\eta + \\sum _{E\\in \\mathcal {E}_{T}}\\omega _{TE}\\int _Ev_{\\mathcal {E}_{T}}(\\eta \\cdot n_E)\\qquad \\forall \\eta \\in \\mathcal {R}^{{\\rm c},k+2}(T).$ A global transverse displacement reconstruction is obtained setting, for all $\\underline{v}_h\\in \\underline{U}_h^k$ , $(P_{U,h}^{k+1}\\underline{v}_h)_{|T}P_{U,T}^{k+1}\\underline{v}_T\\qquad \\forall T\\in \\mathcal {T}_h.$ Finally, we define a global discrete transverse displacement gradient $\\underline{G}_h^{k}:\\underline{U}_h^k\\rightarrow \\underline{\\Theta }_h^k$ as follows: For all $\\underline{v}_h\\in \\underline{U}_h^k$ , $\\underline{G}_h^{k}\\underline{v}_h\\big ((\\pi _{\\mathcal {R},T}^{k-1}G_T^{k}\\underline{v}_T, \\pi _{\\mathcal {R},T}^{{\\rm c},k}G_T^{k}\\underline{v}_T)_{T\\in \\mathcal {T}_h},((v_{\\mathcal {E}_h})_{|E}^{\\prime }t_E)_{E\\in \\mathcal {E}_h}\\big ),$ where the derivative along the edge is taken in the direction of $t_E$ .", "To state the key commutation property used to prove the error estimates for the DDR scheme, we need to introduce a modified version of the interpolator on $\\underline{\\Theta }_h^k$ , which is adjusted to the account for the fact that, on the edges, the discrete gradient only encodes the tangential derivatives.", "The modified interpolator is $\\underline{I}_{\\Theta ,h}^{\\flat ,k}:H^1(\\Omega )^2\\rightarrow \\underline{\\Theta }_h^k$ such that, for all $\\eta \\in H^1(\\Omega )^2$ , $\\underline{I}_{\\Theta ,h}^{\\flat ,k}\\eta \\big ((\\pi _{\\mathcal {R},T}^{k-1}\\eta _{|T},\\pi _{\\mathcal {R},T}^{{\\rm c},k}\\eta _{|T})_{T\\in \\mathcal {T}_h},(\\pi _{\\mathcal {P},E}^{k}(\\eta _{|E}\\cdot t_E)~t_E)_{E\\in \\mathcal {E}_h}\\big ).$ The commutation property is the following, obtained by considering only the face components in the 3D formula [20]: $\\underline{G}_h^{k}(\\underline{I}_{U,h}^kv) = \\underline{I}_{\\Theta ,h}^{\\flat ,k}(\\operatorname{\\bf grad}v)\\qquad \\forall v\\in C^1(\\overline{\\Omega }).$" ], [ "Discrete scalar rotor and rotation reconstruction on $\\underline{\\Theta }_T^k$", "Let a mesh element $T\\in \\mathcal {T}_h$ be fixed.", "The local scalar rotor operator $R_T^k:\\underline{\\Theta }_T^k\\rightarrow \\mathcal {P}_{}^{k}(T)$ is such that, for all $\\underline{\\eta }_T\\in \\underline{\\Theta }_T^k$ , $\\int _TR_T^k\\underline{\\eta }_T q= \\int _T\\eta _{\\mathcal {R},T}\\cdot \\operatorname{\\bf rot}q- \\sum _{E\\in \\mathcal {E}_{T}}\\omega _{TE}\\int _E(\\eta _E\\cdot t_E)~q\\qquad \\forall q\\in \\mathcal {P}_{}^{k}(T).$ This operator enables the reconstruction of a discrete rotation $P_{\\Theta ,T}^{k}:\\underline{\\Theta }_T^k\\rightarrow \\mathcal {P}_{}^{k}(T)$ defined such that, for all $\\underline{\\eta }_T\\in \\underline{\\Theta }_T^k$ and all $(\\tau ,q)\\in \\mathcal {R}^{{\\rm c},k}(T)\\times \\mathcal {P}_{}^{k+1}(T)$ , $\\int _TP_{\\Theta ,T}^{k}\\underline{\\eta }_T\\cdot (\\tau + \\operatorname{\\bf rot}q)=\\int _T\\eta _{\\mathcal {R},T}^{\\rm c}\\cdot \\tau + \\int _TR_T^k\\underline{\\eta }_T~q+ \\sum _{E\\in \\mathcal {E}_{T}}\\omega _{TE}\\int _E(\\eta _E\\cdot t_E)~q.$ The scalar rotor and rotation reconstructions correspond to the face curl and tangential face potential of the DDR method [20].", "We note the following property [20]: For all $\\underline{\\eta }_T\\in \\underline{\\Theta }_T^k$ , $ \\pi _{\\mathcal {R},T}^{k-1}(P_{\\Theta ,T}^{k}\\underline{\\eta }_T)=\\eta _{\\mathcal {R},T}\\quad \\mbox{ and }\\quad \\pi _{\\mathcal {R},T}^{{\\rm c},k}(P_{\\Theta ,T}^{k}\\underline{\\eta }_T)=\\eta _{\\mathcal {R},T}^{\\rm c}.$ In consequence, for all $\\eta \\in H^1(T)^2$ , we have $\\pi _{\\mathcal {R},T}^{k-1}\\big [P_{\\Theta ,T}^{k}(\\underline{I}_{\\Theta ,T}^k\\eta )\\big ]=\\pi _{\\mathcal {R},T}^{k-1}\\eta $ and $\\pi _{\\mathcal {R},T}^{{\\rm c},k-1}\\big [P_{\\Theta ,T}^{k}(\\underline{I}_{\\Theta ,T}^k\\eta )\\big ]=\\pi _{\\mathcal {R},T}^{{\\rm c},k-1}\\eta $ (where we have used $\\mathcal {R}^{{\\rm c},k-1}(T)\\subset \\mathcal {R}^{{\\rm c},k}(T)$ , see (REF ), to write $\\pi _{\\mathcal {R},T}^{{\\rm c},k-1}=\\pi _{\\mathcal {R},T}^{{\\rm c},k-1}\\pi _{\\mathcal {R},T}^{{\\rm c},k}$ ).", "Combining these relations with (REF ) written for $\\ell =k-1$ and [20], we get $\\pi _{\\mathcal {P},T}^{k-1}\\big [P_{\\Theta ,T}^{k}(\\underline{I}_{\\Theta ,T}^k\\eta )\\big ]=\\pi _{\\mathcal {P},T}^{k-1}\\eta \\qquad \\forall \\eta \\in H^1(T)^2.$" ], [ "Discrete symmetric gradient, divergence and stabilisation on $\\underline{\\Theta }_T^k$", "The discretisation of the bilinear form (REF ) requires to define a discrete symmetric gradient (and divergence) on the discrete space of rotations.", "Since vectors in this space have polynomial components inside the elements and on the edges, a natural approach to define such discrete differential operators comes from the Hybrid High-Order (HHO) machinery [19].", "In what follows, we let a mesh element $T\\in \\mathcal {T}_h$ be fixed." ], [ "Gradients and divergence.", "Let us define the local (vector-valued) HHO space, extension of $\\underline{\\Theta }_T^k$ in which the element component is taken in the full polynomial space: $\\underline{\\Theta }^k_{{\\rm HHO},T}=\\left\\lbrace \\underline{w}_T=(w_T,(w_E)_{E\\in \\mathcal {E}_{T}})\\,:\\,w_T\\in \\mathcal {P}_{}^{k}(T)\\,,\\quad w_E\\in \\mathcal {P}_{}^{k}(E)\\quad \\forall E\\in \\mathcal {E}_{T}\\right\\rbrace .$ The discrete rotation enables the definition of the following embedding $\\underline{\\mathfrak {I}}_{{\\rm HHO},T}^k:\\underline{\\Theta }_T^k\\rightarrow \\underline{\\Theta }^k_{{\\rm HHO},T}$ : $\\underline{\\mathfrak {I}}_{{\\rm HHO},T}^k \\underline{\\eta }_T (P_{\\Theta ,T}^{k}\\underline{\\eta }_T,(\\eta _E)_{E\\in \\mathcal {E}_{T}})\\qquad \\forall \\underline{\\eta }_T\\in \\underline{\\Theta }_T^k.$ Owing to (REF ), $\\underline{\\mathfrak {I}}_{{\\rm HHO},T}^k$ is indeed a one-to-one mapping.", "Using HHO techniques (see in particular [19]) on $\\underline{\\mathfrak {I}}_{{\\rm HHO},T}^k\\underline{\\eta }_T$ , we can then design the local discrete gradients (standard and symmetric) and divergence of a discrete rotation $\\underline{\\eta }_T\\in \\underline{\\Theta }_T^k$ .", "Specifically, this leads to defining the rotation gradient $\\mathsf {G}_T^k:\\underline{\\Theta }_T^k\\rightarrow \\mathsf {P}_{}^{k}(T)$ such that, for all $\\underline{\\eta }_T\\in \\underline{\\Theta }_T^k$ , $\\int _T\\mathsf {G}_T^k\\underline{\\eta }_T:\\mathsf {t}=-\\int _TP_{\\Theta ,T}^{k}\\underline{\\eta }_T\\cdot (\\operatorname{\\bf div}\\mathsf {t})+ \\sum _{E\\in \\mathcal {E}_{T}}\\omega _{TE}\\int _E\\eta _E\\cdot (\\mathsf {t}n_E)\\qquad \\forall \\mathsf {t}\\in \\mathsf {P}_{}^{k}(T).$ The local symmetric gradient $\\mathsf {G}_{{\\rm s},T}^k:\\underline{\\Theta }_T^k\\rightarrow \\mathsf {P}_{\\rm s}^{k}(T)$ and divergence $D_T^k:\\underline{\\Theta }_T^k\\rightarrow \\mathcal {P}_{}^{k}(T)$ are obtained setting, for all $\\underline{\\eta }_T\\in \\underline{\\Theta }_T^k$ , $\\mathsf {G}_{{\\rm s},T}^k\\underline{\\eta }_T\\frac{1}{2}\\big [\\mathsf {G}_T^k\\underline{\\eta }_T + (\\mathsf {G}_T^k\\underline{\\eta }_T)^\\intercal \\big ],\\qquad D_T^k\\underline{\\eta }_T\\operatorname{tr}(\\mathsf {G}_T^k\\underline{\\eta }_T).$ In (REF ), since $\\operatorname{\\bf div}\\mathsf {t}\\in \\mathcal {P}_{}^{k-1}(T)$ we can replace $P_{\\Theta ,T}^{k}\\underline{\\eta }_T$ with $\\pi _{\\mathcal {P},T}^{k-1}(P_{\\Theta ,T}^{k}\\underline{\\eta }_T)$ and thus, using (REF ) and following the techniques of [19], we obtain the commutation formula $\\mathsf {G}_T^k(\\underline{I}_{\\Theta ,T}^k\\eta )=\\pi _{\\mathcal {P},T}^{k}(\\operatorname{\\bf grad}\\eta )$ for all $\\eta \\in H^1(T)^2$ ; this shows that $\\mathsf {G}_T^k$ (hence also $\\mathsf {G}_{{\\rm s},T}^k$ and $D_T^k$ ) has optimal approximation properties." ], [ "Stabilisation.", "As usual in numerical methods for polytopal meshes, the discrete counterpart of a bilinear form such as (REF ) involves a consistent component (here, based on $\\mathsf {G}_{{\\rm s},T}^k$ ), and a stabilisation term.", "In HHO methods, the local stabilisation bilinear forms are defined through the introduction of a higher-order reconstruction.", "For elasticity problems involving the discrete symmetric gradient, and accounting for the embedding (REF ), this leads to defining $p_{T}^{k+1}:\\underline{\\Theta }_{T}\\rightarrow \\mathcal {P}_{}^{k+1}(T)$ by: For all $\\underline{\\eta }_T\\in \\underline{\\Theta }_{T}$ and all $w\\in \\mathcal {P}_{}^{k+1}(T)$ , $&\\int _T\\operatorname{\\mathsf {grad}_{\\rm s}}p_{T}^{k+1}\\underline{\\eta }_T:\\operatorname{\\mathsf {grad}_{\\rm s}}w=-\\int _T P_{\\Theta ,T}^{k}\\underline{\\eta }_T\\cdot \\operatorname{\\bf div}(\\operatorname{\\mathsf {grad}_{\\rm s}}w)+ \\sum _{E\\in \\mathcal {E}_{T}}\\int _E \\eta _E\\cdot (\\operatorname{\\mathsf {grad}_{\\rm s}}w~n_{TE}), \\\\&\\int _T\\operatorname{\\mathsf {grad}_{\\rm ss}}p_{T}^{k+1}\\underline{\\eta }_T=\\frac{1}{2} \\sum _{E\\in \\mathcal {E}_{T}}\\int _E (\\eta _E\\otimes n_{TE}-n_{TE}\\otimes \\eta _E)\\,,\\mbox{ and } \\\\&\\int _Tp_{T}^{k+1}\\underline{\\eta }_T=\\int _TP_{\\Theta ,T}^{k}\\underline{\\eta }_T\\quad \\mbox{ if $k\\ge 1$},\\qquad \\int _{\\partial T}p_{T}^{k+1}\\underline{\\eta }_T=\\sum _{E\\in \\mathcal {E}_{T}}\\int _E\\eta _E\\quad \\mbox{ if $k=0$}.$ In a similar way as for $\\mathsf {G}_T^k$ above, in (REF ) the term $P_{\\Theta ,T}^{k}\\underline{\\eta }_T$ can be replaced with $\\pi _{\\mathcal {P},T}^{k-1}(P_{\\Theta ,T}^{k}\\underline{\\eta }_T)$ (because $\\operatorname{\\bf div}(\\operatorname{\\mathsf {grad}_{\\rm s}}w)\\in \\mathcal {P}_{}^{k-1}(T)$ ).", "Hence, using (REF ) and the techniques of [19] we see that, for $k\\ge 1$ , $p_{T}^{k+1}(\\underline{I}_{\\Theta ,T}^k\\eta ) = \\pi _{\\mathsf {\\varepsilon },T}^{k+1}\\eta \\qquad \\forall \\eta \\in H^1(T)^2,$ where $\\pi _{\\mathsf {\\varepsilon },T}^{k+1}:H^1(T)^2\\rightarrow \\mathcal {P}_{}^{k+1}(T)$ is the strain projector of degree $k+1$ , see [19].", "If $k=0$ , the relation (REF ) is still verified with a modified version of the strain projector (still denoted by $\\pi _{\\mathsf {\\varepsilon },T}^{1}$ ), inspired by the modified elliptic projector of [19], whose closure equation involves the average over $\\partial T$ instead of the average over $T$ ; this modified strain projector has the same approximation properties as the standard strain projector.", "The local stabilisation is then defined by: $\\mathrm {s}_T(\\underline{\\tau }_T,\\underline{\\eta }_T)=\\sum _{E\\in \\mathcal {E}_{T}}h_T^{-1}\\int _E (\\delta _{TE}^k-\\delta _T^k)\\underline{\\tau }_T \\cdot (\\delta _{TE}^k-\\delta _T^k)\\underline{\\eta }_T\\quad \\forall \\underline{\\tau }_T,\\underline{\\eta }_T\\in \\underline{\\Theta }_T^k,$ where the difference operators are such that, for all $\\underline{\\eta }_T\\in \\underline{\\Theta }_T^k$ and $E\\in \\mathcal {E}_{T}$ , $\\delta ^k_T\\underline{\\eta }_TP_{\\Theta ,T}^{k}\\big [\\underline{I}_{\\Theta ,T}^k(p_{T}^{k+1}\\underline{\\eta }_T-P_{\\Theta ,T}^{k}\\underline{\\eta }_T)\\big ]\\,,\\quad \\delta ^k_{TE}\\underline{\\eta }_T\\pi _{\\mathcal {P},E}^{k}(p_{T}^{k+1}\\underline{\\eta }_T-\\eta _E).$ Observing that $P_{\\Theta ,T}^{k}\\underline{I}_{\\Theta ,T}^k:H^1(T)^2\\rightarrow \\mathcal {P}_{}^{k}(T)$ is a projector (see [20]) and using (REF ), it can be checked that $\\delta ^k_T(\\underline{I}_{\\Theta ,T}^k\\eta ) = 0$ and $\\delta ^k_{TE}(\\underline{I}_{\\Theta ,T}^k\\eta ) = 0$ for all $E\\in \\mathcal {E}_{T}$ , whenever $\\eta \\in \\mathcal {P}_{}^{k+1}(T)$ .", "As a consequence, we have the following polynomial consistency property for $\\mathrm {s}_T$ : $\\mathrm {s}_T(\\underline{I}_{\\Theta ,T}^k\\eta ,\\underline{\\xi }_T)=0\\qquad \\forall (\\eta ,\\underline{\\xi }_T)\\in \\mathcal {P}_{}^{k+1}(T)\\times \\underline{\\Theta }_T^k.$ Remark 1 (Original HHO stabilisation) In the original HHO stabilisation, the $L^2$ -projector $\\pi _{\\mathcal {P},T}^{k}$ is used instead of $P_{\\Theta ,T}^{k}\\underline{I}_{\\Theta ,T}^k$ in the expression of $\\delta ^k_T$ ; see (REF ).", "The reason for using $P_{\\Theta ,T}^{k}\\underline{I}_{\\Theta ,T}^k$ here lies in the need to satisfy, for the interpolator $\\underline{I}_{\\Theta ,T}^k$ on $\\underline{\\Theta }_T^k$ , the polynomial consistency (REF ).", "Note also that, in $\\mathrm {s}_T$ , the scaling factor $h_T^{-1}$ has been preferred over the original HHO scaling factor $h_E^{-1}$ , as it is proved in [23] to lead to a more robust discretisation in presence of small edges.", "Using the $L^2$ -boundedness of $P_{\\Theta ,T}^{k}\\underline{I}_{\\Theta ,T}^k$ (stemming from the two-dimensional versions of [20]), the commutation property (REF ), and the polynomial consistency (REF ), it is easy to reproduce, with our definitions of $\\mathsf {G}_{{\\rm s},T}^k$ , $p_{T}^{k+1}$ and $\\mathrm {s}_T$ , the standard HHO analysis of [19] and to obtain corresponding boundedness and consistency results (translated through $\\underline{\\mathfrak {I}}_{{\\rm HHO},T}^k$ )." ], [ "Global operators.", "Global symmetric gradient, divergence, and higher-order reconstruction operators are obtained setting, for all $\\underline{\\eta }_h\\in \\underline{\\Theta }_h^k$ , $\\text{$(\\mathsf {G}_{{\\rm s},h}^{k}\\underline{\\eta }_h)_{|T}\\mathsf {G}_{{\\rm s},T}^k\\underline{\\eta }_T$,$(D_h^k\\underline{\\eta }_h)_{|T}D_T^k\\underline{\\eta }_T$,and $(p_h^{k+1}\\underline{\\eta }_h)_{|T}=p_{T}^{k+1}\\underline{\\eta }_T$ for all $T\\in \\mathcal {T}_h$.", "}$ Likewise, denoting by $\\underline{\\Theta }_{{\\rm HHO},h}^k$ the global HHO space obtained patching together the local spaces (REF ) by enforcing the single-valuedness of the edge components, we define the global embedding $\\underline{\\mathfrak {I}}_{{\\rm HHO},h}^k:\\underline{\\Theta }_h^k\\rightarrow \\underline{\\Theta }_{{\\rm HHO},h}^k$ by setting, for all $\\underline{\\eta }_h \\in \\underline{\\Theta }_h^k$ , $(\\underline{\\mathfrak {I}}_{{\\rm HHO},h}^k\\underline{\\eta }_h)_{|T}\\underline{\\mathfrak {I}}_{{\\rm HHO},T}^k\\underline{\\eta }_T$ for all $T\\in \\mathcal {T}_h$ .", "We also let $\\mathrm {s}_h:\\underline{\\Theta }_h^k\\times \\underline{\\Theta }_h^k\\rightarrow \\mathbb {R}$ be the global stabilisation bilinear form such that $\\mathrm {s}_h(\\underline{\\tau }_h,\\underline{\\eta }_h)\\sum _{T\\in \\mathcal {T}_h}\\mathrm {s}_T(\\underline{\\tau }_T,\\underline{\\eta }_T)\\qquad \\forall (\\underline{\\tau }_h,\\underline{\\eta }_h)\\in \\underline{\\Theta }_h^k\\times \\underline{\\Theta }_h^k.$" ], [ "Discrete forms", "Based on the reconstructions introduced in the previous section, we define the discrete counterparts of the forms that appear in the weak formulation (REF ).", "Specifically, we let the bilinear form $\\mathrm {A}_h:\\left[\\underline{\\Theta }_h^k\\times \\underline{U}_h^k\\right]^2\\rightarrow \\mathbb {R}$ and the linear form $\\ell _h:\\underline{U}_h^k\\rightarrow \\mathbb {R}$ be such that, for all $(\\underline{\\tau }_h,\\underline{w}_h),(\\underline{\\eta }_h,\\underline{v}_h)\\in \\underline{\\Theta }_h^k\\times \\underline{U}_h^k$ , $\\mathrm {A}_h((\\underline{\\tau }_h,\\underline{w}_h),(\\underline{\\eta }_h,\\underline{v}_h))\\mathrm {a}_h(\\underline{\\tau }_h,\\underline{\\eta }_h)+ \\mathrm {b}_h((\\underline{\\tau }_h,\\underline{w}_h),(\\underline{\\eta }_h,\\underline{v}_h)),\\qquad \\ell _h(\\underline{v}_h)\\int _\\Omega f~P_{U,h}^{k+1}\\underline{v}_h,$ where the bilinear forms $\\mathrm {a}_h:\\underline{\\Theta }_h^k\\times \\underline{\\Theta }_h^k\\rightarrow \\mathbb {R}$ and $\\mathrm {b}_h:\\left[\\underline{\\Theta }_h^k\\times \\underline{U}_h^k\\right]^2\\rightarrow \\mathbb {R}$ are such that ${\\begin{array}{c}\\mathrm {a}_h(\\underline{\\tau }_h,\\underline{\\eta }_h)\\beta _0\\left(\\int _\\Omega \\mathsf {G}_{{\\rm s},h}^{k}\\underline{\\tau }_h:\\mathsf {G}_{{\\rm s},h}^{k}\\underline{\\eta }_h+ \\mathrm {s}_h(\\underline{\\tau }_h,\\underline{\\eta }_h)+ \\mathrm {j}_h(\\underline{\\tau }_h,\\underline{\\eta }_h)\\right)+ \\beta _1\\int _\\Omega D_h^k\\underline{\\tau }_h~D_h^k\\underline{\\eta }_h,\\\\\\mathrm {b}_h((\\underline{\\tau }_h,\\underline{w}_h),(\\underline{\\eta }_h,\\underline{v}_h))\\frac{\\kappa }{t^2}(\\underline{\\tau }_h - \\underline{G}_h^{k}\\underline{w}_h, \\underline{\\eta }_h - \\underline{G}_h^{k}\\underline{v}_h)_{\\Theta ,h}.\\end{array}}$ Here, $\\mathrm {j}_h$ is an additional stabilisation term appearing only in the case $k=0$ and which penalises the jumps of higher-order reconstructions between elements: $\\mathrm {j}_h(\\underline{\\tau }_h,\\underline{\\eta }_h){\\left\\lbrace \\begin{array}{ll}0 & \\mbox{ if $k\\ge 1$},\\\\\\displaystyle \\sum _{E\\in \\mathcal {E}_h}h_E^{-1}\\int _E [p_h^{1}\\underline{\\tau }_h]_E[p_h^{1}\\underline{\\eta }_h]_E & \\mbox{ if $k=0$},\\end{array}\\right.", "}$ where, for any internal edge $E\\in \\mathcal {E}_h^{{\\rm i}}$ , if $T_1,T_2$ are the two elements (in an arbitrary but fixed order) on each side of $E$ , we set $[p_h^{1}\\underline{\\tau }_h]_E(p_{T_1}^{1}\\underline{\\tau }_{T_1})_{|E}-(p_{T_2}^{1}\\underline{\\tau }_{T_2})_{|E}$ while, for any boundary edge $E\\in \\mathcal {E}_h^{{\\rm b}}\\cap \\mathcal {E}_{T}$ for $T\\in \\mathcal {T}_h$ , $[p_h^{1}\\underline{\\tau }_h]_E(p_{T}^{1}\\underline{\\tau }_T)_{|E}$ .", "We also introduced in (REF ) the DDR $L^2$ -product $(\\cdot ,\\cdot )_{\\Theta ,h}$ on $\\underline{\\Theta }_h^k$ assembled from the following local contributions: For all $\\underline{\\tau }_T,\\underline{\\eta }_T\\in \\underline{\\Theta }_T^k$ , ${\\begin{array}{c}(\\underline{\\tau }_T,\\underline{\\eta }_T)_{\\Theta ,T}\\int _TP_{\\Theta ,T}^{k}\\underline{\\tau }_T\\cdot P_{\\Theta ,T}^{k}\\underline{\\eta }_T + \\mathrm {S}_{\\Theta ,T}(\\underline{\\tau }_T,\\underline{\\eta }_T)\\\\\\text{with $\\mathrm {S}_{\\Theta ,T}(\\underline{\\tau }_T,\\underline{\\eta }_T)\\sum _{E\\in \\mathcal {E}_{T}} h_E\\int _E(P_{\\Theta ,T}^{k}\\underline{\\tau }_T-\\tau _E)\\cdot t_E~(P_{\\Theta ,T}^{k}\\underline{\\eta }_T-\\eta _E)\\cdot t_E$.", "}\\end{array}}$ Remark 2 (Normal components of edge polynomials) A simple inspection of (REF ), (REF ) and (REF ) shows that the normal components of edge unknowns do not enter the definition of $(\\cdot ,\\cdot )_{\\Theta ,h}$ ." ], [ "Discrete problem", "Define the following subspaces of $\\underline{\\Theta }_h^k$ and $\\underline{U}_h^k$ incorporating the clamped boundary condition: $\\underline{\\Theta }_{h,0}^k\\Big \\lbrace \\underline{\\eta }_h\\in \\underline{\\Theta }_h^k\\,:\\,\\text{$\\eta _E = 0$ for all $E\\in \\mathcal {E}_h^{{\\rm b}}$}\\Big \\rbrace ,\\qquad \\underline{U}_{h,0}^k\\Big \\lbrace \\underline{v}_h\\in \\underline{U}_h^k\\,:\\,\\text{$(v_{\\mathcal {E}_h})_{|\\partial \\Omega } = 0$}\\Big \\rbrace .$ The discrete problem reads: Find $(\\underline{\\theta }_h,\\underline{u}_h)\\in \\underline{\\Theta }_{h,0}^k\\times \\underline{U}_{h,0}^k$ such that $\\mathrm {A}_h((\\underline{\\theta }_h,\\underline{u}_h),(\\underline{\\eta }_h,\\underline{v}_h))= \\ell _h(\\underline{v}_h)\\qquad \\forall (\\underline{\\eta }_h,\\underline{v}_h)\\in \\underline{\\Theta }_{h,0}^k\\times \\underline{U}_{h,0}^k.$" ], [ "Analysis", "Let $\\mu \\min (\\kappa ,\\beta _0).$ Throughout the rest of the paper, we use $a\\lesssim b$ as a shorthand notation for the inequality $a\\le Cb$ with multiplicative constant $C$ that possibly depends on $\\Omega $ , the mesh regularity, and on the polynomial degree, but not on $\\beta _0$ , $\\beta _1$ , $\\kappa $ , $\\mu $ , $t$ , or $h$ and, for local inequalities, on the mesh element or edge." ], [ "Discrete norm and stability", "We define the discrete seminorm on $\\underline{\\Theta }_h^k\\times \\underline{U}_h^k$ such that, for all $(\\underline{\\eta }_h,\\underline{v}_h)\\in \\underline{\\Theta }_h^k\\times \\underline{U}_h^k$ , $\\Vert (\\underline{\\eta }_h,\\underline{v}_h)\\Vert _{\\Theta \\times U,h}\\bigg [\\beta _0\\left(\\Vert \\mathsf {G}_{{\\rm s},h}^{k}\\underline{\\eta }_h\\Vert _{L^2(\\Omega )^{2\\times 2}}^2+ |\\underline{\\eta }_h|_{{\\rm s,j},h}^2\\right)+ \\beta _1\\Vert D_h^k\\underline{\\eta }_h\\Vert _{L^2(\\Omega )}^2\\\\+ \\frac{\\kappa }{t^2}\\Vert \\underline{\\eta }_h-\\underline{G}_h^{k}\\underline{v}_h\\Vert _{\\Theta ,h}^2+ \\mu \\left(\\Vert \\underline{\\eta }_h\\Vert _{\\Theta ,h}^2+ \\Vert \\underline{G}_h^{k}\\underline{v}_h\\Vert _{\\Theta ,h}^2\\right)\\bigg ]^{\\frac{1}{2}},$ where $\\Vert {\\cdot }\\Vert _{\\Theta ,h}$ and $|{\\cdot }|_{{\\rm s,j},h}$ denote the seminorms respectively induced by $(\\cdot ,\\cdot )_{\\Theta ,h}$ and $\\mathrm {s}_h+\\mathrm {j}_h$ on $\\underline{\\Theta }_h^k$ .", "Using, respectively, the discrete Korn and Korn–Poincaré inequalities [19] (see also [19] in the case $k=0$ ) and the fact that $h_E\\le 1$ for the terms composing the norm in the left-hand side (see (REF ) for the corresponding local contribution), we readily obtain $\\Vert \\underline{\\eta }_h\\Vert _{\\Theta ,h}\\lesssim \\left(\\Vert \\mathsf {G}_{{\\rm s},h}^{k}\\underline{\\eta }_h\\Vert _{L^2(\\Omega )^{2\\times 2}}^2+ |\\underline{\\eta }_h|_{{\\rm s,j},h}^2\\right)^{\\frac{1}{2}}\\qquad \\forall \\underline{\\eta }_h\\in \\Theta _{h,0}^k.$ Together with the Poincaré inequality for $\\underline{G}_h^{k}$ in $\\underline{U}_{h,0}^k$ , whose proof can be obtained using arguments similar to [20] (leveraging the Poincaré inequality with zero boundary condition stated in [19]), (REF ) proves that the energy seminorm $\\Vert {\\cdot }\\Vert _{\\Theta \\times U,h}$ is actually a norm on $\\underline{\\Theta }_{h,0}^k\\times \\underline{U}_{h,0}^k$ .", "We can now establish the coercivity of $\\mathrm {A}_h$ with respect to this norm.", "Lemma 3 (Coercivity) For all $(\\underline{\\eta }_h,\\underline{v}_h)\\in \\underline{\\Theta }_{h,0}^k\\times \\underline{U}_{h,0}^k$ , it holds $\\Vert (\\underline{\\eta }_h,\\underline{v}_h)\\Vert _{\\Theta \\times U,h}^2\\lesssim \\mathrm {A}_h((\\underline{\\eta }_h,\\underline{v}_h),(\\underline{\\eta }_h,\\underline{v}_h)).$ By the definitions (REF ) of $\\mathrm {A}_h$ and (REF ) of $\\mathrm {a}_h$ and $\\mathrm {b}_h$ , we have $\\beta _0\\left(\\Vert \\mathsf {G}_{{\\rm s},h}^{k}\\underline{\\eta }_h\\Vert _{L^2(\\Omega )^{2\\times 2}}^2{+} |\\underline{\\eta }_h|_{{\\rm s,j},h}^2\\right)+ \\beta _1\\Vert D_h^k\\underline{\\eta }_h\\Vert _{L^2(\\Omega )}^2{+} \\frac{\\kappa }{t^2}\\Vert \\underline{\\eta }_h-\\underline{G}_h^{k}\\underline{v}_h\\Vert _{\\Theta ,h}^2= \\mathrm {A}_h((\\underline{\\tau }_h,\\underline{v}_h),(\\underline{\\tau }_h,\\underline{v}_h)).$ We next write, using a triangle inequality, $\\begin{aligned}\\Vert \\underline{G}_h^{k}\\underline{v}_h\\Vert _{\\Theta ,h}^2+\\Vert \\underline{\\eta }_h\\Vert _{\\Theta ,h}^2&\\le 2\\Vert \\underline{\\eta }_h - \\underline{G}_h^{k}\\underline{v}_h\\Vert _{\\Theta ,h}^2+ 3\\Vert \\underline{\\eta }_h\\Vert _{\\Theta ,h}^2\\\\&\\le 2\\kappa ^{-1}\\frac{\\kappa }{t^2}\\Vert \\underline{\\eta }_h - \\underline{G}_h^{k}\\underline{v}_h\\Vert _{\\Theta ,h}^2+ 3\\beta _0^{-1}\\beta _0\\left(\\Vert \\mathsf {G}_{{\\rm s},h}^{k}\\underline{\\eta }_h\\Vert _{L^2(\\Omega )^{2\\times 2}}^2+|\\underline{\\eta }_h|_{{\\rm s,j},h}^2\\right)\\\\&\\lesssim \\mu ^{-1}\\mathrm {A}_h((\\underline{\\tau }_h,\\underline{v}_h),(\\underline{\\tau }_h,\\underline{v}_h)),\\end{aligned}$ where we have used the fact that $t<1$ along with the discrete Korn inequality (REF ) to pass to the second line and (REF ) together with the definition (REF ) of $\\mu $ to conclude.", "The proof is completed by combining (REF ) and (REF ) with the definition of $\\Vert {\\cdot }\\Vert _{\\Theta \\times U,h}$ ." ], [ "Error estimates", "The regularity assumptions in the error estimates are expressed in terms of the broken Sobolev spaces $H^s(\\mathcal {T}_h)\\left\\lbrace v\\in L^2(\\Omega )\\,:\\,\\text{$v_{|T}\\in H^s(T)$ for all $T\\in \\mathcal {T}_h$}\\right\\rbrace .$ The first error estimate is for an arbitrary polynomial degree $k$ .", "Theorem 4 (Error estimate for arbitrary $k$ ) Denote by $(\\eta ,u)\\in \\Theta \\times U$ and $(\\underline{\\eta }_h,\\underline{u}_h)\\in \\underline{\\Theta }_{h,0}^k\\times \\underline{U}_{h,0}^k$ the solutions to problems (REF ) and (REF ), respectively.", "We assume the additional regularity $u\\in C^1(\\overline{\\Omega })\\cap H^{k+2}(\\mathcal {T}_h)$ for the displacement and $\\theta \\in H^1(\\Omega )^2\\cap H^{k+2}(\\mathcal {T}_h)^2$ for the rotation.", "Then, it holds $\\Vert (\\underline{\\theta }_h - \\underline{I}_{\\Theta ,h}^k\\theta , \\underline{u}_h - \\underline{I}_{U,h}^ku)\\Vert _{\\Theta \\times U,h}\\lesssim h^{k+1}\\left(\\beta _0^{-\\frac{1}{2}}(\\beta _0 + \\beta _1)|\\theta |_{H^{k+2}(\\mathcal {T}_h)^2}+\\mu ^{-\\frac{1}{2}}|\\gamma |_{H^{k+1}(\\mathcal {T}_h)^2}\\right).$ Remark 5 (Regularity of the shear strain $\\gamma $ ) Under the regularity assumptions on $u$ and $\\theta $ in the theorem, the shear strain defined by () satisfies $\\gamma \\in H^1(\\Omega )^2\\cap H^{k+1}(\\mathcal {T}_h)^2$ .", "See Section REF .", "The bound (REF ) shows that the DDR scheme achieves as expected a high-order of accuracy, when the solution is smooth enough and $t$ is not too small.", "When $t\\rightarrow 0$ the higher derivatives of the shear strain $\\gamma $ are known to explode; typically, $|\\gamma |_{H^{k+1}(\\mathcal {T}_h)^2}$ grows as $t^{-k-1}$ , as explained in [5].", "Thus, even though $t$ does not explicitly appear in the right-hand side of (REF ), the dependency of this right-hand side on higher derivatives of the solution means that this estimate is not locking-free.", "Such a dependency is unavoidable for high-order schemes (see, e.g., [4] in the case of continuous/discontinuous Galerkin schemes).", "However, for $k=0$ , one could expect a better estimate than (REF ) in which $|\\gamma |_{H^1(\\mathcal {T}_h)}$ is multiplied by $t$ as in [4], [17], [28], [11]; this ensures that the method is locking-free at least if $\\Omega $ is convex since, on such domains, $t|\\gamma |_{H^1(\\Omega )}$ remains bounded as $t\\rightarrow 0$ .", "Such an error estimate is stated in the next theorem.", "Note that, contrary to most analyses in the aforementioned references and others (a notable exception being [11]), the proof of the following estimate does not use a Helmoltz decomposition of the shear strain.", "Theorem 6 (Locking-free error estimate for $k=0$ ) Assume the hypotheses of Theorem REF , and that $k=0$ .", "Then, it holds $\\Vert (\\underline{\\theta }_h - \\underline{I}_{\\Theta ,h}^0\\theta , \\underline{u}_h - \\underline{I}_{U,h}^0u)\\Vert _{\\Theta \\times U,h}\\\\\\lesssim h\\left(\\beta _0^{-\\frac{1}{2}}(\\beta _0 + \\beta _1)|\\theta |_{H^{2}(\\mathcal {T}_h)^2}+\\kappa ^{-\\frac{1}{2}}t|\\gamma |_{H^1(\\mathcal {T}_h)^2}+\\beta _0^{-\\frac{1}{2}}\\Vert \\gamma \\Vert _{L^2(\\Omega )^2}+\\mu ^{-\\frac{1}{2}}\\Vert f\\Vert _{L^2(\\Omega )}\\right).$ See Section REF .", "Remark 7 (Locking-free property) If $\\Omega $ is convex, all terms in the right-hand side of (REF ) are bounded independently of $t$ [5].", "The techniques used to prove (REF ) can be extended (at the price of some technicalities) to arbitrary values of $k$ to replace, in the right-hand side of (REF ), the term $|\\gamma |_{H^{k+1}(\\mathcal {T}_h)^2}$ with $t|\\gamma |_{H^{k+1}(\\mathcal {T}_h)^2} + |\\gamma |_{H^k(\\mathcal {T}_h)^2} + |f|_{H^k(\\mathcal {T}_h)}$ in the spirit of [4].", "However, since a bound independent of $t$ for the quantity $t|\\gamma |_{H^{k+1}(\\mathcal {T}_h)^2} + |\\gamma |_{H^k(\\mathcal {T}_h)^2} + |f|_{H^k(\\mathcal {T}_h)}$ can only be established for $k=0$ , this would not yield complete robustness of the estimate (REF ) for $k\\ge 1$ .", "For this reason, and also to make the exposition less technical, we have decided to state two separate estimates." ], [ "Proof of the arbitrary-order error estimate", "[Proof of Theorem REF ] 1.", "Basic error estimate.", "Combining the coercivity (REF ) of $\\mathrm {A}_h$ with the Third Strang Lemma [18], we obtain the following basic error estimate: $\\Vert (\\underline{\\theta }_h - \\underline{I}_{\\Theta ,h}^k\\theta , \\underline{u}_h - \\underline{I}_{U,h}^ku)\\Vert _{\\Theta \\times U,h}\\lesssim \\sup _{(\\underline{\\eta }_h,\\underline{v}_h)\\in \\underline{\\Theta }_{h,0}^k\\times \\underline{U}_{h,0}^k\\setminus \\lbrace (\\underline{0},\\underline{0})\\rbrace }\\frac{\\mathcal {E}_h((\\theta ,u);(\\underline{\\eta }_h,\\underline{v}_h))}{\\Vert (\\underline{\\eta }_h,\\underline{v}_h)\\Vert _{\\Theta \\times U,h}},$ where the consistency error linear form $\\mathcal {E}_h((\\theta ,u);\\cdot )$ is such that, for all $(\\underline{\\eta }_h,\\underline{v}_h)\\in \\underline{\\Theta }_{h,0}^k\\times \\underline{U}_{h,0}^k$ , $\\mathcal {E}_h((\\theta ,u);(\\underline{\\eta }_h,\\underline{v}_h))\\ell _h(\\underline{v}_h) - \\mathrm {A}_h((\\underline{I}_{\\Theta ,h}^k\\theta ,\\underline{I}_{U,h}^ku),(\\underline{\\eta }_h,\\underline{v}_h)).$ 2.", "Reformulation of the consistency error.", "To prove (REF ), we need to estimate the dual norm of the consistency error, which corresponds to the right-hand side of (REF ).", "We first recast $\\mathrm {b}_h$ .", "Recall the definition (REF ) of the modified interpolator $\\underline{I}_{\\Theta ,h}^{\\flat ,k}$ and notice that, by Remark REF , it holds $(\\underline{I}_{\\Theta ,h}^k\\eta ,\\underline{\\tau }_h)_{\\Theta ,h}= (\\underline{I}_{\\Theta ,h}^{\\flat ,k}\\eta ,\\underline{\\tau }_h)_{\\Theta ,h}\\qquad \\forall (\\eta ,\\underline{\\tau }_h)\\in H^1(\\Omega )^2\\times \\underline{\\Theta }_{h}^k.$ We can then write $\\mathrm {b}_h((\\underline{I}_{\\Theta ,h}^k\\theta ,\\underline{I}_{U,h}^ku),(\\underline{\\eta }_h,\\underline{v}_h))&= \\frac{\\kappa }{t^2}(\\underline{I}_{\\Theta ,h}^{\\flat ,k}\\theta - \\underline{G}_h^{k}(\\underline{I}_{U,h}^ku), \\underline{\\eta }_h - \\underline{G}_h^{k}\\underline{v}_h)_{\\Theta ,h}\\\\&= \\frac{\\kappa }{t^2}(\\underline{I}_{\\Theta ,h}^{\\flat ,k}(\\theta - \\operatorname{\\bf grad}u), \\underline{\\eta }_h - \\underline{G}_h^{k}\\underline{v}_h)_{\\Theta ,h}\\\\&= (\\underline{I}_{\\Theta ,h}^k\\gamma , \\underline{G}_h^{k}\\underline{v}_h - \\underline{\\eta }_h)_{\\Theta ,h},$ where we have used the definition (REF ) of $\\mathrm {b}_h$ along with (REF ) in the first line, the key commutation property (REF ) to pass to the second line, and the definition () of the shear strain $\\gamma $ followed by (REF ) to conclude.", "Expanding the inner product $(\\cdot ,\\cdot )_{\\Theta ,h}$ according to its definition from the local products (REF ) and using the relation $P_{\\Theta ,T}^{k}\\underline{G}_T^{k}=G_T^{k}$ (see [20]), we infer $\\mathrm {b}_h((\\underline{I}_{\\Theta ,h}^k\\theta ,\\underline{I}_{U,h}^ku),(\\underline{\\eta }_h,\\underline{v}_h))\\\\= \\sum _{T\\in \\mathcal {T}_h} \\int _T\\gamma \\cdot (G_T^{k}\\underline{v}_T - P_{\\Theta ,T}^{k}\\underline{\\eta }_T) -\\mathfrak {T}$ with $\\mathfrak {T}\\sum _{T\\in \\mathcal {T}_h} \\int _T[\\gamma -P_{\\Theta ,T}^{k}(\\underline{I}_{\\Theta ,T}^k\\gamma )]\\cdot P_{\\Theta ,T}^{k}(\\underline{G}_T^{k}\\underline{v}_T - \\underline{\\eta }_T)+ \\sum _{T\\in \\mathcal {T}_h}\\mathrm {S}_{\\Theta ,T}(\\underline{I}_{\\Theta ,T}^k\\gamma ,\\underline{G}_T^{k}\\underline{v}_T - \\underline{\\eta }_T).$ Accounting for the definition of the material tensor $\\mathsf {C}$ , we also have, for all $\\underline{\\tau }_h,\\underline{\\eta }_h\\in \\underline{\\Theta }_h^k$ , $\\mathrm {a}_h(\\underline{\\tau }_h,\\underline{\\eta }_h)= \\int _\\Omega \\mathsf {C}\\mathsf {G}_{{\\rm s},h}^{k}\\underline{\\tau }_h:\\mathsf {G}_{{\\rm s},h}^{k}\\underline{\\eta }_h+ \\beta _0\\mathrm {s}_h(\\underline{\\tau }_h,\\underline{\\eta }_h)+ \\beta _0\\mathrm {j}_h(\\underline{\\tau }_h,\\underline{\\eta }_h).$ Recalling the definitions (REF ) of $\\mathrm {A}_h$ and $\\ell _h$ , the relations $\\gamma =-\\operatorname{\\bf div}(\\mathsf {C}\\operatorname{\\mathsf {grad}_{\\rm s}}\\theta )$ and $f=-\\operatorname{div}\\gamma $ (see (REF ) and ()) along with (REF ) and (REF ) shows that the consistency error (REF ) can be recast as $\\mathcal {E}_h((\\theta ,u);(\\underline{\\eta }_h,\\underline{v}_h))=\\mathcal {E}_{\\operatorname{\\bf grad},h}(\\gamma ;\\underline{v}_h)+\\mathfrak {T}+ \\mathcal {E}_{\\operatorname{\\mathsf {grad}_{\\rm s}},h}(\\mathsf {C}\\operatorname{\\mathsf {grad}_{\\rm s}}\\theta ;\\underline{\\eta }_h),$ where the adjoint consistency errors for the gradient on $\\underline{U}_h^k$ and for the symmetric gradient on $\\Theta _h^k$ are defined as $\\mathcal {E}_{\\operatorname{\\bf grad},h}(\\gamma ;\\underline{v}_h){}& -\\int _\\Omega \\operatorname{div}\\gamma ~P_{U,h}^{k+1}\\underline{v}_h- \\sum _{T\\in \\mathcal {T}_h}\\int _T\\gamma \\cdot G_T^{k}\\underline{v}_T,\\\\\\mathcal {E}_{\\operatorname{\\mathsf {grad}_{\\rm s}},h}(\\mathsf {C}\\operatorname{\\mathsf {grad}_{\\rm s}}\\theta ;\\underline{\\eta }_h){}&- \\sum _{T\\in \\mathcal {T}_h}\\int _T\\operatorname{\\bf div}(\\mathsf {C}\\operatorname{\\mathsf {grad}_{\\rm s}}\\theta )\\cdot P_{\\Theta ,T}^{k}\\underline{\\eta }_T- \\int _\\Omega \\mathsf {C}\\mathsf {G}_{{\\rm s},h}^{k}(\\underline{I}_{\\Theta ,h}^k\\theta ):\\mathsf {G}_{{\\rm s},h}^{k}\\underline{\\eta }_h\\\\&- \\beta _0\\mathrm {s}_h(\\underline{I}_{\\Theta ,h}^k\\theta ,\\underline{\\eta }_h)- \\beta _0\\mathrm {j}_h(\\underline{I}_{\\Theta ,h}^k\\theta ,\\underline{\\eta }_h).$ 3.", "Bound on the consistency error.", "To deal with $\\mathcal {E}_{\\operatorname{\\mathsf {grad}_{\\rm s}},h}$ , we use the estimate in [19] (and [19] if $k=0$ ) which, in the present context, yields $|\\mathcal {E}_{\\operatorname{\\mathsf {grad}_{\\rm s}},h}(\\mathsf {C}\\operatorname{\\mathsf {grad}_{\\rm s}}\\theta ;\\underline{\\eta }_h)|&\\lesssim h^{k+1}(\\beta _0 + \\beta _1)|\\theta |_{H^{k+2}(\\mathcal {T}_h)^2}\\left(\\Vert \\mathsf {G}_{{\\rm s},h}^{k}\\underline{\\eta }_h\\Vert _{L^2(\\Omega )^{2\\times 2}}^2+ |\\underline{\\eta }_h|_{{\\rm s,j},h}^2\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\lesssim h^{k+1}\\beta _0^{-\\frac{1}{2}}(\\beta _0 + \\beta _1)|\\theta |_{H^{k+2}(\\mathcal {T}_h)^2}\\Vert (\\underline{\\eta }_h,\\underline{v}_h)\\Vert _{\\Theta \\times U,h},$ where the conclusion follows from the definition (REF ) of the discrete norm.", "The term $\\mathfrak {T}$ is estimated using Cauchy–Schwarz inequalities: $|\\mathfrak {T}|\\le {}&\\sum _{T\\in \\mathcal {T}_h}\\Vert \\gamma -P_{\\Theta ,T}^{k}(\\underline{I}_{\\Theta ,T}^k\\gamma )\\Vert _{L^2(T)^2}~\\Vert P_{\\Theta ,T}^{k}(\\underline{G}_T^{k}\\underline{v}_T-\\underline{\\eta }_T)\\Vert _{L^2(T)^2} \\\\&+ \\sum _{T\\in \\mathcal {T}_h}\\mathrm {S}_{\\Theta ,T}(\\underline{I}_{\\Theta ,T}^k\\gamma ,\\underline{I}_{\\Theta ,T}^k\\gamma )^{\\frac{1}{2}}~\\mathrm {S}_{\\Theta ,T}(\\underline{G}_T^{k}\\underline{v}_T-\\underline{\\eta }_T,\\underline{G}_T^{k}\\underline{v}_T-\\underline{\\eta }_T)^{\\frac{1}{2}}\\\\\\lesssim {}& \\sum _{T\\in \\mathcal {T}_h} h^{k+1}|\\gamma |_{H^{k+1}(T)^2}\\left(\\Vert P_{\\Theta ,T}^{k}(\\underline{G}_T^{k}\\underline{v}_T-\\underline{\\eta }_T)\\Vert _{L^2(T)^2}+\\mathrm {S}_{\\Theta ,T}(\\underline{G}_T^{k}\\underline{v}_T-\\underline{\\eta }_T,\\underline{G}_T^{k}\\underline{v}_T-\\underline{\\eta }_T)^{\\frac{1}{2}}\\right)\\\\\\lesssim {}& h^{k+1}|\\gamma |_{H^{k+1}(\\mathcal {T}_h)^2}\\Vert \\underline{G}_h^{k}\\underline{v}_h-\\underline{\\eta }_h\\Vert _{\\Theta ,h},$ where we have used, in the second line, the consistency properties of $P_{\\Theta ,T}^{k}\\underline{I}_{\\Theta ,T}^k$ and $\\mathrm {S}_{\\Theta ,T}$ (two-dimensional versions of [20], see Remark REF below), and the conclusion follows from Cauchy–Schwarz inequalities on the sum and the definition of the norm $\\Vert {\\cdot }\\Vert _{\\Theta ,h}$ .", "Using the definition (REF ) of the discrete norm, we infer $|\\mathfrak {T}|\\lesssim h^{k+1} \\kappa ^{-\\frac{1}{2}} t|\\gamma |_{H^{k+1}(\\mathcal {T}_h)^2}\\Vert (\\underline{\\eta }_h,\\underline{v}_h)\\Vert _{\\Theta \\times U,h}.$ The estimate of $\\mathcal {E}_{\\operatorname{\\bf grad},h}$ follows proceeding as in the proof of [20], with straightforward modifications to account for the different boundary conditions, fewer (and simpler) terms to track, and accounting for Remark REF : $|\\mathcal {E}_{\\operatorname{\\bf grad},h}(\\gamma ;\\underline{v}_h)|\\lesssim h^{k+1}|\\gamma |_{H^{k+1}(\\mathcal {T}_h)^2}\\Vert \\underline{G}_h^{k}\\underline{v}_h\\Vert _{\\Theta ,h}\\le h^{k+1}\\mu ^{-\\frac{1}{2}}|\\gamma |_{H^{k+1}(\\mathcal {T}_h)^2}\\Vert (\\underline{\\eta }_h,\\underline{v}_h)\\Vert _{\\Theta \\times U,h}.$ 4.", "Conclusion.", "Plugging the estimates (REF ), (REF ) and (REF ) into (REF ), we arrive at $|\\mathcal {E}_h((\\theta ,u);(\\underline{\\eta }_h,\\underline{v}_h))|\\\\\\lesssim h^{k+1}\\left(\\beta _0^{-\\frac{1}{2}}(\\beta _0 + \\beta _1)|\\theta |_{H^{k+2}(\\mathcal {T}_h)^2}+ \\kappa ^{-\\frac{1}{2}} t |\\gamma |_{H^{k+1}(\\mathcal {T}_h)^2}+ \\mu ^{-\\frac{1}{2}}|\\gamma |_{H^{k+1}(\\mathcal {T}_h)^2}\\right)\\Vert (\\underline{\\eta }_h,\\underline{v}_h)\\Vert _{\\Theta \\times U,h}.$ The estimate (REF ) follows using this bound in (REF ) and recalling that $\\mu \\le \\kappa $ and $t\\le 1$ .", "Remark 8 (Norms of $\\gamma $ in the estimates) In [20], the estimates mentioned above (Theorem 39 and Section 6.1) involve, in the case $k=0$ , weighted $H^2$ -seminorms of $\\gamma $ .", "This is because these estimates are stated in three dimensions, in which interpolating a function on $\\underline{\\Theta }_h^k$ requires a higher minimal regularity (to ensure traces along the edges are well-defined).", "In two dimensions, the local interpolator obtained restricting (REF ) to $T$ is well-defined on $H^1(T)^2$ , and the seminorm in this space is sufficient to state the consistency estimates for $k=0$ ." ], [ "Proof of the low-order locking-free error estimate", "The proof of Theorem REF relies on liftings of elements in $\\underline{U}_T^0$ and $\\underline{\\Theta }_T^0$ , for each $T\\in \\mathcal {T}_h$ .", "The assumption on the mesh yields a conforming simplicial subdivision $\\mathcal {S}_T$ of $T$ that is shape regular (with the same regularity parameter as in the mesh regularity assumption); actually, by [19] each $T\\in \\mathcal {T}_h$ is star-shaped with respect to every point in a ball of radius $\\varrho h_T$ , where $\\varrho $ is the mesh regularity parameter, so $\\mathcal {S}_T$ can be constructed by adding only one vertex (the center of that ball) in $T$ and creating the triangles between this vertex and the edges of $T$ .", "The proof given here, however, applies also to elements that are possibly not star-shaped.", "The coordinate $x_V$ of any vertex $V$ of $\\mathcal {S}_T$ can be written as a convex combination of the coordinates of the vertices $\\mathcal {V}_{T}$ of $T$ : $x_V=\\sum _{W\\in \\mathcal {V}_{T}}\\lambda _{V,W}x_{W}\\,,\\quad \\mbox{ with $\\lambda _{V,W}\\ge 0$ and }\\sum _{W\\in \\mathcal {V}_{T}}\\lambda _{V,W}=1$ (this includes the vertices $V\\in \\mathcal {V}_{T}$ , in which case we choose $\\lambda _{V,V}=1$ and $\\lambda _{V,W}=0$ if $W\\ne V$ ).", "Denoting by $\\mathcal {P}_{\\rm c}^{1}(\\mathcal {S}_T)$ the space of $H^1(T)$ -conforming piecewise $\\mathcal {P}_{}^{1}$ functions on $\\mathcal {S}_T$ , for all $\\underline{z}_T=z_{\\mathcal {E}_{T}}\\in \\underline{U}_T^0$ we define $\\widetilde{z}_T\\in \\mathcal {P}_{\\rm c}^{1}(\\mathcal {S}_T)$ such that $\\widetilde{z}_T(x_V)=\\sum _{W\\in \\mathcal {V}_{T}}\\lambda _{V,W}\\,z_{\\mathcal {E}_{T}}(x_{W}).$ This construction is linearly exact, that is $\\widetilde{\\underline{I}_{U,T}^0\\phi _T}=\\phi _T\\qquad \\forall \\phi _T\\in \\mathcal {P}_{}^{1}(T).$ The next lemma, whose proof is postponed to the end of the section, states useful properties of the lifting $\\underline{U}_T^0\\ni \\underline{z}_T\\mapsto \\widetilde{z}_T\\in \\mathcal {P}_{\\rm c}^{1}(\\mathcal {S}_T)$ .", "Lemma 9 (Properties of the lifting on $\\underline{U}_T^0$ ) The following properties hold: For all $\\underline{z}_T\\in \\underline{U}_T^0$ , $\\pi _{\\mathcal {P},T}^{0}(\\operatorname{\\bf grad}\\widetilde{z}_T)={}&G_T^{0}\\underline{z}_T,\\\\\\Vert \\operatorname{\\bf grad}\\widetilde{z}_T\\Vert _{L^2(T)^2}\\lesssim {}& \\Vert \\underline{G}_T^{0}\\underline{z}_T\\Vert _{\\Theta ,T},\\\\\\Vert \\widetilde{z}_T-P_{U,T}^{1}\\underline{z}_T\\Vert _{L^2(T)}\\lesssim {}& h_T\\Vert \\underline{G}_T^{0}\\underline{z}_T\\Vert _{\\Theta ,T},$ where $\\Vert {\\cdot }\\Vert _{\\Theta ,T}$ is the local seminorm induced by the product (REF ) on $\\underline{\\Theta }_T^0$ .", "Moreover, if $\\underline{z}_h\\in \\underline{U}_{h,0}^0$ and $\\widetilde{z}_h$ is defined such that $(\\widetilde{z}_h)_{|T}=\\widetilde{z}_T$ for all $T\\in \\mathcal {T}_h$ , then $\\widetilde{z}_h\\in H^1_0(\\Omega )$ .", "We now define the lifting on $\\underline{\\Theta }_T^0$ .", "For any $\\underline{\\eta }_T=(\\eta _E)_{E\\in \\mathcal {E}_{T}}\\in \\underline{\\Theta }_T^0$ , let $\\underline{\\eta }_T^\\flat =((\\eta _E\\cdot t_E)t_E)_{E\\in \\mathcal {E}_{T}}$ be the vector comprising only the tangential components to the edges.", "Since $(\\cdot ,\\cdot )_{\\Theta ,T}$ is an inner product when only these components are considered, we can write a unique decomposition $\\underline{\\eta }_T^\\flat =\\underline{G}_T^{0}\\underline{w}_T + \\underline{\\kappa }_T\\mbox{ with $\\underline{w}_T\\in \\underline{U}_T^0$ and $\\underline{\\kappa }_T\\perp \\underline{G}_T^{0}\\underline{U}_T^0$},$ the orthogonality being understood for $(\\cdot ,\\cdot )_{\\Theta ,T}$ .", "We then set $\\widetilde{\\eta }_T\\operatorname{\\bf grad}\\widetilde{w}_T + P_{\\Theta ,T}^{0}\\underline{\\kappa }_T.$ The proof of the properties of the lifting $\\underline{\\Theta }_T^0\\ni \\underline{\\eta }_T\\mapsto \\widetilde{\\eta }_T\\in L^2(T)^2$ stated in the following lemma is postponed to the end of the section.", "Lemma 10 (Properties of the lifting on $\\underline{\\Theta }_T^0$ ) The following properties hold: For all $\\underline{\\eta }_T\\in \\underline{\\Theta }_T^0$ , $\\pi _{\\mathcal {P},T}^{0}\\widetilde{\\eta }_T={}&P_{\\Theta ,T}^{0}\\underline{\\eta }_T,\\\\\\Vert \\operatorname{\\bf grad}\\widetilde{v}_T-\\widetilde{\\eta }_T\\Vert _{L^2(T)^2}\\lesssim {}&\\Vert \\underline{G}_T^{0}\\underline{v}_T-\\underline{\\eta }_T\\Vert _{\\Theta ,T}\\qquad \\forall \\underline{v}_T\\in \\underline{U}_T^0.$ Moreover, for all $\\underline{\\eta }_h\\in \\underline{\\Theta }_{h,0}^0$ , $\\left(\\sum _{T\\in \\mathcal {T}_h}\\Vert \\widetilde{\\eta }_T-P_{\\Theta ,T}^{0}\\underline{\\eta }_T\\Vert _{L^2(T)^2}^2\\right)^{\\frac{1}{2}}\\lesssim h\\left(\\Vert \\mathsf {G}_{{\\rm s},h}^{0}\\underline{\\eta }_h\\Vert _{L^2(\\Omega )^{2\\times 2}}^2+|\\underline{\\eta }_h|_{{\\rm s,j},h}^2\\right)^{\\frac{1}{2}}.$ We are now ready to prove Theorem REF .", "[Proof of Theorem REF ] Given the basic error estimate (REF ), we only have to find a proper upper bound of the consistency error.", "We consider the first term in the expression (REF ) of $\\mathrm {b}_h((\\underline{I}_{\\Theta ,h}^k\\theta ,\\underline{I}_{U,h}^ku),(\\underline{\\eta }_h,\\underline{v}_h))$ .", "Owing to (REF ) and (REF ) we have $\\int _T\\gamma \\cdot (G_T^{0}\\underline{v}_T - P_{\\Theta ,T}^{0}\\underline{\\eta }_T)={}&\\int _T \\gamma \\cdot \\pi _{\\mathcal {P},T}^{0}(\\operatorname{\\bf grad}\\widetilde{v}_T-\\widetilde{\\eta }_T)=\\int _T \\pi _{\\mathcal {P},T}^{0}\\gamma \\cdot (\\operatorname{\\bf grad}\\widetilde{v}_T-\\widetilde{\\eta }_T)\\\\={}&\\int _T \\gamma \\cdot (\\operatorname{\\bf grad}\\widetilde{v}_T-\\widetilde{\\eta }_T)+\\underbrace{\\int _T (\\pi _{\\mathcal {P},T}^{0}\\gamma -\\gamma )\\cdot (\\operatorname{\\bf grad}\\widetilde{v}_T-\\widetilde{\\eta }_T)}_{\\mathfrak {T}_{T,a}}\\\\={}&\\int _T\\gamma \\cdot \\operatorname{\\bf grad}\\widetilde{v}_T-\\int _T \\gamma \\cdot P_{\\Theta ,T}^{0}\\underline{\\eta }_T+\\underbrace{\\int _T \\gamma \\cdot (P_{\\Theta ,T}^{0}\\underline{\\eta }_T-\\widetilde{\\eta }_T)}_{\\mathfrak {T}_{T,b}}+\\mathfrak {T}_{T,a}.$ Summing over $T\\in \\mathcal {T}_h$ , using the fact that $\\widetilde{v}_h\\in H^1_0(\\Omega )$ (see Lemma REF ) to perform an integration by parts, recalling that $f=-\\operatorname{div}\\gamma $ by (), and setting $\\mathfrak {T}_{\\star }\\sum _{T\\in \\mathcal {T}_h}\\mathfrak {T}_{T,\\star }$ for $\\star \\in \\lbrace a,b\\rbrace $ , we infer $\\sum _{T\\in \\mathcal {T}_h}\\int _T\\gamma \\cdot (G_T^{0}\\underline{v}_T - P_{\\Theta ,T}^{0}\\underline{\\eta }_T)&=\\int _\\Omega f~\\widetilde{v}_h-\\sum _{T\\in \\mathcal {T}_h}\\int _T \\gamma \\cdot P_{\\Theta ,T}^{0}\\underline{\\eta }_T+\\mathfrak {T}_a+\\mathfrak {T}_b\\\\&=\\int _\\Omega f~P_{U,h}^{1}\\underline{v}_h-\\sum _{T\\in \\mathcal {T}_h}\\int _T \\gamma \\cdot P_{\\Theta ,T}^{0}\\underline{\\eta }_T+\\underbrace{\\int _\\Omega f(\\widetilde{v}_h-P_{U,h}^{1}\\underline{v}_h)}_{\\mathfrak {T}_c}+\\mathfrak {T}_a+\\mathfrak {T}_b.$ We plug this relation into the expression (REF ) of $\\mathrm {b}_h((\\underline{I}_{\\Theta ,h}^k\\theta ,\\underline{I}_{U,h}^ku),(\\underline{\\eta }_h,\\underline{v}_h))$ and recall (REF ) and the definitions (REF ) of $\\mathrm {A}_h$ and $\\ell _h$ to re-write consistency error (REF ) as $\\mathcal {E}_h((\\theta ,u);(\\underline{\\eta }_h,\\underline{v}_h))= \\mathcal {E}_{\\operatorname{\\mathsf {grad}_{\\rm s}},h}(\\mathsf {C}\\operatorname{\\mathsf {grad}_{\\rm s}}\\theta ;\\underline{\\eta }_h)-\\mathfrak {T}_a-\\mathfrak {T}_b-\\mathfrak {T}_c+\\mathfrak {T}.$ We now estimate $\\mathfrak {T}_a$ , $\\mathfrak {T}_b$ and $\\mathfrak {T}_c$ .", "Using the approximation properties of $\\pi _{\\mathcal {P},T}^{0}$ together with Cauchy–Schwarz inequalities and (), we have $|\\mathfrak {T}_a|\\lesssim h|\\gamma |_{H^1(\\mathcal {T}_h)^2}\\Vert \\underline{G}_h^{k}\\underline{v}_h-\\underline{\\eta }_h\\Vert _{\\Theta ,h}\\lesssim h|\\gamma |_{H^1(\\mathcal {T}_h)^2}\\kappa ^{-\\frac{1}{2}}t\\Vert (\\underline{\\eta }_h,\\underline{v}_h)\\Vert _{\\Theta \\times U,h},$ where we have used the definition (REF ) of $\\Vert {\\cdot }\\Vert _{\\Theta \\times U,h}$ to conclude.", "For $\\mathfrak {T}_b$ , we use again Cauchy–Schwarz inequalities and the estimate (REF ), together with the definition of the norm on $\\underline{\\Theta }_{h,0}^0\\times \\underline{U}_{h,0}^0$ , to write $|\\mathfrak {T}_b|\\lesssim \\Vert \\gamma \\Vert _{L^2(\\Omega )^2} h \\beta _0^{-\\frac{1}{2}}\\Vert (\\underline{\\eta }_h,\\underline{v}_h)\\Vert _{\\Theta \\times U,h}.$ Finally, for $\\mathfrak {T}_c$ , Cauchy–Schwarz inequalities followed by the estimate () yield $|\\mathfrak {T}_c|\\lesssim \\Vert f\\Vert _{L^2(\\Omega )}h\\Vert \\underline{G}_h^{0}\\underline{v}_h\\Vert _{\\Theta ,h}\\lesssim \\Vert f\\Vert _{L^2(\\Omega )}h\\mu ^{-\\frac{1}{2}}\\Vert (\\underline{\\eta }_h,\\underline{v}_h)\\Vert _{\\Theta \\times U,h}.$ Plugging (REF )–(REF ) and the estimates (REF ) on $\\mathfrak {T}$ and (REF ) on $\\mathcal {E}_{\\operatorname{\\mathsf {grad}_{\\rm s}},h}(\\mathsf {C}\\operatorname{\\mathsf {grad}_{\\rm s}}\\theta ;\\underline{\\eta }_h)$ into (REF ), we infer $|\\mathcal {E}_h((\\theta ,u);(\\underline{\\eta }_h,\\underline{v}_h))|\\\\\\lesssim h\\left(\\kappa ^{-\\frac{1}{2}}t|\\gamma |_{H^1(\\mathcal {T}_h)^2}+\\beta _0^{-\\frac{1}{2}}\\Vert \\gamma \\Vert _{L^2(\\Omega )^2}+\\mu ^{-\\frac{1}{2}}\\Vert f\\Vert _{L^2(\\Omega )}+\\beta _0^{-\\frac{1}{2}}(\\beta _0 + \\beta _1)|\\theta |_{H^{2}(\\mathcal {T}_h)^2}\\right)\\Vert (\\underline{\\eta }_h,\\underline{v}_h)\\Vert _{\\Theta \\times U,h}.$ Plugging this estimate into (REF ) concludes the proof.", "To conclude this section, we provide the proofs of the properties of the liftings.", "[Proof of Lemma REF ] 1.", "Proof of (REF ).", "Let $\\underline{z}_T\\in \\underline{U}_T^0$ .", "For all $\\eta \\in \\mathcal {P}_{}^{0}(T)$ , an integration by parts yields $\\int _T \\operatorname{\\bf grad}\\widetilde{z}_T\\cdot \\eta =\\sum _{E\\in \\mathcal {E}_{T}}\\int _E \\widetilde{z}_T (\\eta \\cdot n_{TE})=\\sum _{E\\in \\mathcal {E}_{T}}\\int _E z_{\\mathcal {E}_{T}}(\\eta \\cdot n_{TE})=\\int _TG_T^{0}\\underline{z}_T\\cdot \\eta ,$ where the second equality comes from the definition of $\\widetilde{z}_T$ which ensures that $(\\widetilde{z}_T)_{|E}=(z_{\\mathcal {E}_{T}})_{|E}$ for all $E\\in \\mathcal {E}_{T}$ (both functions are linear on $E$ and match at the edge's vertices), and the last equality is obtained applying the definition (REF ) of $G_T^{0}$ .", "This proves (REF ).", "2.", "Proof of ().", "For any two vertices $V,V^\\star $ of $\\mathcal {S}_T$ we have by construction $\\widetilde{z}_T(x_V)-\\widetilde{z}_T(x_{V^\\star })=\\sum _{W,Z\\in \\mathcal {V}_{T}}\\lambda _{V,W}\\lambda _{V^\\star ,Z}\\left(z_{\\mathcal {E}_{T}}(x_{W})-z_{\\mathcal {E}_{T}}(x_{Z})\\right).$ Integrating the derivative (oriented by each tangent $t_E$ ) of $z_{\\mathcal {E}_{T}}$ on $\\partial T$ between $W$ and $Z$ and using the two-dimensional version of the equivalence stated in [20] between $\\Vert {\\cdot }\\Vert _{\\Theta ,T}$ and the component $L^2$ -norm, we have $|z_{\\mathcal {E}_{T}}(x_{W})-z_{\\mathcal {E}_{T}}(x_{Z})|\\le \\Vert z_{\\mathcal {E}_{T}}^{\\prime }\\Vert _{L^1(\\partial T)}\\le |\\partial T|^{\\frac{1}{2}}\\,\\Vert z_{\\mathcal {E}_{T}}^{\\prime }\\Vert _{L^2(\\partial T)}\\lesssim \\Vert \\underline{G}_T^{0}\\underline{z}_T\\Vert _{\\Theta ,T}.$ Since $\\operatorname{card}(\\mathcal {V}_{T})$ is uniformly bounded by the mesh regularity parameter and $(\\lambda _{V,W})_{W\\in \\mathcal {V}_{T}}$ and $(\\lambda _{V^\\star ,Z})_{Z\\in \\mathcal {V}_{T}}$ are coefficients of convex combinations, we infer from the above relations that $|\\widetilde{z}_T(x_V)-\\widetilde{z}_T(x_{V^\\star })|\\lesssim \\Vert \\underline{G}_T^{0}\\underline{z}_T\\Vert _{\\Theta ,T}.$ Since any edge $e$ of $\\mathcal {S}_T$ has a length comparable to $h_T$ , this shows that $|\\operatorname{\\bf grad}\\widetilde{z}_T\\cdot t_e|\\lesssim h_T^{-1} \\Vert \\underline{G}_T^{0}\\underline{z}_T\\Vert _{\\Theta ,T}$ where $t_e$ is any unit tangent to $e$ .", "Hence, on any triangle $\\tau \\in \\mathcal {S}_T$ , $\\Vert \\operatorname{\\bf grad}\\widetilde{z}_T\\Vert _{L^2(\\tau )^2}= |\\tau |^{\\frac{1}{2}}|(\\operatorname{\\bf grad}\\widetilde{z}_T)_{|\\tau }|\\le |T|^{\\frac{1}{2}}|(\\operatorname{\\bf grad}\\widetilde{z}_T)_{|\\tau }|\\lesssim |T|^{\\frac{1}{2}}h_T^{-1} \\Vert \\underline{G}_T^{0}\\underline{z}_T\\Vert _{\\Theta ,T}.$ Using $|T|^{\\frac{1}{2}}\\lesssim h_T$ , squaring, summing over $\\tau \\in \\mathcal {S}_T$ and taking the square root concludes the proof of ().", "3.", "Proof of ().", "We start from the following Poincaré inequality with trace: $\\Vert w_T\\Vert _{L^2(T)}^2\\lesssim h_T^2\\Vert \\operatorname{\\bf grad}w_T\\Vert _{L^2(T)^2}^2+\\sum _{E\\in \\mathcal {E}_{T}}h_E\\Vert w_T\\Vert _{L^2(E)}^2\\qquad \\forall w_T\\in \\mathcal {P}_{\\rm c}^{1}(\\mathcal {S}_T).$ To prove this estimate, consider a triangle $\\tau \\in \\mathcal {S}_T$ with an edge $e\\subset \\partial T$ .", "Taking $x\\in \\tau $ and $y\\in e$ we have $|w_T(x)|\\le h_T|(\\operatorname{\\bf grad}w_T)_{|\\tau }|+|w_T(y)|$ ; integrating over $y\\in e$ , squaring, integrating over $x\\in \\tau $ and using $|\\tau |/h_e\\lesssim h_e$ (by shape regularity) leads to $\\Vert w_T\\Vert _{L^2(\\tau )}^2\\lesssim h_T^2\\Vert \\operatorname{\\bf grad}w_T\\Vert _{L^2(\\tau )^2}^2+h_e\\Vert w_T\\Vert _{L^2(e)}^2.$ If all triangles in $\\mathcal {S}_T$ have an edge $e\\subset \\partial T$ , summing (REF ) over $\\tau \\in \\mathcal {S}_T$ concludes the proof of (REF ); otherwise, a discrete trace inequality and (REF ) give a bound on the trace of $w_T$ on the other edges of $\\tau $ , and the process can be iterated on the triangles in $\\mathcal {S}_T$ that touch $\\tau $ but do not have an edge on $\\partial T$ .", "Applying (REF ) to $w_T=\\widetilde{z}_T-P_{U,T}^{1}\\underline{z}_T$ , using a triangle inequality, $h_E\\le h_T$ , the estimate (), and the fact that $(\\widetilde{z}_T)_{|E}=(z_{\\mathcal {E}_{T}})_{|E}$ for all $E\\in \\mathcal {E}_{T}$ , we obtain $\\Vert \\widetilde{z}_T-P_{U,T}^{1}\\underline{z}_T\\Vert _{L^2(T)}^2\\lesssim h_T^2\\left(\\Vert \\underline{G}_T^{0}\\underline{z}_T\\Vert _{\\Theta ,T}^2+\\Vert \\operatorname{\\bf grad}P_{U,T}^{1}\\underline{z}_T\\Vert _{L^2(T)^2}^2\\right)+h_T^2\\!\\sum _{E\\in \\mathcal {E}_{T}}\\!h_E^{-1}\\Vert z_{\\mathcal {E}_{T}}-P_{U,T}^{1}\\underline{z}_T\\Vert _{L^2(E)}^2.$ The proof of () is completed by invoking [20] to write $\\Vert \\operatorname{\\bf grad}P_{U,T}^{1}\\underline{z}_T\\Vert _{L^2(T)^2}^2+\\sum _{E\\in \\mathcal {E}_{T}}h_E^{-1}\\Vert z_{\\mathcal {E}_{T}}-P_{U,T}^{1}\\underline{z}_T\\Vert _{L^2(E)}^2\\lesssim \\Vert \\underline{G}_T^{0}\\underline{z}_T\\Vert _{\\Theta ,T}^2.$ [Proof of Lemma REF ] 1.", "Proof of (REF ).", "By (REF ), $\\pi _{\\mathcal {P},T}^{0}\\widetilde{\\eta }_T= G_T^{0}\\underline{w}_T+P_{\\Theta ,T}^{0}\\underline{\\kappa }_T= P_{\\Theta ,T}^{0}(\\underline{G}_T^{0}\\underline{w}_T+\\underline{\\kappa }_T),$ where the last equality follows from the relation $P_{\\Theta ,T}^{0}\\underline{G}_T^{0}=G_T^{0}$ , see [20].", "This proves that $\\pi _{\\mathcal {P},T}^{0}\\widetilde{\\eta }_T=P_{\\Theta ,T}^{0}\\underline{\\eta }_T^\\flat $ .", "Since $P_{\\Theta ,T}^{0}$ depends only on the tangential components of $\\underline{\\eta }_T$ (see (REF ) and (REF )), this concludes the proof of (REF ).", "2.", "Proof of ().", "We use the definition of $\\widetilde{\\eta }_T$ to write $\\operatorname{\\bf grad}\\widetilde{v}_T-\\widetilde{\\eta }_T=\\operatorname{\\bf grad}(\\widetilde{v}_T-\\widetilde{w}_T)-P_{\\Theta ,T}^{0}\\underline{\\kappa }_T$ and thus, by triangle inequality, $\\Vert \\operatorname{\\bf grad}\\widetilde{v}_T-\\widetilde{\\eta }_T\\Vert _{L^2(T)^2}^2\\le {}&2\\Vert \\operatorname{\\bf grad}(\\widetilde{v}_T-\\widetilde{w}_T)\\Vert _{L^2(T)^2}^2+2\\Vert P_{\\Theta ,T}^{0}\\underline{\\kappa }_T\\Vert _{L^2(T)^2}^2\\\\\\lesssim {}& \\Vert \\underline{G}_T^{0}(\\underline{v}_T-\\underline{w}_T)\\Vert _{\\Theta ,T}^2+\\Vert \\underline{\\kappa }_T\\Vert _{\\Theta ,T}^2\\\\={}&\\Vert \\underline{G}_T^{0}(\\underline{v}_T-\\underline{w}_T)-\\underline{\\kappa }_T\\Vert _{\\Theta ,T}^2,$ where the second line follows from () applied to $\\underline{z}_T=\\underline{v}_T-\\underline{w}_T$ and the estimate $\\Vert P_{\\Theta ,T}^{0}\\underline{\\kappa }_T\\Vert _{L^2(T)^2}\\lesssim \\Vert \\underline{\\kappa }_T\\Vert _{\\Theta ,T}$ (see [20]), while the conclusion is obtained using the orthogonality for the $(\\cdot ,\\cdot )_{\\Theta ,T}$ product of $\\underline{\\kappa }_T$ and $\\underline{G}_T^{0}(\\underline{v}_T-\\underline{w}_T)$ .", "This gives $\\Vert \\operatorname{\\bf grad}\\widetilde{v}_T-\\widetilde{\\eta }_T\\Vert _{L^2(T)^2}\\lesssim \\Vert \\underline{G}_T^{0}\\underline{v}_T-\\underline{\\eta }_T^\\flat \\Vert _{\\Theta ,T}$ and the proof of () is complete since $\\Vert {\\cdot }\\Vert _{\\Theta ,T}$ depends only on tangential components of vectors in $\\underline{\\Theta }_T^0$ .", "3.", "Proof of (REF ).", "Let $\\phi _T(x)P_{\\Theta ,T}^{0}\\underline{\\eta }_T\\cdot (x-x_T)\\in \\mathcal {P}_{}^{1}(T)$ .", "By (REF ), we have $P_{\\Theta ,T}^{0}\\underline{\\eta }_T=\\operatorname{\\bf grad}\\phi _T=\\operatorname{\\bf grad}\\widetilde{\\underline{I}_{U,T}^0\\phi _T}$ , and thus () with $\\underline{v}_T=\\underline{I}_{U,T}^0\\phi _T$ yields $\\Vert P_{\\Theta ,T}^{0}\\underline{\\eta }_T-\\widetilde{\\eta }_T\\Vert _{L^2(T)^2}^2&\\lesssim \\Vert \\underline{G}_T^{0}(\\underline{I}_{U,T}^0\\phi _T)-\\underline{\\eta }_T\\Vert _{\\Theta ,T}^2\\\\&\\lesssim \\sum _{E\\in \\mathcal {E}_{T}}h_E\\Vert P_{\\Theta ,T}^{0}\\underline{\\eta }_T\\cdot t_E-\\eta _E\\cdot t_E\\Vert _{L^2(E)}^2\\lesssim h_T^2\\sum _{E\\in \\mathcal {E}_{T}}h_E^{-1}\\Vert P_{\\Theta ,T}^{0}\\underline{\\eta }_T-\\eta _E\\Vert _{L^2(E)}^2,$ where the second inequality follows from the two-dimensional version of the norm equivalence [20] together with the local version of (REF ) which gives $\\underline{G}_T^{0}(\\underline{I}_{U,T}^0\\phi _T)=\\underline{I}_{\\Theta ,T}^{\\flat ,0}(\\operatorname{\\bf grad}\\phi _T)=\\underline{I}_{\\Theta ,T}^{\\flat ,0}(P_{\\Theta ,T}^{0}\\underline{\\eta }_T)$ .", "The estimate (REF ) follows writing $h_T\\le h$ , summing over $T\\in \\mathcal {T}_h$ , and invoking [19] to see that $\\sum _{T\\in \\mathcal {T}_h}\\sum _{E\\in \\mathcal {E}_{T}}h_E^{-1}\\Vert P_{\\Theta ,T}^{0}\\underline{\\eta }_T-\\eta _E\\Vert _{L^2(E)^2}^2\\lesssim \\Vert \\mathsf {G}_{{\\rm s},h}^{0}\\underline{\\eta }_h\\Vert _{L^2(\\Omega )^{2\\times 2}}^2+|\\underline{\\eta }_h|_{{\\rm s,j},h}^2.$" ], [ "Numerical results", "We illustrate the practical behaviour of the DDR scheme (REF ) on two different analytical solutions and three families of meshes of $\\Omega =(0,1)^2$ : (mostly) hexagonal meshes, triangular meshes, and locally refined meshes (with hanging nodes); Figure REF shows a representative member of each family of meshes.", "The DDR tools and the scheme (for both clamped and simply supported boundary conditions) have been implemented in the HArDCore2D C++ framework (see https://github.com/jdroniou/HArDCore), which is based on linear algebra facilities from the Eigen3 library (see http://eigen.tuxfamily.org).", "The resolution of the global sparse linear systems uses the Intel MKL PARDISO library (see https://software.intel.com/en-us/mkl).", "We focus on the $h$ -convergence for the degrees $k\\in \\lbrace 0,1,2,3\\rbrace $ , check the convergence rates, and discuss the robustness of the scheme with respect to the thickness $t$ of the plate.", "In all the tests, the Young modulus is taken as $E=1$ , while the Poisson ratio is $\\nu =0.3$ .", "The error is computed as the (relative) $\\Vert {\\cdot }\\Vert _{\\Theta \\times U,h}$ -norm of the difference between the approximate solution and the interpolate of the exact solution, that is: $E_h\\frac{\\Vert (\\underline{\\theta }_h-\\underline{I}_{\\Theta ,h}^k\\theta ,\\underline{u}_h-\\underline{I}_{U,h}^ku)\\Vert _{\\Theta \\times U,h}}{\\Vert (\\underline{I}_{\\Theta ,h}^k\\theta ,\\underline{I}_{U,h}^ku)\\Vert _{\\Theta \\times U,h}}.$ Figure: Members of mesh families used in numerical tests." ], [ "Polynomial solution", "The first series of tests is run with source term corresponding to the following exact polynomial solution introduced in [17]: $u(x)={}&\\frac{1}{3} x_1^3(1-x_1^3)x_2^3(1-x_2)^3\\\\&-\\frac{2t^2}{5(1-\\nu )}\\left[x_2^3(x_2-1)^3x_1(x_1-1)(5x_1^2-5x_1+1)+x_1^3(x_1-1)^3x_2(x_2-1)(5x_2^2-5x_2+1)\\right],\\\\\\theta (x)={}&\\begin{bmatrix}x_2^3(x_2-1)^3x_1^2(x_1-1)^2(2x_1-1) \\\\[2pt]x_1^3(x_1-1)^3x_2^2(x_2-1)^2(3x_2-1)\\end{bmatrix}.$ The results are presented in Figure REF .", "We notice that, for all considered polynomial degrees $k\\in \\lbrace 0,1,2,3\\rbrace $ and thicknesses $t\\in \\lbrace 10^{-1},10^{-3}\\rbrace $ , the error decays as $h^{k+1}$ (as expected from Theorem REF ) and is mostly independent of $t$ .", "The same observation can be made for $k\\in \\lbrace 0,1\\rbrace $ and $t=10^{-5}$ .", "However, for $k\\ge 2$ we notice an apparent loss of convergence on the finest meshes when $t=10^{-5}$ .", "This loss of convergence is actually not a sign of lack of robustness of the scheme, but rather a consequence of reaching the attainable precision combined with the accumulation of round-off errors.", "Indeed, the considered solution is such that the $H^s$ -norms of the variables (displacement, rotation, shear strain) remain uniformly bounded with respect to $t$ , and Theorem REF thus shows that we should expect a convergence in $\\mathcal {O}(h^{k+1})$ with multiplicative constants that are independent of $t$ .", "This apparent loss of convergence actually comes from unavoidable rounding errors.", "In double precision, the matrices of the local $L^2$ -products $(\\cdot ,\\cdot )_{\\Theta ,T}$ are typically computed with a precision in the range $[10^{-15},10^{-12}]$ , the worst cases corresponding to higher polynomial degrees $k$ and elements with many edges – situations in which the local $L^2$ -products lead to the largest matrices.", "When these local matrices are multiplied by $t^{-2}=10^{-10}$ (for $t=10^{-5}$ ) to assemble the local term in $\\mathrm {b}_h$ , the precision drops to $[10^{-5},10^{-2}]$ .", "Due to this large scaling $t^{-2}$ , the final precision on the global matrix is then rather poor, especially on meshes with a high number of elements; this poor precision prevents an accurate calculation of the approximate solution $(\\underline{\\theta }_h,\\underline{u}_h)$ .", "We notice that the tests we present here are among the few on high-order schemes for the Reissner–Mindlin plate model.", "In [9], isogeometric schemes are considered up to a polynomial degree 5, corresponding to a convergence rate in $h^4$ , and thus to the choice $k=3$ in the DDR scheme.", "The smallest thickness considered in this reference is $t=10^{-3}$ , and the largest mesh has about 300 rectangles; at these levels, no rounding error is noticeable in our tests (to compare, the second locally refined mesh we consider has more than 600 elements, and the largest one has more than 10,000 elements).", "An over-penalised discontinuous Galerkin scheme is presented in [14], and tests are produced with a polynomial degree 4 (convergence rate in $h^3$ ), corresponding to $k=2$ for the DDR scheme.", "Very thin plates are considered in these tests, with $t$ as small as $10^{-6}$ ; however, for this thickness, the largest triangular mesh in the tests of [14] has 512 triangles; our second coarsest triangular mesh has 896 triangles and, as can be seen in Figure REF , for $k=2$ and at this size of mesh the convergence is not affected by round-off errors.", "It therefore seems that those previous tests were carried out under conditions in which round-off errors are not perceptible, and that the tests we present here are the first ones to highlight this phenomenon for high-order schemes and very thin plates.", "Figure: Error E h E_h w.r.t.", "hh for the polynomial solution of Section ." ], [ "Analytical solution with improved physical behaviour", "As explained in [5], as $t\\rightarrow 0$ the shear strain $\\gamma $ is expected to remain bounded in $L^2$ -norm, but to grow unboundedly in $H^1$ -norm.", "The polynomial solution considered in Section REF does not reproduce this behaviour (for this solution, the shear strain is actually independent of $t$ ).", "To test our DDR scheme in a setting which is at least quantitatively closer to the generic physical behaviour of the Reissner–Mindlin model, we design in this section a new analytical solution on $\\Omega =(0,1)^2$ (with non-homogeneous boundary conditions), with the following behaviour as $t\\rightarrow 0$ : ${\\begin{array}{c}\\text{$\\Vert u\\Vert _{H^3(\\Omega )}\\sim 1$,$\\Vert \\theta \\Vert _{H^2(\\Omega )^2}\\sim 1$,$\\Vert \\gamma \\Vert _{L^2(\\Omega )^2}\\sim 1$,$|\\gamma |_{H^s(\\Omega )^2}\\sim t^{-s+\\frac{1}{2}}$for all $s\\ge 1$},\\\\\\text{and $f$ is independent of $t$}.\\end{array}}$ As noticed in [5], the expected growth of $|\\gamma |_{H^1(\\Omega )}$ is in $t^{-1}$ , not $t^{-\\frac{1}{2}}$ as in the solution we construct here.", "This solution could easily be adjusted to produce such a growth, but this would come at the cost of extremely steep dependency on $t$ (in particular, a term $t^6$ in (REF ) below) that would make the solution even more challenging to handle using double-precision arithmetic." ], [ "Design of the solution", "We look for a solution under the form ${\\begin{array}{c}\\text{$u(x)=v(t,x)+t^2w(t,x)$and$\\theta (x)=\\operatorname{\\bf grad}v(t,x)$,}\\\\\\text{where $v(t,x)=t^3 V(t^{-1}x) + g(x)$with $V(y)=y_1e^{-y_1}\\cos (y_2)$and $g(x)=\\sin (\\pi x_1)\\sin (\\pi x_2)$.", "}\\end{array}}$ Defining $\\gamma $ by () gives $\\gamma (t,\\cdot )=\\kappa \\operatorname{\\bf grad}w(t,\\cdot )$ .", "The function $w$ is then selected to ensure that (REF ) holds.", "Since $\\operatorname{\\bf div}(\\mathsf {C}\\operatorname{\\mathsf {grad}_{\\rm s}}\\theta )=\\operatorname{\\bf div}(\\mathsf {C}\\operatorname{\\mathsf {grad}_{\\rm s}}(\\operatorname{\\bf grad}v))=(\\beta _0+\\beta _1)\\operatorname{\\bf grad}\\Delta v$ , (REF ) corresponds to $w(t,\\cdot )=-\\frac{\\beta _0+\\beta _1}{\\kappa }\\Delta v(t,\\cdot ).$ The transverse load $f$ is fixed according to (): $f(t,\\cdot )=(\\beta _0+\\beta _1)\\Delta ^2 v(t,\\cdot ).$ Let us now briefly check that (REF ) holds.", "We first notice that, for any natural numbers $m,n$ , the mapping $(0,\\infty )\\times \\Omega \\ni (t,x)\\mapsto (\\partial _1^m\\partial _2^nV)(t^{-1}x)$ is uniformly bounded.", "This shows that $\\Vert u\\Vert _{H^3(\\Omega )}$ , $\\Vert \\theta \\Vert _{H^2(\\Omega )^2}$ and $\\Vert \\gamma \\Vert _{L^2(\\Omega )^2}$ remain bounded as $t\\rightarrow 0$ ; these norms also do not go to zero owing to the presence of $g$ (which could actually be any smooth function with non-zero derivatives up to order 3).", "The function $V$ satisfies $\\Delta V(y)=-2e^{-y_1}\\cos (y_2)$ and thus $\\Delta ^2V=0$ ; hence, $f=(\\beta _0+\\beta _1)\\Delta ^2 g$ is independent of $t$ .", "This also shows that $\\gamma (x)=-2(\\beta _0+\\beta _1)e^{-t^{-1}y_1}\\left[\\begin{array}{c}\\cos (t^{-1}y_2)\\\\ \\sin (t^{-1}y_2) \\end{array}\\right]-(\\beta _0+\\beta _1)\\operatorname{\\bf grad}\\Delta g(x).$ For a given $s\\ge 1$ , taking any partial derivative of order $s$ of this expression and using $\\Vert e^{-t^{-1}\\bullet }\\Vert _{L^2(0,1)}\\sim t^{1/2}$ shows that $|\\gamma |_{H^s(\\Omega )^2}\\sim t^{-s+\\frac{1}{2}}$ ." ], [ "Results", "The results of the numerical tests with the analytical solution (REF ) are presented in Figure REF .", "We observe a similar behaviour as in the numerical results for the polynomial solution (see Figure REF ).", "The scheme is here completely robust for $k=0$ (as expected from Theorem REF ), and for $k=1$ up to $t=10^{-3}$ (and also for $t=10^{-5}$ up to errors of magnitude $10^{-3}$ ).", "The degradation of convergence occurs however sooner, with respect to increasing $k$ or $1/t$ , than in Section REF : the apparent loss of convergence is here already perceptible for $(k,t)=(1,10^{-5})$ or $(k,t)=(3,10^{-3})$ for example; it also seems more severe for $t=10^{-5}$ and $k\\ge 1$ .", "This is not completely unexpected as the dependency of the analytical solution (REF ) with respect to $t$ is more severe, and the higher-order norms of the shear strain indeed grows with $t$ here.", "Combined with the round-off errors phenomenon previously mentioned, this explains the worse numerical behaviour.", "We however notice that the scheme remains more robust, even for higher degrees, than what the error estimate (REF ) could lead us to believe; considering for example $k=3$ , since $|\\gamma |_{H^{4}(\\mathcal {T}_h)}$ grows as $t^{-3.5}$ , the upper bound on the error in (REF ) grows between $t=10^{-1}$ and $t=10^{-3}$ by a factor $10^{2\\times 3.5}=10^7$ , which is clearly not the case of the error itself (on the finest mesh, the ratio of the errors for these two values of $t$ is at most $10^3$ ).", "Figure: Error E h E_h w.r.t.", "hh for the analytical solution of Section ." ], [ "Acknowledgements", "The authors acknowledge the partial support of Agence Nationale de la Recherche grant NEMESIS (ANR-20-MRS2-0004)." ] ]
2105.11773
[ [ "Searches for continuous gravitational waves from young supernova\n remnants in the early third observing run of Advanced LIGO and Virgo" ], [ "Abstract We present results of three wide-band directed searches for continuous gravitational waves from 15 young supernova remnants in the first half of the third Advanced LIGO and Virgo observing run.", "We use three search pipelines with distinct signal models and methods of identifying noise artifacts.", "Without ephemerides of these sources, the searches are conducted over a frequency band spanning from 10~Hz to 2~kHz.", "We find no evidence of continuous gravitational radiation from these sources.", "We set upper limits on the intrinsic signal strain at 95\\% confidence level in sample sub-bands, estimate the sensitivity in the full band, and derive the corresponding constraints on the fiducial neutron star ellipticity and $r$-mode amplitude.", "The best 95\\% confidence constraints placed on the signal strain are $7.7\\times 10^{-26}$ and $7.8\\times 10^{-26}$ near 200~Hz for the supernova remnants G39.2--0.3 and G65.7+1.2, respectively.", "The most stringent constraints on the ellipticity and $r$-mode amplitude reach $\\lesssim 10^{-7}$ and $ \\lesssim 10^{-5}$, respectively, at frequencies above $\\sim 400$~Hz for the closest supernova remnant G266.2--1.2/Vela Jr." ], [ "Introduction", "Transient gravitational waves (GWs) from compact binary coalescences [8], [13] have been directly observed by the Advanced Laser Interferometer Gravitational-Wave Observatory (Advanced LIGO) detectors [3] and the Advanced Virgo detector [17].", "Continuous gravitational waves (CWs) have not yet been detected.", "The most likely sources of CWs detectable by ground-based interferometers are non-axisymmetric, rapidly rotating neutron stars.", "Searches for CWs have been carried out targeting various isolated sources, including known pulsars with electromagnetic ephemerides [9], [14], neutron stars without ephemerides in the galactic center or in globular clusters [98], [2], [41], [7], neutron stars in binary systems [10], [137], [86], and young supernova remnants (SNRs) [4], [120], [88], [11], [87], [78], [94], [30].", "Searches have also been conducted over the whole sky for CWs instead of targeting at a particular direction [12], [40], [116], [132], [37], [16].", "This work searches for CWs from SNRs in the first half of the third observing run (O3a), which commenced on April 1st, 2019 and ended on March 27th, 2020 [33], [17].", "Young neutron stars in SNRs are one potential source of continuous, quasi-monochromatic GWs.", "If pulsations are observed in electromagnetic emission from the neutron star, one can search for CWs guided by the ephemerides obtained from those observations, as in e.g., [9] and [15].", "Even so, there is no guarantee that the GW-emitting quadrupole is phase locked to the electromagnetic pulsations.", "When there is no phase locking, search algorithms are needed that can track small (and possibly randomly varying) displacements between the gravitational and electromagnetic frequencies [30], [6].", "If the neutron star does not pulsate, it may be observed as an X-ray point source, known as a central compact object [54].", "In the latter scenario, the maximum GW strain can be inferred from the age of the SNR [133], [106], as has been done in recent GW searches [87], [30].", "A rotating, non-axisymmetric neutron star has a time-varying mass quadrupole (from the point of view of a distant observer) and emits GWs at a strain proportional to the stellar ellipticity, which is affected by the nuclear equation of state, the history of strain build up and diffusion in the crust, and the magnetic field configuration [51].", "For an isolated star, young neutron stars may have larger non-axisymmetries than older ones and consequently may produce stronger GW emissions [71], [107].", "As the star ages, Ohmic [56], thermal [52], [99], tectonic, or other relaxation processes work to reduce the asymmetries introduced in the birth process.", "Young neutron stars are therefore promising targets for CW searches.", "The GW frequency is proportional to the stellar spin frequency $f_\\star $ .", "For thermoelastic [127], [64] or magnetic [38], [82], [74] mass quadrupoles, the predicted frequency is either $f_\\star $ or $2f_\\star $ ; $r$ -mode current quadrupoles emit at $\\sim 4f_\\star /3$ [93], [20], [34], with minor equation-of-state dependent corrections; also, pinned superfluids in neutron stars may produce CWs at frequencies proportional to $f_\\star $  [67], [84].", "In young, rapidly-rotating neutron stars, $f_\\star $ evolves quickly under the action of gravitational and electromagnetic torques [71], [106].", "Rapid spin-down in young SNRs creates challenges for traditional CW search methods, especially over a long observation with duration $T_{\\rm obs} \\gtrsim 1\\,{\\rm yr}$ .", "Most previous searches for SNRs have been restricted to short ($\\sim 1$ month) stretches of data [5], [11], limited parameter space [78], or have had a high associated computational cost [94], [120].", "Accounting for spin down in a coherent search requires a very large number of templates, which increases computation cost beyond feasibility.", "Furthermore, $f_\\star $ may wander randomly, a phenomenon known as spin wandering or timing noise [59], [113], [100], [21], [96], [91], [81], due to unknown internal or magnetospheric processes [36], [85].", "One computationally efficient alternative to a coherent search is a semi-coherent search in which the integration is calculated coherently on blocks of short duration $T_{\\rm coh}$ and added incoherently over the full $T_{\\rm obs}$ .", "We apply three semi-coherent methods to search for signals from 15 known young SNRs in the data collected in the first half (six months) of O3: the directed Band-Sampled-Data (BSD) pipeline [97], based on the FrequencyHough (FH) transform [23], [43], and the single-harmonic Viterbi and dual-harmonic Viterbi pipelines, both based on a hidden Markov model (HMM) tracking scheme [121], [119].", "The two Viterbi methods achieve a lower sensitivity compared to the BSD pipeline, but take into consideration the uncertainties associated with the star's stochastic spin evolution, with one of them tracking two harmonics of the star's spin frequency simultaneously [121], [119], making the three methods complementary to each other.", "The structure of the paper is as follows.", "In Section , we introduce the 15 young SNR targets, listing their location, estimated age and distance.", "In Section , we briefly describe the interferometric data analyzed.", "In Section , we review each of the three search methods and the parameter space covered.", "The strain upper limits, estimated sensitivity, and astrophysical interpretation are discussed in Section .", "A conclusion is given in Section .", "The postprocessing procedure applied to the candidates identified in each search is presented in Appendix .", "Technical details on the pipelines are described in Appendix ." ], [ "Targeted sources", "The target SNRs are selected from the Green supernova catalogue [55] and the SNRcat, an online catalogue of high-energy galactic SNRs hosted by the University of Manitoba [44], [1], as SNRs with X-ray point sources are likely to contain neutron stars.", "Of the 15 SNRs in Table REF , seven are searched using all three different pipelines, while the remaining eight are only searched by the single-harmonic Viterbi pipeline.", "The characteristic ages of the neutron stars are inferred from the estimated supernova ages listed in the table.", "In the three pipelines, we cover parts of different parameter spaces, corresponding to slightly different assumptions of the characteristic age of the star.", "See Section  for details for each pipeline.", "The 15 SNRs were previously searched in the earlier LIGO observing runs, but no CW signal was identified [11], [87], [78], [94].", "Additionally, [94] performed a follow-up search for sub-threshold candidates obtained in the first observing run of Advanced LIGO (O1) [89] for three of the SNRs, Cassiopeia A (Cas A), Vela Jr. and G347.3–0.5, using data collected in the second observing run of Advanced LIGO (O2), and reported one possible CW candidate in G347.3–0.5.", "This fully coherent follow-up search uses two stretches of data in O2 ($T_{\\rm coh} \\sim 4$ months each).", "As indicated in Table REF , only the single-harmonic Viterbi pipeline (which allows for stochastic spin wandering) searches G347.3–0.5 semi-coherently using a short $T_{\\rm coh}$ .", "Since the signal-to-noise ratio roughly scales $\\propto T_{\\rm coh}^{1/2}$ , the sensitivity presented in [94] exceeds that presented here for G347.3–0.5, provided that the signal power leaked into adjacent frequency bins due to the spin down and spin wandering over the coherent duration is negligible.", "In addition, the candidate reported in [94] was originally identified as a sub-threshold one.", "Therefore it is not surprising that we do not find a possible candidate in G347.3–0.5.", "Table: The 15 SNRs covered in this analysis.", "Sources in the upper half of the table are searched by all three pipelines described in Section .", "Sources in the bottom half are searched by a single pipeline described in Section .", "The ages and distances listed are consistent with the values used in the previous LIGO analysis ." ], [ "Instrumental overview and data", "The O3 observing run started on April 1st, 2019 at 15:00 UTC and ended on March 27th, 2020 at 17:00 UTC.", "For the search, we use data collected by the two Advanced LIGO detectors in Hanford, Washington (H) and Livingston, Louisiana (L) and Advanced Virgo in the first half of O3, from the start until October 1st, 2019.", "This time period is referred to as “O3a”.", "The data collected by the two LIGO detectors during the second half of O3 (O3b), starting from November 1st, 2019 until the end of O3, are used by the BSD pipeline (Section REF ) and dual-harmonic Viterbi pipeline (Section REF ) to cross-check candidates.", "Data collected by Virgo are only used by the BSD pipeline, which runs the initial search using individual detectors separately (Section REF ).", "In the two Viterbi-based pipelines, the Virgo data are not used due to the detector's relatively lower sensitivity, and the two pipelines both operate on all detectors combined.", "All three pipelines use data collected when the detectors are in the nominal low-noise observing mode [39].", "The BSD pipeline (Section REF ) uses low-latency calibrated data (C00 frames) [122] for H and L detectors and the “online\" calibration version for Virgo, after a procedure of removing significant short-duration noise transients, known as “glitches\" [39], in the Short Fourier Transform Database (SFDB) [26].", "Tests show that the difference between the C00 data, after glitch removal in SFDB, and glitch gated C01 frames is negligible.", "The two Viterbi pipelines (Sections REF and REF ) use the high-latency calibrated data (C01 frames) [122], passed through a procedure of glitch gating [140]." ], [ "BSD", "The BSD directed search pipeline is a hierarchical semi-coherent method based on the FH transform [23], [43].", "A previous search using the BSD directed search pipeline, pointing to the Galactic Center in Advanced LIGO O2, was reported in [98].", "The pipeline descibed in this section is based on the BSD framework, i.e.", "a library of functions which allows the user to freely select a subset of the detector strain data (both in frequency and time domain), starting from a collection of basic files (BSD files) in a special data format.", "All the properties of the framework are described in [97], and here we only remind the reader that the standard format of the BSD files, containing an opportunely down-sampled complex time series, covers a 10-Hz frequency band and $\\sim 1$ month of data.", "For the purpose of this search, where the actual signal frequency is unknown, each BSD file is partially corrected for the Doppler modulation in each 1-Hz frequency sub-band using its central frequency (see [98] for more details).", "From this partially corrected time series, a collection of time-frequency peaks (called “peakmaps\") is obtained, by choosing all the local maxima above a given threshold from equalized spectra [26].", "The equalization is given by the square modulus of the periodogram divided by the average spectrum.", "In this way also narrow peaks are kept.", "This peakmap is the input of the FH transform, which maps each time-frequency peak into the intrinsic source frequency and spin-down $(f_0,\\dot{f}_0)$ plane at a given reference time.", "The resolution of a single FH map is the size of the bins in the template grid $\\delta f_{\\rm FH} = \\frac{1}{T_{\\rm coh}K_{f}},\\\\\\delta \\dot{f}_{\\rm FH}=\\frac{1}{T_{\\rm coh}T_{\\rm obs}K_{\\dot{f}}},$ where $T_{\\rm coh}$ is the coherence time, while $T_{\\rm obs}$ is the observational time.", "The parameters $K_{f}$ and $K_{\\dot{f}}$ are the over-resolution factors as described in [23], here chosen as $K_{f}=10$ and $K_{\\dot{f}}=2$ .", "The coherence time $T_{\\rm coh}$ scales with the maximum frequency of the band as $1/\\sqrt{f_{\\rm max}}$ , and hence the frequency and spin-down bin sizes in Eqs.", "(REF ) and () change for each 10-Hz band.", "For a source with age $t_{\\rm age}$ , the spin-down range is defined as $-{f_{\\rm max}}/{t_{\\rm age}}\\le \\dot{f} \\le 0.1 {f_{\\rm max}}/{t_{\\rm age}}$ , where $f_{\\rm max}$ is the maximum frequency in each 10-Hz band.", "In this analysis, the age of the source affects the parameter space investigated, with a wider spin-down range covered when the source is younger.", "When possible, we use the youngest age estimate available in the SNRcat catalog [1], [44].", "On the other hand, according to the age of the source, we can consider the effects of the second order spin down as negligible or not (a discussion is reported in Appendix REF ).", "In this search, we investigate a frequency band of $[10, 600]$  Hz for targets with assumed $t_{\\rm age}\\le 3$ kyr, and a wider range of $[10, 1000]$  Hz for older sources.", "We remind the reader of the subtle difference when talking about the source age estimates (which is most of the time inferred from the SNR age) and the characteristic age of the star (which is unknown because they have no observed electromagnetic pulsations).", "The maximum coherence time used is 17.8 hr for the frequency band $[10, 20]$  Hz and a minimum of 2.5 hr for $[990, 1000]$  Hz.", "We search both positive and negative $\\dot{f}$ to allow for the possibility of unexpected spin up.", "A summary of the parameter space investigated for each source is shown in Table REF .", "Table: Sources searched in the BSD analysis (Section ) and the parameter space covered.", "The coherence time and the spin-down/up range scale with the maximum frequency in each 10-Hz frequency band.", "For each source, we report the T coh T_{\\rm coh} and spin-down/up range used for the frequency band [90, 100] Hz where f max =100f_{\\rm max}=100 Hz.The first set of candidates is selected from a final FH map, which is the sum of all the single monthly-based FH maps spanning the same frequency and spin-down ranges.", "These candidates are independently selected in each detector, including Virgo, using the ranking procedure of [23] where candidates with the highest FH number count are kept.", "At a later stage, coincidences are calculated between the candidate sets from the two LIGO detectors using a coincidence distance defined as $d=\\sqrt{\\left(\\frac{\\Delta f}{\\delta f_{\\rm FH}}\\right)^2+\\left(\\frac{\\Delta \\dot{f}}{\\delta \\dot{f}_{\\rm FH}}\\right)^2},$ where $\\Delta f$ and $\\Delta \\dot{f}$ are the differences between the candidate parameters in each data set.", "A candidate is then selected when the coincidence distance is below a given threshold distance, $d_{\\rm thr}$ in this search chosen equal to 4.", "The choice of the window size has been widely discussed in [23], using injected simulated signals.", "The coincidence step has been applied first to the pair of LIGO candidates.", "At a later stage, the same coincidence criterion has been applied between the HL coincident candidates and the most significant Virgo candidates.", "Candidates found in triple coincidence were discarded after applying the post-processing methods described in Appendix .", "However, we cannot conclude with certainty that a pair of LIGO candidates are non-astrophysical if they have $d < d_{\\rm thr}$ but are not seen in Virgo data, because Virgo is less sensitive than LIGO.", "For this reason we also postprocessed all the candidates found in coincidence between H and L only.", "Surviving candidates are further investigated through a followup process described in Appendix .", "Also, we apply a threshold to the Critical Ratio (CR) $\\rho _{\\rm CR}$ , which measures the statistical significance of a candidate based on the number count associated with the pixel of the FH map where the candidate lies.", "The threshold $\\rho _{\\rm CR,thr}$ is chosen as the mean $\\rho _{\\rm CR}$ plus one standard deviation of the CR distribution across the candidates excluding those due to known instrumental lines (Appendix REF ) and with an inconsistent significance among the two detectors (Appendix REF ).", "For the targets G65.7+1.2, G189.1+3.0, and G266.2–1.2, we use $\\rho _{\\rm CR,thr}=4.7$ ; for G18.9–1.1 and G93.3+6.9, we use $\\rho _{\\rm CR,thr}=4.6$ ; and for G353.6–0.7 and G39.2–0.3, we use $\\rho _{\\rm CR,thr}=4.5$ .", "The threshold chosen here is less stringent than in [98] where the threshold was $\\approx 6.5$ , corresponding to the probability of picking an average of one false candidate over the total number of points in the parameter space, under the assumption of Gaussian noise.", "For this work, a lower CR threshold is picked since we are using some new postprocessing methods, described in Appendix , which allow us to followup a higher number of candidates, given the low computational cost of each step." ], [ "Single-harmonic Viterbi", "An HMM is an efficient search algorithm capable of handling both spin down and spin wandering.", "Previous searches for young SNRs using an HMM [121] were conducted in the Advanced LIGO O2 data, but no evidence for a GW signal was reported [87].", "An HMM models a time-varying signal with underlying hidden (i.e.", "unobservable) parameters by treating the hidden parameters as links in a Markov chain, with each hidden parameter linked to an observable through a likelihood statistic.", "Given an observed sequence, the goal is to infer the most probable hidden sequence.", "For a set of $N_T$ observations at discrete times $\\lbrace t_0,t_1, ..., t_{N_T-1}\\rbrace $ , the corresponding discrete states $\\lbrace q(t_0), q(t_1), ..., q(t_{N_T-1})\\rbrace $ (chosen from $N_Q$ possible hidden states $\\lbrace q_1, ..., q_{N_Q}\\rbrace $ ) form a Markov chain with transition probabilities from $t_k$ to $t_{k+1}$ defined by $A_{q_iq_j} = P[ q(t_{k+1}) = q_j | q(t_k) = q_i ]$ .", "For this search, we choose $A_{q_iq_i} = A_{q_{i\\pm 1}q_i} = 1/3$ and all other $A_{q_iq_j} = 0$ , allowing the frequency to remain static or wander up or down one bin for each time step.", "This allows us to track both spin down and stochastic spin wandering, which may cause spin up.", "Strictly speaking, spin down is expected to be more rapid than spin up due to spin wandering, but the exact values of $A_{q_i q_j}$ have minimal effect on the performance of an HMM, provided they capture the behaviour of the signal in a broad sense [101], [123].", "We assume a uniform prior over the initial state, i.e.", "$\\Pi [q(t_0)] = N_Q^{-1}$ .", "The observations are denoted $\\lbrace o(t_0), o(t_1), ..., o(t_{N_T-1})\\rbrace $ and are connected to $q(t_k)$ through unknown parameters.", "We call the probability of observing $o(t_k)$ given some state $q(t_k)$ the emission probability $L_{o(t_k) q(t_k)} = P[o(t_k) | q(t_k)]$ .", "Given some observed sequence $O$ , we can then infer the most likely hidden sequence $Q^*$ by maximizing $ P(Q^*|O) =\\Pi [q(t_0)]\\prod _{k = 1}^{N_T-1} L_{o(t_k)q(t_k)}A_{q(t_k) q(t_{k-1})}.$ The Viterbi algorithm is an efficient implementation of the inference step, using dynamic programming to sample and discard unfavourable paths at each time-step [130], [123].", "For our purposes, the hidden state is the true GW frequency and the observable is the value of the $\\mathcal {F}$ -statistic, calculated coherently over a block of duration $T_{\\rm coh}$ and width (in the frequency domain) $(2T_{\\rm coh})^{-1}$ .", "The $\\mathcal {F}$ -statistic is a maximum likelihood filter for a CW signal of frequency $f$ with time derivatives $\\dot{f}$ , $\\ddot{f}$ , etc.", "(for more details on the $\\mathcal {F}$ -statistic, please see [62]).", "In this search, we compute the $\\mathcal {F}$ -statistic as a function of $f$ only, and account for spin down by choosing $T_{\\rm coh} \\propto \\left|\\dot{f}_0^{\\rm max}\\right|^{-1/2}$ (as in [121]), where $\\dot{f}_0^{\\rm max}$ is the maximum $\\dot{f}$ within $T_{\\rm coh}$ , such that the signal should wander by at most one frequency bin per time step.", "We choose our parameter space according to the detectability of a potential signal.", "First, we estimate the maximum expected GW strain for a neutron star at distance $D$ with characteristic age $t_{\\rm age}$ and a principle moment of inertia $I_{zz}$ using $h_0^{\\rm age} = 2.27\\times 10^{-24} \\left( \\frac{1 \\, {\\rm kpc}}{D}\\right) \\left(\\frac{1 \\, {\\rm kyr}}{t_{\\rm age}}\\right)^{1/2} \\left(\\frac{I_{zz}}{10^{38} \\, {\\rm kg \\, m}^2}\\right)^{1/2}$ and assuming purely gravitational spin down [133].", "We also estimate the minimum detectable strain using an analytic estimate of the 95% confidence sensitivity for a semi-coherent search, given by [121], [133] $h_0^{\\rm est}=\\Theta S_n(f)^{1/2}\\left(T_\\mathrm {obs}T_{\\rm coh}\\right)^{-1/4},$ where $S_n(f)$ is the noise amplitude spectral density.", "The statistical threshold $\\Theta $ is defined by the location in parameter space and typically lies in the range $30 \\lesssim \\Theta \\lesssim 40$ .", "Following previous studies for CWs with an HMM, we take $\\Theta = 35$ [121], [133].", "The frequency range for each source is defined by $h_0^{\\rm est} < h_0^{\\rm age}$ .", "The parameter space for each source, including $T_{\\rm coh}$ , is summarized in Table REF , and the process for defining the parameter space is described in Appendix REF .", "We split the data into $N_{\\rm band}$ frequency sub-bands of width 2 Hz to ensure loud, non-Gaussian noise artifacts (e.g.", "lines) are confined to one sub-band and do not affect the whole analysis.", "We overlap the frequency sub-bands by 0.57 Hz, ensuring that any signal corresponding to a rapidly spinning down neutron star can always be contained in a single sub-band.", "For each sub-band, we apply the Viterbi algorithm outlined above and obtain $N_Q$ frequency paths ending in $N_Q$ different bins with associated likelihoods $\\mathcal {L}$ .", "Alternative implementations of Viterbi (including [123] and [121]) used a Viterbi score as their detection statistic (see Section REF ).", "This statistic generally requires $N_T \\ll N_Q$ .", "[87] demonstrated that this statistic fails to identify an injected (or real) path for $N_T \\sim N_Q$ because the score is calculated for the optimal path relative to other paths in the band.", "If most of the paths overlap, the optimal path is similar to other paths in the band.", "In this search, we have a minimum $T_{\\rm coh} = 1$  hr ($N_T = 4391$ , $N_Q = 14400$ ), which is sufficient for almost one third of Viterbi paths to converge over $T_{\\rm obs}$ and consequently lower the sensitivity of the Viterbi score.", "To maintain the search sensitivity with $N_T \\sim N_Q$ , we use the log-likelihood $\\mathcal {L}$ as our detection statistic.", "Using the process outlined in Appendix REF , we determine the 1% false alarm threshold for each source and denote the corresponding likelihood $\\mathcal {L}_{\\rm th}$ .", "We follow up all unique frequency paths with $\\mathcal {L} > \\mathcal {L}_{\\rm th}$ using the procedure described in Appendix and find no CW candidates which cannot be described by non-astrophysical noise.", "Table: Sources searched in the single-harmonic Viterbi analysis (Section ) and the parameter space covered.", "The parameter space for each of the 15 sources is derived using the age and distance estimates in columns two and three." ], [ "Dual-harmonic Viterbi", "Methods in Sections REF and REF assume that the star rotates about one of its principal axes of the moment of inertia, and hence the GWs are emitted at $2 f_\\star $ .", "This assumption is based on the fact that the phenomenon of free precession is not clearly observed in the population of known pulsars [67].", "However, the superfluid interior of a star pinned to the crust along an axis nonaligned with any of its principal axes could allow the star to emit GWs at both $f_\\star $ and $2f_\\star $ , even without free precession [67], [29], [84].", "The dual-harmonic emission mechanism motivates searches combining the two frequency components of a signal to improve signal-to-noise ratio.", "The HMM tracking scheme described in Section REF has been extended to track two frequency components simultaneously [119].", "The signal model considered in this section consists of both $f_\\star $ and $2 f_\\star $ components, given by [62], [119] $ h_{2+} &=& \\frac{1}{2}h_0(1+\\cos ^2\\iota )\\sin ^2\\theta \\cos 2\\Phi ,\\\\h_{2\\times } &=& h_0 \\cos \\iota \\sin ^2\\theta \\sin 2 \\Phi , \\\\h_{1+} &=&\\frac{1}{8}h_0\\sin 2\\iota \\sin 2\\theta \\sin \\Phi ,\\\\ h_{1\\times } &=& \\frac{1}{4} h_0 \\sin \\iota \\sin 2 \\theta \\cos \\Phi ,$ where $\\iota $ is the inclination angle of the source, $\\theta $ is the wobble angle between the star’s rotation axis and its principal axis of the moment of inertia, and $\\Phi $ is the GW signal phase observed at the detector.", "In general, when precession and triaxiality of the star are included, emission occurs at other frequencies too [139], [129], [75].", "In this analysis, the HMM formulation generally follows the description in Section REF , with three major updates.", "First, two different coherent times of $T_{\\rm coh}= 12$  hr and 9 hr are selected for three sources with $t_{\\rm age}\\gtrsim 20$  kyr and four sources with $t_{\\rm age}\\lesssim 5$  kyr, respectively.", "Second, two frequency components are tracked simultaneously.", "The GW signal for each frequency component is assumed to be monochromatic over $T_{\\rm coh}$ .", "The signal power in each frequency bin is computed by the two-component $\\mathcal {F}$ -statistic, denoted by $\\mathcal {F}_1(f_i) + \\mathcal {F}_2(2f_i)$ , where $\\mathcal {F}_1$ and $\\mathcal {F}_2$ are the $\\mathcal {F}$ -statistic outputs computed in two separate frequency bands, and $f_i$ is the frequency value in the $i$ th bin.", "We use $\\Delta f = 1/(4 T_{\\rm coh})$ and $2\\Delta f = 1/(2 T_{\\rm coh})$ as frequency bin sizes when computing $\\mathcal {F}_1$ and $\\mathcal {F}_2$ , respectively, such that both the $f_\\star $ and $2f_\\star $ signal components stay in one bin for each time interval $T_{\\rm coh}$ .", "Third, we assume that the signal frequency evolution is dominated by secular spin down, and can be approximated by a negatively biased random walk.", "The unknown spin-down rate lies in the range between zero and the maximum estimated spin-down rate and can vary over time.", "Hence we use a transition probability matrix $A_{q_{i-1} q_i} = A_{q_i q_i} = 1/2$ , with all other entries being zero.", "The full frequency band is divided into 1-Hz and 1.5-Hz sub-bands for $T_{\\rm coh}= 12$  hr and $T_{\\rm coh}= 9$  hr, respectively, to parallelize computing.", "The detection statistic used in this analysis requires that the number of frequency bins in each sub-band (with bandwidth $B$ ) is significantly larger than the total number of tracking steps (i.e., $2BT_{\\rm coh}\\gg T_{\\rm obs}/T_{\\rm coh}$ ).", "Thus for $T_{\\rm coh}= 9$  hr, we choose a 0.5-Hz wider sub-band such that the requirement is satisfied.", "More details are provided in Appendix REF .", "Seven sources in the top half of Table REF with an assumed age of $t_{\\rm age} \\gtrsim 3$  kyr are searched using this method.", "Due to the fact that two frequency bands are combined, this method is susceptible to noise features present in either band.", "Coherent times shorter than $\\sim 5$  hr and correspondingly, wider $\\Delta f$ , can further degrade the sensitivity.", "Hence we do not search the other eight sources with $t_{\\rm age} \\lesssim 3$  kyr that require a much shorter $T_{\\rm coh}$ .", "The parameter space covered for each source is listed in Table REF .", "The $\\dot{f}_\\star $ range covered in this analysis is hence $|\\dot{f}_\\star | \\in [0, 1/(4T_{\\rm coh}^2)]$ .", "The frequency range is determined as follows.", "For all seven sources, we fix the minimum frequency at 50 Hz and 100 Hz for $f_\\star $ and $2f_\\star $ , respectively.", "We do not search below 50 Hz because the number of instrumental lines in each 1-Hz band significantly increases at low frequencies and the optimal Viterbi paths would be dominated by noise artifacts.", "The maximum frequency is set by the assumed minimum characteristic age of the source, $t_{\\rm age}$ (the second column in Table REF ), assuming $|\\dot{f}_\\star | = f_\\star (n-1)^{-1} t_{\\rm age}^{-1}$  [121], [11], where $n=f_\\star \\ddot{f}_\\star /\\dot{f}_\\star ^2$ is the braking index with $\\ddot{f}_\\star $ being the second time derivative of $f_\\star $ .", "We assume the spin down of the star is dominated by gravitational radiation due to a non-zero ellipticity, i.e., $n=5$ .", "Table: Sources searched in the dual-harmonic Viterbi analysis (Section ) and the parameter space covered.We use the Viterbi score $S$ as the detection statistic in the dual-harmonic search, which indicates the significance of the optimal Viterbi path obtained in each sub-band compared to all other paths in that band at the final step of the tracking.", "Given that the condition $N_T \\ll N_Q$ is generally satisfied with the choices of $T_{\\rm coh}$ in this method, the issue described in Section REF with short $T_{\\rm coh}\\sim 1$  hr does not happen.", "The full mathematical definition of $S$ is given in [119].", "We determine a threshold corresponding to 1% false alarm probability $S_{\\rm th}=5.47$ and $S_{\\rm th}=5.33$ for $T_{\\rm coh}= 12$  hr and $T_{\\rm coh}= 9$  hr, respectively, obtained from Monte-Carlo simulations in Gaussian noise and verified in real O3a data.", "The results obtained from simulations in O3a interferometric noise are consistent with the Gaussian noise thresholds." ], [ "Sensitivity and constraints", "A total of 42464, 9236, and 477 first-stage candidates are identified across all SNRs in BSD, single-harmonic Viterbi, and dual-harmonic Viterbi pipelines.", "We apply a hierarchical veto procedure (Appendix REF ) to the full population and perform dedicated follow-up analyses on 35, 1, and 25 candidates for BSD, single-harmonic Viterbi, and dual-harmonic Viterbi, respectively (Appendix REF ).", "No candidate survives from any pipeline.", "All are consistent with a non-astrophysical origin.", "In this section, we present the sensitivity of each pipeline and the constraints obtained from this analysis." ], [ "BSD constraints", "Surviving candidates are all compatible with noise fluctuations and no evidence of their presence is found in Virgo O3a and/or in the full LIGO O3 data.", "We compute the constraints on the strain amplitude using a well established method used in [98] and described in [42].", "The sensitivity curve is obtained from the 95% confidence level upper limits of 10 randomly selected frequency sub-bands of 1 Hz each for targets in the [10, 1000] Hz frequency band, and 9 sub-bands for the remaining targets.", "The $h_0^{95\\%}$ in the sub-bands is computed with the frequentist approach, i.e., injecting 50 signals with a given amplitude $h_0$ and computing the corresponding detection efficiency.", "The injections are done for each source, assuming the same sky position as the selected source for each injection.", "The spin-down and polarization parameters ($\\cos \\iota $ and $\\psi $ ) are randomly chosen from their uniform distributions.", "We repeat the injections in a given sub-band using 6–18 values of $h_0$ in the interval [1.3$\\times 10^{-26}$ , 3$\\times 10^{-23}$ ].", "The detection efficiency for a given amplitude $h_0$ is given by the fraction of injections recovered.", "The actual $h_0^{95\\%}$ corresponding to a detection efficiency of 0.95 is derived from the sigmoidal fit of the detection efficiency curve versus the injected amplitude.", "Given that the sensitivity to $h_0$ is proportional to $\\sqrt{S_n(f)}$ , which is the noise amplitude spectral density, we compute the Normalized Upper Limit (NUL), $h_{\\rm NUL}(f_i)={h_0^{95\\%}(f_i)}/{\\sqrt{S_n(f_i)}}$ , in each of the randomly chosen sub-bands.", "We remark that it is the inverse of the more widely used “sensitivity depth\" [28].", "Since the NUL values should follow a linear trend, given by the dependence of the coherence time used in each 10 Hz band, we extrapolate the NUL values of the remaining bands with a linear fit of the NUL versus frequency.", "In this way we can translate the NUL values, interpolated from the linear fit for each 1 Hz band, into the $h_0^{95\\%}(f)$ curve.", "The final $h_0^{95\\%}(f)$ curve is then obtained for each detector, by multiplying the NUL values extrapolated from the linear fit in each 1-Hz band with the corresponding value of $\\sqrt{S_n(f)}$ in that band, i.e., $h_0^{95\\%}(f)=h_{\\rm NUL}(f)\\sqrt{S_n(f)}.$ The sensitivity plots are presented in Figure REF where we also report the indirect age-based limit from Eq.", "(REF ) (solid line) for each target.", "The best sensitivity is below the indirect age-based limit for all the sources.", "In particular for G65.7+1.2, G189.1+3.0 and G266.2–1.2/Vela Jr., this happens for the full frequency band analyzed, except for the most disturbed regions, and for all the detectors.", "The difference in sensitivity among the analyzed targets, is caused by the different antenna pattern response due to different sky locations of the sources, even when the same coherence time is used for multiple sources.", "We present different curves for each detector; the combined $h_0^{95\\%}(f)$ result would correspond to the one for the less sensitive LIGO detector.", "The best sensitivity at 95% confidence level occurs at the Livingston detector at $h_0 \\approx 7.8\\times 10^{-26}$ near 200 Hz for G65.7+1.2 and at $h_0 \\approx 7.7\\times 10^{-26}$ for G39.2–0.3 in the same bucket region." ], [ "Single-harmonic Viterbi constraints", "We report no evidence of CWs in the single-harmonic Viterbi search.", "In this section, we estimate the sensitivity of this search across nine of the fifteen sources.", "We estimate the sensitivity first using Eq.", "(REF ) and assume this is a reasonable representation of the key parameters determining the sensitivity, i.e.", "that between sources, the sensitivity of the search is predominantly determined by $T_{\\rm coh}$ .", "So we determine the sensitivity for $T_{\\rm coh} = 1$  hr using G266.2–1.2 and G347.3–0.5 and assume the variation in sky position for other targets with the same $T_{\\rm coh}$ has a negligible effect on sensitivity.", "This assumption has been validated through detailed simulations.", "For each source we set limits on, we inject 100 simulated signals with fixed $h_0$ , and randomly selected $f$ and $\\dot{f}_0$ into five frequency sub-bands, selected at random from a set of bands with no known lines, and which returned $< 2$ unique paths with $\\mathcal {L} > \\mathcal {L}_{\\rm th}$ in the original search.", "We then apply the Viterbi algorithm to each injection.", "We repeat this for 5–10 values of $h_0$ .", "Each set of $N_I = 100$ injections forms a binomial distribution, with each injection and search acting as a Bernoulli trial with a probability of success (efficiency) $p$ .", "We infer the value of $p$ given $s$ successes for each $h_0$ given using the Wilson interval [134] $p \\approx \\frac{s + \\frac{1}{2}(1-\\alpha _F/2)^2}{N_I + (1-\\alpha _F/2)^2} \\pm \\frac{1-\\alpha _F/2}{N_I+(1-\\alpha _F/2)^2}\\sqrt{\\frac{s(N_I-s)}{N_I} + \\frac{(1-\\alpha _F/2)^2}{4}},$ where $\\alpha _F$ is the false alarm probability.", "For each frequency band, we fit a sigmoid curve (as in [27]) to the set of $h_0$ and the corresponding $p$ using the Bayesian inference package Bilby [22] with a uniform prior over the sigmoid parameters.", "We sample the posterior and, for each sample, determine the $h_0^{\\rm 95\\%}$ as the $h_0$ corresponding to $p = 95\\%$ .", "We take the average $h_0^{95\\%}$ of this population to be the 95% frequentist confidence upper limit in that frequency band.", "For each frequency band, we calculate $a = h_0^{95\\%}/h_0^{\\rm est}$ at the appropriate frequency, where $h_0^{\\rm est}$ is estimated by Eq.", "(REF ).", "Lastly, we find the mean $a$ across the five frequency bands and calculate the sensitivity across the full frequency band as $h_0^{95\\%} = a h_0^{\\rm est}$ , plotted as the curves in Figure REF .", "We overplot the age-based limit from Eq.", "(REF ) (dashed line) for each target.", "Our search is more sensitive than the age-based limit for all targets except G18.9-1.1, G39.2-0.3, G330.2+1.0, and G353.6-0.7, despite G353.6-0.7 having the smallest detectable strain in this search, $2.64 \\times 10^{-25}$ at 172 Hz.", "The targets with the poorest overall sensitivity (those with short $T_{\\rm coh}$ ) place the tightest constraints relative to the age-based spin-down limit.", "The constraints obtained in this search are for a random-walk signal model including spin down and spin wandering.", "The random-walk signal model (including spin down and spin wandering) and the range of $\\dot{f}_0$ searched (up to $\\dot{f}_0^{\\rm max} = 3.9\\times 10^{-8}$  Hz/s for $T_{\\rm coh} = 1$  hr) mean the $h_0^{95\\%}$ for this search is less stringent than for the other pipelines in this and other papers, which use a different signal model (e.g.", "Taylor expansion) and smaller range of $\\dot{f}$ .", "For G65.7+1.2, one of the injections at just over 1000 Hz appears to be on a noise spike despite known noise features being filtered out, however, the scale factor obtained for that band is consistent with the other four bands tested.", "Figure: The sensitivity estimate h 0 95% h_0^{95\\%} obtained from the single-harmonic Viterbi search for each source.", "Multiple sources have T coh =1T_{\\rm coh} = 1 hr and have the same sensitivity; these sources are shown on one plot for a representative source, G266.2–1.2.", "The blue curves represent the estimated h 0 95% h_0^{95\\%} in the full band searched by the single-harmonic Viterbi pipeline (Table ).", "The orange crosses represent the h 0 95% h_0^{95\\%} values obtained empirically in the sample sub-bands.", "The black dashed line is the age-based upper limit on the gravitational-wave strain from Eq.", "()." ], [ "Dual-harmonic Viterbi constraints", "No evidence of CWs is found in the dual-harmonic Viterbi search.", "We empirically derive the sensitivity by estimating the signal strain $h_0^{95\\%}$ in each frequency sub-band (as a function of $f_\\star $ ), such that a signal with $h_0 \\ge h_0^{95\\%}$ can be detected on 95% or more occasions.", "Since this pipeline considers a signal model with both $f_\\star $ and $2f_\\star $ components, we use $f_\\star $ instead of the GW frequency to avoid confusion.", "Note that the sensitivity on the strain $h_0$ quoted in this pipeline is based on a different signal model from the other two pipelines [see Eqs.", "(REF )–()].", "Here we assume the special scenario $\\theta =45$  deg and $\\cos \\iota = 0$ , i.e., signals at both $f_\\star $ and $2f_\\star $ are linearly polarized.", "In this scenario, tracking the two frequency bands simultaneously offers the most significant sensitivity improvement from searching a single band, compared to other choices of $\\theta $ and $\\cos \\iota $  [119].", "Figure REF shows $h_0^{95\\%}$ in all sub-bands and a set of frequentist upper limits obtained through injections in a handful of randomly selected sample sub-bands (orange cross markers).", "The procedure to produce these results is as follows.", "First, we derive the frequentist upper limit in one sample sub-band, starting from 112.5 Hz for $f_\\star $ and 225 Hz for $2f_\\star $ .", "A set of 200 synthetic signals are injected into the O3a data at random sky positions in that sub-band with a fixed $h_0$ .", "We use fixed $\\theta =45$  deg and $\\cos \\iota = 0$ .", "The other source parameters, including $f_\\star $ , $\\dot{f}_\\star $ , the polarization angle, and the initial phase, are randomly drawn from their uniform distributions.", "The corresponding detection rate is calculated.", "This process is repeated with different $h_0$ values with step size $1 \\times 10^{-26}$ and $2 \\times 10^{-26}$ in the regions where the detection rate is roughly above and below 50%, respectively.", "With all the injected $h_0$ values and the corresponding detection rates, $h_0^{95\\%}$ is obtained through a sigmoidal fit.", "The $h_0^{95\\%}$ value found in the sample sub-band is $2.9 \\times 10^{-25}$ and $3.2 \\times 10^{-25}$ for $T_{\\rm coh}=12$  hr and 9 hr, respectively.", "Next, we use these values obtained in the sample sub-band (starting from 112.5 Hz for $f_\\star $ and 225 Hz for $2f_\\star $ ) to analytically calculate $h_0^{95\\%}$ in the full frequency band (blue dots), using the scaling [119] $h_0^{95\\%}(f) \\propto \\left(\\frac{S_n(f)S_n(2f)}{S_n(f)+S_n(2f)}\\right)^{1/2},$ where $S_n$ is the effective power spectral density calculated from the harmonic mean of the two detectors over all the 30-min SFTs collected from September 1 to October 1, 2019 (GPS time 1251331218–1253923218).", "Finally, in order to verify the analytical scaling, the simulation procedure in the first step is repeated in several other randomly selected sub-bands, indicated by the orange cross markers.", "The $h_0^{95\\%}$ values obtained empirically in those sample sub-bands agree to $<1.5\\%$ with the analytic sensitivity estimates.", "In the full frequency band searched, the best $h_0^{95\\%}$ values for $T_{\\rm coh}=12$  hr and 9 hr are $2.88 \\times 10^{-25}$ at $f_\\star = 158.75$  Hz and $3.17 \\times 10^{-25}$ at $f_\\star = 123.75$  Hz, respectively.", "These results are obtained from randomized sky positions and hence apply to all sources using the same $T_{\\rm coh}$ .", "In the dual-harmonic Viterbi pipeline, the sensitivity is dominated by the length of $T_{\\rm coh}$ (for a fixed $T_{\\rm obs}$ ) rather than the sky position of the source.", "Additional spot checks validate that the difference between the empirical $h_0^{95\\%}$ values obtained from a fixed sky location and those from randomized sky positions is negligible.", "Assuming that the star's rotational kinetic energy loss is all radiated in GWs, the age-based indirect strain limits $h_0^{\\rm age}$ can be calculated for each source by fixing $\\theta =45$  deg (consistent with the scenario presented in Figure REF ), setting $t_{\\rm age}$ to the value in Table REF , and setting the distance to the minimum value in Table REF .", "Note that the $h_0^{\\rm age}$ value derived explicitly for the dual-harmonic model with $\\theta =45$  deg is a factor of $\\sim 2$ larger than the value calculated from Eq.", "(REF ) [139], [133].", "For five out of the seven sources, the indirect limits are much larger than the constraints obtained in this search across the full band.", "For G39.2–0.3, the $h_0^{95\\%}$ has beaten the indirect limit at most of the frequencies except for the noisy bands around 60 Hz.", "For G353–0.7, the $h_0^{95\\%}$ obtained from the search is close to $h_0^{\\rm age}$ but has not reached it at any frequency.", "We emphasize that the sensitivity in the dual-harmonic Viterbi pipeline, and whether it beats the indirect limit, is not directly comparable to other methods due to the model difference." ], [ "Astrophysical implications", "The sensitivity in terms of the GW strain amplitude can be converted into constraints on the fiducial ellipticity of the neutron star, $\\epsilon $ [62], and the $r$ -mode amplitude parameter, $\\alpha $ [92].", "We first discuss the constraints obtained in the BSD and single-harmonic Viterbi pipelines.", "For ellipticity, we assume the GW frequency $f$ (equivalent to $f_0$ in the single-harmonic Viterbi pipeline) is at $2f_\\star $ , which aligns with the model of a perpendicular biaxial rotor considered in both pipelines.", "Given $h_0^{95\\%}$ (derived with a uniform prior on the $\\cos \\iota $ ), we constrain the ellipticity of the neutron star in terms of the GW frequency $f=2f_\\star $ via [62] $ \\epsilon = 9.46\\times 10^{-5} \\left(\\frac{h_0}{10^{-24}}\\right) \\left(\\frac{D}{1{\\rm ~kpc}}\\right) \\left(\\frac{100{\\rm ~Hz}}{f}\\right)^2,$ assuming the moment of inertia with respect to the rotation axis ($I_{zz}$ for a perpendicular biaxial rotor) is $10^{38}$  kg m$^2$ .", "We can also convert $h_0^{95\\%}$ to limits on the amplitude of $r$ -mode oscillations via [92] $\\alpha \\simeq 0.028\\left(\\frac{h_{0}}{10^{-24}}\\right)\\left(\\frac{D}{1 \\mathrm {~kpc}}\\right)\\left(\\frac{100 \\mathrm {~Hz}}{f}\\right)^{3}.$ Figures REF and REF present the constraints on $\\epsilon $ and $\\alpha $ obtained from the BSD and single-harmonic Viterbi pipelines, respectively.", "The most stringent constraints, $\\epsilon \\lesssim 10^{-7}$ and $\\alpha \\lesssim 10^{-5}$ , come from the BSD pipeline (see Figure REF ), where the values are converted using the $h_0^{95\\%}$ curve from the L detector.", "The results from the single-harmonic Viterbi pipeline, covering more targets and a wider parameter space, are presented in Figure REF .", "We also convert the $h_0^{95\\%}$ obtained in the dual-harmonic Viterbi pipeline to the 95% confidence constraint on the ellipticity of the star.", "Since GW emission is at both $f_\\star $ and $2f_\\star $ , Eq.", "(REF ) can be written in terms of the spin frequency of the star $f_\\star $ , $ \\epsilon = 2.36\\times 10^{-5} \\left(\\frac{h_0}{10^{-24}}\\right) \\left(\\frac{D}{1{\\rm ~kpc}}\\right) \\left(\\frac{100{\\rm ~Hz}}{f_\\star }\\right)^2.$ Figure REF shows the limits on $\\epsilon $ as a function of $f_\\star $ .", "Note that the results here are converted from the $h_0^{95\\%}$ values obtained for a specific scenario with source properties $\\theta =45$  deg and $\\cos \\iota = 0$ , and hence the $\\epsilon $ values in Figure REF are not directly comparable to the results obtained in other conventional searches where $\\theta =90$  deg and the emission is only at $2f_\\star $ .", "The signal model adopted in the dual-harmonic search cannot be interpreted as current quadrupole emission from an $r$ -mode, so we do not infer $r$ -mode amplitudes from $h_0^{95\\%}$ .", "The strictest constraints on the intrinsic GW strain from the BSD pipeline are $h_0^{95\\%}\\approx 7.7 \\times 10^{-26}$ for G39.2–0.3 and $h_0^{95\\%}\\approx 7.8 \\times 10^{-26}$ for G65.7+1.2 near 200 Hz.", "The results obtained by the Viterbi pipelines set the first constraints on CWs which allow for spin wandering in the signal model.", "Note that the authors of [87] conducted a search for 13 out of the 15 sources in Advanced LIGO O2 data using the single-harmonic Viterbi method but did not derive the constraints from the search sensitivity.", "Furthermore, the dual-harmonic Viterbi analysis provides the first results for these SNR sources derived considering two frequency harmonics simultaneously.", "The best constraints on the star's ellipticity obtained at frequencies $f \\gtrsim 100$  Hz reach $\\epsilon < 10^{-6}$ for most of the sources, reaching below the rough theoretical upper limit for normal neutron stars [65], and reach as low as $\\epsilon \\approx 6 \\times 10^{-8}$ for the closest source G266.2–1.2/Vela Jr., well below the theoretical limits.", "However, these limits are model dependent; the uncertainties on the star's geometry and composition, like the internal equation of state, the moment of inertia, and the magnetic field, play a significant role when deriving these limits.", "For example, [135] shows that an ellipticity of $\\epsilon \\sim 10^{-9}$ can be sustained by neutron stars with a buried magnetic field of $\\sim 10^{11}$  G. The most stringent constraints on the $r$ -mode amplitude obtained above $\\sim 100$  Hz arrive at the theoretical prediction level of $\\alpha \\sim 10^{-3}$ , expected for the nonlinear saturation mechanisms [31], and reach as low as $\\alpha \\sim 10^{-5}$ at higher frequencies.", "Figure: Constraints on the (a)–(b) neutron-star ellipticity ϵ 95% \\epsilon ^{95\\%}, and (c)–(d) rr-mode amplitude α 95% \\alpha ^{95\\%}, from the BSD pipeline, converted from the h 0 95% h_0^{95\\%} values in Figure .", "Panels (a) and (c) report the results derived for G189.1+3.0, G65.7+1.2 and G266.2–1.2 (Vela Jr.), covering the [10, 600] Hz frequency band.", "Panels (b) and (d) report the results for G93.3+6.9, G18.9–1.1, G39.2–0.3 and G353.6–0.7, where the [10, 1000] Hz frequency band is investigated.", "Curves have been converted from h 0 95% h_0^{95\\%} derived for the L detector.", "Shaded regions correspond to the inferred ellipticity and rr-mode amplitude using the full range of distances in Table .", "The minimum distance is assumed for the solid dot curves.Figure: Constraints on the neutron star ellipticity ((a) and (b)) and rr-mode amplitude ((c) and (d)) from the single-harmonic Viterbi pipeline converted from the h 0 95% h_0^{95\\%} values in Figure .", "The results are plotted as a function of the GW frequency ff.", "The solid line uses the closest distance estimate in Table  and the shaded area indicates the results across the full distance range.", "Panels (a) and (c) display the results for targets with f max <1500 Hz f_{\\rm max} < 1500 \\, {\\rm Hz}; panels (b) and (d) display results for targets with f max >1500 Hz f_{\\rm max} > 1500 \\, {\\rm Hz}.Figure: Constraints on the neutron star ellipticity with 95% confidence from the dual-harmonic Viterbi search as a function of f ☆ f_\\star for (a) T coh =12T_{\\rm coh}=12 hr and (b) T coh =9T_{\\rm coh}=9 hr, converted from the h 0 95% h_0^{95\\%} values in Figure .", "The shaded region indicates the resulting ellipticity range calculated from the full estimated distance range for each SNR (see Table ), and the solid dots correspond to the minimum distance estimate." ], [ "Conclusion", "In this work, we present the results of the search for CWs from neutron stars in 15 young SNRs by analyzing the data collected in the first half of O3.", "No evidence of CWs is identified.", "We present constraints on the GW strain, as well as the implied mass ellipticity and $r$ -mode amplitude for each source.", "The inferred upper limits on the latter quantities reach below the maximum values allowed on theoretical grounds.", "The strictest constraints on the intrinsic GW strain from the BSD pipeline are $h_0^{95\\%}\\approx 7.7 \\times 10^{-26}$ for G39.2–0.3 and $h_0^{95\\%}\\approx 7.8 \\times 10^{-26}$ for G65.7+1.2, both near 200 Hz.", "The Viterbi pipelines set the first constraints on the signal strain allowing for spin wandering.", "The dual-harmonic Viterbi analysis reports the first results for these SNR sources derived considering two frequency harmonics simultaneously.", "All of the three pipelines are computationally efficient, costing $\\sim 10^2$  core-hr for each source per pipeline (postprocessing excluded).", "The significantly improved search efficiency compared to the existing studies for this type of source mainly comes from two factors: (1) we use a much shorter coherence time $T_{\\rm coh}$ in this analysis compared to other studies [11], [78], [94] and the computing cost scales roughly as $T_{\\rm coh}^{4}$ to $T_{\\rm coh}^{7}$ depending on the orders of the time derivatives of frequency searched; (2) advanced signal processing techniques are used in these three pipelines to further accelerate the search, e.g., the dynamic programming in Viterbi pipelines [130], [121] and the use of sub-sampled data and heterodyne correction in the BSD pipeline [97].", "We briefly compare the constraints derived in this work to existing constraints in the literature.", "The constraints from the two Viterbi-based pipelines are not directly comparable with other analyses because they assume different signal models.", "The dual-harmonic Viterbi pipeline searches simultaneously for emission at $f_\\star $ and $2f_\\star $ .", "In addition, both Viterbi pipelines search a set of frequency random walks, unlike other pipelines that search a set of Taylor expansion coefficients.", "A previous search for these sources in O1 reported $h_0^{95\\%} \\approx 2\\times 10^{-25}$ for most of the sources and $h_0^{95\\%} \\approx 1\\times 10^{-25}$ for one source in the most sensitive band  [11], excluding results on Fomalhaut B [66].", "The best limit on $h_0$ for G39.2–0.3 is $h_0^{95\\%}\\approx 2\\times 10^{-25}$ , approximately 2.5 times higher than the results obtained here for the same source with the BSD pipeline.", "[78] presented a similar search to [11] using the O2 data and reported strain limits slightly above $1 \\times 10^{-25} $ at 90% confidence level, e.g., for G39.2–0.3, which is $\\sim 1.4$ times higher than the results obtained here for the same source with the BSD pipeline.", "Also, using the LIGO data in the full O1 run and a coherent integration duration longer than 10 days, [94] set 90% confidence limits of $h_0^{90\\%} = 1.2\\times 10^{-25}$ , $9.3\\times 10^{-26}$ , and $8.8\\times 10^{-26}$ for Cas A, Vela Jr. and G347.3–0.5 near 172.5 Hz, respectively.", "These limits for Cas A and G347.3–0.5 are lower than those obtained in the single-harmonic Viterbi search for these two SNRs, without any adjustments being made for the different signal models assumed.", "We do not search for these two SNRs using the BSD and dual-harmonic Viterbi pipelines.", "The results on Vela Jr. in this analysis from the BSD search slightly improve the previous constraints set by [94], despite the use of a coherent integration duration $\\sim 20$ times shorter.", "Future data collection and improved analysis methods will further extend the sensitivity of CW searches and increase the probability of a future discovery." ], [ "Acknowledgments", "This material is based upon work supported by NSF’s LIGO Laboratory which is a major facility fully funded by the National Science Foundation.", "The authors also gratefully acknowledge the support of the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector.", "Additional support for Advanced LIGO was provided by the Australian Research Council.", "The authors gratefully acknowledge the Italian Istituto Nazionale di Fisica Nucleare (INFN), the French Centre National de la Recherche Scientifique (CNRS) and the Netherlands Organization for Scientific Research, for the construction and operation of the Virgo detector and the creation and support of the EGO consortium.", "The authors also gratefully acknowledge research support from these agencies as well as by the Council of Scientific and Industrial Research of India, the Department of Science and Technology, India, the Science & Engineering Research Board (SERB), India, the Ministry of Human Resource Development, India, the Spanish Agencia Estatal de Investigación, the Vicepresidència i Conselleria d'Innovació, Recerca i Turisme and the Conselleria d'Educació i Universitat del Govern de les Illes Balears, the Conselleria d'Innovació, Universitats, Ciència i Societat Digital de la Generalitat Valenciana and the CERCA Programme Generalitat de Catalunya, Spain, the National Science Centre of Poland and the Foundation for Polish Science (FNP), the Swiss National Science Foundation (SNSF), the Russian Foundation for Basic Research, the Russian Science Foundation, the European Commission, the European Regional Development Funds (ERDF), the Royal Society, the Scottish Funding Council, the Scottish Universities Physics Alliance, the Hungarian Scientific Research Fund (OTKA), the French Lyon Institute of Origins (LIO), the Belgian Fonds de la Recherche Scientifique (FRS-FNRS), Actions de Recherche Concertées (ARC) and Fonds Wetenschappelijk Onderzoek – Vlaanderen (FWO), Belgium, the Paris Île-de-France Region, the National Research, Development and Innovation Office Hungary (NKFIH), the National Research Foundation of Korea, the Natural Science and Engineering Research Council Canada, Canadian Foundation for Innovation (CFI), the Brazilian Ministry of Science, Technology, and Innovations, the International Center for Theoretical Physics South American Institute for Fundamental Research (ICTP-SAIFR), the Research Grants Council of Hong Kong, the National Natural Science Foundation of China (NSFC), the Leverhulme Trust, the Research Corporation, the Ministry of Science and Technology (MOST), Taiwan, the United States Department of Energy, and the Kavli Foundation.", "The authors gratefully acknowledge the support of the NSF, STFC, INFN and CNRS for provision of computational resources.", "This work was supported by MEXT, JSPS Leading-edge Research Infrastructure Program, JSPS Grant-in-Aid for Specially Promoted Research 26000005, JSPS Grant-in-Aid for Scientific Research on Innovative Areas 2905: JP17H06358, JP17H06361 and JP17H06364, JSPS Core-to-Core Program A.", "Advanced Research Networks, JSPS Grant-in-Aid for Scientific Research (S) 17H06133, the joint research program of the Institute for Cosmic Ray Research, University of Tokyo, National Research Foundation (NRF) and Computing Infrastructure Project of KISTI-GSDC in Korea, Academia Sinica (AS), AS Grid Center (ASGC) and the Ministry of Science and Technology (MoST) in Taiwan under grants including AS-CDA-105-M06, Advanced Technology Center (ATC) of NAOJ, and Mechanical Engineering Center of KEK.", "We would like to thank all of the essential workers who put their health at risk during the COVID-19 pandemic, without whom we would not have been able to complete this work." ], [ "Candidate follow up", "Narrow-band noise features and the non-Gaussianity in the interferometric data can cause outliers with detection statistic above the threshold.", "Consequently, each pipeline requires postprocessing of the results to eliminate candidates originating from noise artifacts.", "We follow up all the first-stage candidates identified in each pipeline with a hierarchy of predefined veto procedures as well as additional manual scrutiny.", "No candidate survives from any pipeline.", "In this section, we detail the postprocessing procedures." ], [ "Vetoes", "We first describe the predefined veto procedures in this section.", "Candidates caused by known instrumental lines are rejected in the first step in all three pipelines.", "For each candidate identified at a starting frequency $f_0$ at $t=0$ , we veto the candidate if the band $[f_0-\\delta f, f_0+\\delta f]$ intersects any known instrumental lines present in either the Hanford or Livingston interferometer, where $\\delta f = 10^{-4} f_0$ is used to account for the Doppler shift due to the Earth's motion.", "Note that there is a subtle difference between the pipelines when applying the Doppler shift effect.", "The BSD and Viterbi pipelines apply $\\delta f$ to the line frequency and the candidate frequency, respectively.", "The three pipelines use a list of known instrumental lines in C01 data [53].", "In the dual-harmonic Viterbi pipeline, candidates caused by instrumental lines in either of the two separate sub-bands, corresponding to the $f_\\star $ and $2f_\\star $ components, are rejected." ], [ "Interferometer veto", "In general, a candidate signal with an astrophysical origin should be present in the data of all detectors.", "If the candidate is louder in one detector than the other, it ought to be louder in the detector with better sensitivity for the source and in the frequency band considered.", "Each pipeline therefore applies a veto to the consistency of the candidate signal strength across each detector.", "For the BSD algorithm, this means vetoing candidates with a weighted CR in the less sensitive detector more than three times higher than the corresponding weighted CR in the more sensitive one.", "This is applied using the CR computed from the statistical distribution of the FH number counts.", "This veto is repeated after the next veto step REF , using another statistic, namely the 5-vector statistic  [25], [24], as the second round of consistency check.", "All the candidates with ${\\rho _{\\rm CR_{1}}}/{\\sqrt{S_{n_1}}} > 3 {\\rho _{\\rm CR_{2}}}/{\\sqrt{S_{n_2}}}$ are vetoed, where $\\sqrt{S_{n_i}}$ is the noise amplitude spectral density in each $i$ -th detector, and assuming that detector 1 is less sensitive than detector 2.", "This is an arbitrary and conservative choice, already used in [12], with a factor of 3 we do not need to consider an eventual weak dependence due to the different detectors orientation.", "For the Viterbi algorithms, we repeat the full search over $T_{\\rm obs}$ in each individual interferometer.", "The candidate is vetoed if the two criteria are both satisfied: a) searching data from a single interferometer yields $\\mathcal {L} \\ge \\mathcal {L}_{\\cup }$ in single-harmonic Viterbi ($S \\ge S_{\\cup }$ in dual-harmonic Viterbi), where $\\mathcal {L}_{\\cup }$ ($S_{\\cup }$ ) is the original statistic obtained with both interferometers combined, while searching the other interferometer yields $\\mathcal {L} < \\mathcal {L}_{\\cup }$ ($S < S_{\\rm th}$ ); and b) the Viterbi path from the interferometer with $\\mathcal {L} \\ge \\mathcal {L}_{\\cup }$ ($S \\ge S_{\\cup }$ ) intersects the original path, i.e., the increased significance in a single detector occurs at the same frequency as the original candidate." ], [ "Doppler-shift (and spin-down-shift) veto", "All three pipelines apply a Doppler correction to transform the observation in the detector frame to the source frame.", "For a true astrophysical signal, this correction should increase the significance of any candidate in the data; for local noise, the significance should decrease or remain unaffected.", "Both the BSD and the dual-harmonic Viterbi pipeline apply a veto based on this correction.", "This is not applied to the single-harmonic Viterbi pipeline because the $T_{\\rm coh}$ are short enough to track the Doppler shift.", "For the BSD algorithm, we compute the significance of the candidate in terms of the CR and signal-to-noise ratio ($\\rho _i$ where $i=\\rm CR, snr$ ) with and without the Doppler and spin-down corrections applied to the time series.", "The Doppler and spin-down corrections are done using a heterodyne phase correction, where the assumed phase evolution of the signal, $\\phi (f_0,\\dot{f}_0)$ , is fully described by the frequency and spin-down parameters of the candidate, given by $f_0$ and $\\dot{f}_0$ , respectively.", "The corrected time series is computed by multiplying the original (uncorrected) time series by the exponential factor $\\exp {j\\phi (f_0, \\dot{f}_0)}$ .", "An easy way to compute the statistical significance of a candidate, once the $\\phi (f_0,\\dot{f}_0)$ is assumed to be known, is to use the 5-vector statistics [25], [24], originally developed for the search of known pulsars.", "Hence, for this step, we use a statistic based on the 5-vector method, whose main properties are described in Appendix REF .", "We use $\\rho _{\\rm CR}$ and $\\rho _{\\rm snr}$ to check the nature of a candidate, but this time we compute them from the 5-vector statistic $\\mathcal {S}$ rather than from the FH number count.", "For this reason, the CR computed in this step is not directly comparable with the CR of the FH map used for the first level selection of candidates.", "If a candidate is from astrophysical origin, we expect, after testing the procedure with simulated signals injected in O3 data, that the significance will increase after the correction, and that it would increase proportionally to the fourth root of the coherence time.", "This comparison is done using two different coherence times, $T_{\\rm sid}$ and $T_{\\rm 4 \\, sid}=4T_{\\rm sid}$ , where $T_{\\rm sid}=86164.0905$  s is the duration of a sidereal day.", "We use the 5-vector $\\rho _{i}$ CR and signal-to-noise ratio to check the change of significance.", "We keep the candidates if a larger 5-vector $\\rho _{\\rm CR,C}$ is obtained with the correction applied than the 5-vector $\\rho _{\\rm CR,NC}$ obtained without the correction applied (in the two cases using $T_{\\rm sid}$ and $T_{\\rm 4 \\, sid}$ ).", "We also veto those candidates which do not show an increased signal-to-noise ratio after the correction.", "Simulation studies show that the false dismissal probability of this veto is below 10% if a tolerance of 5% is used, e.g.", "we keep all those candidates with $\\rho _{i,C}-0.95\\rho _{i,NC}>0$ where $\\rho _{i,\\rm C}$ and $\\rho _{i, \\rm NC}$ refer the corrected and uncorrected case, respectively.", "In the dual-harmonic Viterbi search, for each candidate remaining, we recompute $\\mathcal {F}$ -statistics over the same $T_{\\rm coh}$ as listed in Table REF with Doppler modulation correction turned off (DM-off), and repeat the search using the DM-off $\\mathcal {F}$ -statistics [138].", "If the candidate is of astrophysical origin, it should become undetectable in the DM-off search, returning a score $S_{\\rm DM-off} < S_{\\rm th}$ and a Viterbi path different from the original one.", "This criterion does not apply to high signal-to-noise candidate, i.e., $S\\gg S_{\\rm th}$ .", "However, after previous veto steps, no such high signal-to-noise candidate is left in this search.", "Instead, if a candidate is caused by noise artifacts on Earth, its significance is expected to increase in the follow-up.", "Hence we veto a candidate if the DM-off follow-up yields $S_{\\rm DM-off} \\ge S_\\cup $ and returns a new Viterbi path intersecting the band $[f_0-\\delta f, f_0+\\delta f]$." ], [ "Sky-position veto", "If a candidate is of astrophysical origin, it should yield the highest detection statistic at the sky position of the source [61].", "For candidates surviving previous steps, the Viterbi pipelines conduct another follow-up step by shifting the sky position away from the true position of the SNR [66].", "This off-target veto contains two separate parts: (a) shift right ascension by an offset $\\delta _{\\rm RA}$ while keeping declination fixed at the true location, and (b) shift declination by $\\delta _{\\rm DEC}$ while keeping right ascension fixed at the true location.", "We use $\\delta _{\\rm RA}=3$  hr and $\\delta _{\\rm DEC}=30$  deg.", "These offset values are chosen based on a large number of Monte-Carlo simulations that pass veto safety check.", "For the sources with a declination angle in the range of $[-90, 0]$  deg and $(0, 90]$  deg, we set $\\delta _{\\rm DEC}$ to 30 deg and $-30$  deg, respectively.", "The single-harmonic Viterbi pipeline conducts (a) only.", "The dual-harmonic Viterbi pipeline conducts both (a) and (b).", "For (a), we veto the candidate if the off-target search yields $\\mathcal {L}_{\\rm off-target} > \\mathcal {L}_{\\rm th}$ ($S_{\\rm off-target} > S_\\cup $ ) and returns a new Viterbi path intersecting the band $[f_0-\\delta f, f_0+\\delta f]$.", "For (b), we veto the candidate if the off-target search yields $S_{\\rm off-target} > S_{\\rm th}$ and returns a new Viterbi path intersecting the band $[f_0-\\delta f, f_0+\\delta f]$.", "Note that the veto criterion for (b) is more stringent than that for (a) in the dual-harmonic Viterbi pipeline, because this analysis is more sensitive to the mismatch along the direction of declination.", "Only the candidates surviving both (a) and (b) are kept." ], [ "Cumulative-significance veto", "The significance of a CW signal should be consistent over $T_{\\rm obs}$ , and in presence of stationary noise, there should be no sudden increase or decrease in the significance when more data are used to integrate the signal.", "The BSD algorithm uses the 5-vector statistics [25], [24] to compute the cumulative signal-to-noise ratio and CR on a monthly base, increasing the amount of data used in each iteration by one month.", "We also compute this trend using an heterodyne-corrected time series with a phase correction $\\phi (f_0,\\dot{f}_0)$ obtained from the candidates parameters $(f_0,\\dot{f}_0)$ .", "The two trends, derived from the corrected and uncorrected time series, are then compared.", "The comparison is done looking at the plots of the CR, the signal-to-noise ratio and the 5-vector statistics $\\mathcal {S}$ (defined in Eq.", "REF of Appendix REF ) as a function of the number of months used to compute these quantities.", "We visually inspect these plots by comparing the trend of both curves in the corrected and uncorrected case.", "We veto the candidates if the plot of the corrected case has lower values than the corresponding uncorrected case for the entire duration of the run.", "We also veto candidates when the CR cumulative curve of the most sensitive detector is well below the less sensitive one, which is a clear clue that the candidate is actually due to some noise present in the less sensitive detector.", "For other more complicated cases, we do not automatically veto the candidate but leave them for further investigation in the full O3 H and L data.", "This is a conservative choice, since vetoing all the candidates that simply present a sudden increase or decrease in the significance (in terms of either CR, signal-to-noise ratio and 5-vector statistic value) is not safe when the noise is not Gaussian." ], [ "Sub-band veto", "If a sub-band is heavily contaminated by non-Gaussian noise, it can be challenging to distinguish noise from a candidate signal.", "In the case of the single-harmonic Viterbi pipeline, this renders $\\mathcal {L}_{\\rm th}$ invalid because $\\mathcal {L}_{\\rm th}$ is calculated using the results of Gaussian noise simulations.", "Furthermore, we do not expect multiple CW signals in a single sub-band.", "Consequently, we veto any candidates in a sub-band if the sub-band has more than two unique Viterbi paths with $\\mathcal {L} > \\mathcal {L}_{\\rm th}$ .", "Simulations in Gaussian noise found $<1\\%$ of bands returned two unique paths with $\\mathcal {L} > \\mathcal {L}_{\\rm th}$ , justifying our assumption that a sub-band with multiple candidates is dominated by non-Gaussian noise.", "The dual-harmonic Viterbi pipeline uses the Viterbi score as the detection statistic and only keeps the optimal path, and hence this step does not apply." ], [ "Coherence-time veto", "Both implementations of the Viterbi algorithm use the $\\mathcal {F}$ -statistic computed over each $T_{\\rm coh}$ interval in Tables REF and REF .", "In the original search, we select $T_{\\rm coh}$ assuming a range of $\\dot{f}_0$ .", "Candidates returned with relatively low $|\\dot{f}_0|$ allow us to increase the coherent time in a follow-up search.", "The sensitivity of the $\\mathcal {F}$ -statistic increases with longer $T_{\\rm coh}$ , as long as there is no power leakage over $T_{\\rm coh}$ given the inferred $|\\dot{f}_0|$ ; a more sensitive $\\mathcal {F}$ -statistic facilitates a more sensitive Viterbi search.", "Hence the significance of the candidate should increase with longer $T_{\\rm coh}$ if the candidate is a real astrophysical signal.", "This has been verified using simulations.", "In practice, we first calculate the mean $\\dot{f}_0$ value over the candidate path, then estimate the maximum $T_{\\rm coh}$ capable of tracking the inferred spin down.", "In the single-harmonic Viterbi pipeline, we conduct a follow-up search using $T_{\\rm coh} = 4$  hr for all survivors.", "With the increased $T_{\\rm coh}$ , the ratio $\\mathcal {L}/\\mathcal {L}_{\\rm th}$ should increase for a real signal, so we veto any candidates for which $\\mathcal {L}(T_{\\rm coh} = 4 \\, {\\rm hr})/\\mathcal {L}_{\\rm th}(T_{\\rm coh} = 4 \\, {\\rm hr}) < \\mathcal {L}/\\mathcal {L}_{\\rm th}$ , where $\\mathcal {L}/\\mathcal {L}_{\\rm th}$ are the values from the initial search.", "Similarly, in the dual-harmonic Viterbi pipeline, we veto a candidate if a decreased Viterbi score is returned with the increased $T_{\\rm coh}$ .", "No candidate survives in the dual-harmonic Viterbi pipeline after this step." ], [ "Further verification", "After the hierarchy of well-defined veto steps, we discuss the additional verification conducted in each pipeline." ], [ "BSD follow up", "A total of 35 candidates identified by the BSD pipeline survive the vetoes described in Appendix REF .", "This is consistent given the low CR threshold chosen in Section REF , indeed we are exposed to false alarm candidates and most of them could arise from noise fluctuations.", "The $\\rho _{\\rm CR,thr}$ chosen corresponds to the probability of picking, on average, more than one noise candidate.", "Indeed the CR threshold corresponding to the selection of only one false candidate over the total number of points in the parameter space would be $\\rho _{\\rm CR,thr}\\sim 5.7$ , while for instance, in the search described in [98], the CR threshold used is 6.5.", "In this section, we describe the extra steps taken to investigate and disqualify the surviving candidates (listed in Table REF ).", "We repeat the Doppler-shift (and spin-down-shift) veto (Appendix REF ), the cumulative-significance veto (Appendix REF ), and the interferometer veto (Appendix REF ) using the full O3 (O3a and O3b) C01 data.", "We remind the reader that the CR computed by the FH is based on the number count associated to the pixel in the FH map where the candidate has been found, while the CR in the 5-vector is computed from the $\\mathcal {S}$ statistic.", "In this step, we use the CR from the $\\mathcal {S}$ statistic.", "None of the candidates in Table REF survive the full-O3 vetos and as can be seen the original CR associated to the candidate in the FH map is below the $\\rho _{\\rm CR,thr}\\sim 5.7$ , hence with a very low significance compatible with noise.", "Table: Surviving candidates from the BSD pipeline after vetos in Appendix .", "The candidates investigated were excluded using the full O3 data in the BSD search.", "The columns list the source name, original mean CR from the FH map, the candidate frequency f 0 f_0, and the spin down f ˙ 0 \\dot{f}_0 at the time of the coincidences.", "The initial set of candidates has been selected using ρ CR , thr =4.7\\rho _{\\rm CR,thr}=4.7 for G65.7+1.2, G189.1+3.0 and G266.2–1.2, ρ CR , thr =4.6\\rho _{\\rm CR,thr}=4.6 for G18.9–1.1 and G93.3+6.9, and ρ CR , thr =4.5\\rho _{\\rm CR,thr}=4.5 for G353.6–0.7 and G39.2–0.3." ], [ "Single-harmonic Viterbi follow up", "One candidate identified by the single-harmonic Viterbi pipeline survives the veto process defined in Appendix REF .", "The candidate is associated with G93.3+6.9 and has a frequency path with mean $f_0 = 1025.95 \\, {\\rm Hz}$ and $\\dot{f}_0 = -2.13\\times 10^{ -9 } \\, {\\rm Hz/s}$ and has a likelihood $\\mathcal {L} =18154.0 $ , within 5% of $\\mathcal {L}_{\\rm th}$ .", "In Figure REF , we plot the power spectral density (black curve) for both detectors combined, calculated over the full duration of O3a, and overplot the frequency of the surviving path (blue vertical line).", "Visual inspection plainly indicates that it is associated with noise.", "Though it does not lie on the peak of the noise disturbance, it is in the wings.", "Figure: Power spectral density (black curve) versus frequency and the frequency of the last surviving candidate for source G93.3+6.9 (blue vertical line).", "The power spectral density is built using data collected from both Hanford and Livingston detecters over the full observing time of O3a." ], [ "Dual-harmonic Viterbi cross check", "All candidates identified in this search pipeline are rejected after the veto procedure described in Appendix REF .", "Here we provide more details of the last set of candidates processed in the coherence-time veto (Appendix REF ).", "In Table REF , the original Viterbi score, estimated $2f_\\star $ at the beginning and the end of the observation ($2{f_\\star }_{\\rm start}$ and $2{f_\\star }_{\\rm end}$ ), and the mean $2\\dot{f}_\\star $ values [i.e., $(2{f_\\star }_{\\rm start} - 2{f_\\star }_{\\rm end})/T_{\\rm obs}$ ] of each candidate are provided.", "Note that the pipeline returns estimated frequencies and spin-down rates corresponding to the $2f_\\star $ component.", "We directly report the returned values in this section rather than converting them into the $f_\\star $ component.", "The increased $T_{\\rm coh}$ used for each candidate is listed in the sixth column.", "All of the new scores obtained by increasing $T_{\\rm coh}$ fall below the original score.", "They are all considered vetoed according to the criteria set in Appendix REF .", "To be more conservative, we further discuss the only two candidates with new scores above $S_{\\rm th}$ (although decreased compared to the original score), marked by “$*$ \" and “$\\dagger $ \" in the table.", "The one marked by “$*$ \" can be confidently ruled out since it returns a completely different path.", "The follow-up search for the candidate marked by “$\\dagger $ \" using $T_{\\rm coh}=11$  hr yields a lower mean spin-down rate, $2\\dot{f}_\\star = -1.56 \\times 10^{-10}$  Hz/s, which allows us to further increase $T_{\\rm coh}$ .", "By searching the same sub-band using $T_{\\rm coh}=15$  hr, we find the optimal path overlaps with the original one, but with a further decreased significance, $S=4.97$ .", "Hence it does not survive the further scrutiny.", "Table: Final candidates from the dual-harmonic Viterbi pipeline and the coherence-time veto results (all vetoed).", "The first five columns list the source name, original score, estimated start and end 2f ☆ f_\\star , and the mean 2f ˙ ☆ 2\\dot{f}_\\star .", "Column 6 lists the new T coh T_{\\rm coh} used in the coherence-time veto.", "The last two columns shows the follow-up results: the new score obtained by increasing T coh T_{\\rm coh}, and whether the new optimal path overlaps the original candidate path.", "The top and bottom halves of the table correspond to the searches using original T coh =12T_{\\rm coh}= 12 hr (S th =5.47S_{\\rm th}=5.47) and T coh =9T_{\\rm coh}= 9 hr (S th =5.33S_{\\rm th}=5.33), respectively.", "The GPS times for the start and end of the observation are 1238166353 and 1253975702, respectively.", "The two new scores marked by “*\" and “†\\dagger \" are above S th S_{\\rm th}.", "(Note that the pipeline returns estimated frequencies and spin-down rates corresponding to the 2f ☆ 2f_\\star component.", ")Next we describe additional verification conducted to ensure that we do not veto a weak signal at the final step accidentally.", "For all the final-stage candidates in Table REF , we cross-check them by searching in the data collected over the second half of O3, with the same configuration as in the original search, in the sub-bands where these candidates are found.", "None of the optimal paths returned in O3b data overlaps the original path (taking into consideration the possible spin down during the shutdown time between two halves of observation).", "A further consistency verification is conducted for the candidates in Table REF .", "All of them have low scores of $S \\lesssim 1.2 S_{\\rm th}$ (cf.", "$S \\sim 5 S_{\\rm th}$ vetoed at early steps).", "Hence we examine whether they are false alarms arising from noise given that the threshold chosen corresponds to 1% false alarm probability.", "Since the signal frequency is approximated by a negatively biased random walk, the mean $2|\\dot{f}_\\star |$ value of a path obtained from pure Gaussian noise over $T_{\\rm obs}$ is expected to be around $|\\dot{f}_{\\rm max}|/2$ , where $|\\dot{f}_{\\rm max}|=1/(2T_{\\rm coh}^2)$ is the maximum spin-down rate covered in the search for the $2f_\\star $ component.", "This is because the method attempts to “track\" pure Gaussian noise with a transition probability $A_{q_{i-1} q_i} = A_{q_i q_i} = 1/2$ and $2|\\dot{f}_\\star |$ can take any value in the range of $[0, |\\dot{f}_{\\rm max}|]$ .", "Figure REF shows the distribution of the mean $2|\\dot{f}_\\star |$ obtained by tracking 2000 pure Gaussian noise realizations (gray histograms; fit with black curve) and the values from the remaining candidates (blue vertical lines).", "The left and right edges of each panel are the minimum and maximum spin-down rates covered in the search, respectively.", "Out of all 25 candidates, 18 lie within the interval of $[-\\sigma , \\sigma ]$ (black dashed lines).", "For both $T_{\\rm coh}=12$  hr and $T_{\\rm coh}=9$  hr, all the candidate paths are consistent with pure noise.", "Moreover, the total number of remaining candidates is consistent with the false alarm probability (1% in each sub-band).", "Hence, these candidates with low scores are likely to be false alarms.", "This explains why they are not confidently rejected at early steps.", "Figure: Noise-only distribution of the mean 2|f ˙ ☆ |2|\\dot{f}_\\star | (gray histogram) and the values obtained from the remaining candidates (blue vertical lines) for (a) T coh =12T_{\\rm coh}=12 hr and (b) T coh =9T_{\\rm coh}=9 hr in the dual-harmonic Viterbi search.", "The right edge of each panel is the maximum spin-down rate |f ˙ max |=1/(2T coh 2 )|\\dot{f}_{\\rm max}|=1/(2T_{\\rm coh}^2) for the 2f ☆ 2f_\\star component.", "The red dashed line indicates |f ˙ max |/2|\\dot{f}_{\\rm max}|/2.", "The noise-only distribution is obtained from 2000 Gaussian noise realizations for each panel.", "The black solid curve indicates the Gaussian fit of the noise distribution.", "The two black dashed lines are the ±1σ\\pm 1\\sigma bounds." ], [ "Impact of the age and the second-order spin down", "As mentioned in Section REF , the age of the source sets the range of frequency and spin down/up we can investigate for a given target, and it does not directly affect the sensitivity of the pipeline as it happens e.g.", "for the coherence time.", "In the BSD search the first and second order spin-down/up (typically referred to as spin-down) ranges are defined by the age $t_{\\rm age}$ and the braking index $n$ of the source as $\\left\\lbrace \\begin{array}{l}-f / t_{\\rm age} \\le \\dot{f} \\le 0.1 f / t_{\\rm age} \\\\0 \\mathrm {~Hz} ~\\mathrm {s}^{-2} \\le \\ddot{f} \\le n|\\dot{f}|_{\\max }^{2} / f=n f / t_{\\rm age}^{2}\\end{array}\\right.", ".$ Since we are not explicitly removing the frequency modulation due to the second order spin down, we will limit our search to those sources which second order spin-down range is constrained in a single second order spin-down bin $\\delta \\ddot{f}$ .", "In practice, we require that $n f / t_{\\rm age}^{2} \\le \\delta \\ddot{f}$ .", "Given that the size of the second order spin-down bin is $\\delta \\ddot{f}={\\delta f}{T_{\\rm obs}^{-2}}$ as in [48], [24], where $\\delta f$ is the frequency bin size and proportional to $\\sqrt{f}$ , we can write the maximum frequency allowed for a given source (in the case of a spin down dominated by gravitational emission, hence $n=5$ ) as $f\\le 4.85\\times 10^{-13}\\left(\\frac{t_{\\rm age}}{T_{\\rm obs}}\\right)^4 \\rm Hz.$ This means that, e.g., for a source of $t_{\\rm age}=3$ kyr and $T_{\\rm obs}$ equal to the O3a run length, the maximum frequency covered in the search, neglecting the second order spin-down modulation, is $\\sim 600$ Hz." ], [ "Follow-up based on the 5-vector statistic", "In this section, we briefly recap the 5-vector method and its statistic in order to describe the new follow-up veto steps (Appendices REF and REF ) used in this search by the BSD pipeline.", "The 5-vector detection statistic is built exploiting the feature of the amplitude modulation we observe at the detector.", "This modulation is induced by the detector radiation pattern in response to a CW signal and, given that the interferometers are on Earth, also by the change of the received polarization, called sidereal modulation.", "The response of the detector to a passing CW signal, after removing the Doppler and spin-down frequency modulations, can be described as (see [25] for more details) $h(t)=H_{0}(\\eta )\\left[H_{+}(\\psi , \\eta ) A_{+}(t)+H_{\\times }(\\psi , \\eta ) A_{\\times }(t)\\right],$ where $A_{+/ \\times }(t)$ are the two sidereal responses to plus and cross polarizations and $H_{0}(\\eta )$ is the maximum signal strain.", "The plus and cross amplitudes $H_{+/ \\times }$ are given by $H_{+}=\\frac{\\cos (2 \\psi )-i \\eta \\sin (2 \\psi )}{\\sqrt{1+\\eta ^{2}}},$ $H_{\\times }=\\frac{\\sin (2 \\psi )-i \\eta \\cos (2 \\psi )}{\\sqrt{1+\\eta ^{2}}},$ and depend on the polarization angle $\\psi $ and the parameter $\\eta $ , which denotes the degree of polarization of the CW ($\\eta =0$ for a linearly polarized wave, $\\eta =\\pm 1$ for a circularly polarized wave).", "It can be shown (see e.g.", "[25]) that the frequency components of the signal at the detector are all encoded in the $A_{+/ \\times }(t)$ functions and in particular that the signal is fully described by its Fourier components at the five angular frequencies centered at the intrinsic angular frequency of the source, $\\omega _0,\\omega _0 \\pm \\Omega , \\omega _0 \\pm 2 \\Omega $ , where $\\Omega $ is the Earth’s sidereal angular frequency.", "These five-component complex vectors identify the so called 5-vector space onto which interferometric data can be projected.", "Let us call $\\vec{X}$ and $\\vec{A}_{+/ \\times }$ the 5-vectors for the data and the plus/cross polarization signal templates, respectively.", "The scalar products between $\\vec{X}$ and $\\vec{A}_{+/ \\times }$ correspond to the matched filters between the data and the signal templates, and if opportunely normalized, these two quantities are the estimators of the signal plus and cross amplitudes $\\widehat{H}_{+/ \\times }=\\frac{\\vec{X} \\cdot \\vec{A}_{+} / x}{\\left|\\vec{A}_{+/ \\times }\\right|^{2}},$ from which a detection statistic can be derived as $\\mathcal {S} \\equiv \\left|\\vec{A}_{+}\\right|^{4}\\left|\\widehat{H}_{+}\\right|^{2}+\\left|\\vec{A}_{\\times }\\right|^{4}\\left|\\widehat{H}_{\\times }\\right|^{2}.$ We can use the value of this detection statistic to compute the associated significance and the false alarm probability of a given candidate.", "To do so, we need to estimate also the noise background distribution, by repeating the calculation of $\\mathcal {S}$ in an “off-source\" analysis (far from the signal frequency).", "In the veto step described in Appendix REF , we compute the statistical properties of the data (and noise) over chunks of data of duration $T_{\\rm sid}$ , by summing up the values of the statistic in each iteration.", "In this step, we compare the statistical features of the data twice: first using a time series corrected for the Doppler and the spin-down parameters provided by the candidate, and then using no correction.", "We expect that even if the correction is not precise, if the candidate is of astrophysical origin, the significance with respect to the uncorrected case will be higher.", "The same comparison is repeated using data of longer duration (4 times longer), which should correspond to an increase of the candidate significance proportional to the fourth square root of the coherence time used.", "In this section, we outline the methods used to determine the parameter space and detection thresholds used in the single-harmonic Viterbi pipeline.", "First, we outline the process to determine the frequency range and $T_{\\rm coh}$ for each source.", "We determine the maximum and minimum spin-down rate $\\dot{f}_0$ expected for the source assuming a typical signal model with $f = 2f_\\star $ , $-\\frac{f}{(n_{\\rm min}-1)t_{\\rm age}} \\le \\dot{f}_0 \\le - \\frac{f}{(n_{\\rm max}-1)t_{\\rm age}},$ where $t_{\\rm age} = f_\\star /\\left[(n-1)\\dot{f}_\\star \\right]$ , and $n$ is the braking index.", "Because the braking index is unknown for each source, we use the most extreme plausible values $n_{\\rm min} = 2$ and $n_{\\rm max} = 7$ .", "Also, we neglect stochastic spin wandering when determining the maximum $\\dot{f}$ because we expect the spin-down rate to be much faster than the rate of spin wandering, especially in young SNRs.", "Using the maximum value of $\\dot{f}_0^{\\rm max}$ from equation (REF ), we make our first estimate of the coherence time $T_{\\rm coh} = 2^{-1/2} \\left|\\dot{f}_0^{\\rm max}\\right|^{-1/2},$ and calculate the analytically estimated sensitivity $h_0^{\\rm est}$ using Eq.", "(REF ).", "We also calculate the inferred $h_0^{\\rm max}$ using Eq.", "(REF ).", "The initial estimate of $T_{\\rm och}$ is used only to find the frequency range, with the maximum frequency covered in the search, $f_{\\rm max}$ , set to the maximum frequency for which $h_0^{\\rm max} > h_0^{\\rm est}$ .", "We then recalculate $T_{\\rm coh}$ using $f_{\\rm max}$ to identify the $T_{\\rm coh}$ necessary to track the $\\dot{f}_0^{\\rm max}$ implied by $f_{\\rm max}$ in Eq.", "(REF ).", "If, after recalculating $T_{\\rm coh}$ , we have $T_{\\rm coh} < 1$  hr, we recalculate $\\dot{f}_{\\rm max}$ (by inverting Eq.", "(REF )) using $T_{\\rm coh} = 1$  hr and insert the new $\\dot{f}_{\\rm max}$ into Eq.", "(REF ) to obtain a new ${f}_{\\rm max}$ .", "We do not search using $T_{\\rm coh} < 1$  hr because to do so would require reproducing and cleaning short Fourier transforms (SFTs) explicitly for this search, instead of using the same standard SFTs as other pipelines.", "In addition, using coherent time shorter than an hour would significantly degrade the sensitivity.", "If, after recalculating $T_{\\rm coh}$ , we have $T_{\\rm coh} \\ge 1$  hr, we do not need to recalculate $f_{\\rm max}$ and $\\dot{f}_0^{\\rm max}$ .", "Finally, we determine the minimum search frequency $f_{\\rm min}$ which satisfies $h_0^{\\rm max} > h_0^{\\rm est}$ .", "The values $T_{\\rm coh}$ , $f_{\\rm min}$ , $f_{\\rm max}$ , and $\\dot{f}_0^{\\rm max}$ define the parameter space for the search and are summarized in Table REF .", "Next, we outline the process of setting the detection threshold for each source.", "In each sub-band, the Viterbi algorithm obtains $N_Q$ frequency paths (ending in $N_Q$ different frequency bins), each with a log-likelihood $\\mathcal {L}$ .", "We set a log-likelihood threshold to determine which, if any, of the $N = N_Q N_{\\rm band}$ paths warrant further analysis.", "We require a false alarm probability $\\alpha _{N} = 0.01$ for each source across all sub-bands.", "This is equivalent to requiring a false alarm probability per sub-band of $\\alpha _F = 1-(1-\\alpha _{N})^{1/N}.$ The likelihood threshold $\\mathcal {L}_{\\rm th}$ is then determined by solving $\\alpha _F = \\int _{\\mathcal {L}_{\\rm th}}^\\infty {\\rm d}\\mathcal {L} \\, p(\\mathcal {L}).$ The threshold $\\mathcal {L}_{\\rm th}$ is unique to each source.", "We follow up any path with $\\mathcal {L}>\\mathcal {L}_{\\rm th}$ .", "While the distribution of the log-likelhoods is unknown (see [123] for details), [87] demonstrated that the mean $\\mu _L$ and standard deviation $\\sigma _L$ depend only on $N_T$ and scale according to linear and power-law relationships, respectively.", "We determine the form of these relationships by simulating 100 sub-bands of Gaussian noise for 11 different $N_T$ values in the range $500 \\le N_T \\le 5500$ and conduct a search on each band.", "From the log-likelihoods for each Viterbi path we calculate $\\mu _\\mathcal {L}$ , $\\sigma _\\mathcal {L}$ , and scaling relations of the log-likelihood distribution for each $N_T$ .", "Figure REF displays $\\mu _\\mathcal {L}$ and $\\sigma _\\mathcal {L}$ from simulations (blue dots) and from the scaling relations (orange curves).", "For each source, we use the scaling relations from Figure REF to define a Gaussian log-likelihood distribution $p(\\mathcal {L})$ for $N_T = T_{\\rm obs}/T_{\\rm coh}$ and solve $\\alpha _F = \\int _{\\mathcal {L}_{\\rm th}}^\\infty p(\\mathcal {L})$ to obtain $\\mathcal {L}_{\\rm th}$ (Table REF ).", "We follow up all unique frequency paths with $\\mathcal {L} > \\mathcal {L}_{\\rm th}$ using the procedure described in Appendix .", "Figure: (a) Mean μ ℒ \\mu _\\mathcal {L} and (b) standard deviation σ ℒ \\sigma _\\mathcal {L} of the log-likelihoods of the Viterbi paths as a function of N T N_T.", "Blue dots are values obtained from 100 trials of simulations.", "The orange curves are the linear and power-law fits describing the mean and standard deviation dependence on N T N_T.", "An analogous calibration is presented in .Table: The threshhold ℒ th \\mathcal {L}_{\\rm th} for each SNR in the single-harmonic Viterbi search." ], [ "Dual-harmonic Viterbi", "The HMM formulation in the dual-harmonic search is essentially the same as described in Section REF , with modifications detailed in Section REF .", "Here we briefly review the dual-harmonic formulation [119] and describe the settings used in this analysis.", "We select a coherent time interval, $T_{\\rm coh}$ , and assume that $\\left|\\int _t^{t+T_{\\rm coh}}dt^{\\prime } \\dot{f_\\star }(t^{\\prime })\\right| < \\Delta f$ is always satisfied, where $\\Delta f= 1/(4 T_{\\rm coh})$ is the frequency bin size in the $\\mathcal {F}_1$ output.", "We use $2\\Delta f = 1/(2 T_{\\rm coh})$ as the frequency bin size when computing $\\mathcal {F}_2$ such that the signal is expected to move at most one bin in the outputs of both $\\mathcal {F}_1$ and $\\mathcal {F}_2$ over each discrete time step, i.e., from one coherent time interval to next.", "The log emission probability computed over each $T_{\\rm coh}$ interval is [62], [119] $\\ln L_{o(t_k) q(t_k)} &=& \\ln P [o(t_k)|f_i \\le f_\\star (t_k) \\le f_i+\\Delta f]\\\\&= & \\mathcal {F}_1(f_i) + \\mathcal {F}_2(2f_i),$ where $f_i$ is the frequency value in the $i$ -th bin.", "In this analysis, we assume that the frequency evolution in these young sources is dominated by the secular spin down of the star, and hence the transition probability matrix $A_{q_j q_i}$ becomes [119] $A_{q_{i-1} q_i} = A_{q_i q_i} = \\frac{1}{2},$ with all other entries being zero.", "A uniform prior $\\Pi [q(t_0)] = N_Q^{-1}$ is used." ] ]
2105.11641
[ [ "Efficient generation of turbulent collisionless shocks in laser-ablated\n counter-streaming plasmas" ], [ "Abstract Laser-ablated high-energy-density (HED) plasmas offer a promising route to study astrophysically relevant processes underlying collisionless shock formation, magnetic field amplification, and particle acceleration in the laboratory.", "Using large-scale, multi-dimensional particle-in-cell simulations, we explore the interpenetration of laser-ablated counter-streaming plasmas for realistic experimental flow profiles.", "We find that the shock formation and its structure are substantially different from those of more idealized and commonly considered uniform flows: shock formation can be up to 10 times faster due to the transition from small-angle scattering to magnetic reflection and the shock front develops strong corrugations at the ion gyroradius scale.", "These findings have important consequences for current experimental programs and open exciting prospects for studying the microphysics of turbulent collisionless shocks with currently available high-energy laser systems." ], [ "Introduction", "Collisionless shocks are ubiquitous across a wide range of scales and astrophysical environments, from galaxy clusters [1] and supernova remnants  [2] to the Earth's bow shock [3], [4].", "They are mediated by plasma instabilities, which dissipate the flow energy by heating the plasma, amplifying magnetic fields, and accelerating particles.", "The microphysics governing the nonlinear shock dynamics is complex; it cannot be directly resolved by astrophysical observations and is still not fully understood.", "In particular, how the structure of the shock front and the amplification of turbulent magnetic fields affect particle injection into diffusive shock acceleration remains a very important open question in high-Mach number ($M \\gg 10$ ) astrophysical shocks [5], [6], [7], [8], [9], [10], [11], [12].", "The advent of high-power lasers has opened a new avenue to produce and study high-Mach number collisionless shocks in the laboratory.", "The main goal is to produce shocks with self-generated magnetic turbulence, where the mechanisms of magnetic field amplification and particle injection can be directly studied in a controlled environment.", "In particular, there is a strong interest in studies of the Weibel instability [13], [14], as both simulations [15], [6], [9], [16], [11], [12] and spacecraft observations [17] suggest this instability is dominant in driving magnetic turbulence at shocks in a wide range of scenarios, from planetary shocks to young supernova remnants to gamma-ray bursts.", "Experimental studies can greatly complement spacecraft observations and play a critical role in benchmarking numerical simulations and theoretical models.", "This prospect has led to a large experimental effort in high-energy-density (HED) laser facilities in the last decade.", "In these experiments, high-Mach number counter-streaming flows are produced from the ablation of a solid-density target by kilojoule, nanosecond laser pulses.", "Important progress has been achieved in the characterization of the plasma flow properties and interaction [18], [19], [20], [21], leading to the demonstration of the magnetic field amplification by the Weibel instability [22], [23], [24], [25], and to a better understanding of the interplay between collisional and collisionless effects [26], [27].", "The generation of Weibel-mediated shocks requires a large flow interpenetration distance that poses a significant experimental challenge.", "Theoretical models and kinetic simulations have been largely limited to uniform plasma flows and suggest that shock formation requires system sizes exceeding $10^3$ ion skin depths [15], [28], [29], which are beyond those attained at current experimental facilities.", "However, very recently, experiments at the National Ignition Facility (NIF) have demonstrated the formation of high-Mach number shocks mediated by electromagnetic instabilities [30].", "The shock formation time observed is significantly faster than previous models predict.", "It is thus crucial to understand the physics of shock formation in the laboratory experiments and in which conditions can current systems allow the study of the relevant microphysics of astrophysical turbulent shocks.", "In this paper, we study the formation of collisionless shocks in laser-ablated counter-streaming plasmas using large-scale two- (2D) and three-dimensional (3D) particle-in-cell (PIC) simulations that include the intrinsic density and velocity inhomogeneity of experimental plasma flows.", "We find that realistic plasma profiles impact the interaction in two important ways, which have not been previously recognized: (i) shock formation is significantly faster than previously found for homogeneous plasmas because the coherence length of the amplified magnetic field becomes comparable to the ion gyroradius, leading to a transition from small-angle scattering to magnetic reflection that efficiently slows down the flows, and (ii) enhanced magnetic field advection towards the shock leads to ion-gyroradius-scale corrugations of the shock front.", "An analytical model for the shock formation time and minimum system size required to study collisionless shocks with laser-ablated plasma flows is introduced and shown to be in good agreement with both PIC simulations and recent NIF experiments.", "These results indicate that state-of-the-art HED facilities, such as the NIF and Laser Megajoule (LMJ), can produce collisionless shocks with strong turbulence driven by the Weibel instability and where the shock front uniformity can be controlled, opening the possibility to probe their influence on particle injection and test relevant astrophysical shock models." ], [ "Plasma flow profiles and simulation setup ", "Plasma flows produced by laser-ablation of solid-density targets have inhomogeneous density and velocity profiles given by a well-established self-similar solution [31], consistent with experimental measurements, such as those in Refs.", "[20], [26].", "In a typical configuration, two counter-facing solid targets separated by a distance $2L_0$ are irradiated by kJ-class lasers to produce counter-streaming plasma flows that interact at the mid-plane between the two targets, defined at $x = 0$ .", "The velocity and density of one of the flows is $v(x,t)= c_{\\rm s} + (x+L_0)/(t+\\tau _0)$ and $n(x,t)= \\tilde{n}\\,{\\rm exp} \\left[- (x+L_0)/(c_{\\rm s}(t+\\tau _0)) -1\\right]$ , with $\\tau _0$ the time the flow takes to travel the distance $L_0$ to the mid-plane where the two flows first meet at $t=0$ , $c_{\\rm s}= (\\gamma Zk_{\\rm B}T_e/m_i)^{1/2}$ the sound speed, $\\gamma $ the adiabatic index, $m_i$ and $Z$ the ion mass and charge number, $T_e$ the electron temperature, and $\\tilde{n}$ the target surface density.", "The opposite, counter-propagating flow has the symmetric profile with respect to $x = 0$ .", "In order to study the importance of more realistic plasma profiles on shock formation we have performed 2D and 3D fully kinetic PIC simulations with the code OSIRIS 3.0 [32], [33] for both homogeneous and inhomogeneous counter-streaming non-relativistic plasma flows.", "In the homogeneous case (identified by the index H), plasma flows are initialized with uniform velocity $v_{\\rm H} = v_0 = 0.11\\,c$ , a sonic Mach number $M = v_0/c_s = 21$ , and density $n_{\\rm H} = n_0$ .", "In the inhomogeneous case (identified by the index $\\rm I$ ), the flow velocity profile follows the self-similar theory, with maximum velocity $v_0 = 0.11\\,c$ near the interaction region (mid-plane of the simulation) at $t = 0$ (Fig.", "REF a).", "The density profile also closely resembles the self-similar solution (see Fig.", "REF a and Appendix ), but has been modified such that $n_{\\rm I}(x=0, t) v_{\\rm I}(x=0, t)^2 = n_{\\rm H} v_{\\rm H}^2 = n_0 v_0^2$ .", "This guarantees that the same plasma energy density is delivered to the interaction region in both cases and allows for a direct comparison of the shock formation efficiency between both.", "The baseline simulations have a longitudinal size of $L_x = 135 \\, c/\\omega _{\\rm pi}$ and a transverse size of $L_y = 50\\,c/\\omega _{\\rm pi}$ , with $\\omega _{\\rm pi} = [4 \\pi (2n_0) Z^2 e^2/m_i]^{1/2}$ the ion plasma frequency corresponding to the total plasma density.", "The simulations resolve the electron skin depth at the interaction region ($c/\\omega _{\\rm pe}$ ) with 8 cells, use a time step of 0.087 $\\omega _{\\rm pe}^{-1}$ , and use 36 particles per cell per species.", "The boundary conditions are periodic along the transverse $y$ -direction and open along the $x$ -axis for both fields and particles.", "A third-order interpolation scheme is used for improved numerical accuracy.", "The baseline ion mass to charge ratio used is $m_i/(m_e Z)=128$ .", "The high flow velocity ($\\sim 0.1\\,c$ ) and reduced mass ratio used are typical choices in the PIC modeling of shocks [15], [22], [34], [29] that capture the non-relativistic nature of the flows and a large separation between electron and ion physics, while balancing computational expense.", "We note that the physics of pure electromagnetic instabilities, such as the Weibel instability discussed in this work, can be rigorously scaled between non-relativistic systems with different flow velocities [35].", "The evolution of the magnetic field and plasma density for both the homogeneous and inhomogeneous cases is shown in Fig.", "REF .", "As expected, the early time dynamics is dominated by the development of the ion Weibel instability and is similar between both cases, which is consistent with previous experiments and simulations [22], [23].", "The instability gives rise to filamentary currents with transverse wavelength comparable to the ion skin depth $c/\\omega _{\\rm pi}$ and to the exponential amplification of the magnetic field with growth rate $\\Gamma \\simeq 0.07\\, \\omega _{\\rm pi}$ for both cases, in good agreement with linear theory [14], [13].", "The instability saturates at $\\tau _{\\rm sat}\\simeq 10\\, \\Gamma ^{-1} \\simeq 140\\,\\omega _{\\rm pi}^{-1}$ , with $B_{\\rm sat} \\simeq 0.014 \\, m_i\\omega _{\\rm pi}c/e$ , which is in good agreement with predictions based on magnetic trapping $B_{\\rm sat}=v_0 m_i\\omega _{\\rm pi}/(2\\pi Ze) $  [36].", "After saturation of the Weibel instability, the transverse coherence length of the magnetic field increases in both cases due to filament merging [37], [38], [34], [39].", "However, the longitudinal extent of the filaments starts to be significantly more limited for the inhomogeneous flows, with the stronger magnetic fields being confined to the central region near the mid-plane (Fig.", "REF e,f).", "This is a consequence of different effects associated with the inhomogeneity of the flow profiles.", "The growth rate of the Weibel instability is reduced away from the mid-plane due to the density and velocity asymmetry and due to the ion heating of the flow that has crossed the interaction region (detailed calculations in Appendix ).", "Moreover, the asymmetric flow profiles lead to enhanced magnetic field advection towards the mid-plane region (Fig.", "REF i,j).", "The advection velocity is $v_{\\rm adv}=(n_{+}v_{+}+n_{-}v_{-})/(n_{+}+n_{-})$  [40], where the index $+ (-)$ refers to the flow moving in the positive (negative) $x$ -direction.", "(Note that $v_{\\rm adv} = 0$ for homogeneous symmetric flows.)", "The weaker magnetic fields produced in the upstream region at a distance $L$ from the mid-plane are thus advected to the central region and compressed on a time scale $\\tau _{\\rm adv}= L/v_{\\rm adv}$ .", "Figure: Evolution of the interaction of counter-streaming flows from 2D PIC simulations using laser-ablated inhomogeneous (left column) and homogeneous (right column) profiles.", "a,b) Initial density (blue) and velocity (orange) profiles.", "Magnetic field B z (x,y)B_z(x,y) evolution at c,d) t 1 ≃70ω pi -1 t_1\\simeq 70\\,\\omega _{\\rm pi}^{-1}, e,f) t 2 ≃285ω pi -1 t_2\\simeq 285\\,\\omega _{\\rm pi}^{-1}, and g,h) t 3 ≃1780ω pi -1 t_3\\simeq 1780\\,\\omega _{\\rm pi}^{-1}.", "i,j) Magnetic field advection velocity (v adv v_{\\rm adv}) profile at t 2 ≃285ω pi -1 t_2 \\simeq 285\\,\\omega _{\\rm pi}^{-1}.", "k,l) Density compression ratio n 2 /n 1 n_2/n_1 between the transversely averaged ion density of interacting flows n 2 n_2 and single flow n 1 n_1.At later times, we observe dramatic differences between the two cases, both in terms of fields (Fig.", "REF g,h) and density (Fig.", "REF k,l).", "In the case of laser-ablated inhomogeneous plasmas, the flows are strongly compressed in the interaction region, two collisionless shocks are formed and propagate against the incoming plasma flows.", "Each shock reaches the density compression dictated by the Rankine-Hugoniot jump conditions [41] $n_2/n_1 \\simeq 3$ (for a 2D system) at a shock formation time $\\tau _{\\rm sf}\\simeq 1030\\,\\omega _{\\rm pi}^{-1}$ (Fig.", "REF a), where $n_2$ is the compressed (downstream) density and $n_1$ is the single flow (upstream) density.", "We have verified that in 3D the density compression reaches $n_2/n_1 \\simeq 4$ (see Appendix ).", "In the homogeneous plasma case, the density compression observed is significantly weaker and slower.", "We have run a larger simulation with $L_x = 1200 \\, c/\\omega _{\\rm pi}$ up to $2500\\,\\omega _{\\rm pi}^{-1}$ and the maximum compression obtained was just $n_2/n_1 \\simeq 2.5$ (Fig.", "REF a).", "Figure: a) Temporal evolution of the density compression ratio n 2 /n 1 n_2/n_1 at the simulation mid-plane (x=0x=0) for the inhomogeneous (blue dots) and homogeneous (orange dots) flow profiles.", "The black curve corresponds to the density evolution predicted from Eq.", "for t>τ c t > \\tau _c.", "b) Temporal evolution of the ion gyroradius r i r_{i} (crosses) and dominant magnetic wavelength λ B \\lambda _{B} (dots) for the inhomogenous (blue) and homogeneous (orange) flow profiles.", "The black curve corresponds to the filament merging model for λ B (t)\\lambda _{B}(t) from Ref.", ".The faster shock formation observed for laser-ablated inhomogenous plasma flows is due to the transition from small-angle scattering to magnetic reflection.", "In the case of homogeneous flows, the magnetic fields produced by the Weibel instability have a wavelength that is much smaller than the ion gyroradius of the flows, $\\lambda _{B} \\ll r_{i}$ (Fig.", "REF b).", "Ions slow down via many small-angle scatterings with the magnetic fields [15], [28] and kink-like instabilities of the current filaments [29], which operate on long time scales compared to the ion cyclotron frequency.", "A different dynamics is observed in the case of inhomogeneous flows where ions are efficiently confined near the mid-plane.", "This is a consequence of the temporal decrease of the flow velocity (and associated ion gyroradius) arriving at the interaction region, intrinsic to laser-ablated plasmas, $r_{i}(x=0, t) \\propto v(x=0,t)\\simeq L_0/(\\tau _0+t)$ (the $c_{\\rm s} \\ll v_0$ term is negligible in high-$M$ experiments).", "As the ion gyroradius becomes comparable to the dominant Weibel-mode wavelength, $r_{i} \\sim \\lambda _{B}$ (see Fig.", "REF b), the flows are effectively slowed down and heated by large-angle magnetic reflections.", "As the shock is formed in the inhomogeneous flow interaction, strong corrugations of the magnetic field (and density) are observed at the shock front on the scale of the ion gyroradius.", "The plasma flows primarily in alternating filaments near the mid-plane region.", "Weaker magnetic fields from the upstream region are advected towards the center through these filaments and slowed down on a scale comparable to the ion gyroradius.", "This leads to the compression of the magnetic fields near the mid-plane and to strong anti-symmetric modulations at the shock front, as observed in Fig.", "REF g. Since the magnetic field is frozen-in the electron fluid, the ratio of electron thermal to fluid velocity, $v_{th}/v_0$ , controls the shock front corrugation level.", "Figure REF shows that indeed the corrugations are significantly less pronounced when $v_{th} \\gtrsim v_0$ .", "Experimentally, $v_{th}$ can be controlled by varying the laser intensity and $Z$ (through radiative effects [42]) to probe the impact of shock front corrugations on magnetic field amplification and particle injection [7], [4], [10].", "Figure: Magnetic field B z B_z at t≃600ω pi -1 t\\simeq 600\\,\\omega ^{-1}_{\\rm pi} for a) v th =0.02v flow v_{\\rm th}= 0.02\\, v_{\\rm flow} and for b) v th =1.25v flow v_{\\rm th}=1.25 \\,v_{\\rm flow}.", "The electron temperature can be used to control the level of shock front corrugations." ], [ "Model of shock formation with laser-ablated flows", "A model for shock formation in laser-ablated counter-streaming plasma flows can be derived by estimating the critical time $\\tau _{\\rm c}$ at which the magnetic wavelength of the Weibel mode becomes comparable to the ion gyroradius at the interaction region, $r_{i}(\\tau _{\\rm c}) \\sim \\lambda _{B}(\\tau _{\\rm c})$ .", "After saturation of the Weibel instability, the evolution of the dominant magnetic wavelength due to filament merging follows [34] $\\lambda _{B}(t)\\sim \\lambda _{\\rm sat}\\left[1+((t-\\tau _{\\rm sat})/\\tau _{\\rm m})^2\\right]$ , where $\\lambda _{\\rm sat}\\sim c/\\omega _{\\rm pi}$ is the wavelength at saturation, $\\tau _{\\rm m} \\sim \\frac{2\\pi c}{v_0} \\left(4m_i/Zm_e\\right)^{1/4}\\omega _{\\rm pi}^{-1}$ is the typical merging time, and $v_0=L_0/\\tau _0$ is the initial flow velocity.", "The ion gyroradius is $r_{i}(t) = m_i c v(t)/(Ze B_{\\rm sat}) = 2\\pi (c/\\omega _{\\rm pi}) L_0/(L_0 + v_0 t)$ .", "The critical time $\\tau _{\\rm c}$ is the solution of $\\left(1+ \\frac{\\tau _{\\rm c}v_0}{L_0}\\right) \\left( 1+ \\frac{(\\tau _{\\rm c}-\\tau _{\\rm sat})^2}{\\tau _{\\rm m}^2} \\right)=2\\pi \\, .$ For the conditions of our simulation, this yields $\\tau _{\\rm c} \\simeq 565\\,\\omega _{\\rm pi}^{-1}$ , in good agreement with our results (Fig.", "REF b).", "We note that filament merging ceases at late times for both homogeneous and inhomogeneous flows (Fig.", "REF b) due to the development of the drift-kink instability  [29].", "However, in practice, this occurs at times comparable to or larger than $\\tau _c$ and thus the use of the filament merging model to estimate $\\tau _c$ is appropriate (see Appendix ).", "The critical time $\\tau _{\\rm c}$ marks the transition from small-angle scattering to magnetic reflection.", "At this stage, the ions reaching the center are slowed down and confined over a region of extension $\\sim r_{i}$ .", "The density compression in the mid-plane region between $\\tau _{\\rm c}$ and the shock formation time $\\tau _{\\rm sf}$ can then be described as $n_{\\rm 2}(t) = 2n_{\\rm 1}(\\tau _{\\rm c})+\\int _{\\tau _{\\rm c}} ^{t} 2n_{\\rm 1}(t^{\\prime })v(t^{\\prime })\\frac{{\\rm d}t^{\\prime }}{r_{i}(t^{\\prime })} \\, ,$ where we have assumed that at $t=\\tau _{\\rm c}$ the two flows overlap without yet significant compression, i.e.", "$n_{\\rm 2}(\\tau _{\\rm c})=2n_{\\rm 1}(\\tau _{\\rm c})$ .", "The shock formation time $\\tau _{\\rm sf}$ is obtained from the density jump condition $n_{\\rm 2}(\\tau _{\\rm sf})=(2+\\delta )n_{\\rm 1}(\\tau _{\\rm sf})$ , where $\\delta =1$ in 2D and $\\delta =2$ in 3D.", "We have confirmed that Eq.", "REF describes well the density compression observed in the simulations of laser-ablated plasma flows after $\\tau _{\\rm c}$ , as shown in Fig.", "REF a.", "The predicted shock formation time $\\tau _{\\rm sf}\\simeq 1060\\,\\omega _{\\rm pi}^{-1}$ is also in good agreement with the simulation results.", "In the limit where the initial single flow density varies weakly over an ion gyroperiod, the shock formation time can be approximated by $\\tau _{\\rm sf} = \\tau _{\\rm c} +\\frac{\\delta }{2\\Omega _c}\\,$ with $\\Omega _c = v(t)/r_{i}(t)$ the ion cyclotron frequency.", "For typical experimental flow velocities  [19], [20], [26], [21], [22], [23], [24], [27] $v_0 \\sim 1000-2000\\, {\\rm km/s} \\ll c$ , the critical time $\\tau _{\\rm c}$ largely exceeds the time of flow compression by magnetic reflection, and dominates the shock formation time $\\tau _{\\rm sf} \\sim \\tau _{\\rm c}$ .", "The minimum flow interpenetration distance required for shock formation is $L_{\\rm sf} \\sim \\tau _{\\rm sf} v_0 \\sim \\tau _{\\rm c} v_0$ and thus corresponds to a minimum target separation $2 L_0 \\sim 2 \\tau _{\\rm c} v_0$ .", "By setting $\\tau _{\\rm c} = \\tau _0$ and taking the appropriate limits $\\tau _{\\rm c} \\gg \\tau _{\\rm m}$ and $\\tau _{\\rm c} \\gg \\tau _{\\rm sat}$ we can calculate the minimum shock formation time as $\\tau _{\\rm sf} \\simeq \\tau _{\\rm c} \\simeq 16 (c/v_0)\\left[m_i/(Z m_e)\\right]^{1/4}\\omega _{\\rm pi}^{-1} \\simeq 2.5 \\left[m_i/(Z m_e)\\right]^{1/4}\\Omega _{\\rm c}^{-1}$ .", "In practice, laser-ablated fully ionized plasmas have $A/Z \\simeq 2$ , for which the required target separation to reach shock formation is $2 L_0 \\simeq 250\\, c/\\omega _{\\rm pi}$ and is independent of the flow velocity $v_0$ .", "Figure: Comparison of the shock formation time τ sf \\tau _{\\rm sf} from Eq.", "(green line) for laser-ablated flows with m i /(Zm e )=3672m_i/(Z m_e) = 3672 and L 0 =125c/ω pi L_0 = 125\\,c/\\omega _{\\rm pi} with the results of 2D PIC simulations with m i /(Zm e )=128m_i/(Z m_e) = 128 (+ + ) and m i /(Zm e )=512m_i/(Z m_e) = 512 (▵ \\triangle ), 3D simulations with m i /(Zm e )=32m_i/(Z m_e) = 32 (∘ \\circ ) and with the experimental observation from Ref.", "(★ \\bigstar ).The shock formation times predicted for homogeneous flows τ sf - iso \\tau _{\\rm sf-iso} (black line)  and τ sf - kink \\tau _{\\rm sf-kink} (blue line)  largely exceed that observed for laser-ablated flows.", "The shaded areas indicate the range of parameters achievable at current HED laser facilities.This analysis considered the longitudinal inhomogeneity of laser-ablated planar plasma flows.", "On time scales longer than $L_0/c_s$ , 3D divergence effects can modify the plasma density profiles.", "It is thus important to guarantee that the shock formation time $\\tau _\\mathrm {sf} = L_\\mathrm {sf}/v_0 < L_0/c_s $ or, equivalently, $L_0 > L_\\mathrm {sf}/M$ .", "This condition is always comfortably met for system sizes large enough to study high-Mach number shocks ($L_0 \\ge L_\\mathrm {sf}$ and $M \\gg 1$ ) and thus the profiles considered are reasonably justified.", "The transverse inhomogeneity of the plasma and obliquity of the flows away from their axes (due to divergence) can also affect plasma interpenetration.", "These effects should not impact the physics of shock formation and corrugation (both on the ion gyroradius scale) if $L_0 \\gg r_i \\sim 0.064/\\sqrt{n_0 [10^{19} \\mathrm {cm}^{-3}]}$ cm.", "For the typical system size, $L_0 \\sim 0.4 - 1.25$ cm, and plasma density, $n_0 \\sim 5\\times 10^{18} - 5\\times 10^{19} \\mathrm {cm}^{-3}$ , of current experiments [20], [24], [26], [30], this condition is always met.", "We have performed additional 2D simulations with spherically divergent flows and 3D simulations with modified density profiles and confirmed that the shock formation time and structure predicted by our model are robustly observed in all cases (see Appendix )." ], [ "Comparison of shock formation models with PIC simulations and experiments", "Shock formation in counter-streaming laser-ablated plasmas is significantly faster than that predicted from previous models of Weibel-mediated shocks in homogeneous flows.", "The predicted shock formation time due to flow isotropization during filament merging is [28] $\\tau _{\\rm sf-iso} \\simeq 35(m_i/Zm_e)^{0.4} (c/v_0) \\omega _{\\rm pi}^{-1}$ , corresponding to $2 L_0 \\simeq 1867 c/\\omega _{\\rm pi}$ .", "Additionally, shock formation due to the disruption of the current filaments by kink-type instabilities predicts [29] $\\tau _{\\rm sf-kink} \\simeq 5 \\tau _{\\rm kink} \\simeq (5/3)(m_i/Zm_e)^{3/4} (c/v_0) \\omega _{\\rm pi}^{-1}$ , where $\\tau _{\\rm kink}$ is the growth time of the drift-kink instability, and corresponds to $2 L_0 \\simeq 1572 c/\\omega _{\\rm pi}$ .", "The shock formation model proposed in this work is thus $6-8\\times $ faster than previous predictions.", "Figure REF summarizes the comparison of our model predictions for the shock formation time and those of previous models [28], [29] with the results of 2D and 3D PIC simulations and experimental measurements on NIF [30].", "The PIC simulations performed over a range of flow velocities ($v_0 = 0.03 - 0.1 \\,c$ ) and mass to charge ratios $[m_i/(m_e Z) = 32 - 512]$ show good agreement with our model, confirming the scaling of $\\tau _{\\rm sf} \\propto 1/v_0$ (shock formation distance independent of $v_0$ ) and the weak dependence on mass to charge ratio.", "Moreover, our model is in good agreement with the recent NIF experiments, which observed shock formation after $\\lesssim 3$ ns of interaction.", "For the measured plasma overlapping density $\\simeq 5\\times 10^{19}\\rm cm^{-3}$ and $v_0 \\simeq 2000\\,\\rm km/s$ , the shock formation time predicted by our model is $2\\times 10^4\\,\\omega _{\\rm pi}^{-1}\\simeq 3\\,\\rm ns$ , consistent with the experimental measurements.", "The results presented here are important to guide the development and interpretation of experimental studies of turbulent collisionless shocks and in connecting them with astrophysical simulations and models.", "Figure  REF indicates the parameter space (shaded areas) that can be probed by the current HED facilities (see Appendix  for details).", "For kJ-class OMEGA experiments [19], [20], [26], [21], [22], [23], [24], with measured plasma densities $n_0 \\sim 10^{18} - 10^{19} {\\rm cm}^{-3}$ , we predict $\\tau _{\\rm sf} \\gtrsim 13$ ns and $2 L_0 \\gtrsim 2.5$ cm, which largely exceed the interaction time of $\\sim 5$ ns and system size of $\\sim 0.8$ cm of previous experiments.", "However, our results indicate that shock formation can be comfortably reached at higher energy laser facilities ($> 100$ kJ), such as NIF and LMJ.", "The collisionless shocks will have a transition size dictated by the ion gyroradius and the magnetic turbulence in the shock foot is driven by the Weibel instability, which are critical properties of high-Mach number astrophysical shocks revealed by kinetic simulations [9] and spacecraft observations [17].", "Moreover, by varying the electron temperature it will be possible to control and study the impact of shock front corrugations on magnetic field amplification and particle injection [7], [4], [10]." ], [ "Conclusions", "In summary, we have shown for the first time that the inhomogeneity of laser-ablated plasma flow profiles has a significant impact in accelerating the formation of turbulent collisionless shocks and in controlling their dynamics.", "These findings have important implications as they enable the use of current HED experimental facilities to study the complex interplay between magnetic field amplification, shock front corrugation, and particle injection underlying high-Mach number astrophysical shocks.", "The authors thank D. D. Ryutov, D. P. Higginson, and H.-S. Park for stimulating discussions.", "This work was supported by the U.S. Department of Energy SLAC Contract No.", "DEAC02-76SF00515 and by the U.S. DOE Early Career Research Program under FWP 100331.", "The authors acknowledge the OSIRIS Consortium, consisting of UCLA and IST (Portugal) for the use of the OSIRIS 3.0 framework and the visXD framework.", "Simulations were run on Mira (ALCF) through an ALCC award and on Cori (NERSC)." ], [ "Initialization of the laser-ablated plasma profiles", "The plasma used in our inhomogeneous simulations has a velocity profile $v(x,t)= c_{\\rm s} + (x+L_0)/(t+\\tau _0)$ that follows the self-similar solution for laser-ablated plasmas [31].", "The density profile has been slightly modified from the self-similar solution in order to guarantee that the flow energy density, $n v^2$ , delivered at the mid-plane region (where the flows interact) is constant in time $n v^2=n_0 v_0^2$ .", "This is convenient because it allows for a direct comparison of the shock formation efficiency between inhomogeneous (laser-ablated) and homogeneous flows.", "We compute the initial density profile $n(x,t=0)$ by integrating the continuity equation for the density, with boundary condition $n(x=0,t)$ and assuming ballistic propagation of a density element (i.e.", "${\\rm d}v/{\\rm d}t=0$ ).", "This gives the density profile showed in Fig.", "REF (blue line), which is a reasonable approximation of the self-similar profile (red dashed line).", "We have performed simulations with different density profiles and checked that the main results of this work are relatively insensitive to the exact details of the density profile and determined primarily by the inhomogeneity of the flow velocity, which is responsible for the transition from small-angle scattering to magnetic reflection in the shock formation.", "Figure: Comparison of the density profile of the forward moving inhomogeneous flow initialized in the simulations (blue curve) and that predicted from self-similar theory   (red dashed curve)." ], [ "Growth rate and saturation of the Weibel instability in inhomogeneous counter-streaming flows", "The growth of the ion Weibel instability in inhomogeneous counter-streaming flows can be best described in a frame where the instability is purely growing (termed \"Weibel frame\"), defined as $u_{\\rm wf } = \\frac{\\frac{n_+v_+}{T_+} + \\frac{n_-v_-}{T_-}}{\\frac{n_+}{T_+} + \\frac{n_-}{T_-}}\\, ,\\qquad \\mathrm {(\\rm S1)}$ Figure: Comparison of the longitudinal spatial profile of the magnetic field measured in the inhomogenous flow simulation at t=125ω pi -1 t=125\\,\\omega _{\\rm pi}^{-1} (blue curve) with the prediction of the magnetic trapping mechanism (Eq. )", "for the growth rate obtained from Eq.", "(black curve).with $n_{\\pm },v_{\\pm },T_{\\pm }$ the local densities, velocities, and temperatures of the ion flows moving in the positive (+) and negative (-) direction.", "The corresponding dispersion relation, accounting for the ion dynamics in the non-relativistic regime, reads $\\omega ^2 -c^2k^2 + {\\omega _{\\rm pi}^*}^2 -k^2\\left[ \\frac{\\omega _{+}^2v_+^{\\prime 2}}{\\omega ^2-k^2v_{\\rm th,+}^2}+\\frac{\\omega _{-}^2v_-^{\\prime 2}}{\\omega ^2-k^2v_{\\rm th,-}^2} \\right] \\,=0 ,\\qquad \\mathrm {(\\rm S2)}$ where $\\omega _{\\rm pi}^*$ is the ion plasma frequency at the local density, $v_{\\rm th}$ is the ion thermal velocity and the primed quantities are defined in the Weibel frame.", "The growth rate $\\Gamma =i\\omega $ , obtained from the solution of Eq.", "REF , in the limit of cold flows and $k\\gg \\omega _{\\rm pi}^*/c$ , can be expressed as $\\Gamma = \\frac{\\sqrt{n_+n_-}}{n_++n_-}\\left(v_+-v_-\\right)\\omega _{\\rm pi}^* \\, ,\\qquad \\mathrm {(\\rm S3)}$ which is maximum at the center of interaction ($x=0$ ), where the flows have equal density and opposite velocity, consistent with the simulation results.", "However, it is important to take into account temperature effects, since the heating of the flows as they cross the strong field region near the mid-plane will further decrease the growth rate away from it.", "This will also impact the saturation of the magnetic field, which, following the magnetic trapping mechanism  [36], is given by $B_{\\rm sat}= \\frac{\\Gamma ^2}{2\\pi v_0} \\frac{m_i c^2}{e \\omega _{\\rm pi}} \\, .", "\\qquad \\mathrm {(\\rm S4)}$ We have compared the magnetic field profile obtained in the inhomogeneous simulation with that predicted by Eqs.", "REF and REF , where we have solved Eq.", "REF numerically for the ion temperature, density, and velocity extracted from the simulation.", "The results, shown in Fig.", "REF near the saturation of the instability ($t=125\\,\\omega _{\\rm pi}^{-1}$ ), indicate that the sharp decrease of the magnetic field amplitude away from the mid-plane in inhomogeneous flows can be reasonably well described by our model." ], [ "Interplay between filament merging and kink instability", "  Our PIC simulations (for both homogeneous and inhomogeneous flow profiles) show that right after the saturation of the Weibel instability the dynamics of the current filaments is primarily governed by filament merging, in good agreement with the model of Ref.", "[28] (see Fig.", "2b).", "However, at late times ($t\\simeq 500\\,\\omega _{\\rm pi}^{-1}$ in Fig.", "2b), we observe that filament merging ceases and the dominant magnetic wavelength remains approximately constant.", "We have found that the stopping of the filament merging coincides with the development of longitudinal modulations of the filaments and the onset of magnetic turbulence, consistent with the development of kink-like instabilities described in Ref. [29].", "We observe that this transition occurs at a time $t \\simeq 2\\tau _{\\rm kink}\\simeq (m_i/Zm_e)^{3/4}(c/v_0)\\,\\omega _{\\rm pi}^{-1}$ after the saturation of the Weibel instability ($\\tau _{\\rm sat}$ ), which is consistent across the range of simulations performed for different flow velocities and ion to electron mass ratios (Fig.", "REF ).", "We have also verified that our estimate of the critical time $\\tau _c$ for the transition between small-angle scattering and magnetic reflection (Eq.", "1) is in good agreement with the PIC simulation results (Fig.", "REF ).", "We observe that for realistic ion to electron mass ratios the critical time occurs always before the saturation of filament merging (onset of the kink instability), thus validating the use of the merging model in the estimate of $\\tau _c$ .", "Figure: Comparison of the time at which the filament wavelength matches the ion gyroradius (orange dots) and the filament merging saturates (blue crosses) observed in PIC simulations with the estimates of τ c \\tau _c (Eq.", "1) and τ kink \\tau _{\\rm kink} .", "The PIC simulations with m i /(Zm e )=128m_i/(Z m_e) = 128 have v 0 =0.11cv_0 = 0.11\\,\\rm c and v 0 =0.03cv_0 = 0.03\\,\\rm c, and the simulation with m i /(Zm e )=512m_i/(Z m_e) = 512 has v 0 =0.11cv_0 = 0.11\\,\\rm c." ], [ "Impact of flow divergence and 3D effects on shock formation", "Laser-produced plasma flows will have an associated divergence, i.e.", "they will have a radial velocity component away from the central axis ($r = 0$ ).", "The flow divergence at the interaction region can be controlled by the ratio of the laser focal spot size and the interaction distance $L_0$ .", "Earlier experiments used laser spot sizes significantly smaller than the system size, and the flow divergence was relatively large  [20], [26], [23], [27].", "Very recently, experiments at the National Ignition Facility (NIF)  [30] used a larger spot size comparable to the system size, and showed that the transverse size of the heated plasma region at the flow-flow interaction is comparable to the laser spot at the target, indicating that the flows are relatively well collimated, as considered in our simulations.", "In general, as discussed in the main text, provided that the system size is much larger that the ion gyroradius associated with the Weibel magnetic field, $L_0\\gg r_i$ , the flow divergence is not expected to affect the shock formation physics and the use of collimated flows considered in the simulations is well justified.", "In order to explore the impact of the plasma flow divergence, we performed a simulation with spherically divergent flows.", "The density and velocity profiles along the $y=0$ axis are equal to the ones discussed in the manuscript and constant at a radial distance $R$ from the target irradiated spot.", "The flow is transversely limited at $\\pm 30^{\\circ }$ from the $y=0$ axis to minimize the need of a very large transverse simulation box size.", "As can be seen in Fig.", "REF , the simulation with spherically expanding flows (right panel) shows a magnetic field amplitude and development of corrugations similar to the one presented in the main paper (left panel).", "This confirms that the dynamics is weakly affected by the flow divergence if the ion gyroradius is much smaller than the system size, as in the recent NIF experiments.", "Figure: Magnetic field B z B_z at t≃570ω pi -1 t\\simeq 570\\,\\omega ^{-1}_{\\rm pi}for a) the simulation with collimated flows as in Fig.", "1d and b) a case with spherical flow expansion.Figure: Evolution in time of the compression factor at the shock front for the 2D simulation (blue dots) and the corresponding 3D simulation (orange dots) with m i /(Zm e )=32m_i/(Zm_e)=32.In addition, we have performed 3D simulations to verify that the shock physics and model prediction for the shock formation time is robust in 3D.", "We have used $m_i/(Zm_e)=32$ due to the very high computational cost.", "The velocity and density profiles are the same as in Fig.", "1, with the difference that the density is kept constant after $\\tau _{\\rm plateau}=700\\,\\omega _{\\rm pi}^{-1}$ .", "This plateau in density profile is typically observed experimentally due to the flow divergence and/or finiteness of the laser duration [20], [30].", "In order to confirm that this does not impact significantly the shock formation physics, which is primarily determined by the flow velocity profile, we have simulated a case where $\\tau _{\\rm sf}>\\tau _{\\rm plateau}$ .", "As can be seen in Fig.", "REF , shock formation is reached in 3D between $t \\sim 900 - 1000 \\,\\omega _{\\rm pi}^{-1}$ with a density compression factor of 4, as expected from the shock jump conditions.", "The observed shock formation time is in very good agreement with our model prediction of $\\tau _{\\rm sf}\\simeq 940\\,\\omega _{\\rm pi}^{-1}$ ." ], [ "Plasma conditions of current HED facilities", "  There has been a significant experimental effort in using high-energy-density laser facilities such as OMEGA and NIF to explore the physics of collisionless shocks in counter-streaming laser-ablated plasmas [19], [20], [26], [21], [22], [23], [24], [27], [25], [30].", "These experiments produce typical flow velocity $v_0\\sim 1000-2000$ km/s, with the plasma density being primarily constrained by the laser energy and system size.", "Experiments at OMEGA [19], [20], [26], [21], [22], [23], [24] have been performed with up to 5kJ/target and have measured typical densities at the interaction region in the range $10^{18}-10^{19} \\,\\rm cm^{-3}$ .", "Recent experiments on the NIF used 250-450 kJ/target  [27], [30] and measured a density at the interaction in the range $10^{19} - 10^{20}\\,\\rm cm^{-3}$ .", "While, to our knowledge, collisionless shock experiments on LMJ have not yet been reported, based on the results from NIF, at the currently available energy of 150 kJ/target it is reasonable to expect plasma densities in the range $5\\times 10^{18} - 5\\times 10^{19}\\,\\rm cm^{-3}$ .", "The maximum plasma system sizes that can be produced at these conditions can be estimated from basic energy conservation arguments as $\\epsilon _{\\rm laser} = \\frac{\\eta }{2} n_0 m_i v_0^2 \\frac{L_0^3}{\\tan (\\theta )^2}\\, ,\\qquad \\mathrm {(\\rm S5)}$ where $\\eta $ is the energy conversion efficiency from the laser to the plasma flows and $\\theta $ is the divergence angle of the flows.", "We have considered typical parameter ranges $\\eta \\sim 0.25-0.5$ and $\\theta \\sim 30^\\circ - 45^\\circ $ .", "The range of interaction times $L_0/v_0$ obtained is shown in Fig.", "3.", "To study collisionless shocks, it is important to guarantee that Coulomb collisions will not affect the shock formation process.", "For collisions to have a negligible effect, one should guarantee that the ion-ion mean free path $\\lambda _{\\rm mfp} = m_i^2 v_r^4/(16 \\pi Z^4 e^4 n_i \\log \\Lambda )$ , which is responsible for slowing down the flows, is much larger than the required shock formation distance $L_{\\rm sf} = 125\\, c/\\omega _{\\rm pi}$ (as described in our model).", "Here $n_i = n_0/Z$ is the ion density of each flow, $v_r = 2 v_0$ is the relative velocity between flows, and $\\log \\Lambda $ is the Coulomb logarithm.", "This condition can be written as $\\frac{\\lambda _{\\rm mfp}}{L_{\\rm sf}} \\sim \\frac{74}{Z}\\frac{(v_0 {[\\rm 1000\\, km/s]})^4}{\\sqrt{n_0 [\\rm 10^{19}\\, cm^{-3}]}} \\gg 1, \\qquad \\mathrm {(\\rm S6)}$ where we have used $A/Z=2$ and $\\log \\Lambda =10$ , as typical of laser-ablated plasmas.", "We find that for the typical range of flow density and velocity values of experiments discussed above and $Z = 6$ (Carbon flows), we have $\\lambda _{\\rm mfp}/L_{\\rm sf} \\sim 6 - 280$ and thus collisions are not expected to affect the shock formation process discussed in our work." ] ]
2105.11750
[ [ "Topological entropy of the geodesic flow of non-positively curved metric\n spaces" ], [ "Abstract We extend the classical Otal-Peign\\'e's Theorem to the class of packed, Gromov-hyperbolic spaces with a convex geodesic bicombing.", "Namely we prove that when a group acts discretely and freely by isometries on one of these spaces then its critical exponent equals the topological entropy of the geodesic flow of the quotient metric space.", "An important tool will be a refinement of the classical Bishop-Jones' Theorem." ], [ "Introduction", "The relation between the topological entropy of the geodesic flow of a Riemannian manifold $(M,g)$ with non-positive sectional curvature and the critical exponent of the discrete group of isometries $\\Gamma \\cong \\pi _1(M)$ acting freely on the universal covering $\\tilde{M}$ of $M$ has been intensively studied during the years.", "A.Manning proved that in case of compact manifolds with non-positive sectional curvature the topological entropy of the geodesic flow equals the volume entropy of the universal covering (cp.", "[18]) and so the critical exponent of the group $\\Gamma $ , whose definition will be recalled later.", "This result has been generalized to compact quotients of geodesically complete, CAT$(0)$ -spaces by R.Ricks [23].", "Its proof works in case of Busemann convex metric spaces, but it is actually enough to consider metric spaces satisfying a weaker notion of non-positive curvature: that is spaces supporting a convex geodesic bicombing that is geodesically complete.", "We recall that a geodesic bicombing is a map $\\sigma \\colon X\\times X \\times [0,1]$ such that for all $x,y\\in X$ the map $t\\mapsto \\sigma (x,y,\\cdot ) = \\sigma _{xy}(\\cdot )$ is a geodesic between $x$ and $y$ parametrized proportionally to arc-length.", "Every geodesic $\\sigma _{xy}$ is said a $\\sigma $ -geodesic.", "The bicombing $\\sigma $ is convex if the map $t\\mapsto d(\\sigma _{xy}(t),\\sigma _{x^{\\prime }y^{\\prime }}(t))$ is convex on $[0,1]$ for every $x,y,x^{\\prime },y^{\\prime }\\in X$ , while it is geodesically complete if every $\\sigma $ -geodesic can be extended to a bigger $\\sigma $ -geodesic.", "Examples of these spaces are geodesically complete CAT$(0)$ and Busemann convex metric spaces, but there are also examples that are not uniquely geodesic, like all Banach spaces.", "However we are more interested in the much more complicated case of non-cocompact actions.", "For Riemannian manifolds the following is true: Theorem (Otal-Peigné, [20]) Let $M = \\Gamma \\backslash \\tilde{M}$ be a Riemannian manifold with pinched, negative sectional curvature, i.e.", "$-b^2 \\le \\textup {Sec}_g \\le -a^2 <0$ .", "Then the topological entropy of the geodesic flow on $M$ equals the critical exponent of $\\Gamma $ .", "The price to pay in order to consider any possible group acting discretely and freely on $\\tilde{M}$ is the condition on the sectional curvature.", "While the lower bound is quite natural, since every compact manifold has such a bound, the negative upper bound marks a difference among the cocompact case and the general one.", "The proof of Otal-Peigné's Theorem uses local estimates that are true only in the strict negative curvature setting.", "The purpose of this paper is to extend Otal-Peigné's Theorem to a wider class of metric spaces.", "As in [6] we will consider Gromov-hyperbolic, packed, GCB-spaces $(X,\\sigma )$ .", "The packing condition, that is a uniform upper bound on the cardinality of $2r_0$ -nets inside any ball of radius $3r_0$ for some $r_0>0$ , can be considered as a weak lower bound on the curvature (cp.", "[9], [8], [6]), while Gromov-hyperbolicity is a condition of negative curvature at large scales.", "Finally the GCB-condition, that is the existence of a geodesically complete, convex geodesic bicombing, gives some control on the local geometry, but it is only a very weak notion of non-positive curvature.", "Before stating our generalization of Otal-Peigné's Theorem we need to introduce some terminology, due to the difference from the manifold case to ours.", "If a group $\\Gamma $ acts discretely and freely by $\\sigma $ -isometries on $X$ (this means that $g\\sigma _{xy} = \\sigma _{g(x)g(y)}$ for every $g\\in \\Gamma $ and every $x,y\\in X$ ) then it acts properly discountinously by homeomorphisms on the space of $\\sigma $ -geodesic lines Geod$_\\sigma (X)$ , that is the space of geodesic lines whose finite subsegments are $\\sigma $ -geodesics.", "The quotient by this action is the space Loc-Geod$_\\sigma (\\Gamma \\backslash X)$ , called the space of $\\sigma $ -local geodesic lines of $\\Gamma \\backslash X$ .", "The space Loc-Geod$_\\sigma (\\Gamma \\backslash X)$ has a natural reparametrization flow $\\lbrace \\Phi _t\\rbrace _{t\\in \\mathbb {R}}$ defined by $\\Phi _t \\gamma = \\gamma (\\cdot + t)$ .", "We will be interested to the dynamical system $(\\textup {Loc-Geod}_\\sigma (\\Gamma \\backslash X), \\Phi _1)$ .", "We remark that in case of geodesically complete Busemann convex or CAT$(0)$ -spaces every $\\sigma $ -geodesic line is a geodesic line and $\\textup {Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ is exactly the space of all local geodesics of the quotient space $\\Gamma \\backslash X$ , as in case of manifolds.", "The topological entropy of the dynamical system $(\\textup {Loc-Geod}_\\sigma (\\Gamma \\backslash X), \\Phi _1)$ is by definition: $h_\\text{top} = \\sup _{\\mu \\in \\mathcal {E}_1}h_\\mu ,$ where $\\mathcal {E}_1$ is the set of ergodic invariant measures of the dynamical system and $h_\\mu $ is the Kolmogorov-Sinai entropy of $\\mu $ .", "Other versions of entropy, inspired by the one introduced by Brin-Katok ([4]), are the lower and upper local entropies, namely: $\\underline{h}^\\text{loc} = \\sup _{\\mu \\in \\mathcal {E}_1}\\inf _\\text{ɗ} \\underline{h}^\\text{loc}_{\\mu ,\\text{ɗ}}, \\qquad \\overline{h}^\\text{loc} = \\sup _{\\mu \\in \\mathcal {E}_1}\\inf _\\text{ɗ} \\overline{h}^\\text{loc}_{\\mu ,\\text{ɗ}},$ where the infimum is among all distances on $\\textup {Loc-Geod}_\\sigma (\\Gamma \\backslash X, \\Phi _1)$ inducing its topology, $\\underline{h}^\\text{loc}_{\\mu ,\\text{ɗ}} = \\underset{\\mu }{\\text{ess}\\inf }\\lim _{r\\rightarrow 0}\\liminf _{n \\rightarrow +\\infty } -\\frac{1}{n}\\log \\mu (B_{\\text{ɗ}^n}(\\gamma ,r))$ and $\\overline{h}^\\text{loc}_{\\mu ,\\text{ɗ}} = \\underset{\\mu }{\\text{ess}\\sup }\\lim _{r\\rightarrow 0}\\limsup _{n \\rightarrow +\\infty } -\\frac{1}{n}\\log \\mu (B_{\\text{ɗ}^n}(\\gamma ,r)).$ Here $B_{\\text{ɗ}^n}(\\gamma ,r)$ is the classical $n$ -dynamical ball of center $\\gamma $ and radius $r$ .", "The topological entropy $h_\\text{top}$ has an interpretation à la Bowen in consequence of the variational principle (see [14]): $h_\\text{top}= \\inf _{\\textup {ɗ}}\\sup _{K}\\lim _{r\\rightarrow 0}\\limsup _{n \\rightarrow +\\infty }\\frac{1}{n}\\log \\text{Cov}_{\\text{ɗ}^n}(K,r),$ where the infimum is again among all distances inducing the topology of $\\textup {Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ , the supremum is among every possible compact subsets of $\\textup {Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ and $\\text{Cov}_{\\text{ɗ}^n}(K,r)$ is the covering number at scale $r$ of the set $K$ with respect to the dynamical distance $\\text{ɗ}^n$ .", "If in (REF ) we consider the supremum only among compact $\\Phi _1$ -invariant subsets we obtain another quantity which we call the invariant-topological entropy and that will be denoted by $h_{\\text{inv-top}}$ .", "For a generic dynamical system it holds: $h_\\text{inv-top} \\le h_\\text{top} \\le \\underline{h}^\\text{loc} \\le \\overline{h}^\\text{loc}.$ The main result of the paper is that in our case these inequalities are actually equalities and that the common value equals the critical exponent of $\\Gamma $ .", "Theorem A Let $(X,\\sigma )$ be a $\\delta $ -hyperbolic GCB-space that is $P_0$ -packed at scale $r_0$ .", "Let $\\Gamma $ be a group of $\\sigma $ -isometries of $X$ acting discretely and freely.", "Then $h_{\\textup {inv-top}} = h_\\textup {top} = \\underline{h}^\\textup {loc} = \\overline{h}^\\textup {loc} = h_\\Gamma ,$ where $h_\\Gamma $ is the critical exponent of $\\Gamma $ .", "This result clearly generalizes Otal-Peigné's Theorem.", "The critical exponent of $\\Gamma $ is defined as $h_\\Gamma = \\limsup _{T \\rightarrow +\\infty } \\frac{1}{T} \\log \\#\\Gamma x \\cap B(x,T).$ Using Bishop-Jones' Theorem we will conclude that the limit superior in the definition of $h_\\Gamma $ is a true limit, generalizing Roblin's Theorem [25] holding for CAT$(-1)$ -spaces to Gromov-hyperbolic metric spaces.", "Indeed we have: Theorem B Let $X$ be a proper, $\\delta $ -hyperbolic metric space and let $\\Gamma $ be a discrete group of isometries of $X$ .", "Then $\\limsup _{T \\rightarrow +\\infty } \\frac{1}{T} \\log \\#\\Gamma x \\cap B(x,T) = \\liminf _{T \\rightarrow +\\infty } \\frac{1}{T} \\log \\#\\Gamma x \\cap B(x,T) = h_\\Gamma .$ The proof of Theorem REF is divided in two parts: $h_\\text{inv-top} \\ge h_\\Gamma $ and $\\overline{h}^\\text{loc} \\le h_\\Gamma $ .", "The first inequality is based on the estimate of the topological entropy of the compact $\\Phi _1$ -invariant subsets made by local geodesics never excaping a fixed compact set of $\\Gamma \\backslash X$ .", "Namely we will fix a basepoint $x_0 \\in \\Gamma \\backslash X$ and for every $\\tau \\ge 0$ we will consider the set $K_\\tau $ of $\\sigma $ -local geodesic lines completely contained in the compact ball of radius $\\tau $ around $x_0$ .", "Since the sets $K_\\tau $ are compact and $\\Phi _1$ -invariant we will use them to estimate from below the invariant-topological entropy.", "Using special distances on $K_\\tau $ we will be able to prove that the topological entropy of the system $(K_\\tau , \\Phi _1)$ is at least the lower Lipschitz-topological entropy (see [6] for more properties of this invariant) of the set Geod$_\\sigma (\\Lambda _\\tau )$ of $\\sigma $ -geodesic lines of the universal cover $X$ with both endpoints in the $\\tau $ -uniform radial set $\\Lambda _\\tau $ .", "The $\\tau $ -uniform radial set is the subset of the limit set of $\\Gamma $ made by geodesic rays $[\\tilde{x}_0, z]$ staying at distance at most $\\tau $ from the orbit $\\Gamma \\tilde{x}_0$ , where $\\tilde{x}_0$ is a covering point of $x_0$ .", "Observe that any such geodesic ray defines a local geodesic in the quotient that is entirely contained in the compact ball of radius $\\tau $ around $x_0$ , showing why the topological entropy of $K_\\tau $ is related to dynamical properties of the set $\\Lambda _\\tau $ .", "The lower Lipschitz-topological entropy of the set Geod$_\\sigma (\\Lambda _\\tau )$ (whose definition will be recalled in Section ) equals the lower Minkowski dimension of $\\Lambda _\\tau $ , $\\underline{\\text{MD}}(\\Lambda _\\tau )$ , by a result of [6].", "Finally by Bishop-Jones Theorem we will conclude $h_\\text{inv-top}\\ge \\sup _{\\tau \\ge 0} h_\\text{top}(K_\\tau ,\\Phi _1) \\ge \\sup _{\\tau \\ge 0} \\underline{\\text{MD}}(\\Lambda _\\tau ) \\ge h_\\Gamma .$ The proof of the inequality $\\overline{h}^\\text{loc} \\le h_\\Gamma $ is inspired to Ledrappier's work [15].", "We sketch here the main ideas: to every ergodic measure $\\mu \\in \\mathcal {E}_1$ we associate a (non-canonical) measure $\\nu $ on $\\partial X$ with the following properties: - for every distance ɗ  it holds $h_{\\mu ,\\text{ɗ}}^\\text{loc} \\le \\underset{\\nu }{\\textup {ess}\\sup }\\limsup _{\\rho \\rightarrow 0}\\frac{\\log \\nu (B(z,\\rho ))}{\\log \\rho };$ - $\\nu $ gives full measure to a special subset of the limit set of $\\Gamma $ , that we will call ergodic limit set and we will denote as $\\Lambda _\\text{erg}(\\Gamma )$ .", "The intuition behind the definition of the ergodic limit set is the following.", "For every geodesic $\\gamma \\in \\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ and for every compact $K \\subseteq \\Gamma \\backslash X$ of positive $\\mu $ -measure we can consider the sequence of returning integer times of $\\gamma $ in $K$ .", "By Birkhoff's Ergodic Theorem we deduce that this sequence has a very nice controlled behaviour for $\\mu $ -a.e.$\\gamma $ , that is if $\\vartheta _i(\\gamma )$ is the $i$ -th integer time such that $\\gamma (\\vartheta _i(\\gamma )) \\in K$ then for $\\mu $ -a.e.$\\gamma $ $\\exists \\lim _{i \\rightarrow +\\infty }\\frac{\\vartheta _i(\\gamma )}{i} = \\frac{1}{\\mu (K)} < +\\infty $ .", "Correspondingly the set of ergodic limit points is defined in a similar way: we say that a point $z\\in \\partial X$ belongs to $\\Lambda _\\text{erg}(\\Gamma )$ if one (hence every) geodesic ray $[x,z]$ , where $x$ is a fixed basepoint of $X$ , satisfies this condition: there are points $y_i \\in [x,z]$ , $i\\in \\mathbb {N}$ and a constant $\\tau \\ge 0$ , such that (a) $d(y_i, \\Gamma x) \\le \\tau $ for every $i$ , (b) $d(x,y_i) \\rightarrow +\\infty $ , (c) $\\exists \\lim \\frac{d(x,y_i)}{i} < +\\infty $ .", "Observe that conditions (a) and (b) alone define the well known radial limit points, while (c) captures the idea of well-behaved returning times.", "Now, the right hand side of (REF ) is the classical definition of the upper packing dimension of the measure $\\nu $ (see Section ) and it is always bounded from above by the packing dimension of any set of full measure.", "Therefore, indicating by PD$(\\Lambda _\\text{erg})$ the packing dimension of the ergodic limit set, we conclude: $\\overline{h}^\\text{loc} \\le \\text{PD}(\\Lambda _\\text{erg}).$ Finally the end of the proof of this inequality is based on the following refinement of the easy part of Bishop-Jones' Theorem: Theorem C Let $X$ be a proper, $\\delta $ -hyperbolic metric space and let $\\Gamma $ be a discrete group of isometries of $X$ .", "Then: $\\textup {PD}(\\Lambda _{\\textup {erg}}) = h_\\Gamma .$ The packing dimension was introduced by Tricot ([26]) with a dual construction with respect to the classical Hausdorff dimension (denoted HD).", "For a subset $B$ of a general metric space $Z$ it holds $\\text{HD}(B) \\le \\text{PD}(B) \\le \\underline{\\text{MD}}(B) \\le \\overline{\\text{MD}}(B),$ where the last two quantities are respectively the lower and upper Minkowski dimension of $B$ , where inequalities can well be strict.", "An interesting question is the following: is there an example of a proper $\\delta $ -hyperbolic space with a discrete group of isometries $\\Gamma $ such that PD$(\\Lambda _\\text{rad}(\\Gamma )) > h_\\Gamma $ ?", "Here $\\Lambda _\\text{rad}(\\Gamma )$ denotes the radial limit set of $\\Gamma $ .", "The author thinks the answer should be affirmative." ], [ "Gromov-hyperbolic spaces", "Let $(X,d)$ be a metric space.", "The open (resp.closed) ball of radius $r$ and center $x \\in X$ is denoted by $B(x,r)$ (resp.", "$\\overline{B}(x,r)$ ).", "If we need to specify the metric we will write $B_d(x,r)$ (resp.", "$\\overline{B}(x,r))$ .", "A geodesic segment is an isometry $\\gamma \\colon I \\rightarrow X$ where $I=[a,b]$ is a a bounded interval of $\\mathbb {R}$ .", "The points $\\gamma (a), \\gamma (b)$ are called the endpoints of $\\gamma $ .", "A metric space $X$ is said geodesic if for all couple of points $x,y\\in X$ there exists a geodesic segment whose endpoints are $x$ and $y$ .", "When we will not need to consider a specific geodesic between $x$ and $y$ we will denote any geodesic segment between them, with an abuse of notation, by $[x,y]$ .", "A geodesic ray is an isometry $\\xi \\colon [0,+\\infty )\\rightarrow X$ while a geodesic line is an isometry $\\gamma \\colon \\mathbb {R}\\rightarrow X$ .", "A map $\\gamma \\colon I \\rightarrow X$ , where $I$ is an interval of $\\mathbb {R}$ is a local geodesic if for every $t\\in I$ there exists $\\varepsilon > 0$ such that $\\gamma \\vert _{[t-\\varepsilon , t+\\varepsilon ]}$ is a geodesic segment.", "Let $X$ be a geodesic metric space.", "Given three points $x,y,z \\in X$ , the Gromov product of $y$ and $z$ with respect to $x$ is defined as $(y,z)_x = \\frac{1}{2}\\big ( d(x,y) + d(x,z) - d(y,z) \\big ).$ The space $X$ is said $\\delta $ -hyperbolic if for every four points $x,y,z,w \\in X$ the following 4-points condition hold: $(x,z)_w \\ge \\min \\lbrace (x,y)_w, (y,z)_w \\rbrace - \\delta $ or, equivalently, $d(x,y) + d(z,w) \\le \\max \\lbrace d(x,z) + d(y,w), d(x,w) + d(y,z) \\rbrace + 2\\delta .$ The space $X$ is Gromov hyperbolic if it is $\\delta $ -hyperbolic for some $\\delta \\ge 0$ .", "We recall that Gromov-hyperbolicity should be considered as a negative-curvature condition at large scale: for instance any CAT$(\\kappa )$ metric space, with $\\kappa <0$ is $\\delta $ -hyperbolic for a constant $\\delta $ depending only on $\\kappa $ .", "The converse is false, essentially because the CAT$(\\kappa )$ condition controls the local geometry much better than the Gromov-hyperbolicity due to the convexity of the distance functions in such spaces (see for instance [17], [9] and [8])." ], [ "Gromov boundary", "Let $X$ be a proper, $\\delta $ -hyperbolic metric space $x$ be a point of $X$ .", "The Gromov boundary of $X$ is defined as the quotient $\\partial X = \\lbrace (z_n)_{n \\in \\mathbb {N}} \\subseteq X \\hspace{2.84526pt} | \\hspace{2.84526pt} \\lim _{n,m \\rightarrow +\\infty } (z_n,z_m)_{x} = + \\infty \\rbrace \\hspace{2.84526pt} /_\\sim ,$ where $(z_n)_{n \\in \\mathbb {N}}$ is a sequence of points in $X$ and $\\sim $ is the equivalence relation defined by $(z_n)_{n \\in \\mathbb {N}} \\sim (z_n^{\\prime })_{n \\in \\mathbb {N}}$ if and only if $\\lim _{n,m \\rightarrow +\\infty } (z_n,z_m^{\\prime })_{x} = + \\infty $ .", "We will write $ z = [(z_n)] \\in \\partial X$ for short, and we say that $(z_n)$ converges to $z$ .", "This definition does not depend on the basepoint $x$ .", "There is a natural topology on $X\\cup \\partial X$ that extends the metric topology of $X$ .", "Every geodesic ray $\\xi $ defines a point $\\xi ^+=[(\\xi (n))_{n \\in \\mathbb {N}}]$ of the Gromov boundary $ \\partial X$ : we say that $\\xi $ joins $\\xi (0) = y$ to $\\xi ^+ = z$ .", "Moreover for every $z\\in \\partial X$ and every $x\\in X$ it is possible to find a geodesic ray $\\xi $ such that $\\xi (0)=x$ and $\\xi ^+ = z$ .", "Indeed if $(z_n)$ is a sequence of points converging to $z$ then, by properness of $X$ , the sequence of geodesics $[x,z_n]$ converges to a geodesic ray $\\xi $ which has the properties above (cp.", "Lemma III.3.13 of [2]).", "A geodesic ray joining $x$ to $z\\in \\partial X$ will be denoted by $\\xi _{xz}$ or simply $[x,z]$ .", "The relation between Gromov product and geodesic ray is highlighted in the following lemma.", "Lemma 2.1 ([6], Lemma 5.4) Let $X$ be a proper, $\\delta $ -hyperbolic metric space, $z,z^{\\prime }\\in \\partial X$ and $x\\in X$ .", "Then (i) if $(z,z^{\\prime })_{x} \\ge T$ then $d(\\xi _{xz}(T - \\delta ),\\xi _{xz^{\\prime }}(T - \\delta )) \\le 4\\delta $ ; (ii) for all $b> 0$ , if $d(\\xi _{xz}(T),\\xi _{xz^{\\prime }}(T)) < 2b$ then $(z,z^{\\prime })_{x} > T - b$ .", "The quasiconvex hull of a subset $C$ of $\\partial X$ is the union of all the geodesic lines joining two points of $C$ and it is denoted by QC-Hull$(C)$ .", "The following is a standard computation, see [1] for instance.", "Lemma 2.2 Let $X$ be a proper, $\\delta $ -hyperbolic metric space.", "Then every two geodesic rays $\\xi , \\xi ^{\\prime }$ with same endpoints at infinity are at distance at most $8\\delta $ , i.e.", "there exist $t_1,t_2\\ge 0$ such that $t_1+t_2=d(\\xi (0),\\xi ^{\\prime }(0))$ and $d(\\xi (t + t_1),\\xi ^{\\prime }(t+t_2)) \\le 8\\delta $ for all $t\\in \\mathbb {R}$ ." ], [ "Visual metrics", "When $X$ is a proper, $\\delta $ -hyperbolic metric space it is known that the boundary $\\partial X$ is metrizable.", "A metric $D_{x,a}$ on $\\partial X$ is called a visual metric of parameter $a\\in \\left(0,\\frac{1}{2\\delta \\cdot \\log _2e}\\right)$ and center $x \\in X$ if there exists $V> 0$ such that for all $z,z^{\\prime } \\in \\partial X$ it holds $\\frac{1}{V}e^{-a(z,z^{\\prime })_{x}}\\le D_{x,a}(z,z^{\\prime })\\le V e^{-a(z,z^{\\prime })_{x}}.$ For all $a$ as before and $x\\in X$ there exists always a visual metric of parameter $a$ and center $x$ , see [21].", "As in [21] and [6] we define the generalized visual ball of center $z \\in \\partial X$ and radius $\\rho \\ge 0$ as $B(z,\\rho ) = \\bigg \\lbrace z^{\\prime } \\in \\partial X \\text{ s.t. }", "(z,z^{\\prime })_{x} > \\log \\frac{1}{\\rho } \\bigg \\rbrace .$ It is comparable to the metric balls of the visual metrics on $\\partial X$ .", "Lemma 2.3 Let $D_{x,a}$ be a visual distance of center $x$ and parameter $a$ on $\\partial X$ .", "Then for all $z\\in \\partial X$ and for all $\\rho >0$ it holds $B_{D_{x,a}}\\left(z, \\frac{1}{V}\\rho ^a\\right) \\subseteq B(z,\\rho )\\subseteq B_{D_{x,a}}(z, V\\rho ^a ).$ It is classical that generalized visual balls are related to shadows, whose definition is the following.", "Let $x\\in X$ be a basepoint.", "The shadow of radius $r>0$ casted by a point $y\\in X$ with center $x$ is the set: $\\text{Shad}_x(y,r) = \\lbrace z\\in \\partial X \\text{ s.t. }", "[x,z]\\cap B(y,r) \\ne \\emptyset \\text{ for all rays } [x,z]\\rbrace .$ For our purposes we just need: Lemma 2.4 Let $X$ be a proper, $\\delta $ -hyperbolic metric space.", "Let $z\\in \\partial X$ , $x\\in X$ and $T\\ge 0$ .", "Then for all $r>0$ it holds $\\textup {Shad}_{x}\\left(\\xi _z\\left(T\\right), r\\right) \\subseteq B(z, e^{-T + r}).$" ], [ "Hausdorff and Packing dimensions", "In the first part we recall briefly the definitions of Hausdorff and packing dimensions of a subset of a metric space.", "In the second part we will recall four definitions of local dimensions of measures and we will relate them to Hasudorff and packing dimensions of suitable subsets.", "Finally we will adapt these constructions and results to the case of the boundary at infinity of a $\\delta $ -hyperbolic metric space.", "The facts presented here are classical and can be found easily in literature." ], [ "Definitions of Hausdorff and Packing dimensions", "Let $(X,d)$ be a metric space.", "The $\\alpha $ -Hausdorff measure of a Borelian subset $B\\subset X$ , $\\alpha \\ge 0$ is classically defined as $\\mathcal {H}^\\alpha _d(B) = \\lim _{\\eta \\rightarrow 0}\\inf \\left\\lbrace \\sum _{i\\in \\mathbb {N}} r_i^\\alpha \\text{ s.t. }", "B\\subseteq \\bigcup _{i\\in \\mathbb {N}}B(x_i,r_i) \\text{ and } r_i\\le \\eta \\right\\rbrace .$ The argument of the limit is increasing when $\\eta $ tends to 0, so the limit exists.", "This formula actually defines a measure on $X$ .", "The Hausdorff dimension of a Borelian subset $B$ of $X$ , denoted HD$_d(B)$ is the unique real number $\\alpha \\ge 0$ such that $\\mathcal {H}^{\\alpha ^{\\prime }}_d(B) = 0$ for all $\\alpha ^{\\prime } > \\alpha $ and $\\mathcal {H}^{\\alpha ^{\\prime }}_d(B) = +\\infty $ for all $\\alpha ^{\\prime } < \\alpha $ .", "The packing dimension is defined in a similar way, but using disjoint balls inside $B$ instead of coverings.", "To be precise we define, for all $\\alpha \\ge 0$ and for all Borelian subsets $B$ of $X$ , $\\mathcal {P}^\\alpha _d(B) = \\lim _{\\eta \\rightarrow 0} \\sup \\left\\lbrace \\sum _{i\\in \\mathbb {N}} r_i^\\alpha \\text{ s.t. }", "B(x_i,r_i) \\text{ are disjoint, }x_i\\in B \\text{ and } r_i\\le \\eta \\right\\rbrace .$ This is not a measure on $X$ but only a pre-measure.", "By a standard procedure one can define the $\\alpha $ -Packing measure as $\\hat{\\mathcal {P}}_d^\\alpha (B) = \\inf \\left\\lbrace \\sum _{k=1}^\\infty \\mathcal {P}_d^\\alpha (B_k) \\text{ s.t. }", "B\\subseteq \\bigcup _{k=1}^\\infty B_k\\right\\rbrace .$ The packing dimension of a Borelian subset $B\\subseteq X$ , denoted PD$_d(B)$ , is the unique real number $\\alpha \\ge 0$ such that $\\hat{\\mathcal {P}}^{\\alpha ^{\\prime }}_d(B) = 0$ for all $\\alpha ^{\\prime } > \\alpha $ and $\\hat{\\mathcal {P}}^{\\alpha ^{\\prime }}_d(B) = +\\infty $ for all $\\alpha ^{\\prime } < \\alpha $ .", "The packing dimension has another useful interpretation (see [13], Proposition 3.8): indeed for all Borelian subsets $B\\subseteq X$ we have $\\text{PD}_d(B) = \\inf \\left\\lbrace \\sup _k \\overline{\\text{MD}}_d(B_k) \\text{ s.t. }", "B\\subseteq \\bigcup _{k=1}^\\infty B_k\\right\\rbrace .$ The quantity $\\overline{\\text{MD}}_d$ denotes the upper Minkowski dimension, namely: $\\overline{\\text{MD}}_d(B) = \\limsup _{r \\rightarrow 0}\\frac{\\log \\text{Cov}_d(B,r)}{\\log \\frac{1}{r}},$ where $B$ is any subset of $X$ and $\\text{Cov}_d(B,r)$ denotes the minimal number of $d$ -balls of radius $r$ needed to cover $B$ .", "Taking the limit inferior in place of the limit superior in (REF ) one defines the lower Minkowski dimension of $B$ , denoted $\\underline{\\text{MD}}_d(B)$ ." ], [ "Local dimensions of measures", "Let $(X,d)$ be a metric space, let ${B}$ be the $\\sigma $ -algebra generated by the Borelian subsets of $X$ and let $\\nu $ be a probability measure on the measure space $(X,{B})$ .", "There are several notions of dimension of the measure $\\nu $ , here we recall four of them.", "The lower and upper Hausdorff dimensions of $\\nu $ are respectively: $\\underline{\\text{HD}}_d(\\nu )=\\underset{\\nu }{\\textup {ess}\\inf }\\liminf _{r \\rightarrow 0}\\frac{\\log \\nu (B(y,r))}{\\log r}$ $\\overline{\\text{HD}}_d(\\nu )=\\underset{\\nu }{\\textup {ess}\\sup }\\liminf _{r \\rightarrow 0}\\frac{\\log \\nu (B(y,r))}{\\log r}$ while the lower and upper packing dimension of $\\nu $ are respectively: $\\underline{\\text{PD}}_d(\\nu )=\\underset{\\nu }{\\textup {ess}\\inf }\\limsup _{r \\rightarrow 0}\\frac{\\log \\nu (B(y,r))}{\\log r},$ $\\overline{\\text{PD}}_d(\\nu )=\\underset{\\nu }{\\textup {ess}\\sup }\\limsup _{r \\rightarrow 0}\\frac{\\log \\nu (B(y,r))}{\\log r}.$ The name Hausdorff and packing dimension is justified by the following facts, that are well-known at least in the Euclidean spaces.", "Proposition 3.1 Let $(X,d)$ be a separable metric space and let $\\nu $ be a probability measure on the measure space $(X,{B})$ .", "Then $\\begin{aligned}\\overline{\\textup {PD}}_d(\\nu )&=\\inf \\lbrace \\textup {PD}_d(B) \\textup { s.t. }", "\\nu (B)=1\\rbrace \\\\\\underline{\\textup {PD}}_d(\\nu )&=\\inf \\lbrace \\textup {PD}_d(B) \\textup { s.t. }", "\\nu (B)>0\\rbrace \\\\\\overline{\\textup {HD}}_d(\\nu )&=\\inf \\lbrace \\textup {HD}_d(B) \\textup { s.t. }", "\\nu (B)=1\\rbrace \\\\\\underline{\\textup {HD}}_d(\\nu )&=\\inf \\lbrace \\textup {HD}_d(B) \\textup { s.t. }", "\\nu (B)>0\\rbrace .", "\\\\\\end{aligned}$ The last two equalities are proved in [15], Proposition 2.5 and Section 7.", "We remark that the properness assumption in these references can be dropped to separability.", "Let us now show the first equality, the proof of the second equality is similar and it will be omitted.", "Let $\\alpha = \\overline{\\textup {PD}}_d(\\nu )$ , that is for all $\\varepsilon > 0$ it exists a set $A_\\varepsilon \\subseteq X$ of positive measure such that $\\limsup _{r \\rightarrow 0} \\frac{\\log \\nu (B(x,r))}{\\log r} \\ge \\alpha - \\varepsilon $ for all $x\\in A_\\varepsilon $ .", "Now we apply Egoroff's Theorem as in [15].", "For all $r>0$ and all $x\\in X$ we set $f_r(x)=\\frac{\\log \\nu (B(x,r))}{\\log r},\\qquad g_r(x)=\\sup _{0<r^{\\prime }<r}f_{r^{\\prime }}(x),\\qquad h_n(x)=g_{\\frac{1}{n}}(x).$ By definition $\\limsup _{r \\rightarrow 0}f_r(x) = \\lim _{r\\rightarrow 0}g_r(x) = \\lim _{n\\rightarrow +\\infty }h_n(x) =: \\alpha (x)$ .", "By Egoroff's Theorem there exists a set of positive measure $A_\\varepsilon ^{\\prime }$ such that $\\vert h_n(x) - \\alpha (x) \\vert < \\varepsilon $ for all $x\\in A_\\varepsilon ^{\\prime }\\subseteq A_\\varepsilon $ and all $n\\ge n_\\varepsilon $ .", "This means that for all $r < r_\\varepsilon =\\frac{1}{n_\\varepsilon }$ and for all $x\\in A_\\varepsilon ^{\\prime }$ it holds $\\frac{\\log \\nu (B(x,r))}{\\log r} \\ge \\alpha (x) - \\varepsilon \\ge \\alpha - 2\\varepsilon $ .", "In other words for all $r\\le r_\\varepsilon $ and for all $x\\in A_\\varepsilon ^{\\prime }$ it holds $\\nu (B(x,r))\\le r^{\\alpha - 2\\varepsilon }$ .", "Let now $B$ be a borelian subset of $X$ with $\\nu (B)=1$ .", "By (REF ) we know that $\\text{PD}_d(B) = \\inf \\left\\lbrace \\sup \\overline{\\text{MD}}_d(B_k)\\text{ s.t. }", "\\bigcup _k B_k \\supseteq B\\right\\rbrace .$ Let $B_k$ be subsets of $X$ such that $\\bigcup _kB_k\\supseteq B$ .", "Since $\\nu (B)=1$ there must be some index $k$ such that $\\nu (B_k\\cap A_\\varepsilon ^{\\prime }) > 0$ .", "Observe that from the estimates we proved before we get $\\text{Cov}_d(B_k\\cap A_\\varepsilon ^{\\prime }, r) \\ge \\nu (B_k\\cap A_\\varepsilon ^{\\prime })\\cdot \\left(\\frac{1}{r}\\right)^{\\alpha - 2\\varepsilon }$ for every $r<r_\\varepsilon $ .", "This directly implies $\\overline{\\text{MD}}_d(B_k)\\ge \\overline{\\text{MD}}_d(B_k\\cap A_\\varepsilon ^{\\prime }) \\ge \\alpha - 2\\varepsilon ,$ and so $\\text{PD}_d(B)\\ge \\alpha - 2\\varepsilon $ .", "By the arbitrariness of $\\varepsilon $ we conclude that $\\overline{\\textup {PD}}_d(\\nu )\\le \\inf \\lbrace \\textup {PD}_d(B) \\textup { s.t. }", "\\nu (B)=1\\rbrace .$ For the other inequality for every $\\varepsilon > 0$ we take the set $B_\\varepsilon = \\left\\lbrace x\\in X \\text{ s.t. }", "\\limsup _{r \\rightarrow 0}\\frac{\\log \\nu (B(x,r))}{\\log r} \\le \\alpha + \\varepsilon \\right\\rbrace .$ Clearly it satisfies $\\nu (B_\\varepsilon ) = 1$ .", "Moreover it can be covered by the union of the sets $B_{\\varepsilon , k} = \\left\\lbrace x\\in X \\text{ s.t. }", "\\frac{\\log \\nu (B(x,r))}{\\log r} \\le \\alpha + 2\\varepsilon \\text{ for all } r\\le \\frac{1}{k} \\right\\rbrace ,$ where $k\\in \\mathbb {N}$ .", "By similar estimates as before we can conclude that $\\overline{\\text{MD}}(B_{\\varepsilon , k}) \\le \\alpha + 2\\varepsilon $ for every $k$ , and so PD$_d(B_\\varepsilon ) \\le \\alpha + 2\\varepsilon $ .", "By the arbitrariness of $\\varepsilon $ we get the desired equality." ], [ "Visual dimensions", "Let $X$ be a proper, $\\delta $ -hyperbolic metric space and let $x\\in X$ .", "We know that $\\partial X$ supports several visual metrics $D_{x,a}$ , so the Hausdorff dimension, the packing dimension and the Minkowski dimension of subsets of $\\partial X$ are well defined with respect to $D_{x,a}$ .", "There is a way to define universal versions of these quantities that do not depend neither on $x$ nor on $a$ .", "For a Borelian subset $B$ of $\\partial X$ and for all $\\alpha \\ge 0$ we set, following [21], $\\mathcal {H}^\\alpha (B) = \\lim _{\\eta \\rightarrow 0}\\inf \\left\\lbrace \\sum _{i\\in \\mathbb {N}} \\rho _i^\\alpha \\text{ s.t. }", "B\\subseteq \\bigcup _{i\\in \\mathbb {N}}B(z_i,\\rho _i) \\text{ and } \\rho _i\\le \\eta \\right\\rbrace ,$ where $B(z_i,\\rho _i)$ are generalized visual balls.", "As in the classical case the visual Hausdorff dimension of $B$ is defined as the unique $\\alpha \\ge 0$ such that $\\mathcal {H}^{\\alpha ^{\\prime }}(B) = 0$ for all $\\alpha ^{\\prime } > \\alpha $ and $\\mathcal {H}^{\\alpha ^{\\prime }}(B) = +\\infty $ for all $\\alpha ^{\\prime }<\\alpha $ .", "The visual Hausdorff dimension of the borelian subset $B$ is denoted by HD$(B)$ .", "By Lemma REF , see also [21], we have HD$(B) = a\\cdot \\text{HD}_{D_{x,a}}(B)$ for all visual metrics $D_{x,a}$ of center $x$ and parameter $a$ .", "In the same way we can define the visual $\\alpha $ -packing pre-measure of a Borelian subset $B$ of $\\partial X$ by $\\mathcal {P}^\\alpha (B) = \\lim _{\\eta \\rightarrow 0} \\sup \\left\\lbrace \\sum _{i\\in \\mathbb {N}} \\rho _i^\\alpha \\text{ s.t. }", "B(z_i,\\rho _i) \\text{ are disjoint, }x_i\\in B \\text{ and } \\rho _i\\le \\eta \\right\\rbrace ,$ and also in this case $B(z_i,\\rho _i)$ are generalized visual balls.", "As usual we can define the visual $\\alpha $ -packing measure by $\\hat{\\mathcal {P}}^\\alpha (B) = \\inf \\left\\lbrace \\sum _{k=1}^\\infty \\mathcal {P}^\\alpha (B_k) \\text{ s.t. }", "B\\subseteq \\bigcup _{k=1}^\\infty B_k\\right\\rbrace .$ Consequently it is defined the visual packing dimension of a Borelian set $B$ , denoted by PD$(B)$ .", "Using Lemma REF as in the case of the Hausdorff measure (see [21]) it is easy to check that for every visual metric $D_{x,a}$ of center $x$ and parameter $a$ it holds: $\\frac{1}{V^a} \\hat{\\mathcal {P}}_{D_{x,a}}^{\\frac{\\alpha }{a}}(B) \\le \\hat{\\mathcal {P}}^\\alpha (B) \\le V^a \\hat{\\mathcal {P}}_{D_{x,a}}^{\\frac{\\alpha }{a}}(B)$ for all $\\alpha \\ge 0$ and all Borelian sets $B\\subseteq \\partial X$ .", "Therefore for every Borelian set $B$ it holds PD$(B)=a\\cdot \\text{PD}_{D_{x,a}}(B)$ .", "Using generalized visual balls instead of metric balls with respect to a visual metric one can define also the visual upper and lower Minkowski dimension of a subset $B\\subseteq \\partial X$ , respectively: $\\overline{\\text{MD}}(B) = \\limsup _{\\rho \\rightarrow 0}\\frac{\\log \\text{Cov}(B,\\rho )}{\\log \\rho }, \\qquad \\underline{\\text{MD}}(B) = \\liminf _{\\rho \\rightarrow 0}\\frac{\\log \\text{Cov}(B,\\rho )}{\\log \\rho },$ where $\\text{Cov}(B,\\rho )$ denotes the minimal number of generalized visual balls of radius $\\rho $ needed to cover $B$ .", "Using again Lemma REF (see also [6]) one has $\\overline{\\text{MD}}(B) = a\\cdot \\overline{\\text{MD}}_{D_{x,a}}(B)$ for every Borelian set $B$ and every visual metric of center $x$ and parameter $a$ .", "The same of course holds for the lower Minkowski dimensions.", "Moreover it is easy to check that for every Borelian set $B$ of $\\partial X$ the numbers HD$(B)$ , PD$(B)$ , $\\underline{\\text{MD}}(B)$ , $\\overline{\\text{MD}}(B)$ do not depend on $x$ too, see Proposition 6.4 of [21].", "Clearly, using their comparisons with the classical dimensions defined by a visual metric, we get $\\text{HD}(B) \\le \\text{PD}(B) \\le \\underline{\\text{MD}}(B) \\le \\overline{\\text{MD}}(B)$ and $\\text{PD}(B) = \\inf \\left\\lbrace \\sup _k \\overline{\\text{MD}}(B_k) \\text{ s.t. }", "B\\subseteq \\bigcup _{k=1}^\\infty B_k\\right\\rbrace .$ for all Borelian subsets $B$ of $\\partial X$ .", "Let ${B}$ be the $\\sigma $ -algebra generated by the Borelian subsets of $\\partial X$ and let $\\nu $ be a probability measure on the measure space $(\\partial X,{B})$ .", "We define the visual lower and upper Hausdorff dimensions of $\\nu $ as: $\\underline{\\text{HD}}(\\nu )=\\underset{\\nu }{\\textup {ess}\\inf }\\liminf _{\\rho \\rightarrow 0}\\frac{\\log \\nu (B(z,\\rho ))}{\\log \\rho }$ $\\overline{\\text{HD}}(\\nu )=\\underset{\\nu }{\\textup {ess}\\sup }\\liminf _{\\rho \\rightarrow 0}\\frac{\\log \\nu (B(z,\\rho ))}{\\log \\rho }$ while the visual lower and upper packing dimension of $\\nu $ are respectively: $\\underline{\\text{PD}}(\\nu )=\\underset{\\nu }{\\textup {ess}\\inf }\\limsup _{\\rho \\rightarrow 0}\\frac{\\log \\nu (B(z,\\rho ))}{\\log \\rho },$ $\\overline{\\text{PD}}(\\nu )=\\underset{\\nu }{\\textup {ess}\\sup }\\limsup _{\\rho \\rightarrow 0}\\frac{\\log \\nu (B(z,\\rho ))}{\\log \\rho }.$ Here the difference is that $B(z,\\rho )$ denotes the generalized visual ball instead of metric balls with respect to a visual distance $D_{x,a}$ .", "Using again Lemma REF it is straightforward to prove that $\\underline{\\text{HD}}(\\nu ) = a\\cdot \\underline{\\text{HD}}_{D_{x,a}}(\\nu )$ for every visual metric $D_{x,a}$ , and similarly for the other dimensions.", "Therefore $\\begin{aligned}\\overline{\\text{PD}}(\\nu )&=\\inf \\lbrace \\text{PD}(B) \\text{ s.t. }", "\\nu (B)=1\\rbrace \\\\\\underline{\\text{PD}}(\\nu )&=\\inf \\lbrace \\text{PD}(B) \\text{ s.t. }", "\\nu (B)>0\\rbrace \\\\\\overline{\\text{HD}}(\\nu )&=\\inf \\lbrace \\text{HD}(B) \\text{ s.t. }", "\\nu (B)=1\\rbrace \\\\\\underline{\\text{HD}}(\\nu )&=\\inf \\lbrace \\text{HD}(B) \\text{ s.t. }", "\\nu (B)>0\\rbrace .", "\\\\\\end{aligned}$" ], [ "Bishop-Jones Theorem", "If $X$ is a proper metric space we denote by Isom$(X)$ its group of isometries, endowed with the uniform convergence on compact subsets of $X$ .", "A subgroup $\\Gamma $ of Isom$(X)$ is called discrete if the following equivalent conditions (see [1]) hold: (a) $\\Gamma $ is discrete as a subspace of Isom$(X)$ ; (b) $\\forall x\\in X$ and $R\\ge 0$ the set $\\Sigma _R(x) = \\lbrace g \\in \\Gamma \\text{ s.t. }", "g x\\in \\overline{B}(x,R)\\rbrace $ is finite.", "The critical exponent of a discrete group of isometries $\\Gamma $ acting on a proper metric space $X$ can be defined using the Poincaré series, or alternatively ([6], [7]), as $\\overline{h_\\Gamma }(X) = \\limsup _{T \\rightarrow +\\infty }\\frac{1}{T}\\log \\# \\Gamma x \\cap B(x,T),$ where $x$ is a fixed point of $X$ .", "Clearly this quantity does not depend on the choice of $x$ .", "In the following we will often write $\\overline{h_\\Gamma }(X)=:h_\\Gamma $ .", "Taking the limit inferior instead of the limit superior we define the lower critical exponent, denoted by $\\underline{h_\\Gamma }(X)$ .", "In [25] it is proved that if $\\Gamma $ is a discrete group of isometries of a CAT$(-1)$ space then $\\overline{h_\\Gamma }(X) = \\underline{h_\\Gamma }(X)$ .", "We will generalize this result to proper, $\\delta $ -hyperbolic spaces (see Corollary REF )." ], [ "Limit sets", "We specialize the general situation above to the case of a proper, $\\delta $ -hyperbolic metric space $X$ .", "Every isometry of $X$ acts naturally on $\\partial X$ and the resulting map on $X\\cup \\partial X$ is a homeomorphism.", "The limit set $\\Lambda (\\Gamma )$ of a discrete group of isometries $\\Gamma $ is the set of accumulation points of the orbit $\\Gamma x$ on $\\partial X$ , where $x$ is any point of $X$ ; it is the smallest $\\Gamma $ -invariant closed set of the Gromov boundary (cp.", "[7], Theorem 5.1) and it does not depend on $x$ .", "There are several interesting subsets of the limit set: the radial limit set, the uniformly radial limit set, etc.", "They are related to important sets of the geodesic flow on the quotient space $\\Gamma \\backslash X$ , as we will see in the second part of the paper.", "In order to recall their definiton we need to introduce a class of subsets of $\\partial X$ .", "We fix a basepoint $x\\in X$ .", "Let $\\tau $ and $\\Theta = \\lbrace \\vartheta _i \\rbrace _{i\\in \\mathbb {N}}$ be, respectively, a positive real number and an increasing sequence of real numbers with $\\lim _{i \\rightarrow +\\infty }\\vartheta _i = +\\infty $ .", "We define $\\Lambda _{\\tau , \\Theta }(\\Gamma )$ as the set of points $z\\in \\partial X$ such that there exists a geodesic ray $[x,z]$ satisfying the following: for every $i\\in \\mathbb {N}$ there exists a point $y_i \\in [x,z]$ with $d(x,y_i) \\in [\\vartheta _i, \\vartheta _{i+1}]$ such that $d(y_i,\\Gamma x) \\le \\tau $ .", "We observe that up to change $\\tau $ with $\\tau + 8\\delta $ the definition above does not depend on the choice of the geodesic ray $[x,z]$ .", "Lemma 4.1 In the situation above it holds: (i) $\\Lambda _{\\tau , \\Theta }(\\Gamma ) \\subseteq \\Lambda (\\Gamma )$ ; (ii) the set $\\Lambda _{\\tau , \\Theta }(\\Gamma )$ is closed.", "The first statement is obvious, so we focus on (ii).", "Let $z^k \\in \\Lambda _{\\tau , \\Theta }(\\Gamma )$ converging to $z^\\infty $ .", "For every $k$ let $\\xi ^k = [x,z^k]$ be a geodesic ray as in the definition of $\\Lambda _{\\tau , \\Theta }(\\Gamma )$ .", "We know that, up to a subsequence, the sequence $\\xi ^k$ converges uniformly on compact sets of $[0,+\\infty )$ to a geodesic ray $\\xi ^\\infty = [x,z^\\infty ]$ .", "We fix $i\\in \\mathbb {N}$ and we take points $y_i^k$ with $d(x,y_i^k)\\in [\\vartheta _{i}, \\vartheta _{i+1}]$ and $d(y_i^k,\\Gamma x)\\le \\tau $ .", "The sequence $y_i^k$ converges to a point $y_i^\\infty \\in [x,z^\\infty ]$ with $d(x,y_i^\\infty ) \\in [\\vartheta _{i}, \\vartheta _{i+1}]$ .", "Moreover clearly $d(y_i^\\infty , \\Gamma x) \\le \\tau $ .", "Since this is true for every $i\\in \\mathbb {N}$ we conclude that $z^\\infty \\in \\Lambda _{\\tau , \\Theta }(\\Gamma )$ .", "We can now introduce some interesting subsets of the limit set of $\\Gamma $ .", "Let $\\Theta _\\text{rad}$ be the set of increasing, unbounded sequences of real numbers.", "The radial limit set is classically defined as $\\Lambda _\\text{rad}(\\Gamma ) = \\bigcup _{\\tau \\ge 0}\\bigcup _{\\Theta \\in \\Theta _\\text{rad}}\\Lambda _{\\tau , \\Theta }(\\Gamma ).$ The uniform radial limit set is defined (see [12]) as $\\Lambda _\\text{u-rad}(\\Gamma ) = \\bigcup _{\\tau \\ge 0}\\Lambda _{\\tau }(\\Gamma ),$ where $\\Lambda _\\tau (\\Gamma )=\\Lambda _{\\tau , \\lbrace i\\tau \\rbrace }(\\Gamma )$ .", "Another interesting set that will play an important role in the paper is the ergodic limit set, defined as: $\\Lambda _\\text{erg}(\\Gamma ) = \\bigcup _{\\tau \\ge 0}\\bigcup _{\\Theta \\in \\Theta _\\text{erg}}\\Lambda _{\\tau , \\Theta }(\\Gamma ),$ where a sequence $\\Theta = \\lbrace \\vartheta _i\\rbrace $ belongs to $\\Theta _\\text{erg}$ if $\\exists \\lim _{i\\rightarrow +\\infty } \\frac{\\vartheta _{i}}{i} <+\\infty $ .", "The name of this set will be justified in the second part of the paper: for the moment we just say it is related to properties of ergodic measures of the geodesic flow in the quotient space.", "When $\\Gamma $ is clear in the context we will simply write $\\Lambda _{\\tau , \\Theta }, \\Lambda _{\\text{rad}}, \\Lambda _{\\text{u-rad}}, \\Lambda _{\\textup {erg}}, \\Lambda $ , omitting $\\Gamma $ .", "Lemma 4.2 In the situation above the sets $\\Lambda _{\\textup {rad}}, \\Lambda _{\\textup {u-rad}}$ and $\\Lambda _{\\textup {erg}}$ are $\\Gamma $ -invariant and do not depend on $x$ .", "Let $y$ be another point of $X$ and let $z\\in \\partial X$ .", "By Lemma REF for every geodesic rays $\\xi = [y,z]$ , $\\xi ^{\\prime } = [x,z]$ there are $t_1,t_2\\ge 0$ such that $t_1+t_2\\le d(x,y)$ and $d(\\xi (t+t_1), \\xi ^{\\prime }(t+t_2))\\le 8\\delta $ .", "This means that $d(\\xi (t), \\xi ^{\\prime }(t)) \\le d(x,y) + 8\\delta $ for every $t\\ge 0$ .", "It is then straightforward to see that if $z\\in \\Lambda _{\\tau , \\Theta }$ (as defined with respect to $x$ ) then it belongs to $\\Lambda _{\\tau + d(x,y) + 8\\delta , \\Theta }$ as defined with respect to $y$ .", "This shows the thesis." ], [ "Bishop-Jones' Theorem", "The celebrated Bishop-Jones' Theorem, in the generic version of [12], states the following: Theorem 4.3 ([3], [12]) Let $\\Gamma $ be a discrete group of isometries of a proper, $\\delta $ -hyperbolic metric space $X$ .", "Then: $h_\\Gamma = \\textup {HD}(\\Lambda _{\\textup {rad}}) = \\textup {HD}(\\Lambda _{\\textup {u-rad}}) = \\sup _{\\tau \\ge 0} \\textup {HD}(\\Lambda _{\\tau }).$ Our contribution to Bishop-Jones' Theorem is the following: Theorem 4.4 Let $X$ be a proper, $\\delta $ -hyperbolic metric space and let $\\Gamma $ be a discrete group of isometries of $X$ .", "Then: $\\textup {PD}(\\Lambda _{\\textup {erg}}) = h_\\Gamma $ In order to introduce the techniques we will use in the proof of this theorem we start with another easier result that generalizes Roblin's Theorem, [25].", "We remark that the proof is completely different.", "Proposition 4.5 Let $X$ be a proper, $\\delta $ -hyperbolic metric space and let $\\Gamma $ be a discrete group of isometries.", "Then the critical exponent is a true limit, i.e.", "$h_\\Gamma = \\lim _{T\\rightarrow +\\infty }\\frac{1}{T}\\log \\#\\Gamma x \\cap B(x,T).$ By Bishop-Jones' Theorem we have $h_\\Gamma = \\sup _{\\tau \\ge 0} \\text{HD}(\\Lambda _\\tau ) \\le \\sup _{\\tau \\ge 0} \\underline{\\text{MD}}(\\Lambda _\\tau ).$ So it would be enough to show that $\\sup _{\\tau \\ge 0} \\underline{\\text{MD}}(\\Lambda _\\tau ) \\le \\underline{h_\\Gamma }(X).$ We fix $\\tau \\ge 0$ .", "For every $\\varepsilon > 0$ we take a subsequence $T_j \\rightarrow +\\infty $ such that for all $j$ it holds $\\frac{1}{T_j}\\log \\#\\Gamma x \\cap \\overline{B}(x,T_j) \\le \\underline{h_\\Gamma }(X) + \\varepsilon .$ We define $\\rho _j = e^{-T_j}$ : notice that $\\rho _j \\rightarrow 0$ .", "Let $k_j\\in \\mathbb {N}$ such that $(k_j-1)\\tau < T_j \\le k_j\\tau $ .", "For each $g\\in \\Gamma $ such that $(k_j-1)\\tau \\le d(x,gx)\\le (k_j+1)\\tau $ we consider the shadow Shad$_x(gx,2\\tau )$ .", "First of all if $z\\in \\Lambda _\\tau $ then we know there exists $g\\in \\Gamma $ as before such that $d([x,z], gx) \\le \\tau $ , so $z\\in \\text{Shad}_x(gx,2\\tau )$ .", "In other words these shadows cover $\\Lambda _\\tau $ , and their cardinality is at most $e^{(\\underline{h_\\Gamma }(X) + \\varepsilon )(k_j+1)\\tau }$ .", "So, by Lemma REF , $\\Lambda _\\tau $ is covered by at most $e^{(\\underline{h_\\Gamma }(X) + \\varepsilon )(k_j+1)\\tau }$ generalized visual balls of radius $e^{-T_j + 3\\tau } = e^{3\\tau }\\rho _j$ .", "Therefore $\\begin{aligned}\\underline{\\text{MD}}(\\Lambda _{\\tau })&\\le \\liminf _{j \\rightarrow +\\infty }\\frac{\\log \\text{Cov}(\\Lambda _{\\tau }, e^{2\\tau }\\rho _j)}{\\log \\frac{1}{e^{3\\tau }\\rho _j}} \\\\&\\le \\liminf _{j \\rightarrow +\\infty }\\frac{(\\underline{h_\\Gamma }(X) + \\varepsilon )(k_j+1)\\tau }{-3\\tau + (k_j-1)\\tau } = \\underline{h_\\Gamma }(X)+\\varepsilon .\\end{aligned}$ By the arbitrariness of $\\varepsilon $ we can conclude the proof.", "There are several remarks we can do about this proof: (a) The proof is still valid for every sequence $T_j \\rightarrow + \\infty $ , so it implies also that $\\sup _{\\tau \\ge 0} \\overline{\\textup {MD}}(\\Lambda _\\tau ) \\le h_\\Gamma $ .", "Therefore we have another improvement of Bishop-Jones Theorem, namely: $\\sup _{\\tau \\ge 0} {\\textup {HD}}(\\Lambda _\\tau ) = \\sup _{\\tau \\ge 0} \\underline{\\textup {MD}}(\\Lambda _\\tau ) = \\sup _{\\tau \\ge 0} \\overline{\\textup {MD}}(\\Lambda _\\tau )=h_\\Gamma .$ (b) $\\Lambda _{\\textup {u-rad}} = \\bigcup _{\\tau \\in \\mathbb {N}}\\Lambda _\\tau $ , so by (a) and (REF ) we deduce that $\\textup {PD}(\\Lambda _{\\textup {u-rad}})=h_\\Gamma $ .", "(c) We can get the same estimate of the Minkowski dimensions from above weakening the assumptions on the sets $\\Lambda _\\tau $ .", "Indeed take a set $\\Lambda _{\\tau , \\Theta }$ such that $\\limsup _{i\\rightarrow +\\infty } \\frac{\\vartheta _{i+1}}{\\vartheta _i} = 1$ .", "Then we can cover this set by the $2\\tau $ -shadows casted by points of the orbit $\\Gamma x$ whose distance from $x$ is between $\\vartheta _{i_j}$ and $\\vartheta _{i_j + 1}$ , with $i_j \\rightarrow +\\infty $ when $j \\rightarrow + \\infty $ .", "Therefore arguing as before we obtain $\\underline{\\textup {MD}}(\\Lambda _{\\tau , \\Theta }) \\le \\liminf _{j \\rightarrow +\\infty }\\frac{(\\underline{h_\\Gamma }(X) + \\varepsilon )\\vartheta _{i_j + 1}}{\\vartheta _{i_j - 1}} \\le \\underline{h_\\Gamma }(X)+\\varepsilon ,$ where the last step follows by the asymptotic behaviour of the sequence $\\Theta $ .", "A similar estimate holds for the upper Minkowski dimension.", "(d) One could be tempted to conclude that the packing dimension of the set $\\bigcup _{\\tau \\ge 0}\\bigcup _{\\Theta } \\Lambda _{\\tau , \\Theta }$ , where $\\Theta $ is a sequence such that $\\limsup _{i\\rightarrow +\\infty } \\frac{\\vartheta _{i+1}}{\\vartheta _i} = 1$ , is $\\le h_\\Gamma $ .", "But this is not necessarily true since in (REF ) it is required a countable covering and not an arbitrary covering.", "That is why the estimate of the packing dimension of the ergodic limit set $\\Lambda _\\textup {erg}$ in Theorem REF is not so easy.", "However as we will see in a moment the ideas behind the proof are similar to the ones used in the proposition above.", "[Proof of Theorem REF ] We notice it is enough to prove that PD$(\\Lambda _\\text{erg}) \\le h_\\Gamma $ .", "The strategy is the following: for every $\\varepsilon > 0$ we want to find a countable family of sets $\\lbrace B_k\\rbrace _{k\\in \\mathbb {N}}$ of $\\partial X$ such that $\\Lambda _{\\text{erg}} \\subseteq \\bigcup _{k=1}^\\infty B_k$ and $\\sup _{k\\in \\mathbb {N}}\\overline{\\text{MD}}(B_k)\\le (h_\\Gamma + \\varepsilon )(1+\\varepsilon )$ .", "Indeed if this is true then by (REF ): $\\text{PD}(\\Lambda _\\text{erg}) \\le \\sup _{k\\in \\mathbb {N}}\\overline{\\text{MD}}(B_k)\\le (h_\\Gamma + \\varepsilon )(1+\\varepsilon ),$ and by the arbitrariness of $\\varepsilon $ the thesis is true.", "So we fix $\\varepsilon > 0$ and we proceed to define the countable family.", "For $m,n\\in \\mathbb {N}$ and $l\\in \\mathbb {Q}_{> 0}$ we define $B_{m,l,n} = \\bigcup _{\\Theta } \\Lambda _{m,\\Theta },$ where $\\Theta $ is taken among all sequences such that for all $i\\ge n$ it holds $l-\\eta _l \\le \\frac{\\vartheta _{i}}{i} \\le l + \\eta _l,$ where $\\eta _l = \\frac{\\varepsilon }{2+\\varepsilon }\\cdot l$ .", "First of all if $z\\in \\Lambda _\\text{erg}$ we know that $z\\in \\Lambda _{m,\\Theta }$ for some $m \\in \\mathbb {N}$ and $\\Theta $ satisfying $\\lim _{i\\rightarrow +\\infty }\\frac{\\vartheta _{i}}{i} = L <+\\infty $ , in particular it exists $n\\in \\mathbb {N}$ such that for all $i\\ge n$ it holds $L - \\beta \\le \\frac{\\vartheta _{i}}{i} \\le L + \\beta $ , where $\\beta = \\frac{2+\\varepsilon }{4+3\\varepsilon }\\cdot \\eta _L$ .", "Now we take $l \\in \\mathbb {Q}_{>0}$ such that $\\vert L - l \\vert < \\beta $ .", "Then it is easy to see that $[L-\\beta ,L + \\beta ]\\subseteq [l-2\\beta , l+ 2\\beta ]$ and $\\eta _l \\ge \\eta _L - \\frac{\\varepsilon }{2+\\varepsilon }\\beta \\ge 2\\beta $ .", "So by definition $z\\in B_{m,l,n}$ , therefore $\\Lambda _\\text{erg}\\subseteq \\bigcup _{m,l,n} B_{m,l,n}.$ Now we need to estimate the upper Minkowski dimension of each set $B_{m,l,n}$ .", "We take $T_0$ big enough such that for every $T\\ge T_0$ it holds $\\frac{1}{T}\\log \\#\\Gamma x \\cap \\overline{B}(x,T) \\le h_\\Gamma + \\varepsilon .$ Let us fix any $\\rho \\le e^{-\\max \\lbrace T_0, n(l-\\eta _l) \\rbrace }$ .", "We consider $j\\in \\mathbb {N}$ with the following property: $(j-1)(l-\\eta _l) < \\log \\frac{1}{\\rho }\\le j(l-\\eta _l)$ .", "We observe that the condition on $\\rho $ gives $\\log \\frac{1}{\\rho } \\ge n(l-\\eta _l)$ , implying $j\\ge n$ .", "We consider the set of elements $g\\in \\Gamma $ such that $j(l-\\eta _l) - m \\le d(x,gx) \\le (j+1)(l+\\eta _l) + m.$ For any such $g$ we consider the shadow $\\text{Shad}_{x}(gx,2m)$ .", "We claim that this set of shadows covers $B_{m,l,n}$ .", "Indeed every point $z$ of $B_{m,l,n}$ belongs to some $\\Lambda _{m,\\Theta }$ with $l-\\eta _l \\le \\frac{\\vartheta _{i}}{i} \\le l + \\eta _l$ for all $i\\ge n$ .", "In particular this holds for $i=j$ , and so $j(l-\\eta _l) \\le \\vartheta _j \\le j(l+\\eta _l)$ , so there exists a point $y$ along a geodesic ray $[x,z]$ satisfying: $j(l-\\eta _l)\\le \\vartheta _j \\le d(x,y)\\le \\vartheta _{j+1}\\le (j+1)(l+\\eta _l), \\qquad d(y,\\Gamma x) \\le m.$ So there is $g\\in \\Gamma $ satisfying (REF ) such that $z\\in \\text{Shad}_{x}(gx,2m)$ .", "Moreover these shadows are casted by points at distance at least $j(l-\\eta _l) - m$ from $x$ , so at distance at least $\\log \\frac{1}{e^m\\rho }$ from $x$ .", "We need to estimate the number of such $g$ 's.", "By the assumption on $\\rho $ we get that this number is less than or equal to $e^{(h_\\Gamma +\\varepsilon )[(j+1)(l+\\eta _l) + m]}.$ Hence, using again Lemma REF , we conclude that $B_{m,l,n}$ is covered by at most $e^{(h_\\Gamma +\\varepsilon )[(j+1)(l+\\eta _l) + m]}$ generalized visual balls of radius $e^{3m}\\rho $ .", "Thus $\\begin{aligned}\\overline{\\text{MD}}(B_{m,l,n})&=\\limsup _{\\rho \\rightarrow 0}\\frac{\\text{Cov}(B_{m,l,n}, e^{3m}\\rho )}{\\log \\frac{1}{e^{3m}\\rho }}\\\\&\\le \\limsup _{j\\rightarrow +\\infty }\\frac{(h_\\Gamma +\\varepsilon )[(j+1)(l+\\eta _l) + m]}{-3m + (j-1)(l-\\eta _l)} \\\\&\\le (h_\\Gamma +\\varepsilon )(1+\\varepsilon ),\\end{aligned}$ where the last inequality follows from the choice of $\\eta _l$ ." ], [ "Dynamics on packed GCB spaces", "In this section we study quotients of proper metric spaces in general.", "In the second part we will apply these results to the special case of packed, GCB-spaces.", "Let $X$ be a proper metric space and $\\Gamma $ be a discrete group of isometries of $X$ .", "We consider the quotient space $\\Gamma \\backslash X$ and the standard projection $\\pi \\colon X \\rightarrow \\Gamma \\backslash X$ .", "On the quotient it is defined a standard pseudometric by $d(\\pi x, \\pi y) = \\inf _{g\\in \\Gamma }d(x, gy)$ .", "Since the action is discrete then this pseudometric is actually a metric.", "Indeed if $d(\\pi x, \\pi y)= 0$ then for every $n > 0$ there exists $g_n\\in \\Gamma $ such that $d(x,g_ny)\\le \\frac{1}{n}$ .", "In particular $d(x,g_nx)\\le d(x,g_ny) + d(g_ny,g_nx) \\le d(x,y) + 1$ for every $n$ .", "So the cardinality of these $g_n$ 's is finite.", "Thus there must be one of these $g_n$ 's such that $d(x, g_ny) = 0$ , i.e.", "$x= g_ny$ , and so $\\pi x = \\pi y$ .", "The map $\\pi $ is 1-Lipschitz and moreover if $\\Gamma $ acts freely (i.e.", "$gx = x$ for some $x\\in X$ , $g\\in \\Gamma $ implies $g=\\text{id}$ ) then it is a local isometry.", "In this case for every $x\\in X$ it is defined the injectivity radius at $x$ by $\\iota (x) = \\sup \\lbrace r > 0 \\text{ s.t. }", "\\pi _{\\vert _{ B(x,r)}}\\colon B(x,r) \\rightarrow \\pi (B(x,r)) \\text{ is an isometry}\\rbrace .$ This defines a map $\\iota \\colon X \\rightarrow (0,+\\infty ]$ with the following properties: - $\\iota (x) > 0$ for every $x\\in X$ ; - $\\iota $ is lower-semicontinuous; - $\\iota $ is $\\Gamma $ -equivariant.", "As a consequence it induces a function $\\iota \\colon \\Gamma \\backslash X \\rightarrow (0,+\\infty ]$ and for every compact subset $K \\subseteq \\Gamma \\backslash X$ there exists the (strictly positive) minimum of $\\iota $ on $K$ , which is called the injectivity radius of $K$ .", "The following lemma is standard.", "Lemma 5.1 Let $X$ be a proper metric space and let $\\Gamma $ be a group of isometries of $X$ acting discretely and freely.", "Then (i) for all $x,y\\in X$ such that $d(\\pi x, \\pi y) < \\iota (x)$ there exists a unique $g\\in \\Gamma $ such that $d(x,gy)<\\iota (x)$ .", "In particular $d(x,gy)=d(\\pi x, \\pi y)$ ; (ii) if $X$ is a length space then $\\pi $ is a locally isometric covering." ], [ "The space of local geodesics", "Let $X$ be a proper metric space and let Loc-Geod$(X)$ be its space of local geodesic lines, thus a subset of the set of maps $\\gamma \\colon \\mathbb {R} \\rightarrow X$ , endowed with the topology of uniform convergence on compact subsets of $\\mathbb {R}$ .", "It is metrizable, indeed we consider, as in [6], the class $\\mathcal {F}$ of continuous functions $f\\colon \\mathbb {R} \\rightarrow \\mathbb {R}$ satisfying (a) $f(s) > 0$ for all $s \\in \\mathbb {R}$ ; (b) $f(s) = f(-s)$ for all $s \\in \\mathbb {R}$ ; (c) $\\int _{-\\infty }^{+\\infty } f(s)ds = 1$ ; (d) $\\int _{-\\infty }^{+\\infty } 2\\vert s \\vert f(s) ds = C(f) < + \\infty $ and we define the quantity $f(\\gamma , \\gamma ^{\\prime }) := \\int _{-\\infty }^{+\\infty }d(\\gamma (s),\\gamma ^{\\prime }(s))f(s)ds.$ Lemma 5.2 Let $X$ be a proper, length metric space and let Loc-Geod$(X)$ be its space of local geodesics.", "Then (i) for all $\\gamma \\in \\textup {Loc-Geod}(X)$ and for all $t<s$ it holds $d(\\gamma (t),\\gamma (s))\\le \\ell (\\gamma \\vert _{[t,s]})= \\vert s -t \\vert $ , where $\\ell (\\cdot )$ denotes the length of a curve; (ii) for all $\\gamma , \\gamma ^{\\prime } \\in \\textup {Loc-Geod}(X)$ and for all $f\\in \\mathcal {F}$ it holds $f(\\gamma , \\gamma ^{\\prime }) \\le d(\\gamma (0),\\gamma ^{\\prime }(0)) + C(f),$ and this expression defines a distance on $\\textup {Loc-Geod}(X)$ which induces its topology.", "Moreover $d(\\gamma (0),\\gamma ^{\\prime }(0)) \\le f(\\gamma , \\gamma ^{\\prime }) + C(f).$ The proof of (i) is easy.", "The first inequality in (ii) can be proved exactly as in Lemma 2.2 of [6], using (i) to show that the integral is finite.", "The second inequality in (ii) is easy: for every $s\\in \\mathbb {R}$ it holds $d(\\gamma (s),\\gamma ^{\\prime }(s)) \\ge d(\\gamma (0),\\gamma ^{\\prime }(0)) - 2\\vert s \\vert $ by (i) and this implies the estimate.", "The fact that the distance $f$ induces the topology of Loc-Geod$(X)$ can be proved in the same way of Lemma 2.2 of [6].", "Every isometry $g$ of $X$ acts on Loc-Geod$(X)$ by $(g\\gamma )(\\cdot ) = g\\gamma (\\cdot )$ for all $\\gamma \\in \\text{Loc-Geod}(X)$ .", "The action is clearly continuous, so $g$ defines a homeomorphism of Loc-Geod$(X)$ .", "Thus if $\\Gamma $ is a group of isometries of $X$ then $\\Gamma $ acts by homeomorphisms on Loc-Geod$(X)$ .", "Lemma 5.3 Let $\\Gamma $ be a group of isometries of a proper metric space $X$ acting discretely and freely.", "Then the action of $\\Gamma $ on Loc-Geod$(X)$ is properly discontinuous.", "We fix $\\gamma \\in \\text{Loc-Geod}(X)$ , we choose $\\varepsilon > 0$ such that if $d(g\\gamma (0),\\gamma (0))< \\varepsilon $ then $g=\\text{id}$ and we consider the set $U = \\left\\lbrace \\gamma ^{\\prime }\\in \\text{Loc-Geod}(X) \\text{ s.t. }", "\\gamma ^{\\prime }(0)\\in B\\left(\\gamma (0),\\frac{\\varepsilon }{2}\\right)\\right\\rbrace .$ The set $U$ is clearly an open neighbourhood of $\\gamma $ .", "Now if $g\\in \\Gamma $ is such that $g\\gamma ^{\\prime }\\in U$ for some $\\gamma ^{\\prime }\\in U$ then both $d(\\gamma ^{\\prime }(0),\\gamma (0))$ and $d(g\\gamma ^{\\prime }(0), \\gamma (0))$ are $<\\frac{\\varepsilon }{2}$ , so $d(\\gamma (0),g\\gamma (0))<\\varepsilon $ and $g=\\text{id}$ .", "We fix a group of isometries $\\Gamma $ of a proper metric space $X$ acting discretely and freely.", "There are two natural objects we can consider: the quotient space $\\Gamma \\backslash \\text{Loc-Geod}(X)$ endowed with the quotient topology and the space of parametrized local geodesic lines of the metric space $\\Gamma \\backslash X$ , namely Loc-Geod$(\\Gamma \\backslash X)$ , endowed as usual with the topology of uniform convergence on compact subsets of $\\mathbb {R}$ .", "Lemma 5.4 Let $X$ be a proper, length metric space and let Loc-Geod$(X)$ be its space of local geodesics.", "Let $\\Gamma $ be a group of isometries of $X$ acting discretely and freely and let $f\\in \\mathcal {F}$ .", "Then (i) the natural action of $\\Gamma $ on $(\\textup {Loc-Geod}(X),f)$ is by isometries and discrete; (ii) the space $\\Gamma \\backslash \\textup {Loc-Geod}(X)$ is metrizable.", "The action of $\\Gamma $ on $(\\textup {Loc-Geod}(X),f)$ is clearly by isometries.", "Moreover for every $\\gamma \\in \\textup {Loc-Geod}(X)$ and every $R\\ge 0$ we have, by Lemma REF , $\\#\\lbrace g \\in \\Gamma \\text{ s.t. }", "f(g\\gamma , \\gamma ) \\le R\\rbrace \\le \\#\\lbrace g \\in \\Gamma \\text{ s.t. }", "d(g\\gamma (0), \\gamma (0)) \\le R + C(f)\\rbrace < +\\infty .$ By the same argument at the beginning of Section we conclude that the quotient pseudometric induced by $f$ is actually a metric, which implies (ii) since the quotient metric induces the quotient topology.", "Proposition 5.5 Let $\\Gamma $ be a group of isometries of a proper, length metric space $X$ acting discretely and freely.", "Then the map $\\Pi \\colon \\textup {Loc-Geod}(X)\\mapsto \\textup {Loc-Geod}(\\Gamma \\backslash X), \\quad \\Pi (\\gamma )(\\cdot ) = \\pi \\gamma (\\cdot )$ is continuous, surjective and $\\Gamma $ -equivariant.", "So it induces a map $\\bar{\\Pi }\\colon \\Gamma \\backslash \\textup {Loc-Geod}(X)\\mapsto \\textup {Loc-Geod}(\\Gamma \\backslash X)$ which is a homeomorphism.", "First of all if $\\gamma = g \\eta $ for some $g\\in \\Gamma $ and $\\gamma ,\\eta \\in \\text{Loc-Geod}(X)$ then clearly $\\pi \\gamma (\\cdot ) = \\pi \\eta (\\cdot )$ , so $\\Pi $ is $\\Gamma $ -equivariant.", "Moreover every map $\\gamma \\colon \\mathbb {R} \\rightarrow \\Gamma \\backslash X$ can be lifted to a map $\\tilde{\\gamma }\\colon \\mathbb {R}\\rightarrow X$ since $\\pi \\colon X\\rightarrow \\Gamma \\backslash X$ is a covering map by Lemma REF .", "Clearly if $\\gamma $ is a local geodesic then $\\tilde{\\gamma }$ is a local geodesic of $X$ , again by Lemma REF .", "This shows that $\\Pi $ is surjective.", "Furthermore since $\\pi $ is 1-Lipschitz then $\\Pi $ is continuous.", "It remains to show that the map $\\bar{\\Pi }$ is a homeomorphism.", "Let us show it is injective: suppose $\\Pi (\\gamma ) = \\Pi (\\eta )$ , i.e.", "$\\pi \\gamma (t)=\\pi \\eta (t)$ for every $t\\in \\mathbb {R}$ .", "Therefore for every $t\\in \\mathbb {R}$ there is a unique $g_t\\in \\Gamma $ such that $g_t\\eta (t) = \\gamma (t)$ , Lemma REF .", "We consider the set $A=\\lbrace t \\in \\mathbb {R} \\text{ s.t. }", "g_t = g_0\\rbrace $ .", "It is closed: indeed if $t_k\\rightarrow t_\\infty $ with $t_k\\in A$ then we have $\\begin{aligned}d(g_{t_\\infty }\\eta (t_\\infty ), g_0\\eta (t_\\infty )) &\\le d(g_{t_\\infty }\\eta (t_\\infty ), g_{t_k}\\eta (t_k)) + d(g_{t_k}\\eta (t_k), g_0\\eta (t_\\infty ))\\\\&\\le d(\\gamma (t_\\infty ),\\gamma (t_k)) + d(\\eta (t_k),\\eta (t_\\infty ))\\end{aligned}$ and the last quantity is as small as we want.", "So $g_{t_\\infty } = g_0$ , i.e.", "$t_\\infty \\in A$ .", "Moreover $A$ is open: let $t\\in A$ and suppose there is a sequence $t_k \\rightarrow t$ such that $g_{t_k}\\ne g_0$ .", "Then $d(\\gamma (t), g_{t_k}\\eta (t)) \\le d(\\gamma (t), \\gamma (t_k)) + d(g_{t_k}\\eta (t_k), g_{t_k}\\eta (t)) \\le 2\\vert t - t_k \\vert .$ When $2\\vert t - t_k \\vert < \\iota (\\eta (t))$ we get $d(g_0^{-1}g_{t_k}\\eta (t), \\eta (t)) < \\iota (\\eta (t))$ and by Lemma REF this implies $g_0^{-1}g_{t_k} = \\text{id}$ , i.e.", "$g_{t_k} = g_0$ for every $t_k$ sufficiently close to $t$ .", "Since $A$ is non empty we conclude that $A=\\mathbb {R}$ , i.e.", "$\\gamma (t) = g_0 \\eta (t)$ for every $t\\in \\mathbb {R}$ .", "So $\\gamma = g_0 \\eta $ , that is $\\bar{\\Pi }$ is injective.", "The continuity of $\\bar{\\Pi }^{-1}$ can be checked on sequences since both spaces are metrizable by Lemma REF and REF .", "Let $p\\colon \\text{Loc-Geod}(X) \\rightarrow \\Gamma \\backslash \\text{Loc-Geod}(X)$ be the standard projection map.", "We take $\\gamma _n, \\gamma _\\infty \\in \\text{Loc-Geod}(\\Gamma \\backslash X)$ such that $\\gamma _k \\rightarrow \\gamma _\\infty $ uniformly on compact subsets of $\\mathbb {R}$ .", "Let $T\\ge 0$ , let $\\varepsilon > 0$ be any real number which is less than the injectivity radius of the compact set $\\gamma _\\infty ([-T,T])$ and let $\\tilde{\\gamma }_k, \\tilde{\\gamma }_\\infty $ be any covering local geodesics of $\\gamma _k, \\gamma _\\infty $ respectively.", "We know that for $k$ big enough it holds $d(\\gamma _k(t),\\gamma _\\infty (t)) < \\varepsilon $ for every $t\\in [-T,T]$ , then there is a unique $g_k(t)\\in \\Gamma $ such that $d(g_k(t)\\tilde{\\gamma }_k(t),\\tilde{\\gamma }_\\infty (t)) < \\varepsilon $ by Lemma REF .", "Arguing as before we conclude that for every such $k$ it holds $g_k(t) = g_k(0) =:g_k$ for every $t\\in [-T,T]$ .", "This implies that $p(\\tilde{\\gamma _k})$ converges uniformly on $[-T,T]$ to $p(\\tilde{\\gamma _\\infty })$ .", "Since this is true for every $T\\ge 0$ we get the continuity of $\\bar{\\Pi }^{-1}$ .", "There is a natural action of $\\mathbb {R}$ on $\\text{Loc-Geod}(X)$ defined by reparametrization: $\\Phi _t\\gamma (\\cdot ) = \\gamma (\\cdot + t)$ for every $t\\in \\mathbb {R}$ .", "It is easy to see it is a continuous action, i.e.", "$\\Phi _t \\circ \\Phi _s = \\Phi _{t+s}$ for all $t,s\\in \\mathbb {R}$ and for every $t\\in \\mathbb {R}$ the map $\\Phi _t$ is a homeomorphism of $\\text{Loc-Geod}(X)$ .", "This action is called the local geodesic flow on $X$ ." ], [ "Packed GCB-spaces", "From now on we will add some other assumptions on our metric space $X$ , in terms of weak upper and lower bounds on the curvature.", "As a lower bound we take a bounded packing condition at a fixed scale.", "Let $Y$ be any subset of a metric space $X$ : – a subset $S$ of $Y$ is called $r$ -dense if $\\forall y \\in Y$ $\\exists z\\in S$ such that $d(y,z)\\le r$ ; – a subset $S$ of $Y$ is called $r$ -separated if $\\forall y,z \\in S$ it holds $d(y,z)> r$ .", "The packing number of $Y$ at scale $r$ is the maximal cardinality of a $2r$ -separated subset of $Y$ and it is denoted by $\\text{Pack}(Y,r)$ .", "The covering number of $Y$ is the minimal cardinality of a $r$ -dense subset of $Y$ and it is denoted by $\\text{Cov}(Y,r)$ .", "The packing and the covering functions of $X$ are respectively $\\text{Pack}(R,r)=\\sup _{x\\in X}\\text{Pack}(\\overline{B}(x,R),r), \\qquad \\text{Cov}(R,r)=\\sup _{x\\in X}\\text{Cov}(\\overline{B}(x,R),r).$ We say that a metric space $X$ is $P_0$ -packed at scale $r_0$ if Pack$(3r_0,r_0)\\le P_0$ , that is every ball of radius $3r_0$ contains no more than $P_0$ points that are $2r_0$ -separated.", "As upper curvature bound we take a very weak notion of non-positive curvature.", "Let $X$ be a metric space.", "A geodesic bicombing is a map $\\sigma \\colon X\\times X\\times [0,1] \\rightarrow X$ with the property that for all $(x,y) \\in X\\times X$ the map $\\sigma _{xy}\\colon t\\mapsto \\sigma (x,y,t)$ is a geodesic from $x$ to $y$ parametrized proportionally to arc-length, i.e.", "$d(\\sigma _{xy}(t), \\sigma _{xy}(t^{\\prime })) = \\vert t - t^{\\prime } \\vert d(x,y)$ for all $t,t^{\\prime }\\in [0,1]$ and $\\sigma _{xy}(0)=x, \\sigma _{xy}(1)=y$ .", "When $X$ is equipped with a geodesic bicombing then for all $x,y\\in X$ we will denote by $[x,y]$ the geodesic $\\sigma _{xy}$ parametrized by arc-length.", "A geodesic bicombing is: convex if the map $t\\mapsto d(\\sigma _{xy}(t), \\sigma _{x^{\\prime }y^{\\prime }}(t))$ is convex on $[0,1]$ for all $x,y,x^{\\prime },y^{\\prime } \\in X$ ; consistent if for all $x,y \\in X$ , for all $0\\le s\\le t \\le 1$ and for all $\\lambda \\in [0,1]$ it holds $\\sigma _{pq}(\\lambda ) = \\sigma _{xy}((1-\\lambda )s + \\lambda t)$ , where $p:= \\sigma _{xy}(s)$ and $q:=\\sigma _{xy}(t)$ ; reversible if $\\sigma _{xy}(t) = \\sigma _{yx}(1-t)$ for all $t\\in [0,1]$ .", "For instance every convex metric space in the sense of Busemann (so also every CAT$(0)$ metric space) admits a unique convex, consistent, reversible geodesic bicombing.", "Given a geodesic bicombing $\\sigma $ we say that a geodesic (segment, ray, line) $\\gamma $ is a $\\sigma $ -geodesic (segment, ray, line) if for all $x,y\\in \\gamma $ we have that $[x,y]$ coincides with the subsegment of $\\gamma $ between $x$ and $y$ .", "A geodesic bicombing is geodesically complete if every $\\sigma $ -geodesic segment is contained in a $\\sigma $ -geodesic line.", "A couple $(X,\\sigma )$ is said a GCB-space if $\\sigma $ is a convex, consistent, reversible, geodesically complete geodesic bicombing on the complete metric space $X$ .", "The packing condition has a controlled behaviour in GCB-spaces.", "Proposition 5.6 (Proposition 3.2 of [8]) Let $(X,\\sigma )$ be a GCB-space that is $P_0$ -packed at scale $r_0$ .", "Then: (i) for all $r\\le r_0$ , the space $X$ is $P_0$ -packed at scale $r$ and it is proper; (ii) for every $0<r\\le R$ and every $x\\in X$ it holds: $\\textup {Pack}(R,r)\\le P_0(1+P_0)^{\\frac{R}{r} - 1} \\text{, if } r\\le r_0;$ $\\textup {Pack}(R,r)\\le P_0(1+P_0)^{\\frac{R}{r_0} - 1} \\text{, if } r > r_0.$ Basic examples of GCB-spaces that are $P_0$ -packed at scale $r_0$ are: i) complete and simply connected Riemannian manifolds with sectional curvature pinched between two nonpositive constants $\\kappa ^{\\prime } \\le \\kappa < 0$ ; ii) simply connected $M^\\kappa $ -complexes, with $\\kappa \\le 0$ , without free faces and bounded geometry (i.e., with valency at most $V_0$ , size at most $S_0$ and positive injectivity radius); iii) complete, geodesically complete, CAT$(0)$ metric spaces $X$ with dimension at most $n$ and volume of balls of radius $R_0$ bounded above by $V$ .", "For further details on the second and the third class of examples we refer to [9].", "Let $(X,\\sigma )$ be a proper GCB-space.", "The space of parametrized $\\sigma $ -geodesic lines of $X$ is $\\text{Geod}_\\sigma (X) = \\lbrace \\gamma \\colon \\mathbb {R} \\rightarrow X \\text{ isometry s.t. }", "\\gamma (\\mathbb {R}) \\text{ is a }\\sigma \\text{-geodesic line}\\rbrace ,$ considered as a subset of Loc-Geod$(X)$ .", "By the continuity of $\\sigma $ (see [10], [6]) we have that $\\text{Geod}_\\sigma (X)$ is closed in $\\text{Loc-Geod}(X)$ .", "Moreover the local geodesic flow on Loc-Geod$(X)$ restricts as an action on $\\text{Geod}_\\sigma (X)$ , called the $\\sigma $ -geodesic flow on $X$ .", "The evaluation map $E\\colon \\text{Geod}_\\sigma (X) \\rightarrow X$ , which is defined as $E(\\gamma )=\\gamma (0)$ , is continuous and proper ([5], Lemma 1.10 and [6]).", "Moreover this restriction is surjective since $\\sigma $ is assumed geodesically complete.", "The topology on $\\text{Geod}_\\sigma (X)$ is metrizable by Lemma REF .", "Let $(X,\\sigma )$ be a GCB-space.", "An isometry $g$ of $X$ is a $\\sigma $ -isometry if for all $x,y\\in X$ it holds $\\sigma _{g(x)g(y)} = g(\\sigma _{xy})$ .", "We say that a group of isometries of $X$ is a group of $\\sigma $ -isometries if every element of the group is a $\\sigma $ -isometry.", "If $\\Gamma $ is a group of $\\sigma $ -isometries of $X$ acting discretely and freely then Geod$_\\sigma (X)$ is $\\Gamma $ -invariant and we set Loc-Geod$_\\sigma (\\Gamma \\backslash X):=\\Pi (\\text{Geod}_\\sigma (X))$ , which is homeomorphic to $\\Gamma \\backslash \\text{Geod}_\\sigma (X)$ by Proposition REF .", "Since in Busemann convex (so also CAT$(0)$ ) metric spaces $X$ every local geodesic is a geodesic then it holds Loc-Geod$(X) =$ Geod$_\\sigma (X)$ , where $\\sigma $ is the unique bicombing, so Loc-Geod$_\\sigma (\\Gamma \\backslash X) =$ Loc-Geod$(\\Gamma \\backslash X)$ .", "We end this section with two easy results and a remark.", "Lemma 5.7 Let $(X,\\sigma )$ be a proper GCB-space.", "Then the natural map $\\Pi \\colon (\\textup {Geod}_\\sigma (X),f) \\rightarrow (\\textup {Loc-Geod}_\\sigma (X),f)$ is 1-Lipschitz for all $f\\in \\mathcal {F}$ .", "For all $\\gamma , \\gamma ^{\\prime }\\in \\text{Geod}_\\sigma (X)$ we have $\\begin{aligned}f(\\gamma , \\gamma ^{\\prime }) &= \\int _{-\\infty }^{+\\infty }d(\\gamma (s),\\gamma ^{\\prime }(s))f(s)ds\\\\&\\ge \\int _{-\\infty }^{+\\infty }d(\\pi \\gamma (s),\\pi \\gamma ^{\\prime }(s))f(s)ds = f(\\Pi (\\gamma ), \\Pi (\\gamma ^{\\prime })).\\end{aligned}$ Lemma 5.8 Let $(X,\\sigma )$ be a GCB-space and let $\\Gamma $ be a group of $\\sigma $ -isometries of $X$ acting discretely and freely.", "Then the metric on $\\textup {Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ induced by every $f\\in \\mathcal {F}$ is complete.", "We observe the result is not completely trivial because the metric $f$ on $\\textup {Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ does not coincide with the quotient metric.", "However given a Cauchy sequence $\\lbrace \\gamma _k\\rbrace $ we know that for every $t\\in \\mathbb {R}$ the sequence $\\lbrace \\gamma _k(t)\\rbrace $ is bounded, so we can apply Ascoli-Arzela's Theorem to find a limit map $\\gamma _\\infty \\colon \\mathbb {R} \\rightarrow \\Gamma \\backslash X$ .", "We need to show $\\gamma _\\infty $ is a local geodesic.", "Let $t\\in \\mathbb {R}$ and let $2\\iota >0$ be smaller than the injectivity radius at $\\gamma _\\infty (t)$ .", "Then by Lemma REF for every $k$ big enough there is a subsegment of length $2\\iota $ of $\\gamma _k$ which converges to $\\gamma _\\infty ([t-\\iota ,t+\\iota ])$ and that is a true geodesic.", "Therefore $\\gamma _\\infty ([t-\\iota ,t+\\iota ])$ is a true geodesic, that is $\\gamma _\\infty $ is a local geodesic.", "Let $\\tilde{\\gamma }_k, \\tilde{\\gamma }_\\infty $ be any covering local geodesic of $\\gamma _k, \\gamma _\\infty $ respectively.", "Arguing as in the proof of Proposition REF we know that for every $T\\in \\mathbb {R}$ there exists $g_T\\in \\Gamma $ such that $g_T\\tilde{\\gamma }_k$ converges uniformly to $\\tilde{\\gamma }_\\infty $ on $[-T,T]$ .", "It is then clear by Lemma REF that if $T^{\\prime }\\ge T$ then $g_{T^{\\prime }} = g_{T}$ .", "In particular we can find $g\\in \\Gamma $ such that $g\\tilde{\\gamma }_k$ converges uniformly on every compact subset of $\\mathbb {R}$ to $\\tilde{\\gamma }_\\infty $ .", "Observe that each $g\\tilde{\\gamma }_k$ is a $\\sigma $ -geodesic line, so it is $\\tilde{\\gamma }_\\infty $ , i.e.", "$\\gamma _\\infty \\in \\textup {Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ .", "Remark 5.9 We observe that the proof shows also that Loc-Geod$_\\sigma (\\Gamma \\backslash X)$ is closed in Loc-Geod$(\\Gamma \\backslash X)$ .", "In our situation the space $\\Gamma \\backslash X$ is naturally a locally GCB-space with local convex bicombing $\\sigma $ as defined in [19].", "A $\\sigma $ -local geodesic can be defined as a local geodesic $\\gamma $ such that for all close enough $t<t^{\\prime }$ it holds $\\gamma \\vert _{[t,t^{\\prime }]} = \\sigma _{\\gamma (t),\\gamma (t^{\\prime })}$ .", "Of course this equality has a meaning only locally, where $\\sigma $ -geodesics are defined.", "Clearly every $\\sigma $ -local geodesic can be lifted to a $\\sigma $ -local geodesic of $X$ and by convexity it is easy to check that every $\\sigma $ -local geodesic of $X$ is actually a $\\sigma $ -geodesic.", "So $\\textup {Loc-Geod}_\\sigma (X)$ can be seen exactly as the space of $\\sigma $ -local geodesics of $\\Gamma \\backslash X$ .", "This picture can be generalized: if $X$ is a complete, connected, locally GCB-space then by Theorem 1.1 of [19] it admits a universal cover $\\tilde{X}$ that can be endowed with a unique structure of GCB-space $(\\tilde{X}, \\sigma )$ and $X=\\Gamma \\backslash \\tilde{X}$ , where $\\Gamma $ is a group of $\\sigma $ -isometries of $\\tilde{X}$ acting discretely and freely.", "In this setting the natural space of $\\sigma $ -local geodesics of $X$ coincides with the space $\\textup {Loc-Geod}_\\sigma (\\Gamma \\backslash \\tilde{X})$ defined above." ], [ "Upper bound of the entropy", "We consider the dynamical system $(\\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X), \\Phi _1)$ , where $(X,\\sigma )$ is a $\\delta $ -hyperbolic GCB-space that is $P_0$ -packed at scale $r_0$ , $\\Gamma $ is a group of $\\sigma $ -isometries of $X$ acting discretely and freely and $\\Phi _1$ is the classical time one reparametrization flow, i.e.", "$\\Phi _1(\\gamma )(\\cdot ) = \\gamma (\\cdot + 1)$ for all $\\gamma \\in \\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ .", "In this section we prove that the topological entropy of the dynamical system above is less than or equal to the critical exponent of $\\Gamma $ .", "Before we recall the definition of the topological entropy and the local entropies of a dynamical system.", "For us a dynamical system is a couple $(Y,T)$ where $Y$ is a topological space and $T\\colon Y \\rightarrow Y$ is a continuous map.", "We will always denote by ${B}$ the $\\sigma $ -algebra of Borelian subsets of $Y$ .", "We denote by $\\mathcal {M}_1(Y,T)$ the set of $T$ -invariant probability measure on $(Y,{B})$ .", "We recall that a measure $\\mu \\in \\mathcal {M}_1(Y,T)$ is ergodic if every subset $A$ of $Y$ such that $T^{-1}A \\subseteq A$ has $\\mu $ -measure 0 or 1.", "We denote the set of ergodic measures by $\\mathcal {E}_1(Y,T)$ .", "For the rest of the section we will always assume that $Y$ is a locally compact, metrizable, topological space.", "The topological entropy of the dynamical system $(Y,T)$ is by definition: $h_\\text{top}(Y,T) := \\sup _{\\mu \\in \\mathcal {M}_1(Y,T)}h_\\mu (Y,T) = \\sup _{\\mu \\in \\mathcal {E}_1(Y,T)}h_\\mu (Y,T),$ where $h_\\mu (Y,T)$ denotes the Kolmogorov-Sinai entropy of the measure $\\mu $ .", "For our purposes we will not need to recall its definition.", "The second equality is classical and follows from the convexity of the entropy functional.", "We introduce two other possible definitions of entropy of a measure $\\mu \\in \\mathcal {E}_1(Y,T)$ .", "We restrict to ergodic measures as we have seen that it is enough to consider them in order to compute the topological entropy of $(Y,T)$ .", "Let $\\mu \\in \\mathcal {E}_1(Y,T)$ and ɗ  be a metric on $Y$ inducing its topology.", "The upper local entropy of $\\mu $ with respect to ɗ  is defined as $\\overline{h}_{\\mu ,\\text{ɗ}}^\\text{loc}(Y,T) = \\underset{\\mu }{\\text{ess}\\sup }\\lim _{r\\rightarrow 0}\\limsup _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\mu (B_{\\text{ɗ}^n}(y,r)),$ while the lower local entropy is $\\underline{h}_{\\mu ,\\text{ɗ}}^\\text{loc}(Y,T) = \\underset{\\mu }{\\text{ess}\\inf }\\lim _{r\\rightarrow 0}\\liminf _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\mu (B_{\\text{ɗ}^n}(y,r)).$ Here $\\text{ɗ}^n$ is the dynamical distance defined as usual as: $\\text{ɗ}^n(y,y^{\\prime }) = \\max _{i=0,\\ldots ,n-1}\\text{ɗ}(T^iy,T^iy^{\\prime }), \\qquad \\forall y,y^{\\prime }\\in Y.$ Observe that this definition of the lower local entropy agrees with the one given in [15] while it is a bit different from the one of [24].", "However it is certainly bigger than or equal to this last one.", "So we conclude: Lemma 6.1 ([24], Theorem 1.28) Let $\\mu \\in \\mathcal {E}_1(Y,T)$ and let ɗ  be a complete metric on $Y$ inducing its topology.", "Then $h_\\mu (Y,T) \\le \\underline{h}_{\\mu ,\\textup {ɗ}}^\\textup {loc}(Y,T).$ It is natural to define the upper and lower local entropy of the dynamical system $(Y,T)$ , respectively, as $\\overline{h}^\\text{loc}(Y,T) = \\sup _{\\mu \\in \\mathcal {E}_1(Y,T)}\\inf _{\\text{ɗ}}\\overline{h}_{\\mu ,\\text{ɗ}}^\\text{loc}(Y,T), \\qquad \\underline{h}^\\text{loc}(Y,T) = \\sup _{\\mu \\in \\mathcal {E}_1(Y,T)}\\inf _{\\text{ɗ}}\\underline{h}_{\\mu ,\\text{ɗ}}^\\text{loc}(Y,T),$ where the infimum are among all complete metrics on $Y$ inducing its topology.", "Clearly by Lemma REF we have $h_\\text{top}(Y,T) \\le \\underline{h}^\\text{loc}(Y,T) \\le \\overline{h}^\\text{loc}(Y,T).$ As said at the beginning of the section we consider the dynamical system $(\\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X), \\Phi _1)$ .", "To simplify the notations we denote by $\\mathcal {M}_1$ , $\\mathcal {E}_1$ the sets $\\mathcal {M}_1(\\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X), \\Phi _1)$ and $\\mathcal {E}_1(\\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X), \\Phi _1)$ .", "In the same way the topological entropy, the lower and the upper local entropy of $(\\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X), \\Phi _1)$ will be simply denoted by, respectively, $h_\\text{top}$ , $\\underline{h}^\\text{loc}$ and $\\overline{h}^\\text{loc}$ .", "In order to prove $h_\\text{top}\\le h_\\Gamma $ the strategy is to show $\\overline{h}^\\textup {loc} \\le h_\\Gamma $ .", "This would be enough by (REF ).", "Before proceeding in this direction we need to simplify the expression of the upper local entropy.", "By Lemma REF every $f\\in \\mathcal {F}$ defines a complete metric on Loc-Geod$_\\sigma (\\Gamma \\backslash X)$ which induces its topology, so by definition it is clear that $\\overline{h}^\\text{loc} \\le \\sup _{\\mu \\in \\mathcal {E}_1} \\overline{h}^\\text{loc}_{\\mu ,f}, \\qquad \\underline{h}^\\text{loc} \\le \\sup _{\\mu \\in \\mathcal {E}_1} \\underline{h}^\\text{loc}_{\\mu ,f}.$ Our aim is to simplify the computation of $\\overline{h}^\\text{loc}_{\\mu ,f}$ and $\\underline{h}^\\text{loc}_{\\mu ,f}$ .", "In order to do that we need the key lemma of [6].", "Lemma 6.2 (Key lemma, [6]) Let $(X,\\sigma )$ be a GCB-space that is $P_0$ -packed at scale $r_0$ .", "Let $f\\in \\mathcal {F}$ , $\\gamma \\in \\textup {Geod}_\\sigma (X)$ and $0<r\\le R$ .", "Then $\\lim _{T\\rightarrow +\\infty }\\frac{1}{T}\\log \\textup {Cov}_{f^T}(B_{f^T}(\\gamma , R), r) = 0.$ We are now ready to prove: Lemma 6.3 Let $\\mu \\in \\mathcal {E}_1$ anf $f\\in \\mathcal {F}$ .", "Then for every $R>0$ and for $\\mu $ -a.e.$\\gamma \\in \\textup {Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ it holds: $\\underline{h}_{\\mu ,f}^\\textup {loc} = \\liminf _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\mu (B_{f^n}(\\gamma ,R)).$ The same corresponding conclusion is true for the upper local entropy.", "We fix $R>0$ and we take $0<r\\le R$ .", "Clearly $B_{f^n}(\\gamma , R) \\supseteq B_{f^n}(\\gamma , r)$ for all $\\gamma \\in \\textup {Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ , so $\\liminf _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\mu (B_{f^n}(\\gamma ,R)) \\le \\lim _{r\\rightarrow 0}\\liminf _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\mu (B_{f^n}(\\gamma ,r))$ for all $\\gamma \\in \\textup {Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ .", "In the other direction we notice that for all $\\gamma \\in \\textup {Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ it holds $\\mu (B_{f^n}(\\gamma , R)) \\le \\text{Cov}_{f^n}(B_{f^n}(\\gamma , R), r)\\cdot \\underset{\\mu }{\\textup {ess}\\sup } \\mu (B_{f^n}(\\gamma ^{\\prime }, r)).$ So we deduce $\\begin{aligned}\\liminf _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\mu (B_{f^n}(\\gamma ,R)) &\\ge \\liminf _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\text{Cov}_{f^n}(B_{f^n}(\\gamma , R), r)\\cdot \\underset{\\mu }{\\textup {ess}\\sup } \\mu (B_{f^n}(\\gamma ^{\\prime }, r)) \\\\&= \\liminf _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\text{Cov}_{f^n}(B_{f^n}(\\gamma , R), r) \\\\&\\quad + \\liminf _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\underset{\\mu }{\\textup {ess}\\sup } \\mu (B_{f^n}(\\gamma ^{\\prime }, r)) \\\\&= \\liminf _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\underset{\\mu }{\\textup {ess}\\sup } \\mu (B_{f^n}(\\gamma ^{\\prime }, r)) \\\\&\\ge \\underset{\\mu }{\\textup {ess}\\sup } \\liminf _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\mu (B_{f^n}(\\gamma ^{\\prime }, r)),\\end{aligned}$ where the second inequality follows directly by Lemma REF .", "By the arbitrariness of $r$ we conclude: $\\begin{aligned}\\liminf _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\mu (B_{f^n}(\\gamma ,R)) &\\ge \\lim _{r\\rightarrow 0} \\underset{\\mu }{\\textup {ess}\\sup } \\liminf _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\mu (B_{f^n}(\\gamma ^{\\prime }, r)) \\\\&\\ge \\underset{\\mu }{\\textup {ess}\\sup }\\lim _{r\\rightarrow 0}\\liminf _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\mu (B_{f^n}(\\gamma ^{\\prime }, r)).\\end{aligned}$ Since this is true for all $\\gamma \\in \\textup {Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ we obtain: $\\begin{aligned}\\underset{\\mu }{\\textup {ess}\\inf }\\lim _{r\\rightarrow 0}\\liminf _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\mu (B_{f^n}(\\gamma ,r)) &\\ge \\underset{\\mu }{\\textup {ess}\\inf }\\liminf _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\mu (B_{f^n}(\\gamma ,R)) \\\\&\\ge \\underset{\\mu }{\\textup {ess}\\sup }\\lim _{r\\rightarrow 0}\\liminf _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\mu (B_{f^n}(\\gamma ^{\\prime }, r)).\\end{aligned}$ This implies that for $\\mu $ -a.e.$\\gamma $ the function $\\gamma \\mapsto \\lim _{r\\rightarrow 0}\\liminf _{n\\rightarrow +\\infty }-\\frac{1}{n}\\log \\mu (B_{f^n}(\\gamma ^{\\prime }, r))$ is constantly equal to $\\underline{h}_{\\mu ,f}^\\text{loc}$ , by definition of lower local entropy.", "So for $\\mu $ -a.e.$\\gamma $ we conclude $\\underline{h}_{\\mu ,f}^\\text{loc} = \\liminf _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\mu (B_{f^n}(\\gamma ,R)).$ The proof for the upper entropy is the same.", "We are ready to prove the first part of Theorem REF , namely Theorem 6.4 Let $(X,\\sigma )$ be a $\\delta $ -hyperbolic GCB-space that is $P_0$ -packed at scale $r_0$ .", "Let $\\Gamma $ be a group of $\\sigma $ -isometries of $X$ acting discretely and freely.", "Then $\\overline{h}^\\textup {loc} \\le h_\\Gamma .$ The proof is inspired by [15].", "Essentially to every measure $\\mu \\in \\mathcal {E}_1$ we associate a measure $\\nu $ on $\\partial X$ with the following properties: (i) $\\overline{h}^\\textup {loc}_{\\mu ,f} \\le \\overline{\\text{PD}}(\\nu )$ , where $f\\in \\mathcal {F}$ is arbitrary; (ii) $\\nu (\\Lambda _{\\text{erg}}) = 1$ .", "Then we will use (REF ) and Theorem REF to conclude the estimate.", "It is enough to show that for all $\\mu \\in \\mathcal {E}_1$ it holds $\\overline{h}^\\textup {loc}_{\\mu ,f} \\le h_\\Gamma $ for some $f\\in \\mathcal {F}$ .", "So we fix $f\\in \\mathcal {F}$ and $\\mu \\in \\mathcal {E}_1$ .", "For every $x\\in X$ we consider the ball $B(x,\\iota (x))$ , where $\\iota (x)$ is the injectivity radius at $x$ .", "Since $X$ is proper, so separable, we can find a countable set $\\lbrace x_i \\rbrace _{i\\in \\mathbb {N}} \\subseteq X$ such that $X=\\bigcup _{i\\in \\mathbb {N}} B(x_i,\\iota (x_i))$ .", "We consider the sets $U_i = \\lbrace \\gamma \\in \\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X) \\text{ s.t. }", "\\gamma (0) \\in B(\\pi x_i, \\iota (x_i))\\rbrace .$ Clearly $\\bigcup _{i\\in \\mathbb {N}}U_i = \\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ , so there must be some index $i_0$ such that $\\mu (U_{i_0}) = c > 0$ .", "By Lemma REF for all $\\gamma \\in U_{i_0}$ there exists a unique $\\tilde{\\gamma }\\in \\text{Geod}_\\sigma (X)$ such that $\\Pi \\tilde{\\gamma } = \\gamma $ and $d(\\tilde{\\gamma }(0), x_{i_0}) < \\iota (x_{i_0})$ .", "Therefore we can define the map $+ \\colon U_{i_0} \\rightarrow \\partial X$ by $+\\gamma = \\tilde{\\gamma }^+$ , where $\\tilde{\\gamma }$ is the unique covering geodesic of $\\gamma $ with $d(\\tilde{\\gamma }(0), x_{i_0}) < \\iota (x_{i_0})$ .", "Let us show it is continuous: let $\\gamma _k \\in U_{i_0}$ converging uniformly on compact subsets to $\\gamma _\\infty \\in U_{i_0}$ .", "We showed in Lemma REF that there exist covering $\\sigma $ -geodesics of $\\gamma _k$ that converge uniformly on compact subsets to $\\tilde{\\gamma }_\\infty $ .", "It is then clear that these covering $\\sigma $ -geodesics must be the $\\sigma $ -geodesics $\\tilde{\\gamma }_k$ .", "So $\\tilde{\\gamma }_k^+$ converges to $\\tilde{\\gamma }_\\infty ^+$ .", "Hence we can define the measure $\\nu $ on $(\\partial X, {B})$ by $\\nu = +_\\ast \\mu $ , i.e.", "$\\nu (A) = \\mu (+^{-1}A)$ for every Borelian subset $A$ of $\\partial X$ .", "Step 1: $\\overline{h}^\\textup {loc}_{\\mu ,f} \\le \\overline{\\text{PD}}(\\nu )$ .", "We fix $R= 28\\delta + 3\\iota (x_{i_0}) + 2C(f)$ .", "By Lemma REF we know that $\\overline{h}_{\\mu ,f}^\\textup {loc} = \\limsup _{n\\rightarrow +\\infty } -\\frac{1}{n}\\log \\mu (B_{f^n}(\\gamma ,R))$ for $\\mu $ -a.e.$\\gamma \\in U_{i_0}$ .", "Clearly the measure $\\nu $ is concentrated on the image of the map $+$ , so in order to compute its upper packing dimension we can just consider the points of this set.", "So for all $\\gamma \\in U_{i_0}$ we consider the point $z = \\tilde{\\gamma }^+ \\in \\partial X$ and the generalized visual ball $B(z,e^{-n})$ , where we choose $x_{i_0}$ as basepoint of $X$ .", "We want to estimate $\\nu (B(z,e^{-n})) = \\mu (+^{-1} B(z,e^{-n}))$ .", "We observe that every $\\eta \\in +^{-1} B(z,e^{-n})$ satisfies: $d(\\tilde{\\eta }(0), \\tilde{\\gamma }(0)) \\le 2\\iota (x_{i_0}), \\qquad d(\\tilde{\\eta }(n), \\tilde{\\gamma }(n)) \\le 14\\delta + \\iota (x_{i_0}).$ Indeed the first inequality is by definition of $\\tilde{\\gamma }$ and $\\tilde{\\eta }$ , while the second one follows by Lemma REF and Lemma REF .", "By standard arguments, see [6], this implies $f^n(\\tilde{\\eta }, \\tilde{\\gamma })\\le 28\\delta + 3\\iota (x_{i_0}) + 2C(f) = R$ and so, by Lemma REF , $f^n(\\eta ,\\gamma )\\le R$ .", "In other words $+^{-1} B(+\\gamma ,e^{-n}) \\subseteq B_{f^n}(\\gamma , R)$ for all $\\gamma \\in U_{i_0}$ .", "We conclude that $\\begin{aligned}\\overline{\\text{PD}}(\\nu ) &= \\underset{\\nu }{\\text{ess}\\sup }\\limsup _{\\rho \\rightarrow 0}\\frac{\\log \\nu (B(z,\\rho ))}{\\log \\rho } \\\\&\\ge \\underset{\\nu }{\\text{ess}\\sup }\\limsup _{n\\rightarrow +\\infty }-\\frac{1}{n}\\log \\nu (B(z,e^{-n})) \\\\&\\ge \\underset{\\mu }{\\text{ess}\\sup }\\limsup _{n\\rightarrow +\\infty }-\\frac{1}{n}\\log \\mu (B_{f^n}(\\gamma ,R)) = \\overline{h}_{\\mu ,f}^\\textup {loc}.\\end{aligned}$ The first inequality is clear choosing $\\rho =e^{-n}$ and using the property that the limit superior along a selected subsequence is smaller than or equal to the limit superior along any subsequence.", "The second follows from what we said before: indeed $\\nu $ gives full measure to the set of points $z \\in \\partial X$ that are in the image of the map $+$ .", "This, together with the fact that $\\nu (B(+\\gamma ,e^{-n})) = \\mu (+^{-1} B(+\\gamma ,e^{-n})) \\le \\mu (B_{f^n}(\\gamma , R))$ , implies the second inequality.", "The last equality is direct consequence of (REF ) and the fact that $U_{i_0}$ has positive $\\mu $ -measure.", "Step 2: $\\nu $ gives full measure to the set $\\Lambda _{\\text{erg}}$ .", "In this step we will use for the first (and last) time the ergodicity of the measure $\\mu $ .", "We associate to $\\gamma \\in U_{i_0}$ the set of integers $\\Theta (\\gamma ) = \\lbrace \\vartheta _i(\\gamma )\\rbrace $ defined recursively by $\\vartheta _0(\\gamma )=0, \\qquad \\vartheta _{i+1}(\\gamma ) = \\inf \\lbrace n > \\vartheta _i(\\gamma ) \\text{ s.t. }", "\\Phi _n(\\gamma ) \\in U_{i_0}\\rbrace .$ We notice that an integer $n$ satisfies $d(\\Phi _{n}\\gamma (0), \\pi x_{i_0}) = d(\\gamma (n), \\pi x_{i_0}) < \\iota ({x_{i_0}})$ if and only if $n \\in \\Theta (\\gamma )$ , i.e.", "$\\Theta (\\gamma )$ is exactly the set of (integer) returning times of $\\gamma $ to the ball $B(\\pi x_{i_0}, \\iota (x_{i_0}))$ .", "We apply Birkhoff's Ergodic Theorem to the indicator function of the set $U_{i_0}$ , namely $\\chi _{U_{i_0}}$ , obtaining that for $\\mu $ -a.e.$\\gamma \\in \\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ it holds $\\exists \\lim _{N\\rightarrow +\\infty }\\frac{1}{N}\\sum _{j=0}^{N-1} \\chi _{U_{i_0}} \\circ \\Phi _j(\\gamma ) = \\mu (U_{i_0}) = c > 0.$ We remark that $\\chi _{U_{i_0}} \\circ \\Phi _j(\\gamma ) = 1$ if and only if $j\\in \\Theta (\\gamma )$ and it is 0 otherwise.", "So $\\lim _{N\\rightarrow +\\infty }\\frac{1}{N}\\sum _{j=0}^{N-1} \\chi _{U_{i_0}} \\circ \\Phi _j(\\gamma ) = \\lim _{N\\rightarrow +\\infty }\\frac{\\#\\Theta (\\gamma ) \\cap [0,N-1]}{N},$ and the right hand side is by definition the density of the set $\\Theta (\\gamma )$ .", "It is classical that, given the standard increasing enumeration $\\lbrace \\vartheta _0(\\gamma ), \\vartheta _1(\\gamma ),\\ldots \\rbrace $ of $\\Theta (\\gamma )$ , it holds $\\lim _{N\\rightarrow +\\infty }\\frac{\\#\\Theta (\\gamma ) \\cap [0,N-1]}{N} = \\lim _{N\\rightarrow +\\infty }\\frac{N}{\\vartheta _N(\\gamma )}.$ Putting all together we conclude that for $\\mu $ -a.e.$\\gamma \\in \\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ the following is true $\\exists \\lim _{N\\rightarrow +\\infty }\\frac{\\vartheta _N(\\gamma )}{N} = \\frac{1}{c} < +\\infty .$ Now we claim that for every $\\gamma \\in U_{i_0}$ satisfying (REF ) it holds $+\\gamma \\in \\Lambda _\\text{erg}$ .", "For every such $\\gamma $ we take its covering geodesic $\\tilde{\\gamma }$ whose limit point $\\tilde{\\gamma }^+$ defines $+\\gamma = z$ .", "We recall that our basepoint of $X$ is $x_{i_0}$ .", "We fix any geodesic ray $\\xi = [x_{i_0}, z]$ .", "First of all we know that for all $t\\ge 0$ it holds $d(\\xi (t), \\tilde{\\gamma }(t)) \\le 8\\delta + \\iota (x_{i_0})$ by Lemma REF .", "Moreover we know that for every $N\\in \\mathbb {N}$ the geodesic $\\gamma $ returns to $B(\\pi x_{i_0}, \\iota (x_{i_0}))$ at time $\\vartheta _N(\\gamma )$ .", "This implies that $d(\\tilde{\\gamma }(\\vartheta _N(\\gamma )), \\Gamma x_{i_0}) < \\iota ({x_{i_0}})$ , and so $d(\\xi (\\vartheta _N(\\gamma )), \\Gamma x_{i_0}) < 8\\delta + 2\\iota (x_{i_0})$ .", "By definition this means that $z\\in \\Lambda _{\\tau , \\Theta (\\gamma )}$ , where $\\tau = 8\\delta + 2\\iota (x_{i_0})$ .", "Finally we observe that the sequence $\\Theta (\\gamma )=\\lbrace \\vartheta _N(\\gamma )\\rbrace $ satisfies (REF ), that is exactly the condition that defines a sequence involved in the definition of $\\Lambda _\\text{erg}$ .", "This shows the claim and so that $\\nu (\\Lambda _\\text{erg}) = 1$ .", "Step 3.", "By Step 1, Step 2, (REF ) and Theorem REF we conclude that $h_\\Gamma = \\text{PD}(\\Lambda _\\text{erg}) \\ge \\overline{\\text{PD}}(\\nu ) \\ge \\overline{h}_{\\mu ,f}^\\textup {loc} \\ge \\overline{h}_{\\mu }^\\textup {loc}.$ Finally, from the arbitrariness of $\\mu $ we get the thesis." ], [ "Lower bound of the entropy", "We consider again the dynamical system $(\\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X), \\Phi _1)$ , where $(X,\\sigma )$ is a $\\delta $ -hyperbolic GCB-space that is $P_0$ -packed at scale $r_0$ and $\\Gamma $ is a group of $\\sigma $ -isometries of $X$ acting discretely and freely.", "By Remark REF and Lemma REF the space $\\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ is locally compact and metrizable, so the topological entropy $h_\\text{top}$ can be computed via the variational principle, see [14], as $h_\\textup {top}=\\inf _{\\text{ɗ}}\\sup _{K}\\lim _{r\\rightarrow 0}\\limsup _{n \\rightarrow +\\infty }\\frac{1}{n} \\log \\textup {Cov}_{\\text{ɗ}^n}(K,r),$ where the infimum is among all metrics ɗ  inducing the natural topology of $\\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ and the supremum is among all compact subsets of $\\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ .", "Observe that in the right hand side of (REF ) the compact subsets $K$ are not required to be $T$ -invariant, where by $T$ -invariant we mean $T(K) = K$ .", "For a general dynamical system the restriction to $T$ -invariant compact subsets gives a strictly smaller quantity, see the remark below Lemma 1.6 of [14].", "We remark that when $K$ is a compact $T$ -invariant subset then the dynamical system $(K,T)$ is supported on a compact metrizable space, so the quantity $\\lim _{r\\rightarrow 0}\\limsup _{n \\rightarrow +\\infty }\\frac{1}{n} \\log \\textup {Cov}_{\\text{ɗ}^n}(K,r)$ does not depend on the choice of the metric ɗ  and it coincides with the topological entropy of the dynamical system $(K,T)$ by the variational principle for compact dynamical systems.", "We define $h_\\textup {inv-top}=\\sup _{K\\,\\,\\, T\\text{-inv.", "}}\\lim _{r\\rightarrow 0}\\limsup _{n \\rightarrow +\\infty }\\frac{1}{n} \\log \\textup {Cov}_{\\text{ɗ}^n}(K,r) = \\sup _{K\\,\\,\\, T\\text{-inv.}}", "h_\\text{top}(K,T),$ called the invariant topological entropy of $(\\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X),\\Phi _1)$ .", "Here the supremum is among all compact $T$ -invariant subsets $K$ and ɗ  is any metric on the compact subset $K$ inducing its topology.", "Clearly $h_\\textup {inv-top} \\le h_\\textup {top}$ .", "The strategy to conclude the proof of Theorem REF is to show $h_\\textup {inv-top} \\ge h_\\Gamma $ .", "First of all we need to identify the invariant compact subsets of $\\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ .", "We fix a basepoint $x\\in X$ and we call $\\pi x$ the projection of $x$ in $\\Gamma \\backslash X$ , as usual.", "Then the following are the prototype of compact $\\Phi _1$ -invariant subsets: $K_\\tau = \\lbrace \\gamma \\in \\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X) \\text{ s.t. }", "\\gamma (n)\\in \\overline{B}(\\pi x, \\tau ) \\text{ for all } n\\in \\mathbb {Z} \\rbrace ,$ where $\\tau \\ge 0$ is a real number.", "The set $K_\\tau $ is essentially the set of local geodesic lines whose image is completely contained in a fixed compact set of the quotient $\\Gamma \\backslash X$ .", "Lemma 7.1 $K_\\tau $ is a compact $\\Phi _1$ -invariant subset for all $\\tau \\ge 0$ .", "Moreover any compact $\\Phi _1$ -invariant subset is contained in $K_\\tau $ for some $\\tau \\ge 0$ .", "For all $\\tau \\ge 0$ the set $K_\\tau $ is clearly $\\Phi _1$ -invariant and closed by Remark REF .", "Let now $\\gamma _k\\colon \\mathbb {R} \\rightarrow \\overline{B}(\\pi x, \\tau )$ be a sequence of elements of $K_\\tau $ .", "By Ascoli-Arzela's Theorem there exists a map $\\gamma _\\infty \\colon \\mathbb {R} \\rightarrow \\overline{B}(\\pi x, \\tau )$ such that $\\gamma _k \\rightarrow \\gamma _\\infty $ uniformly on compact subsets of $\\mathbb {R}$ and since $K_\\tau $ is closed clearly $\\gamma _\\infty \\in K_\\tau $ .", "This completes the proof of the first statement.", "Now let $K$ be a compact $\\Phi _1$ -invariant subset of $\\text{Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ .", "We need to show it exists $\\tau \\ge 0$ such that $K$ is contained in $K_\\tau $ .", "Suppose it is not true.", "Then for every $k \\ge 0$ there exists $\\gamma _k \\in K$ such that $\\gamma _k(n_k) \\notin \\overline{B}(\\pi x,k)$ for some $n_k\\in \\mathbb {Z}$ .", "For every $k$ we reparametrize $\\gamma _k$ so that $\\gamma _k(0) \\notin \\overline{B}(\\pi x,k)$ .", "Since $K$ is $\\Phi _1$ -invariant then the reparametrized geodesics belong again to $K$ .", "Moreover $K$ is compact, then there exists a subsequence, denoted again by $\\gamma _k$ , that converges to $\\gamma _\\infty \\in K$ .", "In particular the sequence $\\gamma _k(0)$ converges to $\\gamma _\\infty (0)$ , but $d(\\gamma _k(0), \\pi x) > k$ for every $k$ , which is a contradiction.", "As a corollary we directly have: $h_\\text{inv-top} = \\sup _{\\tau \\ge 0} h_{\\text{top}}(K_\\tau , \\Phi _1).$ Now the idea is to relate the topological entropy of the dynamical system $(K_\\tau , \\Phi _1)$ to the Minkowski dimension of the set $\\Lambda _\\tau $ , computed with respect to the basepoint $x$ .", "The tool we need is the Lipschitz-topological entropy of a closed subset of the boundary at infinity, studied in [6].", "We recall briefly its definition.", "Let $(X,\\sigma )$ be a $\\delta $ -hyperbolic, GCB-space that is $P_0$ -packed at scale $r_0$ .", "For a subset $C$ of $\\partial X$ and $Y\\subseteq X$ we set $\\text{Geod}_\\sigma (Y,C)= \\lbrace \\gamma \\in \\text{Geod}_\\sigma (X) \\text{ s.t. }", "\\gamma ^{\\pm } \\in C \\text{ and } \\gamma (0)\\in Y\\rbrace .$ If $Y = X$ we simply write $\\text{Geod}_\\sigma (C)$ .", "Clearly $\\text{Geod}_\\sigma (C)$ is a $\\Phi $ -invariant subset of $\\text{Geod}_\\sigma (X)$ , so the geodesic flow is well defined on it.", "The lower Lipschitz-topological entropy of $\\text{Geod}_\\sigma (C)$ is defined as $\\underline{h_{\\text{Lip-top}}}(\\text{Geod}_\\sigma (C)) = \\inf _{\\text{ɗ}}\\sup _K \\lim _{r\\rightarrow 0} \\liminf _{T\\rightarrow + \\infty }\\frac{1}{T}\\log \\text{Cov}_{\\text{ɗ}^T}(K,r),$ where the infimum is taken among all geometric metrics on $\\text{Geod}_\\sigma (C)$ , the supremum is among all compact subsets of $\\text{Geod}_\\sigma (C)$ , $\\text{ɗ}^T$ denotes the metric $\\text{ɗ}^T(\\gamma ,\\gamma ^{\\prime })=\\sup _{t\\in [0,T]}\\text{ɗ}(\\Phi _t \\gamma , \\Phi _t \\gamma ^{\\prime })$ and $\\text{Cov}_{\\text{ɗ}^T}(K,r)$ denotes the covering number of the set $K$ at scale $r$ with respect to the metric $\\text{ɗ}^T$ .", "Taking the limit superior instead of the limit inferior we define the so-called upper Lipschitz-topological entropy of Geod$_\\sigma (C)$ , denoted $\\overline{h_{\\text{Lip-top}}}(\\text{Geod}_\\sigma (C))$ .", "The computation of these invariants can be really simplified.", "Proposition 7.2 ([6]) Let $(X,\\sigma )$ be a $\\delta $ -hyperbolic GCB-space that is $P_0$ -packed at scale $r_0$ .", "Let $C$ be a subset of $\\partial X$ and $x\\in X$ .", "Then there exists a constant $L$ depending only on $x$ such that $\\underline{h_\\textup {Lip-top}}(\\textup {Geod}_\\sigma (C)) = \\sup _{C^{\\prime }\\subseteq C}\\liminf _{T \\rightarrow +\\infty }\\frac{1}{T}\\log \\textup {Cov}_{f^T}(\\textup {Geod}_\\sigma (\\overline{B}(x,L), C^{\\prime }), r)$ for every $f\\in \\mathcal {F}$ and every $r>0$ , where the supremum is among all closed subsets $C^{\\prime }$ of $C$ .", "Moreover if $C$ is closed then $\\underline{h_\\textup {Lip-top}}(\\textup {Geod}_\\sigma (C)) = \\underline{\\textup {MD}}(C).$ In particular for every subset $C$ of $\\partial X$ it holds $\\underline{h_\\textup {Lip-top}}(\\textup {Geod}_\\sigma (C)) = \\sup _{C^{\\prime }\\subseteq C} \\underline{\\textup {MD}}(C^{\\prime }),$ where the supremum is among all closed subsets of $C$ .", "The corresponding results are true for the upper Lipschitz-topological entropy.", "We are now ready to show: Theorem 7.3 Let $(X,\\sigma )$ be a $\\delta $ -hyperbolic GCB-space that is $P_0$ -packed at scale $r_0$ .", "Let $\\Gamma $ be a group of $\\sigma $ -isometries of $X$ acting discretely and freely.", "Then $h_{\\textup {inv-top}} \\ge h_\\Gamma .$ We already know that $h_\\text{inv-top} = \\sup _{\\tau \\ge 0} h_{\\text{top}}(K_\\tau , \\Phi _1).$ Moreover $K_\\tau $ is compact, so we can use every metric on $K_\\tau $ inducing its topology in order to compute its topological entropy.", "We are going to select a specific metric in the following way: first of all we call $\\iota _\\tau $ the positive injectivity radius of the compact set $\\overline{B}(x,\\tau )$ .", "Secondly we consider $f_\\tau (s) = \\frac{1}{2}\\frac{4}{\\iota _\\tau }e^{-\\frac{4}{\\iota _\\tau }\\vert s \\vert }.$ It is easy to check that $f_\\tau \\in \\mathcal {F}$ and $C(f_\\tau )=\\int _{-\\infty }^{+\\infty }2\\vert s \\vert f_\\tau (s)ds = \\frac{\\iota _\\tau }{2}.$ Moreover we have: $\\begin{aligned}h_\\text{top}(K_\\tau ,\\Phi _1) &= \\lim _{r\\rightarrow 0}\\limsup _{n \\rightarrow +\\infty }\\frac{1}{n}\\log \\text{Cov}_{f^n_\\tau }(K_\\tau , r) \\ge \\lim _{r\\rightarrow 0}\\liminf _{n \\rightarrow +\\infty }\\frac{1}{n}\\log \\text{Cov}_{f^n_\\tau }(K_\\tau , r) \\\\&\\ge \\liminf _{n \\rightarrow +\\infty }\\frac{1}{n}\\log \\text{Cov}_{f^n_\\tau }\\left(K_\\tau , \\frac{\\iota _\\tau }{2}\\right) \\ge \\liminf _{T \\rightarrow +\\infty }\\frac{1}{T}\\log \\text{Cov}_{f^T_\\tau }\\left(K_\\tau , \\frac{\\iota _\\tau }{2}\\right).\\end{aligned}$ Our aim is to show $\\small \\liminf _{T \\rightarrow +\\infty }\\frac{1}{T}\\log \\text{Cov}_{f^T_\\tau }\\left(K_\\tau , \\frac{\\iota _\\tau }{2}\\right) \\ge \\liminf _{T\\rightarrow +\\infty }\\frac{1}{T}\\log \\text{Cov}_{f_\\tau ^T}\\left(\\text{Geod}_\\sigma (\\overline{B}(x,L), \\Lambda _{\\tau - 8\\delta - L}), \\frac{3}{2}\\iota _\\tau \\right),$ where $L$ is the constant of Proposition REF that depends only on $x$ and not on $\\tau $ .", "Indeed the right hand side equals $\\underline{h_\\text{Lip-top}}(\\text{Geod}_\\sigma (\\Lambda _{\\tau - 8\\delta - L}))$ by Proposition REF , so we would have $h_\\text{inv-top} = \\sup _{\\tau \\ge 0} h_{\\text{top}}(K_\\tau , \\Phi _1) \\ge \\sup _{\\tau \\ge 0} \\underline{h_{\\text{Lip-top}}}(\\Lambda _{\\tau - 8\\delta - L}) = \\sup _{\\tau \\ge 0} \\underline{\\text{MD}}(\\Lambda _{\\tau }) = h_\\Gamma $ again by Proposition REF and (REF ), concluding the proof.", "It remains to show (REF ).", "For every $T \\ge 0$ let $\\gamma _1,\\ldots ,\\gamma _N$ be a set of $\\sigma $ -local geodesics realizing $\\text{Cov}_{f_\\tau ^T}(K_\\tau ,\\frac{\\iota _\\tau }{2}).$ For every $i=1,\\ldots ,N$ we consider the set $A_i = \\lbrace \\tilde{\\gamma }\\in \\text{Geod}_\\sigma (X) \\text{ s.t. }", "\\Pi \\tilde{\\gamma }= \\gamma _i \\text{ and } \\tilde{\\gamma }(0)\\in \\overline{B}(x,L + \\iota _\\tau )\\rbrace .$ Observe that for every two elements $\\tilde{\\gamma },\\tilde{\\gamma }^{\\prime }\\in A_i$ we have $\\Pi \\tilde{\\gamma }(0) = \\Pi \\tilde{\\gamma }^{\\prime }(0)$ , so by Lemma REF either $\\tilde{\\gamma }=\\tilde{\\gamma }^{\\prime }$ or $d(\\tilde{\\gamma }(0),\\tilde{\\gamma }^{\\prime }(0)) \\ge \\iota _\\tau $ since $\\iota _\\tau $ is smaller than or equal to the injectivity radius at $\\gamma _i(0)$ .", "We conclude that the set $\\lbrace \\tilde{\\gamma }(0)\\rbrace _{\\tilde{\\gamma }\\in A_i}$ is a $\\frac{\\iota _\\tau }{2}$ -separated subset of $\\overline{B}(x,L+\\iota _\\tau )$ , implying $\\#A_i \\le \\text{Pack}(L + \\iota _\\tau ,\\frac{\\iota _\\tau }{2})$ .", "We consider the set $A_T = \\bigcup _{i=1}^N A_i$ whose cardinality satisfies $\\#A_T \\le \\text{Cov}_{f_\\tau ^T}(K_\\tau ,r)\\cdot \\text{Pack}\\left(L + \\iota _\\tau ,\\frac{\\iota _\\tau }{2}\\right).$ We claim that $A_T$ covers the set $\\text{Geod}_\\sigma (\\overline{B}(x,L), \\Lambda _{\\tau - 8\\delta - L})$ at scale $\\frac{3}{2}\\iota _\\tau $ with respect to the metric $f_\\tau ^T$ .", "Indeed let $\\tilde{\\gamma }\\in \\text{Geod}_\\sigma (\\overline{B}(x,L), \\Lambda _{\\tau - 8\\delta - L})$ .", "This means that a geodesic ray $\\xi = [x,\\tilde{\\gamma }^+]$ is contained in the $(\\tau - 8\\delta - L)$ -neighbourhood of the orbit $\\Gamma x$ .", "Moreover by Lemma REF we know that $d(\\tilde{\\gamma }(t), \\xi (t)) \\le 8\\delta + L$ for every $t\\ge 0$ .", "The same consideration holds for the negative ray $\\tilde{\\gamma }\\vert _{(-\\infty ,0]}$ , so $d(\\tilde{\\gamma }(t), \\Gamma x) \\le \\tau $ for all $t\\in \\mathbb {R}$ : this implies that $\\Pi \\tilde{\\gamma }$ belongs to $K_\\tau $ .", "By definition there exists $i \\in \\lbrace 1,\\ldots , N\\rbrace $ such that $f_\\tau ^T(\\gamma _i, \\Pi \\tilde{\\gamma })\\le \\frac{\\iota _\\sigma }{2}.$ Therefore for every $t\\in [0,T]$ we know that $\\int _{-\\infty }^{+\\infty } d(\\gamma _i(t+s), \\Pi \\tilde{\\gamma }(t+s))f_\\tau (s)ds \\le \\frac{\\iota _\\tau }{2}.$ Moreover from the choice of $f_\\tau $ we have $\\begin{aligned}\\frac{\\iota _\\tau }{2}&\\ge \\int _{-\\infty }^{+\\infty } d(\\gamma _i(t+s), \\Pi \\tilde{\\gamma }(t+s))f_\\tau (s)ds \\\\&\\ge \\int _{-\\infty }^{+\\infty } (d(\\gamma _i(t), \\Pi \\tilde{\\gamma }(t)) - 2\\vert s \\vert )f_\\tau (s)ds \\ge d(\\gamma _i(t), \\Pi \\tilde{\\gamma }(t)) - \\frac{\\iota _\\tau }{2}.\\end{aligned}$ In conclusion $d(\\gamma _i(t), \\Pi \\tilde{\\gamma }(t)) \\le \\iota _\\tau $ for every $t\\in [0,T]$ .", "By definition of injectivity radius, since $\\Pi \\tilde{\\gamma }(t)\\in \\overline{B}(x,\\tau )$ , we conclude that for every $t\\in [0,T]$ there exists a covering geodesic $\\tilde{\\gamma }_t$ of $\\gamma _i$ such that $d(\\tilde{\\gamma }_t(t), \\tilde{\\gamma }(t)) = d(\\gamma _i(t), \\Pi \\tilde{\\gamma }(t)) \\le \\iota _\\tau .$ For $t=0$ we have the covering geodesic $\\tilde{\\gamma }_0$ of $\\gamma _i$ that satisfies $d(\\tilde{\\gamma }_0(0), x)\\le d(\\tilde{\\gamma }_0(0), \\tilde{\\gamma }(0)) + d(\\tilde{\\gamma }(0), x)\\le L + \\iota _\\tau ,$ so $\\tilde{\\gamma }_0 \\in A_i$ .", "Moreover for every $t\\in [0,T]$ there exists an element $g_t \\in \\Gamma $ such that $\\tilde{\\gamma }_t = g_t \\tilde{\\gamma }_0$ .", "But arguing as in the proof of Proposition REF we conclude that $g_t = \\text{id}$ for every $t\\in [0,T]$ and so $d(\\tilde{\\gamma }_0(t), \\tilde{\\gamma }(t)) \\le \\iota _\\tau $ for every $t \\in [0,T]$ .", "We can now estimate the distance between $\\tilde{\\gamma }_0$ and $\\tilde{\\gamma }$ with respect to $f_\\tau ^T$ .", "For every $t\\in [0,T]$ we have $\\begin{aligned}f^t(\\tilde{\\gamma }_0, \\tilde{\\gamma }) &= \\int _{-\\infty }^{+\\infty } d(\\tilde{\\gamma }_0(t + s), \\tilde{\\gamma }(t + s)) f_\\tau (s)ds \\\\&\\le \\int _{-\\infty }^{+\\infty } (d(\\tilde{\\gamma }_0(t), \\tilde{\\gamma }(t)) + 2\\vert s\\vert )f_\\tau (s)ds \\le \\iota _\\tau + \\frac{\\iota _\\tau }{2} = \\frac{3}{2}\\iota _\\tau .\\end{aligned}$ Since $\\tilde{\\gamma }_0 \\in A_T$ we conclude that the set $A_T$ covers Geod$_\\sigma (\\overline{B}(x,L),\\Lambda _{\\tau - 8\\delta - L})$ at scale $\\frac{3}{2}\\iota _\\tau $ with respect to the metric $f_\\tau ^T$ .", "We can finish the proof of the theorem since $\\liminf _{T\\rightarrow +\\infty }\\frac{1}{T}\\log \\text{Cov}_{f_\\tau ^T}\\left(\\text{Geod}_\\sigma (\\overline{B}(x_0,L), \\Lambda _{\\tau - 8\\delta -L}), \\frac{3}{2}\\iota _\\tau \\right)$ is less than or equal to $\\begin{aligned}\\liminf _{T\\rightarrow +\\infty }\\frac{1}{T}\\log \\# A_T &\\le \\liminf _{T\\rightarrow +\\infty }\\frac{1}{T}\\log \\text{Cov}_{f_\\tau ^T}\\left(K_\\tau ,\\frac{\\iota _\\tau }{2}\\right)\\cdot \\text{Pack}\\left(L + \\iota _\\tau ,\\frac{\\iota _\\tau }{2}\\right) \\\\&= \\liminf _{T\\rightarrow +\\infty }\\frac{1}{T}\\log \\text{Cov}_{f_\\tau ^T}\\left(K_\\tau ,\\frac{\\iota _\\tau }{2}\\right),\\end{aligned}$ where the last equality follows from Proposition REF ." ], [ "Additional remarks", "The definition of upper and lower local entropy of a dynamical system $(Y,T)$ as $\\overline{h}^\\text{loc}(Y,T) = \\sup _{\\mu \\in \\mathcal {E}_1(Y,T)}\\inf _{\\text{ɗ}}\\overline{h}_{\\mu ,\\text{ɗ}}^\\text{loc}(Y,T), \\quad \\underline{h}^\\text{loc}(Y,T) = \\sup _{\\mu \\in \\mathcal {E}_1(Y,T)}\\inf _{\\text{ɗ}}\\underline{h}_{\\mu ,\\text{ɗ}}^\\text{loc}(Y,T)$ could seem arbitrary, in the sense that the order of the supremum and the infimum could be switched.", "In our case there is no difference.", "Corollary 8.1 Let $(X,\\sigma )$ be a $\\delta $ -hyperbolic GCB-space that is $P_0$ -packed at scale $r_0$ .", "Let $\\Gamma $ be a group of $\\sigma $ -isometries of $X$ acting discretely and freely.", "Then $\\inf _{\\textup {ɗ}}\\sup _{\\mu \\in \\mathcal {E}_1}\\overline{h}_{\\mu ,\\textup {ɗ}}^\\textup {loc} = \\inf _{\\textup {ɗ}}\\sup _{\\mu \\in \\mathcal {E}_1}\\underline{h}_{\\mu ,\\textup {ɗ}}^\\textup {loc} = h_\\Gamma .$ Clearly it always holds $\\inf _{\\textup {ɗ}}\\sup _{\\mu \\in \\mathcal {E}_1}\\overline{h}_{\\mu ,\\textup {ɗ}}^\\textup {loc} \\ge \\sup _{\\mu \\in \\mathcal {E}_1}\\inf _{\\textup {ɗ}}\\overline{h}_{\\mu ,\\textup {ɗ}}^\\textup {loc}.$ But, for $f\\in \\mathcal {F}$ , we have also $h_\\Gamma = \\sup _{\\mu \\in \\mathcal {E}_1}h_\\mu \\le \\sup _{\\mu \\in \\mathcal {E}_1}\\inf _{\\textup {ɗ}}\\overline{h}_{\\mu ,\\textup {ɗ}}^\\textup {loc} \\le \\sup _{\\mu \\in \\mathcal {E}_1}\\overline{h}_{\\mu ,f}^\\textup {loc} \\le h_\\Gamma .$ So $h_\\Gamma \\le \\sup _{\\mu \\in \\mathcal {E}_1}\\inf _{\\textup {ɗ}}\\overline{h}_{\\mu ,\\textup {ɗ}}^\\textup {loc} \\le \\inf _{\\textup {ɗ}}\\sup _{\\mu \\in \\mathcal {E}_1}\\overline{h}_{\\mu ,\\textup {ɗ}}^\\textup {loc} \\le \\sup _{\\mu \\in \\mathcal {E}_1}\\overline{h}_{\\mu ,f}^\\textup {loc} \\le h_\\Gamma ,$ implying the equalities.", "The same proof holds for the lower entropies.", "The second observation is about measure of maximal entropy.", "Corollary 8.2 Let $(X,\\sigma )$ be a $\\delta $ -hyperbolic GCB-space that is $P_0$ -packed at scale $r_0$ .", "Let $\\Gamma $ be a group of $\\sigma $ -isometries of $X$ acting discretely and freely.", "If there exists a measure $\\mu \\in \\mathcal {E}_1$ of maximal entropy, that is $h_\\mu = h_\\textup {top}$ , then for all $f\\in \\mathcal {F}$ it holds: $h_\\Gamma = h_\\mu = \\underline{h}_\\mu ^\\textup {loc} = \\underline{h}_{\\mu ,f}^\\textup {loc} = \\overline{h}_\\mu ^\\textup {loc} = \\overline{h}_{\\mu ,f}^\\textup {loc}.$ Moreover for $\\mu $ -a.e.$\\gamma \\in \\textup {Loc-Geod}_\\sigma (\\Gamma \\backslash X)$ it holds $h_\\Gamma = \\lim _{n \\rightarrow +\\infty }-\\frac{1}{n}\\log \\mu (B_{f^n}(\\gamma ,R))$ for every $R>0$ .", "We recall that it can happen that a measure of maximal entropy does not exist.", "For instance if $M$ is a Riemannian manifold with pinched negative curvature it is known by Otal-Peigné's Theorem (cp.", "[20]) that such a measure exists if and only if the Bowen-Margulis measure is finite, and it that case this measure, once normalized, is the unique measure of maximal entropy.", "Probably the same is true in the setting of Gromov-hyperbolic, packed, GCB-spaces but the construction of the Bowen-Margulis entropy is less direct.", "However using the tecnhiques of [22] and [16] it is plausible that Otal-Peigné's result can be generalized to our setting.", "We conclude with a corollary and an example.", "Corollary 8.3 Let $(X,\\sigma )$ be a $\\delta $ -hyperbolic GCB-space that is $P_0$ -packed at scale $r_0$ , $\\Gamma $ be a discrete group of $\\sigma $ -isometries of $X$ and $x$ be a fixed basepoint of $X$ .", "Then $h_\\Gamma = \\overline{h_{\\textup {Lip-top}}}(\\textup {Geod}_\\sigma (\\Lambda _\\textup {u-rad})) = \\underline{h_{\\textup {Lip-top}}}(\\textup {Geod}_\\sigma (\\Lambda _\\textup {u-rad})).$ The set $\\Lambda _\\text{u-rad}$ is the increasing union of the closed subsets $\\Lambda _\\tau $ , $\\tau \\ge 0$ .", "Let $C^{\\prime }$ be a closed subset of $\\Lambda _{\\text{u-rad}}$ .", "There are two possibilities: either $C^{\\prime } = \\Lambda _{\\text{u-rad}}$ or it is contained in $\\Lambda _\\tau $ for some $\\tau \\ge 0$ .", "In the first case by Theorem 12.2.7 of [12] we get $\\Lambda _{\\text{u-rad}} = \\Lambda _\\tau $ for some $\\tau \\ge 0$ .", "Therefore in both cases $\\overline{\\text{MD}}(C^{\\prime })\\le \\overline{\\text{MD}}(\\Lambda _{\\tau })$ , and the same holds for the lower Minkowski dimensions.", "This implies the thesis by Proposition REF .", "Now we show an example of subset $C$ of $\\partial X$ such that $\\overline{h_{\\textup {Lip-top}}}(\\textup {Geod}_\\sigma (C))<\\overline{h_{\\textup {Lip-top}}}(\\textup {Geod}_\\sigma (\\overline{C})),$ where $\\overline{C}$ is the closure of $C$ .", "In [11] there is an example of a pinched negatively curved Riemannian manifold $(M,g)$ admitting a non-uniform lattice $\\Gamma $ (i.e.", "the volume of $\\Gamma \\backslash M$ is finite) such that $h_\\Gamma < h_\\text{vol}(M)$ , where $h_\\text{vol}(M)$ is the volume entropy of $M$ .", "By [6] we know that $h_\\text{vol}(M) = \\overline{h_{\\textup {Lip-top}}}(\\textup {Geod}(\\partial M)) = \\overline{h_{\\textup {Lip-top}}}(\\textup {Geod}(\\Lambda (\\Gamma ))),$ while $h_\\Gamma = \\overline{h_{\\textup {Lip-top}}}(\\textup {Geod}(\\Lambda _\\textup {u-rad}(\\Gamma )))$ by the corollary above.", "Since the closure of $\\Lambda _{\\text{u-rad}}(\\Gamma )$ is $\\Lambda (\\Gamma )$ then (REF ) holds." ] ]
2105.11774
[ [ "ScalaBFS: A Scalable BFS Accelerator on HBM-Enhanced FPGAs" ], [ "Abstract High Bandwidth Memory (HBM) provides massive aggregated memory bandwidth by exposing multiple memory channels to the processing units.", "To achieve high performance, an accelerator built on top of an FPGA configured with HBM (i.e., FPGA-HBM platform) needs to scale its performance according to the available memory channels.", "In this paper, we propose an accelerator for BFS (Breadth-First Search) algorithm, named as ScalaBFS, that builds multiple processing elements to sufficiently exploit the high bandwidth of HBM to improve efficiency.", "We implement the prototype system of ScalaBFS and conduct BFS in both real-world and synthetic scale-free graphs on Xilinx Alveo U280 FPGA card real hardware.", "The experimental results show that ScalaBFS scales its performance almost linearly according to the available memory pseudo channels (PCs) from the HBM2 subsystem of U280.", "By fully using the 32 PCs and building 64 processing elements (PEs) on U280, ScalaBFS achieves a performance up to 19.7 GTEPS (Giga Traversed Edges Per Second).", "When conducting BFS in sparse real-world graphs, ScalaBFS achieves equivalent GTEPS to Gunrock running on the state-of-art Nvidia V100 GPU that features 64-PC HBM2 (twice memory bandwidth than U280)." ], [ "Introduction", "BFS (Breadth-First Search) is one of the most fundamental algorithms of Graph theory and is widely used in application domains, such as navigation [1], social network [2], and many others.", "The computation of BFS, however, is famous for its irregular memory accesses and low computation-to-memory ratio [3].", "To boost the performance of BFS, a computer system needs a large memory bandwidth, more specifically, a large memory bandwidth for random accesses.", "For such reason, BFS is chosen as one of the key benchmarks in Graph500 [4] to measure the capability of handling graph computing workloads of a computer system.", "For the low cost and massive storage capacity, DRAMs (i.e., DDR RAMs) are widely used as the storage devices in most computer systems nowadays.", "But DRAMs yield much lower bandwidth when handling random accesses than sequential accesses, and their single-channel performance is limited (e.g., 19.2GB/s for DDR4).", "To improve the memory bandwidth, newly emerging memory techniques, such as HMC (Hyper Memory Cube) [5] and HBM (High Bandwidth Memory) [6], are proposed.", "HMC stacks multiple DRAM dies and logic dies into a single stack and connects with computing units (e.g., CPU/GPU/FPGA) with a serial link providing bandwidth up to 240GB/s.", "Similarly, the HBM technique stacks multiple DRAM dies and a logic die into a single stack, but exposes multiple memory channels to the computing units.", "By this, HBM can easily scale its bandwidth with more stacks (thus more memory channels).", "For example, two stacks of HBM2 (the second generation of HBM) provide up to 460GB/s aggregated bandwidth in Xilinx U280 [7], while four stacks of HBM2 provide up to 900GB/s aggregated bandwidth in Nvidia V100 GPU [8].", "The unprecedented huge memory bandwidth, especially the scaling memory channels provided by HBM, makes it possible to build efficient accelerators for bandwidth-critical workloads like BFS.", "Since with multiple memory channels, HBM can compensate the weakness on random memory accessing of the underlying DRAM devices.", "However, it is still challenging to design an FPGA-based BFS accelerator that sufficiently exploits the increasing number of memory channels, since the accelerator itself needs to be scalable in design to match HBM's massive memory bandwidth.", "To investigate this research direction, we design a BFS accelerator, named as ScalaBFS, in this paper.", "By attaching each memory channel with configurable amounts of processing elements, ScalaBFS can sufficiently exploits the memory bandwidth provided by an HBM to improve the performance of BFS.", "To the best of our knowledge, our work is the first system design that accelerates BFS on the FPGA-HBM platform.", "Our paper makes the following contributions: $\\bullet $ proposes a design of BFS accelerator, that scales its performance according to the available memory channels of the FPGA-HBM platform.", "$\\bullet $ provides an open-source Available at: https://github.com/lizardll/ScalaBFS implementation that works on a real FPGA accelerator card (Xilinx Alveo U280) for data centers.", "$\\bullet $ extensively evaluates our prototype system and compares its performance with state-of-art FPGA approaches as well as those on GPUs.", "The rest of paper is organized as follows: Section presents the background knowledge required to understand this work.", "Section motivates the building of ScalaBFS.", "Section elaborates the design of our accelerator.", "We discuss the performance scaling issues of ScalaBFS in Section , and evaluate our prototype system in Section .", "Section concludes the paper and discusses the future works." ], [ "Background and Related Works", "In this section, we will first introduce the background knowledge on the BFS algorithm and HBM technology, and then briefly discuss the related works to our paper." ], [ "BFS algorithm", "A directed Undirected graphs can be easily converted into directed ones by treating each of its edges as two edges pointing to opposite directions.", "graph $G = (V,E)$ consists of a set of a finite number of vertices $V$ , and an edge set $E$ containing edges connecting two vertices from $V$ .", "The BFS algorithm computes the distances of vertices in $V$ from a given $root$ vertex.We omit the computing of parents of the vertices in BFS algorithms, as it is trivial when compared with the computing of distance values.", "[h] +5pt -12pt BFS algorithm (push mode/pull mode).", "Array $Level[0...|V|-1]$ , distances of vertices from $r$ .", "$i\\in [0,|V|-1]$ $Level[i] \\leftarrow \\infty $ ; $Level[r] \\leftarrow 0$ ; $bfs\\_level \\leftarrow 0$ ; *[l]push mode (beginning and ending iterations).", "$\\exists i \\in V$ , such that $Level[i]==bfs\\_level$ vertex $i$ whose $Level[i]==bfs\\_level$ outgoing neighbor $v$ of $i$ $Level[v] > bfs\\_level$ $Level[v] \\leftarrow bfs\\_level + 1$ ; $bfs\\_level \\leftarrow bfs\\_level + 1$ ; *[l]pull mode (mid-term iterations).", "$\\exists i \\in V$ , such that $Level[i] == \\infty $ vertex $i$ whose $Level[i] == \\infty $ incoming neighbor $u$ of $i$ $Level[u] == bfs\\_level$ $Level[i] \\leftarrow bfs\\_level + 1$ ; $bfs\\_level \\leftarrow bfs\\_level + 1$ ; Our work considers the vertex-centric level synchronous implementation of BFS as listed in Algorithm REF .", "The computing is organized into multiple iterations, where each while-loop in line 4$\\sim $ 9 or 10$\\sim $ 15 in the listing is treated as one iteration.", "Regarding to the direction of message passing, there are two modes of processing: push mode and pull mode.", "By push mode (line 4$\\sim $ 9), an iteration browses the level array to locate all the active (i.e., a vertex $i$ , whose $Level[i]==bfs\\_level$ ) vertices, and then for each of them, visits their outgoing (child) neighbors to set their level value to the current level if they are not visited before (i.e., pushing messages to the children).", "In pull mode (line 10$\\sim $ 15), an iteration browses the level array to locate all the unvisited vertices.", "Then for each of them, the algorithm visits their incoming (parent) neighbors, and sets its level value to the current level if one of its parents is active (i.e., pulling messages from the parents).", "Generally, during the computation of the BFS algorithm, there will be small amounts of active vertices during the beginning and ending iterations, and large quantities of active vertices during the mid-term iterations [3], [9].", "Therefore, choosing the push mode during the beginning and ending iterations, and switch to the pull mode during the mid-term iterations can help to reduce the number of memory access, and improve the performance of computation.", "For a scale-free graph (degrees of vertices comply with the power-law [10]) containing millions of vertices, a push-mode (or pull-mode) iteration of BFS may contain hundreds of thousands of active (or unvisited) vertices.", "Although conflicts (e.g., different active vertices push level values to the same child vertex) exist, the algorithm logic of these active (or unvisited) vertices can be conducted in full parallel.", "The massively parallel nature of FPGAs makes them very suitable for the processing of the BFS algorithm.", "At the same time, the conflicts raised during computation can be comfortably solved using the on-chip memory resources (e.g., double pump BRAM used in our work) of the FPGAs.", "Consider the case that graph data are stored in the off-chip memory (e.g., HBM studied in this paper), which is regarded as the performance bottleneck of the system.", "The performance of BFS is then mainly decided by the time paid on reading the child (or parent) neighbors of the hundreds of thousands of active (or unvisited) vertices.", "With this regard, the throughput (not the latency) of the off-chip memory device plays a vital role in deciding the performance of BFS." ], [ "High Bandwidth Memory (HBM)", "As a typical example, Figure REF illustrates the HBM subsystem of Xilinx Alveo U280 [7].", "The HBM subsystem contains two HBM stacks, each of which is divided into 16 pseudo channels (PCs).", "As each PC provides 2Gbit storage capacity, 32 HBM PCs provide 8GB ($2\\times 16\\times 2/8$ ) storage capacity totally.", "Figure: The HBM subsystem of Xilinx Alveo U280Above the 32 PCs, there are 16 memory channels (MCs), each of which controls the accesses towards two adjacent PCs.", "The switch network consists of 8 4$\\times $ 4 mini-switches, each of which connects two adjacent MCs and exposes 4 AXI interfaces to the FPGA.", "Thus, the HBM subsystem of U280 exposes 32 AXI ports totally to the user logic built in the FPGA.", "Every two adjacent mini-switches are connected with a bus to provide global addressing, such that a memory access towards an arbitrary PC can be issued from any AXI interface.", "By conducting random (but regular memory) accessing workloads, Shuhai [11] measured the performance of the HBM subsystem of U280, and observed that: 1) A memory access to HBM suffers higher latency than that towards DDR4.", "2) Although the bandwidth of a single HBM PC is smaller than DDR4, the aggregated bandwidth of HBM arrives at 425 GB/s when handling sequential accesses.", "As BFS is in-essence a throughput-critical workload, the aggregated bandwidth of HBM can greatly help on boosting its performance, despite its longer latency for individual memory accesses." ], [ "Graph data", "We use Compressed Sparse Row (CSR) and its transpose, i.e., Compressed Sparse Column (CSC), to represent the graph data.", "The CSR consists of an offset array and an edge array, where the offset array stores the offsets of the outgoing (child) neighbor lists.", "The CSC has a similar structure as CSR, except that its edge array stores the incoming (parent) neighbor lists.", "The reason for ScalaBFS to use both CSR and CSC is that they facilitate the PEs to obtain both the outgoing and incoming neighbor lists of a selected vertex.", "Figure REF gives an example graph and its CSR and CSC representations in Figure REF .", "Figure: Partitioning an example graph into two subgraphs in CSC and CSR." ], [ "Related Works", "As an important algorithm, the computation of BFS is extensively studied in various platforms, including CPUs [12], [13], GPUs [14], [15], and FPGAs [16], [17], [18], [19], [20], [21], [3], [5], [22], [9], [23].", "We briefly survey the relevant works in FPGA-based BFS accelerators below.", "For a small graph that can be fully loaded in the on-chip memory of an FPGA, BFS can be conducted in it with very high performance using GraphStep [16].", "However, for a large graph stored in the off-chip memory device (e.g., DRAM, HMC, or HBM studied in this paper), the key to improve the performance of BFS is to exploit the bandwidth provided by the off-chip memory devices, and at the same time, increase the parallelism of processing.", "With the advent of the big-data era, conducting BFS in the large graphs is of the research interest in most works.", "Works in [17] and [21] use bitmaps stored in the on-chip memory devices (e.g., BRAM) of FPGAs to track the status changing (active/unvisited) of vertices and focus on studying the interconnect networks connecting the processing elements.", "[17] proposes to use a fully-connected network and FIFOs to buffer the messages.", "The design requires $P^2$ FIFOs for $P$ processing elements.", "[21] constructs the network into a 2-D torus to reduce the resource consumption on FIFOs to $5\\times P$ .", "The work in [18] and CyGraph [19] are two BFS accelerators build on the Convey HC-1 and HC-2 respectively.", "Convey HC-1 and HC-2 are customized machines [24] whose coprocessor has a hardware 4$\\times $ 8 full crossbar connecting 16 memory channels (8 memory controllers) with 4 Virtex-5 LX330 FPGAs.", "Both of these two accelerators repeatedly read the level array stored in off-chip DRAMs to track the status changing of vertices, and issue writes to the level array directly to the DRAMs (according to Algorithm REF ).", "By sufficiently using all memory channels and all logic resources of the 4 FPGAs, these two accelerators achieve the highest performance of 2.5 GTEPS on processing scale-free graphs.", "The work in [3] proposes a hybrid BFS accelerator on the FPGA-CPU platform, where the CPU conducts the push-mode processing, and the FPGA executes the pull-mode processing.", "The work shows that choosing proper modes for different stages processing can effectively improve the performance of BFS.", "Recent work in [23] proposes an approach (named as Dr.BFS) that enhances the BFS performance by compressing the vertex data by using bitmaps.", "Dr.BFS can even achieve better performance than [3], with only push-mode processing.", "The methods on accelerating BFS on FPGA-HMC platform are studied in [5], [22] and [9], where the main focuses are accelerating bitmap operations conducted in BFS with HMC memory devices.", "Besides the BFS accelerators studied in this paper, there are FPGA-based designs for general-purpose graph processing (i.e., supporting other graph algorithms, such as PageRank [25], besides BFS), such as ForeGraph [26] and the work in [27].", "Aiming at supporting various graph algorithms, they generally adopt the edge-centric processing model to smooth the irregularity of graph processing workloads.", "The edge-centric processing model, however, limits their performances on BFS.", "For example, even after improvements on vertex caching [28], ForeGraph achieves only about 410 MTEPS on the soc-LiveJournal graph (parameters listed in Table REF ) according to the metrics of Graph500 by using a single DDR4 channel." ], [ "Opportunity and Challenge", "In synchronous vertex-centric BFS algorithm REF , the vertices of the current level can be processed in parallel.", "However, the neighbor update of every vertex can have overlaps, thus need to be combined before written into level array or visited map.", "For example, the state-of-the-art implementation Dr. BFS [23] builds a status recombiner to combine those updates after every iteration.", "When the parallelism increases, the recombiner will need a longer time to process the updates due to limited external memory bandwidth, which becomes the performance bottleneck of the overall system [23].", "The underlying reason is that previous FPGA boards always feature up to two DDR4 channels, providing up to 40GB/s memory bandwidth.", "Therefore, exploiting more computing parallelism fails to increase the overall performance and then the FPGA resources are underutilized even in a small FPGA [23].", "For example, Dr. BFS [23] utilizes 27% of the FPGA logic and Fabgraph [28] uses only 19%." ], [ "Opportunity from HBM-enhanced FPGAs", "FPGA vendors such as Xilinx have already provided HBM-enhanced FPGA boards like U280 to provide up to 460GB/s memory bandwidth [11], which opens up new opportunity to accelerate graph processing in the context of FPGA.", "Therefore, HBM-enhanced FPGAs allow more computing parallelism to fully leverage FPGA resources.", "However, it is not trivial to efficiently utilize HBM bandwidth in the context of BFS, which introduces random memory access." ], [ "Challenge when Leveraging HBM", "Due to the random memory access property of BFS, we need a crossbar to globally access all the memory channels of HBM.", "There are two types of crossbar, which turns out to fail to satisfy the memory bandwidth requirement of our BFS engine.", "A Full Xilinx AXI Interconnect.", "In our practice, the full Xilinx AXI Interconnect IP [29] only supports up to 16$\\times $ 16 AXI interconnect, which is unacceptable as U280 has 32 HBM PCs.", "Xilinx's Switch Network.", "We examine the effect of the built-in Xilinx's switch network when memory accesses cross HBM channels.", "We employ Shuhai [11] to benchmark the performance of each AXI channel by reading data from neighboring HBM channels.", "Our data width is 256 bits, with the outstanding buffer size 256, and the burst length 64.", "Intuitively, the more number of HBM channels each AXI channel reads from, the higher pressure the switch network encounters.", "Figure REF illustrates the throughput of each AXI channel that reads across $2^k$ neighboring HBM channels, where $k=0,1,...,5$ .", "For example, \"4\" in the figure legend indicates that the $i^{th}$ $(0 \\le i \\le 31)$ AXI channel in Shuhai visits 4 consecutive HBM PCs in the range of [$\\lfloor i/4\\rfloor \\cdot 4$ , $\\lfloor i/4\\rfloor \\cdot 4+3$ ] in every 4 memory accesses.", "We observe that reading across HBM channels significantly affects the achievable throughput of each AXI channel.", "For example, the case that crosses 32 HBM channels achieves less than 0.5GB/s, more than 20 times less than the case that does not cross any HBM channels.", "It means that globally random memory access leads to dramatically low throughput using Xilinx's built-in switch network.", "Figure: Effect of switch network when the application reads cross HBM channels." ], [ "System Design", "In this section, we present the system design of ScalaBFS.", "We present the design methodology behind and the overall hardware architecture of ScalaBFS, followed by detailed hardware design of each hardware component." ], [ "Design Methodology", "In this subsection, we present two design goals of ScalaBFS, followed by our solution.", "$\\\\$ G1: Minimize Off-chip Memory Accesses Given Limited On-chip Memory Capacity.", "The size of on-chip memory, i.e., BRAMs and URAMs, is always limited, especially on FPGAs, where memory blocks are uniformed distributed.", "For example, the RAM size on state-of-the-art FPGA board U280 is 41MB [7].", "In the context of BFS, we are not able to fit all the data (vertex data and graph data) in FPGA's on-chip memory, which means we have to read data from external memory.", "Since external memory access is quite expensive, we aim at reducing external memory accesses as many as possible.", "G2: Maximize Computing Parallelism to Match High HBM Bandwidth.", "Previous work [23] can only achieve limited computing parallelism due to limited external memory bandwidth.", "In other words, it cannot fully utilize FPGA resources on a small-size FPGA as external memory bandwidth has already been saturated.", "On an HBM-enhanced FPGA, we are able to explore high computing parallelism to match high HBM bandwidth.", "However, it is not trivial to achieve high computing parallelism, because the parallel implementation of BFS is not able to guarantee execution locality, so we need data shuffling that increases design complexities and impedes us from easily achieving linear scalability." ], [ "Our Solution", "$\\\\$ In order to satisfy G1, we need to minimize 1) reference input graph from external memory M1, and 2) the on-chip memory overhead for intermediate states M2.", "Satisfying M1.", "For an input graph, we separate its data into two types: vertex data and graph data.", "Vertex data refers to the level array, while graph data refers to the neighbor lists.", "We leverage precious on-chip memory resources (i.e., BRAMs and URAMs) to store all the vertex data, which is similar to [17] and [21].", "The underlying reason behind this design choice is two-fold.", "First, Algorithm REF illustrates that vertex data is frequently modified to track the status (i.e., active or visited) of vertices, and at the same time, store the result level values.", "On the contrary, graph data never change.", "Second, modern FPGAs now enjoy larger and larger on-chip memory capacity, in terms of BRAM and URAM, as FPGA technique advances.", "Larger on-chip memory capacity allows storing millions of vertices.Discussing the case the size of vertex data is larger than the on-chip memory capacity is beyond the scope of this paper.", "Satisfying M2.", "We present a new BFS algorithm to minimize intermediate state size, as listed in Algorithm REF .", "The key idea of the proposed algorithm is to employ three bitmaps, i.e., current frontier, next frontier and visited map, to track the statuses of vertex data during the execution of BFS.", "Each vertex occupies a single bit from each of these three bitmaps, which means that a vertex only consumes three bits.", "A bit in the current frontier indicates whether its corresponding vertex is active (1 for active, and 0 for inactive) in the current iteration.", "Similarly, a bit in the next frontier indicates whether its corresponding vertex will be activated in the next iteration.", "A bit in the visited map indicate whether the corresponding vertex is visited before (1 for visited, and 0 for un-visited).", "These three bitmaps are stored in double pump BRAMs, such that two operations can be conducted on them within a clock cycle.", "[h] BFS algorithm using three bitmaps +5pt -15pt Array $Level[0...|V|-1]$ , distances of vertices from $r$ .", "$i\\in [0,|V|-1]$ $Level[i] \\leftarrow \\infty $ ; $current\\_frontier[i] \\leftarrow 0$ ; $next\\_frontier[i] \\leftarrow 0$ ; $visited\\_map[i] \\leftarrow 0$ ; $Level[r] \\leftarrow 0$ ; $bfs\\_level \\leftarrow 0$ ; $current\\_frontier[r] \\leftarrow 1$ ; $visited\\_map[r] \\leftarrow 1$ ; *[l]push mode (beginning and ending iterations).", "$\\exists i \\in V$ , such that $current\\_frontier[i]=1$ vertex $i$ whose $current\\_frontier[i]=1$ outgoing neighbour $v$ of $i$ $visited\\_map[v] == 0$ $next\\_frontier[v] \\leftarrow 1$ ; $visited\\_map[v] \\leftarrow 1$ ; $Level[v] \\leftarrow bfs\\_level + 1$ ; $bfs\\_level \\leftarrow bfs\\_level + 1$ ; $swap(current\\_frontier, next\\_frontier)$ *[l]pull mode (mid-term iterations).", "$\\exists i \\in V$ , such that $visited\\_map[i]\\ne 1$ vertex $i$ whose $visited\\_map[i]\\ne 1$ incoming neighbour $u$ of $i$ $current\\_frontier[v] == 1$ $next\\_frontier[i] \\leftarrow 1$ ; $visited\\_map[i] \\leftarrow 1$ ; $Level[i] \\leftarrow bfs\\_level + 1$ ; $bfs\\_level \\leftarrow bfs\\_level + 1$ ; $swap(current\\_frontier, next\\_frontier)$ In order to satisfy G2, ScalaBFS partitions the graph data into multiple subgraphs, and places each subgraph in an HBM PC to enforce locality of accessing (prevent the crossing shown in Figure REF ).", "Figure REF illustrates how to divide the vertex ID space of an example graph into two PEs: first, IDs are divided into two intervals, i.e., [0,2,4] and [1,3,5], due to the load-balancing reason.", "Then, the graph data are partitioned according to the partitioning results on the vertex ID space: neighbor lists of the vertices belonging to the same partition will be placed in the same subgraph as shown in Figure REF .", "From the viewpoint of the adjacency matrix, our partitioning scheme partitions a graph horizontally, which is different from [23] where the input graph is partitioned vertically.", "The reason for this horizontal partition scheme is that it does not breakdown the neighbor lists, and longer neighbor lists mean more chances of sequential accesses towards the HBM, which helps on improving the memory bandwidth usage rates during processing.", "Further, ScalaBFS employs multiple computing engines to provide massive computing parallelism, and the details are shown in the following hardware design.", "Figure: Architecture of ScalaBFSFigure: Processing logic of a PE in both push and pull modes.According to Algorithm REF , we present the hardware architecture of ScalaBFS, which consists of multiple Processing Groups (PGs), a Scheduler, and a Vertex dispatcher.", "A PG consists of an HBM reader and one or more Processing Elements (PEs).", "The Scheduler in Figure REF controls the processing mode (either pull or push model) of each PE, and informs its decisions to the PEs at the beginning of each iteration on the fly.", "Each PG is assigned exclusively to a single HBM pseudo channel (PC).", "The HBM reader is shared by all the PEs in a PG to issue memory requests to its corresponding HBM PC via AXI port to read the neighbor lists from HBM.", "Therefore, ScalaBFS has no more than 32 PGs on Xilinx Alveo U280, as its HBM subsystem has only 32 PCs (shown in Figure REF ).", "Each PE processes an interval of vertices of an input graph.", "In case of total $Q$ PEs, we divide the vertex ID space of an input graph into $Q$ non-overlapping intervals.", "In order to achieve load balancing between PEs, the vertex IDs are hashed before assigning to the intervals, such that the $i$ -th PE is in charge of processing the vertex whose $VID$ satisfies $VID \\% Q = i$ .", "Moreover, Each PE works in a hybrid (push-pull) processing mode according to the stages of BFS, to improve hardware utilization rate.", "The design details of a PE is shown in Subsection REF .", "After HBM readers issue memory requests, the Vertex dispatcher gathers all responses of neighbor lists from all the PCs and then scatters the vertices in the neighbor lists to the destination PEs according to their vertex IDs (denoted as $VID$ ).", "We will present the details of these two modules in Subsection REF .", "Figure: Converting full-crossbar into multi-layer crossbar." ], [ "Hybrid-Mode PE", "The goal of a hybrid-model PE is to allow both push and pull models within a PE.", "Intuitively, a PE works in the push mode in the beginning and ending iterations, and work in the pull mode during the mid-term iterations as listed in Algorithm REF .", "A hybrid-model PE consists of on-chip intermediate states and three execution components (i.e., stages).", "Working Mechanism of Intermediate States.", "During a push-mode iteration, the current frontier is scanned to locate all the active vertices.", "And then for each active vertex, vertices in its outgoing (child) neighbor list will be checked.", "If a vertex in the list has not been visited before (by checking the visited map), the corresponding bit in the visited map and next frontier will be set, and its level value will be written back to the level array stored in URAMs.", "Whereas during a pull-mode iteration, the visited map will be scanned to locate all unvisited vertices.", "And then for each unvisited vertex, vertices in its incoming (parent) neighbor list will be checked.", "If a parent vertex in the list is active (by checking the current frontier), the corresponding bits in the visited map and next frontier of the aforementioned unvisited vertex will be set, and its level value will be written back to URAMs.", "Workload Preparing (P1).", "This stage prepares the workloads to be processed.", "When working in the push mode, the active vertices are first obtained from the current frontier, and then passed to the Read CSR module to read their outgoing neighbor lists from the CSR data.", "As the CSR data are stored in HBM, the module will pass its requests to the HBM reader.", "When working in the pull mode, the unvisited vertices are obtained from the visited map, and the remaining processes are the same as in the push mode except that the Read CSC module will read the CSC data for incoming neighbor lists from the HBM.", "Neighbor Checking (P2).", "This stage accepts messages (i.e., the neighboring vertices) from the Vertex dispatcher.", "When working in the push mode, the messages will be the outgoing (child) neighbors of the active vertices prepared in P1.", "For each child vertex, the processing logic will proceed to check the visited map, and pass it to the next stage of processing (i.e., P3) if it is not visited before, or drop it if otherwise.", "When working in the pull mode, the messages will be the incoming (parent) neighbors of the unvisited vertices prepared by the stage of P1.", "For each parent vertex, the processing logic will then proceed to check the current frontier, and pass its child vertex (i.e., the unvisited vertex prepared by P1) to the next stage of processing (i.e., P3) if it (the parent vertex) is active, or drop it if otherwise.", "Note that in the pull mode, the child vertex to be passed to P3 may not be in the same PE as that processing its parent, since the incoming neighbor lists in CSC contains vertices of other partitions as shown in Figure REF .", "In such a case, the child vertex will be passed from one PE to another PE via a soft crossbar, which will be discussed in detail in the next subsection.", "Result Writing (P3).", "This stage accepts result messages from P2 and conducts the modifications in the bitmaps according to Algorithm REF .", "In both processing modes, the results will be written to the next frontier and visited map.", "Besides, the results will also be written to the Level array stored in URAMs (not shown in Figure REF for brevity).", "Due to the similarities of the logic for these two processing modes, except for the soft crossbar used in P2 in the pull mode, we use the same sets of circuits with parameters in registers to implement the similar modules, to save logic resources of the FPGA.", "Moreover, as the above processing stages are driven by signals (from the Scheduler) or data streams from other modules, the circuits work asynchronously in a pipelined fashion to maximize processing efficiency." ], [ "HBM Reader and Vertex Dispatcher", "HBM Reader.", "The functionality of an HBM reader is to receive requests from the Read CSR or Read CSC, convert these requests into memory accessing AXI commands, and then issue them to its corresponding HBM PC.", "When PEs are now working in the push mode.", "A request (reading the outgoing neighbor list of an active vertex) is sent to an HBM reader, which will first assemble an AXI command to read the corresponding values in the offset array of CSR from HBM.", "After receiving the offset values, the HBM reader will then assemble another AXI command to read the outgoing neighbor list from the edge array.", "Procedures are similar in the pull mode, except that each PE will read the offsets in the CSC and incoming neighbor lists.", "Vertex Dispatcher.", "The Vertex dispatcher intercepted read the neighbor lists read from the HBM.", "particularly, The functionality of the Vertex dispatcher is to scrutinize the vertices in the input neighbor-list streams, classify them according to intervals that they belong to, and send them back to the corresponding PEs.", "The most straightforward approach that achieves such an objective is to use a full $N\\times N$ crossbar where N is the number of PEs (and accordingly, $N$ subgraphs).", "However, such a full crossbar requires $N^2$ FIFOs to implement [17], which are hard to fit within an FPGA when $N$ is sufficiently large.", "For example, when the length of FIFO is 16 and $N=64$ , the Vertex dispatch employs a full 64$\\times $ 64 crossbar that consumes more than half of the LUTs in the U280 FPGA card, leaving very limiting number of LUTs for PEs.", "Efficient Crossbar Design.", "Inspired by the fact that the switching logic of our Vertex dispatcher is unidirectional (and thus simpler), we propose a multi-layer crossbar that requires much fewer FPGA resources to implement while keeping the same functionality.", "Efficient Crossbar Design.", "Inspired by the fact that the switching logic of our Vertex dispatcher is unidirectional (and thus simpler), we propose a multi-layer crossbar that requires much fewer FPGA resources to implement while keeping the same functionality.", "Our multi-layer crossbar design in ScalaBFS shares some similarities as the recent work of butterfly crossbar in [30].", "The differences between our design and that in [30] are that: 1) our crossbar design supports general RTL (as ScalaBFS), while that in [30] mainly supports HLS (High Level Synthesis); 2) the focus of our crossbar design is to cope with random and irregular memory accesses, while that in [30] mainly addresses sequential memory accesses; 3) our crossbar design studies the trade-off between performance and on-chip resource consumption (explained in the following), while there is no such concern in [30].", "To explain our multi-layer crossbar, we discuss a case that dispatches 16 neighbor-list streams to 16 PEs as shown in Figure REF .", "Figure REF depicts the 16$\\times $ 16 full crossbar that contains 16 input and 16 output ports.", "The switching logic of this 16$\\times $ 16 full crossbar is to send a vertex to the $(VID\\%16)^{th}$ port, where $VID$ denotes the vertex's ID.", "Equivalently, we can convert the 16$\\times $ 16 full crossbar into a two-layer crossbar, which consists of two layers (input layer and output layer) and each layer consists of four 4$\\times $ 4 crossbar, as shown in Figure REF .", "The switching logic of each 4$\\times $ 4 crossbar of the input layer is to send a vertex to its $(VID\\%4)^{th}$ port, which connects to the $(VID\\%4)^{th}$ 4$\\times $ 4 crossbar of the output layer.", "The $i^{th}$ 4$\\times $ 4 crossbar of the output layer connects to the PEs such that $PE_{ID}\\%4 = i$ , where $PE_{ID}$ denotes the ID of a PE.", "By this for the output layer, the first (0th) 4$\\times $ 4 crossbar connects to $PE_0$ , $PE_4$ , $PE_8$ and $PE_{12}$ , while the second (1th) 4$\\times $ 4 crossbar connects to $PE_1$ , $PE_5$ , $PE_9$ and $PE_{13}$ , and so on.", "The switching logic of each 4$\\times $ 4 crossbar of the output layer is thus to send an incoming vertex to the $(VID\\%16)^{th}$ PE (equals to the effectiveness of the 16$\\times $ 16 full crossbar).", "To generalize our approach on converting an $N\\times N$ full crossbar into a multi-layer crossbar: we need first decompose $N$ into multiple (say $k$ ) factors, such that $N = C_1\\times C_2 \\times ... \\times C_k$ .", "The first (input) layer uses $N/C_1$ $C_1\\times C_1$ crossbars, and classifies the input vertices into $C1$ groups according to $VID\\%C_1$ , where $VID$ denotes the ID of an input vertex.", "The second (relay) layer uses $N/C_2$ $C_2\\times C_2$ crossbars, and further classifies the input vertices into $C1\\times C2$ groups according to $VID\\%(C_1 \\times C2)$ .", "Such a process continues till the vertices arrive at the last (output) layer, which uses $N/C_k$ $C_k\\times C_k$ crossbars, and classifies the vertices into $C_1\\times C_2\\times ... \\times C_k = N$ groups.", "The ($N$ ) vertex groups coming out of the last layer are finally sent to the ($N$ ) PEs, each of which is in charge of processing one vertex group (i.e., a vertex interval).", "Figure REF merely illustrates a simple case where $N=16$ , $k=2$ , and $C_1=C_2=4$ .", "We now compare these two approaches (full-crossbar vs. multi-layer crossbar) from the angles of resource consumption and efficiency.", "An $N\\times N$ full crossbar consumes $N^2$ FIFOs, while the number of FIFOs required by its equivalent $k$ -layer crossbar is $(N/C_1) \\times C_1^2 + (N/C_2) \\times C_2^2 + ... + (N/C_k) \\times C_k^2$ .", "It is easy to prove that the number of FIFOs consumed by our $k$ -layer crossbar is smaller than that of its equivalent $N\\times N$ full crossbar, as $N = C_1\\times C_2 \\times ... \\times C_k$ .", "Consider the example in Figure REF , the 16$\\times $ 16 full crossbar consumes $16\\times 16 = 256$ FIFOs, while the two-layer crossbar consume only $2\\times 4\\times 4\\times 4 = 128$ FIFOs, meaning half resource consumption.", "From the efficiency point of view, the $N\\times N$ full-crossbar achieves 1-hop latency on message passing, while the equivalent $k$ -layer crossbar approach requires k-hop latency for vertex dispatching.", "Obviously, the $k$ -layer crossbar approach leads to higher latency than the full crossbar approach as $k\\ge 2$ .", "Nevertheless, based on the observation that BFS is a throughput-critical workload as discussed in subsection REF , it is appropriate for us to use multi-layer crossbars to trade latency for resource.", "In Section , we will use 3-layer 4$\\times $ 4 crossbars to replace the 64$\\times $ 64 full crossbar to help our accelerator to achieve the scale of 64 PEs on U280." ], [ "Performance Model", "ScalaBFS can scale its performance in two directions: 1) increasing the number of HBM PCs (thus PGs) with a fixed number of PEs in a PG, and 2) increasing the number of PEs in a PG with a fixed number of HBM PCs.", "The first direction of scaling is limited by the number of available HBM PCs of a given HBM subsystem, while scaling in the second direction may be limited by the amount of logic resources of an FPGA.", "If we regard the HBM as the slower device, and the processing circuits (PEs, HBM readers, and the Vertex dispatcher) of our accelerator works in a pipelined fashion, the performance of ScalaBFS should scale linearly along the first direction (i.e., increasing the number of HBM PCs).", "We study the scalability of our accelerator in the first direction empirically by experiments conducted in Section , and focus our discussions of this section in the second direction (i.e., increasing PEs).", "By a model-based study, we intend to investigate the answer to the following question: Given a fixed number of HBM PCs, how many PEs in a PG should we choose to achieve the optimal performance?", "To simplify discussions, we consider a single HBM PC, on top of which a PG is constructed, and the number of PEs in the PG vary.", "We assume all processing units, i.e., PEs, the HBM reader, and the Vertex dispatcher, work in a pipelined fashion to mask delays in processing stages.", "As our PEs use double pump BRAMs as their local store on processing the bitmaps, each PE is capable of conducting two operations at each clock cycle.", "Therefore, when there are $N_{pe}$ PEs, we configure the data width of AXI bus (denoted as $DW$ ) in Equation REF .", "$\\begin{aligned}DW = 2 \\cdot N_{pe} \\cdot S_v\\end{aligned}$ where $S_v$ denotes the storage size of a vertex.", "As each neighbor list read from an HBM PC is treated as a stream of vertices, which are classified and then sent to all participating PEs by the Vertex dispatcher, assuming each PE receives the same number of vertices during processing (perfect load balancing), to feed more PEs, we need longer AXI buses.", "Denote the PE's frequency as $F$ , the bandwidth of a single HBM PC (denoted as $BW$ ) can be computed according to Equation REF .", "$\\begin{aligned}BW =&\\begin{dcases}DW \\cdot F,&{DW \\cdot F < BW_{MAX}}\\\\BW_{MAX},& {DW \\cdot F \\ge BW_{MAX}}\\end{dcases}\\end{aligned}$ where $BW_{MAX}$ denotes the maximum physical bandwidth of a single HBM PC.", "According to [11], $BW_{MAX}$ is about 13.27GB/s.", "As HBM works in much higher frequency than PGs (e.g., the frequency of HBM in U280 is about 900 MHz, which is much higher than the FPGA circuits of ScalaBFS), we assume it produces a datum whose length is $DW$ for each cycle, but at the same time, $BW$ can not exceed its physical limit of $BW_{MAX}$ .", "Now consider conducting a push-mode BFS iteration (pull-mode iterations exhibit similar patterns) in a given graph, a set of active vertices, each of which has a list of neighboring vertices stored in the edge array in Figure REF , is going to be processed.", "We assume that each active vertex has $Len_{nl}$ neighboring vertices on average.", "Before retrieving the neighboring vertices, the HBM reader needs to read the offset values from the HBM.", "For each subgraph, assuming the length of data read from the offset array for each active vertex equals to the data width (i.e., $DW$ ), the percentage of HBM bandwidth paid on reading the neighbor lists $P_{nl}$ can be computed using Equation REF .", "$\\begin{aligned}P_{nl} = \\frac{Len_{nl} \\cdot S_v}{DW + Len_{nl} \\cdot S_v}\\end{aligned}$ Accordingly, the portion of bandwidth paid on reading the neighbor lists can be computed using Equation REF .", "$\\begin{aligned}BW_{nl} = BW\\cdot P_{nl} =&\\begin{dcases}DW \\cdot F \\cdot P_{nl},&{DW \\cdot F < BW_{MAX}}\\\\BW_{MAX} \\cdot P_{nl},& {DW \\cdot F \\ge BW_{MAX}}\\end{dcases}\\end{aligned}$ Regard each neighbor vertex as an edge connecting itself with a corresponding active vertex, by combining Equation REF , REF and REF , the theoretical performanceof a single PG (denoted as $Perf_{pg}.$ , in number of traversed edges per second (TEPS)) can then be computed approximately in Equation REF .", "$\\begin{aligned}&\\mathrel {\\phantom{=}} Perf_{pg}.", "\\approx \\frac{BW_{nl}}{S_v} = \\\\&\\begin{dcases}\\frac{2 N_{pe}\\cdot F\\cdot Len_{nl}}{2 N_{pe} + Len_{nl}},&{2 N_{pe} \\cdot S_v \\cdot F < BW_{MAX}}\\\\\\frac{BW_{MAX}\\cdot Len_{nl}}{2 N_{pe}\\cdot S_v+Len_{nl}\\cdot S_v},& {2 N_{pe} \\cdot S_v \\cdot F \\ge BW_{MAX}}\\end{dcases}\\end{aligned}$ Assuming that the vertex dispatcher is not the bottleneck (which means adding PC/PGs can improve the performance linearly, and our experiment can prove this), We can come up with our overall performance $Perf.$ in Equation REF , where $N_{pc}$ is the number of PC or PGs.", "$\\begin{aligned}&\\mathrel {\\phantom{=}} Perf.", "= Perf_{pg} \\cdot N_{pc}\\\\&\\begin{dcases}\\frac{2 N_{pe}\\cdot F\\cdot Len_{nl} \\cdot N_{pc}}{2 N_{pe} + Len_{nl}},&{2 N_{pe} \\cdot S_v \\cdot F < BW_{MAX}}\\\\\\frac{BW_{MAX}\\cdot Len_{nl} \\cdot N_{pc}}{2 N_{pe}\\cdot S_v+Len_{nl}\\cdot S_v},& {2 N_{pe} \\cdot S_v \\cdot F \\ge BW_{MAX}}\\end{dcases}\\end{aligned}$ Also we can take the resource consumption into the consideration.", "For simplicity, only LUT usage is considered.", "Assuming each FIFO in our $k$ level vertex dispatcher costs $R_{FIFO}$ LUTs, each PE costs $R_{PE}$ LUTs, and the overall limit of LUTs is $R_{limit}$ .", "Then we have the constraints as inequality .", "Notice that $N_{pe}$ must be power of 2 in our project.", "$\\begin{aligned}kN_{pe}^{\\frac{1}{k}+1} \\cdot R_{FIFO} + N_{PE} \\cdot R_{PE} < R_{limit}\\end{aligned}$ If we know $R_{limit}$ , then we can figure out the maximum number of PEs we can have.", "For example on Xilinx Alveo U280, our maximum number of PE is 64.", "Figure: Theoretical performance on a single HBM PC when S v =32S_v = 32 bits, F=100MHzF = 100MHz, BW MAX =13.27GB/sBW_{MAX} = 13.27GB/sLet $S_v = 32$ bits, $F = 100MHz$ and $BW_{MAX} = 13.27GB/s$ , we compute and illustrate $Perf.$ in Figure REF .", "From Figure REF , we can observe that: 1) with an equal number of PEs, the accelerator achieves better performance in graphs with larger $Len_{nl}$ .", "2) for a given graph (fixed $Len_{nl}$ ), increasing the number of PEs can improve the performance of the accelerator, and the performance gains are much higher for graphs with larger $Len_{nl}$ s. Nevertheless, there is a break-point (i.e., 16 PEs), after which the performance will degrade when the number of PEs increases.", "The second observation is somewhat counter-intuitive since in the worst case, increasing the PEs should result in no detrimental effects on performance except for wasting logic resources.", "The underlying reason for such a phenomenon is that we configure the data width (i.e., $DW$ ) of the AXI bus according to the number of PEs.", "A larger number of PEs leads to larger $DW$ according to Equation REF , which further leads to smaller $P_{nl}$ in Equation REF , and consequently results in smaller $BW_{nl}$ and $Perf.$ , if the HBM PC becomes saturated (i.e., the second conditions of Equation REF and REF ).", "Such an observation reveals the fact that when given a fixed number of HBM PCs, there exists a configuration regarding the number of PEs for ScalaBFS to achieve its upper-bound performance.", "We will further investigate this configuration by experiments in Section and verify the model of this section." ], [ "System Evaluation", "In this section, we will give the setups for our experiments in subsection REF , present the resource consumption of ScalaBFS in subsection REF , discuss hybrid-mode processing of the PEs in subsection REF , address the scaling issues in subsection REF , study the HBM bandwidth utilization in subsection REF , and compare the performances of ScalaBFS with other state-of-art systems in subsection REF ." ], [ "Experiment Setups", "Hardware.", "We run our experiments in a COTS PC server, which features two Xeon Silver 4110 CPU running at 2.10GHz, and a Xilinx Alveo U280 accelerator card [7] attached to the PC server via PCIe bus.", "The U280 card has an HBM subsystem (shown in Figure REF ) of 8GB storage capacity.", "The HBM subsystem contains 32 pseudo channels, and provides a theoretical aggregated memory bandwidth up to 460GB/s.", "The Ultrascale+ FPGA in the U280 card contains 9.072MB BRAMs, 34.56MB URAMs and 1304K LUTs for implementing ScalaBFS.", "We implement ScalaBFS with Xilinx Vitis 2019.2.", "The host part of ScalaBFS is coded using OpenCL, and device part is programmed as an RTL kernel using Chisel language [31].", "Thanks to the productivity and flexibility (with which we can easily vary the number of PCs and PEs in ScalaBFS by simply changing the parameters) of Chisel language, the RTL kernel design of ScalaBFS consists of only about 1700 lines of Chisel code.", "Workloads.", "We choose four real-world graphs taken from [32] and ten synthetic RMAT graphs created using the Kronecker Generator (with parameters: A=0.57, B=0.19, C=0.19) from Graph 500 [4] as listed in Table REF to evaluate the performance of ScalaBFS.", "In the names of RMAT graphs, the first number stands for the scale (i.e., the number of vertices) of the graph, and the second number stands for the average degree (i.e., dividing number of edges by the number of vertices).", "For example, “RMAT18-8” represents a synthetic graph with $2^{18}=256K$ vertices and $2^{18}\\times 8 = 2M$ edges.", "For an undirected graph, we convert each of its edges (except for the loop that connects the same vertex) into two directed edges with opposite directions.", "On evaluating the performances, we use the notion of GTEPS (Giga Traversed Edges Per Second), which is computed by dividing the sum of outgoing or incoming neighbor list lengths of all visited vertices by the execution time of BFS.", "If an edge is “visited” more than once, it is counted only once.", "Table: Graph datasets" ], [ "Resource Consumption", "Table REF lists the amounts of FPGA resources consumed by ScalaBFS in typical configurations (place and route results).", "When configured with 64 PEs and using all 32 PCs from HBM (i.e., the third configuration), the Vertex dispatcher of ScalaBFS uses a 3-layer crossbar, each layer of which consists of 16 4$\\times $ 4 crossbars.", "For other configurations, the Vertex dispatcher uses full crossbars.", "Comparing the 16-PC/32-PE configuration with the 32-PC/32-PE configuration in Table REF , we can observe that the Vertex dispatchers of these two configurations consume almost the same amount of resources from U280.", "This is because both these two configurations of ScalaBFS use the same 32$\\times $ 32 full crossbars with 32 PEs.", "There are two subgraphs stored in each HBM PC in the 16-PC/32-PE configuration, while there is only one subgraph stored in each HBM PC in the 32-PC/32-PE configuration.", "We can also observe that the PGs in the 32-PC/32-PE configuration consume more resources than those in the 16-PC and 32-PE configuration.", "This is because the 16-PC/32-PE configuration has only 16 PGs, each of which contains 2 PEs and an HBM reader (totally 16 HBM readers).", "In contrast, the 32-PC/32-PE configuration has 32 PGs, each of which contains one PE and also an HBM reader (totally 32 HBM readers).", "Comparing the 32-PC/32-PE configuration with the 32-PC/64-PE configuration in Table REF , we can observe that the Vertex dispatcher of the former configuration even consumes more resources than the latter configuration.", "This is because the 3-layer 4$\\times $ 4 crossbars of the 32-PC/64-PE configuration consumes 3 (layer) $\\times 16$ (crossbars per layer) $\\times 4 \\times 4$ (FIFOs per crossbar) = 768 FIFOs, while the 32$\\times $ 32 full crossbar of the 32-PC/32-PE configuration consumes $32^2 = 1024$ FIFOs.", "These numbers partially explain the differences in resource consumption of these two configurations.", "From Table REF , we can observe that the PEs of ScalaBFS consume relatively small amounts of resources, even they are capable of hybrid (push-pull) mode processing.", "This is because the PEs reuse the circuits for push and pull modes, and the logic for memory accessing is decoupled from processing.", "Nevertheless, ScalaBFS stops at 64 PEs in the direction of scaling with more PEs in Table REF , since at our current stage of developing, higher numbers (e.g., 128) of PEs still lead to timing problems during the place and route phase.", "We will report performance results of ScalaBFS with more than 64 PEs in our future work." ], [ "Hybrid-mode Processing", "In this section, We leverage the 32-PC/64-PE configuration that yields the highest GTEPS to examine the effect of hybrid-model processing, compared with pull and push models, where two PEs are associated with a PC.", "Figure REF illustrates the absolute throughput, in terms of GTEPS, of three models.", "We have two observations.", "First, the hybrid mode leads to 1.20$\\sim $ 2.10x (or 3.65$\\sim $ 11.52x) throughput improvement over the push (or pull) model, Second, ScalaBFS is able to achieve higher performance improvement when processing denser graphs, as the BFS processing in the hybrid mode effectively reduces the number of unnecessary memory accesses to the neighbor lists, coincident with previous works [33], [3] and [9].", "For example, when processing the dense RMAT22-64 graph, ScalaBFS achieves its peak performance of 19.7 GTEPS.", "Therefore, we let PEs of ScalaBFS work in the hybrid mode in the following experiments." ], [ "Performance Scaling", "For ScalaBFS, there are two scaling directions: increasing HBM PCs and increasing PEs.", "We first study the performance scaling of ScalaBFS by increasing the number of HBM PCs, and then study the performance scaling by increasing the number of PEs.", "Figure: Performances of ScalaBFS using increasing numbers of HBM PCs.Effect of HBM PC.", "We examine the effect of HBM PC with the configuration of a single PE for each PG (thus on each HBM PC).", "Figure REF illustrates the throughput improvement with a increasing number of HBM PCs.", "We observe that ScalaBFS achieves almost linear speedup with respect to the number of HBM PCs, indicating that more the decoupled design of ScaleBFS that allows decoupling memory accessing from processing leads to high efficency and effectiveness when performing BFS.", "Effect of PE.", "We examine the effect of PE within a PG (i.e., on a HBM PC).", "Since a single HBM PC provides only 2 Gbits storage capacity that limits the size of the graph data, we use small synthetic graphs with a scale of $2^{18}$ vertices, as illustrated in Figure REF .", "We observe that more PEs lead to higher performance of ScalaBFS, especially when the vertices of the input graph have higher average degrees.", "However, the performance scaling stops at specific break-points.", "For example, the performances for the RMAT18-8 and RMAT18-16 graph stop at 4 PEs, that for the RMAT18-32 graph stops at 8PEs.", "Comparing Figure REF and Figure REF , we can observe that the trends are similar, except that the break-points show up earlier for graphs whose vertices have smaller numbers of neighboring vertices in real systems than in theory.", "This is because in the theoretical study in Section , we assume the load-balancing status is perfect, i.e., the Vertex dispatcher evenly distributes vertices among the PEs.", "Nevertheless, such an assumption may be compromised by the structure of the input graphs in real systems.", "Whereas, experimental results from Figure REF still suggests that the optimal configuration on the number of PEs configured to each HBM PC in ScalaBFS is about 4 to 8 for sparse graphs, and it is about 8 to 16 for dense graphs.", "By this observation, our 32-PC/64-PE configuration listed in Table REF has not fully exploited the HBM subsystem of U280, since each HBM PC is configured with only two PEs in such configuration.", "Deductively, we should configure about 128 (32$\\times $ 4) PEs or 512 (32$\\times $ 16) PEs for ScalaBFS to achieve the optimal performance in the sparse or dense graph respectively on U280.", "Comparing Figure REF with Figure REF , we can observe that the performance gain from increasing the number of PEs is much less than from increasing the number of HBM PCs.", "For example, increasing the number of PEs from one to two or one to four leads to about 1.68x or 2.48x speedup, even on the (dense) RMAT18-64 graph.", "Such an observation suggests that ScalaBFS should prioritize the scaling in the direction of using more HBM PCs, rather than increasing the number of PEs, especially when the amount of FPGA logic resources is limited.", "It is also the reason why the 32-PC/64-PE configuration of ScalaBFS can achieve the best performance so far on U280.", "Figure: Performances with different numbers of PEs within a single HBM PC" ], [ "HBM Bandwidth Utilization", "We measure the aggregated bandwidth (summation of bandwidths of 32 PCs) of HBM during the execution of BFS on various graph datasets with ScalaBFS.", "For comparison, we introduce the baseline case.", "Contrary to ScalaBFS where the edge data of an input graph are partitioned (shown in Figure REF ) to evenly distributed to all HBM PCs, in the baseline case, the edge data are not partitioned (shown in Figure REF ), and placed in the HBM PCs sequentially (starting from PC0).", "For the data distribution, the HBM readers of the PGs in ScalaBFS only need to access its corresponding PC, while an HBM reader in the baseline case have to cross (possibly) multiple PCs to access the adjacency lists of the vertices assigned to its PG.", "Figure REF reports the aggregated bandwidths of HBM, as well as the performances of both ScalaBFS and the baseline case, when both systems are employed to process the same graphs.", "From Figure REF , we can observe that the baseline cases poorly use the HBM (small aggregated bandwidths) and consequently produce poor performances on BFS.", "The reasons for the small aggregated bandwidths in these baseline cases are that: 1) the PGs have to read the required edge data from remote PCs during computation, which places pressure on the switch network of HBM in Figure REF , and further leads to poor bandwidths as in Figure REF ; 2) the edge data of our chosen graphs are relatively small, and are thus stored in the PCs with small suffixes.", "During execution, such data placement scheme leads to unbalanced accesses and further limits the achievable bandwidths of HBM.", "Compared with the baseline cases, ScalaBFS achieves much higher aggregated bandwidths by evenly distributing the edge data in HBM PCs, and at the same time, confining the PGs to access their corresponding PCs.", "From Figure REF , we can observe that the maximal achieved bandwidth is about 46GB/s.", "Since the RTL of ScalaBFS runs at 90MHz, the burst length is 128bits (2 PEs to each PC), the theoretical upper-bound aggregated bandwidth is about: 90MHz $\\times $ 128/8 Byte $\\times $ 32 PCs $\\approx $ 46.08GB/s (i.e., the measured bandwidth is close to the upper-bound in theory).", "Nevertheless, such bandwidth is still much smaller than the theoretical bandwidth (up to 460GB/s) of HBM in U280.", "The reasons are that: 1) the access pattern of BFS is random and irregular, which limits the achievable bandwidths of each HBM PC using DRAM; 2) the relatively low clock frequency (i.e., 90MHz) limits the maximum achievable bandwidth as computed above." ], [ "Comparison with Other Existing Systems", "Compared with Other FPGA-based Systems.", "In the existing works regarding FPGA-based BFS accelerators, [18] and [19] achieve 2.5 GTEPS by using 16xDDR2 on the Convey machines and Dr. BFS [23] that achieves about 470 MTEPS by using 2xDDR4, ScalaBFS achieves its peak performance of 19.7 GTEPS and yields about 7.9x over [18] and [19], since ScalaBFS exploits HBM's high memory bandwidth and massive FPGA resources for PE-level parallelism.", "AS different accelerators use different FPGA platforms with different numbers of DRAM channels, to be fair, Figure REF compares the BFS performance of ScalaBFS and other similar systems using a single DRAM channel.", "From Figure REF , we can observe that ScalaBFS is much faster than existing systems even in the context of single memory channel performance, which further explains the high BFS performance speedups over existing systems.", "Note that the performance of ScalaBFS on U280 is still far from the theoretical performance of 45.8 GTEPS [9] on bitmap operations (key operations of BFS) using the HMC device that supports the processing-in-memory (PIM) technology.", "Nevertheless, we should notice that the above experiments are conducted on the real FPGA board U280, which limits the performance of ScalaBFS with its fixed amount of physical resources, e.g., the number of HBM PCs and LUTs.", "We believe with the technology progresses, ScalaBFS will continuously achieve higher performance on future FPGA cards, that feature more HBM stacks and more logic resources, with its scalability.", "Compared with GPU-based Systems.", "We further compare the performances of ScalaBFS on U280 with those of Gunrock [15], which is a popular accelerator for graph processing on GPUs, running on Nvidia V100 GPU (model SXM2, configured with 640 tensor cores and 5120 CUDA cores running at 1.53 GHz, 4 HBM2 stacks (totally 64 PCs, 16 GB storage space), consuming 300 watts [34]), when conducting BFS in real-world graphs in Table REF .", "Gunrock also adopts the hybrid processing mode as in ScalaBFS when conducting BFS.", "We fine-tune the parameters of Gunrock during the experiments such that Gunrock achieves its best performance on V100.", "From Table REF , we can observe that for sparse real-world graphs (i.e., PK and LJ) whose vertices have short neighbor lists, the performances of ScalaBFS are very close to those of Gunrock.", "This is because, when conducting BFS in these sparse graphs, the memory accesses towards the HBM are of smaller burst lengths, which makes the HBM to be the bottleneck of the system.", "ScalaBFS uses Algorithm REF where the bitmaps stored in the BRAM absorb large amount of memory accesses to conduct BFS, and thus achieves equivalent performances by using only 32 HBM PCs to those achieved in Gunrock using 64 HBM PCs.", "Moreover, the power efficiencies of ScalaBFS on U280 are about 5.68$\\sim $ 10.19x better than those of Gunrock on V100, due to the low power consumption of U280.", "We read the on-board power meter by using the Xilinx Board Utility (xbutil) [35], which reports 32 watts during the processing of all graphs in Table REF .", "Table: Performance comparison between GunRock and ScalaBFS (32-PC/64-PE configuration)From Table REF , we can also observe that for dense real-world graphs (i.e., OR and HO) whose vertices have long neighbor lists, the performances of ScalaBFS are only about 0.13$\\sim $ 0.22x of the performances of Gunrock.", "The reason is that the dense graphs lead to memory accesses with much larger burst lengths during processing, which makes the processing elements to be the real bottlenecks of the system.", "To this end, Gunrock enjoys the huge memory bandwidth provided by the 64 PCs of the V100's HBM and takes advantage of the large number of high-frequency hardware cores of the V100 to win the battle.", "Such a performance disadvantage of ScalaBFS also suggests that when conducting BFS in dense graphs, we need to build enough processing elements in an FPGA-based BFS accelerator to achieve comparable or even higher performance than GPUs." ], [ "Conclusions and Future Works", "In this paper, we propose ScalaBFS, an open-source BFS accelerator built on the FPGA-HBM platform.", "ScalaBFS features hybrid-mode processing elements that sufficiently exploit the memory bandwidth of HBM, and decoupled circuits for processing and memory accessing that scales its performance with the increasing HBM PCs.", "By fully using the 32 HBM pseudo channels (PCs), the performance of ScalaBFS arrives at 19.7 GTEPS when conducting BFS in real-world and synthetic scale-free graphs on Xilinx Alveo U280 Data Center Accelerator card.", "The future works of ScalaBFS include further fine-tuning of the system, supporting more processing elements on U280, processing larger graphs, and extending itself to a general graph-processing framework that is capable of conducting other graph algorithms." ] ]
2105.11754
[ [ "Avoiding Dense and Dynamic Obstacles in Enclosed Spaces: Application to\n Moving in Crowds" ], [ "Abstract This paper presents a closed-form approach to constrain a flow within a given volume and around objects.", "The flow is guaranteed to converge and to stop at a single fixed point.", "We show that the obstacle avoidance problem can be inverted to enforce that the flow remains enclosed within a volume defined by a polygonal surface.", "We formally guarantee that such a flow will never contact the boundaries of the enclosing volume and obstacles, and will asymptotically converge towards an attractor.", "We further create smooth motion fields around obstacles with edges (e.g.", "tables).", "Both obstacles and enclosures may be time-varying, i.e.", "moving, expanding and shrinking.", "The technique enables a robot to navigate within an enclosed corridor while avoiding static and moving obstacles.", "It was applied on an autonomous robot (QOLO) in a static complex indoor environment, and also tested in simulations with dense crowds.", "The final proof of concept was performed in an outdoor environment in Lausanne.", "The QOLO-robot successfully traversed a marketplace in the center of town in presence of a diverse crowd with a non-uniform motion pattern." ], [ "Introduction", "Robots navigating in human-inhabited environments will encounter disturbances constantly, for instance when pedestrians walk around autonomous delivery robots.", "To avoid collisions, the robot must have a flexible control scheme.", "As the number of obstacles increase and their motion becomes less predictable, the robot needs to reevaluate its path within milliseconds to avoid a crash, while moving actively towards its goal.", "Control using dynamical systems (DS) is ideal to address such situations.", "In contrast to classical path planning, the control law is closed-form, hence requires no re-planning, and can ensure impenetrability of obstacles , .", "DS, thus, offer stability and convergence guarantees in addition to the desired on-the-fly re-activity.", "Sampling algorithms such as probabilistic road map (PRM) or the rapidly exploring random trees (RRT) can find a path in cluttered environments, but they are computationally slow and limited to static environments.", "Online (partial) replanning and elastic-band methods deform locally the path have extended the approaches to dynamic environments , .", "However, this switching comes with lose of global convergence .", "Recent work use customized circuitry on a chips for on-board global sampling and evaluation of all paths .", "Path sampling methods are not suitable for dynamic environments due to computational time.", "We introduce an closed-form algorithm which is able to evaluates and reacts to the environment in real-time.", "With improvements in hardware and computational speed, optimization algorithm such as model predictive control (MPC) have become feasible for on-board use in dynamic path planning and obstacle avoidance .", "MPC has been applied on non-holonomic obstacles for environment which have to be described as a union of convex obstacles / A recent approach uses power diagrams to identify the robot's collision free, convex neighbourhood and an associated, well-known convex optimization problem generates a continuous flow .", "This method is limited to convergence for convex obstacles almost spherical curvature.", "Both methods guarantee convergence in only simple, local environments.", "In the past years, machine learning algorithms have been applied to sensor data to infer data-driven control , but later cannot ensure impenetrability.", "Other approaches use neural networks on a circular representation of crowds to create steering laws .", "Neither is able to ensure convergence.", "Many local methods for obstacle avoidance are based on artificial potential fields , .", "These algorithm often create (topologically) avoidable local minimum .", "Using quadratic potential functions allowed to obtain full convergence around ellipse obstacles .", "Learning methods have be used to tune the hyper-parameters of potential fields to obtain human-inspired behavior for obstacle avoidance of learned motion .", "Navigation functions allow to transform static environments of trees of stars-shapes into simpler environments where all obstacle-trees are reduced to spheres .", "In these transformed world, the artificial potential fields lead to converging vector-fields .", "Other approaches further reduces the space to a point-world and apply navigation function based on harmonic potentials in planar space .", "Navigation function were able to elevate the concept of potential functions to complex environments, but their construction is hard and limited to static environments.", "Harmonic potential functions are particularly interesting as they guarantee that no topologically critical points arise in free space.", "However, as they are hard to find, potential functions are often evaluated numerically .", "Closed-form harmonic potential functions can be generated by approximating the obstacles through linear panels .", "This allows to treat concave obstacles, but is limited to static environments .", "In other attempts, harmonic potential function are found for convex obstacles by using sliding mode between them .", "This can be extended to linearly moving and rotating obstacles, but is limited to two dimensions .", "A closed-form solution of harmonic potential flow around simple obstacles was presented in .", "While the work ensures to avoid moving obstacles, it was restricted to convex obstacles.", "An extension to concave obstacles using discrete, sensor-based representation was offered in .", "Many previous implementations of obstacle avoidance algorithm are simplified to circular world or require high (close to circular) curvature.", "The velocity obstacle approach allows to navigate in dynamic environments by creating analysing the possible velocities with respect to another agent.", "This could is often used in multi-agent cases and could be extended to consider include acceleration and non-holonomic constraints of the agent .", "The velocity obstacle approach often limits the work space, we propose a less conservative approach by evaluating in the frame of the obstacle.", "Dynamic reference points placement to establish state dependent potential field help increase the convergence for potential fields .", "In combination with dynamical systems this could be extended to star-shaped obstacles with smooth boundaries .", "This work allowed a fast and closed-form description for many environments, but it does not handle boundaries of the space.", "Contributions: We address the need to have a reactive obstacle avoidance approach, in closed-form with formal guarantees of non-penetrability, to handle highly dynamic environments and realistic obstacles, such as obstacles with sharp edges.", "To this end, this paper extends our previous work , in which we presented a closed-form obstacle avoidance approach guaranteed to not penetrate smooth concave, albeit star-shaped, obstacles.", "We present three novel theoretical contributions: 1) We invert the obstacle description and by so doing ensure that the robot moves within the enclosed space defined by the boundary, while preserving stability guarantees at an attractor.", "This boundary may represent walls, furniture or joint limits of manipulators (Sec. ).", "2) We extend the approach to handle to non-smooth concave and convex surfaces, i.e.", "with sharp edges.", "The novelty comes from creating a smooth dynamical system around an obstacle without approximation of the curvature (Sec. ).", "3) We show that the approach can be extended to tackle dynamic environments, with obstacles that have deforming shapes (Sec. )", "We validate these contributions with a wheel-chair robot moving in a simulated crowd of pedestrians and in an office world with real furniture.", "To enable real-world implementation of the approach, we propose two additional technical contributions: we introduce a general directional weighting for (ensured) non-trivial summing of vector fields (Sec.", "), and a corresponding gradient descend (Sec. ).", "We also propose new modulation parameters (surface friction and repulsive value) which allow an agent to move slower and further away from obstacles, respectively.", "This results in more cautious behavior, which we believe is desirable when robot moves around pedestrians.", "While the experimental implementation is executed on a autonomous wheel chair, a two dimensional scenario, this work provides a theoretical solution, which is not limited to 2D world and could be applied to higher dimensional spaces." ], [ "Preliminaries", "$\\xi \\in \\mathbb {R}^d$ is the state of the robotic system whose dynamics is governed by a dynamical system (DS) which is autonomous (time invariant) and linear with a single attractor $\\xi ^a$ of the form: $\\mathbf {f}(\\xi ) = - k(\\xi ) \\, (\\xi - \\xi ^a) $ with $k(\\xi ) \\in \\mathbb {R}_{>0}$ a position dependent scaling factor." ], [ "Obstacle Description", "Similar to , we define for each obstacle a continuous distance function $\\Gamma (\\xi ): \\mathbb {R}^d \\setminus \\mathcal {I} \\mapsto \\mathbb {R}_{\\ge 1}$ , which allows to distinguish three regions: $&\\text{Free points:}& \\qquad & \\mathcal {F} = \\lbrace \\xi \\in \\mathbb {R}^d: \\Gamma (\\xi ) > 1 \\rbrace \\nonumber \\\\ &\\text{Boundary points:}& \\qquad & \\mathcal {B} = \\lbrace \\xi \\in \\mathbb {R}^d: \\Gamma (\\xi ) = 1 \\rbrace \\\\&\\text{Interior points:}& \\qquad & \\mathcal {I} = \\lbrace \\xi \\in \\mathbb {R}^d \\setminus ( \\mathcal {F} \\cup \\mathcal {B} ) \\rbrace \\nonumber $ For each obstacle $i$ a reference point inside the obstacle $\\xi ^r \\in \\mathcal {I}$ can be defined.", "The reference direction is the radial direction towards the state of the robotic system with respect to the reference point: ${\\mathbf {r}}(\\xi ) = {\\left( \\xi - \\xi ^r \\right)}/{\\Vert \\xi - \\xi ^r \\Vert }\\quad \\forall \\xi \\in \\mathbb {R}^d \\setminus \\xi ^r$ By construction, the distance function $\\Gamma (\\cdot )$ increases monotonically in radial direction and has a continuous first-order partial derivative ($C^1$ smoothness).", "Here, we define it for general obstacles as: $\\Gamma ^o( \\xi ) = \\left( \\Vert \\xi -\\xi ^r\\Vert / R(\\xi )\\right)^{2p} \\quad \\forall \\xi \\in \\mathbb {R}^d \\setminus \\xi ^r $ with the power $p \\in \\mathbb {N}_+$ with the local radius $R(\\xi ) = \\Vert \\xi ^b - \\xi ^r \\Vert $ as a function of the surface point: $\\xi ^b=b \\mathbf {r} (\\xi ) + \\xi ^r \\quad \\text{such that} \\quad b>0 \\, , \\;\\; \\xi ^b \\in \\mathcal {B} $ The present algorithm uses star shaped obstacles, i.e.", "an obstacle which has a reference point within its boundary, such that the reference direction is linearly independent of the tangent for any point on the surface, i.e.", "$\\langle \\mathbf {r} (\\xi ), \\, \\mathbf {n} (\\xi ) \\rangle > 0 \\hspace{14.22636pt}\\forall \\, \\xi \\in \\mathcal {E}, \\;\\; \\exists \\, \\xi ^r \\in \\mathcal {I} $ where $\\mathbf {n}(\\xi )$ the normal is evaluated at the edge of a surface $\\mathcal {E}$ as in (REF )." ], [ "Directional Weighted Mean", "The weighted summation of vectors can result in a zero sum, e.g.", "two vectors opposing each other with equal weight.", "If the sum of the vectors is a dynamical system for a robot motion, this can lead to local minima and undesired stopping of the agent.", "If the vector is used in an algorithm, the system might not be defined anymore.", "This problem can be resolved by summing in directional space, similarly as in .", "Let us define a general direction as: $\\lbrace \\mathbf {v} \\in \\mathbb {R}^d \\, :\\, \\Vert \\mathbf {v}\\Vert = 1\\rbrace $ The transformation into direction space $\\mathcal {K}^{\\pi }$ is given by: $\\mathcal {K}^{\\pi } = \\lbrace \\kappa \\in \\mathbb {R}^{d-1} \\; : \\; \\Vert \\kappa \\Vert < \\pi \\rbrace $ The direction space is with respect to a reference vector $\\mathbf {r}^0$ which is the first column of the orthonormal transformation matrix: $\\mathbf {R}^0 = \\left[ \\mathbf {r}^0 \\;\\; \\mathbf {e}^0_1 \\;\\; \\hdots \\;\\; \\mathbf {e}^0_{d-1} \\right] $ This allows the transformation into the new basis: $\\hat{\\mathbf {v}}^{i}= \\left( \\mathbf {R}^0\\right)^T \\mathbf {v}^{i} $ To evaluate the direction space, only the projections orthogonal to a reference vector are considered.", "The magnitude of the transformed vector in direction space is equal to the angle between the original vector and the reference vector.", "The transformation of the initial vector $\\mathbf {v}^i$ in the direction-space is: $\\hspace{-7.96674pt}\\kappa ^i(r^0) = \\mathbf {k}(\\mathbf {v}^i, \\mathbf {r}^0) ={\\left\\lbrace \\begin{array}{ll}\\arccos \\left(\\hat{ \\mathbf {v}}^{ i}_1 \\right)\\frac{\\left[ \\hat{\\mathbf {v}}_{2}^{i} \\;\\; .. \\;\\;\\; \\hat{\\mathbf {v}}_{d}^{i} \\right]^T }{\\Vert \\left[ \\hat{\\mathbf {v}}_{2}^{i} \\;\\; .. \\;\\;\\; \\hat{\\mathbf {v}}_{d}^{i} \\right]^T \\Vert } & \\text{if} \\;\\; \\hat{ \\mathbf {v}}^{ i}_1 \\ne 1 \\\\\\mathbf {0}^T & \\text{if} \\;\\; \\hat{ \\mathbf {v}}^{ i}_1 =1\\end{array}\\right.", "}$ The mean is evaluated as a function of the weight $w^i$ of all $N^v$ vectors: $\\bar{\\kappa }= \\sum _{i=1}^{N^v} w^i \\kappa ^i $ The reconstruction into original space is evaluated as: $\\bar{\\mathbf {v}} ={\\left\\lbrace \\begin{array}{ll}\\mathbf {R}^0\\begin{bmatrix}\\cos {\\Vert \\bar{ \\kappa } \\Vert } &\\sin {\\Vert \\bar{\\kappa }} \\Vert \\frac{\\bar{\\kappa }}{ \\Vert \\bar{\\kappa } \\Vert }\\end{bmatrix}^T & \\Vert \\bar{\\kappa } \\Vert \\ne 0\\\\\\mathbf {R}^0\\begin{bmatrix}1 & 0 & .. & 0\\end{bmatrix}^T & \\text{if} \\; \\Vert \\bar{\\kappa } \\Vert = 0\\end{array}\\right.", "}$" ], [ "Intuition", "In the two-dimensional case, this hyper-sphere is a line which represents the angle between the initial DS $f(\\xi )$ and the modulated DS $\\dot{\\xi }_k$ .", "It has a magnitude strictly smaller than $\\pi $ .", "In higher dimensions (e.g.", "three dimensions in Fig.", "REF ) the directional space is a vector-space, where the weighted mean is taken.", "This transformation ensures that any weighted mean calculation of directions outputs a direction as defined in (REF ).", "Figure: Various directions (here three obstacles) are described with respect to a basis direction 𝐫 o \\mathbf {r}^o.(a).", "The directions are transformed to κ\\kappa -space (b) where the weighted mean, κ ¯\\bar{\\kappa }, is obtained.Theorem 1 Consider a unit vector $\\mathbf {r}^0$ as the basis for the projection given in (REF ) and the corresponding reconstruction function defined in (REF ).", "The resulting transformation of unit vector $k(\\mathbf {v}, \\mathbf {r}^0) \\; : \\; \\lbrace \\mathbf {v} \\in \\mathbb {R}^d \\setminus - \\mathbf {r}^0 \\;\\; : \\;\\; \\Vert \\mathbf {v}\\Vert = 1 \\rbrace \\rightarrow \\mathcal {K}^{\\pi }$ defined in (REF ) is a bijection and the basis vector projects to the origin, i.e.", "$\\mathbf {r}^0 \\rightarrow \\mathbf {0}$ .", "${}$ Proof: see Appendix REF ." ], [ "Smooth Vector field", "Any path in a vector field is smooth, if the whole velocity flow is continuous, i.e.", "$\\lim _{\\xi _1 \\rightarrow \\xi _2 } \\mathbf {M}(\\xi _1) \\mathbf {f}(\\xi _1) = \\mathbf {M}(\\xi _2) \\mathbf {f}(\\xi _2)$ since $\\mathbf {f}(\\cdot )$ is continuous it is sufficient for $\\mathbf {M}(\\cdot )$ to be continuous.", "The modulation matrix is a function of three variables: the pseudo normal $\\hat{\\mathbf {n}}(\\xi )$ , the null-direction $\\mathbf {r}(\\xi )$ , and the distance function $\\Gamma (\\xi )$ .", "If all of them are continuously defined outside, continuity of the modulated DS follows.", "Real-time obstacle avoidance is obtained by applying a dynamic modulation matrix to the original DS given in (REF ): $\\dot{\\xi } = \\mathbf {M}(\\xi ) \\mathbf {f}(\\xi )\\qquad \\text{with} \\quad \\mathbf {M} ( \\xi ) = \\mathbf {E}(\\xi ) \\mathbf {D}(\\xi ) \\mathbf {E}(\\xi )^{-1} $ It is composed of a basis matrix: $\\mathbf {E} (\\xi ) =\\left[ {\\mathbf {r} }(\\xi ) \\;\\; \\mathbf {e}_1(\\xi ) \\;\\; .. \\;\\; \\mathbf {e}_{d-1}(\\xi ) \\right]$ which has the orthonormal tangent vectors $\\mathbf {e}_i(\\xi )$ with $i=1..d-1$ evaluated at the boundary point $\\xi ^b$ given in (REF ).", "The basis matrix $\\mathbf {E}(\\xi )$ has full rank (but is not necessarily orthonogonal).", "The diagonal eigenvalue matrix is given as: $\\mathbf {D}(\\xi ) =\\textbf {diag}\\left(\\lambda _r(\\xi ) ,\\lambda _e(\\xi ) ,\\hdots ,\\lambda _{e}( \\xi )\\right)$ Using the existence of $\\Gamma (\\xi )$ see (REF ), that measures the distance to the obstacle's surface, we set: $\\lambda _r(\\xi ) = 1 - {1}/{\\Gamma (\\xi )}^{1/\\rho } \\qquad \\lambda _e(\\xi ) = 1 + {1}/{\\Gamma (\\xi )}^{1/\\rho }$ with the reactivity factor $\\rho \\in \\mathbb {R}_{>0}$ and where $\\Gamma (\\xi )$ from (REF ) is defined and finite." ], [ "Surface Friction Imitation", "The choice of eigenvalues in (REF ) are inspired by the harmonic potential flow, as it is to describe fluid motion.", "The eigenvalues in tangent direction are increasing as is observed in incompressible flow.", "This can have undesired acceleration close to the surface (see Fig.", "REF ).", "We therefore propose to additionally include a notion of friction on the surface, i.e.", "slowing down in tangent direction close to an obstacle ($\\lim _{\\Gamma \\rightarrow 1} \\xi _e \\rightarrow 0$ ).", "A friction factor $\\lambda _f(\\xi )$ ensures the slowing down towards the surface.", "It has to be applied in tangent and reference direction: $\\dot{\\xi }_f = \\lambda _f(\\xi ) \\frac{\\Vert \\mathbf {f}(\\xi ) \\Vert }{\\Vert \\dot{\\xi } \\Vert } \\dot{xi} \\qquad \\text{with} \\quad $ Figure: The isometric inspired eigenvalue (left) induce unexpected accelerations in free space.", "The modulation inspired by surface friction (right) reduces the velocity with decreasing distance to the obstacle." ], [ "Repulsive Eigenvalue", "Positive eigenvalues in reference direction (as described above) decrease the flow towards an obstacle.", "A reversion of the flow, i.e.", "a repulsive effect, can be achieved by negative eigenvalue $\\lambda (\\xi )_r$ .", "This increases the distance by which the robot avoids the obstacle (Fig.", "REF ).", "This differs from from simply increasing obstacle margin, as this does not create any dead space in clustered environments.", "The eigenvalue in normal direction is defined as: $\\lambda _r(\\xi ) ={\\left\\lbrace \\begin{array}{ll}1 - {c_{rep}}/{\\Gamma (\\xi )}^{1/\\rho } & \\text{if} \\; \\langle \\mathbf {f}(\\xi ), \\mathbf {r} \\rangle < 0 \\\\1 & \\text{otherwise}\\end{array}\\right.", "}$ with the repulsive coefficient $c_{rep} \\ge 1$ .", "A repulsive coefficient of one corresponds to no repulsion.", "The undesirable influence of the eigenvalue in reference direction behind the obstacle can be canceled using the tail-effect as introduced by .", "Figure: A repulsion coefficient c rep =1c_{rep}=1 results in strictly positive eigenvalues (top).", "This allows the flow to move closer to the obstacle than with a repulsion coefficient c rep =2c_{rep}=2 where we encounter active repulsion in front of the obstacle (bottom)." ], [ "Reference Point", "The algorithm is based on a reference point $\\mathbf {r}(\\xi )$ .", "It is a point within a star-shaped obstacle, from which in each direction there is only one surface point.", "For convex obstacles, it can be placed anywhere within the hull." ], [ "Multiple Obstacles", "In the presence of multiple obstacles, the nominal DS is modified by taking the weighted mean of the modulated DS $\\dot{\\xi }^o$ created by each obstacle $o=1..N^{o}$ ; separately for the magnitude $\\Vert \\dot{\\xi }^o \\Vert $ and direction $\\mathbf {v}^{\\dot{\\xi }^o}$ .", "The directional weighted mean is evaluated using the algorithm described in Sec.", "REF , with the reference direction being along the initial DS $\\mathbf {f}(\\xi )$ .", "The weight of each obstacle is a decreasing function of the distance function $\\Gamma (\\xi )$ ." ], [ "Inverted Obstacle Avoidance", "Autonomous robot often encounter scenarios where it has boundaries which it can not pass.", "This might be a wall for wheeled robot or a flying drone inside.", "Conversely, it could also be joint limits for a robot arm.", "This problem can be seen as staying within an obstacle, where the boundary of the obstacle represent the limits of the free space." ], [ "Distance Inversion", "The distance function $\\Gamma (\\xi )$ from (REF ) can be evaluated within the obstacle $\\xi \\in \\mathcal {I}$ .", "In the classic obstacle avoidance case, this is of no use, since theoretically the obstacle does never reach the boundary , and practically an emergency control has to be applied if such a case happens.", "For interior points, our boundary function is monotonically decreasing along the radial direction and bounded: $\\frac{\\partial \\Gamma (\\xi )}{\\partial \\mathbf {r}(\\xi )} > 1, \\hspace{19.91684pt}\\Gamma (\\xi ) \\in \\left[0,1\\right[ \\qquad \\forall \\; \\mathcal {I} \\setminus \\xi ^r$ If we consider the obstacle boundary $\\mathcal {X}$ as the description of an enclosing hull, the interior points of the classical obstacle become points of free space of the enclosing hull and vise versa.", "Boundary points stay boundary points.", "Defining the distance function of obstacles as the inverse of the obstacle distance function: $\\Gamma ^w(\\xi ) = 1/\\Gamma ^o = \\left( R(\\xi )/ \\Vert \\xi -\\xi ^r \\Vert \\right)^{2p} \\qquad \\forall \\; \\mathbb {R}^d \\setminus \\xi ^r $ results in a new distance function which again fulfills the condition for the three regions as given in (REF ) since on boundary $\\Gamma =1$ .", "The distance function $\\Gamma $ is now monotonically decreasing along radial direction and reaches infinity at the reference point, i.e.", "$\\lim _{\\xi \\rightarrow \\xi ^r}\\Gamma ^w(\\xi ) \\rightarrow \\infty $ (Fig.", "REF )." ], [ "Modulation Matrix", "The modulation matrix in (REF ) consists of two parts.", "The diagonal eigenvalue matrix defined in (REF ), which can be evaluated using the Inverted distance function from (REF ).", "Conversely, the basis matrix is constant along radial direction, hence it is defined within the free space of enclosing walls except the reference point.", "Theorem 2 Consider a star-shaped enclosing wall in $\\mathbb {R}^d$ with respect to a reference point inside the obstacle $\\xi ^r$ as in (REF ) and a boundary $\\Gamma ^w(\\xi )=1$ as in (REF ).", "Any trajectory $\\lbrace \\xi \\rbrace _t$ , that starts within the free space of an enclosing wall, i.e.", "$ \\Gamma (\\lbrace \\xi \\rbrace _0) > 1$ and evolves on a smooth path according to (REF ), will never reach the wall, i.e.", "$\\Gamma (\\lbrace \\xi \\rbrace _t) > 1, t = 0..\\infty $ and converges towards an attractor in free space $\\xi ^a \\in \\mathcal {F}$ , i.e.", "$\\lim _{t \\rightarrow \\infty }\\xi \\rightarrow \\xi ^a$ .", "Proof: see Appendix REF .", "${}$ $\\blacksquare $ Figure: A smooth flow with full convergence towards the attractor (black star) can be observed within any star-shaped wall with reference point ξ r \\xi ^r (black plus)." ], [ "Guiding Reference Point to Pass Wall Gaps", "In practical scenarios, it may happen that the hull entails gaps or holes (e.g.", "door in a room), through which we may want the agent to enter or exit the space enclosed by the boundary.", "While the obstacle avoidance approach we have develop would allow the agent to escape through the gap, it would lead the agent to slow down massively when approaching the exit as its velocity is meant to vanish when reaching the boundary.", "To counter this effect, we introduce a guiding reference point for boundary obstacles.", "For simplicity, we only look at convex gaps.", "In Sec.", "REF , it was shown that at the center of the obstacle where the reference point and the agent's position align ($\\xi _r = \\xi $ ), there is no influence of the modulation, i.e.", "$\\dot{\\xi } = M(\\xi ) \\mathbf {f}(\\xi ) = \\mathbf {f}(\\xi )$ .", "The newly introduced guiding reference point $\\mathbf {g}_{r} (\\xi )$ changes position during the execution, similarly to a dynamic reference point.", "However, the guiding reference point is a function of the position (not of time).", "Let us define the set of gap points $\\mathcal {G}$ , which includes all points which are enclosed by the connection of the gap edges and the reference point $\\xi _r$ (see Fig.", "REF ).", "The guiding reference point $\\mathbf {g}_{r}$ is equal to the reference point $\\xi _r$ when the robot is far away from the gap, so as to generate the same behavior as in a boundary without gap.", "In the region close to the gap, the guiding reference point $\\mathbf {g}_{r}$ is equal to the position of the evaluation, hence no influence of the modulation.", "In between the two regions, the guiding reference point is projected onto the gap region $\\mathcal {G}$ .", "This can be written as: $\\xi _{r, g} ={\\left\\lbrace \\begin{array}{ll}\\xi _r \\quad &\\text{if} \\;\\;\\; \\Vert \\xi _{c,g} - \\xi \\Vert > \\Vert \\xi _{c,g} - \\xi _r \\Vert \\\\\\xi \\quad &\\text{else if} \\;\\;\\; \\xi \\in \\mathcal {G} \\\\\\text{argmin}_{\\xi _{r,g} \\in \\mathcal {G}} \\Vert \\xi - \\xi _{r,g} \\Vert &\\text{otherwise}\\end{array}\\right.", "}$ Figure: Inside a boundary with gap and the use of a guiding reference point, the dynamical system (left) is not modulated in front of the gap since the Gamma-function reaches infinity (right).", "The edge of the set of gap points 𝒢\\mathcal {G} is limited by blue lines blue lines.", "The influence of the gap is limited by the green circle." ], [ "Non-Smooth Surfaces", "Human designed environment often contain obstacles and enclosing walls with non-smooth surfaces; from edged of tables to corners of rooms and buildings.", "While they could be approximated with a smooth surface of high gradient, this is often not desired.", "These edges are often the most prone to collision as agents need to turn around those.", "A smoothing of the edge would increase this risk.", "An increased hull with smooth boundaries would remove this problem, but add unnecessary conservatism around the edge where certain free parts of the space can not be reached by the autonomous agent.", "Moreover, a high gradient surface used with the obstacle avoidance algorithm we offered in leads to fast change of flow.", "Even though the trajectories are smooth the curvature of the flow can be high which might be undesired as it comes with fast accelerations.", "This may lead to dangerous behavior in the presence of humans or simply exceed the robot's torque limits.", "Instead, we propose an algorithm to avoid obstacles with non-smooth surfaces without smoothing the boundary.", "A polygonial obstacle consists of $i = 1.. N^s$ individually smooth surface planes and of a set of continuous points which form a star shape in $d-1$ , as given in (REF ), such that: $\\mathcal {S}_i = \\lbrace &\\xi , \\hat{\\xi } \\in \\mathcal {B}, \\; \\exists \\, \\mathbf {n} \\, : \\,\\mathbf {n}^T (\\xi - \\hat{\\xi }) = 0 \\rbrace $ Figure: The variables for the evaluation of the pseudo normal 𝐧 ^(ξ)\\hat{\\mathbf {n}}(\\xi ) of a non-smooth star-shaped obstacle are displayed for the first three surfaces.", "For the others only reference point (cross) and the surface normal (red arrow) are visualized.", "The angle ψ w \\psi _{}^w is always evaluated at the edge-point of each surface which is the closest to the surface." ], [ "Pseudo Normal Vector", "The normal to the surface of the obstacle is not defined continuously.", "Hence, the modulation with the basis matrix from (REF ) is not smoothly defined and the modulated system results in a non-continuous flow.", "For this reason, the basis matrix is redefined for non-smooth surfaces as $\\mathbf {E} (\\xi ) =\\left[ {\\mathbf {r} }(\\xi ) \\;\\; \\hat{\\mathbf {e}}_1(\\xi ) \\;\\; .. \\;\\; \\hat{\\mathbf {e}}_{d-1}(\\xi ) \\right] $ with $\\hat{\\mathbf {e}}_i(\\xi )$ for $i=1..d-1$ begin the orthonormal basis to the pseudo normal $\\hat{\\mathbf {n}}(\\xi )$ which is continuously defined in free space.", "The pseudo normal is evaluated as the directional mean (Sec.", "REF ) of all normal vectors $\\mathbf {n}_i(\\xi )$ of the individual surfaces surfaces.", "The orthonormal transformation matrix $\\mathbf {R}^0$ is created based on the reference vector, i.e.", "$\\mathbf {r}^o = \\mathbf {r} (\\xi )$ .", "The weights are chosen such that the pseudo norm is equal to the actual norm on the surface of the obstacle, while away the obstacle the pseudo normal needs to be smoothly defined.", "Here we chose it based on the angle distance to the surface with respect to a edge point .", "At first the vector from the closest surface point $\\mathbf {p}_i^s$ (Fig.", "REF ) to the agent's state is created: $\\mathbf {v}_i^\\angle (\\xi ) = \\xi - \\mathbf {p}_i^s\\;\\;\\; \\text{with} \\;\\;\\;\\mathbf {p}_i^s = \\underset{\\chi \\in \\mathcal {E}_i}{\\mathrm {argmin}} \\Vert \\xi - \\chi \\Vert $ the point being at the edge of a source $\\mathcal {E}_i$ .", "This vector is projected onto the surface plane: $\\hat{\\mathbf {e}}_i(\\xi ) = \\left( \\mathbf {v}^\\angle _i(\\xi ) - \\langle \\mathbf {n}_i (\\xi ), \\, \\mathbf {v}_i^\\angle (\\xi )\\rangle \\, \\mathbf {n}_i (\\xi ) \\right) \\mathrm {sign} \\langle \\mathbf {v}_i^\\angle (\\xi ), \\, \\xi - \\xi ^{r,s}_i \\rangle \\nonumber $ The angle to the plane (Fig.", "REF ) in the range $[0, \\pi ]$ is evaluated as: $\\phi ^{w}_i(\\xi ) = \\arccos \\frac{\\langle \\hat{\\mathbf {e}}_i(\\xi ), \\, \\mathbf {v}_i^\\angle (\\xi ) \\rangle }{\\Vert \\hat{\\mathbf {e}}_i(\\xi )\\Vert \\, \\Vert \\mathbf {v}_i^\\angle (\\xi )\\Vert } \\text{sign} \\langle \\mathbf {n}_i(\\xi ), \\, \\mathbf {v}_i^\\angle (\\xi ) \\rangle $ Which leads to the non-normalized edge weight is evaluated as: $\\tilde{w}^{s}_i(\\xi ) ={\\left\\lbrace \\begin{array}{ll}\\left( \\frac{\\phi ^{w,max}}{\\phi ^{w}_i(\\xi )} \\right)^{p}-1 & \\text{if} \\;\\; \\phi ^w_i(\\xi ) \\in ]0, \\phi ^{w,max}] \\\\0 & \\text{if} \\;\\; \\phi ^w_i(\\xi ) \\notin [0, \\phi ^{w,max}]\\end{array}\\right.", "}$ with the limit angle$\\phi ^{w,max}\\le \\pi $ and the weight power $p \\in \\mathbb {R}$ .", "We choose $\\phi ^{w,max}=\\pi $ and $p=3$ .", "In the case of $\\phi ^w_i(\\xi )=0$ , which implies $\\xi \\in \\mathcal {S}_i$ the weight results from (REF ) in one.", "The final step is the normalization: $w^s_i(\\xi ) ={\\left\\lbrace \\begin{array}{ll}\\tilde{w}^s_i(\\xi ) / \\sum _j \\tilde{w}^s_j(\\xi ) & \\text{if} \\;\\; \\xi \\in \\mathbb {R}^d \\setminus \\mathcal {S}_i \\\\1 & \\text{otherwise}\\end{array}\\right.", "}$" ], [ "Inverted Obstacles", "This continuous pseudo normal and hence the basis matrix $\\mathbf {E}(\\xi )$ for non-smooth surfaces given in (REF ) is not uniquely a function of the direction.", "In order to evaluate the pseudo normal for an Inverted obstacle, current robot state $\\xi $ is mirrored along the reference direction $\\mathbf {r}(\\xi ) = \\xi -\\xi ^r$ to a position $\\xi ^{mir}$ on the other side of the boundary: (REF ) as: $\\xi ^{mir} = \\Gamma (\\xi )^2 \\left( \\xi -\\xi ^r \\right) + \\xi ^r$ The mirrored position allows to evaluate the distance function as described in Sec.", "REF .", "Further, the inverted obstacle is treated as described in Sec. .", "This allows to avoid obstacles in Fig.", "REF .", "Theorem 3 Consider a polygon composed of $N^s$ surfaces as given in $(\\ref {eq:non_smooth_planes})$ or alternatively a inverted polygon as described Sec.", "REF .", "Any trajectory $\\lbrace \\xi \\rbrace _t$ , that starts in free space, i.e.", "$ \\Gamma (\\lbrace \\xi \\rbrace _0) > 1$ and evolves on a smooth path according to (REF ), will never reach the surface, i.e.", "$\\Gamma (\\lbrace \\xi \\rbrace _t) > 1, t = 0..\\infty $ and converge towards an attractor in free space, i.e.", "$\\lim _{t \\rightarrow \\infty }\\xi \\rightarrow \\xi ^a \\in \\mathcal {F}$ .", "Proof: see Appendix REF .", "Figure: Noonsmoth Inverted obstacles representing rooms or boundary conditions." ], [ "Dynamic Environments", "In changing environments (moving or deforming obstacles) the system is modulated with respect to a local, relative velocity such as: $\\dot{\\mathbf {\\xi }} = { \\mathbf {M}(\\xi )} \\left( \\mathbf {f} (\\xi ) - \\dot{ \\xi }_o \\right) + \\dot{\\xi }_o $ The relative velocity consists of the obstacle velocity $\\dot{\\tilde{\\xi }}_v$ and the part from the deforming obstacle $\\dot{\\tilde{\\xi }}_d$ : $\\dot{{\\xi }}_o = \\dot{\\tilde{\\xi }}_v + \\dot{\\tilde{\\xi }}_d$" ], [ "Moving Obstacles", "For moving obstacles the modulation is performed in the obstacle reference frame, and then transformed to the inertial frame : $\\quad \\text{with} \\; \\; \\dot{\\tilde{\\xi }}_v = \\dot{\\mathbf {\\xi }}^{L,o}_v + \\dot{ \\mathbf {\\xi }}^{R,o}_v \\times \\tilde{\\mathbf {\\xi }}_v $ with linear and angular velocity of the obstacle with respect to its center point $\\dot{\\xi }^{L,o}$ and $\\dot{\\xi }^{R,o}$ , respectively and the relative position $\\tilde{\\xi }= \\xi -\\xi ^c$ .", "Avoiding moving obstacles is not a pure modulation of the DS in form of a matrix multiplication anymore, hence topographically critical points, including the attractor, can be displaced." ], [ "Deforming Obstacle", "The environment of the robot can in many cases have not only moving obstacles, but they may also be deformable, e.g.", "changing self-collision limits in joint space, moving body for a surgery robot.", "The deformation can also be the result of estimating an obstacle surface and updating this estimate in real time.", "We introduce a factor related to the surface deformation: $\\dot{\\tilde{\\xi }}_d = \\dot{\\mathbf {\\xi }}^{L,o}_d + \\dot{ \\mathbf {\\xi }}^{R,o}_d \\times \\tilde{\\mathbf {\\xi }}_d$ Where the linear and angular velocity are at the surface position in reference direction.", "If the expansion is now as a model-parameter of the obstacle (e.g.", "expanding heart for a surgery setup) this can be evaluated from the model.", "In environments which are constructed from the surrounding this With moving obstacle and moving obstacle this can be complex to evaluate.", "For a circular object, we have: $\\dot{\\tilde{\\xi }}_d = \\dot{\\mathbf {\\xi }}^{L,o}_d ={\\left\\lbrace \\begin{array}{ll}\\dot{r} \\; \\mathbf {n}(\\xi ) \\hspace{14.22636pt} & \\dot{r}) > 0 \\\\0 & \\text{otherwise}\\end{array}\\right.", "}$ where $\\dot{r}$ is the rate of change in time of the circle radius." ], [ "Impenetrability with Respect to Maximum Velocity", "In critical situations, i.e.", "when the agent is close to an obstacle, the agent needs to move away from the obstacle.", "The relative velocity of the surface must hence point towards the agent.", "When projected onto the normal direction $v_{o, e}$ , this relative velocity is therefore limited to: $v_{o, n} := \\langle \\dot{ \\xi }_o, \\mathbf {n}(\\xi ) \\rangle < v_{max} \\qquad \\forall \\; \\Gamma (\\xi ) \\rightarrow 1$ where $v_{max}$ is the maximum velocity of the agent (see Fig.", "REF ).", "If the agent is close to the obstacle's surface, its final velocity $\\dot{\\xi }_s$ is evaluated as: $\\dot{\\xi }_{s} ={\\left\\lbrace \\begin{array}{ll}v_{o, n} \\mathbf {n}(\\xi ) + \\sqrt{v_{max}^2 -v_{o, n}^2} \\; \\mathbf {e} (\\xi ) & \\text{if} \\;\\;\\langle \\dot{\\xi }, \\mathbf {n}(\\xi ) \\rangle \\frac{v_{max}}{\\Vert \\dot{\\xi }\\Vert } \\Vert < v_{o, n} \\\\v_{max} / \\Vert \\dot{\\xi }\\Vert \\; \\dot{\\xi }& \\text{else if} \\;\\; \\Vert \\dot{\\xi }\\Vert > v_{max} \\\\\\dot{\\xi }& \\text{otherwise}\\end{array}\\right.", "}$ The evaluation is based on the obstacle's surface velocity $\\dot{ \\xi }_o$ and the modulated velocity in dynamic environments $\\dot{\\xi }$ as given in (REF ).", "Figure: In order to comply with a velocity limit of the robot while avoiding a collision with an obstacle of velocity ξ ˙ o \\dot{\\xi }_o, the modulated velocity ξ ˙\\dot{\\xi } might be stretched only in tangent direction to obtain the save velocity command xi ˙ s \\dot{xi}_s.Theorem 4 Consider the dynamic environment including one obstacle which has a surface velocity $\\dot{\\xi }_o$ , including by the obstacle moving or expanding, given as (REF ).", "An agent is moving in this space and has a maximum velocity of $v_{max}$ , further the obstacle's surface velocity is limited by (REF ).", "The agent which starts in free space, i.e.", "$ \\Gamma (\\lbrace \\xi \\rbrace _0) > 1$ and moves according to (REF ), will stay in free space for infinite time, i.e.", "$\\Gamma (\\lbrace \\xi \\rbrace _t) > 1, t = 0..\\infty $ .", "Proof: see Appendix REF ." ], [ "Dynamic Reference Points Placement", "The reference point plays an important role in the behavior of the algorithm.", "In a dynamic environment it has to be evaluated and adapted in real time to allow for the robot to converge to the attractor.", "We propose an algorithm which finds the closest point for each pair of obstacles separately.", "This is then expanded to find the optimal position of more convex environments.", ".", "The algorithm focuses on convex obstacles.", "More star-shapes can be formed through the combination of convex obstacles." ], [ "Pairwise Closest Distance in Direction Space", "First step is to find the closest distance between two (convex) obstacles.", "This is done by moving along the surface of the obstacle in direction space.", "The optimization problem is such that: $\\min \\, \\mathbf {f}_b(\\mathbf {\\xi }) = \\min _{\\Vert \\mathbf {v}_1\\Vert , \\Vert \\mathbf {v}_2\\Vert \\le \\pi /2} \\Vert \\mathbf {s}_1(v_1) - \\mathbf {s}_2(v_2) \\Vert $ where the $\\mathbf {s}_i(\\mathbf {v}_i)$ denotes the surface point in direction $\\mathbf {v}_i$ .", "The direction space of each obstacle is created such that the null-direction points towards the other obstacles center.", "With the gradient step being defined as: $\\mathbf {v}^{k+1} = \\mathbf {v}^{k} + \\alpha _{b, k} \\; \\nabla \\mathbf {f}(\\mathbf {v}^k)$ with $\\mathbf {v} = \\left[ \\mathbf {v}_1^T \\; \\mathbf {v}_2^T \\right]^T $ .", "The direction of the null-space and initial direction as described above leads to a non-convex optimization problem in direction space, where the local minima of (REF ) is the local minimum (see Fig.", "REF for an example).", "Figure: The minimum distance problem for a squared object (with boundary) and an ellipsoid.", "The boundary-reference-point which corresponds to the closes point is marked in red (a) and the corresponding gradient descent problem in direction space (b).In many scenarios, the obstacles can intersect with the enclosing space boundary.", "This case requires the placement such that we have still full convergence.", "We will assume that the curvature of the obstacle $c_o$ is larger than the curvature of the boundary $c_b$ at any position: $c_o (\\xi _1) < c_b(\\xi _2) \\qquad \\forall \\, \\xi _1, \\, \\xi _2 $ The local curvature is given as: $\\quad c_{(\\cdot )} = \\lim _{\\mathbf {d}\\xi \\rightarrow \\mathbf {0}} \\frac{R(\\xi ) - R(\\xi + \\mathbf {d} \\xi )}{\\mathbf {d} \\xi } \\quad \\forall \\xi \\in \\mathcal {X}^b, \\;\\; \\mathbf {d} \\xi ^T \\mathbf {n} (\\xi ) = 0 \\nonumber $ Note for non-circular obstacles condition (REF ) might locally not hold true.", "Especially if the space contains polygon obstacles with local flat regions ($c=0$ ).", "This can lead to locally non-optimal solution in special cases (e.g.", "the obstacle is close to the boundary's edge, where the curvature approaches infinity).", "In general scenarios, the obtained solution show the desired output." ], [ "Intersecting Obstacle Descent", "Two intersecting obstacles have one common reference point.", "It is additionally not on the surface anymore.", "The simplification of the space is therefore not an advantage anymore.", "In the case, that the surface reference point is found in the other obstacle, the optimization problem is changed to fined an optimal common point: $\\min \\, \\mathbf {f}_i(\\mathbf {\\xi }) = \\min _{\\xi \\in \\mathcal {X}_1^b \\cap \\mathcal {X}_2^b} \\frac{1}{1-\\Gamma _1(\\xi )} + \\frac{1}{1-\\Gamma _2(\\xi )}$ A gradient descent step is performed in direction space: $\\mathbf {\\xi }^{k+1} = \\mathbf {\\xi }^{k} + \\alpha _{i, k} \\; \\nabla \\mathbf {f}_i(\\xi )$ The step size can be optimized based on the gradient.", "The optimization problem is convex and point starting within the intersection region will stay inside due to the infinite repulsion at the boundary as $\\Gamma \\rightarrow 1$ .", "The value function of two ellipse obstacles can be found in Fig.", "REF .", "Figure: The sum of the value functions from (a) and (b) allows to fine an optimal common point of the two obstacles (c)" ], [ "Reference Point Evaluation", "At each time step, the gradient descent is performed to evaluate two boundary-reference-point for each pair of obstacles (within the margin region).", "This point is used as initialization at the next time step.", "Since the obstacles can only move a small amount at each time step, the needed gradient descent to the global minimum is small.", "Additionally, as the update rate increased, the movement distance at each time step decreases.", "The reference point is evaluated each time step as a weighted sum of all boundary reference point of an obstacle." ], [ "Dynamic Extension of Hull", "In many scenarios the obstacles can form clusters which cannot be approximated by star-shapes anymore.", "We propose the dynamic extension of the surfaces.", "Since the algorithm can work with deforming hulls (Sec.", "REF ) the shape of the obstacle can be extended to meet the obstacle avoidance requirements.", "The reference point is expanded such that each obstacle forms a convex obstacle with it.", "This can be done by extending the hull with a cone that is tangent to the obstacles surface, but has the tip at the reference point (Fig.", "REF ).", "Figure: Dynamic extension of the hull for an ellipsoid object without margin (a)-(c) and a non-smooth polygon object with constant margin in all directions (d)-(e).Note that the reference point is placed for a globally optimal solution.", "Future work, we will try to extend this work to allow the reference point to adapt locally." ], [ "Mixed Scenarios", "Real world implementation have often a mix of obstacles and boundaries moving.", "Often scenarios can occur where the obstacles intersect with the boundaries (see Fig.", "REF ).", "The reference point of the boundary obstacle only be displaced in a limited region.", "Furthermore, moving it has little influence to the convergence of the obstacles.h The reference point of the obstacles which are intersecting with the wall can be placed inside the wall, i.e.", "$\\Gamma _b(\\xi _{ref}) < 1$ .", "This enforces that all trajectories avoid the obstacles around the same side which also the boundary enforces them and leads to full avoidance of the whole DS.", "This is true for a boundary-obstacle with a positive (local) curvature, i.e.", ": $c_b (\\xi ) > 0 \\qquad \\forall \\, \\xi $ In such a scenario, there is full convergence of the dynamical system towards the attractor.", "Figure: Full convergence towards the attractor in an environment of three obstacles intersecting with the boundary." ], [ "Comparison Algorithms", "The method for the modulation algorithm in dynamic environments as presented in this paper (referred as Dynamic during this section).", "It is compared to , which uses modulation matrix which uses an orthogonal decomposition matrix $\\mathbf {E}(\\xi )$ (referred as Orthogonal) and the potential field algorithm and to , a potential field algorithm (referred as Repulsion).", "The comparison algorithms have been chosen on the base, that it had to be a local-collision avoidance algorithm, and able to handle external hulls.", "The comparison is done in an environment as Fig.", "REF .", "The two ellipse shaped obstacle are randomly changing shape during the motion, and move in a random walk manner.", "The combined maximum expansion velocity and obstacle velocity is lower than the maximum speed of the three agents of $1 m/s$ .", "The Dynamic algorithm is observed to have the most convergences (Tab.", "REF ).", "This is the result of the fact, that it has with the knowledge of the reference point a better knowledge of the environment.", "The Repulsion has a preferable behavior on avoiding collisions.", "This is the result of a conservative behavior around obstacles (moving far away and only approaching them slowly).", "This has has an effect on the distance, but especially the time needed to reach a goal (Tab.", "REF ).", "The mean of the velocity is lower for the Dynamic algorithm.", "This is expected to be the result of the reduced influence behind the obstacles (see Sec.", "REF ).", "The variance of the velocity is similar for the three algorithms.", "Table: The percentage out of 300 trials of success full runs, once where the trajectory collided with an obstacle and runs which ended up in a local minimum.Table: The mean and the standard deviation (after the ±\\pm ) are compared for the three algorithms from the 54 trials where all three agents converged.", "The metrics of distance (d), duration of the run (t), the mean velocity (v ¯\\bar{v}) and the standard deviation of the velocity (σ v \\sigma _v) are listed.Figure: An example of a simulation with the three different algorithms.", "Full convergence is observed due to the additional knowledge about the environment for the algorithm presented in this work." ], [ "Empirical Validation", "The empirical validation is performed first in simulation and then on a real robot platform.", "We use the mobile robot QOLO , see Fig.", "REF .", "The mobile platform QOLO, a semi-autonomous wheel chair, is designed to navigate in pedestrian environments and indoors.", "This platform is hence ideally suited to test our algorithm's ability to avoid rapidly many moving obstacles (pedestrians) and non-convex obstacles (walls, indoor furniture) containing sharp edges (tables, shelf).", "Figure: A picture of the semi-autonomous wheelchair real (left) and the simulated with a person operating it (right).In all our experiments, we assume that QOLO has received a high-level command from its user to set the desired end-goal, i.e.", "the attractor $x^*$ of our nominal DS.", "A video of the experiments is available at www.To Be Uploaded.com." ], [ "Static Environment", "We task QOLO to navigate in an office-like environment.", "The room is a square (5m x 5m), modelled as a boundary obstacle with 4 walls including the closed door.", "Further, there are two clusters of tables which disturb the path of the robot (one at the side and one at the center of the room).", "The robot starts from one of the wall (the simulates the robot entering the room) and is tasked to reach the opposite diagonal position (illustrated with a cross).", "All objects are static and known, the localization is performed using SLAM algorithm.", "The robot gets an input dynamical system which is further modulated as described in this paper.", "We observe following two scenarios: A) QOLO is in the room, and there are two possible paths to go around the center table.", "The dynamical systems splits freely.", "The robot is able to find a path in the two scenarios, even without global planning but only local adaptation.", "The table in the middle as in Fig.", "REF , the dynamical system is split around the center table.", "The robot chooses it's preferred trajectory on the go.", "B) Additionally there is a (static) person in the room, which blocks the center passage.", "Due to the optimization of the reference points, at the common placement at the wall, the robot finds its way around.", "It is to notice, that due to the fact that the modulated flow is tangent on the surface, we can observe behavior in front of and behind the obstacles which is not optimal.", "Figure: Two static office environments including two tables in a rectangular room.", "There exist two passages (a), which is bocked by a static person in (b).", "The reference point (black cross) guide the flow to the attractor (black star).", "The experimental path is displayed in red." ], [ "Dense Crowd (Simulation)", "The robot is navigating in a corridor jointly with in a dense, simulated crowd.", "The motion of the crowd is created according to .", "To navigate successfully, the robot must avoid all pedestrians and the walls.", "A crowd of 200 people is moving along (same and opposite direction as the robot) in a 6 meter wide corridor.", "QOLO is tasked to travel from one end of the corridor to the other end, where we set the attractor of the nominal DS.", "All pedestrians are modeled as circular obstacles with radius of 0.6 meters (Fig.", "REF ).", "At each timestep, the problem is reduced to avoiding a subset of the pedestrians.", "Indeed, due to the density of the crowd, the robot could realistically perceive only a subset of the pedestrians in real-time.", "We, thus, set that only a set number of closest people ($n_{c}=10$ ) are perceived.", "We additionally introduce a circular wall.", "All remaining obstacles are hidden behind this virtual wall.", "The center of the circular wall $\\xi _{c,w}$ is displacement from the position of the robot $\\xi _0$ based on the remaining obstacles: $\\xi _{c,w} = \\xi _0 + \\sum _{i=n_c+1}^{N_{obs}} \\frac{\\xi _{c,i} - \\xi _0}{\\Vert \\xi _{c,i} - \\xi _0\\Vert } e^{-(\\Vert \\xi _{c,i} - \\xi _0\\Vert - r_{p} - m_{q})}$ where $i$ is iterating over the ordered list based on the distance.", "The displacement factor is with respect to the radius of a each pedestrian ($r_p=0.6 m$ ) and the robot radius $m_q$ .", "Note, that for a real implementation sensory distance measurements in the horizontal plane can be used to create the circular boundary.", "The radius of the hull is chosen such that the next closest obstacle $n_c+1$ is fully within the hull.", "The resulting environment has a hull with changing center-position and radius (Fig.", "REF ).", "The reduction of the environment to only sphere obstacles allows to speed up the computational time, since there exist a solution for the closest distance between two spheres.", "This allowed to evaluate the local world at a frequency of over 1000 Hz, even with eleven obstacles in close proximity (including the wall).", "While the agent remains far away from the wall, it is still crucial to guide the robot around local crowds.", "This is done as the reference point is placed at the wall, if a crowd-cluster is touching it (see small cluster at the bottom in Fig.", "REF ).", "Further, fast contraction of the boundary (in case of many people in the surrounding) forces the obstacle to stay away from surrounding obstacles.", "Figure: QOLO moving in a crowd.Figure: The environments with many agents (left) is reduced to a scenario with 10 obstacles and an enclosing hull (right)." ], [ "Quantitative Analysis", "We evaluate the effect of the crowd size on the time it takes for the robot to travel through the corridor.", "The crowd's motion is generated through a realistic simulator created by .", "Each crowd flow is meant to move from one end of the corridor to the other end.", "The crowd has an average velocity of $1 m/s$ .", "When all agents have the same goal, that is when they move towards the same corridor's end, the crowd tends to structure into a uniform flow.", "We run our simulation in such a steady-state crowd-flow, with the QOLO-agent either integrating and moving in the same direction, or moving into the opposite direction of the crowd.", "The robot has a desired velocity of $1 m/s$ .", "We assess the time, speed and distance travelled by the robot when moving with and opposite to the flow, see Fig.", "REF .", "When moving with the flow (i.e.", "In parallel-flow), the crowd has no significant effect on the distance travelled by the agent and its velocity.", "When moving against the crowd (counter-flow), a decrease of the robot's velocity can be observed for crowds denser than 20 agents per 1000 square-meters.", "(The effect on the crowd was be observed, since the effect is small as only a single robot moving against a large crowd.)", "The distance travelled increases significantly for densities above 100 agents per 1000 square-meters.", "As a result, the time mean time needed to reach the goal more than doubles for a crowd-size of hundred people.", "In counter flow scenarios, the standard deviation of the flow increases.", "This results from situations where the robot has to slow down or stop in order to avoid the upcoming agents.", "Figure: The QOLO agent is moving in parallel (red) and opposite direction (blue) to the crowd.", "When the robot moves with the crowd, the density of the crowd has negligible effect.", "When the robot moves in opposite direction, the denser the crowd, the larger the cumulative distance (D) and mean velocity (V) as well as time (T) needed to reach the end of the corridor (T).", "The standard deviation of the velocity (V Std.)", "increases in counter flow for more dense crowds." ], [ "Proof of concept: Outdoor Environment", "A qualitative proof of concept was performed in an outdoor environment.", "We brought the QOLO robot in the center of Lausanne, Switzerland cityAppropriate ethics and safety approval were obtained from the EPFL Ethics board and Policy of Lausanne city.", "A driver was on-board of the robot.", "He could start and stop the robot by pressing on the on-board button, as required for pedestrians' safety.", "A second experimenter was watching the scene and verified the output of the Lidar-vision tracker on a separate laptop.", "This was necessary if needed to guarantee proper safety, in case the detector/tracker dis-functioned..", "The robot was tasked to travel back and forth across a small market place (Fig.", "REF ).", "The location is restricted to pedestrians only, and a total of six streets meet at the crossing.", "This results in a large diversity in both the pedestrian's velocities and directions of movement.", "The path is about 20 meters long and the robot's controller is initialized with a linear DS to reach a goal 20 meters away from the onset position.", "Pedestrians are detected with a camera and LIDAR-based tracker published in as can be seen in Fig.", "REF .", "Recordings were taken on Saturday morning when the market is running full blow and the crowd was the densest.", "The dynamical system is transferred into the controller of the non-holonomic, by placing the evaluation point 0.53 m in front of the center of the wheel-axes (in the reference frame of the robot).", "The dynamical system if evaluated at this point, and the linear velocity is the part parallel in moving direction, the angular velocity is perpendicular.", "The geometry of QOLO is taken into account, by placing a margin around each pedestrian of with 0.5 m. The different method from the simulator was chosen in order to reduce the angular acceleration.", "A total of five runs were executed with the detector.", "The robot reached its goal without intervention of the drive.", "The driver reported that in some cases the robot had high angular acceleration.", "Post-hoc analysis of the video recordings revealed that the crowd density varied with a mean between 5 and 7.5 people per square meter (Fig.", "REF ).", "The agent completed the runs within 115 and 150 seconds.", "No correlation was observed between the density of the crowd and the time taken to reach the goal.", "This is to expect, since many additional factors influence the duration of the run, such as distribution and the motion of the crowd.", "We see this as a successful proof-of-concept test of the obstacle avoidance in real crowd scenarios.", "Compared to the stream-line simulation the crowd motion was more complex, as people would come from all directions and would not group in steady flow.", "Moreover, the crowd included a large diversity in the type of pedestrians, from families with small children to elders.", "Figure: The desired path of the robot in the outdoor environment is the direct line from the initial position of the robot (right) to the target position in the left.Figure: The camera (a) and the LIDAR of the robot are interpreted by the detector (b), which is used for the obstacle avoidance algorithm.Figure: The crowd-density is highly varying during the five successful runs." ], [ "Discussion and Future Work", "The algorithms have theoretical proof and possible application in higher dimensional space.", "Current application has been focusing on the avoidance in two dimensional application such as navigation of mobile robots.", "We plan to apply the algorithm further on in three dimensional task space, but also use it for control in joint space.", "The development of faster transformation such as will make such algorithms more interesting.", "Many sampling methods are not easily salable to higher dimensions, due to increasing computational cost.", "We expect the present method to be of advantage there.", "The concept of dividing dynamical systems into direction and magnitude and the presented method summing vectors to avoid local minima has been used throughout the paper.", "It was used in three areas: (1) creating a smooth pseudo-normal for polygon obstacles and walls, and from that a smooth flow in their presence, (2) it allowed the summation of the flow created by several obstacles without creating local minima and (3) to solve the optimisation problem to find the closest distance for pairs of obstacles.", "We plan to explore the direction space further in motion control, but also learning by demonstration.", "Further having a unified description will allow to extend obstacle avoidance to initially nonlinear motion.", "Since many human-made environments contain non-smooth surfaces (e.g.", "tables, corners of rooms), the solution for providing a smooth flow without creating an additional hull around is of great value for practical implementations.", "The inverted obstacle has proposed itself as a great representation for limitations of mobile robots, such as walls of a room or the local window in dense crowd navigation.", "We will further explore these boundaries to control constraints similar to control barrier functions (CFD).", "Recent development have used such control barrier function in the context of safe learning by demonstration , or reinforcement learning .", "Other than existing methods, we propose closed form solution for star-shaped barriers.", "Future work could include the application of the presented algorithm as barrier function and further extend it to the velocity space.", "The algorithm has been successfully moved on a non-holonomic robot in a static indoor environment and a dynamic outdoor crowd.", "The robot's behavior could however be improved in three areas: Low-level controller: The low-level controller displaced the evaluation of the DS from the center of the robot.", "This resulted into an increased (conservative margin around the robot).", "The design of the controller could be improved by taking into the consideration the local evolution of the DS.", "Environment Recognition: The update rate of the tracker has been around 5 Hz.", "The algorithm is running at a frequency between 50 Hz to 100 Hz, and hence often evaluates with old environment information.", "An intermediate predictor/estimator of how the crowds move in-between could help bridge this gap.", "High-level Planning: The combination of the fast obstacle avoidance controller with a slower, planning algorithms could allow to handle more complex environments, i.e.", "including the avoidance of surrounding (non-starshaped) environments." ], [ "Conclusion", "A dynamical system based algorithm for local navigation under convergence constraint is presented in this paper.", "The present method provides a good solution for local crowd navigation.", "It ensures certain convergence constraint to not only safely navigate, but also reach goal in local scenarios.", "The advantage of presented concepts come from the low complexity and speed of the algorithm.", "This allows the extension to scale to higher dimensions and transfer to various scenarios." ], [ "ACKNOWLEDGEMENT", "We would like to thank the support of Diego Paez-Granados and David Gonon for the running of the experiments.", "Their effort and insight hes helped a lot in the qualitative analysis of these results." ], [ "Proof of Theorem 1", "We show the existence of a bijection in three steps: (1) showing that transformation and reconstruction function are the inverse function of each other, (2) any unit vector is transformed to the vector space domain, and (3) any vector space vector is reconstructed to a unit vector.", "For the transformation and reconstruction functions to generate a bijection, applying one after the other result in the initial value.", "Let us apply the transformation function from (REF ) and the reconstruction function (REF ) to any unit vector (except $\\pm \\mathbf {r}^0$ , as they will be treated separately bellow).", "$\\kappa ^1(r^0) = \\arccos (\\langle \\mathbf {r}^0, \\mathbf {v}^1 \\rangle ) \\frac{\\left( \\hat{\\mathbf {R}}^0\\right)^T \\mathbf {v}^1}{\\Vert \\left( \\hat{\\mathbf {R}}^0\\right)^T {\\mathbf {v}}^1\\Vert }$ with $\\hat{ \\mathbf {R}}^0$ being the partial matrix of $\\mathbf {R}^0$ without the first column.", "We apply it to a single vector (i.e.", "$N^v=1$ ), hence the corresponding weight is $w^i=1$ , as a result we have $\\bar{\\kappa }= \\kappa ^1$ .", "The reconstruction follows as: $\\bar{\\mathbf {v}} & = \\mathbf {R}^0\\begin{bmatrix}\\cos {\\Vert \\bar{ \\kappa } \\Vert } &\\sin {\\Vert \\bar{\\kappa }} \\Vert \\frac{\\bar{\\kappa }}{ \\Vert \\bar{\\kappa } \\Vert }\\end{bmatrix}^T \\\\& = \\mathbf {r}^0 \\cos (\\arccos (\\langle \\mathbf {r}^0 , \\mathbf {v}^i \\rangle )) + \\frac{\\hat{\\mathbf {R}}^0 \\sin (\\arccos (\\langle \\mathbf {r}^0, \\mathbf {v}^1 \\rangle )) \\left( \\hat{\\mathbf {R}}^0\\right)^T {\\mathbf {v}}^1}{\\Vert \\left( \\hat{\\mathbf {R}}^0\\right)^T {\\mathbf {v}}^1\\Vert } \\\\& = \\langle (\\mathbf {r}^0 )^ T, \\mathbf {v}^1 \\rangle \\mathbf {r}^0 + \\hat{\\mathbf {R}}^0 \\left( \\hat{\\mathbf {R}}^0\\right)^T {\\mathbf {v}}^1 = \\mathbf {v}^1$ because of $\\sin (\\arccos (\\langle \\mathbf {r}^0, \\mathbf {v}^1 \\rangle )) = \\sqrt{1-\\langle (\\mathbf {r}^0 )^ T, \\mathbf {v}^1 \\rangle ^2} = {\\Vert \\left( \\hat{\\mathbf {R}}^0\\right)^T {\\mathbf {v}}^1\\Vert }$ .", "This follows from the fact $|\\\\matr R^0 \\mathbf {v}^1|| = 1$ (rotated unit vector).", "The special case of $\\mathbf {r}^0$ is projected on the trivial vector $\\kappa = \\mathbf {0}$ (REF ).", "Finally it is reconstructed (REF as $\\mathbf {r}^0$ ." ], [ "Transformation Domain", "Any possible direction vector is transformed with (REF ) the set of direction space $\\mathcal {K}^\\pi $ given in (REF ).", "The limitation of the set is by definition in (REF ) the magnitude of $\\kappa $ .", "Using (REF ), the maximum of the magnitude can be evaluated: $\\Vert \\kappa \\Vert = \\arccos (\\hat{\\mathbf {v}} _1^i) =\\arccos \\left( (\\mathbf {r}^0)^T \\mathbf {v}^i \\right){\\left\\lbrace \\begin{array}{ll}= \\pi & \\text{if} \\;\\; \\mathbf {v}^i = - \\mathbf {r}^0\\\\< \\pi & \\text{otherwise}\\end{array}\\right.}", "\\nonumber $" ], [ "Reconstruction Domain", "Any vector $\\kappa \\in \\mathcal {K}^\\pi $ reconstructed with (REF ) is in the set of possible unit vectors.", "From (REF ) we can get that $\\Vert \\bar{\\mathbf {v}}\\Vert ^2 = 1$ .", "Hence, we get any unit directional vector but $-\\mathbf {r}^0$ , since $\\cos \\Vert \\bar{\\kappa }\\Vert > -1$ , since $\\Vert \\bar{\\kappa }\\Vert < \\pi $ by definition of the set $\\mathcal {K}^\\pi $ ." ], [ "Applicability of General Proofs", "In a corresponding theorem has been proven for star-shaped obstacles.", "The proof was developed based on the distance function $\\Gamma (\\xi )$ the basis matrix.", "Due to inverting the distance function for enclosing wall obstacles and a continuous definition of the modulation, the proof of star-shaped obstacles applies to the case of enclosing walls." ], [ "Discontinuity Across $\\xi ^r$", "In Sec.", "REF the Inverted distance function $\\Gamma ^w(\\xi )$ was not defined at the reference point, as it reaches an infinite value.", "The continuous definition for the eigenvalue is a unit value, i.e.", "$\\lambda _e(\\xi ^r)=\\lambda _r(\\xi ^r)=1$ , it follows that the diagonal matrix is equal to the identity matrix $D(\\xi ^r) = I$ .", "From this we get for the modulated dynamical system REF : $\\xi =\\xi ^r \\;\\; \\rightarrow \\;\\; \\dot{\\xi }= \\mathbf {E} \\, \\mathbf {D}\\, \\mathbf {E}^{-1} \\mathbf {f}(\\xi ^r)= \\mathbf {E} \\, \\mathbf {I} \\, \\mathbf {E}^{-1} \\mathbf {f}(\\xi ^r) = \\mathbf {f}(\\xi ^r) $ i.e.", "no modulation of the initial DS.", "In fact, this is equivalent to the case far away for a classical obstacle with $\\lim _{\\Vert \\xi -\\xi ^r\\Vert \\rightarrow \\infty }\\Gamma ^o(\\xi ) \\rightarrow \\infty $ .", "Even though the basis matrix $\\mathbf {E}(\\xi )$ is not defined at $\\xi ^r$ ,the DS is continuously defined across this point since the modulation has no effect.", "Note that the trajectory which goes through the reference point $\\xi ^r$ of the inverted obstacle is corresponding to the trajectory which gets stuck in a saddle point for the normal obstacle.", "As a result, there is full convergence for the inverted obstacles." ], [ "Proof of Theorem 3", "We show first that the modulation has full rank and hence that the dynamics does not vanish outside the attractor and that it is smooth." ], [ "Full Rank", "The basis matrix from (REF ) has full rank everywhere outside of the obstacle, if the following condition holds: $\\arccos \\langle \\mathbf {r}(\\xi ), \\, \\hat{\\mathbf {n}}(\\xi ) \\rangle < \\pi /2 $ The angle between the normal to each surface i, $\\mathbf {n}^i(\\xi )$ and the reference direction $\\mathbf {r} (\\xi )$ can be evaluated by defining an vector $\\tilde{\\mathbf {n}}^i(\\xi ) = \\mathbf {n}^i(\\xi ) + \\sum _{j=1}^{d-1} k^e_i\\mathbf {e}_j(\\xi )$ with $\\langle \\mathbf {e}_j(\\xi ), \\, \\mathbf {r}(\\xi ) \\rangle = 0$ , $k^e_i \\in \\mathbb {R}$ such that $\\mathbf {p}_i^s(\\xi ) + \\tilde{\\mathbf {n}}^i(\\xi )$ intersects with $\\xi ^r + k^r \\mathbf {r}(\\xi )$ at $\\mathbf {q}^s_i(\\xi )$ with $k^r \\in \\mathbb {R}$ (Fig.", "REF ).", "This allows to create a triangle spanned by the lines $\\xi $ , $\\mathbf {p}_i^s(\\xi )$ and $\\mathbf {q}^s_i(\\xi )$ , colored in blue in Fig.", "REF .", "Using the associative law of the dot product, the geometry constraint of the blue triangle and (REF ), the maximum angle results in: $\\langle \\mathbf {n}^i, \\mathbf {r} \\rangle = \\langle \\tilde{\\mathbf {n}}^i, \\mathbf {r} \\rangle \\ge \\langle \\xi -\\mathbf {p}_i^s, \\mathbf {r} \\rangle \\ge 0 \\quad \\forall \\, w_i(\\xi ) > 0$ Figure: Visualization of variables used for the weighted directional mean.Hence the directional transformation of (REF ) results in $\\Vert \\kappa _i \\Vert < \\pi /2\\Vert , \\; \\forall \\, w_i(\\xi ) > 0$ .", "Using additionally the triangle equality for vectors: $ \\Vert \\kappa ^1 + \\kappa ^2 \\Vert \\le \\Vert \\kappa ^1 \\Vert + \\Vert \\kappa ^2 \\Vert $ applied to all surface directions, it follows with (REF ) that: $\\Vert \\bar{\\kappa }\\Vert & = \\Vert \\sum _{i=1}^{N^v} w^i \\kappa ^i \\Vert \\le \\sum _{i=1}^{N^v} w^i \\Vert \\kappa ^i \\Vert \\le \\left( \\sum _{i=1}^{N^v} w^i \\right)\\Vert \\max _{i \\; \\text{with} \\; w^i > 0}\\kappa ^i \\Vert \\le \\frac{\\pi }{2} \\nonumber $ Since the basis vector of the directional mean is $\\mathbf {r}(\\xi )$ , with (REF ) condition (REF ) holds true." ], [ "Smooth Vector Field", "The continuous extension across the reference point is defined in Appendix REF and applicable, too.", "The reference direction $\\mathbf {r}(\\xi )$ and the distance function $\\Gamma (\\xi )$ does not have any other discontinuity.", "The pseudo normal $\\hat{\\mathbf {n}}(\\xi )$ is smoothly defined across space.", "Even the case when the edge point with the minimum is switching (REF ) no discontinuity occurs, since the angle will stay the same due to the flat surface." ], [ "Applicability of General Proofs", "Since we have a smooth field of normal vectors $\\mathbf {n}(\\xi )$ , we further need to define any smooth distance function which decreases its value with increasing distance.", "The two properties are sufficient to comply with the proof of Sec.", "REF" ], [ "Proof Theorem 4", "To ensure impenetrability, the three cases described in (REF ) must be considered:" ], [ "Evaluation in Moving Frame: ", "$\\dot{\\xi } = \\dot{\\xi } $ The simplest case comes with no stretching, but the evaluation in the local frame of the moving boundary of the obstacle.", "It follows that the Neuman-boundary condition for impenetrability holds (see also )." ], [ "Contraction within Margin", "$\\dot{\\xi } = v_{max} \\Vert \\dot{\\xi } \\Vert \\dot{\\xi }$ This contraction is only performed, if it results in a normal velocity which is larger than the velocity of the obstacle $\\dot{\\xi }_o$ , i.e.", "the evaluation in the moving frame results $\\langle (\\dot{\\xi }-\\dot{\\xi }_o), \\mathbf {n}(\\xi ) \\rangle \\ge 0$ , hence ensuring impenetrability." ], [ "Contraction in Tangent Direction", "$\\dot{\\xi } = v_{o,n} \\mathbf {n}(\\xi ) + \\sqrt{v_{max}^2 -\\Vert \\dot{\\xi }_n\\Vert ^2} \\; \\mathbf {e} (\\xi ) $ This limited contraction along the normal direction ensures that the velocity in normal direction remains equal to the obstacles' velocity.", "The evaluation of the Neuman boundary condition in the moving frame leads to: $\\langle (v_{o,n} \\mathbf {n}(\\xi ) + \\sqrt{v_{max}^2 -\\Vert \\dot{\\xi }_n\\Vert ^2} \\; \\mathbf {e} (\\xi ) )-\\xi _o, \\mathbf {n}(\\xi ) \\rangle = v_{o,n} - v_{o,n} = 0$ using the definition of (REF ) and the fact that the normal $\\mathbf {n}(\\xi )$ and the tangent $\\mathbf {e}(\\xi )$ are orthogonal." ] ]
2105.11743
[ [ "Fast and Accurate Scene Parsing via Bi-direction Alignment Networks" ], [ "Abstract In this paper, we propose an effective method for fast and accurate scene parsing called Bidirectional Alignment Network (BiAlignNet).", "Previously, one representative work BiSeNet~\\cite{bisenet} uses two different paths (Context Path and Spatial Path) to achieve balanced learning of semantics and details, respectively.", "However, the relationship between the two paths is not well explored.", "We argue that both paths can benefit each other in a complementary way.", "Motivated by this, we propose a novel network by aligning two-path information into each other through a learned flow field.", "To avoid the noise and semantic gaps, we introduce a Gated Flow Alignment Module to align both features in a bidirectional way.", "Moreover, to make the Spatial Path learn more detailed information, we present an edge-guided hard pixel mining loss to supervise the aligned learning process.", "Our method achieves 80.1\\% and 78.5\\% mIoU in validation and test set of Cityscapes while running at 30 FPS with full resolution inputs.", "Code and models will be available at \\url{https://github.com/jojacola/BiAlignNet}." ], [ "Introduction", "Semantic Segmentation is a fundamental vision task that aims to classify each pixel in the images correctly.", "Some earlier approaches , use structured prediction operators such as conditional random fields (CRFs) to refine segmentation results.", "Recent methods for semantic segmentation are predominantly based on FCNs .", "Current state-of-the-art methods , , apply atrous convolutions  at the last several stages of their networks to yield feature maps with strong semantic representation while at the same time maintaining the high resolution, as shown in Fig.", "REF (a).", "Moreover, there are also several methods based on Feature Pyramid Network (FPN)-like , , models which leverage the lateral path to fuse feature maps in a top-down manner.", "In this way, the deep features of the last several layers strengthen the shallow features with high resolution.", "Therefore, the refined features are possible to keep high resolution and meanwhile catch semantic representation, which is beneficial to the accuracy improvement, as shown in Fig.", "REF (b).", "However, both designs are not practical for real-time settings.", "The former methods , require extra computation since the feature maps in the last stages can reach up to 64 times bigger than those in FCNs.", "Meanwhile, the latter one  has a heavier fusion operation in their decoder.", "For example, under a single GTX 1080Ti GPU, the previous model PSPNet  has a frame rate of only 1.6 FPS for $1024 \\times 2048$ input images.", "As a consequence, this is very problematic for many time-critical applications, such as autonomous driving and robot navigation, which desperately demand real-time online data processing.", "There are several specific designed real-time semantic segmentation models , , , handling above issues.", "However, these methods can not achieve satisfactory segmentation results as accurate models.", "The representative works BiSeNets , propose to use two different paths for learning spatial details and coarse context information respectively, shown in Fig.", "REF (c).", "However, they have not explored the interaction between two data flows explicitly.", "We believe such two data flows contain complementary content that can benefit each other.", "In this paper, we propose a new network architecture for real-time scene parsing settings.", "As shown in Fig.", "REF (d), two paths interact with each other through specific design modules before the fusing.", "Motivated by a recent alignment module  which deforms the entire feature map using a learned flow field, we propose a Gated Flow Alignment Module to avoid noise during the fusing since two paths contain diverse information.", "The proposed module is light-weight and can be inserted on each path before fusion.", "The features are aligned to each other through the learned flow fields.", "Moreover, to make the spatial path learn detailed information, we supervise it using the edge-guided hard pixel mining loss  to further improve the performance.", "We term our network as BiAlignNet for short.", "Finally, we evaluate BiAlignNet on two datasets, i.e., Cityscapes  and CamVid .", "The results demonstrate the effectiveness of the proposed components.", "Specifically, our methods improve the origin BiSegNet baseline by about 2% mIoU on the test set of Cityscapes with only 3 FPS drop.", "Our method can achieve 78.5% mIoU while running at 32 FPS on single 1080Ti without acceleration." ], [ "Method", "We present the overall network architecture in Fig.", "REF .", "BiAlignNet includes the following three parts: two pathways, which are Spatial Path and Context Path, and Bidirectional Alignment using Gated Flow Alignment Module to align features in both directions.", "We also specially design the loss functions explained in Sec.", "REF to supervise different sorts of information in two paths at last." ], [ "Spatial Path and Context Path", "We briefly review the spatial and context path in BiSeNet .", "The spatial path is designed to capture the low-level information from the input image.", "We only use shallow layers to preserve spatial details.", "It only consists of three convolution layers with batch normalization and ReLU.", "Each layer has a stride of 2, so the final feature map of the spatial path is $\\frac{1}{8}$ of the input size.", "The context path is responsible for extracting high-level information using a deeper network with more downsample operation.", "For implementation, we employ lightweight backbone DFNet  series for context path.", "Pyramid Pooling Module (PPM) , which has shown a strong ability to catch contextual information, is also added to our model.", "All backbones have four stages of residual blocks, and the first layer of each stage has a stride of 2.", "Thus, the final output of the context path is $\\frac{1}{32}$ of the input size." ], [ "Bidirectional Alignment", "In this section, we present a Gated Flow Alignment Module (GFAM) to align features with each other.", "The original FAM  is proposed to align adjacent features in the decoder.", "However, directly using such a module may lead to inferior results because of the huge semantic gap between the two paths.", "Thus, we plug a gate into the FAM to avoid the noises and highlight the important information.", "Suppose $\\mathbf {F}_s$ is the source feature, and we want to align the information from $\\mathbf {F}_s$ to target feature $\\mathbf {F}_t$ .", "Inspired by original FAM , we first generate a flow field grid $G$ : $G = conv(cat(\\mathbf {F}_s || \\mathbf {F}_t)),$ where $\\mathbf {F}_s$ and $\\mathbf {F}_t$ can be features from the spatial path and the context path respectively, and vice versa.", "The feature map that has a smaller size is bilinearly upsampled to reach the same size as the larger one.", "After flow field grid generation, we adopt a pixel-wise gate to emphasize the important part in current data flow: $\\hat{G} = \\sigma (conv(\\mathbf {F}_t)) \\odot G,$ where $\\hat{G}$ is the gated flow field grid, $\\sigma $ means the sigmoid layer and $\\odot $ represents element-wise product.", "Each position $p$ in target feature $\\mathbf {F}_t$ can be mapped to a position $p^\\prime $ , according to the values in gated flow field grid $\\hat{G}$ .", "Note that the mapping result is not an integer, so the value at $\\mathbf {F}_t(p^\\prime )$ is interpolated by the values of the 4-neighbors $\\mathcal {N}\\left(p^\\prime \\right)$ (top-left, top-right, bottom-left, and bottom-right): $\\hat{\\mathbf {F}_t}\\left(p\\right)=\\sum _{i \\in \\mathcal {N}\\left(p^\\prime \\right)} w_{p} \\mathbf {F}_t(p^\\prime ),$ where $w_{p}$ is the bilinear kernel weights estimated by the distance of warped grid, $\\hat{\\mathbf {F}_t}$ is the target feature aligned with information from source feature $\\mathbf {F}_s$ .", "In BiAlignNet, we take both spatial feature and context feature as source features to align with each other bidirectionally.", "In this way, different pieces of information can complement each other, as shown in the orange box of Fig.", "REF ." ], [ "Loss Function", "The spatial path gives priority to spatial details while context path focuses on high-level semantic context.", "To force spatial path to focus on detailed information, we introduce an edge-guided hard pixel indicator map $d$ to supervise the learning.", "$d$ is predicted from the spatial path feature and normalized by a sigmoid layer.", "Since most of the fine information are concentrated in the boundaries, the edge map $b$ is derived from the segmentation labels through algorithm  which retrieves contours from the binary image.", "We utilize the edge map $b$ to guide the prediction of indicator $d$ .", "As for context path, we use cross-entropy loss with online hard example mining (OHEM) , .", "We jointly supervise two paths with a loss function $L$ : $L = L_{spatial}(d, b, s, g) + L_{context}(s, g),$ where $s$ is the predicted segmentation output of the model and $g$ is the ground truth segmentation labels, and $L_{context}$ is the OHEM loss.", "$L_{spatial}$ is calculated from the following equation.", "$L_{spatial} =\\lambda L_{bce}(d, b) + L_{hard}(s, g, d),$ $L_{hard} = -\\frac{1}{K} \\sum _{i=1}^{N} \\mathbb {1}\\left[s_{i, g_{i}}<t_{K} \\& d_{i}>t_{b}\\right] \\cdot \\log s_{i, g_{i}},$ where $L_{bce}$ is the binary cross-entropy loss for edge-guided hard pixel indicator $d$ , $L_{hard}$ mines the hard pixels with high probability in $d$ and calculate the cross-entropy loss.", "$N$ is the total number of pixels.", "$\\mathbb {1}[x]=1$ if $x=1$ otherwise 0.", "First Eq.", "REF filters the positions that have a higher probability than threshold $t_b$ =0.8 in $d$ .", "Then it picks positions within top $K$ losses, where $t_K$ is the threshold for top $K$ loss.", "Empirically, we set $\\lambda = 25$ to balance the losses in all experiments.", "In this way, the spatial path learns more detailed information during the training." ], [ "Datasets", "We carry out experiments on Cityscapes and Camvid datasets.", "Cityscapes  is a large street scene dataset which contains 2,975 fine-annotated images for training, 500 images for validation and a testing set without annotations of 1,525 images.", "All images in this dataset have a high resolution of 1,024$\\times $ 2,048.", "CamVid  is another road scene dataset.", "This dataset contains 367 training images, 101 validation images and 233 testing images with a resolution of $720 \\times 960$ ." ], [ "Speed and Accuracy Analysis", "Implementation Details.", "Our experiments are done with the PyTorch framework.", "We use stochastic gradient descent (SGD) with a batch size of 16 and a momentum of 0.9 and weight decay of 5e-4.", "The initial learning rate is 0.01 with a \"poly\" learning rate strategy in which the initial rate is multiplied by $\\left(1-\\frac{\\text{ iter }}{\\text{total\\_iter}}\\right)^{0.9}$ .", "As for data augmentation, we randomly horizontally flip the images and randomly resize them with a scale of [0.5, 2.0], and crop images to a size of 1024$\\times $ 1024 (720$\\times $ 720 for CamVid).", "We use the single scale inference and report the speed with one 1080Ti GPU.", "Table: Conclusion" ] ]
2105.11651
[ [ "An explicit Maclaurin series solution to a classic non-autonomous\n abstract evolution equation" ], [ "Abstract It is well known that the non-autonomous scalar differential equation of evolution has a unique solution given by an elementary exponential function.", "In general there is no such analogous solution to the corresponding non-autonomous evolution equation for square matrices.", "In this paper we propose and justify an explicit Maclaurin series solution to a classic non-autonomous abstract evolution equation for bounded linear operators on Banach space." ], [ "Introduction", "Let $a, c \\in {\\mathbb {C}}$ be complex constants.", "The exponential function $r(t) = c e^{a t} \\in {\\mathbb {C}}$ is the unique solution on the interval $t \\in [0, \\infty )$ to the scalar differential equation $dr(t)/dt = a r(t)$ with constant coefficient $a \\in {\\mathbb {C}}$ and initial condition $r(0) = c$ .", "This solution can be generalized to the equation $dr(t)/dt = a(t) r(t)$ where the coefficient $a(t) \\in {\\mathbb {C}}$ is a time-dependent analytic function on some interval $(-b,b)$ .", "In this case the solution is given by $r(t) = c e^{\\int _{[0,t]} a(s) ds}$ for all $t \\in [0,b)$ .", "However a similar generalization does not always hold for the analogous matrix differential equation $dR(t)/dt = A(t) R(t)$ with $R(0) = C \\in {\\mathcal {B}}({\\mathbb {C}}^k)$ where $A(t), R(t), C \\in {\\mathcal {B}}({\\mathbb {C}}^k)$ are bounded linear mappings on ${\\mathbb {C}}^k$ .", "These mappings are represented relative to the standard orthonormal basis of ${\\mathbb {C}}^k$ as square matrices $A(t), R(t), C \\in {\\mathbb {C}}^{k \\times k}$ .", "For instance if $A(t) = \\left[ \\begin{array}{cc}0 & 1 \\\\t & 0 \\end{array} \\right] \\in {\\mathcal {B}}({\\mathbb {C}}^2)$ for all $t \\in [0,\\infty )$ then $e^{\\int _{[0,t]} A(s)ds} & = & \\left[ \\begin{array}{cc}1 & 0 \\\\0 & 1 \\end{array} \\right] + \\left[ \\begin{array}{cc}0 & t \\\\\\frac{t^2}{2} & 0 \\end{array} \\right] + \\frac{1}{2} \\left[ \\begin{array}{cc}\\frac{t^3}{2} & 0 \\\\0 & \\frac{t^3}{2} \\end{array} \\right] + \\frac{1}{6} \\left[ \\begin{array}{cc}0 & \\frac{t^4}{2} \\\\\\frac{t^5}{4} & 0 \\end{array} \\right] + \\cdots \\\\& = & \\left[ \\begin{array}{cc}1 + \\frac{t^3}{4} + \\cdots & t + \\frac{t^4}{12} + \\cdots \\\\\\rule {0cm}{0.5cm} \\frac{t^2}{2} + \\frac{t^5}{24} + \\cdots & 1 + \\frac{t^3}{4} + \\cdots \\end{array} \\right]$ and $\\frac{d}{dt} \\left[ e^{\\int _{[0,t]} A(s)ds} \\right] & = & \\left[ \\begin{array}{cc}\\frac{3t^2}{4} + \\cdots & 1 + \\frac{t^3}{3} + \\cdots \\\\\\rule {0cm}{0.5cm} t + \\frac{5t^4}{24} + \\cdots & \\frac{3t^2}{4} + \\cdots \\end{array} \\right] \\\\& \\ne & \\left[ \\begin{array}{cc}0 & 1 \\\\t & 0 \\end{array} \\right]\\left[ \\begin{array}{cc}1 + \\frac{t^3}{4} + \\cdots & t + \\frac{t^4}{12} + \\cdots \\\\\\rule {0cm}{0.5cm} \\frac{t^2}{2} + \\frac{t^5}{24} + \\cdots & 1 + \\frac{t^3}{4} + \\cdots \\end{array} \\right].$ The differential equation $dR(t)/dt = A(t)R(t)$ is commonly referred to as an evolution equation with evolution coefficient $A(t)$ .", "In this paper we will use Maclaurin series to find an explicit solution to the classic non-autonomous abstract evolution equation for bounded linear operators on Banach space.", "As a motivation for what follows we begin by considering an elementary first order scalar differential equation with a time-dependent coefficient." ], [ "A scalar differential equation with a time-dependent coefficient", "The scalar differential equation $dr(t)/dt = (a_0 + a_1 t) r(t)$ with $a_0, a_1 \\in {\\mathbb {C}}$ , $r(t):[0,\\infty ) \\rightarrow {\\mathbb {C}}$ and $r(0) = 1$ has an elementary solution given by $r(t) = e^{\\int _{[0,t]} (a_0 + a_1 s) ds} = e^{a_0t + a_1t^2/2}$ for all $t \\in [0,\\infty )$ .", "This solution can be found by separation of variables.", "It seems plausible that we might also find the solution by substituting a Maclaurin series $r(t) = 1 + r_1t + r_2t^2 + \\cdots $ where $r_j \\in {\\mathbb {C}}$ for each $j \\in {\\mathbb {N}}$ and equating coefficients of the various powers of $t$ .", "Thus we obtain a system of fundamental equations $r_1 = a_0, \\ 2 r_2 = a_0r_1 + a_1, \\ \\cdots , \\ nr_n = a_0r_{n-1} + a_1r_{n-2}, \\ \\cdots .$ Now, with forward substitution, we can easily read off the solution ${ r_1 = a_0, \\ r_2 = a_0^2/2 + a_1/2, \\ r_3 = a_0^3/6 + a_0a_1/2, \\ r_4 = a_0^4/24 + a_0^2a_1/4 + a_1^2/8, } \\\\& & r_5 = a_0^5/120 + a_0^3a_1/12 + a_0 a_1^2/8, \\ r_6 = a_0^6/720 + a_0^4 a_1/48 + a_0^2 a_1^2/16 + a_1^3/48, \\\\& & \\hspace{28.45274pt} r_7 = a_0^7/5040 + a_0^5a_1/240 + a_0^3a_1^2/48 + a_0a_1^3/48, \\cdots .$ With a little inspired guesswork we can see that $r_n = \\sum _{q = 0}^{\\lfloor n/2 \\rfloor } \\frac{a_0^{n-2q}\\, a_1^q\\,}{(n-2q)!\\, q!\\, 2^q\\,}$ for each $n \\in {\\mathbb {N}}$ where we have used the notation $\\lfloor x \\rfloor $ for the largest integer less than or equal to $x$ .", "If $c = \\max \\lbrace |a_0|, |a_1| \\rbrace $ then we can use an inductive argument to show that there exists some $d > 0$ such that $|r_n| < d c^{\\lfloor n/2 \\rfloor } / \\lfloor n/2 \\rfloor !$ which means the series $r(t) = 1 + r_1t + r_2t^2 + \\cdots $ converges absolutely for all $t \\in [0, \\infty )$ .", "Hence our solution is given by $r(t) & = & 1 + \\sum _{n \\in {\\mathbb {N}}} \\left[ \\sum _{q = 0}^{\\lfloor n/2 \\rfloor } \\frac{a_0^{n-2q}\\, a_1^q\\,}{(n-2q)!\\, q!\\, 2^q\\,} \\right] t^n \\\\& = & 1 + \\sum _{n \\in {\\mathbb {N}}} \\left[\\, \\sum _{\\langle \\mbox{$\\mbox{\\scriptsize $k$}$}, \\mbox{$\\mbox{\\scriptsize $n$}$}\\rangle = n} \\frac{a_0^{k_1}\\, a_1^{k_2}\\,}{k_1!\\, k_2!\\, 2^{k_2}} \\right] t^n$ where ${\\mathbb {N}} = \\lbrace 1,2,\\ldots \\rbrace $ is the set of natural numbers.", "Let ${\\mathbb {N}}-1 = \\lbrace n-1 \\mid n \\in {\\mathbb {N}} \\rbrace $ and define $\\mbox{$\\mbox{$k$}$}= (k_1, k_2 , 0, 0, \\ldots ) \\in ({\\mathbb {N}}-1)^{\\infty }$ and $\\mbox{$\\mbox{$n$}$}= (1,2,3,\\ldots ) \\in ({\\mathbb {N}}-1)^{\\infty }$ .", "Now we identify $k_1 \\cong n-2q$ and $k_2 \\cong q$ and write $\\langle \\mbox{$\\mbox{$k$}$}, \\mbox{$\\mbox{$n$}$}\\rangle = k_1 + 2k_2$ .", "If we define $\\mbox{$\\mbox{$1$}$}= (1,1,1, \\ldots ) \\in ({\\mathbb {N}}-1)^{\\infty }$ and change the dummy variable of summation to $m = \\langle \\mbox{$\\mbox{$k$}$}, \\mbox{$\\mbox{$1$}$}\\rangle = k_1 + k_2$ then $n = m + k_2$ and the solution takes the form $r(t) = 1 + \\sum _{m \\in {\\mathbb {N}}} \\, \\left[ \\sum _{\\langle \\mbox{$\\mbox{\\scriptsize $k$}$}, \\mbox{$\\mbox{\\scriptsize $1$}$}\\rangle = m} \\frac{a_0^{k_1}\\, a_1^{k_2}\\,}{k_1!\\, k_2!\\, 2^{k_2}} \\cdot t^{m + k_2} \\right]$ where the terms are grouped according to the total exponent in the constant coefficient.", "Now it follows from the binomial expansion that $r(t) & = & 1 + \\sum _{m \\in {\\mathbb {N}}} \\left[\\, \\sum _{\\langle \\mbox{$\\mbox{\\scriptsize $k$}$}, \\mbox{$\\mbox{\\scriptsize $1$}$}\\rangle = m} \\frac{(a_0 t)^{k_1}\\, (a_1t^2/2)^{k_2}\\,}{k_1!\\, k_2!\\,} \\right] \\\\& = & 1 + \\sum _{m \\in {\\mathbb {N}}} (a_0t + a_1t^2/2)^m/m!", "\\\\& = & e^{a_0t + a_1t^2/2}$ for all $t \\in [0,\\infty )$ .", "Thus our Maclaurin series solution agrees with the elementary solution." ], [ "A matrix differential equation with a time-dependent coefficient", "The standard solution procedures for the scalar differential equation fail for the analogous linear matrix differential equation.", "However the direct calculation of a power series solution is still valid—albeit with some algebraic complications due to the absence of the commutative property for matrix multiplication.", "Nevertheless the solution to the scalar equation will prove useful when we consider convergence of the series solution to the matrix equation $dR(t)/dt = (A_0 + A_1t) R(t)$ where $A_0, A_1 \\in {\\mathcal {B}}({\\mathbb {C}}^k)$ and $R(0) = I \\in {\\mathcal {B}}({\\mathbb {C}}^k)$ is the unit matrix.", "If we substitute $R(t) = I + R_1t + R_2t^2 + \\cdots $ where $R_j \\in {\\mathcal {B}}({\\mathbb {C}}^k)$ for each $j \\in {\\mathbb {N}}$ and equate coefficients for the various powers of $t$ then we obtain a system of fundamental equations $R_1 = A_0, \\ 2 R_2 = A_0R_1 + A_1, \\ \\cdots , \\ nR_n = A_0R_{n-1} + A_1R_{n-2}, \\ \\cdots .$ Once again we can read off the solution ${R_1 = A_0, \\ R_2 = A_0^2/2 + A_1/2, \\ R_3 = A_0^3/6 + [A_0 A_1/6 + A_1 A_0/3], } \\\\& & R_4 = A_0^4/24 + [A_0^2A_1/24 + A_0A_1A_0/12 + A_1A_0^2/8] + A_1^2/8, \\\\& & R_5 = A_0^5/120 + [A_0^3 A_1/120 + A_0^2A_1A_0/60 + A_0A_1A_0^2/40 + A_1 A_0^3/30] \\\\& & \\hspace{28.45274pt} + [A_0 A_1^2/40 + A_1A_0A_1/30 + A_1^2A_0/15], \\\\& & R_6 = A_0^6/720 \\\\& & \\hspace{28.45274pt} + [A_0^4A_1/720 + A_0^3A_1A_0/360 + A_0^2A_1A_0^2/240 + A_0A_1A_0^3/180 + A_1A_0^4/144] \\hspace{14.22636pt} \\\\& & \\hspace{28.45274pt} + [A_0^2A_1^2/240 + (A_0A_1)^2/180 + A_0A_1^2A_0/90 \\\\& & \\hspace{85.35826pt} + A_1A_0^2A_1/144 + (A_1A_0)^2/72 + A_1^2A_0^2/48] + A_1^3/48, \\cdots .$ The number of terms in $R_n$ forms a Fibonacci sequence because the recursion generates one new term from each of $R_{n-1}$ and $R_{n-2}$ and all such terms are distinct.", "Can we find a formula for the coefficients?", "We make the following general definitions.", "For each $n \\in {\\mathbb {N}}$ and $q \\in \\lbrace 0,1,\\ldots , n-1 \\rbrace $ define the index set $S_{n,q} = \\lbrace \\mbox{$\\mbox{$m$}$}= (m_1,m_2,\\ldots ,m_{n-q}) \\subseteq ({\\mathbb {N}}-1)^{n-q} \\mid m_1 + \\cdots + m_{n-q} = q\\rbrace .$ If we wish to restrict our attention to indices $\\mbox{$\\mbox{$m$}$}\\in \\lbrace 0,1,\\ldots ,p\\rbrace ^{n-q}$ for some $p \\in {\\mathbb {N}}$ then for each $q \\in \\lbrace 0,1,\\ldots ,\\lfloor pn/(p+1) \\rfloor \\rbrace $ the restricted index set of order $p$ is defined by $S_{n,q,p} = \\lbrace \\mbox{$\\mbox{$m$}$}= (m_1,m_2,\\ldots ,m_{n-q}) \\subseteq \\lbrace 0,1,\\ldots ,p\\rbrace ^{n-q} \\mid m_1 + \\cdots + m_{n-q} = q\\rbrace \\subseteq S_{n,q}.$ For each index $\\mbox{$\\mbox{$m$}$}\\in S_{n,q}$ we also define the coefficient $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} = [m_{n-q} +1]^{-1} \\cdot [m_{n-q-1} + m_{n-q} + 2]^{-1} \\cdot \\cdots \\cdot [m_1 + \\cdots + m_{n-q} + (n-q)]^{-1} \\in {\\mathbb {R}}.$ If we write $A_{\\mbox{$\\mbox{\\scriptsize $m$}$}} = A_{m_1} A_{m_2} \\cdots A_{m_{n-q}}$ for each index $\\mbox{$\\mbox{$m$}$}\\in S_{n,q}$ then $m_1+\\cdots +m_{n-q} = q$ is the total index for the product $A_{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ and $n-q$ is the total exponent.", "In the special case where $A(t) = A_0 + A_1t$ and $p=1$ we will show that $R_n = \\sum _{q=0}^{\\lfloor n/2 \\rfloor } \\left[ \\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q,1}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} A_{\\mbox{$\\mbox{\\scriptsize $m$}$}} \\right] \\in {\\mathcal {B}}({\\mathbb {C}}^k)$ and that $\\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q,1}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} = \\frac{1}{(n-2q)!\\, q!\\, 2^q\\,}$ for each $n \\in {\\mathbb {N}}$ and $q \\in \\lbrace 0,1,\\ldots , \\lfloor n/2 \\rfloor \\rbrace $ .", "We will also show that $\\Vert R_n\\Vert \\le d c^{\\lfloor n/2 \\rfloor }/ \\lfloor n/2 \\rfloor !$ for some $d > 0$ and $c = \\max \\lbrace \\Vert A_0\\Vert , \\Vert A_1\\Vert \\rbrace $ .", "Thus the series $R(t) = I + R_1t + R_2t^2 + \\cdots $ converges absolutely for all $t \\in [0,\\infty )$ ." ], [ "Preview of the main results", "Let $X$ be a complex Banach space and let $A:(-b, b) \\rightarrow {\\mathcal {B}}(X)$ be an analytic function where $b \\in (0, \\infty ) \\cup \\lbrace \\infty \\rbrace $ is a positive extended real number and ${\\mathcal {B}}(X)$ is the space of bounded linear operators on $X$ .", "We wish to solve the evolution equation $dR(t)/dt = A(t) R(t)$ for $t \\in [0,b)$ given that $R(0) = I$ where $R(t) \\in {\\mathcal {B}}(X)$ for all $t \\in [0,b)$ and $I \\in {\\mathcal {B}}(X)$ is the identity operator.", "The main results are a general theorem which describes the solution when the evolution coefficient is any analytic function and an important corollary which describes a more specific form of the solution when the evolution coefficient is a polynomial.", "Theorem 1 Let $\\lbrace A_j\\rbrace _{j\\in {\\mathbb {N}}-1} \\in {\\mathcal {B}}(X)$ be an infinite collection of bounded linear operators on the complex Banach space $X$ with $\\limsup _{j \\rightarrow \\infty } \\Vert A_j\\Vert ^{1/j} = 1/b$ for some $b \\in (0,\\infty ) \\cup \\lbrace \\infty \\rbrace $ .", "Define the analytic function $A(t) = A_0 + A_1t + A_2t^2 + \\cdots $ for all $t \\in (-b,b)$ and suppose that $R(t) \\in {\\mathcal {B}}(X)$ satisfies the differential equation $dR(t)/dt = A(t) R(t)$ for all $t \\in [0,b)$ with $R(0) = I$ .", "For each $n \\in {\\mathbb {N}}$ and $q \\in \\lbrace 0,1,\\ldots ,n-1\\rbrace $ let $S_{n,q}$ be the index set defined by (REF ) and for each $\\mbox{$\\mbox{$m$}$}\\in S_{n,q}$ let $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ be the coefficient defined by (REF ).", "Then $R(t) = I + R_1t + R_2t^2 + R_3t^3 + \\cdots $ for all $t \\in [0,b)$ where $R_n = \\sum _{q=0}^{n-1} \\left[ \\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} A_{\\mbox{$\\mbox{\\scriptsize $m$}$}} \\right] \\in {\\mathcal {B}}(X)$ for all $n \\in {\\mathbb {N}}$ .", "$\\hfill \\Box $ Corollary 1 Let $\\lbrace A_j\\rbrace _{j=1}^p \\in {\\mathcal {B}}(X)$ be a finite collection of bounded linear operators on the complex Banach space $X$ .", "Define the polynomial function $A(t) = A_0 + A_1t + \\cdots + A_pt^p$ for all $t \\in {\\mathbb {R}}$ and suppose that $R(t) \\in {\\mathcal {B}}(X)$ satisfies the differential equation $dR(t)/dt = A(t) R(t)$ for all $t \\in [0,\\infty )$ with $R(0) = I$ .", "For each $n \\in {\\mathbb {N}}$ and $q \\in \\lbrace 0,1,\\ldots ,\\lfloor pn/(p+1) \\rfloor \\rbrace $ let $S_{n,q,p}$ be the restricted index set of order $p$ defined by (REF ) and for each $\\mbox{$\\mbox{$m$}$}\\in S_{n,q,p}$ let $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ be the coefficient defined by (REF ).", "Then $R(t) = I + R_1t + R_2t^2 + R_3t^3 + \\cdots $ for all $t \\in [0,\\infty )$ where $R_n = \\sum _{q=0}^{ \\lfloor pn/(p+1) \\rfloor } \\left[ \\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q,p}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} A_{\\mbox{$\\mbox{\\scriptsize $m$}$}} \\right] \\in {\\mathcal {B}}(X)$ for all $n \\in {\\mathbb {N}}$ .", "$\\hfill \\Box $" ], [ "Structure of the paper", "In Section  we review the literature with particular attention to the work on existence and uniqueness of solutions to abstract evolution equations and to earlier work on the development of iterative solution procedures for matrix evolution equations.", "Our first task is to justify the explicit formulæ proposed in Section  for the Maclaurin series solution to evolution equations where the evolution coefficient depends linearly on time.", "This is done in Section  where we restrict our attention to matrix equations.", "However we also use the discussion to develop an intuitive insight into the recursive construction of the series solutions in more general problems.", "In Section  we derive an explicit Maclaurin series solution to the classic abstract evolution equation on Banach space when the evolution coefficient is an analytic function of time.", "We also show how this solution can be reduced to a more specific solution when the evolution coefficient is a polynomial.", "We conclude our general considerations in Section  by relating the Maclaurin series solutions to the original Peano–Baker series solutions.", "In Section  we present two specific examples—a finite-dimensional evolution and a special birth and death process with time-dependent rates on an infinite-dimensional space.", "In Section  we look more closely at the general structure of the coefficients in the special birth and death process.", "In Section  we draw some brief conclusions." ], [ "Literature review", "Let $X$ be a complex Banach space and suppose that $A(t) \\in {\\mathcal {B}}(X)$ for all $t \\in [a,b]$ .", "Consider the differential equation of evolution $d\\mbox{$\\mbox{$x$}$}(t)/dt = A(t) \\mbox{$\\mbox{$x$}$}(t)$ for $\\mbox{$\\mbox{$x$}$}(t) \\in X$ on an interval $[s,t] \\subseteq [a,b] \\subseteq {\\mathbb {R}}$ with the initial condition $\\mbox{$\\mbox{$x$}$}(s) = \\mbox{$\\mbox{$x$}$}_s$ .", "Kato and Tanabe [13] showed that a unique solution exists if $A(t)$ satisfies the conditions: the inverse operator $A(t)^{-1}$ is once strongly differentiable for $t \\in [a,b]$ and for some real positive constants $K$ and $\\alpha $ we have $\\Vert d A(t)^{-1}/dt - d A(s)^{-1}/ds \\Vert \\le K |t-s|^{\\alpha }$ for all $[s,t] \\subseteq [a,b]$ ; and there exist real positive constants $N$ and $\\beta \\in (0,1)$ such that $\\Vert \\partial ( \\lambda I - A(t))^{-1}/ \\partial t \\Vert \\le N |\\lambda |^{\\beta -1}$ for all $t \\in [a,b]$ and all $\\lambda \\in \\rho [A(t)] \\supseteq \\lbrace \\lambda \\in {\\mathbb {C}} \\mid |\\arg (\\lambda )| < \\pi /2 + \\theta \\rbrace $ for some $\\theta \\in (0,\\pi /2)$ where $\\rho [A(t)]$ is the resolvent set for $A(t)$ .", "If these conditions are satisfied the iteration described by Yosida [21], with initial approximation $\\mbox{$\\mbox{$x$}$}_0(t) = e^{(t-s) A(s)} \\mbox{$\\mbox{$x$}$}_s$ and subsequent approximations defined inductively by $\\mbox{$\\mbox{$x$}$}_{n}(t) = e^{(t-s) A(s)} \\mbox{$\\mbox{$x$}$}_s + \\int _{[s,t]} e^{(t-\\sigma )A(\\sigma )} (A(\\sigma ) - A(s)) \\mbox{$\\mbox{$x$}$}_{n-1}(\\sigma ) d\\sigma $ for each $n \\in {\\mathbb {N}}$ , is guaranteed to converge to a solution for all $t \\in [s,b]$ .", "The formal solution is $\\mbox{$\\mbox{$x$}$}(t) = \\left[ e^{(t-s) A(s)} + \\int _{[s,t]} e^{(t-\\sigma )A(\\sigma )}{\\mathfrak {R}}(\\sigma ,s) d\\sigma \\right] \\mbox{$\\mbox{$x$}$}_s$ where ${\\mathfrak {R}}(t,s) = \\sum _{m \\in {\\mathbb {N}}} {\\mathfrak {R}}_m(t,s)$ is defined by ${\\mathfrak {R}}_1(t,s) = \\left\\lbrace \\begin{array}{ll}[A(t) - A(s)]e^{t-s}A(s) & \\mbox{for}\\ t > s \\\\0 & \\mbox{for}\\ t \\le s, \\end{array} \\right.$ and ${\\mathfrak {R}}_{m+1}(t,s) = \\int _{[s,t]} {\\mathfrak {R}}_1(t,\\sigma ){\\mathfrak {R}}_m(\\sigma , s) d\\sigma $ for each $m \\in {\\mathbb {N}}$ .", "If we define the evolutionary operator ${\\mathfrak {U}}(t,s) = e^{(t-s)A(s)} + \\int _{[s,t]} e^{(t - \\sigma )A(\\sigma )} {\\mathfrak {R}}(\\sigma , s) d\\sigma $ it can be shown [21] that the solution to (REF ) is $\\mbox{$\\mbox{$x$}$}(t) = {\\mathfrak {U}}(t,s)\\mbox{$\\mbox{$x$}$}(s)$ for all $[s,t] \\subseteq [a, b]$ .", "Note that the evolutionary operator satisfies ${\\mathfrak {U}}(t,s){\\mathfrak {U}}(s,r) = {\\mathfrak {U}}(t,r)$ for $s \\in [r,t] \\subseteq [a,b]$ with ${\\mathfrak {U}}(t,t) = I$ .", "It is clearly true that ${\\mathfrak {U}}(t,s)$ satisfies the evolutionary operator equation $\\partial {\\mathfrak {U}}(t,s)/ \\partial t = A(t) {\\mathfrak {U}}(t,s)$ subject to the initial condition ${\\mathfrak {U}}(s,s) = I$ .", "For an extended discussion of the conditions under which (REF ) is valid see [21].", "The conditions (REF ) and (REF ) given above by Yosida [21] were developed by Kato [12] and Kato and Tanabe [13] to simplify conditions proposed in the original work by Tanabe [20] in 1960.", "Sobolevski [19], working independently at around the same time, used conditions that were similar to the original conditions proposed by Tanabe and obtained analogous results.", "Other more recent papers have continued this work by investigating various aspects relating to the existence and uniqueness of solutions to evolution equations in abstract spaces.", "See, for instance, the papers by Acquistapace and Terrini [1], Acquistapace [2], Baskakov [5], di Giorgio et al.", "[10], Giscard et al.", "[11] and Latushkin et al.", "[15] and references therein.", "The purpose of this paper is somewhat different.", "In essence we assume that a solution exists and that it can be represented by an absolutely convergent Maclaurin series.", "We substitute the series into the evolution equation and equate coefficients of the various powers of $t$ in order to find a general formula for the unknown coefficients.", "Once these formulæ are known we use a retrospective argument to prove that the series converges and that it does indeed provide a solution on some interval $[0,b)$ .", "Our only explicit assumption is that the evolution coefficient $A(t)$ is analytic on the interval $(-b,b)$ .", "We do not assume that $A(t)^{-1}$ is well defined.", "Nor do we make any explicit assumptions about the resolvent operator $(\\lambda I - A(t))^{-1}$ .", "The Peano–Baker series discussed in a recent paper by Baake and Schlägel [3] is a long-standing computational technique for solution of non-autonomous matrix evolution equations.", "This series dates back to early work by Baker [4] and Campbell [6], [7] and to the original paper by Peano [18].", "For $A \\in {\\mathcal {B}}({\\mathbb {C}}^k)$ we define $\\Vert A\\Vert = \\sup _{\\Vert \\mbox{$\\mbox{\\scriptsize $x$}$}\\Vert \\le 1} \\Vert A \\mbox{$\\mbox{$x$}$}\\Vert $ where $\\Vert \\cdot \\Vert $ is any norm on ${\\mathbb {C}}^k$ and assume that $A(t) \\in {\\mathcal {B}}({\\mathbb {C}}^k)$ for all $t \\in [a,b]$ .", "The Peano–Baker series is given by ${\\mathfrak {U}}(t,s) = \\sum _{n\\in {\\mathbb {N}}-1}{\\mathfrak {U}}_n(t,s)$ where ${\\mathfrak {U}}_0(t,s)) = I$ for all $t \\in [a,b]$ and ${\\mathfrak {U}}_n(t,s) = \\int _{[s,t]} A(\\sigma ) {\\mathfrak {U}}_{n-1}(\\sigma ,s) d\\sigma $ for all $[s,t] \\subseteq [a,b]$ and all $n \\in {\\mathbb {N}}$ .", "Baake and Schlägel [3] show that if $A(t) \\in {\\mathcal {B}}({\\mathbb {C}}^k)$ is continuous for $t \\in [a,b]$ and $\\int _{[a,b]} \\Vert A(t)\\Vert dt < \\infty $ then the series (REF ) is compactly convergent on $[a,b]$ and ${\\mathfrak {U}}(t,s)$ satisfies the matrix form of (REF ).", "An obvious application of this work is in the derivation of the time-dependent distribution of a non-homogeneous continuous-time Markov chain.", "For a detailed discussion we cite the 1998 paper by Massey and Whitt [16].", "We also mention in passing the note by Keeler and Taylor [14] where some specific examples were used to show that the matrix exponential $e^{\\int _{[0,t]} A(s) ds}$ is not a valid solution to the non-autonomous matrix equation of evolution." ], [ "An explicit Maclaurin series solution when the evolution coefficient is a linear function of time", "In this section we justify the Maclaurin series solutions derived in Section .", "We begin by considering the scalar equation (REF ) and the proposed series solution (REF ).", "Lemma 1 If $r(t) = 1 + r_1t + r_2 t^2 + \\cdots $ is a solution to $dr(t)/dt = (a_0 + a_1t)r(t)$ with $r(0) = 1$ then the coefficients $\\lbrace r_n\\rbrace _{n \\in {\\mathbb {N}}}$ are necessarily given by $r_n = \\sum _{q=0}^{\\lfloor n/2 \\rfloor } \\frac{ a_0^{n-2q}\\, a_1^q\\,}{ (n - 2q)!\\, q!\\, 2^q\\,}$ for all $n \\in {\\mathbb {N}}$ .", "$\\hfill \\Box $ Proof.", "The proof is by induction.", "For $n = 1$ we know that $\\lfloor n/2 \\rfloor = 0$ and so (REF ) gives $r_1 = a_0$ .", "For $n=2$ we can see that $\\lfloor n/2 \\rfloor = 1$ and so (REF ) gives $r_1 = a_0^2/2 + a_1/2$ .", "Thus (REF ) is true for $n \\le 2$ .", "Suppose (REF ) is true for $n \\le 2m$ .", "It follows that $r_{2m+1} & = & \\frac{1}{2m+1} \\left[ a_0 \\sum _{q=0}^m \\frac{a_0^{2m-2q} a_1^q}{(2m-2q)!", "q!", "2^q} + a_1 \\sum _{k=0}^{m-1} \\frac{a_0^{2m-1-2k} a_1^k}{(2m-1-2k)!", "k!", "2^k} \\right] \\\\& = & \\frac{a_0^{2m+1}}{(2m+1)!}", "+ \\sum _{q=1}^m \\frac{a_0^{2m+1-2q}a_1^q }{(2m+1)(2m+1-2q)!q!2^q} \\left[ (2m+1-2q) + 2q \\right] \\\\& = & \\sum _{q=0}^{m} \\frac{ a_0^{2m+1-2q} a_1^q}{ (2m+1 - 2q)!", "q!", "2^q}$ which means (REF ) is true for $n = 2m+1$ .", "Now $r_{2m+2} & = & \\frac{1}{2m+2} \\left[ a_0 \\sum _{q=0}^{m} \\frac{ a_0^{2m+1-2q} a_1^q}{ (2m+1 - 2q)!", "q!", "2^q} + a_1 \\sum _{k=0}^m \\frac{a_0^{2m-2k} a_1^k}{(2m-2k)!", "k!", "2^k} \\right] \\\\& = & \\frac{a_0^{2m+2}}{(2m+2)!}", "+ \\sum _{q=1}^m \\frac{a_0^{2m+2-2q}a_1^q}{(2m+2)(2m+2-2q)q!2^q} \\left[ (2m+2-2q) + 2q \\right] \\\\& & \\hspace{284.52756pt} + \\frac{a_1^{m+1}}{(2m+2)m!2^m} \\\\& = & \\sum _{q=0}^{m+1 } \\frac{ a_0^{2m+2-2q} a_1^q}{ (2m+2- 2q)!", "q!", "2^q}.$ Thus (REF ) is also true for $n = 2m+2 = 2(m+1)$ .", "Hence it is true for all $n \\le 2(m+1)$ .", "Therefore, by induction, the formula (REF ) is true for all $n \\in {\\mathbb {N}}$ .", "$\\hfill \\Box $ In order to establish the absolute convergence of the series solution we need to find a suitable bound on the magnitude of the coefficients.", "We have the following result.", "Lemma 2 If $r(t) = 1 + r_1t + r_2 t^2 + \\cdots $ is a solution to $dr(t)/dt = (a_0 + a_1t)r(t)$ with $r(0) = 1$ and we define positive constants $c = \\max \\lbrace |a_0|, |a_1| \\rbrace $ and $d = \\max _{n < 2c} \\lbrace r_n \\lfloor n/2 \\rfloor !", "/c^{\\lfloor n/2 \\rfloor }\\rbrace $ then $|r_n| \\le d c^{ \\lfloor n/2 \\rfloor }/ \\lfloor n/2 \\rfloor !$ for each $n \\in {\\mathbb {N}}$ .", "$\\hfill \\Box $ Proof.", "Once again the proof is by induction.", "Choose $m \\in {\\mathbb {N}}$ such that $m + 1/2 > c$ .", "Since $c = \\max \\lbrace |a_0|, |a_1| \\rbrace $ and $d = \\max _{n < 2c} \\lbrace |r_n| \\lfloor n/2 \\rfloor !", "/c^{\\lfloor n/2 \\rfloor }\\rbrace $ it follows that (REF ) is true for $n \\le 2m$ .", "Now choose $k \\ge m$ and assume that (REF ) is true for all $n \\le 2k$ .", "When $n = 2k+1$ we have $|r_{2k+1}| & \\le & \\frac{1}{2k+1} \\left[ |a_0||r_{2k}| + |a_1||r_{2k-1}| \\right] \\\\& \\le & \\frac{1}{2k+1} \\left[ c \\cdot dc^k/ k!", "+ c \\cdot dc^{k-1}/ (k-1)!", "\\right] \\\\& \\le & dc^k/(2 \\cdot k!)", "+ dc^k/ (2 \\cdot k!)", "\\\\& = & dc^k/k!$ where we have used the fact that $2k+1 \\ge 2m+1 > 2c$ .", "Therefore (REF ) is also true for $n = 2k+1$ .", "When $n = 2k+2$ we can see that $|r_{2k+2}| & \\le & \\frac{1}{2k+2} \\left[ |a_0||r_{2k+1}| + |a_1||r_{2k}| \\right] \\\\& \\le & \\frac{1}{2k+2} \\left[ c \\cdot dc^k/ k!", "+ c \\cdot dc^k/ k!", "\\right] \\\\& = & dc^{k+1}/(k+1)!.$ Therefore (REF ) is also valid for $n = 2k+2 = 2(k+1)$ .", "It follows by induction that (REF ) is true for all $n \\in {\\mathbb {N}}$ .", "$\\hfill \\Box $ Lemma REF shows that the series (REF ) converges absolutely for all $t \\in [0,\\infty )$ .", "Thus (REF ) can be differentiated term by term.", "Hence (REF ) is a solution to (REF ).", "We will now consider the matrix equation (REF ).", "The derivation and justification of a formula for the Maclaurin series is necessarily more complex for the matrix equation than it was for the scalar equation.", "Nevertheless we will show that the recursive definition of the coefficients and the relationship of the matrix series (REF ) to a corresponding scalar series are key steps in understanding the solution process.", "For each $n \\in {\\mathbb {N}}$ our calculation of $R_n$ is based on systematically identifying all terms in the form $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}}A_{\\mbox{$\\mbox{\\scriptsize $m$}$}} = \\pi _{(m_1,\\ldots ,m_{n-q})}A_{m_1}\\cdots A_{m_{n-q}}$ that contribute to $R_n$ .", "We will do this by collecting these terms according to the total index $m_1+\\cdots +m_{n-q} = q$ in the product $A_{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ .", "For each $n \\in {\\mathbb {N}}$ and each $q \\in \\lbrace 0,1,\\ldots , \\lfloor n/2 \\rfloor \\rbrace $ we assume that $S_{n,q,1}$ and $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ are defined by (REF ) and (REF ) respectively.", "We also define $R_{n,q} = \\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q,1}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} A_{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ and write $R_n = \\sum _{q=0}^{\\lfloor n/2 \\rfloor } R_{n,q} = \\sum _{q=0}^{\\lfloor n/2 \\rfloor } \\left[ \\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q,1}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} A_{\\mbox{$\\mbox{\\scriptsize $m$}$}} \\right].$ The key to calculating the individual terms in $R_n$ is to consider their ancestry—the lineage defined by the recursive relationships that express $R_n$ in terms of the lower order coefficients.", "In this case there are only two possibilities—either $m_1=0$ and $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}}A_{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ is generated by a term in $R_{n-1}$ or $m_1 = 1$ and $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}}A_{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ is generated by a term in $R_{n-2}$ .", "In the first case the term $\\pi _{(0,m_2,\\ldots ,m_{n-q})} A_0A_{m_2} \\cdots A_{m_{n-q}}$ with $0 + m_2 + \\cdots + m_{n-q} = q$ is generated from the term $\\pi _{(m_2,\\ldots ,m_{n-q})} A_{m_2} \\cdots A_{m_{n-q}}$ with $m_2 + \\cdots + m_{n-q} = q$ according to the formula $\\pi _{(0,m_2,\\ldots ,m_{n-q})} A_0A_{m_2} \\cdots A_{m_{n-q}} = \\frac{A_0}{n} \\cdot \\pi _{(m_2,\\ldots ,m_{n-q})} A_{m_2} \\cdots A_{m_{n-q}}.$ The condition $0 + m_2 + \\cdots + m_{n-q} = q$ means $\\mbox{$\\mbox{$m$}$}= (0,m_2,\\ldots ,m_{n-q}) \\in S_{n,q}$ and the term $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}}A_{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ contributes to $R_{n,q}$ .", "If we write $\\mbox{$\\mbox{$m$}$}^{\\prime } = (m_1^{\\prime },\\ldots ,m_{n-1-q}^{\\prime }) = (m_2,\\ldots ,m_{n-q})$ then the condition $m_1^{\\prime } + \\cdots + m_{n-1-q}^{\\prime } = q$ means $\\mbox{$\\mbox{$m$}$}^{\\prime } \\in S_{n-1,q}$ and the term $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}^{\\prime }} A_{\\mbox{$\\mbox{\\scriptsize $m$}$}^{\\prime }}$ contributes to $R_{n-1,q}$ .", "In the second case the term $\\pi _{(1,m_2,\\ldots ,m_{n-q})} A_1A_{m_2} \\cdots A_{m_{n-q}}$ with $1 + m_2 + \\cdots + m_{n-q} = q$ is generated from the term $\\pi _{(m_2,\\ldots ,m_{n-q})} A_{m_2} \\cdots A_{m_{n-q}}$ with $m_2 + \\cdots + m_{n-q} = q-1$ according to the formula $\\pi _{(1,m_2,\\ldots ,m_{n-q})} A_1A_{m_2} \\cdots A_{m_{n-q}} = \\frac{A_1}{n} \\cdot \\pi _{(m_2,\\ldots ,m_{n-q})} A_{m_2} \\cdots A_{m_{n-q}}.$ The condition $1 + m_2 + \\cdots + m_{n-q} = q$ means $\\mbox{$\\mbox{$m$}$}= (1,m_2,\\ldots ,m_{n-q}) \\in S_{n,q}$ and the term $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}}A_{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ contributes to $R_{n,q}$ .", "If we write $\\mbox{$\\mbox{$m$}$}^{\\prime } = (m_1^{\\prime },\\ldots ,m_{n-2-[q-1]}^{\\prime }) = (m_2,\\ldots ,m_{n-q})$ then the condition $m_1^{\\prime } + \\cdots + m_{n-2-[q-1]}^{\\prime } = q-1$ means $\\mbox{$\\mbox{$m$}$}^{\\prime } \\in S_{n-2,q-1}$ and the term $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}^{\\prime }} A_{\\mbox{$\\mbox{\\scriptsize $m$}$}^{\\prime }}$ contributes to $R_{n-2,q-1}$ .", "In general we calculate the term $\\pi _{(m_1,\\ldots ,m_{n-q})} A_{m_1} \\cdots A_{m_{n-q}}$ where $(m_1,\\ldots ,m_{n-q}) \\in S_{n,q,1}$ using the recursive relationship $\\pi _{(m_1,\\ldots ,m_{n-q})} A_{m_1} \\cdots A_{m_{n-q}} = \\frac{A_{m_1}}{n} \\cdot \\pi _{(m_2,\\ldots ,m_{n-q})} A_{m_2} \\cdots A_{m_{n-q}}$ where $(m_2,\\ldots ,m_{n-q}) \\in S_{n-1-m_1,q-m_1}$ .", "If we apply this recursive relationship again we obtain $\\pi _{(m_1,\\ldots ,m_{n-q})} A_{m_1} \\cdots A_{m_{n-q}} = \\frac{A_{m_1}}{n} \\cdot \\frac{A_{m_2}}{n-m_1+1} \\cdot \\pi _{(m_3,\\ldots ,m_{n-q})} A_{m_3} \\cdots A_{m_{n-q}}$ where $(m_3,\\ldots ,m_{n-q}) \\in S_{n-2-m_1-m_2,q-m_1-m_2}$ .", "An inductive argument ultimately shows that $\\pi _{(m_1,\\ldots ,m_{n-q})} A_{m_1} \\cdots A_{m_{n-q}} = \\frac{A_{m_1}}{n} \\cdot \\frac{A_{m_2}}{n-1-m_1} \\cdot \\cdots \\cdot \\frac{A_{m_{n-q}}}{n - (n-q+1) -m_1 - \\cdots - m_{n-q+1}}.$ Thus we obtain $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} = n^{-1} \\cdot [n - 1- m_1]^{-1} \\cdot \\cdots \\cdot [n - (n-q+1) - m_1 - \\cdots -m_{n-q+1}]^{-1}.$ for $\\mbox{$\\mbox{$m$}$}\\in S_{n,q}$ .", "Since $m_1 + \\cdots + m_{n-q} = q$ we can rewrite this expression in the form $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} = [m_1 + \\cdots + m_{n-q} + (n-q)]^{-1} \\cdot [m_2 + \\cdots + m_{n-q} +(n-q-1)]^{-1} \\cdot \\cdots \\cdot [m_{n-q}+1]^{-1}.$ This justifies the definitions of $S_{n,q,1}$ and $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ given respectively in (REF ) and (REF ).", "Remark 1 In the above argument we have assumed that $A(t) = A_0 + A_1t$ .", "Nevertheless we can can apply essentially the same argument more generally when $A(t) = A_0 + A_1t + A_2t^2 + \\cdots $ is any analytic function represented by a Maclaurin series on some interval $(-b,b)$ .", "The idea that $R_n$ can be determined explicitly by recursively considering the ancestry of each individual term in $R_{n,q}$ is a key intuitive idea.", "$\\hfill \\Box $ The next example illustrates the relevant formulæ in a particular case.", "Example 1 As a demonstration of the formulæ (REF ) and (REF ) when $A(t) = A_0 + A_1t$ we set $n = 7$ and $q=2 \\Rightarrow n-q=5$ .", "We list the indices $\\mbox{$\\mbox{$m$}$}\\in S_{7,2,1}$ in numerical order, $S_{7,2,1}\\hspace{4.2679pt} = \\left[ \\begin{array}{c}\\mbox{$\\mbox{$m$}$}_{7,2,1,1} \\\\\\mbox{$\\mbox{$m$}$}_{7,2,1,2} \\\\\\vdots \\\\\\mbox{$\\mbox{$m$}$}_{7,2,1,10} \\end{array} \\right] = \\left[ \\begin{array}{c}(0, 0, 0, 1, 1) \\\\(0, 0, 1, 0, 1) \\\\(0, 0, 1, 1, 0) \\\\(0, 1, 0, 0, 1) \\\\(0, 1, 0, 1, 0) \\\\(0, 1, 1, 0, 0) \\\\(1, 0, 0, 0, 1) \\\\(1, 0, 0, 1, 0) \\\\(1, 0, 1, 0, 0) \\\\(1, 1, 0, 0, 0) \\end{array} \\right]$ and then list the corresponding product terms and their coefficients, $A_0^3A_1^2: \\quad \\pi _{[0,0,0,1,1]} & = & (2 \\cdot 4 \\cdot 5 \\cdot 6 \\cdot 7)^{-1} = 1/1680 \\\\A_0^2A_1A_0A_1: \\quad \\pi _{[0,0,1,0,1]} & = & (2 \\cdot 3 \\cdot 5 \\cdot 6 \\cdot 7)^{-1} = 1/1260 \\\\A_0^2A_1^2A_0: \\quad \\pi _{[0,0,1,1,0]} & = & (1 \\cdot 3 \\cdot 5 \\cdot 6 \\cdot 7)^{-1} = 1/630 \\\\A_0A_1A_0^2A_1: \\quad \\pi _{[0,1,0,0,1]} & = & (2 \\cdot 3 \\cdot 4 \\cdot 6 \\cdot 7)^{-1} = 1/1008 \\\\(A_0A_1)^2A_0: \\quad \\pi _{[0,1,0,1,0]} & = & (1 \\cdot 3 \\cdot 4 \\cdot 6 \\cdot 7)^{-1} = 1/504 \\\\A_0 A_1^2 A_0^2: \\quad \\pi _{[0,1,1,0,0]} & = & (1 \\cdot 2 \\cdot 4 \\cdot 6 \\cdot 7)^{-1} = 1/336 \\\\A_1A_0^3A_1: \\quad \\pi _{[1,0,0,0,1]} & = & (2 \\cdot 3 \\cdot 4 \\cdot 5 \\cdot 7)^{-1} = 1/840 \\\\A_1A_0^2A_1A_0: \\quad \\pi _{[1,0,0,1,0]} & = & (1 \\cdot 3 \\cdot 4 \\cdot 5 \\cdot 7)^{-1} = 1/420 \\\\(A_1A_0)^2A_0: \\quad \\pi _{[1,0,1,0,0]} & = & (1 \\cdot 2 \\cdot 4 \\cdot 5 \\cdot 7)^{-1} = 1/280 \\\\A_1^2A_0^3: \\quad \\pi _{[1,1,0,0,0]} & = & (1 \\cdot 2 \\cdot 3 \\cdot 5 \\cdot 7)^{-1} = 1/210.$ In this example the general formula becomes $A_{m_1} A_{m_2} \\cdots A_{m_5}: \\quad \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} = [m_5+1]^{-1} [m_4 + m_5 + 2]^{-1} \\cdots [m_1+\\cdots +m_5 + 5]^{-1}$ for each $\\mbox{$\\mbox{$m$}$}= (m_1,m_2,\\ldots ,m_5) \\in S_{7,2,1}$ .", "Note that $m_1 + \\cdots + m_5 + 5 = 7$ .", "The coefficient $R_{7,2}$ is obtained by summing the above terms.", "Thus $R_{7,2} = A_0^3A_1^2/1680 + A_0^2A_1A_0A_1/1260 + \\cdots + (A_1A_0)^2A_0/280 + A_1^2A_0^3/210$ .", "$\\hfill \\Box $ In order to progress we note that contributions to the scalar coefficients $\\lbrace r_n\\rbrace _{n \\in {\\mathbb {N}}}$ in the solution to (REF ) can be calculated in exactly the same way as the matrix coefficients $\\lbrace R_n\\rbrace _{n \\in {\\mathbb {N}}}$ .", "The only difference is that the order of multiplication is immaterial for the scalar coefficients.", "We have the following result.", "Lemma 3 For $n \\in {\\mathbb {N}}$ and $q \\in \\lbrace 0,1,\\ldots ,\\lfloor n/2 \\rfloor \\rbrace $ the sum of the coefficients $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ for $\\mbox{$\\mbox{$m$}$}\\in S_{n,q,1}$ is given by $\\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q,1}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} = \\frac{1}{(n-2q)!\\, q!\\, 2^q\\,}$ where $S_{n,q,1}$ is defined by (REF ) and $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ is defined by (REF ).", "$\\hfill \\Box $ Proof.", "The proof is simply a matter of recognising that for each $q =0,1,\\ldots , \\lfloor n/2 \\rfloor $ both sides of (REF ) are calculating the coefficient of the term $a_0^{n-2q}a_1^q$ .", "We know from (REF ) that the only term in $r_n$ containing a power of $a_1^q$ is $r_{n,q} = \\frac{1}{ (n-2q)!\\, q!\\, 2^q\\,}\\, a_0^{n-2q}\\,a_1^q$ for each $q = 0,1,\\ldots , \\lfloor n/2 \\rfloor $ .", "Thus we have $r_n = r_{n,0} + \\cdots + r_{n, \\lfloor n/2 \\rfloor }$ for each $n \\in {\\mathbb {N}}$ .", "It follows from (REF ), (REF ) and (REF ) that for each $\\mbox{$\\mbox{$m$}$}\\in S_{n,q,1}$ there is a corresponding product term $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}}a_{m_1}a_{m_2} \\cdots a_{m_{n-q}}$ with $m_1 + \\cdots + m_{n-q} = q$ and $m_k \\in \\lbrace 0,1 \\rbrace $ for each $k=1,\\ldots ,n-1$ .", "Hence there must be $q$ indices with $m_k = 1$ and $n-2q$ indices with $m_k = 0$ in each term.", "Thus we also have $r_{n,q} = \\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q,1}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} a_{m_1} a_{m_2} \\cdots a_{m_{n-q}} = \\left[ \\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q,1}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} \\right] a_0^{n-2q} a_1^q$ for each $q=0,1,\\ldots ,\\lfloor n/2 \\rfloor $ .", "This completes the proof.", "More generally, in anticipation of things to come, we note that expansion of the term $(a_0t + a_1t^2/2)^{n-q}$ by the binomial theorem shows that $\\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q,1}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} = \\frac{1}{ (n-2q)!\\, q!\\, 2^q\\,} = \\sum _{\\langle \\mbox{$\\mbox{\\scriptsize $k$}$}, (\\mbox{$\\mbox{\\scriptsize $n$}$}, \\mbox{$\\mbox{\\scriptsize $1$}$}) \\rangle = (n, n-q)} \\frac{1}{k_1!\\, k_2!\\, 2^{k_2}}$ for each $q = 0,1,\\ldots \\lfloor n/2 \\rfloor $ .", "In (REF ) we have written $\\langle \\mbox{$\\mbox{$k$}$}, (\\mbox{$\\mbox{$n$}$}, \\mbox{$\\mbox{$1$}$}) \\rangle = (\\langle \\mbox{$\\mbox{$k$}$}, \\mbox{$\\mbox{$n$}$}\\rangle , \\langle \\mbox{$\\mbox{$k$}$}, \\mbox{$\\mbox{$1$}$}\\rangle )$ for convenience.", "The identity (REF ) adds some additional context to the equivalence in (REF ).", "$\\hfill \\Box $ Lemma 4 If $R(t) = I + R_1t + R_2 t^2 + \\cdots \\in {\\mathcal {B}}({\\mathbb {C}}^k)$ is a solution to $dR(t)/dt = (A_0 + A_1t)R(t)$ with $R(0) = I$ where $I \\in {\\mathcal {B}}({\\mathbb {C}}^k)$ is the identity matrix then there exist positive constants $c = \\max \\lbrace \\Vert A_0\\Vert , \\Vert A_1\\Vert \\rbrace $ and $d = \\max _{n < 2c} \\lbrace \\Vert R_n\\Vert \\lfloor n/2 \\rfloor !", "/c^{\\lfloor n/2 \\rfloor }$ such that $\\Vert R_n\\Vert \\le dc^{\\lfloor n/2 \\rfloor }/ \\lfloor n/2 \\rfloor !$ for each $n \\in {\\mathbb {N}}$ .", "$\\hfill \\Box $ Proof.", "We have $\\Vert R_n\\Vert & \\le & \\sum _{q=0}^{\\lfloor n/2 \\rfloor } \\Vert R_{n,q}\\Vert \\le \\sum _{q=0}^{\\lfloor n/2 \\rfloor } \\left[ \\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q,1}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} \\Vert A_{m_1}\\Vert \\Vert A_{m_2} \\Vert \\cdots \\Vert A_{m_{n-q}} \\Vert \\right] \\\\& & \\hspace{28.45274pt} = \\sum _{q=0}^{\\lfloor n/2 \\rfloor } \\left[ \\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q,1}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} \\right] \\Vert A_0\\Vert ^{n-2q} \\Vert A_1\\Vert ^q = \\sum _{q=0}^{\\lfloor n/2 \\rfloor } \\frac{a_0^{n-2q} a_1^q}{(n-2q)!", "q!", "2^q} = \\sum _{q=0}^{\\lfloor n/2 \\rfloor } r_{n,q} = r_n$ where $r(t) = 1 + r_1t + r_2t^2 + \\cdots $ is the solution to (REF ) in the case where $a_0 = \\Vert A_0\\Vert $ and $a_1 = \\Vert A_1\\Vert $ .", "If we set $c = \\max \\lbrace |a_0|, |a_1| \\rbrace $ then it follows from Lemma REF that (REF ) is true for all $n \\in {\\mathbb {N}}$ .", "Therefore $\\Vert R_n\\Vert \\le |r_n| < d c^{\\lfloor n/2 \\rfloor }/ \\lfloor n/2 \\rfloor !$ as required.", "$\\hfill \\Box $ Therefore the Maclaurin series (REF ) and (REF ) with coefficients defined respectively by (REF ) and (REF ) each converge absolutely for all $t \\in [0,\\infty )$ .", "Both series can be differentiated term by term and hence satisfy the respective evolution equations (REF ) and (REF )." ], [ "An explicit Maclaurin series solution when the evolution coefficient is a general analytic function of time", "Consider the scalar differential equation $dr(t)/dt = (a_0 + a_1t + a_2t^2 + \\cdots ) r(t)$ where $a_j \\in {\\mathbb {C}}$ for each $j \\in {\\mathbb {N}}$ are known constants with $\\limsup _{j \\rightarrow \\infty }|a_j|^{1/j} = 1/b$ for some $b \\in (0, \\infty ) \\cup \\lbrace \\infty \\rbrace $ and $r:[0,b) \\rightarrow {\\mathbb {C}}$ with $r(0) =1$ .", "If we write $\\mbox{$\\mbox{$k$}$}= (k_1,k_2,k_3, \\ldots ) \\in ({\\mathbb {N}}-1)^{\\infty }$ and $\\mbox{$\\mbox{$n$}$}= (1,2,3,\\ldots ) \\in ({\\mathbb {N}}-1)^{\\infty }$ and apply the multinomial expansion to the standard solution we have $r(t) & = & e^{a_0t + a_1t^2/2 + a_2t^3/3 + \\cdots } \\\\& = & 1 + \\sum _{m \\in {\\mathbb {N}}} (a_0t + a_1t^2/2 + a_2t^3/3 + \\cdots )^m/m!", "\\\\& = & 1 + \\sum _{m \\in {\\mathbb {N}}} \\left[\\, \\sum _{ \\langle \\mbox{$\\mbox{\\scriptsize $k$}$}, \\mbox{$\\mbox{\\scriptsize $1$}$}\\rangle = m} \\frac{ (a_0t)^{k_1}\\, (a_1t^2/2)^{k_2}\\, (a_2t^3/3)^{k_3}\\, \\cdots (a_{m-1}t^m/m)^{k_m}}{k_1!\\, k_2!\\,k_3!\\, \\cdots k_m!}", "\\right] \\\\& = & 1 + \\sum _{n \\in {\\mathbb {N}}} \\left[\\, \\sum _{ \\langle \\mbox{$\\mbox{\\scriptsize $k$}$}, \\mbox{$\\mbox{\\scriptsize $n$}$}\\rangle = n} \\frac{ a_0^{k_1}\\, a_1^{k_2}\\, a_2^{k_3} \\cdots a_{n-1}^{k_n}}{k_1!\\, k_2!\\, k_3!\\, \\cdots k_n!\\,\\, 2^{k_2}\\, 3^{k_3}\\, \\cdots n^{k_n}} \\right] t^n$ for all $t \\in [0,b)$ .", "If we collect all terms with $\\langle \\mbox{$\\mbox{$k$}$}, \\mbox{$\\mbox{$n$}$}\\rangle = n$ and $\\langle \\mbox{$\\mbox{$k$}$}, \\mbox{$\\mbox{$1$}$}\\rangle = n-q = m$ and define $r_{n,q} = \\sum _{\\langle \\mbox{$\\mbox{\\scriptsize $k$}$}, (\\mbox{$\\mbox{\\scriptsize $n$}$}, \\mbox{$\\mbox{\\scriptsize $1$}$}) \\rangle = (n, n-q)} \\frac{a_0^{k_1}\\, a_1^{k_2}\\, a_2^{k_3}\\, \\cdots a_{n-1}^{k_n}}{ k_1!\\, k_2!\\, k_3!\\, \\cdots \\, k_n!\\, 2^{k_2}\\, 3^{k_3} \\cdots \\, n^{k_n}}$ for each $q \\in \\lbrace 0,1,\\ldots ,n-1\\rbrace $ then we can write $r_n = \\sum _{q=0}^{n-1} r_{n,q} = \\sum _{\\langle \\mbox{$\\mbox{\\scriptsize $k$}$}, \\mbox{$\\mbox{\\scriptsize $n$}$}\\rangle = n} \\frac{ a_0^{k_1}\\, a_1^{k_2}\\, a_2^{k_3}\\, \\cdots a_{n-1}^{k_n} }{k_1!\\, k_2!\\, k_3!\\, \\cdots k_n!\\, 2^{k_2}\\, 3^{k_3}\\, \\cdots n^{k_n}}$ for each $n \\in {\\mathbb {N}}$ .", "The terms in $r_{n,q}$ each have total index $\\langle \\mbox{$\\mbox{$k$}$}, \\mbox{$\\mbox{$n$}$}- \\mbox{$\\mbox{$1$}$}\\rangle = \\langle \\mbox{$\\mbox{$k$}$}, \\mbox{$\\mbox{$n$}$}\\rangle - \\langle \\mbox{$\\mbox{$k$}$}, \\mbox{$\\mbox{$1$}$}\\rangle = q$ .", "Now let $X$ be a complex Banach space and consider the operator equation $d{R}(t)/dt = (A_0 + A_1t + A_2t^2 + \\cdots ) R(t)$ where $A_0, A_1, A_2, \\cdots \\in {\\mathcal {B}}(X)$ with $\\limsup _{j \\rightarrow \\infty } \\Vert A_j\\Vert ^{1/j} = 1/b$ for some $b \\in (0,\\infty ) \\cup \\lbrace \\infty \\rbrace $ and $R:[0,b) \\rightarrow {\\mathcal {B}}(X)$ with $R(0) = I$ where $I \\in {\\mathcal {B}}(X)$ is the identity operator.", "We substitute $R(t) = I + R_1t + R_2t^2 + \\cdots $ where $R_j \\in {\\mathcal {B}}(X)$ for each $j \\in {\\mathbb {N}}$ and equate coefficients of the various powers of $t$ to obtain a system of fundamental equations $R_1 & = & A_0 \\\\2 R_2 & = & A_0R_1 + A_1 \\\\\\vdots & \\vdots & \\vdots \\\\nR_n & = & A_0R_{n-1} + A_1R_{n-2} + \\cdots A_{n-2}R_1 + A_{n-1}$ and so on for all $n \\in {\\mathbb {N}}$ .", "For each $n \\in {\\mathbb {N}}$ our calculation of the coefficient $R_n$ is based on identifying all terms in the form $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}}A_{\\mbox{$\\mbox{\\scriptsize $m$}$}} = \\pi _{(m_1,\\ldots ,m_{n-q})}A_{m_1}\\cdots A_{m_{n-q}}$ that contribute to $R_n$ .", "Once again we collect these terms systematically according to the total index $m_1+\\cdots +m_{n-q} = q$ .", "For each $q \\in \\lbrace 0,1,\\ldots ,n-1 \\rbrace $ define $R_{n,q} = \\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} A_{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ and write $R_n = \\sum _{q=0}^{n-1} R_{n,q} = \\sum _{q=0}^{n-1} \\left[ \\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} A_{\\mbox{$\\mbox{\\scriptsize $m$}$}} \\right].$ The key to calculating the individual terms is to consider their ancestry—the lineage defined by the recursive relationships that express $R_n$ in terms of $R_{n-1},R_{n-2},\\ldots ,R_1, I$ .", "The term $\\pi _{(m_1,m_2,\\ldots ,m_{n-q})} A_{m_1}A_{m_2} \\cdots A_{m_{n-q}}$ with $m_1 + m_2 + \\cdots + m_{n-q} = q$ is generated from the term $\\pi _{(m_2,\\ldots ,m_{n-q})} A_{m_2} \\cdots A_{m_{n-q}}$ with $m_2 + \\cdots + m_{n-q} = q - m_1$ according to the formula $\\pi _{(m_1,m_2,\\ldots ,m_{n-q})} A_{m_1}A_{m_2} \\cdots A_{m_{n-q}} = \\frac{A_{m_1}}{n} \\cdot \\pi _{(m_2,\\ldots ,m_{n-q})} A_{m_2} \\cdots A_{m_{n-q}}.$ The condition $m_1 + m_2 + \\cdots + m_{n-q} = q$ means that $\\mbox{$\\mbox{$m$}$}= (m_1,m_2,\\ldots ,m_{n-q}) \\in S_{n,q}$ and the term $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}}A_{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ contributes to $R_{n,q}$ .", "If we write $\\mbox{$\\mbox{$m$}$}^{\\prime } = (m_1^{\\prime },\\ldots ,m_{n-m_1-1- [q - m_1]}^{\\prime }) = (m_2,\\ldots ,m_{n-q})$ then the condition $m_1^{\\prime } + \\cdots + m_{n-m_1-1-[q-m_1]}^{\\prime } = q - m_1$ means that $\\mbox{$\\mbox{$m$}$}^{\\prime } \\in S_{n-1-m_1,q - m_1}$ and the term $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}^{\\prime }} A_{\\mbox{$\\mbox{\\scriptsize $m$}$}^{\\prime }}$ contributes to $R_{n-m_1-1,q - m_1}$ .", "An inductive argument can now be used to justify the definitions in (REF ) and (REF ).", "The argument is similar to the argument used in Section .", "In order to prove convergence of the operator series we need the following result.", "Lemma 5 For each fixed $n \\in {\\mathbb {N}}$ and each $q \\in \\lbrace 0,1,\\ldots ,n-1\\rbrace $ the sum of the coefficients $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ for $\\mbox{$\\mbox{$m$}$}\\in S_{n,q}$ is $\\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} = \\sum _{\\langle \\mbox{$\\mbox{\\scriptsize $k$}$}, (\\mbox{$\\mbox{\\scriptsize $n$}$}, \\mbox{$\\mbox{\\scriptsize $1$}$}) \\rangle = (n, n-q)} \\frac{1}{k_1!\\, k_2!\\, \\cdots k_n!\\, 2^{k_2}\\, 3^{k_3}\\, \\cdots n^{k_n}}$ where $S_{n,q}$ is defined by (REF ) and $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ is defined by (REF ).", "$\\hfill \\Box $ Proof.", "The proof is similar to the proof of Lemma REF .", "$\\hfill \\Box $ Since $\\limsup _{j \\rightarrow \\infty } \\Vert A_j\\Vert ^{1/j} = 1/b$ we can find some $d > 0$ such that $\\Vert A_j\\Vert < d b^{\\, j}$ for all $j \\in {\\mathbb {N}}$ .", "Therefore $\\Vert R_n\\Vert & \\le & \\sum _{q=0}^{n-1} \\Vert R_{n,q} \\Vert \\le \\sum _{q=0}^{n-1} \\left[ \\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} \\Vert A_{m_1}\\Vert \\Vert A_{m_2} \\Vert \\cdots \\Vert A_{m_{n-q}} \\Vert \\right] \\\\& & \\hspace{21.33955pt} \\le \\sum _{q=0}^{n-1}\\, d^{\\,n-q} b^{\\,n} \\left[ \\sum _{m \\in S_{n,q}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} \\right] = \\sum _{q=0}^{n-1} \\left[ \\sum _{\\langle \\mbox{$\\mbox{\\scriptsize $k$}$}, (\\mbox{$\\mbox{\\scriptsize $n$}$}, \\mbox{$\\mbox{\\scriptsize $1$}$}) \\rangle = (n, n-q) }\\, \\frac{ d^{\\,k_1} (db)^{k_2} \\cdots (db^{n-1})^{k_n}}{k_1!\\, k_2!\\, \\cdots k_n!\\, 2^{k_2}\\, \\cdots n^{k_n}} \\right] \\\\& & \\hspace{42.67912pt} = \\sum _{q=0}^{n-1} \\left[ \\sum _{\\langle \\mbox{$\\mbox{\\scriptsize $k$}$}, (\\mbox{$\\mbox{\\scriptsize $n$}$}, \\mbox{$\\mbox{\\scriptsize $1$}$}) \\rangle = (n, n-q) }\\, \\frac{ a_0^{\\,k_1} a_1^{k_2} \\cdots a_{n-1}^{k_n}}{k_1!\\, k_2!\\, \\cdots k_n!\\, 2^{k_2}\\, \\cdots n^{k_n}} \\right] = \\sum _{q=0}^{n-1} r_{n,q} = r_n$ where $r(t) = 1 + r_1t + r_2t^2 + \\cdots $ is the solution to (REF ) in the case where $a_j = db^{\\,j}$ for all $j \\in {\\mathbb {N}}$ .", "Since the scalar series converges absolutely for all $t \\in [0,b)$ it follows that the operator series $R(t) = I + R_1t + R_2t^2 + \\cdots $ does too.", "This proves Theorem REF .", "If $A(t) = A_0 + A_1t + \\cdots + A_pt^p$ is a polynomial of degree $p$ for some $p \\in {\\mathbb {N}}$ we can apply the general results by writing $A(t) = A_0 + A_1t + \\cdots $ as an infinite series with $A_j \\in {\\mathcal {B}}(X)$ for all $j \\in {\\mathbb {N}}-1$ but with $A_j = \\mbox{\\large $0$}\\in {\\mathcal {B}}(X)$ for all $j \\in {\\mathbb {N}}+1$ .", "However this approach means that key formulæ such as (REF ), (REF ) and (REF ) will often include indices $\\mbox{$\\mbox{$m$}$}= (m_1,\\ldots ,m_{n-q}) \\in S_{n,q}$ where $m_j \\notin \\lbrace 0,1,\\ldots ,p\\rbrace $ and $A_{m_j} = \\mbox{\\large $0$}$ for some $j \\in \\lbrace 0,1,\\ldots , n-q\\rbrace $ .", "Hence we also have $A_{\\mbox{$\\mbox{\\scriptsize $m$}$}} = \\mbox{\\large $0$}$ for the corresponding products.", "These key formulæ would be more meaningful if the indices that correspond to the artificial products were excluded a priori.", "Before addressing this issue we illustrate the problem with an elementary example.", "Example 2 Consider the evolution equation $dR(t)/dt = (A_0 + A_1t)R(t)$ with $R(0) = I$ and investigate the calculations relating to $R_4$ .", "It follows from (REF ) that $S_{4,0} = \\lbrace (0,0,0,0) \\rbrace $ , $S_{4,1} = \\lbrace (0,0,1), (0,1,0) (1,0,0) \\rbrace $ , $S_{4,2} = \\lbrace (0,2), (2,0) \\rbrace $ , $S_{4,3} = \\lbrace 3 \\rbrace $ .", "Now (REF ) and (REF ) give $R_{4,0} & = & \\frac{1}{4} \\cdot \\frac{1}{3} \\cdot \\frac{1}{2} \\cdot \\frac{1}{1} \\cdot A_0^4, \\\\R_{4,1} & = & \\frac{1}{4} \\cdot \\frac{1}{3} \\cdot \\frac{1}{2} \\cdot A_0^2A_1 + \\frac{1}{4} \\cdot \\frac{1}{3} \\cdot \\frac{1}{1} \\cdot A_0 A_1 A_0 + \\frac{1}{4} \\cdot \\frac{1}{2} \\cdot \\frac{1}{1} \\cdot A_1A_0^2, \\\\R_{4,2} & = & \\frac{1}{4} \\cdot \\frac{1}{3} \\cdot A_0A_2 + \\frac{1}{4} \\cdot \\frac{1}{1} \\cdot A_2A_0, \\\\R_{4,3} & = & \\frac{1}{4} A_3$ and (REF ) gives $R_4 = R_{4,0} + R_{4,1} + R_{4,2} + R_{4,3}$ .", "Since $A_2 = A_3 = \\mbox{\\large $0$}$ we have $R_{4,2} = R_{4,3} = \\mbox{\\large $0$}$ too.", "However the corresponding coefficients are nonzero with $\\pi _{(0,2)} = 1/12$ and $\\pi _{(2,0)} = 1/4$ when $\\mbox{$\\mbox{$m$}$}\\in \\lbrace (0,2), (2,0)\\rbrace = S_{4,2}$ and with $\\pi _3 = 1/4$ when $\\mbox{$\\mbox{$m$}$}\\in \\lbrace 3 \\rbrace = S_{4,3}$ .", "Thus (REF ) is essentially irrelevant for $q=3,4$ .", "More meaningful relationships can be obtained if the indices that correspond to zero products are eliminated beforehand.", "$\\hfill \\Box $ If $A_j = \\mbox{\\large $0$}$ for $j \\in {\\mathbb {N}}+p$ then for each $n \\in {\\mathbb {N}}$ we wish to find the largest value of $q$ for which the coefficient $R_{n,q}$ is nonzero.", "Every nonzero term that contributes to $R_{n,q}$ must be identified by an index $\\mbox{$\\mbox{$m$}$}\\in S_{n,q}$ with $m_1 + \\cdots + m_{n-q} = q$ and $m_j \\in \\lbrace 0,1,\\ldots ,p\\rbrace $ for all $j \\in \\lbrace 0,1,\\ldots ,n-q\\rbrace $ .", "The largest value of $q$ , or equivalently the smallest possible value of $n - q$ that allows a nonzero contribution, will occur when $(p+1)(n-q-1) + r + 1 = n$ for some $r \\in \\lbrace 1,\\ldots ,p\\rbrace $ .", "This equality can be explained by noting that the minimum value of $n-q$ will occur when we have the largest possible number of indices $\\lbrace m_j\\rbrace _{j=1}^{n-q}$ equal to $p$ .", "If all indices are equal to $p$ then the corresponding operator product will take the form $(A_p t^{p+1})^{n-q} = A_p^{n-q} t^{(p+1)(n-q)}$ and so we must have $(p+1)(n-q) = n$ .", "If all indices bar one are equal to $p$ and one index is less than $p$ then the corresponding operator product will take the form $(A_p t^{p+1})^{\\ell -1} A_r t^{r+1} (A_pt^{p+1})^{n-q-\\ell }$ for some $\\ell \\in \\lbrace 1,\\ldots ,n-q\\rbrace $ and $r \\in \\lbrace 1,\\ldots ,p-1\\rbrace $ and so we must have $(p+1)(n-q-1) + r + 1 = n$ .", "We can cover both cases at once if we also allow $r = p$ in (REF ).", "Now it follows that $q = \\left(p(n-1) + r \\right)/ (p+1) \\le pn/(p+1).$ Thus we can assume $q \\in \\lbrace 0,1,\\ldots , \\lfloor pn/(p+1) \\rfloor \\rbrace $ .", "This justifies the definition of the restricted index set $S_{n,q,p}$ in (REF ).", "For all $n \\in {\\mathbb {N}}$ and $q \\in \\lbrace 0,1,\\ldots , \\lfloor pn/(p+1) \\rfloor \\rbrace $ we can replace the general formulæ (REF ) and (REF ) by the more discriminating formulæ $R_{n,q} = \\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q,p}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} A_{\\mbox{$\\mbox{\\scriptsize $m$}$}},$ and $R_n = \\sum _{q=0}^{\\lfloor pn/(p+1) \\rfloor } R_{n,q} = \\sum _{q=0}^{\\lfloor pn/(p+1) \\rfloor } \\left[ \\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q,p}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} A_{\\mbox{$\\mbox{\\scriptsize $m$}$}} \\right].$ If we apply the multinomial expansion to deduce that $e^{(a_0t + a_1t^2/2 + \\cdots + a_pt^{p+1}/(p+1))} = 1 + \\sum _{n \\in {\\mathbb {N}}} \\left[ \\sum _{\\langle \\mbox{$\\mbox{\\scriptsize $k$}$}, \\mbox{$\\mbox{\\scriptsize $n$}$}\\rangle = n} \\frac{ a_0^{k_1}\\, a_1^{k_2}\\, \\cdots \\, a_p^{k_{p+1}} }{k_1!\\, k_2!\\, \\cdots \\, k_{p+1}!\\, 2^{k_2}\\, \\cdots \\, (p+1)^{k_{p+1}}\\,} \\right] t^n$ then a straightforward argument shows that (REF ) can also be replaced by $\\sum _{\\mbox{$\\mbox{\\scriptsize $m$}$}\\in S_{n,q,p}} \\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}} = \\sum _{\\langle \\mbox{$\\mbox{\\scriptsize $k$}$}, (\\mbox{$\\mbox{\\scriptsize $n$}$}, \\mbox{$\\mbox{\\scriptsize $1$}$}) \\rangle = (n, n-q)} \\frac{1}{k_1!\\, k_2!\\, \\cdots k_{p+1}!\\, 2^{k_2}\\, \\cdots (p+1)^{k_{p+1}}}.$ It is now a matter of elementary algebra to justify Corollary REF .", "Of course it is still true with a polynomial evolution coefficient $A(t) = A_0 + A_1t + \\cdots + A_pt^p$ that there may be some values of $j \\in \\lbrace 0,1,\\ldots ,p-1\\rbrace $ with $A_j = \\mbox{\\large $0$}$ .", "In such cases there will still be some zero products in the reduced expressions $(\\ref {eq:ecda7})$ and $(\\ref {eq:ecda8})$ and some coefficients $\\pi _{\\mbox{$\\mbox{\\scriptsize $m$}$}}$ in (REF ) for indices $\\mbox{$\\mbox{$m$}$}\\in S_{n,q,p}$ where $A_{\\mbox{$\\mbox{\\scriptsize $m$}$}} = \\mbox{\\large $0$}$ .", "Remark 2 We could eliminate all zero products by defining a list ${\\mathcal {J}} = \\lbrace j_1,j_2,\\ldots \\rbrace \\subseteq {\\mathbb {N}}-1$ of all nonzero coefficients $A_j$ and a corresponding restricted index set $S_{n,q,{\\mathcal {J}}} = \\lbrace \\mbox{$\\mbox{$m$}$}= (m_1,\\ldots ,m_{n-q}) \\in {\\mathcal {J}}^{n-q} \\mid m_1 + \\cdots + m_{n-q} = q \\rbrace \\subseteq S_{n,q}$ for each $n \\in {\\mathbb {N}}$ and each $q \\in \\lbrace 0,1,\\ldots ,n-1\\rbrace $ .", "We will not pursue a detailed analysis.", "$\\hfill \\Box $" ], [ "Relationship between the Maclaurin series solution and the Peano–Baker series solution", "We begin with the scalar equation.", "Suppose that $r:[0,b) \\rightarrow {\\mathbb {R}}$ satisfies $dr(t)/dt = a(t)r(t)$ with $r(0) = 1$ where the series $a(t) = a_0 + a_1t + a_2t^2 + \\cdots $ converges for all $t \\in (-b,b)$ .", "Define $u_0(t) = 1$ for all $t \\in [0,b)$ .", "The Peano-Baker iteration proceeds as follows.", "Define $u_1(t) = \\int _{[0,t]} a(\\tau ) u_0(\\tau ) d \\tau = a_0t + a_1t^2/2 + a_2t^3/3 + \\cdots $ for all $t \\in [0,b)$ .", "Next we have $u_2(t) = \\int _{[0,t]} a(\\tau ) u_1(\\tau ) d\\tau = \\int _{[0,t]} u_1^{\\, \\prime }(\\tau ) u_1(\\tau ) d\\tau = u_1(t)^2/2!,$ $u_3(t) = \\int _{[0,t]} a(\\tau ) u_2(\\tau ) d\\tau = \\int _{[0,t]} u_1^{\\, \\prime }(\\tau ) [u_1(\\tau )^2/2!]", "d\\tau = u_1(t)^3/3!,$ and so on for all $t \\in [0,b)$ .", "In general $u_n(t) = \\int _{[0,t]} a(\\tau ) u_{n-1}(\\tau ) d\\tau = \\int _{[0,t]} u_1^{\\, \\prime }(\\tau ) [u_1(\\tau )^{n-1}/(n-1)!]", "d\\tau = u_1(t)^n/n!$ for each $n \\in {\\mathbb {N}}$ and so the solution obtained by the Peano–Baker series is given by $r(t) = 1 + \\sum _{n \\in {\\mathbb {N}}} u_n(t) = 1 + \\sum _{n \\in {\\mathbb {N}}} u_1(t)^n/n!", "= e^{u_1(t)} = e^{\\int _{[0,t]} a(s) ds}$ for all $t \\in [0,\\infty )$ as expected.", "The details of the iteration are more difficult to describe for the operator equation.", "We begin with the linear evolution coefficient.", "Let $X$ be a complex Banach space where $R:[0,\\infty ) \\rightarrow X$ satisfies $dR(t)/dt = A(t)R(t)$ with $R(0) = I \\in X$ and $A(t) = A_0 + A_1t \\in {\\mathcal {B}}(X)$ for all $t \\in [0,\\infty )$ .", "Define $U_0(t) = I$ .", "Now set $U_1(t) = \\int _{[0,t]} A(\\tau ) U_0(\\tau ) d \\tau = \\int _{[0,t]} (A_0 + A_1 \\tau ) d \\tau = A_0t + A_1t^2/2$ for all $t \\in [0,\\infty )$ .", "Next we have $U_2(t) & = & \\int _{[0,t]} A(\\tau )U_1(\\tau ) d\\tau \\\\& = & \\int _{[0,t]} (A_0 + A_1\\tau )(A_0 \\tau + A_1 \\tau ^2/2) d \\tau \\\\& = & A_0^2t^2/2 + A_0A_1t^3/6 + A_1A_0t^3/3 + A_1^2t^4/8,$ $U_3(t) & = & \\int _{[0,t]} A(\\tau )U_2(\\tau ) d\\tau \\\\& = & \\int _{[0,t]} (A_0 + A_1\\tau )(A_0^2\\tau ^2/2 + A_0A_1\\tau ^3/6 + A_1A_0\\tau ^3/3 + A_1^2\\tau ^4/8) d \\tau \\\\& = & A_0^3t^3/6 + A_0^2A_1t^4/24 + A_0A_1A_0t^4/12 + A_0A_1^2t^5/40 \\cdots \\\\& & \\hspace{56.9055pt} + A_1A_0^3t^4/8 + A_1A_0A_1t^5/30 + A_1^2A_0t^5/15 + A_1^4t^6/48$ for all $t \\in [0,\\infty )$ and so on.", "It is intuitively clear that the expression for $U_n(t)$ will only contain terms in $t^k$ for $k \\ge n$ .", "Hence, from our calculations thus far, we have $R(t) & = & \\sum _{n \\in {\\mathbb {N}}-1} U_n(t) \\\\& = & I + A_0t + (A_0^2/2+A_1/2)t^2 + (A_0^3/6+A_0A_1/6+A_1A_0/3)t^3 + \\cdots $ for all $t \\in [0,\\infty )$ .", "If we wish to find a general pattern it is easier to add all of the iterations together which gives $R(t) & = & I + \\sum _{n \\in {\\mathbb {N}}-1} U_{n+1}(t) \\\\& = & I + \\sum _{n \\in {\\mathbb {N}}-1} \\int _{[0,t]} A(\\tau )U_n(\\tau ) d\\tau \\\\& = & I + \\int _{[0,t]} A(\\tau ) \\left[ \\sum _{n \\in {\\mathbb {N}}-1}U_n(\\tau ) \\right] d\\tau \\\\& = & I + \\int _{[0,t]} A(\\tau ) R(\\tau ) d\\tau .$ Now the substitutions $A(t) = A_0 + A_1t + \\cdots $ and $R(t) = I + R_1t + \\cdots $ and subsequent equation of coefficients yield the general recursion $R_n = \\left(A_0R_{n-1} + A_1R_{n-2} + \\cdots + A_{n-2}R_1 + A_{n-1}\\right)/n$ for all $n \\in {\\mathbb {N}}$ .", "This is, of course, a return to our original starting point." ], [ "Examples", "In this section we present two examples to illustrate the proposed Maclaurin series solution.", "Example 3 (A linear coefficient) Let $A_0 = \\left[ \\begin{array}{ccc}1 & -1 & 2 \\\\1 & -2 & 1 \\\\2 & 1 & 1 \\end{array} \\right] \\quad \\mbox{and} \\quad A_1 = \\left[ \\begin{array}{ccc}2 & 1 & 3 \\\\-2 & 1 & 2 \\\\-3 & 2 & 1 \\end{array} \\right].$ We calculate $R_1 = A_0 = \\left[ \\begin{array}{ccc}1 & -1 & 2 \\\\1 & -2 & 1 \\\\2 & 1 & 1 \\end{array} \\right],$ $R_2 & = & \\frac{A_0}{2} \\cdot R_1 + \\frac{A_1}{2} \\cdot R_0 \\\\& = & \\frac{1}{2} \\left[ \\begin{array}{ccc}1 & -1 & 2 \\\\1 & -2 & 1 \\\\2 & 1 & 1 \\end{array} \\right] \\left[ \\begin{array}{ccc}1 & -1 & 2 \\\\1 & -2 & 1 \\\\2 & 1 & 1 \\end{array} \\right] + \\frac{1}{2} \\left[ \\begin{array}{ccc}2 & 1 & 3 \\\\-2 & 1 & 2 \\\\-3 & 2 & 1 \\end{array} \\right] \\left[ \\begin{array}{ccc}1 & 0 & 0 \\\\0 & 1 & 0 \\\\0 & 0 & 1 \\end{array} \\right] \\\\& = & \\left[ \\begin{array}{ccc}3 & 2 & 3 \\\\-0.5 & 2.5 & 1.5 \\\\1 & -0.5 & 3.5 \\end{array} \\right],$ $R_3 & = & \\frac{A_0}{3} \\cdot R_2 + \\frac{A_1}{3} \\cdot R_1 \\\\& = & \\frac{1}{3} \\left[ \\begin{array}{ccc}1 & -1 & 2 \\\\1 & -2 & 1 \\\\2 & 1 & 1 \\end{array} \\right] \\left[ \\begin{array}{ccc}3 & 2 & 3 \\\\-0.5 & 2.5 & 1.5 \\\\1 & -0.5 & 3.5 \\end{array} \\right] + \\frac{1}{3} \\left[ \\begin{array}{ccc}2 & 1 & 3 \\\\-2 & 1 & 2 \\\\-3 & 2 & 1 \\end{array} \\right] \\left[ \\begin{array}{ccc}1 & -1 & 2 \\\\1 & -2 & 1 \\\\2 & 1 & 1 \\end{array} \\right] \\\\& = & \\left[ \\begin{array}{ccc}4.8333 & -0.8333 & 5.5 \\\\2.6667 & -0.5 & 0.8333 \\\\2.5 & 2 & 2.6667 \\end{array} \\right],$ $R_4 & = & \\frac{A_0}{4} \\cdot R_3 + \\frac{A_1}{4} \\cdot R_2 \\\\& = & \\frac{1}{3} \\left[ \\begin{array}{ccc}1 & -1 & 2 \\\\1 & -2 & 1 \\\\2 & 1 & 1 \\end{array} \\right] \\hspace{-5.69054pt} \\left[ \\begin{array}{ccc}4.8333 & -0.8333 & 5.5 \\\\2.6667 & -0.5 & 0.8333 \\\\2.5 & 2 & 2.6667 \\end{array} \\right] + \\frac{1}{3} \\left[ \\begin{array}{ccc}2 & 1 & 3 \\\\-2 & 1 & 2 \\\\-3 & 2 & 1 \\end{array} \\right] \\hspace{-5.69054pt} \\left[ \\begin{array}{ccc}3 & 2 & 3 \\\\-0.5 & 2.5 & 1.5 \\\\1 & -0.5 & 3.5 \\end{array} \\right] \\\\& = & \\left[ \\begin{array}{ccc}3.9167 & 2.1667 & 7 \\\\-0.625 & -0.0833 & 2.25 \\\\1.4583 & -0.4167 & 3 \\end{array} \\right],$ and so on.", "The solution is given by $R(t) = I + R_0t + R_1t^2 + \\cdots $ for all $t \\in [0,\\infty )$ .", "These recursive calculations are easily performed in Matlab when the matrices are not too large.", "Hence the Maclaurin series should be well suited to finding accurate solutions to matrix evolution equations.", "If we attempt to find a solution using spectral methods we can see from $\\det (\\lambda I - A(t) ) & = & \\det \\left[ \\begin{array}{ccc}\\lambda - 2t - 1 & - t - 1 & -3t - 2 \\\\-1 + 2t & \\lambda - t + 2 & -2t - 1 \\\\-2 + 3t & -2t - 1 & \\lambda - t - 1 \\end{array} \\right] \\\\& = & \\rule {0cm}{0.6cm} \\lambda ^3 - 4t \\lambda ^2 + (12t^2 - 8t - 7)\\lambda + 13t^3 + 33t^2 + 13t - 6$ that calculation of eigenvalues and eigenvectors will involve complicated algebraic expressions that may be valid only on certain restricted intervals.", "One might expect that several different intervals would need to be considered to find solutions on the entire positive real axis.", "$\\hfill \\Box $ Example 4 (A special birth and death process with variable rates) Let $\\ell ^1$ denote the Banach space of real-valued absolutely summable sequences.", "We will follow the usual convention for Markov chains and write the elements of $\\ell ^1$ as row vectors.", "Thus if $\\mbox{$\\mbox{$x$}$}\\in \\ell ^1$ we write $\\mbox{$\\mbox{$x$}$}= (x_1,x_2,\\ldots ) = x_1\\mbox{$\\mbox{$e$}$}_1 + x_2 \\mbox{$\\mbox{$e$}$}_2 + \\cdots $ where $\\lbrace \\mbox{$\\mbox{$e$}$}_j\\rbrace _{j \\in {\\mathbb {N}}}$ is the standard orthonormal basis for $\\ell ^1$ .", "The basis vectors are also written as row vectors.", "If $Q \\in {\\mathcal {B}}(\\ell ^1)$ is a bounded linear operator we define $Q(\\mbox{$\\mbox{$x$}$}) = \\mbox{$\\mbox{$x$}$}Q \\in \\ell ^1$ for each $\\mbox{$\\mbox{$x$}$}\\in \\ell ^1$ where $Q = [q_{ij}] \\in {\\mathbb {R}}^{\\infty \\times \\infty }$ is the infinite matrix representation of $Q$ relative to the standard basis.", "In this notation $\\mbox{$\\mbox{$q$}$}_i = (q_{i1},q_{i2},\\ldots ) = \\mbox{$\\mbox{$e$}$}_i Q \\in \\ell ^1$ denotes the $i^{\\mbox{\\scriptsize th}}$ row of $Q$ .", "If $Q$ is the transition operator for a continuous-time Markov chain we insist that $q_{ij} \\ge 0$ for $i \\ne j$ and that $\\langle \\mbox{$\\mbox{$e$}$}_i Q, \\mbox{$\\mbox{$1$}$}\\rangle = q_{i1} + q_{i2} + \\cdots = 0$ where $\\mbox{$\\mbox{$1$}$}= (1,1,\\ldots ) \\in \\ell ^{\\infty }$ for all $i \\in {\\mathbb {N}}$ and $\\ell ^{\\infty }$ is the dual space to $\\ell ^1$ consisting of all real-valued bounded row vectors.", "Thus we insist that the row sums of $Q$ are all zero.", "Let $\\lambda _j, \\mu _j \\in {\\mathbb {R}}$ be positive constants and define $A_j = \\left[ \\begin{array}{ccccc}-\\lambda _j & \\lambda _j & 0 & 0 & \\cdots \\\\\\mu _j & -(\\lambda _j + \\mu _j) & \\lambda _j & 0 & \\cdots \\\\0 & \\mu _j & -(\\lambda _j + \\mu _j) & \\lambda _j & \\cdots \\\\0 & 0 & \\mu _j & -(\\lambda _j + \\mu _j) & \\cdots \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\end{array} \\right] \\in {\\mathcal {B}}(\\ell ^1)$ for each $j=0,1$ .", "The evolution equation $dR(t)/dt = R(t)(A_0 + A_1t)$ for $t \\in [0,\\infty )$ where $R(t) \\in {\\mathcal {B}}(\\ell ^1)$ and $R(0) = I$ is a special case of a continuous-time birth and death process.", "It is convenient to define auxiliary operators $U = \\left[ \\begin{array}{rrrrr}-1 & 1 & 0 & 0 & \\cdots \\\\0 & -1 & 1 & 0 & \\cdots \\\\0 & 0 & -1 & 1 & \\cdots \\\\0 & 0 & 0 & -1 & \\cdots \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\end{array} \\right], \\quad L = \\left[ \\begin{array}{rrrrr}0 & 0 & 0 & 0 & \\cdots \\\\1 & -1 & 0 & 0 & \\cdots \\\\0 & 1 & -1 & 0 & \\cdots \\\\0 & 0 & 1 & -1 & \\cdots \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\end{array} \\right] \\in {\\mathcal {B}}(\\ell ^1)$ and the left-shift and right-shift operators $S_{\\ell }, S_r \\in {\\mathcal {B}}(\\ell ^1)$ by setting $S_{\\ell } = \\left[ \\begin{array}{rrrrr}0 & 0 & 0 & 0 & \\cdots \\\\1 & 0 & 0 & 0 & \\cdots \\\\0 & 1 & 0 & 0 & \\cdots \\\\0 & 0 & 1 & 0 & \\cdots \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\end{array} \\right], \\quad S_r= \\left[ \\begin{array}{rrrrr}0 & 1 & 0 & 0 & \\cdots \\\\0 & 0 & 1 & 0 & \\cdots \\\\0 & 0 & 0 & 1 & \\cdots \\\\0 & 0 & 0 & 0 & \\cdots \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\end{array} \\right] \\in {\\mathcal {B}}(\\ell ^1).$ Thus we may write $U = -I + S_r$ and $L = - S_{\\ell }U$ .", "Left multiplication by $S_{\\ell }$ on $M \\in {\\mathcal {B}}(\\ell ^1)$ to form the product $S_{\\ell }M$ will shift the rows of $M$ down one unit while introducing a zero row at the top.", "Thus $\\mbox{$\\mbox{$e$}$}_1 S_{\\ell } M = \\mbox{$\\mbox{$0$}$}$ and $\\mbox{$\\mbox{$e$}$}_{j+1}S_{\\ell } M = \\mbox{$\\mbox{$e$}$}_j M = \\mbox{$\\mbox{$m$}$}_j$ for all $j \\in {\\mathbb {N}}$ .", "Left multiplication by $S_r$ on $M \\in {\\mathcal {B}}(\\ell ^1)$ to form the product $S_rM$ will shift the rows of $M$ up one unit while discarding the top row.", "Thus $\\mbox{$\\mbox{$e$}$}_j S_rM = \\mbox{$\\mbox{$e$}$}_{j+1} M = \\mbox{$\\mbox{$m$}$}_{j+1}$ for all $j \\in {\\mathbb {N}}$ .", "Hence we might just as well refer to $S_{\\ell }$ as the down-shift or delay operator and $S_r$ as the up-shift or advance operator.", "The point spectra of the left and right shift operators are quite complicated which means that spectral methods are unlikely to provide a viable alternative solution procedure.", "The book by Naylor and Sell [17] contains a complete treatment.", "We can write $A_j = \\lambda _j U + \\mu _j L = \\lambda _j U - \\mu _j S_{\\ell }U \\in {\\mathcal {B}}(\\ell ^1)$ for each $j=0,1$ .", "If we substitute $R(t) = I + R_1t + R_2t^2 + \\cdots $ where $R_j \\in {\\mathcal {B}}(\\ell ^1)$ for each $j \\in {\\mathbb {N}}$ and equate coefficients of the various powers of $t$ we obtain the fundamental equations with $R_1 = A_0$ , $R_2 = (R_1A_0 + A_1)/2 = (A_0^2 + A_1)/2$ and $R_n = (R_{n-1}A_0 + R_{n-2}A_1)/n$ for each $n \\in {\\mathbb {N}} + 2$ .", "These equations can be solved progressively to give $R_1 & = & A_0 \\\\R_2 & = & A_0^2/2 + A_1/2 \\\\R_3 & = & A_0^3/6 + \\left[ A_0 A_1/3 + A_1 A_0/6 \\right] \\\\R_4 & = & A_0^4/24 + \\left[ A_0^2A_1/8 + A_0A_1A_0/12 + A_1A_0^2/24 \\right] + A_1^2/8 \\\\\\vdots & & \\vdots $ and so on.", "The numerical calculation of these coefficients is a simple matter of setting $R_1 = A_0$ and $R_2 = (A_0^2 + A_1)/2$ and applying the recursive formula (REF ) with an appropriate restriction on the number of states.", "Nevertheless there are some interesting theoretical questions regarding calculation of these coefficients that relate to the shift operators $S_{\\ell }$ and $S_r$ .", "We will consider these questions separately.", "$\\hfill \\Box $" ], [ "Some general observations about the shift operators in the special birth and death process", "Consider the coefficients $A_j = (\\lambda _j U - \\mu _jS_{\\ell }U)$ for each $j = 1,2$ defined by (REF ), (REF ) and (REF ) in Example REF .", "We investigate the possibility of finding general formulæ for calculating ${A_j}^r$ for each $r \\in {\\mathbb {N}}$ .", "We have the following result.", "Proposition 1 The operators $U, S_{\\ell }U \\in {\\mathcal {B}}(\\ell ^1)$ satisfy $(i)$ $U{S_{\\ell }} = (I - S_{\\ell })$ , $(ii)$ $(S_{\\ell }U)^q = S_{\\ell }(I - S_{\\ell })^{q-1}U$ , and $(iii)$ $U^p(S_{\\ell }U)^q = U [ U^{p-1}(I + U^{-1})^{-q} ]_{\\mbox{\\scriptsize reg}} + S_{\\ell } [ {S_{\\ell }}^{-p}(I - S_{\\ell })^{p+q-1} ]_{\\mbox{\\scriptsize reg}} U$ for each $p, q \\in {\\mathbb {N}}$ where we have written $[f(X)]_{\\mbox{\\scriptsize reg}} = \\sum _{k \\in {\\mathbb {N}}-1} a_kX^k$ to denote the regular part of the Laurent series $f(X) = \\sum _{k \\in {\\mathbb {Z}}} a_k X^k$ and have used the formal notation $U [ U^{p-1}(I + U^{-1})^{-q} ]_{\\mbox{\\scriptsize reg}} = \\mbox{$\\sum _{k=0}^{p-1} (-1)^k \\binom{q+k-1}{k} U^{p-k}$}$ and $S_{\\ell }[ S_{\\ell }^{-p}(I - S_{\\ell })^{q+p-1} ]_{\\mbox{\\scriptsize reg}} = \\mbox{$ \\sum _{k=p}^{q+p-1} (-1)^k \\binom{q+p-1}{k} S_{\\ell }^{k-p+1}$}$ to denote the finite sums on the right-hand side despite the fact that the respective inverse operators $U^{-1}$ and $S_{\\ell }^{-1}$ do not exist.", "$ \\hfill \\Box $ Proof.", "To begin we note that for $0 < |z| < 1$ we have $z [ z^{p-1}(1 + z^{-1})^{-q} ]_{\\mbox{\\scriptsize reg}} & = & z \\mbox{$[z^{p-1} \\sum _{k=0}^{\\infty } (-1)^k \\binom{q+k-1}{k} z^{-k} ]_{\\mbox{\\scriptsize reg}}$} \\nonumber \\\\& = & z \\mbox{$[\\sum _{k=0}^{\\infty } (-1)^k \\binom{q+k-1}{k} z^{p-k-1} ]_{\\mbox{\\scriptsize reg}}$} \\nonumber \\\\& = & \\mbox{$\\sum _{k=0}^{p-1} (-1)^k \\binom{q+k-1}{k} z^{p-k}$}$ and for $z \\ne 0$ we have $z[ z^{-p}(1 - z)^{q+p-1} ]_{\\mbox{\\scriptsize reg}} & = & \\mbox{$z [ z^{-p} \\sum _{k=0}^{q+p-1} (-1)^k \\binom{q+p-1}{k} z^k ]_{\\mbox{\\scriptsize reg}}$} \\nonumber \\\\& = & \\mbox{$z [ \\sum _{k=0}^{q+p-1} (-1)^k \\binom{q+p-1}{k} z^{k-p} ]_{\\mbox{\\scriptsize reg}}$} \\nonumber \\\\& = & \\mbox{$ \\sum _{k=p}^{q+p-1} (-1)^k \\binom{q+p-1}{k} z^{k-p+1}$}$ The identities (REF ) and (REF ) explain the formal notation in (REF ) and (REF ).", "For $(i)$ we have $US_{\\ell } = (-I + S_r)S_{\\ell } = - S_{\\ell } + S_rS_{\\ell }$ and $S_rS_{\\ell } = \\left[ \\begin{array}{rrrrr}0 & 1 & 0 & 0 & \\cdots \\\\0 & 0 & 1 & 0 & \\cdots \\\\0 & 0 & 0 & 1 & \\cdots \\\\0 & 0 & 0 & 0 & \\cdots \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\end{array} \\right] \\left[ \\begin{array}{rrrrr}0 & 0 & 0 & 0 & \\cdots \\\\1 & 0 & 0 & 0 & \\cdots \\\\0 & 1 & 0 & 0 & \\cdots \\\\0 & 0 & 1 & 0 & \\cdots \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\end{array} \\right] = \\left[ \\begin{array}{rrrrr}1 & 0 & 0 & 0 & \\cdots \\\\0 & 1 & 0 & 0 & \\cdots \\\\0 & 0 & 1 & 0 & \\cdots \\\\0 & 0 & 0 & 1 & \\cdots \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\end{array} \\right] = I.$ Therefore $US_{\\ell } = I - S_{\\ell }$ .", "This establishes $(i)$ .", "For $(ii)$ we have $(S_{\\ell }U)^q = S_{\\ell } \\cdot (US_{\\ell })^{q-1} \\cdot U = S_{\\ell }(I - S_{\\ell })^{q-1}U$ for all $q \\in {\\mathbb {N}}$ .", "Therefore $(ii)$ is also true.", "The proof of $(iii)$ is by induction.", "It follows from $(i)$ and $(ii)$ that $U(S_{\\ell }U)^q = U \\cdot {S_{\\ell }}(I - S_{\\ell })^{q-1}U = US_{\\ell } \\cdot (I - S_{\\ell })^{q-1}U = (I - S_{\\ell })^q U$ When $p=1$ the formula $(iii)$ gives $U(S_{\\ell }U)^q & = & U [(I + U^{-1})^{-q}]_{\\mbox{\\scriptsize reg}} + S_{\\ell }[S_{\\ell }^{-1}(I - S_{\\ell })^q]_{\\mbox{\\scriptsize reg}} U \\\\& = & U[I - q U^{-1} + \\cdots ]_{\\mbox{\\scriptsize reg}} + S_{\\ell }[ S_{\\ell }^{-1} ( I - qS_{\\ell } + \\cdots + (-1)^qS_{\\ell }^q]_{\\mbox{\\scriptsize reg}} U \\\\& = & U [\\,I\\,] + S_{\\ell }[ S_{\\ell }^{-1}(I - S_{\\ell })^q - S_{\\ell }^{-1} ] U \\\\& = & (I - S_{\\ell })^q U.$ Thus $(iii)$ is true for $p=1$ .", "Suppose $(iii)$ is true for $p = r$ .", "That is, suppose $U^r(S_{\\ell }U)^q = \\mbox{$\\sum _{k=0}^{r-1} (-1)^k \\binom{q+k-1}{k} U^{r-k}$} + \\mbox{$ \\sum _{k=r}^{q+r-1} (-1)^k \\binom{q+r-1}{k} {S_{\\ell }}^{k-r+1}U$}.$ Therefore $U^{r+1}(S_{\\ell }U)^q & = & \\mbox{$\\sum _{k=0}^{r-1} (-1)^k \\binom{q+k-1}{k} U^{r+1-k}$} + \\mbox{$ \\sum _{k=r}^{q+r-1} (-1)^k \\binom{q+r-1}{k} U{S_{\\ell }}^{k-r+1}U$} \\\\& = & \\mbox{$\\sum _{k=0}^{r-1} (-1)^k \\binom{q+k-1}{k} U^{r+1-k}$} + \\mbox{$ \\sum _{k=r}^{q+r-1} (-1)^k \\binom{q+r-1}{k} S_{\\ell }^{k-r}(I - {S_{\\ell }})U$} \\\\& = & \\mbox{$\\sum _{k=0}^{r-1} (-1)^k \\binom{q+k-1}{k} U^{r+1-k}$} + \\mbox{$ \\sum _{k=r}^{q+r-1} (-1)^k \\binom{q+r-1}{k} S_{\\ell }^{k-r}U$} \\\\& & \\hspace{142.26378pt} - \\mbox{$\\sum _{k=r}^{q+r-1} (-1)^k \\binom{q+r-1}{k} S_{\\ell }^{k-r+1}U$} \\\\& = & \\mbox{$\\sum _{k=0}^r (-1)^k \\binom{q+k-1}{k} U^{r+1-k}$} + \\mbox{$ \\sum _{k=r+1}^{q+r-1} (-1)^k \\binom{q+r-1}{k} S_{\\ell }^{k-r}U$} \\\\& & \\hspace{85.35826pt} - \\mbox{$\\sum _{k=r}^{q+r-2} (-1)^k \\binom{q+r-1}{k} S_{\\ell }^{k-r+1}U$} + (-1)^{q+r} {S_{\\ell }}^qU \\\\& = & \\mbox{$\\sum _{k=0}^r (-1)^k \\binom{q+k-1}{k} U^{r+1-k}$} + \\mbox{$ \\sum _{k=r+1}^{q+r-1} (-1)^k \\binom{q+r-1}{k} S_{\\ell }^{k-r}U$} \\\\& & \\hspace{113.81102pt} - \\mbox{$\\sum _{j=r+1}^{q+r-1} (-1)^{j-1} \\binom{q+r-1}{j -1} S_{\\ell }^{j-r}U$} + (-1)^{q+r} {S_{\\ell }}^qU$ where we have replaced $k$ by $j = k+1$ in the third summation.", "Since $j$ is just a dummy variable of summation we can combine the second and third sums to give $U^{r+1}(S_{\\ell }U)^q & = & \\mbox{$\\sum _{k=0}^r (-1)^k \\binom{q+k-1}{k} U^{r+1-k}$} \\\\& & \\hspace{28.45274pt} + \\mbox{$ \\sum _{k=r+1}^{q+r-1} (-1)^k \\left[ \\binom{q+r-1}{k} + \\binom{q+r-1}{k -1} \\right] S_{\\ell }^{k-r}U$} + (-1)^{q+r} {S_{\\ell }}^qU\\\\& = & \\mbox{$\\sum _{k=0}^r (-1)^k \\binom{q+k-1}{k} U^{r+1-k}$} + \\mbox{$ \\sum _{k=r+1}^{q+r-1} (-1)^k \\binom{q+r}{k} S_{\\ell }^{k-r}U$} + (-1)^{q+r} {S_{\\ell }}^qU \\\\& = & \\mbox{$\\sum _{k=0}^r (-1)^k \\binom{q+k-1}{k} U^{r+1-k}$} + \\mbox{$ \\sum _{k=r+1}^{q+r} (-1)^k \\binom{q+r}{k} S_{\\ell }^{k-r}U$}$ where we have used the identity $\\mbox{$\\binom{q+r-1}{k} + \\binom{q+r-1}{k -1} = \\binom{q+r}{k}$}$ and included the final term in the second summation.", "Thus the formula $(iii)$ is also true for $p = r+1$ .", "Hence $(iii)$ is true for all $p \\in {\\mathbb {N}}$ .", "$\\hfill \\Box $ The primary purpose of Proposition REF is to write the powers of the coefficients $A_j$ as a sum of polynomials ${\\mathcal {P}}(U)$ and delayed polynomials $S_{\\ell }^p {\\mathcal {Q}}(U)$ .", "This means that the solution to the evolution equation can also expressed as a sum of polynomials and delayed polynomials.", "Any term that is expressed in the form $S_{\\ell }^p {\\mathcal {Q}}(U)$ has no effect on states $1,\\ldots ,p$ because $\\mbox{$\\mbox{$e$}$}_j S_{\\ell }^p = \\mbox{$\\mbox{$0$}$}$ for $j=1,\\ldots ,p$ .", "If we are primarily interested in what happens in states $1,\\ldots ,p$ we can neglect all terms in the solution prefixed by the operator $S_{\\ell }^{p+n}$ where $n \\in {\\mathbb {N}}-1$ .", "The next example shows how Proposition REF can be applied to move the operator $S_{\\ell }$ to the left within the product terms.", "Example 5 Consider the coefficient $A_j^3$ in Example REF for each $j = 0,1$ .", "We have $A_j^3 & = & (\\lambda _j U - \\mu _j S_{\\ell }U)^3 \\\\& = & \\lambda _j^3 U^3 - \\lambda _j^2\\mu _j [ U^2(S_{\\ell }U) + U(S_{\\ell }U)U + (S_{\\ell }U)U^2] \\\\& & \\hspace{14.22636pt} + \\lambda _j\\mu _j^2[ U (S_{\\ell }U)^2 + (S_{\\ell }U)U(S_{\\ell }U) + (S_{\\ell }U)^2U] - \\mu _j^3 (S_{\\ell }U)^3 \\\\& = & \\lambda _j^3U^3 - \\lambda _j^2\\mu _j[ U(I - S_{\\ell })U + (I - S_{\\ell })U^2 + S_{\\ell }U^3] \\\\& & \\hspace{14.22636pt} + \\lambda _j \\mu _j^2[ (I - S_{\\ell })^2U + S_{\\ell }U(I - S_{\\ell })U + S_{\\ell }(I - S_{\\ell })U^2] - \\mu _j^3 S_{\\ell }(I - S_{\\ell })^2U \\\\& = & \\lambda _j^3 U^3 - \\lambda _j^2\\mu _j [ U^2 - (I - S_{\\ell })U + (I - S_{\\ell })U^2 + S_{\\ell }U^3] \\\\& & \\hspace{14.22636pt} + \\lambda _j \\mu _j^2[ (I - S_{\\ell })^2U + S_{\\ell }U^2 - S_{\\ell }(I - S_{\\ell })U + S_{\\ell }(I - S_{\\ell })U^2] \\hspace{85.35826pt} \\\\& & \\hspace{256.0748pt} - \\mu _j^3 S_{\\ell }(I - S_{\\ell })^2U \\\\& = & \\lambda _j^3 U^3 - \\lambda _j^2\\mu _j [ (2U^2 - U) + S_{\\ell }(U^3 - U^2 + U)] \\\\& & \\hspace{14.22636pt} + \\lambda _j\\mu _j^2(U + S_{\\ell }[2U^2 - 3U] + S_{\\ell }^2[- U^2 + 2U]) - \\mu _j^3(S_{\\ell }U - 2S_{\\ell }^2U + S_{\\ell }^3U)$ for each $j =0,1$ .", "The terms in $A_j^k$ are either polynomials in $U$ or polynomials in $U$ multiplied on the left by a power of $S_{\\ell }$ .", "For a polynomial ${\\mathcal {P}}(U) \\in {\\mathcal {B}}(\\ell ^1)$ the mapping ${\\mathcal {P}}(U) \\mapsto S_{\\ell }{\\mathcal {P}}(U)$ causes one unit of delay or equivalently one unit of downward shift in the rows of ${\\mathcal {P}}(U)$ .", "Thus we categorize the polynomials in $A_j^k$ into two categories—the head polynomials with no delay and the tail polynomials of order $s$ with $s$ units of delay where $s \\in \\lbrace 1,\\ldots ,k\\rbrace $ .", "In the case of $A_j^3$ we write $A_j^3 & = & \\lambda _j^3 {\\mathfrak {H}}_{3,0}(U) - \\lambda _j^2 \\mu _j [ {\\mathfrak {H}}_{2,1}(U) + S_{\\ell } {\\mathfrak {T}}_{1,2,1}(U)] \\\\& & \\hspace{42.67912pt} + \\lambda _j \\mu _j^2 [{\\mathfrak {H}}_{1,2}(U) + S_{\\ell } {\\mathfrak {T}}_{1,1,2}(U) + S_{\\ell }^2 {\\mathfrak {T}}_{2,1,2}(U)] \\\\& & \\hspace{85.35826pt} - \\mu _j^3 [ S_{\\ell }{\\mathfrak {T}}_{1,0,3}(U) + S_{\\ell }^2 {\\mathfrak {T}}_{2,0,3}(U) + S_{\\ell }^3 {\\mathfrak {T}}_{3,0.3}(U)]$ where ${\\mathfrak {H}}_{m,k-m}(U)$ is the head polynomial for the term in $(-1)^n \\lambda _j^m \\mu _j^{k-m}$ and ${\\mathfrak {T}}_{s,m,k-m}(U)$ is the tail polynomial for the term in $(-1)^{k-m} \\lambda _j^m \\mu _j^{k-n}$ with $s$ units of delay for $s \\in \\lbrace 1,\\ldots ,k\\rbrace $ .", "In this case the head polynomials are ${\\mathfrak {H}}_{3,0}(U) = U^3, \\quad {\\mathfrak {H}}_{2,1}(U) = 2U^2 - U, \\quad {\\mathfrak {H}}_{1,2}(U) = U,$ and the tail polynomials are ${ {\\mathfrak {T}}_{1,2,1}(U) = U^3 - U^2 + U, \\quad {\\mathfrak {T}}_{1,1,2}(U) = 2U^2 - 3U, } \\hspace{28.45274pt} \\\\& & \\quad {\\mathfrak {T}}_{2,1,2} = -U^2 + 2U, \\quad {\\mathfrak {T}}_{1,0,3}(U) = U, \\quad {\\mathfrak {T}}_{2,0,3}(U) = - 2U, \\quad {\\mathfrak {T}}_{3,0,3}(U) = U.$ To determine these polynomials we consider the individual evolutionary processes.", "$\\hfill \\Box $ In general we have an expression in the form $A_j^k & = & \\lambda _j^k {\\mathfrak {H}}_{k,0}(U) + \\mbox{$\\sum _{p=1}^{k-1}$} (-1)^p \\lambda _j^{k-p} \\mu _j^{\\,p} \\left[ {\\mathfrak {H}}_{k-p,p}(U) + \\mbox{$\\sum _{q = 1}^{p}$} S_{\\ell }^{\\,q}\\, {\\mathfrak {T}}_{q,k-p,p}(U) \\right] \\hspace{56.9055pt} \\\\& & \\hspace{241.84842pt} + (-1)^k \\sum _{q=1}^k S_{\\ell }^{\\, q} {\\mathfrak {T}}_{k,0,k}(U)$ and the task is to determine the individual polynomials.", "The general idea of the required reduction is to move the delay operators to the left.", "The process finishes when there is no power of $U$ preceding a power of $S_{\\ell }$ .", "The following formulæ will be useful.", "Proposition 2 We have $(iv)$ $U^q {S_{\\ell }}^{\\,q+r-1} = (I - S_{\\ell })^q S_{\\ell }^{\\,r-1} $ ; $(v)$ $U^{q+r}{S_{\\ell }}^{\\,q} = U^r(I - S_{\\ell })^q$ ; $(vi)$ $U^r(I-S_{\\ell })^q = U^r(I - S_{\\ell })^{q-1} - U^{r-1}(I - S_{\\ell })^q$ ; for $q, r \\in {\\mathbb {N}}$ .", "It is possible to give a general recursive formula for the polynomials ${\\mathfrak {H}}_{m,k-m}(U)$ and ${\\mathfrak {T}}_{s,m,k-m}(U)$ for each $s \\in \\lbrace 1,\\ldots ,k\\rbrace $ but the complexities of the recursion mean that direct calculation is much more efficient.", "Rather than consider all $2^p$ individual products in the binomial expansion $A_j^k = (\\lambda _j U - \\mu _j S_{\\ell }U)^k$ we group the terms according to the overall powers $(-1)^{k-m} \\lambda _j^m \\mu _j^{k-m}$ .", "For each $k \\in {\\mathbb {N}}$ we define the binomial group ${\\mathfrak {B}}_{m,k-m}(U)$ of polynomials ${\\mathfrak {B}}_{m,k-m}(U) = {\\mathfrak {H}}_{m,k-m}(U) + \\left[ {\\mathfrak {T}}_{1,m,k-m}(U) + \\cdots + {\\mathfrak {T}}_{k-1,m,k-m}(U) \\right]$ for each $m=1,\\ldots ,k-1$ .", "We also have ${\\mathfrak {B}}_{k,0} = {\\mathfrak {H}}_{k,0}$ and ${\\mathfrak {B}}_{0,k} = {\\mathfrak {T}}_{1,0,k} + \\cdots + {\\mathfrak {T}}_{k,0,k}$ .", "Example 6 Calculate ${\\mathfrak {B}}_{2,2}(U)$ and the associated head polynomial and tail polynomials using the formulæ in Propositions REF and REF .", "We have ${\\mathfrak {B}}_{2,2}(U) & = & U^2(S_{\\ell }U)^2 + U(S_{\\ell }U)U(S_{\\ell }U) + U(S_{\\ell }U)^2U + (S_{\\ell }U)U^2(S_{\\ell }U) \\\\& & \\hspace{199.16928pt} + (S_{\\ell }U)U(S_{\\ell }U)U + (S_{\\ell }U)^2 U^2 \\\\& = & \\lbrace U[ U(I + U^{-1})^{-2} ]_{\\mbox{\\rm \\scriptsize reg}} + S_{\\ell } [ S_{\\ell }^{-2}(I - S_{\\ell })^3]_{\\mbox{\\rm \\scriptsize reg}}U \\rbrace + US_{\\ell } \\cdot U^2S_{\\ell } \\cdot U \\\\& & \\hspace{42.67912pt} + \\lbrace U [ (I + U^{-1})^{-2} ]_{\\mbox{\\rm \\scriptsize reg}} + S_{\\ell } [ {S_{\\ell }}^{-1} (I - S_{\\ell })^2 ]_{\\mbox{\\rm \\scriptsize reg}} U \\rbrace U \\\\& & \\hspace{85.35826pt} + S_{\\ell } \\cdot U^3 (S_{\\ell }U) + S_{\\ell } \\cdot U^2S_{\\ell } \\cdot U^2 + S_{\\ell }(I - S_{\\ell })U \\cdot U^2 \\\\& = & \\lbrace U[ U - 2I] + S_{\\ell } [- S_{\\ell } + 3I ] U \\rbrace + (I - S_{\\ell }) \\cdot U(I - S_{\\ell }) \\cdot U \\\\& & \\hspace{28.45274pt} + \\lbrace U [I] + S_{\\ell } [-2I + S_{\\ell }] U \\rbrace U \\\\& & \\hspace{56.9055pt} + S_{\\ell } \\lbrace U [ U^2(I + U^{-1})^{-1}]_{\\mbox{\\rm \\scriptsize reg}} + S_{\\ell } [ {S_{\\ell }}^{-3}(I - S_{\\ell })^3 ]_{\\mbox{\\rm \\scriptsize reg}} U \\rbrace \\\\& & \\hspace{85.35826pt} + S_{\\ell } \\cdot U(I - S_{\\ell }) \\cdot U^2 + S_{\\ell }U^3 - {S_{\\ell }}^2 U^3 \\\\& = & \\lbrace U^2 - 2U + 3S_{\\ell }U - {S_{\\ell }}^2U \\rbrace + \\lbrace (I - S_{\\ell })U^2 - (I - S_{\\ell })^2 U \\rbrace \\\\& & \\hspace{56.9055pt} + \\lbrace U^2 - 2S_{\\ell } U^2 + {S_{\\ell }}^2U^2 \\rbrace + S_{\\ell } \\lbrace [U^3 - U^2 + U] - S_{\\ell }U \\rbrace \\\\& & \\hspace{113.81102pt} + \\lbrace S_{\\ell } U^3 - S_{\\ell }(I - S_{\\ell }) U^2\\rbrace + S_{\\ell }U^3 - {S_{\\ell }}^2 U^3 \\\\& = & [3U^2 - 3U] + S_{\\ell } [ 3U^3 - 5U^2 + 6U] + {S_{\\ell }}^2 [ -U^3 + 2U^2 -3U].$ Therefore ${\\mathfrak {H}}_{2,2}(U) = 3U^2 - 3U$ , ${\\mathfrak {T}}_{1,2,2}(U) = 3U^3 - 5U^2 + 6U$ and ${\\mathfrak {T}}_{2,2,2}(U) = -U^3 + 2U^2 - 3U$ .", "$ \\hfill \\Box $" ], [ "Conclusions", "We have shown that the method of equation of coefficients can be used to solve the non-autonomous equation of evolution for bounded linear operators on Banach space.", "The proposed method provides a viable alternative to the standard Peano–Baker iteration when the evolutionary operator is analytic at the origin.", "The method could also be used to find an analytic approximation to evolution equations where the evolutionary operator is continuous but not analytic on some closed interval.", "In such cases one might use a uniform analytic approximation to the evolutionary operator and solve the corresponding analytic evolutionary equation.", "Our future research will consider the effectiveness of such approximations.", "We would also like to investigate effective calculation procedures for more general birth and death processes." ] ]
2105.11670
[ [ "On WL-rank of Deza Cayley graphs" ], [ "Abstract The WL-rank of a digraph $\\Gamma$ is defined to be the rank of the coherent configuration of $\\Gamma$.", "We construct a new infinite family of strictly Deza Cayley graphs for which the WL-rank is equal to the number of vertices.", "The graphs from this family are divisible design and integral." ], [ "Introduction", "Let $V$ be a finite set and $|V|=n$ .", "A coherent configuration $\\mathcal {X}$ on $V$ can be thought as a special partition of $V\\times V$ for which the diagonal of $V\\times V$ is a union of classes (see [4]).", "The number of classes is called the rank of $\\mathcal {X}$ .", "Let $\\Gamma =(V,E)$ be a digraph with vertex set $V$ and arc set $E$ .", "The WL-rank (the Weisfeiler-Leman rank) of $\\Gamma $ is defined to be the rank of the smallest coherent configuration on the set $V$ for which $E$ is a union of classes.", "The term “WL-rank of a digraph” was introduced in [1].", "This term was chosen because the coherent configuration of a digraph can be found using the Weisfeiler-Leman algorithm [20].", "Since the diagonal of $V\\times V$ is a union of classes of any coherent configuration on $V$ , we conclude that $\\operatorname{rk_{WL}}(\\Gamma )\\ge 2$ unless $|V|=1$ .", "One can verify that $\\operatorname{rk_{WL}}(\\Gamma )\\le 2$ if and only if $\\Gamma $ is complete or empty.", "On the other hand, obviously, $\\operatorname{rk_{WL}}(\\Gamma )\\le n^2$ .", "From [1] it follows that if $\\Gamma $ is vertex-transitive then $\\operatorname{rk_{WL}}(\\Gamma )\\le n$ .", "Let $G$ be a finite group, $|G|=n$ , and $S$ an identity-free subset of $G$ .", "The Cayley digraph $\\operatorname{Cay}(G,S)$ is defined to be the digraph with vertex set $G$ and arc set $\\lbrace (g,sg):~s\\in S,~g\\in G\\rbrace $ .", "If $S$ is inverse-closed then $\\operatorname{Cay}(G,S)$ is a Cayley graph.", "If $\\Gamma $ is a Cayley digraph over $G$ then $\\operatorname{Aut}(\\Gamma )\\ge G_{\\mathrm {right}}$ , where $G_{\\mathrm {right}}$ is the subgroup of $\\operatorname{Sym}(G)$ induced by right multiplications of $G$ .", "This implies that $\\Gamma $ is vertex-transitive and hence $\\operatorname{rk_{WL}}(\\Gamma )\\le n$ .", "A $k$ -regular graph $\\Gamma $ is called strongly regular if there exist nonnegative integers $\\lambda $ and $\\mu $ such that every two adjacent vertices have $\\lambda $ common neighbors and every two nonadjacent vertices have $\\mu $ common neighbors.", "The following generalization of the notion of a strongly regular graph was introduced in [6] and goes back to [5].", "A $k$ -regular graph $\\Gamma $ on $n$ vertices is called a Deza graph if there exist nonnegative integers $\\alpha $ and $\\beta $ such that any pair of distinct vertices of $\\Gamma $ has either $\\alpha $ or $\\beta $ common neighbors.", "The numbers $(n,k,\\beta ,\\alpha )$ are called the parameters of $\\Gamma $ .", "Clearly, if $\\alpha >0$ and $\\beta >0$ then $\\Gamma $ has diameter 2.", "A Deza graph is called a strictly Deza graph if it is nonstrongly regular and has diameter 2.", "The WL-rank of a strongly regular graph is at most 3 (see [1]).", "It is a natural question how large the WL-rank of a Deza graph $\\Gamma $ can be.", "In this paper we are interested in the WL-rank of Deza Cayley graphs.", "The WL-rank of a nonstrictly Deza Cayley graph can be sufficiently large.", "For example, an undirected cycle on $n$ vertices is a nonstrictly Deza graph of WL-rank $[\\frac{n}{2}]+1$ (see [1]).", "However, strictly Deza graphs seem close to strongly regular graphs.", "All known strictly Deza Cayley graphs over cyclic groups have WL-rank at most 6 [1].", "As it was said before, the WL-rank of any Cayley graph does not exceed the number of vertices of this graph.", "It turns out that there exists an infinite family of strictly Deza Cayley graphs whose WL-rank is equal to the number of vertices.", "This follows from the theorem below which is the main result of this paper.", "The cyclic and dihedral groups of order $n$ are denoted by $C_n$ and $D_n$ respectively.", "Theorem 1 Let $k\\ge 3$ be an odd integer, $G\\cong D_{2k}\\times C_2\\times C_2$ , and $n=|G|$ .", "There exists a strictly Deza Cayley graph $\\Gamma $ over $G$ such that $\\operatorname{rk_{WL}}(\\Gamma )=n$ .", "Note that the graphs from Theorem  are divisible design integral graphs (see Section 4).", "We finish the introduction with the brief outline of the paper.", "If $\\Gamma =\\operatorname{Cay}(G,S)$ then the WL-rank of $\\Gamma $ is equal to the rank of the smallest $S$ -ring over $G$ for which $S$ is a union of basic sets.", "The necessary background of $S$ -rings and Cayley graphs is provided in Section 2.", "In Section 3 we construct the required family of strictly Deza Cayley graphs and prove Theorem .", "In Section 4 we prove that each graph from the constructed family is an integral divisible design graph (Lemma REF ), has the same parameters as the grid graph but not isomorphic to it (Lemma REF ), and can be identified efficiently (Lemma REF ).", "The authors would like to thank prof.", "I. Ponomarenko for the valuable comments which help us to improve the text significantly." ], [ "Preliminaries", "In this section we provide a background of $S$ -rings and Cayley graphs.", "In general, we follow to [1], [16], [17], where the most of definitions and statements is contained." ], [ "$S$ -rings", "Let $G$ be a finite group and $\\mathbb {Z}G$ the integer group ring.", "The identity element of $G$ and the set of all nonidentity elements of $G$ are denoted by $e$ and $G^\\#$ respectively.", "If $X\\subseteq G$ then the element $\\sum \\limits _{x\\in X} {x}$ of the group ring $\\mathbb {Z}G$ is denoted by $\\underline{X}$ .", "An easy straightforward computation implies that $\\underline{G}^2=|G|\\underline{G}$ .", "The set $\\lbrace x^{-1}:x\\in X\\rbrace $ is denoted by $X^{-1}$ .", "A subring $\\mathcal {A}\\subseteq \\mathbb {Z} G$ is called an $S$ -ring (a Schur ring) over $G$ if there exists a partition $\\mathcal {S}=\\mathcal {S}(\\mathcal {A})$ of $G$ such that: $(1)$ $\\lbrace e\\rbrace \\in \\mathcal {S}$ ; $(2)$ if $X\\in \\mathcal {S}$ then $X^{-1}\\in \\mathcal {S}$ ; $(3)$ $\\mathcal {A}=\\operatorname{Span}_{\\mathbb {Z}}\\lbrace \\underline{X}:\\ X\\in \\mathcal {S}\\rbrace $ .", "The notion of an $S$ -ring goes back to Schur [18] and Wielandt [19].", "The elements of $\\mathcal {S}$ are called the basic sets of $\\mathcal {A}$ and the number $\\operatorname{rk}(\\mathcal {A})=|\\mathcal {S}|$ is called the rank of $\\mathcal {A}$ .", "The group ring $\\mathbb {Z}G$ is an $S$ -ring over $G$ corresponding to the partition of $G$ into singletons and $\\operatorname{rk}(\\mathbb {Z}G)=|G|$ .", "The following lemma provides a well-known property of $S$ -rings (see, e.g. [16]).", "Lemma 2.1 Let $\\mathcal {A}$ be an $S$ -ring over a group $G$ .", "If $X,Y\\in \\mathcal {S}(\\mathcal {A})$ then $XY\\in \\mathcal {S}(\\mathcal {A})$ whenever $|X|=1$ or $|Y|=1$ .", "Lemma 2.2 Let $\\mathcal {A}$ be an $S$ -ring over a group $G$ and $X\\subseteq G$ such that $\\langle X \\rangle =G$ .", "Suppose that $\\lbrace x\\rbrace \\in \\mathcal {S}(\\mathcal {A})$ for every $x\\in X$ .", "Then $\\mathcal {A}=\\mathbb {Z}G$ .", "Let us prove that $\\lbrace g\\rbrace \\in \\mathcal {S}(\\mathcal {A})$ for every $g\\in G$ .", "Since $\\langle X \\rangle =G$ , there exist $x_1,\\ldots ,x_k\\in X$ and $\\varepsilon _1,\\ldots \\varepsilon _k\\in \\lbrace -1,1\\rbrace $ such that $g=x_1^{\\varepsilon _1}\\ldots x_k^{\\varepsilon _k}$ .", "We proceed by induction on $k$ .", "Let $k=1$ .", "If $\\varepsilon _1=1$ then $\\lbrace g\\rbrace \\in \\mathcal {S}(\\mathcal {A})$ by the assumption of the lemma; if $\\varepsilon _1=-1$ then $\\lbrace g\\rbrace \\in \\mathcal {S}(\\mathcal {A})$ by the assumption of the lemma and the second property from the definition of an $S$ -ring.", "Now let $k\\ge 2$ .", "By the induction hypothesis, we have $\\lbrace x_1^{\\varepsilon _1}\\ldots x_{k-1}^{\\varepsilon _{k-1}}\\rbrace \\in \\mathcal {S}(\\mathcal {A})$ and $\\lbrace x_k^{\\varepsilon _k}\\rbrace \\in \\mathcal {S}(\\mathcal {A})$ .", "So $\\lbrace g\\rbrace =\\lbrace x_1^{\\varepsilon _1}\\ldots x_{k-1}^{\\varepsilon _{k-1}}\\rbrace \\lbrace x_k^{\\varepsilon _k}\\rbrace \\in \\mathcal {S}(\\mathcal {A})$ by Lemma REF .", "A set $X \\subseteq G$ is called an $\\mathcal {A}$ -set if $\\underline{X}\\in \\mathcal {A}$ or, equivalently, $X$ is a union of some basic sets of $\\mathcal {A}$ .", "The set of all $\\mathcal {A}$ -sets is denoted by $\\mathcal {S}^*(\\mathcal {A})$ .", "Obviously, if $X\\in \\mathcal {S}^*(\\mathcal {A})$ and $|X|=1$ then $X\\in \\mathcal {S}(\\mathcal {A})$ .", "It is easy to check that if $X,Y\\in \\mathcal {S}^*(\\mathcal {A})$ then $X\\cap Y,X\\cup Y, X\\setminus Y, Y\\setminus X, XY\\in \\mathcal {S}^*(\\mathcal {A}).~\\qquad \\mathrm {(1)}$ A subgroup $H \\le G$ is called an $\\mathcal {A}$ -subgroup if $H\\in \\mathcal {S}^*(\\mathcal {A})$ .", "For every $\\mathcal {A}$ -set $X$ , the groups $\\langle X \\rangle $ and $\\operatorname{rad}(X)=\\lbrace g\\in G:~Xg=gX=X\\rbrace $ are $\\mathcal {A}$ -subgroups.", "Lemma 2.3 [19] Let $\\mathcal {A}$ be an $S$ -ring over $G$ , $\\xi =\\sum \\limits _{g\\in G} c_g g\\in \\mathcal {A}$ , where $c_g\\in \\mathbb {Z}$ , and $c\\in \\mathbb {Z}$ .", "Then $\\lbrace g\\in G:~c_g=c\\rbrace \\in \\mathcal {S}^*(\\mathcal {A})$ .", "Let $L \\unlhd U\\le G$ .", "A section $U/L$ is called an $\\mathcal {A}$ -section if $U$ and $L$ are $\\mathcal {A}$ -subgroups.", "If $S=U/L$ is an $\\mathcal {A}$ -section then the module $\\mathcal {A}_S=Span_{\\mathbb {Z}}\\left\\lbrace \\underline{X}^{\\pi }:~X\\in \\mathcal {S}(\\mathcal {A}),~X\\subseteq U\\right\\rbrace ,$ where $\\pi :U\\rightarrow U/L$ is the canonical epimorphism, is an $S$ -ring over $S$ .", "Let $S=U/L$ be an $\\mathcal {A}$ -section of $G$ .", "The $S$ -ring $\\mathcal {A}$ is called the $S$ -wreath product or generalized wreath product of $\\mathcal {A}_U$ and $\\mathcal {A}_{G/L}$ if $L\\trianglelefteq G$ and $L\\le \\operatorname{rad}(X)$ for each basic set $X$ outside $U$ .", "In this case we write $\\mathcal {A}=\\mathcal {A}_U\\wr _{S}\\mathcal {A}_{G/L}$ .", "If $L>\\lbrace e\\rbrace $ and $U<G$ then the $S$ -wreath product is called nontrivial.", "The notion of the generalized wreath product of $S$ -rings was introduced in [7].", "Since $L\\le \\operatorname{rad}(X)$ for each basic set $X$ outside $U$ , the basic sets of $\\mathcal {A}$ outside $U$ are in one-to-one correspondence with the basic sets of $\\mathcal {A}_{G/L}$ outside $S$ .", "Therefore $\\operatorname{rk}(\\mathcal {A}_U\\wr _{S}\\mathcal {A}_{G/L})=\\operatorname{rk}(\\mathcal {A}_U)+\\operatorname{rk}(\\mathcal {A}_{G/L})-\\operatorname{rk}(\\mathcal {A}_S).~\\qquad \\mathrm {(2)}$ The automorphism group $\\operatorname{Aut}(\\mathcal {A})$ of $\\mathcal {A}$ is defined to be the group $\\bigcap \\limits _{X\\in \\mathcal {S}(\\mathcal {A})} \\operatorname{Aut}(\\operatorname{Cay}(G,X)).$ Since $\\operatorname{Aut}(\\operatorname{Cay}(G,X))\\ge G_{\\mathrm {right}}$ for every $X\\in \\mathcal {S}(\\mathcal {A})$ , we conclude that $\\operatorname{Aut}(\\mathcal {A})\\ge G_{\\mathrm {right}}$ .", "It is easy to check that $\\operatorname{Aut}(\\mathcal {A})=G_{\\mathrm {right}}$ if and only if $\\mathcal {A}=\\mathbb {Z}G$ ." ], [ "Cayley graphs", "Let $S\\subseteq G$ , $e\\notin S$ , and $\\Gamma =\\operatorname{Cay}(G,S)$ .", "The WL-closure $\\operatorname{WL}(\\Gamma )$ of $\\Gamma $ can be thought as the smallest $S$ -ring over $G$ such that $S\\in \\mathcal {S}^*(\\mathcal {A})$ (see [1]).", "If $\\mathcal {A}=\\operatorname{WL}(\\Gamma )$ then $\\operatorname{rk_{WL}}(\\Gamma )=\\operatorname{rk}(\\mathcal {A})$ by [1].", "From [4] it follows that $\\operatorname{Aut}(\\Gamma )=\\operatorname{Aut}(\\mathcal {A})$ .", "Lemma 2.4 [1] Let $G$ be a group of order $n$ , $S\\subseteq G$ such that $e\\notin S$ , $S=S^{-1}$ , and $|S|=k$ , and $\\Gamma =\\operatorname{Cay}(G,S)$ .", "The graph $\\Gamma $ is a Deza graph with parameters $(n,k,\\beta ,\\alpha )$ if and only if $\\underline{S}^2=ke+\\alpha \\underline{X_{\\alpha }}+\\beta \\underline{X_{\\beta }}$ , where $X_{\\alpha }\\cup X_{\\beta }=G^\\#$ and $X_{\\alpha }\\cap X_{\\beta }=\\varnothing $ .", "Moreover, $\\Gamma $ is strongly regular if and only if $X_{\\alpha }=S$ or $X_{\\beta }=S$ ." ], [ "Proof of Theorem 1", "Let $k\\ge 3$ be an integer, $G=(\\langle a \\rangle \\rtimes \\langle b \\rangle )\\times \\langle c \\rangle \\times \\langle d \\rangle $ , where $|a|=k$ , $|b|=|c|=|d|=2$ , and $bab=a^{-1}$ , and $n=|G|$ .", "The groups $\\langle a \\rangle $ , $\\langle c\\rangle $ , and $\\langle a \\rangle \\rtimes \\langle b \\rangle $ are denoted by $A$ , $C$ , and $H$ respectively.", "Clearly, $H\\cong D_{2k}$ , $G\\cong D_{2k}\\times C_2 \\times C_2$ , $|H|=2k$ , and $|G|=8k$ .", "Put $S=b(A\\setminus \\lbrace a^{-1}\\rbrace )\\cup c(A\\cup \\lbrace b\\rbrace )\\cup \\lbrace db,dcba^{-1}\\rbrace .$ One can see that $S=S^{-1}$ and $|S|=2(k+1)$ .", "Put $\\Gamma =\\operatorname{Cay}(G,S)$ .", "Note that $\\Gamma $ is $2(k+1)$ -regular.", "Lemma 3.1 In the above notations, the graph $\\Gamma $ is a strictly Deza graph with parameters $(8k,2(k+1),2(k-1),2)$ .", "The straightforward computation in the group ring $\\mathbb {Z}G$ using the equalities $\\underline{A}^2=k\\underline{A}$ , $b\\underline{A}=\\underline{A}b$ , $bab=a^{-1}$ , $cg=gc$ , and $dg=gd$ , where $g\\in G$ , implies that $\\underline{S}^2=2(k+1)e+2(k-1)(\\underline{A}^\\#+cb\\underline{A})+2(b+c)\\underline{A}+2d\\underline{C}\\underline{H}.~\\qquad \\mathrm {(3)}$ Indeed, $\\underline{S}^2=(b\\underline{A}+c\\underline{A}-ba^{-1}+cb+db+dcba^{-1})^2=$ $=4e+(2(k-1)e+2(k-1)cb+2b+2c+2d+2dc+2db+2dcb)\\underline{A}=$ $=2(k+1)e+2(k-1)(\\underline{A}^\\#+cb\\underline{A})+2(b+c)\\underline{A}+2d\\underline{C}\\underline{H}.$ From Lemma REF and Eq.", "(3) it follows that $\\Gamma $ is a nonstrongly regular Deza graph with parameters $(8k,2(k+1),2(k-1),2)$ , $X_{2(k-1)}=A^\\#\\cup cbA$ , and $X_2=bA\\cup cA\\cup d(C\\times H)$ .", "This means that $\\Gamma $ is a strictly Deza graph.", "All Deza Cayley graphs with at most 60 vertices, including the graphs from the constructed family for $k\\le 7$ , were enumerated in [10].", "Put $A_1=\\langle a^2 \\rangle $ .", "If $k$ is odd then $A_1=A$ ; if $k$ is even then $|A:A_1|=2$ .", "The group $A_1$ is normal in $G$ .", "So one can form the group $L=A_1\\rtimes \\langle cb \\rangle $ which is isomorphic to $D_{2k}$ if $k$ is odd and to $D_k$ if $k$ is even.", "It can be verified in a straightforward way that $L$ is normal in $G$ .", "Put $U=\\langle L,ca,da\\rangle $ and $S=U/L$ .", "Since $L\\cap \\langle ca \\rangle =L\\cap \\langle da \\rangle =A_1$ , we obtain $|U:L|=4$ .", "Lemma 3.2 In the above notations, $\\operatorname{WL}(\\Gamma )=\\mathbb {Z}G$ if $k$ is odd and $\\operatorname{WL}(\\Gamma )=\\mathbb {Z}U\\wr _S \\mathbb {Z}(G/L)$ if $k$ is even.", "Let $\\mathcal {A}=\\operatorname{WL}(\\Gamma )$ .", "Put $V=A^\\#\\cup cbA$ .", "From Eq.", "(3) it follows that every element of $V$ enters the element $\\underline{S}^2$ with coefficient $2(k-1)$ and any other element of $G$ enters $\\underline{S}^2$ with coefficient distinct from $2(k-1)$ .", "Together with $S\\in \\mathcal {S}^*(\\mathcal {A})$ and Lemma REF , this implies that $V\\in \\mathcal {S}^*(\\mathcal {A})$ .", "So $V\\cap S=\\lbrace cb\\rbrace \\in \\mathcal {S}(\\mathcal {A})~\\qquad \\mathrm {(4)}$ by Eq. (1).", "Since $S,\\lbrace cb\\rbrace \\in \\mathcal {S}^*(\\mathcal {A})$ , Eq.", "(1) implies that $cbS,Scb\\in \\mathcal {S}^*(\\mathcal {A})$ .", "So $S_1=(cbS\\setminus Scb)\\cap S=\\lbrace ca\\rbrace \\in \\mathcal {S}(\\mathcal {A})~\\qquad \\mathrm {(5)}$ by Eq. (1).", "Now from Eqs.", "(1) and (5) it follows that $S_2=(cbS\\setminus Scb)\\setminus S_1=\\lbrace da^{-1}\\rbrace \\in \\mathcal {S}(\\mathcal {A})~\\qquad \\mathrm {(6)}$ Due to Eqs.", "(1) and (5), we obtain $S_1S_1=\\lbrace a^2\\rbrace \\in \\mathcal {S}(\\mathcal {A})$ .", "Since $a^2$ is a generator of $A_1$ , Lemma REF yields that $\\mathcal {A}_{A_1}=\\mathbb {Z}A_1.~\\qquad \\mathrm {(7)}$ Let $k$ be odd.", "Then $A_1=A$ and $G=\\langle A, cb, ca, da^{-1} \\rangle $ .", "From Eqs.", "(4)-(7) and Lemma REF it follows that $\\mathcal {A}=\\mathbb {Z}G$ .", "Let $k$ be even.", "The partition of $G$ into sets $\\lbrace g\\rbrace ,~g\\in U,~La,~Lc,~Ld,~Lcda$ defines the $S$ -ring $\\mathcal {B}$ over $G$ such that $\\mathcal {B}=\\mathbb {Z}U\\wr _S \\mathbb {Z}(G/L)$ .", "Note that $S=Lc\\cup S_U$ , where $S_U=b(A\\setminus (A_1\\cup \\lbrace a^{-1}\\rbrace ))\\cup c((A\\setminus A_1)\\cup \\lbrace b\\rbrace )\\cup \\lbrace db,dcba^{-1}\\rbrace \\subseteq U.$ So $S\\in \\mathcal {S}^*(\\mathcal {B})$ and hence $\\mathcal {B}\\ge \\mathcal {A}$ .", "Let us prove that $\\mathcal {B}\\le \\mathcal {A}$ .", "Observe that $da\\in da^{-1}A_1$ .", "So $\\lbrace da\\rbrace \\in \\mathcal {S}(\\mathcal {A})~\\qquad \\mathrm {(8)}$ by Eqs.", "(6)-(7) and Lemma REF .", "Eqs.", "(4), (5), (7), (8), and Lemma REF imply that $U$ is an $\\mathcal {A}$ -subgroup and $\\mathcal {A}_U=\\mathbb {Z}U=\\mathcal {B}_U.~\\qquad \\mathrm {(9)}$ Since $S\\in \\mathcal {S}^*(\\mathcal {A})$ and $U\\in \\mathcal {S}^*(\\mathcal {A})$ , Eq.", "(1) implies that $S\\setminus U=Lc\\in \\mathcal {S}^*(\\mathcal {A}).~\\qquad \\mathrm {(10)}$ From Eqs.", "(1), (5), (6), (8), and (10) it follows that $La=Lc\\lbrace ca\\rbrace ,~Ld=Lc\\lbrace ca\\rbrace \\lbrace da^{-1}\\rbrace ,~Lcda=Lc\\lbrace da\\rbrace \\in \\mathcal {S}^*(\\mathcal {A}).$ Together with Eq.", "(9), this implies that every basic set of $\\mathcal {B}$ is an $\\mathcal {A}$ -set and hence $\\mathcal {B}\\le \\mathcal {A}$ .", "Thus, $\\mathcal {B}=\\mathcal {A}$ and we are done.", "Remark If $k$ is odd then $\\operatorname{Aut}(\\Gamma )=\\operatorname{Aut}(\\mathbb {Z}G)=G_{\\mathrm {right}}$ .", "If $k$ is even then $\\operatorname{Aut}(\\Gamma )=\\operatorname{Aut}(\\mathbb {Z}U\\wr _S \\mathbb {Z}(G/L))$ is the canonical generalized wreath product of $U_{\\mathrm {right}}$ by $(G/L)_{\\mathrm {right}}$ (see [8] for the definitions).", "Lemma 3.3 In the above notations, $\\operatorname{rk_{WL}}(\\Gamma )=8k=n$ if $k$ is odd and $\\operatorname{rk_{WL}}(\\Gamma )=4k+4=\\frac{n}{2}+4$ if $k$ is even.", "If $k$ is odd then $\\operatorname{WL}(\\Gamma )=\\mathbb {Z}G$ by Lemma REF and hence $\\operatorname{rk_{WL}}(\\Gamma )=\\operatorname{rk}(\\mathbb {Z}G)=8k$ .", "Let $k$ be even.", "Then $\\operatorname{WL}(\\Gamma )=\\mathbb {Z}U\\wr _S \\mathbb {Z}(G/L)$ by Lemma REF .", "Since $|L|=k$ and $|U:L|=4$ , we have $|U|=4k$ and hence $\\operatorname{rk}(\\mathbb {Z}U)=4k$ .", "Observe that $|G/L|=8$ and $|S|=|U/L|=4$ .", "So $\\operatorname{rk}(\\mathbb {Z}(G/L))=8$ and $\\operatorname{rk}(\\mathbb {Z}S)=4$ .", "Therefore $\\operatorname{rk_{WL}}(\\Gamma )=\\operatorname{rk}(\\mathbb {Z}U\\wr _S \\mathbb {Z}(G/L))=\\operatorname{rk}(\\mathbb {Z}U)+\\operatorname{rk}(\\mathbb {Z}(G/L))-\\operatorname{rk}(\\mathbb {Z}S)=4k+4$ by Eq. (2).", "Theorem  follows from Lemma REF and Lemma REF ." ], [ "Some properties of $\\Gamma $", "In this section we collect some properties of the graph $\\Gamma $ constructed in the previous section.", "A $k$ -regular graph on $n$ vertices is called a divisible design graph (DDG) with parameters $(n,k,\\alpha ,\\beta ,m,l)$ if its vertex set can be partitioned into $m$ classes of size $l$ , such that every two distinct vertices from the same class have $\\alpha $ common neighbors and every two vertices from different classes have $\\beta $ common neighbors.", "For a divisible design graph, the partition into classes is called a canonical partition.", "The notion of a divisible design graph was introduced in [12] as a generalization of $(v,k,\\lambda )$ -graphs [15].", "For more information on divisible design graphs, we refer the readers to [12], [14].", "A graph is called integral if all eigenvalues of its adjacency matrix are integers.", "The investigations on integral graphs goes back to [13].", "More information on spectra of graphs and integral graphs can be found in [3].", "Lemma 4.1 The graph $\\Gamma $ is an integral divisible design graph.", "From [14] it follows that $\\Gamma $ is a divisible design graph if and only if, in the notations of Lemma REF , $X_2\\cup \\lbrace e\\rbrace $ or $X_{2(k-1)}\\cup \\lbrace e\\rbrace $ is a subgroup of $G$ .", "Moreover, the canonical partition of $G$ is a partition into the right cosets by this subgroup.", "Eq.", "(3) implies that $X_{2(k-1)}\\cup \\lbrace e\\rbrace =A\\cup cbA$ .", "Since $A$ is normal in $G$ , $|cb|=2$ , and $a^{cb}=a^{-1}$ , the set $X_{2(k-1)}\\cup \\lbrace e\\rbrace $ is a subgroup of $G$ isomorphic to $D_{2k}$ .", "Therefore $\\Gamma $ is a divisible design graph with parameters $(8k,2(k+1),2(k-1),2,4,2k)$ .", "Since $\\Gamma $ is a divisible design graph, one can calculate eigenvalues of its adjacency matrix from its parameters by the formulas from [12].", "It turns out that the set of eigenvalues of the adjacency matrix of $\\Gamma $ is equal to $\\lbrace 2(k+1),\\pm 2(k-1),\\pm 2\\rbrace $ .", "This implies that $\\Gamma $ is integral.", "Recall that the $(l\\times m)$ -grid is the line graph of the complete bipartite graph $K_{l,m}$ (see [2]).", "Lemma 4.2 The graph $\\Gamma $ has the same parameters as the $(4 \\times 2k)$ -grid but it is not isomorphic to the $(4 \\times 2k)$ -grid.", "Let $\\Gamma ^{\\prime }$ be the graph isomorphic to the $(4\\times 2k)$ -grid.", "The graph $\\Gamma ^{\\prime }$ has parameters $(8k,2(k+1),2(k-1),2)$ by [12].", "However, due to [4], we obtain $\\operatorname{rk_{WL}}(\\Gamma ^{\\prime })=4$ and $\\operatorname{Aut}(\\Gamma ^{\\prime })\\cong \\operatorname{Sym}(4) \\times \\operatorname{Sym}(2k)$ .", "So $\\Gamma $ is not isomorphic to $\\Gamma ^{\\prime }$ .", "The Weisfeiler-Leman dimension $\\operatorname{dim_{WL}}(\\Delta )$ of a graph $\\Delta $ is defined to be the smallest positive integer $m$ for which $\\Delta $ is identified by the $m$ -dimensional Weisfeiler-Leman algorithm  [11].", "If $\\operatorname{dim_{WL}}(\\Delta )\\le m$ then the isomorphism between $\\Delta $ and any other graph can be verified in time $n^{O(m)}$ using the Weisfeiler-Leman algorithm [20].", "The Weisfeiler-Leman dimension of Deza circulant graphs was studied in [1].", "Lemma 4.3 The Weisfeiler-Leman dimension of $\\Gamma $ is equal to 2.", "The $S$ -ring $\\operatorname{WL}(\\Gamma )$ is separable in the sense of [1].", "Indeed, if $k$ is odd then $\\operatorname{WL}(\\Gamma )=\\mathbb {Z}G$ by Lemma REF and the required follows from [4].", "If $k$ is even then $\\operatorname{WL}(\\Gamma )=\\mathbb {Z}U\\wr _S \\mathbb {Z}(G/L)$ by Lemma REF and the required follows from [4].", "The separability of $\\operatorname{WL}(\\Gamma )$ and [9] imply that $\\operatorname{dim_{WL}}(\\Gamma )\\le 2$ .", "Since $\\Gamma $ is regular but nonstrongly regular, $\\operatorname{dim_{WL}}(\\Gamma )\\ne 1$ by [1].", "Thus, $\\operatorname{dim_{WL}}(\\Gamma )=2$ ." ] ]
2105.11746
[ [ "Impatient Queuing for Intelligent Task Offloading in Multi-Access Edge\n Computing" ], [ "Abstract Multi-access edge computing (MEC) emerges as an essential part of the upcoming Fifth Generation (5G) and future beyond-5G mobile communication systems.", "It adds computational power towards the edge of cellular networks, much closer to energy-constrained user devices, and therewith allows the users to offload tasks to the edge computing nodes for low-latency applications with very-limited battery consumption.", "However, due to the high dynamics of user demand and server load, task congestion may occur at the edge nodes resulting in long queuing delay.", "Such delays can significantly degrade the quality of experience (QoE) of some latency-sensitive applications, raise the risk of service outage, and cannot be efficiently resolved by conventional queue management solutions.", "In this article, we study a latency-outage critical scenario, where users intend to limit the risk of latency outage.", "We propose an impatience-based queuing strategy for such users to intelligently choose between MEC offloading and local computation, allowing them to rationally renege from the task queue.", "The proposed approach is demonstrated by numerical simulations to be efficient for generic service model, when a perfect queue status information is available.", "For the practical case where the users obtain only imperfect queue status information, we design an optimal online learning strategy to enable its application in Poisson service scenarios." ], [ "Introduction", "While the maturity of the current microelectronic technology enables portable devices to accomplish severe and complex tasks, there is a hype on the cloud computing services as enabler to offload such a burden onto ad-hoc cloud server—usually located at the edge of the network—based on the specific service requirements.", "This has recently fostered a novel paradigm shift towards the Edge Computing concept [1].", "Edge computing (or Multi-access Edge Computing, MEC) represents the first-mover advantage for the upcoming generation of network design (5G) and the core feature of beyond-5G (B5G) networks, as it opens up to a number of business opportunities for the service provider that allocates shared computational resources to a diverse set of customers, dubbed as tenants, in an on-demand fashion [2].", "With the ever-increasing number of tenants and corresponding computing resource requests, there is an impelling need to properly and efficiently manage computing service queues due to the randomness of arriving tenant requests and very-stringent latency requirements expected for B5G use cases: waiting long for being served may result in a low business profitability or, in the worst case, in a service disruption, e.g., when dealing with “age-of-information” or “age-of-task” sensitive services [3], [4].", "To address the queue congestion issue, a number of techniques have been proposed in the literature, such as the active queue management (AQM) solution based on queue truncation (limited queue length) and active dropping.", "AQM is proven to be the most effective approach in data networking applications such as packet routing and switching: it is capable to guarantee a full utilization of network resources, even when specified with a simple fair dropping policy [5].", "Nevertheless, despite its mature development over three decades and wide success in data networking, AQM has been rarely deployed in edge/cloud computingOur analysis and proposals here presented are insensitive to the position of cloud service deployments.", "Therefore, the terms edge and cloud are interchangeably used throughout the text., due to the fundamental difference between the natures of data packets and service requests.", "Although packet dropping in data stream can be generally overcome with automatic repeat request (ARQ) without compromising the quality of data traffic, denial of an awaiting computational service request will usually terminate the service and it might even reduce the tenant's future interest in such a computing service provider.", "This calls for a perspective change where a deep analysis of tenant behaviors may benefit the overall service queue management, as recently suggested by [6] that relies on the tenant impatience.", "Specifically, rational tenants may naturally behave impatiently if the waiting service time exceeds reasonable levels (and thus failing to bring the expected end-profit) thereby hesitating to enter the queue or reneging on its entrance into the queue and leave before being served.", "In this way, service sessions with higher profitability will be more likely completed than those with lower profitability, leaving the full mechanism control to each tenant decision: a full or partial information of the queue status will enable an effective exploitation of the tenant impatience.", "However, such information if available might break privacy concerns and might not be completely disclosed by the service provider.", "In such a case, if only imperfect queue status information is available, a revenue-vs-cost approach cannot be applied straightforwardly due to a lack of flexibility in the risk management.", "In other words, tenants may not be able to exactly estimate whether the revenue coming from successfully delivering the service pays off the queue waiting cost, and thus it may need to learn incomplete status information via advanced learning strategy.", "Apart from simple and static learning approaches, this problem has been rarely addressed in the literature.", "Conversely, in this paper we pioneer a solution that relies on the tenant behavior analysis.", "To summarize, i) we have considered a flexible option with own risk profile instead of the zero-revenue task cancellation as alternative to the target computing service, ii) we have proposed a risk-based impatience mechanism that allows the tenants to flexibly balance between the chance of being served and the risk of profit loss, iii) we have devised an optimal online learning strategy that reduces both the inappropriate tenant decisions due to inadequate learning, and the waste of cost in unnecessary over-learning and, $iv)$ we have validated our proposal by means of an exhaustive numerical evaluation campaign.", "The remainder of this paper is organized as follows: we begin with a brief review to different fields related to our study in Section , then in Section  we set up the investigated problem, and formerly discuss about the decision model of risk-based tenant impatience, showing that the balking behavior can be merged with reneging.", "We then analyze the tenant risk preference in more depth in Section , proving that non-regretting tenants will only balk but never renege from Poisson service queues, when a perfect queue status information is available.", "We lead the discussion by one step further with Section  into the case of imperfect queue status information, proposing our optimal online Poisson learning strategy.", "The proposed approaches are then numerically evaluated in Section  and compared against baseline solutions, before providing concluding remarks in Section ." ], [ "Related work", "The simplest mechanism to eliminate the queue divergence, namely the queue truncation, has been widely deployed in practical systems, and extensively studied in classical queuing theory [7].", "Combining queue truncation with an active task dropping scheduled by the server, AQM was initially introduced in 1993 as Random Early Detection (RED) [8].", "Over the past three decades since then, a rich collection of AQM variations, such as reported in [9], [10], [11], have been developed.", "It provides in [12] a complete review of queuing theory-based cloud computing approaches differentiating between accurate implementations, performance and QoS guarantees whereas the authors of [13] detail an approximate analytical model to accurately estimate the probability distribution of the request response time of the cloud computing queue.", "Research on the client impatience in queuing systems dates back to the 1950s, when the client behavior of hesitating to join long queues was first noticed [14], and afterwards rigorously investigated [15].", "The most important pioneering work in this field was conducted by Haight, where the aforementioned phenomenon was named as “balking” [16], and later generalized to the “reneging” concept [17].", "In 1963, Ancker and Gafarian pushed the work of Haight further, proposing several most widely used statistical models of balking and reneging, and therewith analyzing their impacts on $M/M/1$ queues [18], [19].", "A generalization of these analyses was presented in [20] regarding $GI/G/1$ queues.", "Recently, in the context of network slicing, which is an emerging use case of public cloud computing in the Fifth Generation (5G) wireless networks, Han et al.", "have revisited the topic of client impatience from the microeconomic perspective [6].", "They demonstrated how the statistic models of balking and reneging in $M/M/1$ queues are determined by the distribution of reward generated in the cloud service, and how this behavior will impact the resource utilization of multiple heterogeneous queues sharing a same server.", "Additionally, they also discussed the impact of clients' knowledge level in their decisions of balking and reneging." ], [ "Mechanism design", "In this study, we focus on a single-queuing system, where all pending service requests wait in the queue according to “First Come, First Served” (FCFS) policy, and consider the dynamic features to be globally homogeneous for all tenants, such as request arrival rate and revenue distribution.", "Alternative queuing policies such like “Last Come, First Served” (LCFS), random selection for service (RSS), and priority-based (PR) are not common in cloud service management, and therefore not discussed in this work.", "On the other hand, the cases of multi-queuing systems and heterogeneous service requests can be easily generalized from our analysis, which is omitted in this manuscript." ], [ "Queueing system of cloud service requests", "Without loss of generality, we assume that: New service requests arrive as a consistent Poisson process with arriving rate $\\lambda $ , i.e.", "the inter-arrival interval is exponentially distributed with a mean of $1/\\lambda $ .", "When a tenant issues its service request, it is charged by the service provider an initial payment $u_0$ for appending its request the queue, dubbed as entrance fee.", "The tenant can also refuse to pay this fee and therewith balk from the queue.", "The server sequentially processes the pending requests awaiting in the queue, the inter-service interval is a consistent exponential random variable with mean of $1/\\mu $ .", "Every tenant has to wait in the queue with a certain rate $u$ , while its service request is pending in the queue, until the request is finally processed.", "Note that the waiting cost is considered to model the tenant's own regular expenses, instead of any charge required by the service provider.", "Alternatively, the tenant can also choose to renege from the queue at arbitrary time, and therewith stop paying the waiting cost.", "Upon balking or reneging, the tenant will get an alternative option, e.g.", "switching to another service provider, relying on its local implementation instead of the cloud service, or simply canceling the task.", "The balked/reneged tenant therewith obtains a random revenue upon the selection of alternative option.", "This will be discussed in more details in Section REF .", "Note that this is a main extension to the environment design proposed in the literature, which considers only the specific case of simple task cancellation with a fixed zero revenue.", "When a request is finally processed by the server, it generates a revenue for the tenant.", "We consider the revenue to be distributed as an exponential random variable with a mean of $\\bar{\\nu }$ for all service requests, while the assigned specific value $\\nu $ of every request is known to the tenant beforehand, but unknown to the cloud service provider.", "An overview to the above described mechanism design is illustrated in Fig.", "REF .", "Figure: An overview of the proposed risk-based tenant impatience mechanism" ], [ "Risk-based reneging", "First, we investigate the impatient behavior of an awaiting tenant with a certain risk preference.", "Consider the $k^\\text{th}$ request in the queue with service revenue $\\nu $ , which has waited for time $t$ , we use $w_k(t)$ to denote the time it still needs to wait until being processed, so the profit for this tenant of waiting becomes $\\phi _{k,t}=\\nu -uw_k(t).$ Literature [6] shows that when the service follows a Poisson process and the balking/reneging events are sparse, $w_k(t)$ can be approximately considered as memoryless.", "We take this assumption here for analysis simplification, so that both the remaining waiting time $w_k(t)$ and the waiting profit $\\phi _{k,t}$ are independent of $t$ , and can be rewritten as $\\phi _{k}=\\nu -uw_k,$ which is linear to the random variable $w_k$ .", "In contrast, if the tenant chooses to renege, it obtains a random profit $\\psi $ from an alternative option, which is independent of $k$ or $t$ .", "While the mean profit $\\mathbb {E}\\left\\lbrace \\phi _k\\right\\rbrace $ has been adequately studied in [6], here we take a novel point of view, as one of our main contributions, that considers the full distribution of $\\phi _k$ .", "More specifically, to characterize the risk-profit trade-off of waiting in the queue, we can model its risk profile (RP) as the maximal profit achievable with certainty of $p\\in [0,1)$ by waiting at position $k$ : $V_k(p)=\\arg \\max \\limits _a\\left\\lbrace \\text{Pr}\\left(\\phi _k\\geqslant a\\right)\\geqslant p\\right\\rbrace .$ Similarly, we can also characterize the RP of the alternative option: $U(p)=\\arg \\max \\limits _a\\left\\lbrace \\text{Pr}\\left(\\psi \\geqslant a\\right)\\geqslant p\\right\\rbrace .$ Remark that their inverse functions $V^{-1}_k(a)&=\\text{Pr}\\left(\\phi _k\\geqslant a\\right),\\\\\\text{and }U^{-1}(a)&=\\text{Pr}\\left(\\psi _k\\geqslant a\\right)$ construct the inverse complementary cumulative distribution function (ICCDF) of decision profits $\\phi _k$ and $\\psi $ , respectively.", "As a special case, a simple task cancellation that generates a fixed zero profit has a constant RP $U(p)=0$ for all $p\\in [0,1)$ .", "We consider the tenant to make its reneging decision on a basis of risk assessment: $R(k,t)={\\left\\lbrace \\begin{array}{ll}\\text{True}&V_k(p_t)\\leqslant U(p_t),\\\\\\text{False}&\\text{otherwise},\\end{array}\\right.", "}$ where $p_t$ is the certainty level at time $t$ .", "The tenant chooses to renege from the queue at position $k$ and time $t$ if $R(k,t)$ is true, and waits in queue when it is false.", "Since it always holds $\\frac{\\partial \\text{Pr}\\left(V _k^{-1}(a)\\right)}{\\partial a}=\\frac{\\partial \\text{Pr}\\left(\\phi _k\\geqslant a\\right)}{\\partial a}\\leqslant 0,$ Eq.", "(REF ) is equivalent to $R(k,t)={\\left\\lbrace \\begin{array}{ll}\\text{True}&V^{-1}_k\\left[U(p_t)\\right]\\leqslant p_t\\\\\\text{False}&\\text{otherwise}.\\end{array}\\right.", "}$" ], [ "Risk-based balking", "Now we study the balking decision concerning the same risk controlling principle as per the reneging discussed in the last subsection.", "Consider a tenant that issues a request when the queue is filled by $k-1$ awaiting requests, i.e., it needs to decide if it shall enter the queue as the $k^\\text{th}$ .", "Since an entrance fee $u_0\\geqslant 0$ must be paid to the service provider if the tenant does not balk, this fee must be taken into account, which makes the profit of entering the queue at $k$ : $\\phi ^{\\prime }_{k}=\\nu -u_0-uw_k(t),$ and the RP of entering queue is characterized by $V^{\\prime }_k(p)=\\arg \\max \\limits _a\\left\\lbrace \\text{Pr}\\left(\\phi ^{\\prime }_k\\geqslant a\\right)\\geqslant p\\right\\rbrace .$ On the other hand, since no entrance fee will be charged in case of balking, the RP of alternative option remains $U(p)$ as defined by Eq.", "(REF ).", "Similarly to Eq.", "(REF ), the decision rule of balking shall be $B(k,t)={\\left\\lbrace \\begin{array}{ll}\\text{True}&V^{\\prime }_k(p_t)\\leqslant U(p_t),\\\\\\text{False}&\\text{otherwise}.\\end{array}\\right.", "}$ Note that since $u_0\\geqslant 0$ it always holds $\\phi ^{\\prime }_{k}\\leqslant \\phi _{k}$ and $V^{\\prime }_k(p)<V_k(p)$ , this implies that $\\lnot B(k,t)\\Rightarrow \\lnot R(k,t),$ which guarantees the rationality of tenant that never balks right after entering the queue.", "So far we have shown that a risk-sensitive tenant makes its decision of balking in the same way as it does for reneging at its entrance position, only being offset by the entrance fee $u_0$ .", "Especially, when $u_0=0$ , the balking phenomenon is completely merged into reneging as the special case at queue entrance.", "Therefore, in the remaining part of this work we will mainly focus only on the reneging mechanism.", "Nevertheless, it is still worth to remark here that in the scenario of public cloud computing, the entrance fee $u_0$ can be employed as a simple but practical measure to counter Denial-of-Service (DoS) attacks, which may paralyze the queuing system by maliciously submitting numerous service requests.", "Charging a reasonable $u_0>0$ for request submission will effectively raise the cost of conducting such DoS attacks, while costing little implementation effort and making only slight impact to regular tenants." ], [ "Risk-based reneging with perfect queue status information", "We start our investigation considering queue status information available, where the tenant knows $\\mu $ (and eventually other tenants' reneging rate) a priori, and searches for a rational reneging policy.", "Without loss of generality, we assume that the impact of tenant impatience is negligible w.r.t.", "the Poisson service.", "Thus, for the $k^\\text{th}$ request in queue, if a tenant does not renege, the probability that its request is processed within time $W$ is $\\text{Pr}\\left(w_k\\leqslant W\\right)=1-\\sum \\limits _{i=0}^{k-1}\\frac{(\\mu W)^ie^{-\\mu W}}{i!", "},\\quad \\forall k\\in \\mathbb {N}^+.$ Recalling Eq.", "(REF ), there is $\\begin{split}&\\text{Pr}\\left(\\phi _k\\geqslant a\\right)=\\text{Pr}\\left(\\nu -uw_k\\geqslant a\\right)=\\text{Pr}\\left(w_k\\leqslant \\frac{\\nu -a}{u}\\right)\\\\=&1-\\exp \\left(\\frac{\\mu (a-\\nu )}{u}\\right)\\sum \\limits _{i=0}^{k-1}\\frac{1}{i!", "}\\left(\\frac{\\mu (\\nu -a)}{u}\\right)^i.\\end{split}$ For convenience of notation we define $m_t\\triangleq \\frac{\\nu -U(p_t)}{u}$ and $F_k(W)\\triangleq \\text{Pr}\\left(w_k\\leqslant W\\right)=V_k^{-1}(\\nu -uW)$ , so that the reneging condition Eq.", "(REF ) turns into $R(k,t)={\\left\\lbrace \\begin{array}{ll}\\text{Ture}&F_k(m_t)\\leqslant p_t;\\\\\\text{False}&\\text{otherwise}.\\end{array}\\right.", "}$ Note that $m_t$ actually identifies a critical waiting time $t$ when the tenant gets a waiting profit of $U(p_t)$ .", "Especially, if a tenant has $p_t< F_k(m_t), \\forall (k,t)$ , it will never renege, and we call it therefore definitely patient.", "Otherwise, i.e.", "when there is a solution to the marginal reneging condition $p_t=F_k(m_t),$ then client needs to leverage its a prior knowledge of $\\mu $ and its own impatience function $p_t$ to make reneging decision at every $(k,t)$ .", "Remark that the first decision to renege will terminate the waiting process without opportunity to regret, so a client can be generically characterized as, upon its belief in $w_k$ , not reneging at $k$ until $t_k$ ." ], [ "Non-regretting tenants with time-invariant certainty", "Obviously, the decision of reneging strongly depends on the tenant's certainty level $p_t$ , so it is worth discussing the time-domain feature of $p_t$ .", "In short we can assert that, in common cloud computing scenarios, the tenants shall be considered as as non-regretting, i.e., $p_t\\equiv p$ is time-invariant, as discussed hereafter.", "To better understand $p_t$ , it takes an economical point of view, in which the decision rule as per Eq.", "(REF ) illustrates an idea of risk management.", "Since there is always $\\partial U(p_t)/\\partial p_t\\leqslant 0$ , a higher $p_t$ implies that the tenant has a stronger preference in certainty of achieving the target profit floor $U(p_t)$ , at a cost of reducing the profit floor itself.", "In contrast, when the certainty level decreases, the tenant's preference drifts towards higher profit floor with more risk.", "It is important to remark here that the profit floor evaluates only profit in future, and ignores the waiting cost already generated in the past (known as the “sunk cost”) $ut$ , hence it cannot be steadily coupled with the end-profit.", "Furthermore, since the sunk cost $ut$ is linear with respect to $t$ , the dependency of $p_t$ on $t$ implies the risk preference w.r.t.", "the sunk cost of a tenant, or w.r.t.", "the total end profit/loss, which has been studied in depth by literature in finance [21].", "As a short summary: For non-regretting tenants that are neutral to the sunk cost, $p_t$ is independent of $t$ , and therefore $U(p_t)$ is independent of $t$ as well.", "For convenience we denote them in this case as $p$ and $U(p)$ , respectively.", "For risk-conservative regretting tenants that are sensitive to end-loss (taking into account the sunk cost), $\\partial p_t/\\partial t>0$ .", "A typical sample of this case is a tenant that has to terminate its task session when the sunk cost exceeds a hard threshold (e.g.", "a fixed amount of budget).", "For risk-seeking regretting tenants that are more sensitive to end-profit than to end-loss, $\\partial p_t/\\partial t<0$ .", "This is usually observed in business scenarios where the tenant is allowed to overspend its budget (e.g.", "by borrowing from external funding sources) and insensitive to extreme end-loss (e.g.", "when a bankruptcy mechanism is valid).", "Since the latter two cases are rare in typical cloud computing scenarios, in the remainder of this work we only focus on non-regretting tenants." ], [ "Condition of definite patience", "As discussed in Section REF , no tenant will renege immediately after submitting its request to the queue.", "Furthermore, any tenant that has entered the queue (i.e.", "without choosing to balk) can be identified as definitely patient, if both the following conditions are fulfilled: Non-decreasing patience upon position: if the tenant does not renege at $(k,t)$ , it is not reneging at $(k-\\Delta k,t)$ for all $\\Delta k\\in \\lbrace 1,2,\\dots ,k-1\\rbrace $ , i.e., $\\begin{split}F_k(m_t)>p_t\\Rightarrow F_{k-\\Delta k}(m_t)>f_t,\\\\\\forall t\\in \\mathbb {R}^+, k\\in \\mathbb {N}^+, \\Delta k\\in \\lbrace 1,2,\\dots ,k-1\\rbrace .\\end{split}$ Non-decreasing patience upon time: if the client does not renege at $(k,t)$ , it is not reneging at $(k,t+\\Delta t)$ for all $\\Delta t>0$ , i.e., $\\begin{split}F_k(m_t)>p_t\\Rightarrow F_k(m_{t+\\Delta t})>p_{t+\\Delta t},\\\\\\forall t\\in \\mathbb {R}^+, \\Delta t\\in \\mathbb {R}^+, k\\in \\mathbb {N}^+;\\end{split}$ These conditions can be reformulated as: $&F_{k+1}(m_t)\\leqslant F_k(m_t),\\\\&F_{k}(m_{t+\\Delta t})-F_k(m_t)\\geqslant p_{t+\\Delta t}-p_t,$ for all $t\\in \\mathbb {R}^+, \\Delta t\\in \\mathbb {R}^+, k\\in \\mathbb {N}^+$ .", "Here, Eq.", "(REF ) is automatically guaranteed by Eq.", "(REF ), so that Eq.", "() can be taken as the sufficient and essential condition of definite patience.", "Especially, we can obtain the following Theorem: Theorem 1 When a perfect knowledge about the queue status is available a priori, non-regretting tenants are always definitely patient in Poisson queues.", "For non-regretting tenants with perfect knowledge about $\\mu $ , there is $p_t\\equiv p$ for all $t$ , which makes both $U(p_t)\\equiv U(p)$ and $m_t\\equiv m=[\\nu -U(p)]/u$ also time-invariant.", "In this case, Eq.", "() becomes $F_k(m)-F_k(m)\\geqslant p-p,$ which always holds with the equality.", "Theorem REF can also be understood in another point of view: a non-regretting tenant with consistent risk threshold $p$ has a maximal waiting position in queue: $K_{\\text{max}}=\\mathtt {invpois}\\left(1-p,\\frac{\\mu \\left[\\nu -U(p)\\right]}{u}\\right),$ where $\\mathtt {invpois}(\\cdot )$ is the inverse Poisson cumulative distribution function.", "The tenant will renege, when and only when its position $k$ in queue is larger than $K_{\\text{max}}$ .", "Since $K_{\\text{max}}$ is independent of $k$ or $t$ , the reneging decision $R$ will also be consistent.", "This implies that if a non-regretting tenant will choose to renege at any time, it must also choose to renege at its entrance of the queue, and therefore must choose to balk.", "On the other hand, any tenant that did not balk at the queue entrance will never renege, i.e., it is always definitely patient.", "The paramount importance of Theorem REF shall be identified as follows: if the privacy policy of the cloud service provider allows a full queue status information transparency (i.e., queue information available), the risk-based tenant impatience can be accomplished by the balking decision alone, while the reneging phenomenon is completely eliminated.", "This will significantly simplify the system implementation and reduce the signaling cost." ], [ "Rational reneging with imperfect knowledge", "We have demonstrated that the reneging phenomenon is fully eliminated by balking under a full transparency of queue status information, if the tenants are non-regretting.", "Nevertheless, due to various concerns such as privacy and confidentiality, cloud service providers can be practically hesitating or unable to share the perfect queue status information with all tenants.", "On the other hand a complete blindness of tenants to the queue status can easily lead to inappropriate decisions of balking/reneging, and therewith damage the business interests of both tenants and cloud service providers.", "As a compromised solution, an imperfect queue status information is usually fed to tenants, so as to enable learning-based tenant decisions while preventing privacy leakage." ], [ "Impatience due to estimation errors", "This mechanism of imperfect queue knowledge sharing is usually implemented by periodically informing every tenant about its position $k$ in the queue.", "A common alternative is to inform the tenant only when its $k$ is updated.", "Sometimes, a rough estimation $\\hat{\\mu }_0$ can also be provided to all clients as an initial belief, but the accurate instantaneous value of $\\mu $ remains unknown and needs to be estimated.", "Generally, with such mechanisms, a tenant is able to learn $\\mu $ from its observations on $k$ , denoted by $\\hat{\\mu }(\\Omega _t)$ , where $\\Omega _t=\\left(\\omega _{t_1},\\omega _{t_2},\\dots ,\\omega _{t}\\right)$ is the sequence of observation entries at $t$ , each entry $\\omega _{t_n}=\\left(k(t_n),t_n\\right)$ consists of the position and the time at observation.", "A consistent estimator $\\hat{\\mu }$ is guaranteed to converge to $\\mu $ in probability: $\\operatornamewithlimits{plim}\\limits _{t\\rightarrow +\\infty }\\hat{\\mu }(\\Omega _t)=\\mu ,$ where $\\operatornamewithlimits{plim}$ is the probability limit operator.", "Nevertheless, within a finite time $t$ , despite of the consistency of $\\hat{\\mu }$ there is always a random error between the estimation $\\hat{\\mu }(\\Omega _t)$ and the ground truth $\\mu $ , : $\\varepsilon _\\mu (\\Omega _t)=\\hat{\\mu }(\\Omega _t)-\\mu ,$ which introduces an error to the decision of reneging.", "More specifically, given the certainty level $p$ and the RP $U$ of alternative option, we recall the reneging condition as per Eq.", "(REF ) to see that the correctness of decision relies on the estimation of $m=\\frac{\\nu -U(p)}{u}$ .", "If a tenant overestimates the service rate $\\mu $ with $\\varepsilon _\\mu (\\Omega _0)<0$ when issuing its request (without loss of generality, we consider it to be $t=0$ ), it will also overestimate $F_k(m)$ , and may therewith choose to enter the queue, although it should have chosen to balk, which we name as over-patient; on the other hand, it can also underestimate $F_k(m)$ with $\\varepsilon _\\mu (\\Omega _0)>0$ and therefore choose to balk, although it should have been definitely patient, which we refer to as under-patient.", "Such imperfect decisions will, naturally, lead to practical business loss.", "A tenant can rely on nothing but a longer observation sequence $\\Omega _t$ to effectively reduce the estimation error, while this learning process itself (implicitly means waiting in the queue) generates a cost with rate of $u$ .", "It becomes therefore a critical task to select an optimal time $\\tau $ to stop learning and make the reneging decision, which is discussed in the following." ], [ "Optimal online Poisson learning: balance between the learning gain and the waiting cost", "To optimize the stopping time of learning process, we need to construct a penalty function to indicate the profit loss caused by inaccurate estimation of $\\mu $ .", "As discussed above, there are two classes of error in the decision of reneging, namely over-patient and under-patient.", "In the earlier case, the tenant will obtain a waiting profit $\\phi _k=\\nu -w_k\\approx \\nu -\\frac{uk}{\\mu }$ , although the correct decision will lead to a reneging profit $\\psi $ from the alternative option instead.", "In contrast, a tenant in the latter case obtains the reneging profit $\\psi $ , although a rational decision, i.e.", "to wait, will return $\\phi _k$ instead.", "Thus, the expected profit loss upon decision can be correspondingly defined as $\\begin{split}&L(k,t)=\\\\&{\\left\\lbrace \\begin{array}{ll}\\int \\limits _{0}^{\\mu _{\\text{c}}}\\left(\\mathbb {E}\\left\\lbrace \\psi \\right\\rbrace -\\nu +\\frac{uk}{x}\\right)f_{\\hat{\\mu }}(x\\vert \\Omega _t)\\text{d}{x}&\\mu >\\mu _{\\text{c}},\\\\\\int \\limits _{\\mu _{\\text{c}}}^{+\\infty }\\left(\\nu -\\frac{uk}{x}-\\mathbb {E}\\left\\lbrace \\psi \\right\\rbrace \\right)f_{\\hat{\\mu }}(x\\vert \\Omega _t)\\text{d}{x}&\\text{otherwise},\\end{array}\\right.", "}\\end{split}$ where $\\mu _\\text{c}$ is the critical arrival rate, which makes $F_k(m)=p$ , and $f_{\\hat{\\mu }}(x\\vert \\Omega _t)$ is the conditional belief of estimation $\\hat{\\mu }$ given observations $\\Omega _t$ .", "As the tenant remains waiting in the queue, $\\Omega _t$ keeps being updated, and $f_{\\hat{\\mu }}(x\\vert \\Omega _t)$ converges to $\\delta (x-\\mu )$ , and $L(k,t)$ also therewith converges to 0.", "The stochastic features of $L(k,t)$ is determined by the specific queue status information updating mechanism.", "As we have named in Section REF , for example, $\\Omega _t$ can be updated periodically at every $t=nT$ where $n\\in \\mathbb {N}$ , or updated at every change of $k$ (e.g., when the service to request at server is accomplished, or other tenants ahead has reneged).", "According to [22], the latter one performs as the optimal dynamic information acquisition for Poisson random processes.", "Therefore, in this work we consider the queue status information updating upon service accomplishment and tenant reneging.", "Thus, $L$ depends only on $k$ , and can be rewritten as $L_k$ for convenience.", "Thus, for a certain awaiting request, when its position in queue is updated from $k+1$ to $k$ , it obtains a marginal learning gain of $G_k^\\text{learn}=\\mathbb {E}\\left\\lbrace L_{k+1}\\right\\rbrace -\\mathbb {E}\\left\\lbrace L_k\\right\\rbrace ,$ which converges to $\\lim \\limits _{k\\rightarrow +\\infty }G_k=0$ .", "Meanwhile, the expected learning cost to observe such an update is $C^\\text{learn}=\\frac{u}{\\mu },$ which is positive and independent of $k$ .", "Thus, we can set the optimal decision position at $k_\\text{opt}=\\inf \\left\\lbrace k: G_k^\\text{learn}-C^\\text{learn}\\leqslant 0\\right\\rbrace .$ Note that the ground truth of $\\mu $ is essential for the calculation of $F_k(m_t)$ , $L_k$ and $C^\\text{learn}$ according to Eqs.", "(REF ) and (REF ), but unknown to the tenant.", "So we propose to use its estimation $\\hat{\\mu }$ to approximate it.", "More specifically, we apply the bias-corrected maximum likelihood estimator $\\hat{\\mu }(\\Omega _t)={\\left\\lbrace \\begin{array}{ll}+\\infty &N_t=1,\\\\\\frac{N_t-1}{\\sum \\limits _{n=2}^{N_t}\\Delta t_n}&N_t=2,\\\\\\frac{\\left(N_t-1\\right)\\left(N_t-3\\right)}{\\left(N_t-2\\right)\\sum \\limits _{n=2}^{N_t}\\Delta t_n}&N_t\\geqslant 3,\\end{array}\\right.", "}$ where $N_t=\\left|\\Omega _t\\right|_0$ is the number of observations at $t$ , and $\\Delta t_n=t_n-t_{n-1}$ denotes the $\\left(n-1\\right)^\\text{th}$ inter-update interval.", "Remark that since $\\Delta t_n\\sim \\text{Exp}\\left(\\frac{1}{\\mu }\\right),\\quad \\forall n\\geqslant 2,$ we have $\\sum \\limits _{n=2}^{N_t}\\Delta t_n\\sim \\text{Erlang}\\left(N_t-1,\\frac{1}{\\mu }\\right),\\quad \\forall N_t\\geqslant 2,$ which can be exploited to calculate $f_{\\hat{\\mu }}(x\\vert \\Omega _t)$ .", "Our proposed risk-based dynamic reneging mechanism with online learning can be briefly described by the algorithm in Fig.", "REF .", "Figure: The risk-based dynamic tenant reneging algorithmFigure: End-profit distribution of all tenant computation tasks achieved by different queue congestion mechanisms.", "10-T stands for queue truncation at maximal queue length 10; “mean” stands for the mean-profit-based impatience described in ; and pp-C represents the proposed risk-based impatience with certainty level pp.", "For all tests, the alternative option C is applied." ], [ "Numerical evaluation", "To evaluate our proposed approaches, we carried out numerical tests.", "A simulation framework was conducted to provide a benchmark to our solution in a consistent environment: i) against baseline methods, ii) with different alternative options, and iii) among various service scenarios, as specified in Table REF ." ], [ "Simulation setup", "Besides the proposed risk-based impatience approach, we have implemented two baseline mechanisms of queue congestion control: Queue truncation: no impatient tenant behavior is considered, while the cloud service provider truncates the request queue at a certain maximal length.", "Mean-profit-based impatience: tenant requests make decisions of balking and reneging so as to maximize the mean profit, as introduced by [6].", "In addition, we have designed three risk profiles of different alternative solutions: C(ancellation): by simply canceling the computing task, no revenue or further cost is generated for the tenant, leading to a zero-reward.", "E(xponential cost): by submitting its task to another service provider or relying on local computation, the tenant obtains its desired revenue $\\nu $ , at an exponentially distributed random cost that $\\sim \\text{Exp}(1)$ .", "G(aussian cost): by investing its budget in other tasks, the tenant obtains a Gaussian random profit that $\\sim \\mathcal {N}(\\nu -1,1)$ .", "Furthermore, we have designed three reference service scenarios, which share the same request arrival rate $\\lambda =1$ , but with different service rates $\\mu $ in order to represent the cases of: B(alance): the service capacity is in long-term balanced to the load, while short-term queue congestion may randomly occur; C(ongestion): the server is significantly overloaded so that a serious queue congestion is generally existing; and D(isaster): the server is extremely overloaded so that most requests cannot be served.", "Table: System specifications" ], [ "Benchmarking queue congestion control mechanisms", "To validate our proposed risk-based impatience mechanism, we tested it with three different certainty levels: $90\\%$ , $99\\%$ , and $99.9\\%$ .", "A perfect queue knowledge is provided to every tenant.", "Additionally, two baseline solutions were also tested for comparison: the mean-profit-based impatience mechanism with perfect queue status knowledge for all tenants, and a simple queue truncation mechanism with maximal queue length of 10.", "The end-profit distribution of all service requests with alternative option C (task cancellation) in all three reference scenarios are illustrated in Fig.", "REF .", "It can be observed that although queue truncation is effective in avoiding net loss, it also rejects most service requests, especially in dense queue congestion, without any selectivity in service revenue.", "This reduces the request admission rate together with the profitability.", "In contrast, the tenant reneging mechanism encourages the requests with higher service revenue to wait, while discouraging those with lower revenue to do so.", "Technically, it therewith achieves a selective task dropping in a distributed manner of decision making.", "Furthermore, compared to the inflexible mean-profit based solution, the risk-based impatience mechanism allows the tenants to flexibly adjust their preference between the request admission chance and the certainty in profitability, where a higher certainly level generally implies lower admission probability.", "Overall, our proposed risk-based impatience approach is able to outperform the mean-profit-based impatience baseline in all scenarios.", "Figure: End-profit distribution of all tenant computation tasks achieved by risk-based reneging with different learning mechanisms.", "“Transparent” stands for the perfect queue knowledge case where the service rate μ\\mu and reneging rate per request is available a priori to all tenants; kk-static stands for the static learning strategy where tenants do not renege until kk observations on service/reneging events are obtained; “dynamic” represents the proposed optimal online Poisson learning method.", "For all tests, the alternative option C is applied and the certainty level is 90%.Additionally, it is also worth remarking that the impatience is only represented by balking in this simulation, while reneging is completely eliminated, which proves our Theorem REF .", "Table: Service statistics of different queue congestion control mechanisms, alternative option C." ], [ "Benchmarking Poisson learning mechanisms", "To demonstrate the effectiveness of our proposed optimal online Poisson learning procedure, we apply it with a risk-based impatience mechanism at the certainty level of $90\\%$ .", "As benchmarks, we also test the perfect queue status information case, and the static learning strategy, where the reneging decision cannot be made until a fixed minimal number of observations are obtained.", "The results are provided in Fig.", "REF and Table REF .", "As expected, a transparency of queue status information enables the tenants to optimize its reneging decisions, leading to both a high average profit and a high request admission rate.", "Once again, the reneging phenomenon is totally absent in this situation.", "Conversely, when only an imperfect queue knowledge is available, the tenants cannot balk but only renege, and the average/median values of end-profit are firmly dependent on the learning strategy.", "Generally, when the congestion is mild (Scenario B), a longer static learning phase leads to better profitability since it reduces unnecessary reneging decisions by reducing the estimation error.", "In contrast, under dense congestion conditions (Scenarios C and D), the learning gain is in average overwhelmingly countered by the waiting cost, so that the end profit decreases along the increasing static learning phase.", "Nevertheless, by adapting the learning phase to every specific request, our proposed optimal online Poisson learning method manages to achieve a better learning efficiency, and therewith remain performing close to the best static learning strategy in all scenarios.", "Table: Service statistics of RBR with different learning mechanisms, alternative option C, certainty level 90%." ], [ "Sensitivity to the alternative option", "Remark that for both the benchmark tests described above, within the limited manuscript length, we only showed the results with alternative option C, since it builds a reasonable benchmark for our evaluation.", "Indeed, similar results were also obtained, when we set the alternative option to E or G instead.", "Some sample tests of this kind are provided in Figs.", "REF –REF and Tables REF –REF .", "Figure: Different queue congestion control solutions considering Scenario D with variant alternative options.Figure: Different tenant learning strategies considering Scenario D with certainty level 90%90\\% and variant alternative options.Table: Service statistics of different queue congestion mechanisms, scenario C.Table: Service statistics of risk-based reneging with different learning mechanisms, scenario C, 90% certainty." ], [ "Conclusion", "In this work we have studied the risk-based tenant reneging mechanism, to realize a profitability-sensitive and privacy-intolerant distributed task dropping solution to control queue congestion in cloud services.", "We have investigated the tenants' choice between waiting for the cloud service and reneging for an alternative solution, and identified the conditions of balking and reneging in Poisson service queues.", "Our analysis proves that a time consistency of risk preference will sufficiently eliminate the reneging phenomenon, when a perfect knowledge about queue status is available a priori to the tenants.", "Regarding the data privacy of cloud service provider, we have also investigated the case where tenants do not possess the full queue status information, but only being able to online observe its position in queue.", "In this context, we have proposed an optimal online Poisson learning strategy that balances between learning gain and waiting cost.", "The effectiveness of our proposed methods is successfully verified in various service scenarios through extensive numerical simulations." ] ]
2105.11727
[ [ "SBEVNet: End-to-End Deep Stereo Layout Estimation" ], [ "Abstract Accurate layout estimation is crucial for planning and navigation in robotics applications, such as self-driving.", "In this paper, we introduce the Stereo Bird's Eye ViewNetwork (SBEVNet), a novel supervised end-to-end framework for estimation of bird's eye view layout from a pair of stereo images.", "Although our network reuses some of the building blocks from the state-of-the-art deep learning networks for disparity estimation, we show that explicit depth estimation is neither sufficient nor necessary.", "Instead, the learning of a good internal bird's eye view feature representation is effective for layout estimation.", "Specifically, we first generate a disparity feature volume using the features of the stereo images and then project it to the bird's eye view coordinates.", "This gives us coarse-grained information about the scene structure.", "We also apply inverse perspective mapping (IPM) to map the input images and their features to the bird's eye view.", "This gives us fine-grained texture information.", "Concatenating IPM features with the projected feature volume creates a rich bird's eye view representation which is useful for spatial reasoning.", "We use this representation to estimate the BEV semantic map.", "Additionally, we show that using the IPM features as a supervisory signal for stereo features can give an improvement in performance.", "We demonstrate our approach on two datasets:the KITTI dataset and a synthetically generated dataset from the CARLA simulator.", "For both of these datasets, we establish state-of-the-art performance compared to baseline techniques." ], [ "Introduction", "Layout estimation is an extremely important task for navigation and planning in numerous robotics applications such as autonomous driving cars.", "The bird's eye view (BEV) layout is a semantic occupancy map containing per pixel class information, e.g.", "road, sidewalk, cars, vegetation, etc.", "The BEV semantic map is important for planning the path of the robot in order to prevent it from hitting objects and going to impassable locations.", "In order to generate a BEV layout, we need 3D information about the scene.", "Sensors such as LiDAR (Light Detection And Ranging) can provide accurate point clouds.", "The biggest limitations of LiDAR are high cost, sparse resolution, and low scan-rates.", "Also, as an active sensor LiDAR is more power hungry, more susceptible to interference from other radiation sources, and can affect the scene.", "Cameras on the other hand, are much cheaper, passive, and capture much more information at a higher frame-rate.", "However, it is both hard and computationally expensive to get accurate depth and point clouds from cameras.", "The classic approach for stereo layout estimation contains two steps.", "The first step is to generate a BEV feature map by an orthographic projection of the point cloud generated using stereo images.", "The second step is bird's eye view semantic segmentation using the projected point cloud from the first step.", "This approach is limited by the estimated point cloud accuracy because the error in it will propagate to the layout estimation step.", "In this paper, we show that explicit depth estimation is actually neither sufficient nor necessary for good layout estimation.", "Estimating accurate depth is not sufficient because many areas in the 3D space can be occluded partially, e.g.", "behind a tree trunk.", "However, these areas can be estimated by combining spatial reasoning and geometric knowledge in bird's eye view representation.", "Explicitly estimating accurate depth is also not necessary because layout estimation can be done without estimating the point cloud.", "Point cloud coordinate accuracy is limited by the 3D to 2D BEV projection and rasterization.", "For these reasons, having an effective bird's eye view representation is very important.", "SBEVNet is built upon recent deep stereo matching paradigm.", "These deep learning based methods have shown tremendous success in stereo disparity/depth estimation.", "Most of these models [11], [9], [21], [19], [6], [26], [2], [8] generate a 3-dimensional disparity feature volume by concatenating the left and right images shifted at different disparities, which is used to make a cost volume containing stereo matching costs for each disparity value.", "Given a location in the image and the disparity, we can get the position of the corresponding 3D point in the world space.", "Hence, every point in the feature volume and cost volume corresponds to a 3D location in the world space.", "The innovation in our approach comes from the observation: it is possible to directly use the feature volume for layout estimation, rather than a two step process, which uses the point cloud generated by the network.", "We propose SBEVNet, an end-to-end neural architecture that takes a pair of stereo images and outputs the bird's eye view scene layout.", "We first project the disparity feature volume to the BEV view, creating a 2D representation from the 3D volume.", "We then warp it by mapping different disparities and the image coordinates to the bird's eye view space.", "In order to overcome the loss of fine grained information imposed by our choice of the stereo BEV feature map, we concatenate a projection of the original images and deep features to this feature map.", "We generate these projected features by applying inverse perspective mapping (IPM) [13] to the input image and its features, choosing the ground as the target plane We feed this representation to a U-Net in order to estimate the BEV semantic map of the scene.", "In order to perform inverse perspective mapping, we require information about the ground in the 3D world space.", "Hence we also consider the scenario where we perform IPM during the training time and not the inference time.", "Here, during the training time, we use cross modal distillation to transfer knowledge from IPM features to the stereo features.", "SBEVNet is the first approach to use an end-to-end neural architecture for stereo layout estimation.", "We show that SBEVNet achieves better performance than existing approaches.", "SBEVNet outperforms all the baseline algorithms on KITTI [5] dataset and a synthetically generated dataset extracted from the CARLA simulator [4].", "In summary, our contributions are the following: We propose SBEVNet, an end-to-end neural architecture for layout estimation from a stereo pair of images.", "We learn a novel representation for BEV layout estimation by fusing projected stereo feature volume and fine grained inverse perspective mapping features.", "We evaluate SBEVNet and demonstrate state-of-the-art performance over other methods by a large margin on two datasets – KITTI dataset and our synthetically generated dataset using the CARLA simulator." ], [ "Related Work", "To the best of our knowledge, there is no published work for estimating layout given a pair of stereo images.", "However, there are several works tackling layout estimation using a single image or doing object detection using stereo images.", "In this section, we review the most closely related approaches.", "Monocular Layout Estimation MonoLayout [14] uses an encoder-decoder model to estimate the bird's eye view layout using a monocular input image.", "They also leverage adversarial training to produce sharper estimates.", "MonoOccupancy [12] uses a variational encoder-decoder network to estimate the layout.", "Both MonoLayout and MonoOccupancy do not use any camera geometry priors to perform the task.", "[16] uses depth estimation to project the image semantics to bird's eye view.", "They also use Open Street Maps data to refine the BEV images via adversarial learning.", "[22] uses [16] to estimate the parameters of the road such as lanes, sidewalks, etc.", "Monocular methods learn strong prior, which does not generalize well when there is a significant domain shift.", "Stereo methods learn weak-prior plus geometric relationship, which can generalize better.", "Deep Stereo Matching Several methods like [11], [9], [21], [19], [6], [26], [2], [8] extract the features of the stereo images and generate a 3D disparity feature volume for disparity/depth estimation.", "They use a 3D CNN on the feature volume to get cost volume to perform stereo matching.", "PSMNet [2] uses a spatial pyramid pooling module and a stacked hourglass network to further improve the performance.", "High-res-stereo [24] uses a hierarchical model, creating cost volumes at multiple resolutions, performing the matching incrementally from over a coarse to fine hierarchy.", "Bird's Eye View Object Detection Several approaches like [23], [17] use LiDAR to perform 3D object detection.", "Pseudo-lidar [20] and pseudo-LiDAR++ [25] use stereo input to first generate a 3D point cloud and then use a 3D object detection network [10], [23], [17] on top.", "BirdGAN [18] maps the input image to bird's eye view using adversarial learning.", "The closest work to our approach is DSGN [3] which constructs a depth feature volume and map it to the 3D space which is then projected to bird's eye view to perform object detection.", "The task of object detection is of sparse prediction, whereas layout estimation is of dense fine granularity prediction.", "Hence we introduced IPM to fuse low level detail with the stereo information to improve the performance of layout estimation." ], [ "Our Method", "This section describes the detailed architecture of our proposed framework.", "SBEVNet is built upon recent deep stereo matching paradigms and follows the rules of multi-view camera geometry.", "An overview of the SBEVNet is summarized in Figure REF ." ], [ "Problem Formulation", "In this paper, we address the problem of layout estimation from a pair of stereo images.", "Formally, given a reference camera image $I_R$ and a target camera image $I_T$ both of size $H \\times W \\times 3$ , the camera intrinsics $K$ , and the baseline length $T_b$ , we aim to estimate the bird's eye view layout of the scene.", "In particular, we estimate the BEV semantic map of size $N_x\\times N_y \\times N_{C} $ within the rectangular range of interest area $(x_{min},x_{max},y_{min},y_{max})$ in front of the camera.", "Here $H$ is image height, $W$ is image width, and 3 indicates RGB channels.", "$N_x$ and $N_y$ are the number of horizontal cells and vertical cells respectively in bird's eye view.", "$N_C$ is the number of semantic classes.", "This BEV semantic map contains the probability distribution among all semantic classes at each cell of the layout.", "We assume that the input images are rectified." ], [ "Feature extraction", "The first step for SBEVNet is to extract features $F_R$ and $F_T$ of size $H^{\\prime } \\times W^{\\prime } \\times C$ for the reference image and the target image respectively.", "This is done by passing $I_R$ and $I_T$ through a convolutional encoder with shared weights.", "This produces multi-channel down-sized feature representations which are next used for building disparity feature volumes." ], [ "Disparity Feature Volume Generation", "Similar to [11], [9], [21], [19], [6], [26], [2], [8] we form a disparity feature volume $V$ by concatenating the features $F_R$ and $F_T^d$ , where $F_T^d$ is $F_T$ shifted horizontally by a disparity of $d$ pixel, resulting in a 3D volume of size $H^{\\prime } \\times W^{\\prime } \\times D \\times 2C $ .", "We then pass the feature volume through a series of 3D convolution layers with skip connections to learn higher level features.", "This feature volume at each $d \\in \\lbrace 0, 1, \\cdots , D-1\\rbrace $ contains a representation of the 3D world at the depth corresponding to the disparity $d$ .", "Rather than using this feature volume to do disparity estimation, we project and warp it to form a bird's eye view representation in the next step." ], [ "Bird's Eye View Representation", "The bird's eye view representation is composed of two parts – 1) The stereo BEV representation which is derived from the disparity feature volume, 2) The IPM BEV representation which is the result of applying inverse perspective mapping on the reference image and the features of the reference image.", "These two parts are concatenated to form the final bird's eye view representation." ], [ "Stereo BEV Representation", "The disparity feature volume generated is widely used to estimate depth/disparity in the stereo image pairs.", "But this feature volume contains a lot of information about the 3D scene which can be used for other tasks as well.", "Each point in the disparity feature volume corresponds to a point in the 3D world space.", "We first need to map the 3D feature volume to a 2D feature map containing information of the bird's eye view.", "If we do max/average pooling along height dimension, a certain degree of the height information is lost quickly before being extracted for our task, which is not desirable.", "Considering height information a good prior for layout estimation but we don't need to recover it explicitly, we concatenate the feature volume along the height, creating a 2D image of size $ W^{\\prime } \\times D \\times 2CH^{\\prime } $ .", "We then use 2D convolutions to generate the reduced feature volume of size $W^{\\prime \\prime } \\times D^{\\prime \\prime } \\times C^{\\prime } $ .", "This reduced feature volume does not spatially match with the bird's eye view layout.", "Hence, we warp the reduced feature volume, transforming it to a feature map of size $N_x\\times N_y \\times C^{\\prime }$ in the bird's eye view space.", "Given the disparity $d$ , position in the image along width $u$ , camera parameters $f$ , $c_x$ , $c_y$ , and stereo baseline length $T_x$ , we can find the coordinates in the bird's eye space $x^{\\prime }$ and $y^{\\prime }$ as follows: $x^{\\prime } = \\frac{ (u - c_x)\\cdot T_x }{d}$ $y^{\\prime } = \\frac{ f\\cdot T_x }{d}$ The 2D origin of the bird's eye view is co-located with the reference camera.", "An example visualization of the layout in the disparity volume space is shown in Figure REF .", "After mapping the coordinates to the BEV space, we map them to a grid of size $N_x \\times N_y$ giving us the stereo BEV representation $R_\\textrm {stereo}$ ." ], [ "IPM BEV Representation", "The stereo BEV representation contains structural information for the bird's eye view space.", "Due to the refinement and reduction of the feature volume, the fine grained details are excluded by design.", "To circumvent that, we need to fuse the low level features to the stereo BEV features, while maintaining geometric consistency.", "In order to fuse the image features to the stereo BEV features at the correct locations, we need to warp the image features to the BEV space.", "We apply inverse perspective mapping on the reference image and the features of the reference image to do that.", "A point in the image $I_R$ can correspond to multiple points in the 3D world space due to perspective projection, but there is a single point which also intersects with the ground plane.", "Let $z = ax + by + c$ be the equation of the ground plane in the world space.", "Given the input image coordinates $(u,v)$ and camera parameters $f$ ,$c_x$ , $c_y$ , we can find the coordinates in the bird's eye space $x^{\\prime }$ and $y^{\\prime }$ as follows: $x^{\\prime } = \\frac{c u - c c_x}{a c_x - a u - b f - c_y + v}$ $y^{\\prime } = \\frac{c f}{a c_x - a u - b f - c_y + v}$ This can be easily derived by combining the camera projection equation with the equation of the ground plane.", "For many applications, the ground is either planar or can be approximated by a plane.", "This is also equivalent to computing a homography $H$ between the ground plane and the image plane of the layout and then applying the transformation.", "We can have the parameters of the plane $a$ , $b$ , and $c$ pre-determined if the placement of the camera with respect to the ground is known, which is the case for many robotics applications.", "We can also determine $a$ , $b$ and $c$ by using stereo depth and a semantic segmentation network for the road/ground class.", "Examples of IPM on the input images is shown in Figure REF .", "We apply the inverse perspective transform on both the input image and the features of the input image to transform them to the bird's eye view space: $R_{\\textrm {IPM\\_feat}} = \\textrm {IPM}(F_R)$ $R_{\\textrm {IPM\\_img}} = \\textrm {IPM}(I_R)$ They are then concatenated with the stereo BEV representation to form the combined BEV representation: $R_{\\textrm {BEV}} = [ R_{\\textrm {IPM\\_feat}} ; R_{\\textrm {IPM\\_img}} ; R_\\textrm {stereo} ]$" ], [ "IPM for cross modal distillation ", "There can be use-cases where we cannot do inverse perspective mapping during inference time, due to the unavailability of the ground information.", "Hence, we consider the case where IPM is only available during the training time.", "We can think of the IPM features and the stereo features as different modalities and apply cross modal distillation ([7]) across them, and transfer knowledge from IPM features to the stereo features.", "Hence, we use the IPM BEV representation as a supervisory signal for the stereo BEV features.", "This forces the stereo branch of the model to implicitly learn the fine grained information learned by the IPM features.", "Rather than concatenating the IPM BEV features with the stereo features, we minimize the distance between them.", "We call this variant of SBEVNet as SBEVNet-CMD (SBEVNet cross modal distillation).", "During the training time, the IPM BEV features and the stereo features are used to generate the BEV semantic maps.", "$C^\\text{IPM} = \\text{U-Net}(R_{\\textrm {IPM\\_feat}})$ $C^\\text{stereo} = \\text{U-Net}(R_{\\textrm {stereo}})$ This ensures both IPM BEV features and stereo BEV features learn meaningful information.", "We jointly minimise the $L_1$ distance between first $K$ channels of the features.", "$L_\\text{KT} = \\left\\Vert R_{\\textrm {IPM\\_feat}}[:K] - R_{\\textrm {stereo}}[:K]\\right\\Vert _{L_{1}}$ By this, we ensure that the stereo model can learn information that is not in the IPM features.", "In our experiments, we found this to yield better results compared to the approach of minimizing the $L_1$ distance between all the channels of the features.", "During test time, we only use the stereo features to get the BEV layout.", "Our experiments show that the stereo model with cross modal distillation performs better than the stereo model without cross modal distillation." ], [ " Layout Generation", "We can generate the semantic map by inputting the BEV features to a semantic segmentation network.", "We pass the concatenated stereo BEV feature map and IPM BEV feature map to a U-Net [15] network to generate the semantic map $C$ .", "$C = \\text{U-Net}(R_{\\textrm {BEV}})$ Some areas in the layout may not be in the view of the front camera, e.g.", "things behind a wall.", "That is why it is not a good idea to penalize the model for the wrong prediction for those areas.", "Hence, we use a visibility mask to mask the pixel-wise loss, applying it only on the pixels which are in the field of view.", "This mask is generated during the ground truth generation process by using ray-tracing on the point cloud to determine which are in the field of view.", "For a visibility mask $V$ , $V_i$ is 1 if the pixel $i$ is in the view of the input image, and 0 otherwise.", "For the loss, we use a pixel-wise categorical cross entropy loss as follows: $L_r = \\sum _{i \\in P } V_i \\cdot \\textrm {CCE}( C_i , C_i^{h} )$ where $C_i^{h}$ is ground truth.", "The total loss for SBEVNet-CMD is the sum of supervision loss from the two feature maps and the $L_1$ distance minimization.", "$L_c = \\sum _{i \\in P } V_i \\cdot \\textrm {CCE}( C_i^\\text{IPM} , C_i^{h} ) + \\sum _{i \\in P } V_i \\cdot \\textrm {CCE}( C_i^\\text{stereo} , C_i^{h} ) + L_\\text{KT}$" ], [ " Datasets", "CARLA dataset: We use the CARLA [4] simulator to generate a synthetic dataset.", "A pair of stereo cameras are placed on a car moving along a set of waypoints.", "We also randomly change the weather/lighting conditions.", "We use the point cloud of the simulator's city model to generate the ground truth semantic map.", "To ensure that we evaluate the generalizability of the models, the training and testing are done on entirely different city models in CARLA.", "Town01, Town02, Town03, and Town04 are used for training, and Town05 is used for testing.", "The training set contains 4,000 pairs of stereo images and the testing set contains 926 pairs of stereo images.", "The classes we use for the semantic map are road, vegetation, car, sidewalk, and building.", "The bounds of the layout with respect to the camera are -19 to 19 meters in $x$ direction and 1 to 39 meters in the $y$ direction.", "KITTI dataset: We also evaluate SBEVNet on the publically available KITTI [5] dataset.", "Similar to [14], we use the KITTI odometery subset and use the SemanticKITTI [1] dataset for labeled ground truth point clouds.", "We use the same training/testing split as used by [14], where separate sequences are used for training and testing.", "The training set contains 3,278 stereo image pairs and the testing set contains 1,371 stereo image pairs.", "The classes we use for the semantic map are road, vegetation, car, sidewalk, and building.", "The bounds of the layout with respect to the camera are -19 to 19 meters in $x$ direction and 5 to 43 meters in the $y$ direction." ], [ " Evaluation Metrics", "As not all the regions of the ground truth layout are visible from the camera, we only consider pixels of the layout which are in the field of view.", "For evaluating the semantic map, we use macro averaged intersection over union (IoU) scores for the layout pixels which are in the visibility mask.", "We report the IoU scores for each semantic class separately.", "Table: Quantitative results of semantic layout estimation on the CARLA dataset.Figure: Qualitative results on the test set of the CARLA and the KITTI dataset.", "The major mistakes in the predictions are annotated by a blue rectangle.Table: Quantitative results of semantic layout estimation on the KITTI dataset." ], [ " Compared Methods", "There are no previously reported quantitative results for the task of stereo layout estimation in our setting.", "Thus, we evaluate appropriate baselines which are prior works extended to our task.", "Pseudo-LiDAR [20] + segmentation: Uses Pseudo-lidar with PSMNet to generate a 3D point cloud from the input stereo images which is used to project the semantic segmentation of the front view to the bird's eye view.", "The PSMNet is trained separately on the respective datasets for better performance.", "Pseudo-LiDAR [20] + BEV U-Net: The RGB 3D point projected in the BEV aligned with the ground truth layout is used to train a U-Net segmentation network.", "IPM + BEV U-Net: Inverse perspective mapping is applied to the input image to project it to the BEV space which is used to train a U-Net segmentation network.", "MonoLayout [14]: This baseline uses MonoLayout to generate BEV semantic map from a single image.", "Rather than using OpenStreetMap data for adversarial training, we used random samples from the training set itself.", "MonoLayout [14] + depth: The input RGB image concatenated with the depth is used as an input to the MonoLayout Model.", "MonoOccupancy [12] + depth: The input RGB image concatenated with the depth is used as an input to the MonoOccupancy Model.", "We also evaluate some variations of our model to perform ablation studies.", "In SBEVNet only stereo we exclude the IPM features and only use features derived from the feature volume.", "To gauge the importance of IPM on RGB images and features, we also try applying IPM only on RGB images (SBEVNet stereo + RGB IPM) and IPM only on the features of the input image (SBEVNet stereo + features IPM).", "We also evaluate the cross modal distillation model SBEVNet-CMD.", "Finally, we evaluate our complete model (SBEVNet ) where we use stereo features and IPM on both RGB image and its features.", "We also evaluate SBEVNet Ensemble where we take an ensemble of SBEVNet with the same architecture but different initialization seeds." ], [ " Implementation Details", "We implemented SBEVNet using Pytorch.", "We use Adam optimizer with the initial learning rate of 0.001 and betas (0.9, 0.999) for training.", "We use a batch-size of 3 on a Titan X Pascal GPU.", "We use the same base network which is used in the basic model of PSMNet.", "The input image size for the CARLA dataset is 512$\\times $ 288 and the input image size for the KITTI dataset is 640$\\times $ 256.", "We report the average scores according to 8 runs to account for the stochasticity due to random initialization and other non-deterministic operations in the network." ], [ " Experimental results ", "We report the IoU scores of all the methods on the CARLA and KITTI [5] dataset in Table REF and Table REF respectively.", "As we can see from the tables, SBEVNet achieves superior performance on both the datasets.", "We also observe the increase in performance if we use both stereo information and inverse perspective mapping.", "IPM yields a greater increase in performance in the CARLA [4] dataset because the ground is perfectly flat.", "If we use only RGB IPM along with stereo, the results are slightly worse on the KITTI dataset because the ground is not perfectly planar.", "We see that degradation does not persist if we also use IPM on the image features.", "For the KITTI dataset, we see a sharp improvement over pseudo-LiDAR approaches because of inaccurate depth estimation.", "On the other hand, our model does not depend on explicit depth data/model.", "The results of MonoLayout [14] and MonoOccupancy [12] are inferior due to lack of any camera geometry priors in the network.", "We also show the qualitative results on the test set of CARLA [4] and KITTI [5] dataset in Figure REF .", "We see that in certain regions SBEVNet gives outputs closer to the ground truth.", "For example, Psuedo-lidar fails to segment cars in the KITTI dataset.", "We also observe a drop in quality in the estimated layout as we move further from the camera." ], [ "Ablation Study", "IPM on RGB image For the CARLA dataset, we observe an increase of 3.67 in the mIoU score, on concatenating IPM RGB with the stereo features.", "We observe an increase in IoU scores for all the classes, with the biggest increase of 8.12 in the cars class.", "For the KITTI dataset, there is a small decrease of 0.45 in the mIoU score.", "This is because, the ground is not perfectly planar, hence the IPM RGB images do not exactly align with the ground truth layout.", "IPM on image features If we apply IPM on the features of the input image and concatenate it with the features from the stereo branch, we see an improvement in both the datasets.", "The improvements in mIoU scores are 6.19 and 0.59 for the CARLA and KITTI dataset.", "The improvement is higher compared to the RGB IPM because image features contain higher level information which is transformed to the BEV space.", "IPM on both RGB image and image features We see the greatest improvement if we apply IPM on both the RBG image and the features of the RGB image.", "The improvements in mIoU scores are 8.26 and 1.35 for the CARLA and KITTI dataset respectively.", "This is because the model is able to exploit the different information present in $R_{\\textrm {IPM\\_feat}}$ and $R_{\\textrm {stereo}}$ .", "Cross modal distillation The performance of SBEVNet-CDM is in between of stereo only SBEVNet and full SBEVNet.", "We see an improvement of 4.00 and 0.72 in the mIoU scores on the CARLA and KITTI dataset, if we train the stereo model using cross modal distillation via IPM features.", "During inference, the architecture of SBEVNet-CMD is the same as the stereo only SBEVNet.", "This shows that CMD is able to transfer most of the IPM knowledge to the stereo branch.", "Minimizing distance between first $K$ features We also evaluate the approach, where we try minimizing the L1 distance between all the channels of the IPM features and stereo features.", "We observe mIoU scores of 32.27 and 50.03 for the CARLA and KITTI dataset respectively.", "This is worse than the mIoU scores achieved by minimizing the distance between first $K$ channels.", "This is because, if we enforce all stereo branch channels to be the same as IPM branch channel, the stereo branch is unable to learn information that is not present in the IPM features." ], [ "3D feature volume analysis", "One claim of our approach is that our model learns 3D information without any explicit depth/disparity supervision.", "To validate this claim, we use the learned 3D feature volume to perform disparity estimation.", "We freeze all the weights and add a small 3D convolution layer to perform disparity regression on the learned feature volume.", "We also observe that the feature volume which is trained with cross modal distillation via IPM performs better at the task of disparity estimation.", "For the CARLA dataset, we find that the SBEVNet only stereo model has a 3-pixel error of 7.92 and with cross model distillation the 3-pixel error goes down to 6.84." ], [ "Distance from camera", "We wish to quantify how our system performs as we move away from the camera.", "Hence, we plot the IoU scores for the pixels in the BEV layout which are more than a given distance from the camera (Figure REF ) and for the pixels which are less than a given distance from the camera (Figure REF ).", "For both the KITTI and the CARLA dataset, we observe that there is a drop in performance as the distance from the camera increases.", "We also observe that SBEVNet outperforms the stereo only SBEVNet at all distances from the camera.", "Figure: Performance as a function of maximum distance from the camera.", "We consider the pixels in the BEV layout which are atmost a certain distance away from the camera.Figure: Performance as a function of minimum distance from the camera.", "We consider the pixels in the BEV layout which are atleast a certain distance away from the camera." ], [ "Amount of training data", "We also quantify how the performance of SBEVNet changes with the number of data-points in the training set, while keeping the test set same.", "On the CARLA dataset, with just 10% of the training data, we get mIoU score of 26.10 compared to the mIoU score of 44.36 with all the training data.", "For both the datasets, performance starts to saturate when we use 100% of the training data.", "This shows that 3,000-4,000 training data points are sufficient for getting the optimal performance from SBEVNet.", "Figure: Performance of the system as a function of amount of training data used." ], [ "Performance evolution during training", "We also perform a qualitative analysis (Figure REF ) of how the performance of the model changes during training.", "We observe, during the initial stages of training, the model learns to identify very course grained attributes such as the direction of the road.", "However, there is ambiguity in detailed attributes such as size and exact position.", "During the later stages of training, the model learns to identify the smaller objects and exact positioning in the BEV space.", "Figure: Evolution of predicted BEV layouts for different epochs during training" ], [ "Ensemble", "We observe some variance in the performance of the models on training with different random seeds.", "For SBEVNet we observe a standard deviation of 2.16 and 2.46 in the mIoU scores for the KITTI and CARLA dataset respectively.", "Due to the diversity in outputs of the individual models ([27]), we see an improvement in the performance, if we take an ensemble of individual models.", "We observe an absolute improvement of 2.49 and 3.56 in the mIoU scores for the KITTI and CARLA dataset respectively." ], [ "Inference time", "On NVIDIA Titan X GPU, with the batch size equal to 1, the inference time of stereo only SBEVNet for one input pair is 0.1307s on the average.", "For the full SBEVNet the inference time is 0.1449s on the average, which is slightly higher than the stereo only model.", "This inference speed is sufficient for majority of robotics applications.", "The majority of computation is done in processing the 3D feature volume with 3D convolutions." ], [ "Conclusion", "In this paper we proposed SBEVNet, an end-to-end network to estimate the bird's eye view layout using a pair of stereo images.", "We observe improvement in the IoU scores compared with approaches that are not end-to-end or do not use geometry.", "We also showed that combining inverse perspective mapping with the projected disparity feature volume gives better performance.", "We also show that, using cross modal distillation to transfer knowledge from IPM features to the stereo features gives us an improvement in results." ], [ "Results on KITTI object dataset", "We also compare our method with the published numbers on the KITTI Object dataset.", "We use the dataset and annotations provided by [14].", "Here, there is only a single cars class.", "We compare our approach with published monocular approaches and 3D object detection approach on pseudo lidar with stereo input.", "AVOD + pseudo lidar is an object detection method which also uses the large sceneflow dataset for pre-training.", "More details of these baseline models can be found in [14].", "Table REF shows the numbers on the KITTI object split which are provided by [14].", "We observe that our method achieves an improvement in the mIoU scores over all the other methods.", "For segmentation, pixel level mAP is not a good metric as is does not consider false negatives.", "We still report the pixel level mAP scores for reference.", "Table: Quantitative results of BEV car segmentation on the KITTI object dataset.", "All the results except SBEVNet are excerpted from" ] ]
2105.11705
[ [ "Cross-match between the latest Swift-BAT and Fermi-LAT catalogs" ], [ "Abstract We report the results of a cross-match study between the hard X-ray and GeV gamma-ray catalogs, by making use of the latest 105-month Swift-BAT and 10-yr Fermi-LAT catalogs, respectively.", "The spatial cross-matching between the two catalogs results in the matching of 132 point-like sources, including ~5% of false-match sources.", "Additionally, 24 sources that have been identified as the same identifications are matched.", "Among the 75 extended sources in the Fermi-LAT catalog, 31 sources have spatial coincidences with at least one Swift-BAT source inside their extent.", "All the matched sources consist of blazars (>60%), pulsars and pulsar wind nebulae (~13%), radio galaxies (~7%), binaries (~5%), and others.", "Compared to the original catalogs, the matched sources are characterized by a double-peaked photon index distribution, higher flux, and larger gamma-ray variability index.", "This difference arises from the different populations of sources, particularly the large proportion of blazars (i.e., FSRQ and BL Lac).", "We also report 13 cross-matched and unidentified sources.", "The matched sources in this study would be promising in the intermediate energy band between the hard X-ray and GeV gamma-ray observations, that is the unexplored MeV gamma-ray domain." ], [ "Introduction", "The sky in the MeV gamma-ray energy range has remained unexplored for almost 30 years since the first devoted MeV detector, the Imaging Compton Telescope COMPTEL onboard the Compton Gamma-Ray Observatory (CGRO) mission [34] launched in April 5, 1991, was in operation.", "However, there are promising discoveries to be made in this energy band [36], which is the main motivation for sensitive and improved observations in the next decades.", "While MeV observations await the next-generation instruments, the neighboring energy bands, the hard X-ray and the GeV gamma ray, have been well studied for the last decade by, for example, Swift/Burst Alert Telescope (BAT) [10] and Fermi/Large Area Telescope (LAT) [8], respectively.", "These two observatories provide us with a legacy of observational data, including source catalogs, in the corresponding energy channels.", "Therefore, by using the latest Swift-BAT and Fermi-LAT catalogs, one can perform catalog cross-match and somewhat predict the currently unavailable information in the MeV band.", "The importance of the catalog cross-match is to list promising objects in the MeV gamma-ray band.", "Sources that have been detected both in the hard X-ray and GeV gamma ray would be plausible MeV gamma-ray emitting sources unless the X-ray and gamma-ray photon indices are extremely soft and hard, respectively.", "This new catalog of the cross-matched sources is useful for ongoing projects for the MeV observations [14], [30], [38], [6].", "The cross-match between the hard X-ray and GeV gamma-ray catalogs is also meaningful in high energy astrophysics.", "Both these energy ranges point to non-thermal radiation processes, as we expect that the thermal X-ray emission does not have a substantial contribution to the hard X-ray.", "Thus, the hard X-rays originate from synchrotron radiation or ic scattering from accelerated electrons, while the gamma-rays are produced by a leptonic process (i.e., ic scattering from high-energy electrons) or a hadronic process (e.g., hadronuclear interaction).", "An alternative is nonthermal bremsstrahlung from accelerated particles.", "If a source emits both the hard X-rays and GeV gamma rays that originate from accelerated particles (electrons or protons) via the same or different radiation mechanisms, the broadband energy spectrum gives us an important clue to understand the particle acceleration and/or the emission mechanisms.", "[29] previously performed a catalog cross correlation using the 54-month Swift-BAT catalog (2PBC; [13]) and the 1-yr Fermi-LAT catalog (1FGL; [1]), which had 1256 and 1451 entries, respectively.", "In this paper, we revisit to the cross-matching by making use of the latest catalogs; the 105-month Swift-BAT catalog [31] and the 10-yr Fermi-LAT catalog (4FGL-DR2; [9]).", "With the more accumulated data and better flux sensitivity, the number of sources in the latest catalogs were improved.", "Both catalogs were based on the observational data of all sky surveys.", "INTEGRAL [40] also performed hard X-ray observations and provided us with a hard X-ray catalog [11].", "However, because of its non-uniform exposure toward the sky (e.g., INTEGRAL has deeper exposure on the Galactic plane), we complementarily use the INTEGRAL catalog in this study.", "In this work, we present a catalog cross-match using the latest Swift-BAT and Fermi-LAT catalogs.", "Section  briefly summarizes the two catalogs.", "The matching method is given in Section .", "The results of the matched sources are presented in Section .", "In Section , we compare the matched catalog with other existing catalogs in the energy bands from hard X-ray to MeV gamma ray, investigate properties of the matched sources, and discuss the unidentified sources.", "The conclusions are presented in Section ." ], [ "Catalogs", "This work makes use of the Swift-BAT 105-month [31] and the Fermi-LAT fourth (Data Release-2) [2], [9] catalogs of hard X-ray and GeV gamma-ray sources, respectively." ], [ "The Neil Gehrels Swift Observatory (Swift) started its operation after the spacecraft was launched on November 20, 2004 [19].", "There are three scientific instruments onboard, UV/Optical Telescope (UVOT; 170–650 nm), X-ray Telescope (XRT; 0.2–10 keV), and Burst Alert Telescope (BAT; 14–195 keV).", "BAT consists of a coded-aperture mask and a large-area solid state detector (CdZnTe) array, enabling us to detect hard X-rays in the 15–150 keV energy band with a large fov of 1.4 sr and a psf of 17$^\\prime $ [10].", "Although BAT is primarily designed for detecting grb, the accumulated data allows the BAT team to perform a uniform all-sky survey and produce a hard X-ray source catalog.", "The latest catalog, the Swift-BAT 105-month catalog [31], made use of data taken from December of 2004 to August of 2013.", "Using the 105-month data, the all sky in the 14–195 keV band was uniformly covered with sensitivities of $8.40 \\times 10^{-12}$  ${\\rm erg}~{\\rm cm}^{-2}~{\\rm s}^{-1}$ and $7.24 \\times 10^{-12}$  ${\\rm erg}~{\\rm cm}^{-2}~{\\rm s}^{-1}$ for over 90% and 50% of the sky, respectively.", "This resulted in detection of 1632 sources at $>4.8 \\sigma $ .", "Images, 8-channel energy spectra, and month-scale light curves of the sources in the catalog are availablehttps://swift.gsfc.nasa.gov/results/bs105mon/.", "In the Swift-BAT 105-month catalog, the largest proportion is Seyfert galaxies (827 in total; including 379 Seyfert I and 448 Seyfert II), the second one is X-ray binaries (225 in total; 109 lmxb, 108 hmxb, and 8 others), and the third one is beamed agn (158 in total; including fsrq and BL Lac types (BLLs))." ], [ "The Fermi satellite, launched on June 11, 2008, consists of two scientific instruments, Large Area Telescope (LAT) and Gamma-ray Burst Monitor (GBM).", "The Fermi-LAT is a pair-conversion gamma-ray telescope with a precision tracker and calorimeter, each consisting of a 4$\\times $ 4 array of 16 modules, a segmented anti-coincidence detector that covers the tracker array, and a programmable trigger and data acquisition system [8].", "Fermi-LAT enables us to perform spectroscopy in gamma-ray energies ranging from 20 MeV to more than 300 GeV with a wide fov of 20% of the sky.", "The psf of Fermi-LAT is approximately 3.5$^\\circ $ at 100 MeV and 0.1$^\\circ $ at 10 GeV.", "The other instrument, GBM, covers two thirds of the sky at a moment and detects grb in the 8 keV–40 MeV band.", "Fermi-LAT 4th Catalog Data Release 2 (4FGL-DR2https://fermi.gsfc.nasa.gov/ssc/data/access/lat/10yr_catalog/; [9]) is the latest catalog based on 10-yr observational data taken from August 4, 2008 to August 2, 2018.", "The previous catalog, the 8-yr Fermi-LAT 4th catalog (4FGLhttps://fermi.gsfc.nasa.gov/ssc/data/access/lat/8yr_catalog/), was described in detail in [2].", "These catalogs made use of the data of the all-sky survey with the flux sensitivity of $10^{-11}$ –$10^{-12}$  ${\\rm erg}~{\\rm cm}^{-2}~{\\rm s}^{-1}$ in the energy range of 50 MeV to 1 TeV, depending on the source location and the energy of gamma rays.", "4FGL-DR2 has 5788 sources detected at $>4\\sigma $ , while 4FGL has 5065 sources.", "In both catalogs, 75 sources were reported to have spatial extension.", "The catalogs provide us with the locations, 7-band energy spectra, and lightcurves in 2-month and 1-yr time binsNote that 2-month lightcurves are available only in 4FGL (the 8-yr catalog)., which are useful for cross-matching in this paper.", "We mainly made use of 4FGL-DR2 for the following analyses and used 2-month lightcurves of 4FGL for reference since 4FGL-DR2 did not include 2-month lightcurves.", "The three biggest source types in 4FGL-DR2 are blazars (60%), unknown or unindentified sources (30%), and pulsars (5%).", "Here we note that the source category defined in the Fermi-LAT catalog has two cases, an upper case (e.g., FSRQ) and a lower case (e.g.", "fsrq), which respectively indicate a firm association and an association.", "Throughout this paper, we also adopt the same definition for the source category of 4FGL-DR2, otherwise mentioned." ], [ "cross-match – method", "We cross-match the 1632 Swift-BAT sources and the 5788 Fermi-LAT sources by a spatially matching for point-like sources (Section REF ) and extended sources (Section REF ) and carry out an identification matching (Section REF ).", "It should be noted that we use coordinates of the detected sources, not coordinates of the associated sources, in order to calculate the angular separation between the BAT and LAT sources." ], [ "Spatial cross-match of point sources", "The separation threshold for spatial cross-match (0.08$^\\circ $ ) was determined in the same way proposed in [26].", "First, we produced a distance profile, which is a sum of the number of the Fermi-LAT sources located between $r$ and $r+dr$ centered at each Swift-BAT source as a function of the distance $r$ (Figure REF ).", "In Figure REF , $dr$ is set to be 0.02$^\\circ $ , and the profile is generated up to $r=2.0$$^\\circ $ .", "The distance profile contained a spike around $r=0$$^\\circ $ and a linear increase for $r>0.2$$^\\circ $ .", "The former feature indicates plausible associations, while the latter could correspond to false matches.", "We thus fit the linearly increasing profile at $r>0.2$$^\\circ $ with an empirical model of $N = a rdr$ , where $a$ is a constant.", "The best-fit parameter of $a$ was obtained to be 2500 counts deg$^{-2}$ .", "In order to suppress the false associations (i.e., the background level) down to 5%, we set the separation threshold, $r_{\\rm sep}$ , to 0.08$^\\circ $ .", "Note that the background level of 10% corresponded to $r_{\\rm sep}$ of 0.12$^\\circ $ .", "We checked that the choices of $dr$ and the $r$ range for the distance profile did not have effects on determination of $a$ and $r_{\\rm sep}$ .", "The obtained $r_{\\rm sep}$ (=0.08$^\\circ $ ) is much smaller than the psf of the detectors and comparable with the average positional uncertainty that is 0.062$^\\circ $ for Swift-BAT and 0.06$^\\circ $ –0.08$^\\circ $ for Fermi-LAT.", "Applying $r_{\\rm sep}$ =0.08$^\\circ $ , 132 sources were found to be cross-matched (i.e., that had counterparts within the separation).", "Note that the number of the matched sources increased to 161 sources if we adjusted $r_{\\rm sep}$ =0.12$^\\circ $ , including possible 10% false matches.", "The 132 spatially matched sources are listed in Table , in which we show the source name, source type, position, spectral information (flux and photon index), and gamma-ray time variability index, taken from the original two catalogs.", "We also show the derived separation and Flag which indicates the status of the matched source (see Section REF for detail).", "The results are presented in Section REF .", "It should be noted that the position determination accuracy of both Swift-BAT and Fermi-LAT depends on brightness of sources.", "Therefore we also carried out a spatial cross-match by setting the separation threshold to $\\sigma _{\\rm BAT} + \\sigma _{\\rm LAT}$ , where $\\sigma _{\\rm BAT}$ and $\\sigma _{\\rm LAT}$ indicate the positional error of each source in the Swift-BAT and Fermi-LAT catalogs, respectively.", "This results in detection of 182 matched sources, which includes all the 132 spatially matched sources.", "Among the 50 sources that are missed in the spatial matching by $r_{\\rm sep}$ =0.08$^\\circ $ , 27 sources are matched extended sources (Section REF ) or identification-matched sources (Section REF ), and the remains are 7 unidentified sources and 16 false matches.", "Figure: Distance profile in the range of r=0r=0–2 ∘ ^\\circ with dr=0.02dr=0.02 ∘ ^\\circ .The red line shows the best-fit background model, with aa being 2500 counts deg -2 ^{-2}.The black dashed vertical line indicates the separation threshold of 0.08 ∘ ^\\circ , which suppresses the background level to 5%." ], [ "Spatial cross-match of extended sources", "The 4FGL-DR2 catalog confirmed 75 extended sources, whose properties, including morphology, were provided in the catalog.", "The source extensions range from 0.03$^\\circ $ to 3.5$^\\circ $ .", "We cross-matched the two catalogs based on the assumption that the extended LAT sources had BAT sources within their extension.", "29 sources were matched with $d \\le \\sigma _\\gamma $ , where $d$ is the angular separation between the center of the LAT source and the nearest BAT source, and $\\sigma _\\gamma $ is the gamma-ray spatial extent [2].", "Additional 2 sources (lmc 30 Dor.", "West and HESS J1420$-$ 607) were matched with $d + \\sigma _{\\rm BAT} \\le \\sigma _\\gamma $ , taking the positional error of the BAT source ($\\sigma _{\\rm BAT}$ ) into consideration.", "In this paper, we defined these 31 sources as extended cross-matched sources.", "It is notable that MSH 15$-$ 52 and Crab nebula (IC component), which were extended sources in 4FGL-DR2, were also positionally matched in Section REF ." ], [ "Source identification cross-match", "We also used an identification matching method to cross-match sources.", "When we cross-matched by the source names provided in the catalogs, 123 were matched between the Swift-BAT and Fermi-LAT catalogs.", "94 of these 123 sources were already included in the spatial match of point sources (Section REF and Table ), and another 5 sources were already presented in the spatial match of extended sources (Section REF and Table ), so we do not include them here.", "The remaining 24 sources were not contained in our method of spatial cross-match.", "Among the spatially unmatched and name-matched 24 sources, 10 were spatially matched if we adopted $r_{\\rm sep}$ =0.12$^\\circ $ in Section REF .", "The remaining 14 sources may have been positionally unmatched because they had relatively large position errors because of the faint flux and had slightly larger separation than $r_{\\rm sep}$ .", "The separation was remarkably large for the galactic two pulsars, PSR J1420$-$ 6048 and PSR J1723$-$ 2837, and they had large position uncertainties because of their location in a complex region on the Galactic plane.", "To search for associated sources in 4FGL-DR2, the 105-month Swift-BAT catalog was utilized as well as the many other catalogs listed in Table 6 of [2].", "In fact 4FGL-DR2 included 5 sources which were registered solely from the Swift-BAT catalog and not from the other catalogs in Table 6 in [2].", "They were included in our matched catalog, No.", "131, 132, and 154–156 in Table .", "The former two were spatially matched with in $r_{\\rm sep}$ =0.08$^\\circ $ , while the latter three were matched by the identifications.", "It should be noted that the latter three sources had small association probability, $P< 0.6$ [2], except for SWIFT J1808.5$-$ 3655." ], [ "Results — Cross-matched catalog", "The catalog of the cross-matched sources between the Swift-BAT and Fermi-LAT is provided here.", "The spatial cross-match resulted in 132 matched sources, while the identification cross-match resulted in 24 more matched sources.", "All the 156 matched point-like sources are summarized in Table  and discussed in Section REF , and the cross-matched extended sources are listed in Table  (Section REF ).", "Section REF presents the summary of source types of the matched sources.", "It should be noted that Crab (No.", "116 in Table ) has three entries in 4FGL-DR2 (i.e., emission from the Crab pulsar, synchrotron emission from the Crab nebula, and inverse Compton scattering from the Crab nebula).", "In this paper, we have listed only the synchrotron component that represents the three entries, because it corresponds to the hard X-ray emission seen by BAT." ], [ "Cross-matched point sources", "The obtained 156 sources in Table  were divided into five groups: firmly matched source (with Flag being M in Table ), false-matched source (F), source with different source categories between the two catalogs (D), unidentified source or unknown association (U), and ambiguous source (A).", "Brief descriptions of each group are given in the following.", "The matched source was defined as a source which was identified as the same source name and the same source type between the Swift-BAT and Fermi-LAT catalogs." ], [ "False match (Flag=F)", "The false match indicates that a spatially matched source had different identifications and different source types in the two catalogs.", "Because $r_{\\rm sep}$ =0.08$^\\circ $ was determined as the level of false matching was reduced to 5%, the 132 spatially matched sources would contain roughly 7 falsely matched sources.", "Indeed, Table  includes 8 sources where the two associated sources are not identical in the two catalogs.", "Among the 8 sources, 3 sources were pulsars in 4FGL-DR2 but different point sources in the BAT catalog (No.", "108, 109, and 114 in Table ).", "One source was classified as a pwn in 4FGL-DR2 but a molecular cloud in the BAT catalog (No.", "117), resulting from the fact that both sources are located in the radio arc near the complex galactic center.", "The rest 4 false-match sources were globular clusters in 4FGL-DR2, but the corresponding BAT sources were lmxb in the globular clusters (No.", "125–127 and 129).", "These sources were likely false-matched because (1) they were confused by the emission from the Galactic plane ($|b| < 10$$^\\circ $ for No.", "108, 109, 117, 127, and 129), (2) they were relatively faint and had large uncertainty in position determination accuracy (No.", "125 and 126), or (3) they had slightly smaller separation than $r_{\\rm sep}$ (No.", "114)." ], [ "Different source type (Flag=D)", "The different-type source is identified as a source which has the same source name, but has different source types defined in the two catalogs.", "Table  includes 11 of these sources.", "7 sources were agn with different subclasses defined in the two catalogs: they were Seyfert galaxies in the BAT catalog, but in 4FGL-DR2 they were classified as blazar candidate of uncertain type (bcu) (No.", "81 in Table ), radio galaxies (No.", "95, 96, 149 and 150), or starburst galaxies (No.", "98 and 100).", "They had the different subclasses because the hard X-ray and GeV gamma-ray radiation would originate from the same AGN but from the different mechanism.", "We can naturally expect such associations.", "The X-ray emission in Seyfert galaxies originates in agn coronae, which do not emit intense GeV gamma-ray emission due to internal $\\gamma \\gamma $ annihilation [25], [24].", "Since Seyfert galaxies also have star-formation activity, we see GeV emission from some of nearby Seyfert galaxies [5].", "In radio galaxies, the X-ray emission originates in the same way as in Seyfert galaxies, while AGN jet can dominate the gamma-ray emission [27].", "Another different-category sources were snr in the BAT catalog but pulsars in 4FGL-DR2 (No.", "106 and 107), and they were known snr hosting pulsars [18], [7], [23].", "1RXS J122758.8-485343 (No.", "110) was classified as a CV and pulsar in the Swift-BAT and Fermi-LAT catalogs, respectively.", "Although the BAT catalog labeled it as a CV, it is also known as a peculiar hard X-ray source possibly associated with the Fermi-LAT source.", "[15], based on the multiwavelength observations from the radio to gamma-ray energy bands, suggested that the system would be a gamma-ray emitting lmxb.", "Despite the extensive study, the nature of source No.", "110 remains undetermined, and thus we labeled this source as Flag=D.", "The other source, the Galactic center (No.", "83), was classified as SGR A$^\\star $ (source type is Galactic Center) in the BAT catalog and Galactic Centre (source type is bcu) in 4FGL-DR2." ], [ "Unidentified association (Flag=U)", "There were 9 sources with unknown associations, of which the source type was unclear either in the Swift-BAT or Fermi-LAT catalogs (No.58, 65, 75, 130–132, and 154–156).", "It should be noted that 4DFL-DR2 has two-type definitions of uncertain sources; unidentified type (i.e., sources without any firm associations) and unknown type (i.e., low Galactic-latitude sources associated solely by the Likelihood-Ratio method [2]).", "4FGL-DR2 has 1679 unidentified sources and 115 sources of unknown type.", "In this paper, we merged both types and referred to them as the unidentified sources.", "These sources, with their sed, are discussed in Section REF ." ], [ "Ambiguous sources (Flag=A)", "Three sources, No.", "7, 13, and 76 in Table , were flagged as ambiguous, although their source types were AGNs in a broad meaning (i.e., Seyfert galaxy in the BAT catalog, but bll or bcu in 4FGL-DR2).", "If the associations defined in the two catalogs are correct, these 3 sources would be false-matched.", "However, the separation was smaller than the accuracy of position determination, and it might be better not to conclude that they were false-matched sources.", "We, therefore, left them being ambiguous sources, and they need more investigations in the future to determine if they could be false matches or AGNs with different subclass." ], [ "Cross-matched extended sources", "All the BAT sources located inside the 31 LAT extended sources are listed in Table , and the angular separation for each source from the LAT source is also shown.", "12 LAT sources have more than one BAT source within the extent.", "It should be noted that among the 31 sources, MSH 15$-$ 52 was also matched by the spatial matching method (Section REF ), and RX J1713.7$-$ 3946, HESS J1837$-$ 069, and HESS J1632$-$ 478 were also matched by the identification-matching (Section REF ).", "Since they were extended LAT sources, they are omitted in Section REF and discussed in this section.", "The breakdown of the 31 matched extended sources is as follows.", "In the Fermi-LAT catalog we had 2 galaxies (smc and lmc) and 3 unidentified subregions of lmc (Far West, 30 Dor West, and North of lmc).", "Although they were positionally coincident with some HMXBs and a pulsar, the extended gamma rays are not associated with these point sources, thus setting them to false matches.", "Additionally, the lobes in Centaurus A detected by LAT were also matched as Centaurus A (radio galaxy) in BAT.", "10 pwn in 4FGL-DR2 were matched with the associated pulsars in the Swift-BAT catalog, which are the central compact object of those pwn.", "There were 7 extended SNRs matched in our study.", "Only two of them (RX J1713.7$-$ 3946 and RX J0852.0$-$ 4622) were known associations, while the other 5 included 3 false-matches (SNR G150.3$+$ 04.5, Monoceros, and gamma Cygni), one unknown association (Sim 147), and one ambiguous source (SNR G337.0$-$ 00.1 which hosted SGR 1627$-$ 41 (a magnetar) and IGR J16358-4726 (a pulsar) within its extent).", "Cygnus X was the only one star forming region among the matched extended sources, and within the gamma-ray extent it contained Cyg X-3 (HMXB) and 2 AGNs.", "This, however, was falsely matched because the extended gamma-ray emission from the star forming region did not originate from those point sources.", "Among the five matched spp`spp' is defined as a possible SNR or PWN in 4FGL-DR2., 3 (HESS J1632$-$ 478, HESS J1813$-$ 178, and Kes 73) were plausible associations between SNR or PWN in gamma-ray and SNR or pulsar in X-ray.", "W 41, having a star SWIFT J1834.9$-$ 0846 measured by BAT, could be a possible false-match source.", "We left HESS J1809$-$ 193 as an ambiguous source because of the association with PSR J1811$-$ 1925, according to the spatial coincidence reported in [20].", "Furthermore, there were 3 unidentified extended Fermi-LAT sources (FGES J1036.3$-$ 5833, FGES J1409.1$-$ 6121, and HESS J1808$-$ 204), which had BAT counterparts within their extended sources radii.", "As mentioned above, Sim 147 that was matched with an unknown BAT source, SWIFT J053457.91$+$ 282837, could be also an unidentified source.", "These 4 unidentified sources will be discussed in ." ], [ "Summary of the matched sources", "The source type summary of the matched sources is presented in the form of the Swift-BAT and Fermi-LAT definitions, respectively, in Table  and Table .", "Figure REF indicates the source type fraction of the matched sources compared to the original catalogs.", "Note that only firmly matched sources (i.e., Flag is M or D in Table  and Table ) are shown in Figure REF .", "In the Swift-BAT 105-month catalog, the biggest population was Seyfert galaxy, which however was not a common source category in 4FGL-DR2, resulting in a few cases of the matched Seyfert galaxies in this study.", "8 BAT Seyfert galaxies were matched, while the number reduced to 2 in the source definition of Fermi-LAT.", "Most of Seyfert galaxies defined in the Swift-BAT catalog were matched with other types of AGNs, such as bcu, radio galaxy, or starburst galaxy, as labeled as Flag=D (see Section REF ).", "The second largest proportion in the Swift-BAT catalog was X-ray binaries (HMXB, LMXB, and XRB`XRB' in the Swift-BAT catalog indicates other type of X-ray binary (i.e., wind-colliding binary system, such as Eta Carina).).", "In this work, the fraction of the matched HMXBs was roughly comparable with that of the original catalog, although LMXBs which occupied the same fraction in the original catalog were hardly matched.", "However, the numbers of the matched HMXB and LMXB were small (i.e., five HMXBs and one LMXB), and thus it did not allow us further discussion about the fraction.", "We note that the matched HMXBs were well known binary systems, such as LS 5039 and Cyg X-1, and two LMXBs classified as the unidentified sources (SAX J1808.4$-$ 3658 and XTE J1652$-$ 453) could be possible candidates of the matched sources (see Section REF for details).", "The beamed AGNs, which were the third largest population in the original catalog, dominate in this matched catalog.", "It is worth noting that the second biggest population in our catalog was pulsars, which was a minor class in the Swift-BAT catalog.", "Some of the Swift-BAT pulsars were matched with their nebulae in 4FGL-DR2.", "In both the Fermi-LAT and our matched catalogs, the most predominant source class was blazars.", "Particularly in our catalog, the fraction of BLLs was compatible with that of the original catalog, while more FSRQs were matched.", "This is ascribed to that FSRQs could be easily detected by Swift-BAT because of the typically hard spectrum in the X-ray energy range blue[37].", "The number of the matched bcu appeared small compared to the original catalog.", "In 4FGL-DR2, the number of the unidentified sources was remarkably numerous, but they were not included in our catalog.", "We found 9 cross-matched unidentified sources in total, most of which needed more investigation to confirm the association with the hard X-ray (see Section REF and Section REF ).", "The third largest population in 4FGL-DR2 was pulsars, and we also had similar fraction of pulsars in our catalog.", "It should be noted that PWNe and radio galaxies constituted a larger fraction in our catalog, while these two source categories were minor components in the original catalog.", "All of the matched PWNe, however, were matched with the pulsars in the X-ray but not matched with the nebulae.", "107 beamed AGNs in the Swift-BAT definition and 98 blazars (FSRQ, BLL, and bcu) in 4FGL-DR2 are firmly identified in our matched catalog.", "These numbers were roughly consistent with that in [32], which reported that 101 BAT blazars were gamma-ray emitting and significantly detected with Fermi-LAT.", "Since [32] selected the BAT blazars not based on the original definition of beamed AGN, the number of the blazars were not exactly same with our study.", "Indeed, 12 blazars in [32] did not appear in the our catalog.", "Figure: Top: Source type fraction of the matched catalog and the Swift-BAT catalog.Bottom: Same as top for the Fermi-LAT catalog.", "Note that the source category includes associations with small letters (i.e., BLL includes BLL and bll).Only source types with the number of the matched sources of ≥6\\ge 6 and ≥9\\ge 9 are shown for the Swift-BAT and Fermi-LAT catalogs, respectively." ], [ "Discussion", "We compared our catalog to existing catalogs in the energy range from hard X-ray to sub-GeV gamma-ray, such as the COMPTEL catalog, the INTEGRAL catalog, the first Fermi-LAT low energy catalog (1FLE), and the previous work by [29], in Section REF .", "In Section REF , we investigate the property of physical parameters (i.e., photon index, flux, and time variability) of our cross-matched sources.", "The unidentified point-like and extended sources are discussed in Section REF and Section REF , respectively.", "Finally, we address the meaning of this work toward the future projects of satellites or balloon experiments in Section REF ." ], [ "Comparison with COMPTEL catalog", "The COMPTEL catalog [35] was produced based on the first five-year data in the 0.75–30 MeV energy range.", "It includes 25 steady sources, 7 line gamma-ray sources, and 31 grb.", "In this paper, we consider the 25 sources that were significantly detected at $>3\\sigma $ , excluded two of them (High-velocity cloud (HVC) complexes M and A area and HVC complex C) due to the large extent of 20–30$^\\circ $ , and added 4 pulsars in Table 3 of [35].", "The 27 COMPTEL sources in total are shown in Table .", "When matching with the COMPTEL catalog, the identification match was the most reasonable, and the spatial match (described in Section REF ) cannot be applicable because the coordinates of most of COMPTEL sources were taken from their counterparts.", "However, the position of sources discovered by the CGRO mission (source name starting with `GRO') was determined by the COMPTEL observations.", "We, thus, can apply the spatial match method to these sources.", "First, we conducted a name-match method to the all COMPTEL sources and searched for counterparts in the Swift-BAT and Fermi-LAT catalogs.", "For the identification-unmatched sources, we also picked up the nearest sources from the Swift-BAT and Fermi-LAT catalogs, and then set a separation threshold of 1$^\\circ $ for positional matching.", "It should be noted that COMPTEL has the source location accuracy of $\\sim $ 1$^\\circ $ and the angular resolution of 3–5$^\\circ $ .", "The results of cross-matching are described in Table .", "Among the 27 COMPTEL sources, 16 sources were included in our Swift-BAT and Fermi-LAT cross-matching and the corresponding source No.", "of Table  and Table  is given in Table .", "The following 5 sources were matched with 4FGL-DR2 but not with the BAT catalog: PSR J0633$+$ 1746 (a.k.a.", "Geminga; No.", "3 in Table ), PSR B0656$+$ 14 (No.4), PSR B1055$-$ 52 (No.", "6), Vela/Carina (an unidentified extended emission; No.", "14), and PKS 0208$-$ 512 (No.", "22).", "The former 3 pulsars appeared faint in hard X-ray energy band.", "For Nova Per 1992 (No.", "12), an X-ray transient, there was no Swift-BAT and Fermi-LAT counterparts.", "The remaining 5 sources were ambiguous: GRO J2227$+$ 61 (No.", "10), GRO J0516$-$ 609 (No.", "20), GRO J1753$+$ 57 (No.", "25), GRO J1040$+$ 48 (No.", "26), and GRO J1214$+$ 06 (No.", "27).", "Since there were no Swift-BAT and Fermi-LAT counterparts within 1$^\\circ $ , the position determination accuracy of COMPTEL, around GRO J1753$+$ 57 (No.", "25) and GRO J1040$+$ 48 (No.", "26), these two sources would be unmatched.", "Indeed, [35] suggested that the emission from GRO J1753$+$ 57 could be modelled as a combination of emission from both GRO J1837$+$ 59 (a bright unidentified EGRET source) and the steep spectrum EGRET blazar QSO 1739$+$ 522.", "GRO J2227$+$ 61 (No.", "10) had SWIFT J2221.6$+$ 5952 and PSR J2229$+$ 6114 located 1.7$^\\circ $ and 0.16$^\\circ $ away from the COMPTEL emission.", "GRO J0516$-$ 609 (No.", "20) that was an unknown flaring source [12] had a Fermi-LAT source, PMN J0507$-$ 6104, within 1.03$^\\circ $ .", "GRO J1214$+$ 06 (No.", "27) had two possible counterparts, 2MASX J12150077$+$ 0500512 and SDSS J12168$+$ 0541 located 0.495$^\\circ $ and 0.567$^\\circ $ away from the COMPTEL emission, respectively." ], [ "Comparison with ", "The INTEGRAL observatory, launched on October 17 of 2002, consists of two main scientific instruments, the gamma-ray spectrometer SPI and the gamma-ray imager IBIS, and two sub instruments, the two X-ray monitors JEM-X and the optical monitoring camera OMC [40].", "The accumulated data taken by one of the main instruments, the coded mask telescope IBIS (particularly ISGRI, the low energy array on IBIS with a pixelated CdTe detector; [39]), allows us a survey in the energy range from 15 keV to 1 MeV.", "Using the 1000-orbit data taken from 2002 to 2010 ($\\sim $ 110 Ms), [11] provided the 4th INTEGRAL-IBIS catalog, which contained 939 sources detected at $>4.5 \\sigma $ in the 17–100 keV energy range.", "The latest IBIS catalog (version 43https://www.isdc.unige.ch/integral/science/catalogue released on September 13 of 2019) contains 1227 entries with `ISGRI_FLAG' of $>1$ , and it was used in the following.", "First, we matched the latest IBIS catalog with the 105-month Swift-BAT catalog.", "Using the same method as in Section REF resulted in $r_{\\rm sep}$ =0.26$^\\circ $ , which is relatively large compared to the position uncertainty of BAT and IBIS.", "The large value of $r_{\\rm sep}$ could be attributed to the fact that the distance profile of the BAT-IBIS catalog cross-match has characteristic features of a sharpened peak (i.e., the angular separation between each BAT source and the closest IBIS source is more concentrated to $r\\sim 0$$^\\circ $ ) and a low level of the background (linear increase), making the background ratio increase smoothly and $r_{\\rm sep}$ larger.", "Indeed, the peak in the distance profile has a e-folding width of 0.024$^\\circ $ in the BAT-IBIS catalog cross-match, while it is 0.082$^\\circ $ in the BAT-LAT catalog cross-match (Figure REF ).", "With the separation threshold of $r_{\\rm sep}$ =0.22$^\\circ $ , roughly 700 sources were matched.", "This indicates that we had about 900 sources detected with BAT but not with IBIS (i.e., the Swift-BAT catalog has roughly 1600 sources, of which 700 are also detected by INTEGRAL), and most of these sources were extragalactic, where the Swift-BAT had better sensitivity.", "On the other hand, there were about 500 sources detected with IBIS but not with BAT, and they were distributed more on the Galactic plane, of which INTEGRAL had deeper exposure.", "Therefore, the IBIS catalog can compensate for the sky region that has not been deeply covered by Swift-BAT.", "We cross-matched the IBIS catalog and 4FGL-DR2 in the same way as described in Section .", "The spatial match with $r_{\\rm sep}$ =0.06$^\\circ $ resulted in 77 matched point-like sources, including 11 new sources that were not matched in the Swift-BAT and Fermi-LAT catalog match.", "Among the 11 sources, 4 were false matches, and 1 was unidentified (NVSS J175948$-$ 230944 in 4FGL-DR2 and IGR J17596$-$ 2315 in the IBIS catalog).", "The remaining 6 sources were 3 FSRQs (PKS 1451$-$ 375, PKS 1730$-$ 13, PKS 1933$-$ 400), a bll (MS 1458.8$+$ 2249), an agn (PKS 1821$-$ 327), and a radio galaxy (M 87).", "The identification match added two more sources (a radio galaxy (Can B) and an fsrq (PKS 1741$-$ 03)).", "39 extended LAT sources were also matched, however, including 27 sources overlapped with the Swift-BAT catalog in Table , 2 false-matched sources, and 6 unidentified sources.", "This led to 4 firmly matched extended sources: an SNR (IC 433), a PWN (HESS J1825$-$ 137), and 2 spp sources (Ken 73 and HESS J1632$-$ 478).", "In summary, in addition to the matched sources between the Swift-BAT and Fermi-LAT catalogs (Table  and Table ), we found 8 point-like sources and 4 extended sources which were newly and firmly matched between the IBIS catalog and 4FGL-DR2.", "Finally, we report on a comparison with INTEGRAL-SPI sources.", "The INTEGRAL catalog contains 277 SPI sources in the 20 keV–8 MeV band (with `SPI_FLAG' being 1) in the latest version.", "29 SPI sources are matched with 4FGL-DR2 by adopting $r_{\\rm sep}$ =0.06$^\\circ $ , which is determined in the same way presented in Section REF .", "Among them, 26 are included in the BAT-LAT matching, one is a IBIS-LAT matched source, and the remaining two sources are false matches or ambiguous associations." ], [ "Comparison with 1FLE", "[33] provided the first Fermi-LAT low energy catalog (1FLE).", "This catalog was based on the 8.7-yr Fermi-LAT data taken from August 4, 2008 to May 3, 2017 in the energy range of 30–100 MeV.", "It should be noted that the psf of even PSF3 eventsGamma rays in Pass 8 data are separated into 4 PSF event types, 0, 1, 2, and 3, where PSF0 has the largest PSF and PSF3 has the best.", "is larger than 3$^\\circ $ at $\\le $ 100 MeV, which is comparable with that of COMPTEL, 3–5$^\\circ $ .", "In the 1FLE catalog, 198 sources were detected at above 3$\\sigma $ .", "Among these 198 sources, 11 sources were not associated with the previous 4-yr Fermi-LAT catalog (3FGL; [4]), 4FGL, and 4FGL-DR2.", "A spatial cross-match between the Swift-BAT 105-month catalog and 1FLE with $r_{\\rm sep}$ =0.25$^\\circ $ , which is comparable with the positional error of the 1FLE catalog, resulted in 19 matched point-like sources, of which 5 sources (AX J1639.0$-$ 4642, Mrk 766, Mrk 841, AX J1639.0$-$ 4642, and SWIFT J1521.6$+$ 3204) were not included in Table .", "A cross-matching by the source names resulted in 35 sources being matched.", "For the name-matched sources, the separation of the source coordinate between the Swift-BAT catalog and 1FLE was at most 1.3$^\\circ $ , which is smaller than the psf of 1FLE of $\\ge 3$$^\\circ $ .", "Note that 14 sources are overlapped between the positionally matched sources and the name-matched sources, and thus the total number of point-like sources matched between the Swift-BAT catalog and 1FLE is 40.", "In our cross-matched catalog (Table ), we show these sources which have counterparts in 1FLE by labelling as `1FLE'.", "Additionally, two extended sources, RX J1713.7$-$ 3946 and HESS J1632$-$ 478, have counterparts in 1FLE.", "The BAT-1FLE matched sources have photon indices $\\lesssim 2$ in the energy band of Swift-BAT and $\\gtrsim 3$ in the energy band of Fermi-LAT except for Mrk 421 with $\\Gamma _{\\rm BAT} >2$ and $\\Gamma _{\\rm Fermi} >2$ , NGC 1275 with $\\Gamma _{\\rm BAT} >2$ , and RX J0115.7$+$ 2519 with $\\Gamma _{\\rm Fermi} <3$ .", "It should be noted that all the 1FLE sources matched here had associations with sources of 3FGL, and the unidentified 11 1FLE sources were not matched with the BAT sources." ], [ "Comparison with Maselli et al. 2011", "In a previous study, [29] performed a catalog cross-match by using the 54-month Swift-BAT catalog (2PBC; 1256 sources; a flux sensitivity of (0.92–1.0)$\\times 10^{-11}$  ${\\rm erg}~{\\rm cm}^{-2}~{\\rm s}^{-1}$ ; [13]) and the 1-yr Fermi-LAT catalog (1FGL; 1451 sources; a flux sensitivity of $10^{-11}$ –$10^{-10}$  ${\\rm erg}~{\\rm cm}^{-2}~{\\rm s}^{-1}$ ; [1]).", "They reported 62 sources as firmly cross-matched sources which had the same identifications between the two catalogs.", "Furthermore, 46 sources were positionally matched if the $Q$ parameter (defined as $ (r_{\\rm BAT} + r_{\\rm LAT}) / r_{\\rm BL} $ where $r_{\\rm BAT}$ , $r_{\\rm LAT}$ , and $r_{\\rm BL}$ are respectively the position uncertainty of a BAT source, that of a LAT source, and the higher value between $r_{\\rm BAT}$ and $r_{\\rm LAT}$ ) was set to be $<1.0$ .", "87 sources in total were matched by the aforementioned positional and identification matching, since 21 sources were overlapped in the two methods.", "By decreasing the X-ray detection threshold to 3$\\sigma $ from 4.8$\\sigma $ , the number of the hard X-ray emitting BAT sources in the direction of 1FGL sources increased to 104, which include all the 87 cross-correlated sources.", "Among the firmly associated 62 sources in [29], 8 were not included in our analysis (Table ).", "However, this discrepancy is attributed to the fact that these 8 sources were excluded either in the latest Swift-BAT or Fermi-LAT catalogs.", "The following 6 sources are included in 4FGL-DR2, but omitted in the latest BAT catalog probably due to flux time variation: OI $+$ 280 in the Swift-BAT 54-month catalog (PKS 0748$+$ 126 in 1FGL), RX J0948.8$+$ 0022 (CGRaBS J0948$+$ 0022), RBS 1420 (1ES 1440$+$ 122), Ap Lib, PG 1553$+$ 113, and PG 0727$-$ 11 (PKS 0727$-$ 11).", "ESO 323$-$ 77 is in the BAT 105-month catalog, but not included in 4FGL-DR2 ([29] also mentioned that this source is a confused LAT source).", "The remaining one source, 1RXS J033913.4$-$ 173553 (PKS 0336$-$ 177) had $Q>1$ (i.e., spatially unmatched) in [29], and thus was not matched in our study.", "In conclusion, all the firmly matched sources in [29] resulted in being matched in this paper, unless the sources were not excluded in the later Swift-BAT or Fermi-LAT catalogs.", "The number of the firmly matched sources roughly doubled in this study owing to the developed flux sensitivity of the observations, particularly that of Fermi-LAT  which was almost one order of magnitude better." ], [ "Property of matched sources", "In the following, we compare the photon index, flux, and time variability of the matched and unmatched sources in order to investigate the properties of the matched sources.", "Figure: Correlation between Γ\\Gamma and flux and their distributions of the Swift-BAT (left) and Fermi-LAT (right) sources.The grey and blue points are distributions of all the sources in the catalog and the spatially matched sources, respectively.The distributions of the marched BLLs and FSRQs are shown in red and green, respectively.The histograms are shown in logarithmic scale.The figure includes the 136 sources which are firmly matched by the coordinates and the identifications, with the Flag being M or D.Figure REF shows a correlation between a photon index ($\\Gamma $ ) and flux and their distributions for the matched sources in this catalog and all sources in the original catalog.", "Here we used the firmly matched point-like sources (136 in total) with Flag being M or D in Table .", "Even when including the firmly matched extended sources, the following results did not largely change.", "For the BAT sources (the left panel of Figure REF ), the distribution of $\\Gamma $ for the matched sources was slightly shifted to the harder side compared to that of all sources, while the distribution of the flux was shifted to the brighter side.", "By using Kolmorogov-Smirnov (KS) statistic, we evaluated the difference of the distributions of $\\Gamma $ and the flux between the matched sources and all sources in the original catalog.", "The $\\Gamma $ distribution showed the value of KS statistic of 0.196 and the p-value of 0.000160, which corresponded to 3.8$\\sigma $ , while the flux distribution showed the value of KS statistic of 0.147 and the p-value of 0.00973, which corresponded to 2.6$\\sigma $ .", "Hence, the distributions of $\\Gamma $ and the flux had different properties at the level of $\\sim 3 \\sigma $ .", "For LAT sources (the right panel of Figure REF ), the $\\Gamma $ distribution shows an apparent bimodal feature, and the distribution of the flux was clearly shifted to the brighter side, compared to all sources in the original catalog.", "Similar to the aforementioned results of the BAT sources, we also found that the distributions of $\\Gamma $ and the flux of the matched sources were different from those of the original catalog.", "The KS statistics and the corresponding p-value were respectively 0.192 and 0.000140 ($3.8\\sigma $ ) in the $\\Gamma $ distribution, while they were respectively 0.427 and $1.49 \\times 10^{-21}$ (over $5\\sigma $ ) in the flux distribution.", "The difference in the $\\Gamma $ and flux properties can be explained as follows.", "Among the matched point sources, the two largest populations were FSRQs (50 sources) and BLLs (33 sources).", "These two classes of blazars might be part of the blazar sequence, with the synchrotron and high-energy peak at different energy bands: in the energy range of Swift-BAT and Fermi-LAT, FSRQs have concave-structure (i.e., hard in X-ray and soft in gamma-ray), while BLLs have convex-structure (i.e., soft in X-ray and hard in gamma-ray).", "Indeed, the double-peak feature in the $\\Gamma $ distribution was ascribed to the $\\Gamma $ distributions of the FSRQs and BLLs (Figure REF ).", "It also should be noted that the fraction of the FSRQs in the matched sources was notably larger than that of the original catalog (Figure REF ), making the $\\Gamma $ distributions modified.", "The difference in the flux distributions can arise from the difference in flux sensitivity between Swift-BAT and Fermi-LAT.", "Particularly the flux distribution of the Fermi sources showed the remarkable distinction between the matched and all sources.", "The better sensitivity of Fermi-LAT resulted in the difference in the flux distributions, recalling that the flux sensitivity of Swift-BAT and Fermi-LAT are respectively $8\\times 10^{-12}$  ${\\rm erg}~{\\rm cm}^{-2}~{\\rm s}^{-1}$ and $\\sim 1 \\times 10^{-12}$  ${\\rm erg}~{\\rm cm}^{-2}~{\\rm s}^{-1}$ .", "Figure: Same as Figure  for correlation between the 1-yr variability index and flux of the Fermi-LAT sources.The 1-yr variability index of >>21.67, shown by the vertical dashed line, means <<1% chance of a steady source.We also investigated property of time variation of the matched sources.", "4FGL-DR2 provides us with `Variability Index', which is defined as a sum of 2$\\times $ Log(Likelihood) difference between flux of each time and the averaged one.", "For the 10-yr lightcurve with 1-yr bin, the variability index of $>21.67$ indicates a $<1$ % chance for a steady source.", "It should be noted that lightcurves with 1-yr bin and 2-month bin were available in 4FGL (the previous 8-yr Fermi-LAT catalog), while only lightcurves with 1-yr bin were provided in 4FGL-DR2.", "We made sure that the variability indices of the 1-yr and 2-month lightcurves were correlated, and the following results produced by 4FGL-DR2 were consistent with when using the corresponding variability index of the 2-month lightcurves in 4FGL.", "Figure REF shows a correlation between the variability index and the flux and their distributions of the matched sources in our catalog and the all sources in the Fermi-LAT catalog.", "There seem to be two groups in the scatter plot in Figure REF : the correlated variability index and flux (i.e., the time variation can be easily detected for the brighter source) and the smaller variability index with the widely ranged flux (i.e., possible steady source).", "The distribution of the variability index of the matched sources was also different from that of the original catalog, inferred from the KS statistics and p-value of 0.414 and $3.29\\times 10^{-20}$ ($>5\\sigma $ ), respectively.", "Our matched sources turned out to be more variable than the sources in the original catalog.", "This discrepancy arised from the fact that the matched sources mainly consisted of FSRQs and BLLs (Figure REF ), which tended to have large variability indices.", "In the original catalog, 80% of FSRQs are variable with the index of $>21.67$ , and 43% of BLLs are so.", "The difference in the distribution of the variability index could also be attributed to the fact that the brighter sources, correlated to the larger variability index, were more matched in this study.", "We present the correlation of the photon indices between the firmly matched Swift-BAT and Fermi-LAT sources in Figure REF .", "As seen in Figure REF , the $\\Gamma _{\\rm BAT}$ -$\\Gamma _{\\rm LAT}$ diagram also confirmed two distinct populations, BLLs and FSRQs.", "The right panel of Figure REF shows the correlation of the flux of the firmly matched Swift-BAT and Fermi-LAT sources.", "In the hard X-ray band, the flux of the matched BLLs tends to be smaller than that of the matched FSRQs.", "Figure: Scatter plots of the Γ\\Gamma (left) and flux (right) of the firmly matched Swift-BAT and Fermi-LAT sources.The red and green respectively show those of BLLs and FSRQs, and the blue indicates those of the rest sources.To summarize, Figure REF and Figure REF suggest that our matched sources can be characterized by the double peak in the $\\Gamma $ distribution, the higher flux, and the larger variability index, compared to the all sources in the original catalogs.", "This difference would be reflected by the features of the two main populations, FSRQs and BLLs." ], [ "Unidentified point-like sources", "Here we report on the unidentified point-like sources found in our analysis and discuss possible associations.", "The unidentified source is defined as the positionally matched source with its source type being unclear either in the Swift-BAT or Fermi-LAT catalogs.", "Figure REF shows sed of the 9 unidentified sources.", "Each source is briefly described in the following.", "No.", "58 in Table : SWIFT J1254.9$+$ 1165 (U3`U3' indicates unknown sources without soft X-ray counterparts in the Swift-BAT 105-month catalog.)", "in the BAT catalog was matched with ON 187 (fsrq) in 4FGL-DR2.", "They are possibly associated, inferred from the FSRQ-like SED and the small separation of 0.006$^\\circ $ .", "No.", "65: SWIFT J0949.1$+$ 4057 (confused source) in the BAT catalog was matched with 4C $+$ 40.24 (fsrq) in 4FGL-DR2.", "This association needs more investigation to be confirmed, particularly in the hard X-ray energy range that was uncertain due to the large errors.", "Deeper observations would give us a clue for such a faint source.", "No.", "75: PMN J0145-2733 (Unknown AGN) in the BAT catalog was matched with PKS 0142-278 (fsrq) in 4FGL-DR2.", "This could be likely an association, inferred from the FSRQ-like sed.", "However, more X-ray observations would be necessary to precisely measure the upturn-like feature seen at $\\sim 70$  keV in order to determine its origin and the association with the GeV gamma-ray radiation.", "No.", "130: GX 340$+$ 0 (LMXB) in the BAT catalog was matched with 4U 1642-45 (unk) in 4FGL-DR2.", "The association between these two sources is promising, since they have the same identification.", "The GeV emission with Fermi-LAT, however, is unknown due to being located in a complex TeV gamma-ray emitting region, HESS J1648$-$ 458 [3].", "Beside the accreting neutron star 4U 1642$-$ 45, HESS J1648$-$ 458 contained PSR J1648$-$ 4611 and a star cluster Westerlund 1.", "4U 1642$-$ 45 was unlikely responsible for the TeV gamma rays, inferred from the spatial extent and time variation.", "They argued that a single source scenario would favor the hadronic gamma-ray radiation produced by collisions of cosmic rays from Westerlund 1 with the ism.", "No.", "131: SAX J1808.4$-$ 3658 (LMXB) in the BAT catalog was matched with SWIFT J1808.5$-$ 3655 (unknown) in 4FGL-DR2.", "Note that the counterpart of the Fermi source is not a firm association (i.e., SWIFT J1808.5$-$ 3655 was labeled with `ASSOC2').", "This association —the gamma-ray emission from the LMXB— was previously reported and discussed in [16].", "No.", "132: XTE J1652$-$ 453 (LMXB) in the BAT catalog was matched with SWIFT J1652.3$-$ 4520 (unknown) in 4FGL-DR2.", "Note that the counterpart of the Fermi source is not a firm association (i.e., SWIFT J1652.3$-$ 4520 was labeled with `ASSOC2').", "They might be associated as the former case of SAX J1808.4$-$ 3658, although further investigation is needed to confirm the association.", "No.", "154–156: CGCG 147$-$ 020 (Sy2; No.", "154), 2MASX J14080674$-$ 3023537 (Sy1.9; No.", "155), and XTE J1817$-$ 330 (LMXB; No.", "156) are the matched Swift-BAT sources, and they are unknown sources in 4FGL-DR2.", "These were faint, and thus the position uncertainty was large both in the BAT and LAT observations.", "The dedicated deeper observations are necessary for them to unveil the association and the nature.", "We conducted a time variation analysis of the unidentified point-like sources using 1-month lightcurves of the Swift-BAT catalog and 2-month lightcurves of 4FGL (the 8-yr Fermi-LAT catalog).", "No significant correlation between the hard X-ray and GeV gamma-ray radiation in the 2-month scale was found in any unidentified source, probably because of the poor statistics.", "In the case of the binary system, timing analyses folded by the orbital period are necessary to track the variability correlation.", "This is beyond the scope of this paper and will be performed in the future publication.", "Figure: sed of the unidentified point-like sources in the Swift-BAT (14–195 keV) and Fermi-LAT (50 MeV–300 GeV) energy bands, shown in red and blue, respectively.The red solid and blue dashed lines indicate the model spectrum taken from the Swift-BAT and Fermi-LAT catalogs, respectively." ], [ "Unidentified extended sources", "We briefly describe the current status of the unidentified and extended sources in our study.", "Their sed are illustrated in Figure REF .", "No.", "17 in Table : Sim 147 (SNR) was spatially matched with SWIFT J053457.91$+$ 282837.9 (U2U2 indicates a source of which its soft X-ray emission is detected from archival X-ray observation with S/N greater than 3.)", "in the Swift-BAT catalog.", "Sim 147 is a middle-aged SNR, including a known PSR-PWN association inside its GeV gamma-ray extent of 1.5$^\\circ $ [28].", "The matched source, SWIFT J053457.91$+$ 282837.9, was revealed to be a possible intermediate polar (i.e., a cataclysmic variable binary star system) by a periodic analysis of optical observations [21].", "Therefore we suggest that these two sources are not associated and are false-matched.", "This would also be supported by the fact that the BAT source is located near the edge of the gamma-ray emission, and there exists the aforementioned PSR-PWN association close to the center of the SNR.", "No.", "29: FGES J1036.3$-$ 5833 (unidentified) hosts inside the extent Eta Carina (XRB), 4U 1036$-$ 56 (HMXB), and 2MASS J10445192$-$ 6025115 (star).", "This gamma-ray emission is largely extended with $\\sim $ 2.5$^\\circ $ in radius, and is remarkably variable in the 1-yr scale with the variability index of $\\sim 75$ .", "The time variation could result from a variable source inside the gamma-ray extent (i.e., Eta Carina or 4U 1036$-$ 56).", "No.", "30: FGES J1409.1$-$ 6121 (unidentified) has spatial coincidences with SWIFT J1408.2$-$ 6113 (CV), [CG2001] G311.45$-$ 0.13 (U2), and MAXI J1409$-$ 619 (Pulsar).", "The gamma-ray extent is $\\sim $ 0.73$^\\circ $ .", "The gamma-ray emission might be associated with [CG2001] G311.45$-$ 0.13, which could be a possible counterpart of a radio SNR G12.4$-$ 0.4 [17].", "However, the hard spectrum in the Swift-BAT energy regime ($\\Gamma \\sim 2$ ) is not likely of origin of the X-ray radiation from the remnant.", "An alternative is MAXI J1409$-$ 619, a pulsar, which is located in the vicinity of SNR G12.4$-$ 0.4.", "Further investigation would be necessary to confirm the association.", "No.", "31: HESS J1808$-$ 204 (unidentified) was spatially matched with SGR 1806$-$ 20 (a pulsar, more like a magnetar) in the Swift-BAT catalog.", "[41] reported the possible association between the gamma-ray radiation with Fermi-LAT and the magnetar, and later the origin (i.e., the gamma-ray emission powered by magnetic dissipation from SGR 1806$-$ 20) was discussed in [22].", "These studies, however, could not reach to a robust conclusion due to other plausible scenarios to account for the gamma-ray radiation.", "Figure: sed of unidentified extended sources.The blue solid and dashed lines indicate the model spectra provided in the Swift-BAT and Fermi-LAT catalogs, respectively.For FGES J1036.3--5833 and FGES J1409.1--6121, sed of all BAT counterparts and the total flux are also shown." ], [ "Future prospect", "Over 20 years ago, COMPTEL confirmed 25 steady MeV gamma-ray emitting sources based on the observational data with the flux sensitivity of $\\sim 10 ^{-10}$  ${\\rm erg}~{\\rm cm}^{-2}~{\\rm s}^{-1}$ [35].", "In the last decade, the sensitivity of the detectors in the neighboring energy bands (i.e., the hard X-ray and GeV gamma ray) has improved to $< 10 ^{-11}$  ${\\rm erg}~{\\rm cm}^{-2}~{\\rm s}^{-1}$ .", "This work reports 151 sources firmly matched between the latest Swift-BAT and Fermi-LAT catalogs.", "We present these cross-matched sources in the all-sky map in Figure REF .", "The matched catalog (Table  and Table ) contains promising objects that are bright in the MeV energy range and are detectable with future instruments with a sensitivity being over one order of magnitude better than COMPTEL.", "This catalog would be a helpful resource when devising a strategy for the ongoing projects of the MeV observation, such as e-ASTROGAM [14], AMEGO [30], COSI [38], and GRAMS [6].", "The cross-matched sources, combined with a simulation of diffuse emission, can be useful to predict the all sky image in the MeV energy channel.", "This will be presented in a future publication.", "Figure: The cross-matched sources shown on the galactic coordinate.The firmly matched point sources and extended sources are shown in blue and red, respectively, while the false matches are shown in grey.The solid line indicates the declination of 0 ∘ ^\\circ ." ], [ "Conclusions", "We performed a cross-matched between the Swift-BAT 105-month catalog and the 4FGL-DR2 catalog.", "We confirmed (1) 132 sources (115 firmly matched sources) by the spatial cross-match with the separation threshold of $r_{\\rm sep}$ =0.08$^\\circ $ , (2) 31 sources (15 firmly matched sources) by the spatial cross-match for extended sources, and (3) 24 sources (21 firmly matched sources) by the identification match.", "The firmly matched sources (151 in total) predominantly consisted of blazars.", "Particularly, the proportion of FSRQs in the matched catalog was over twice as large as that of the 4FGL-DR2.", "We found that most of COMPTEL sources were included in this study, and the cross-match with INTEGRAL-IBIS catalog could add 8 point-like and 4 extended sources.", "Compared to the original catalogs, the distributions of physical parameters of the matched sources were characterized by the bimodal feature in the $\\Gamma $ distribution, a higher flux, and larger variability index, resulting from the different source fractions.", "We thank the anonymous referee for the fruitful comments and advise.", "This work made use of data from the Swift and Fermi observatories.", "N.T.", "acknowledges RIKEN iTHEMS Program.", "H.Y.", "is supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI grant No.", "20K22355, Y.I.", "is supported by JSPS KAKENHI grant Nos.", "JP16K13813, JP18H05458, and JP19K14772, and H.O.", "is supported by JSPS KAKENHI grant Nos.", "19H05185 and 19H01906.", "Swift-BAT, Fermi-LAT =2cm cccccccccccccccccc Cross-matched point-like sources No.", "Swift-BAT name Type Fermi-LAT name Type Sep.", "Flag 1FLE BAT coord.", "$\\Gamma _{\\rm BAT}$ $\\Gamma _{\\rm LAT}$ $F_{\\rm BAT}$ $F_{\\rm LAT}$ VarIndex (deg) ($\\alpha _{\\rm J200}$ , $\\delta _{\\rm J2000}$ ) 1 [HB89] 0537-441 Beamed AGN PKS 0537-441 BLL 0.063 M 1 (84.63, -44.12) 0.88 2.11 1.5 17 9.8e+03 2 [HB89] 0716+714 Beamed AGN S5 0716+71 BLL 0.01 M 1 (110.5, 71.33) 1.15 2.08 1.9 21 2.7e+03 3 Mrk 421 Beamed AGN Mkn 421 BLL 0.014 M 1 (166.1, 38.21) 2.76 1.78 14 42 1.3e+03 4 Mrk 501 Beamed AGN Mkn 501 BLL 0.001 M 0 (253.5, 39.76) 2.39 1.76 7.2 13 5.4e+02 5 BL Lac Beamed AGN BL Lac BLL 0.018 M 1 (330.7, 42.27) 1.76 2.2 3.5 23 3.5e+03 6 1ES 0033+595 Beamed AGN 1ES 0033+595 bll 0.008 M 0 (8.989, 59.84) 2.81 1.76 2.6 4 2e+02 7 2MASX J01155048+2515369 Sy1 RX J0115.7+2519 bll 0.018 A 0 (18.97, 25.33) 1.97 1.92 1.1 2 3.1e+02 8 SHBL J012308.7+342049 Beamed AGN 1ES 0120+340 bll 0.021 M 0 (20.78, 34.37) 2.94 1.7 1.1 0.3 15 9 B3 0133+388 Beamed AGN B3 0133+388 bll 0.052 M 0 (24.13, 39.05) 1.99 1.72 0.79 5.1 61 10 RBS 259 Beamed AGN 1RXS J015658.6-530208 bll 0.062 M 0 (29.13, -53.04) 2.31 1.74 0.73 0.59 45 11 2MASX J02141794+5144520 Beamed AGN TXS 0210+515 bll 0.03 M 0 (33.55, 51.77) 2.58 1.9 1.4 0.53 13 12 QSO B0229+200 Beamed AGN 1ES 0229+200 bll 0.036 M 0 (38.19, 20.29) 2.28 1.79 2.3 0.37 6.8 13 ESO 416-G002 Sy1.9 PHL 1389 bll 0.068 A 0 (38.83, -29.63) 1.67 2.03 2.1 0.17 14 14 BZB J0244-5819 Beamed AGN RBS 0351 bll 0.031 M 0 (41.19, -58.3) 2.43 1.75 1 0.67 40 15 QSO B0347-121 Beamed AGN 1ES 0347-121 bll 0.018 M 0 (57.37, -11.98) 2.2 1.76 1.6 0.39 13 16 PKS 0352-686 Beamed AGN PKS 0352-686 bll 0.004 M 0 (58.28, -68.53) 2.52 1.67 1.2 0.33 8.3 17 PKS 0426-380 Beamed AGN PKS 0426-380 bll 0.059 M 0 (67.14, -37.89) 2.56 2.1 0.38 21 3.5e+03 18 PKS 0548-322 Beamed AGN PKS 0548-322 bll 0.054 M 0 (87.69, -32.27) 3.23 1.89 1.8 0.39 9 19 PMN J0640-1253 Beamed AGN TXS 0637-128 bll 0.056 M 0 (100.1, -12.87) 2.56 1.65 0.79 0.56 4.9 20 2MASX J07103005+5908202 Beamed AGN 1H 0658+595 bll 0.008 M 0 (107.6, 59.14) 2.28 1.69 2.4 0.67 27 21 2MASS J09303759+4950256 Beamed AGN 1ES 0927+500 bll 0.069 M 0 (142.5, 49.88) 2.59 1.82 0.74 0.23 16 22 2MASS J09343014-1721215 Beamed AGN RXC J0934.4-1721 bll 0.039 M 0 (143.6, -17.34) 2.73 1.94 0.79 0.21 8 23 2MASX J10311847+5053358 Beamed AGN 1ES 1028+511 bll 0.02 M 0 (157.9, 50.9) 2.85 1.74 0.78 1.2 13 24 2MASX J11033765-2329307 Beamed AGN 1ES 1101-232 bll 0.045 M 0 (165.9, -23.47) 2.53 1.77 1.1 0.55 13 25 2MASX J11363009+6737042 Beamed AGN RX J1136.5+6737 bll 0.033 M 0 (174.1, 67.64) 2.33 1.75 1.3 0.63 28 26 FBQS J1221+3010 Beamed AGN PG 1218+304 bll 0.008 M 0 (185.3, 30.16) 2.94 1.71 1.1 4.4 60 27 [HB89] 1415+259 Beamed AGN 1E 1415.6+2557 bll 0.041 M 0 (214.4, 25.72) 2.61 1.45 0.59 0.33 8.1 28 1ES 1426+428 Beamed AGN H 1426+428 bll 0.027 M 0 (217.1, 42.66) 2.56 1.63 2.1 0.88 16 29 [HB89] 1803+784 Beamed AGN S5 1803+784 bll 0.057 M 1 (269.9, 78.47) 1.93 2.21 0.91 5.4 2.8e+02 30 QSO B1959+650 Beamed AGN 1ES 1959+650 bll 0.019 M 0 (300, 65.16) 2.67 1.82 2.9 11 1.5e+03 31 2MASX J23470479+5142179 Beamed AGN 1ES 2344+514 bll 0.005 M 0 (356.8, 51.69) 2.66 1.81 1 3.2 60 32 H 2356-309 Beamed AGN H 2356-309 bll 0.058 M 0 (359.8, -30.58) 2.28 1.82 1.5 0.53 12 33 3C 454.3 Beamed AGN 3C 454.3 FSRQ 0.011 M 1 (343.5, 16.15) 1.5 2.4 16 1e+02 5.6e+04 34 [HB89] 2230+114 Beamed AGN CTA 102 FSRQ 0.031 M 1 (338.2, 11.71) 1.49 2.29 3 49 7.5e+04 35 PKS 2227-088 Beamed AGN PKS 2227-08 FSRQ 0.072 M 1 (337.5, -8.492) 1.46 2.59 1.7 4.2 6.4e+02 36 [HB89] 2142-758 Beamed AGN PKS 2142-75 FSRQ 0.074 M 1 (327.1, -75.58) 1.41 2.44 1.5 5.6 2.8e+03 37 QSO B2013+370 Beamed AGN MG2 J201534+3710 FSRQ 0.036 M 0 (303.9, 37.21) 2.13 2.45 1.8 7.5 1.3e+02 38 PKS 1830-21 Beamed AGN PKS 1830-211 FSRQ 0.017 M 1 (278.4, -21.07) 1.47 2.53 8.7 19 2.5e+03 39 3C 345 Beamed AGN 3C 345 FSRQ 0.026 M 0 (250.8, 39.81) 1.17 2.4 2.1 3 2e+02 40 PKS 1622-29 Beamed AGN PKS B1622-297 FSRQ 0.034 M 1 (246.6, -29.86) 1.32 2.56 1.6 3.9 4.3e+02 41 PKS 1510-08 Beamed AGN PKS 1510-089 FSRQ 0.025 M 1 (228.2, -9.081) 1.32 2.38 6.7 42 5.9e+03 42 3C 279 Beamed AGN 3C 279 FSRQ 0.015 M 1 (194.1, -5.799) 1.32 2.29 3.9 45 3e+04 43 3C 273 Beamed AGN 3C 273 FSRQ 0.009 M 1 (187.3, 2.047) 1.75 2.7 42 11 7e+03 44 PG 1222+216 Beamed AGN 4C +21.35 FSRQ 0.021 M 1 (186.2, 21.4) 1.7 2.34 2.5 20 2e+04 45 4C +49.22 Beamed AGN 4C +49.22 FSRQ 0.017 M 1 (178.3, 49.5) 1.83 2.41 1.3 1.6 1.2e+03 46 [HB89] 0836+710 Beamed AGN 4C +71.07 FSRQ 0.004 M 1 (130.3, 70.89) 1.7 2.82 7 4.8 3.4e+03 47 PMN J0641-0320 Beamed AGN PMN J0641-0320 FSRQ 0.046 M 0 (100.5, -3.362) 0.96 2.68 2.2 2.5 3.6e+02 48 PKS 0528+134 Beamed AGN PKS 0528+134 FSRQ 0.032 M 0 (82.74, 13.57) 1.25 2.56 1.8 2.2 4.6e+02 49 PKS 0402-362 Beamed AGN PKS 0402-362 FSRQ 0.017 M 1 (60.97, -36.07) 1.91 2.53 1.1 7.4 6.1e+03 50 PKS 2325+093 Beamed AGN PKS 2325+093 fsrq 0.012 M 1 (351.9, 9.663) 1.4 2.69 3 1.8 3.7e+02 51 87GB 215950.2+503417 Beamed AGN NRAO 676 fsrq 0.016 M 1 (330.4, 50.82) 1.78 2.66 1.9 5.1 3.1e+03 52 PKS 2149-306 Beamed AGN PKS 2149-306 fsrq 0.015 M 1 (328, -30.46) 1.61 2.85 8.9 3 9.6e+02 53 [HB89] 1921-293 Beamed AGN PKS B1921-293 fsrq 0.063 M 0 (291.2, -29.18) 2.04 2.39 1.6 2.3 2.8e+02 54 2MASS J16561677-3302127 Beamed AGN 2MASS J16561677-3302127 fsrq 0.024 M 0 (254.1, -33.04) 1.55 2.79 6.2 1.5 1.3e+02 55 [HB89] 1354+195 Beamed AGN 4C +19.44 fsrq 0.079 M 0 (209.3, 19.29) 2.02 2.76 0.87 0.4 39 56 [HB89] 1334-127 Beamed AGN PKS 1335-127 fsrq 0.013 M 0 (204.4, -12.95) 2.19 2.42 1.3 1.8 95 57 PKS 1329-049 Beamed AGN PKS 1329-049 fsrq 0.032 M 1 (203, -5.153) 1.51 2.51 1.5 3.5 2.2e+03 58 SWIFT J1254.9+1165 U3 ON 187 fsrq 0.006 U 0 (193.7, 11.65) 1.72 2.79 1.3 0.52 28 59 4C +04.42 Beamed AGN 4C +04.42 fsrq 0.047 M 0 (185.6, 4.219) 1.45 2.79 3.6 1.9 1.6e+02 60 FBQS J1159+2914 Beamed AGN Ton 599 fsrq 0.045 M 1 (179.9, 29.23) 1.84 2.19 0.75 12 7.7e+03 61 7C 1150+3324 Beamed AGN B2 1150+33A fsrq 0.053 M 0 (178.2, 33.09) 1.83 3.01 1 0.28 19 62 PKS 1143-696 Beamed AGN PKS 1143-696 fsrq 0.077 M 0 (176.5, -69.9) 1.69 2.69 1.5 0.56 35 63 PKS 1127-14 Beamed AGN PKS 1127-14 fsrq 0.065 M 0 (172.5, -14.8) 1.88 2.69 2.9 0.89 4.1e+02 64 [HB89] 1039+811 Beamed AGN S5 1039+81 fsrq 0.022 M 0 (161.2, 80.92) 1.67 2.78 1.2 0.92 47 65 SWIFT J0949.1+4057 confused source 4C +40.24 fsrq 0.063 U 0 (147.3, 40.57) 2.46 2.61 0.73 0.31 32 66 CGRaBS J0805+6144 Beamed AGN TXS 0800+618 fsrq 0.048 M 0 (121.3, 61.75) 1.35 2.8 1.8 0.67 2.2e+02 67 B2 0743+25 Beamed AGN B2 0743+25 fsrq 0.042 M 0 (116.6, 25.81) 1.43 2.86 3.6 0.67 88 68 PKS 0637-752 Beamed AGN PKS 0637-75 fsrq 0.038 M 0 (99.02, -75.28) 2.0 2.7 1.7 1.3 4.5e+02 69 [HB89] 0552+398 Beamed AGN B2 0552+39A fsrq 0.02 M 0 (88.9, 39.81) 1.54 2.76 2.1 1.8 1.5e+02 70 [HB89] 0537-286 Beamed AGN PKS 0537-286 fsrq 0.045 M 0 (84.97, -28.7) 1.33 2.73 2.9 1.6 1.1e+02 71 PKS 0524-460 Beamed AGN PKS 0524-460 fsrq 0.035 M 0 (81.32, -46.01) 1.37 2.38 1.6 0.35 7.3 72 [HB89] 0403-132 Beamed AGN PKS 0403-13 fsrq 0.061 M 0 (61.36, -13.14) 1.78 2.55 1.1 0.87 80 73 4C +50.11 Beamed AGN NRAO 150 fsrq 0.017 M 0 (59.9, 50.97) 1.51 2.66 1.7 3.3 7.7e+02 74 [HB89] 0212+735 Beamed AGN S5 0212+73 fsrq 0.073 M 1 (34.38, 73.81) 1.55 2.94 3.5 1.4 41 75 PMN J0145-2733 Unknown AGN PKS 0142-278 fsrq 0.042 U 1 (26.21, -27.54) 1.43 2.6 1.1 0.98 8.1e+02 76 ESO 354- G 004 Sy1.9 PMN J0151-3605 bcu 0.007 A 0 (27.87, -36.13) 1.98 2.28 1.1 0.22 8 77 4C +33.06 Beamed AGN 4C +33.06 bcu 0.049 M 0 (46.19, 33.8) 1.86 2.57 1.2 0.3 33 78 PKS 0706-15 Beamed AGN PKS 0706-15 bcu 0.024 M 0 (107.3, -15.44) 3.42 1.81 0.74 0.46 19 79 PKS 0723-008 Beamed AGN PKS 0723-008 bcu 0.03 M 0 (111.5, -0.942) 1.75 2.06 1.5 1.1 22 80 2MASX J07332681+5153560 Beamed AGN NVSS J073326+515355 bcu 0.057 M 0 (113.4, 51.93) 2.32 1.81 0.82 0.28 12 81 IGR J13109-5552 Sy1 PMN J1310-5552 bcu 0.03 D 0 (197.7, -55.91) 1.56 2.82 2.5 0.72 78 82 PMN J1508-4953 Beamed AGN PMN J1508-4953 bcu 0.013 M 0 (227.2, -49.87) 1.15 2.84 2.9 1.9 36 83 SGR A* Galactic Center Galactic Centre bcu 0.016 D 0 (266.4, -29.01) 2.69 2.34 11 42 2.6 84 PKS 1936-623 Beamed AGN PKS 1936-623 bcu 0.017 M 1 (295.3, -62.16) 1.32 2.43 1.8 3.6 1.2e+03 85 SWIFT J1943536.21+211822.9 Beamed AGN MG2 J194359+2118 bcu 0.028 M 0 (296, 21.31) 2.11 1.51 2.8 0.74 15 86 B2 2023+33 Beamed AGN B2 2023+33 bcu 0.008 M 0 (306.3, 33.68) 1.49 2.63 1.3 2.9 1.2e+02 87 RX J2056.6+4940 Beamed AGN RGB J2056+496 bcu 0.007 M 0 (314.2, 49.66) 2.62 1.86 1.3 2.1 25 88 RBS 1895 Beamed AGN RBS 1895 bcu 0.014 M 0 (341.7, -52.12) 2.51 1.71 0.7 0.2 19 89 1RXS J225146.9-320614 Beamed AGN 1RXS J225146.9-320614 bcu 0.044 M 0 (342.9, -32.1) 2.01 1.79 1.4 0.15 11 90 PKS 0521-36 Beamed AGN PKS 0521-36 AGN 0.018 M 1 (80.76, -36.46) 1.92 2.46 3.5 5.5 6.8e+02 91 Cen A Beamed AGN Cen A RDG 0.012 M 1 (201.4, -43.02) 1.88 2.64 1.3e+02 6.3 18 92 3C 120 Beamed AGN 3C 120 RDG 0.039 M 0 (68.3, 5.356) 2.01 2.74 9.5 1.5 2.8e+02 93 NGC 1275 Beamed AGN NGC 1275 RDG 0.009 M 1 (49.95, 41.51) 3.82 2.11 8.3 33 4.8e+03 94 PKS 2300-18 Beamed AGN PKS 2300-18 rdg 0.069 M 0 (345.8, -18.7) 2.03 2.18 1.5 0.25 9.9 95 3C 303 Sy1 3C 303 rdg 0.046 D 0 (220.7, 52.06) 2.39 2.05 0.57 0.17 8.5 96 PICTOR A Sy2 Pictor A rdg 0.045 D 0 (79.95, -45.77) 2.05 2.43 3.7 0.41 7.3 97 QSO B0309+411 Beamed AGN B3 0309+411B rdg 0.039 M 0 (48.26, 41.29) 1.58 2.69 1.5 0.41 25 98 NGC 4945 Sy2 NGC 4945 sbg 0.003 D 0 (196.4, -49.47) 1.5 2.27 28 1.1 10 99 M 82 Starburst galaxy M 82 sbg 0.029 M 0 (148.9, 69.64) 3.39 2.22 0.48 1.1 6.1 100 NGC 1068 Sy1.9 NGC 1068 sbg 0.004 D 0 (40.66, -0.004) 1.82 2.33 3.8 0.65 18 101 Circinus Galaxy Sy2 Circinus galaxy sey 0.01 M 0 (213.3, -65.34) 2.09 2.26 27 0.67 14 102 1H 0323+342 Beamed AGN 1H 0323+342 nlsy1 0.046 M 0 (51.21, 34.17) 1.62 2.82 2.7 1.9 2e+02 103 3C 380 Beamed AGN 3C 380 css 0.028 M 1 (277.4, 48.74) 1.52 2.42 1.5 1.8 69 104 Cas A SNR Cas A snr 0.008 M 0 (350.8, 58.82) 3.33 1.97 6.5 6.2 4.5 105 Tycho SNR SNR Tycho snr 0.01 M 0 (6.326, 64.14) 3.03 2.18 1.3 0.98 4.2 106 SNR G068.8+02.6 SNR PSR J1952+3252 PSR 0.016 D 0 (298.3, 32.89) 2.27 2.28 0.92 15 6 107 SNR G21.5-00.9 SNR PSR J1833-1034 PSR 0.01 D 0 (278.4, -10.57) 2.26 2.5 7.4 8.2 3 108 4U 1820-30 LMXB PSR J1823-3021A PSR 0.025 F 0 (275.9, -30.36) 5.2 2.21 95 1.4 2.6 109 SLX 1744-299 LMXB PSR J1747-2958 PSR 0.047 F 0 (266.9, -30) 3.25 2.56 13 16 15 110 1RXS J122758.8-485343 CV PSR J1227-4853 PSR 0.007 D 0 (187, -48.89) 1.85 2.38 3.7 2.3 74 111 PSR J1124-5916 Pulsar PSR J1124-5916 PSR 0.054 M 0 (171.1, -59.31) 2.47 2.46 0.83 6.1 8.6 112 Vela Pulsar Pulsar PSR J0835-4510 PSR 0.006 M 1 (128.8, -45.18) 1.97 2.23 18 9.4e+02 4.6 113 PSR B0540-69 Pulsar PSR J0540-6919 PSR 0.027 M 0 (85.02, -69.35) 1.93 2.47 4.9 2.8 8.1 114 2MASX J04372814-4711298 Sy1 PSR J0437-4715 PSR 0.079 F 0 (69.41, -47.21) 1.96 2.35 0.99 1.7 9.6 115 PSR J1811-1925 Pulsar PSR J1811-1925 psr 0.014 M 0 (272.9, -19.42) 2.07 2.14 3.5 1.1 8.6 116 Crab Pulsar Crab Nebula$^\\dagger $ PWN 0.003 M 0 (83.63, 22.02) 2.17 3.8 2.3e+03 16 6.1e+02 117 IGR J17461-2853 molecular cloud PWN G0.13-0.11 pwn 0.079 F 0 (266.5, -28.89) 1.7 2.46 3.3 7 7.4 118 Cyg X-3 HMXB Cyg X-3 HMB 0.072 M 0 (308.1, 40.96) 3.0 2.66 2.5e+02 3.5 83 119 RX J1826.2-1450 HMXB LS 5039 HMB 0.002 M 0 (276.6, -14.85) 1.62 2.61 3.3 27 9 120 2XMM J130247.6-635008 HMXB PSR B1259-63 HMB 0.069 M 0 (195.6, -63.85) 1.2 2.75 2 1.6 1.9e+02 121 LS I +61 303 HMXB LSI +61 303 HMB 0.014 M 1 (40.16, 61.24) 1.73 2.38 3.2 47 2.5e+02 122 Cyg X-1 HMXB Cyg X-1 hmb 0.035 M 0 (299.6, 35.2) 1.9 2.14 1.7e+03 0.65 12 123 V395 Car LMXB 2S 0921-630 lmb 0.053 M 0 (140.5, -63.31) 5.38 2.23 0.56 0.19 2.7 124 Eta Carina XRB Eta Carinae BIN 0.005 M 0 (161.3, -59.68) 3.76 2.31 0.78 19 47 125 4U 2129+12 LMXB NGC 7078 glc 0.026 F 0 (322.5, 12.17) 2.66 2.62 7.7 0.42 23 126 XB 1832-330 LMXB NGC 6652 glc 0.015 F 0 (278.9, -32.99) 2.26 2.35 18 0.48 13 127 4U 1746-37 LMXB NGC 6441 glc 0.021 F 0 (267.6, -37.05) 5.45 2.4 8.3 1.5 8.2 128 ESO 520-27 GC Terzan 5 glc 0.009 M 0 (267, -24.78) 4.32 2.37 2.6 7.9 5.3 129 4U 1722-30 LMXB Terzan 2 glc 0.049 F 0 (261.9, -30.8) 2.51 2.37 43 0.67 7.4 130 GX 340+0 LMXB 4U 1642-45 unk 0.058 U 0 (251.4, -45.61) 5.59 2.61 86 4.5 5.4 131 SAX J1808.4-3658 LMXB (SWIFT J1808.5-3655) unk 0.043 U 0 (272.1, -36.99) 2.34 2.44 3.7 0.44 12 132 XTE J1652-453 LMXB (SWIFT J1652.3-4520) unk 0.061 U 0 (253.1, -45.34) 2.51 2.58 3.1 1.9 9.2 133 PKS 2005-489 Beamed AGN PKS 2005-489 BLL 0.088 M 0 (302.5, -48.87) 2.42 1.83 0.6 3.2 1.5e+02 134 87GB 050246.4+673341 Beamed AGN 1ES 0502+675 bll 0.094 M 0 (76.92, 67.53) 2.5 1.58 0.9 2.4 56 135 2MASX J03252346-5635443 Beamed AGN 1RXS J032521.8-563543 bll 0.082 M 0 (51.47, -56.53) 2.06 1.95 0.85 0.58 36 136 PKS 0607-549 Beamed AGN PKS 0607-549 bcu 0.137 M 0 (92.21, -55.08) 2.19 2.68 0.62 0.68 77 137 B2 0920+39 Beamed AGN B2 0920+39 bcu 0.107 M 0 (140.8, 38.78) 1.44 2.69 1.2 0.59 91 138 8C 1849+670 Beamed AGN S4 1849+67 FSRQ 0.087 M 0 (282.1, 67.05) 2.72 2.29 0.64 4.4 1.9e+03 139 RBS 0315 Beamed AGN TXS 0222+185 fsrq 0.089 M 0 (36.26, 18.8) 1.73 2.95 3.1 0.5 52 140 4C +32.14 Beamed AGN NRAO 140 fsrq 0.121 M 0 (54.12, 32.29) 1.67 2.8 4.4 1.3 46 141 PKS 2008-159 Beamed AGN PKS 2008-159 fsrq 0.083 M 0 (302.8, -15.75) 2.41 2.82 1.3 0.48 18 142 PKS 2052-47 Beamed AGN PKS 2052-47 fsrq 0.183 M 1 (313.8, -47.16) 2.19 2.45 1.8 6 1.3e+03 143 PKS 2145+06 Beamed AGN PKS 2145+06 fsrq 0.142 M 0 (327, 6.936) 1.9 2.8 1.7 0.54 39 144 [HB89] 0834-201 Beamed AGN PKS 0834-20 fsrq 0.197 M 0 (129.2, -20.25) 1.43 2.91 1.4 0.58 25 145 1RXS J174036.3+521155 Beamed AGN 4C +51.37 fsrq 0.222 M 1 (265.2, 51.97) 1.9 2.47 0.87 2.1 8e+02 146 2MASX J06230765-6436211 Beamed AGN RX J062308.0-643619 fsrq 0.115 M 0 (95.85, -64.61) 1.98 3.04 1.2 0.25 7.9 147 87GB 162418.8+435342 Beamed AGN MG4 J162551+4346 fsrq 0.249 M 0 (246.5, 43.81) 2.04 2.91 1.2 0.23 14 148 PKS 2331-240 Beamed AGN PKS 2331-240 agn 0.262 M 0 (353.5, -23.69) 1.4 2.5 1.6 0.44 20 149 PKS 2153-69 Sy2 PKS 2153-69 rdg 0.125 D 0 (329.4, -69.7) 1.59 2.87 1.5 0.37 5.2 150 3C 111.0 Sy1.2 3C 111 rdg 0.1 D 1 (64.59, 38.02) 2.0 2.74 12 1.5 40 151 3C 309.1 Beamed AGN 3C 309.1 css 0.098 M 0 (224.5, 71.72) 1.8 2.5 0.77 0.43 2.1e+02 152 PSR J1420-6048 Pulsar PSR J1420-6048 PSR 0.272 M 0 (215.3, -60.58) 2.24 2.42 1 14 14 153 PSR J1723-2837 Pulsar PSR J1723-2837 psr 0.361 M 0 (260.8, -28.64) 0.88 2.58 1.9 0.58 12 154 CGCG 147-020 Sy2 (SWIFT J0725.8+3000) unk 0.282 U 0 (111.4, 30.02) 2.19 1.61 0.92 0.094 15 155 2MASX J14080674-3023537 Sy1.9 (SWIFT J1408.1-3024) unk 0.15 U 0 (212.1, -30.38) 2.16 2.59 1.6 0.24 16 156 XTE J1817-330 LMXB (SWIFT J1817.8-3301) unk 0.144 U 0 (274.3, -32.98) 1.65 2.5 1.8 0.22 8.8 Flux is in units of $10^{-11}$  ${\\rm erg}~{\\rm cm}^{-2}~{\\rm s}^{-1}$ .", "In 4FGL-DR2, source types with capital letters (e.g., BLL) indicate firm associations, while those with small letters (e.g., bll) are associations.", "The Fermi-LAT name in parenthesis indicate the identifications of ASSOC2 (less firm associations).", "No.", "133–156 are matched by their names, not spatially matched.", "$^\\dagger $ Crab in 4FGL-DR2 has three entries, PSR J0534+2200, Crab nebula (synchrotron radiation), and Crab nebula (ic scattering).", "Only the synchrotron component of the nebula is shown here.", "ccc cccc | ccc cc | c Cross-matched extended sources No.", "Fermi-LAT name Type LAT coord.", "Ext.", "$\\Gamma _{\\rm LAT}$ $F_{\\rm LAT}$ Swift-BAT name Type Sep. $\\Gamma _{\\rm BAT}$ $F_{\\rm BAT}$ Flag ($\\alpha _{\\rm J200}$ , $\\delta _{\\rm J2000}$ ) (deg) (deg) 1 LMC GAL (80, -68.75) 3.0 2.19 11 2MASX J05052442-6734358 Unknown AGN 1.7 2.02 1 F SWIFT J045106.8-694803 HMXB 2.8 2.5 3.3 IGR J05007-7047 HMXB 2.6 2.4 2.2 LMC X-4 HMXB 2.6 2.8 32.7 RX J0531.2-6609 HMXB 2.7 2.8 1.35 LMC X-1 HMXB 2.0 2.6 3.21 PSR B0540-69 Pulsar 1.9 1.9 4.93 XMMU J054134.7-682550 HMXB 2.0 2.9 2.42 [RSG2010] A HMXB 1.9 2.8 0.566 2 LMC-FarWest ... (75.25, -69.75) 0.9 2.1 2 SWIFT J045106.8-694803 HMXB 0.8 2.48 3.3 F 3 LMC-30DorWest ... (82.5, -69) 0.9 2.12 4.4 RSG2010 A HMXB 1.0 2.82 0.57 F 4 LMC-North ... (82.97, -66.65) 0.6 2.0 2.3 LMC X-4 HMXB 0.3 2.83 33 F RX J0531.2-6609 HMXB 0.4 2.8 1.35 5 SMC GAL (14.5, -72.75) 1.5 2.2 2.5 IGR J01054-7253 HMXB 0.3 3.46 0.34 F RX J0052.1-7319 HMXB 0.7 2.7 1.58 RX J0053.8-7226 HMXB 0.5 2.5 1.06 XTE J0103-728 HMXB 0.4 3.2 1.34 SXP 202 HMXB 0.4 2.9 0.153 6 Cen A Lobes RDG (201, -43.5) 2.5 2.51 5.2 Cen A Beamed AGN 0.5 1.88 1.4e+02 M 7 Vela X PWN (128.3, -45.19) 0.9 2.18 13 Vela Pulsar Pulsar 0.4 1.97 18 M SWIFT J0837.8-4440 U2 1.0 2.5 1.16 8 HESS J1420-607 PWN (215.1, -60.78) 0.1 2.0 3.7 Rabbit Pulsar 0.2 1.53 0.8 M 9 MSH 15-52 PWN (228.6, -59.16) 0.2 1.83 5.3 PSR B1509-58 Pulsar 0.0 1.85 26 M 10 HESS J1616-508 PWN (244.1, -50.91) 0.3 2.05 12 PSR J1617-5055 Pulsar 0.2 2.05 1.5 M 11 HESS J1825-137 PWN (276.1, -13.85) 0.8 1.75 14 IGR J18246-1425 Pulsar 0.6 2.8 1.9 M XMMSL1 J182155.0-134719 HMXB 0.6 2.7 1.52 12 HESS J1841-055 PWN (280.2, -5.55) 0.6 1.98 13 AX J1841.0-0535 HMXB 0.1 1.91 2.5 M 1E 1841-045 Pulsar 0.6 1.3 10.7 13 HESS J1632-478 pwn (247.9, -47.94) 0.3 1.76 3 AX J1631.9-4752 Pulsar 0.1 2.84 31 M 14 HESS J1837-069 pwn (279.1, -6.866) 0.5 2.04 22 PSR J1838-0655 Pulsar 0.3 1.71 6.9 M 15 HESS J1837-069 pwn (279.7, -7.067) 0.5 1.85 7 PSR J1838-0655 Pulsar 0.3 1.71 6.9 M 16 SNR G150.3+04.5 SNR (66.82, 55.55) 1.5 1.68 4.1 XTE J0421+560 HMXB 1.2 2.27 1.2 F 17 Sim 147 SNR (85.1, 27.94) 1.5 2.18 5.7 SWIFT J053457.91+282837.9 U2 1.3 2.35 1.2 U 18 Monoceros SNR (99.86, 6.93) 3.5 2.3 9.1 2MASX J06262702+0727287 Unknown AGN 3.2 1.87 1.6 F 19 RX J0852.0-4622 SNR (133, -46.34) 1.0 1.79 12 PSR J0855-4644 Pulsar 0.8 2.06 1 M 20 SNR G337.0-00.1 SNR (249.1, -47.52) 0.1 2.34 10 SGR 1627-41 Gamma-ray source 0.1 1.81 1.4 A IGR J16358-4726 Pulsar 0.1 2.1 0.39 21 gamma Cygni SNR (305.3, 40.52) 0.6 1.96 10 2MASX J20183871+4041003 Sy2 0.5 2.03 2.6 F 22 RX J1713.7-3946 SNR (258.4, -39.76) 0.6 1.71 7 SWIFT J1712.9-4002 U1 0.3 3.25 1.3 M SNR G347.3-0.5 SNR 0.3 3.1 1.99 23 Cygnus X SFR (307.2, 41.17) 3.0 2.09 1.2e+02 Cyg X-3 HMXB 0.7 3.0 2.5e+02 F 2MASX J20183871+4041003 Sy2 1.9 2.0 2.62 SSTSL2 J203705.58+415005.3 Beamed AGN 1.7 5.2 1.36 24 HESS J1632-478 spp (248.3, -47.77) 0.6 2.17 25 AX J1631.9-4752 Pulsar 0.2 2.84 31 M 4U 1630-47 LMXB 0.4 2.7 30.3 IGR J16328-4726 HMXB 0.4 3.1 2.38 SGR 1627-41 Gamma-ray source 0.5 1.8 1.38 IGR J16358-4726 Pulsar 0.6 2.1 0.39 25 HESS J1809-193 spp (272.6, -19.43) 0.5 2.36 5.1 PSR J1811-1925 Pulsar 0.3 2.07 3.5 A XTE J1810-189 LMXB 0.4 2.2 8.21 26 HESS J1813-178 spp (273.3, -17.62) 0.6 2.34 15 IGR J18135-1751 SNR 0.2 1.92 4.1 M GX 13+1 LMXB 0.6 5.7 37.9 27 W 41 spp (278.6, -8.78) 0.2 2.13 11 Swift J1834.9-0846 star 0.2 2.13 0.96 F 28 Kes 73 spp (280.2, -4.89) 0.3 2.37 7.2 1E 1841-045 Pulsar 0.1 1.33 11 M 29 (FGES J1036.3-5833) ... (159.1, -58.56) 2.5 1.93 29 Eta Carina XRB 1.6 3.76 0.78 U 4U 1036-56 HMXB 1.8 2.7 2.82 2MASS J10445192-6025115 star 2.2 1.9 1.49 30 (FGES J1409.1-6121) ... (212.3, -61.35) 0.7 2.16 25 SWIFT J1408.2-6113 CV 0.2 2.68 1.1 U [CG2001] G311.45-0.13 U2 0.7 2.1 1.7 MAXI J1409-619 Pulsar 0.6 3.1 1.17 31 (HESS J1808-204) ... (272, -20.48) 0.6 2.57 4.9 SGR 1806-20 Pulsar 0.1 1.66 5.3 U $^\\dagger $ In 4FGL, HESS J1837$-$ 069 and HESS J1632$-$ 478 are extended and have two entries.", "Flux is in units of $10^{-11}$  ${\\rm erg}~{\\rm cm}^{-2}~{\\rm s}^{-1}$ .", "The nearest BAT source is listed at the top for the source matched with more than one BAT sources.", "The LAT extent of a major axis is shown here.", "ccc | cc | cc | l p4cm Cross-match with COMPTEL sources No.", "COMPTEL Type Swift-BAT Type Fermi-LAT Type No.", "in Table  Note 1 PSR B1951+32 PSR SNR G068.8+02.6 SNR PSR J1952+3252 PSR 106 2 PSR B0531+21 PSR Crab Pulsar PSR J0534+2200 PSR 114 3 PSR J0633+1746 PSR ... ... PSR J0633+1746 PSR ...", "Matched with only Fermi.", "Faint in the hard X-ray.", "4 PSR B0656+14 PSR ... ... PSR J0659+1414 PSR ...", "Matched with only Fermi.", "Faint in the hard X-ray.", "5 PSR B0833-45 PSR Vela Pulsar Pulsar PSR J0835-4510 PSR 112 6 PSR B1055-52 PSR ... ... PSR J1057-5226 PSR ...", "Matched with only Fermi.", "Faint in the hard X-ray.", "7 PSR B1509-58 PSR PSR B1509-58 Pulsar MSH 15-52 PWN 10 in Table  8 GRO J1823-12 Galactic RX J1826.2-1450 HMXB LS5039 HMB 120 9 Cygnus X-1 Galactic Cyg X-1 HMXB Cyg X-1 hmb 123 10 GRO J2227+61 Galactic ... ... PSR J2229+6114 PSR ...", "It has a BAT source (SWIFT J2221.6$+$ 5952) at 1.7$^\\circ $ .", "11 GT 0236+610 Galactic LS I +61 303 HMXB LSI +61 303 HMB 122 12 Nova Per 1992 Galactic ... ... ... ... ... No match.", "X-ray transient.", "13 Crab Unpulsed Galactic Crab Pulsar PSR J0534+2200 PSR 114 14 Vela/Carina Galactic$^\\dagger $ ... ... 4FGL J0853.9-5501 unk ... 15 3C 273 AGN 3C 273 Beamed AGN 3C 273 FSRQ 43 16 3C 279 AGN 3C 279 Beamed AGN 3C 279 FSRQ 42 17 3C 454.3 AGN 3C 454.3 Beamed AGN 3C 454.3 FSRQ 33 18 CTA 102 AGN [HB89] 2230+114 Beamed AGN CTA 102 FSRQ 34 19 Centaurus A AGN Cen A Beamed AGN Cen A RDG 91 20 GRO J0516-609 AGN ... ... ... ... ...", "Unknown flaring source.", "It has a Fermi source (PMN J0507$-$ 6104) at 1.03$^\\circ $ .", "21 GRO J1224+2155 AGN PG 1222+216 Beamed AGN 4C +21.35 FSRQ 44 22 PKS 0208-512 AGN ... ... PKS 0208-512 FSRQ ...", "Matched with only Fermi.", "23 PKS 0528+134 AGN PKS 0528+134 Beamed AGN PKS 0528+134 FSRQ 48 24 PKS 1622-297 AGN PKS 1622-29 Beamed AGN PKS B1622-297 FSRQ 40 25 GRO J 1753+57 Unknown$^\\dagger $ ... ... ... ... ... No match.", "26 GRO J 1040+48 Unknown ... ... ... ... ... No match.", "27 GRO J 1214+06 Unknown 2MASX J12150077+0500512 Sy1.8 (SDSS J12168+0541) (unk) ... Association?", "$^\\dagger $ Extended.", "c | cc | cc | cc | cc | cccc [ht] Classes of cross-matched sources (Swift-BAT definition) Source 2cOriginal 2cMatched point sources 2cExtended 2cID-matched 4cTotal-matched # % # Firm # Firm # Firm # % Firm # Firm % Total 1632 132 115 31 15 24 21 187 151 Beamed AGN 158 9.7 89 89 1 1 17 17 107 57.2 107 70.9 Starburst galaxy 1 0.1 1 1 0 0 0 0 1 0.5 1 0.7 Seyfert galaxy 827 50.7 10 6 1 0 4 2 15 8.0 8 5.3 LINER 6 0.4 0 0 0 0 0 0 0 0.0 0 0.0 Unknown AGN 114 7.0 1 0 2 0 0 0 3 1.6 0 0.0 Compact group of galaxies 1 0.1 0 0 0 0 0 0 0 0.0 0 0.0 Galaxy Cluster 26 1.6 0 0 0 0 0 0 0 0.0 0 0.0 Galactic Center 1 0.1 1 1 0 0 0 0 1 0.5 1 0.7 HMXB 108 6.6 5 5 6 0 0 0 11 5.9 5 3.3 LMXB 109 6.7 10 1 0 0 1 0 11 5.9 1 0.7 XRB 8 0.5 1 1 1 0 0 0 2 1.1 1 0.7 Pulsar 25 1.5 5 5 14 12 2 2 21 11.2 19 12.6 SNR 7 0.4 4 4 2 2 0 0 6 3.2 6 4.0 Nova 6 0.4 0 0 0 0 0 0 0 0.0 0 0.0 CV 75 4.6 1 1 1 0 0 0 2 1.1 1 0.7 Symbiotic star 4 0.3 0 0 0 0 0 0 0 0.0 0 0.0 star 12 0.7 0 0 1 0 0 0 1 0.5 0 0.0 Open star cluster 1 0.1 0 0 0 0 0 0 0 0.0 0 0.0 molecular cloud 2 0.1 1 0 0 0 0 0 1 0.5 0 0.0 GC 1 0.1 1 1 0 0 0 0 1 0.5 1 0.7 Gamma-ray source 1 0.1 0 0 1 0 0 0 1 0.5 0 0.0 confused source 10 0.6 1 0 0 0 0 0 1 0.5 0 0.0 U1 36 2.2 0 0 0 0 0 0 0 0.0 0 0.0 U2 55 3.4 0 0 1 0 0 0 1 0.5 0 0.0 U3 38 2.3 1 0 0 0 0 0 1 0.5 0 0.0 Firm matches indicate sources with Flag being M or D, and do not include false-matched, unidentified, and ambiguous sources for safety.", "The nearest source was used for the counterpart of the extended Fermi sources.", "Here Seyfert galaxy includes all Seyfert 1 and 2 types.", "c | cc | cc | cc | cc | cccc [ht] Classes of cross-matched sources (4FGL-DR2 definition) Source 2cOriginal 2cMatched point sources 2cExtended 2cID-matched 4cTotal-matched # % # Firm # Firm # Firm # % Firm # Firm % Total 5788 132 115 31 15 24 21 187 151 BLL 1190 21 32 30 0 0 3 3 35 18.7 33 21.9 FSRQ 730 13 43 40 0 0 10 10 53 28.3 50 33.1 BCU 1517 26 14 13 0 0 2 2 16 8.6 15 9.9 AGN 11 0.19 1 1 0 0 1 1 2 1.1 2 1.3 RDG 44 0.76 7 7 1 1 2 2 10 5.3 10 6.6 SBG 8 0.14 3 3 0 0 0 0 3 1.6 3 2.0 SEY 1 0.017 1 1 0 0 0 0 1 0.5 1 0.7 NLSY1 9 0.16 1 1 0 0 0 0 1 0.5 1 0.7 css 5 0.086 1 1 0 0 1 1 2 1.1 2 1.3 ssrq 2 0.035 0 0 0 0 0 0 0 0.0 0 0.0 GAL 5 0.086 0 0 2 0 0 0 2 1.1 0 0.0 SNR 43 0.74 2 2 7 2 0 0 9 4.8 4 2.6 PSR 259 4.5 10 7 0 0 2 2 12 6.4 9 6.0 PWN 18 0.31 2 1 9 9 0 0 11 5.9 10 6.6 spp 95 1.6 0 0 5 3 0 0 5 2.7 3 2.0 BIN 9 0.16 1 1 0 0 0 0 1 0.5 1 0.7 HMB 8 0.14 5 5 0 0 0 0 5 2.7 5 3.3 LMB 4 0.069 1 1 0 0 0 0 1 0.5 1 0.7 glc 30 0.52 5 1 0 0 0 0 5 2.7 1 0.7 SFR 5 0.086 0 0 1 0 0 0 1 0.5 0 0.0 NOV 1 0.017 0 0 0 0 0 0 0 0.0 0 0.0 unidentified 1794 31 3 0 6 0 3 0 12 6.4 0 0.0 Firm matches indicate sources with Flag being M or D, and do not include false-matched, unidentified, and ambiguous sources for safety." ] ]
2105.11791
[ [ "PyTorch, Explain! A Python library for Logic Explained Networks" ], [ "Abstract \"PyTorch, Explain!\"", "is a Python module integrating a variety of state-of-the-art approaches to provide logic explanations from neural networks.", "This package focuses on bringing these methods to non-specialists.", "It has minimal dependencies and it is distributed under the Apache 2.0 licence allowing both academic and commercial use.", "Source code and documentation can be downloaded from the github repository: https://github.com/pietrobarbiero/pytorch_explain." ], [ "Introduction", "The lack of transparency in the decision process of some machine learning models, such as neural networks, limits their application in many safety-critical domains [4].", "Employing black-boxIn the context of this paper, a black-box classifier is any classifier that cannot provide human understandable explanations about its decisions models may be unacceptable in contexts such as industry, medicine or courts, where the potential economical or ethical repercussions are calling for lawmakers to discourage from a reckless application of non-interpretable models [8], [15], [10], [11].", "As a consequence, research in Explainable Artificial Intelligence (XAI) has become strategic and has been massively encouraged, leading to the development of a variety of techniques that aim at explaining black-box models [7], [2] or at developing effective interpretable models [3], [20].", "The need for high-quality human-friendly explanations is one of the main reasons why concept-based explanations are receiving ever-growing consideration.", "Explanations are given in terms of human-understandable symbols (the concepts) rather than raw features such as pixels or characters [13], [9], [14].", "As a consequence, they seem more suitable to serve many strategic human purposes such as decision making tasks.", "For instance, a concept-based explanation may describe a high-level category through its attributes as in “a human has hands and a head”.", "While collecting high-quality evidences for an explanation is a common feature of concept-based techniques, there are very few approaches formulating hypothesis and even less providing synthetic descriptions whose validity can be quantitatively assessed [7].", "A possible solution is to rely on a formal language that is very expressive and closely related to natural language and reasoning, such as First-Order Logic (FOL).", "A FOL explanation can be considered a special kind of a concept-based explanation, where the description is given in terms of logical predicates, connectives and quantifiers, such as “$\\forall x:\\ \\textit {is\\_human}(x) \\rightarrow \\textit {has\\_hands}(x) \\wedge \\textit {has\\_head}(x)$ ”.", "However, FOL formulas generally express much more complex relationships among the concepts involved in a certain explanation.", "Compared to other concept-based techniques, logic-based explanations provide many key advantages, that we briefly described in what follows.", "An explanation reported in FOL is a rigorous and unambiguous statement (clarity).", "This formal clarity may serve cognitive-behavioral purposes such as engendering trust, aiding bias identification, or taking actions/decisions.", "For instance, dropping quantifiers and variables for simplicity, the formula “snow $\\wedge $ tree $\\leftrightarrow $ wolf” may easily outline the presence of a bias in the collection of training data.", "Different logic-based explanations can be combined to describe groups of observations or global phenomena (modularity).", "For instance, for an image showing only the face of a person, an explanation could be “(nose $\\wedge $ lips) $\\rightarrow $ human”, while for another image showing a person from behind a valid explanation could be “(feet $\\wedge $ hair $\\wedge $ ears) $\\rightarrow $ human”.", "The two local explanations can be combined into “(nose $\\wedge $ lips) $\\vee $ (feet $\\wedge $ hair $\\wedge $ ears) $\\rightarrow $ human”.", "The quality of logic-based explanations can be quantitatively measured to check their correctness and completeness (measurability).", "For instance, once the explanation “(nose $\\wedge $ lips) $\\vee $ (feet $\\wedge $ hair $\\wedge $ ears)” is extracted for the class human, this logic formula can be applied on a test set to check its generality in terms of quantitative metrics like accuracy, fidelity and consistency.", "Finally, FOL-based explanations can be rewritten in different equivalent forms such as in Disjunctive Normal Form (DNF) and Conjunctive Normal Form (CNF) (simplifiability).", "Further, techniques such as the Quine–McCluskey algorithm can be used to compact and simplify logic explanations [17], [19], [16].", "For instance, the explanation “(person $\\wedge $ nose) $\\vee $ ($\\lnot $person $\\wedge $ nose)” can be easily simplified in “nose”.", "This work presents a Python library for XAI enabling neural networks to solve and explain a categorical learning problem integrating elements from deep learning and logic.", "Differently from vanilla neural architectures, these models can be directly interpreted by means of a set of FOL formulas.", "In order to implement such a property, such models require their inputs to represent the activation scores of human-understandable concepts.", "Then, specifically designed learning objectives allow them to make predictions in a way that is well suited for providing FOL-based explanations that involve the input concepts.", "In order to reach this goal, LENs exploit parsimony criteria aimed at keeping their structure simple as described in recent works [14], [6], [5]." ], [ "Background", "Classification is the problem of identifying a set of categories $y \\in Y\\subset [0,1]^r$ an observation $x \\in X\\subset \\mathbb {R}^d$ belongs to.", "A standard neural network is a black-box model $f: X \\mapsto Y$ predicting for any sample $x\\in X$ the corresponding class membership $\\hat{y}\\in Y$ .", "In case the input features $x$ are not easily interpretable (low-level feature, image pixels) concept-based classifiers have been introduced to predict class memberships $Y$ from human-understandable categories (a.k.a.", "concepts) $C\\subset [0,1]^k$ : $f: C \\mapsto Y$ to improve the understanding of black boxes and their decision process.", "Concepts can either correspond to the predictions of a classifier (i.e.", "$g: X \\mapsto C$ ) [14] or simply to a re-scaling of the inputs space from the unbounded $\\mathbb {R}^d$ to the unit interval $[0, 1]^k$ such that input features can be treated as logic predicates.", "Concept-based classifiers improve human understanding as their input and output spaces consists of interpretable symbols.", "Recent work on concept-based neural networks has led to the development of models like the $\\psi $ network [5] or the entropy-based network [1] i.e., concept-based classifiers explaining their own decision process.", "These models are designed to compute concise logic formulas representing how the network combines input concepts in order to arrive to a prediction.", "This library implements the core learning criteria and methods allowing a customized implementation of neural models providing logic explanations." ], [ "Application Programming Interfaces (APIs)", "The code library is designed with intuitive APIs requiring only a few lines of code to train and get explanations from deep neural networks.", "The library currently provides the APIs for three explainable neural architectures: $\\psi $ networks described in [6], multi-layer neural networks with rectified linear units, and entropy-based networks [1].", "The library supports a fine-grained customization of the neural networks as shown in the following code example .", "The architecture of the model and the training loop is defined by means of standard PyTorch APIs [18].", "The entropy_logic_loss allows the network to get rid of less relevant input concepts and to generate concise explanations for each prediction [6].", "The explain_class method allows the extraction of logic formulas from the trained model.", "Once extracted, formulas can be tested on an unseen set of test samples using the test_explanation method providing the accuracy of the logic formula.", "Explanations will be logic formulas in disjunctive normal form.", "In the example below, the explanation will be $y=1 \\leftrightarrow (x_1 \\wedge \\lnot x_2) \\vee (x_2 \\wedge \\lnot x_1)$ corresponding to $y=1 \\leftrightarrow x_1 \\oplus x_2$ .", "mystyle import torch from torch.nn.functional import one_hot import torch_explain as te from torch_explain.nn.functional import entropy_logic_loss from torch_explain.logic.nn import entropy from torch_explain.logic.metrics import test_explanation, complexity   # train data x_train = torch.tensor([     [0, 0],     [0, 1],     [1, 0],     [1, 1], ], dtype=torch.float) y_train = torch.tensor([0, 1, 1, 0], dtype=torch.float).unsqueeze(1)   # instantiate an \"entropy-based network\" layers = [     te.nn.EntropyLinear(x_train.shape[1], 10, n_classes=2),     torch.nn.LeakyReLU(),     torch.nn.Linear(10, 4),     torch.nn.LeakyReLU(),     torch.nn.Linear(4, 1), ] model = torch.nn.Sequential(*layers)   # fit the model optimizer = torch.optim.AdamW(model.parameters(), lr=0.01) loss_form = torch.nn.CrossEntropyLoss() model.train() for epoch in range(1001):     optimizer.zero_grad()     y_pred = model(x_train).squeeze(-1)     loss = loss_form(y_pred, y_train) + 0.00001 * entropy_logic_loss(model)     loss.backward()     optimizer.step()   # get first-order logic explanations for a specific y1h = one_hot(y_train.squeeze().long()) explanation = entropy.explain_class(model, x_train)   # compute explanation accuracy and complexity accuracy, preds = test_explanation(explanation, x_train, y1h, target_class=1) explanation_complexity = complexity(explanation)" ], [ "Software and documentation availability", "In order to make state-of-the-art approaches accessible to the whole community, we released this library as a Python package on PyPI: https://pypi.org/project/torch-explain/.", "An extensive documentation on methods is available on read the docshttps://pytorch-explain.readthedocs.io/en/latest/.", "and unit tests results on TravisCIhttps://travis-ci.org/pietrobarbiero/pytorch_explain/..", "The Python code and the scripts used for benchmarking against state-of-the-art white-box models, including parameter values and documentation, is freely available under Apache 2.0 Public License from a GitHub repositoryhttps://github.com/pietrobarbiero/logic_explainer_networks.." ], [ "Limitations and future research directions", "The extraction of a first-order logic explanation requires symbolic input and output spaces.", "This constraint is the main limitation of our framework, as narrows the range of applications down to symbolic I/O problems.", "In some contexts, such as computer vision, the use of LENs may require additional annotations and attribute labels to get a consistent symbolic layer of concepts.", "However, recent work on automatic concept extraction may partially solve this issue leading to more cost-effective concept annotations [9], [12].", "The improvement of LENs models is an open research area.", "The efficiency and the classification performances of fully interpretable LENs, i.e.", "$\\psi $ network, is still quite limited due to the extreme pruning strategy." ], [ "Broader impact", "Current legislation in US and Europe binds AI to provide explanations especially when the economical, ethical, or financial impact is significant [8], [15].", "This work contributes to a lawful and safer adoption of some of the most powerful AI technologies allowing deep neural networks to have a greater impact on society.", "Extracting first-order logic explanations from deep neural networks enables satisficing [21] knowledge distillation while achieving performances comparable with the state of the art.", "The formal language of logic provides clear and synthetic explanations, suitable for laypeople, managers, and in general for decision makers outside the AI research field.", "Thanks to their explainable nature, LENs can be effectively used to understand the behavior of an existing algorithm, to reverse engineer products, to find vulnerabilities, or to improve system design.", "From a scientific perspective, formal knowledge distillation from state-of-the-art networks may enable scientific discoveries or confirmation of existing theories." ], [ "Acknowledgments and Disclosure of Funding", "We thank Stefano Melacci, Pietro Lió, and Marco Gori for useful feedback and suggestions.", "This work was partially supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 848077." ] ]
2105.11697
[ [ "An Upper Limit of Decaying Rate with Respect to Frequency in Deep Neural\n Network" ], [ "Abstract Deep neural network (DNN) usually learns the target function from low to high frequency, which is called frequency principle or spectral bias.", "This frequency principle sheds light on a high-frequency curse of DNNs -- difficult to learn high-frequency information.", "Inspired by the frequency principle, a series of works are devoted to develop algorithms for overcoming the high-frequency curse.", "A natural question arises: what is the upper limit of the decaying rate w.r.t.", "frequency when one trains a DNN?", "In this work, our theory, confirmed by numerical experiments, suggests that there is a critical decaying rate w.r.t.", "frequency in DNN training.", "Below the upper limit of the decaying rate, the DNN interpolates the training data by a function with a certain regularity.", "However, above the upper limit, the DNN interpolates the training data by a trivial function, i.e., a function is only non-zero at training data points.", "Our results indicate a better way to overcome the high-frequency curse is to design a proper pre-condition approach to shift high-frequency information to low-frequency one, which coincides with several previous developed algorithms for fast learning high-frequency information.", "More importantly, this work rigorously proves that the high-frequency curse is an intrinsic difficulty of DNNs." ], [ "Introduction", "The study of generalization in deep learning attracts much attention in recent years due to the contradiction to the traditional wisdom [1], [2], that is, over-parameterized DNNs often generalize well in real dataset.", "To study the generalization, one has to be cautious of the no-free-lunch theorem, which hints that for any method one can find a dataset the method generalizes badly.", "Therefore, to study the generalization puzzle of over-parameterized DNNs in real dataset, it is necessary to separately study the DNN algorithm and the real dataset.", "If the characteristics of the algorithm are consistent with those of the real dataset, then, the algorithm generalizes well, otherwise, badly.", "Usually, the training of DNNs are enforced with no explicit constraints, therefore, the implicit bias of the training of DNNs is important.", "Recently, a series of works have demonstrated an implicit bias in Fourier domain, that is, a DNN tends to learn a target function from low to high frequencies during the training [3], [4], [5].", "[6], [4] propose that the low-frequency bias is due to that a function with a certain regularity decays w.r.t.", "frequency in the Fourier domain with a certain rate.", "This mechanism is further confirmed by a series of theoretical works [7], [8], [9], [10], [11], [12], [13], [14].", "The frequency principle implies a rational that DNNs generalize well for real datasets, which are often low-frequency dominant [4].", "Meanwhile, such low-frequency bias also suggests a high-frequency curse, i.e., DNNs are difficult to learn high-frequency information.", "To overcome the high-frequency curse, various approaches are proposed [4], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28].", "A natural question is that what is the upper limit of the decaying rate w.r.t.", "frequency in DNN.", "Such an upper limit characterizes the boundary of the frequency bias, providing a better understanding of the implicit bias of DNNs in Fourier domain.", "In addition, it also provides a guidance for algorithm design of DNNs which could be more efficient in capturing high-frequency information.", "In this work, we prove that there is a critical decaying rate w.r.t.", "frequency.", "Below the upper limit of the decaying rate, the DNN interpolates the training data by a function with a certain regularity.", "However, above the upper limit, the DNN interpolates the training data by a trivial function, i.e., a function is only non-zero at training data points.", "Theoretical works have estimated the decaying rate w.r.t.", "frequency follows a power law for DNNs with a certain regularity activation function [7], [8], [9], [10], [12], [13] in the gradient descent training.", "The long-time limit solution of such gradient descent training is proved to be equivalent to solving a Fourier-domain variational problem [8], [9].", "Inspired by above works about the F-Principle, in this paper, we propose a general Fourier-domain variational formulation for supervised learning problem, including DNNs, and study its well-posedness.", "In continuum modelling, it is often difficult to impose the constraint of given values on isolated data points in a function space without sufficient regularity, e.g., a $L^p$ space.", "We circumvent this difficulty by regarding the Fourier-domain variation as the primal problem and the constraint of isolated data points is imposed through a linear operator.", "Under a necessary and sufficient condition within our unified framework, we establish the well-posedness of the Fourier-domain variational problem.", "We show that the well-posedness depends on a critical exponent, which equals to the data dimension.", "This is a stark difference compared with a traditional partial differential equation (PDE) problem.", "For example, in a boundary value problem of any PDE in a $d$ -dimensional domain, the boundary data should be prescribed on the $(d-1)$ -dimensional boundary of the domain, where the dimension $d$ plays an important role.", "However, in a well-posed supervised learning problem, the constraint is always on isolated points, which are 0-dimensional independent of $d$ , while the model has to satisfy a well-posedness condition depending on the dimension.", "In practice, common DNNs is a convenient way to implement our formulation.", "Therefore, the convergence rate of high-frequency has a upper limit.", "An algorithm with too fast high-frequency learning would lead to a learned function only non-zero at training data points.", "Such understanding of the upper limit of decaying rate indicates a better way to overcome the high-frequency curse is to design a proper pre-condition approach to shift high-frequency information to low-frequency one, which coincides with several previous developed algorithms for fast learning high-frequency information [17], [19], [20], [21], [22], [23], [24], [25].", "The rest of the paper is organized as follows.", "Section  shows some related work.", "In section , we propose a Fourier-domain variational formulation for supervised learning problems.", "The necessary and sufficient condition for the well-posedness of our model is presented in section .", "Section  is devoted to the numerical demonstration in which we solve the Fourier-domain variational problem using band-limited functions.", "Finally, we present a short conclusion and discussion in section ." ], [ "Related Works", "It has been an important approach to study machine learning from the perspective of implicit bias [29], such as the implicit bias of training algorithms [30], [31], dropout [32], linear network [33] and DNNs under different initializations [34].", "The low-frequency implicit bias is named as frequency principle (F-Principle) [3], [4] or spectral bias [5] and can be robustly observed no matter how overparameterized NNs are.", "[6], [4] propose a key mechanism of the F-Principle that the regularity of the activation function converts into the decay rate of a loss function in the frequency domain.", "Theoretical studies subsequently show that the F-Principle holds in general setting with infinite samples [7] and in the regime of wide NNs (Neural Tangent Kernel (NTK) regime [35]) with finite samples [8], [9] or samples distributed uniformly on sphere [10], [11], [12], [13].", "[14] show that the integral equation would naturally leads to the F-Principle.", "In addition to characterizing the training speed of DNNs, the F-Principle also implicates that DNNs prefer low-frequency function and generalize well for low-frequency functions [4], [8], [9].", "To accelerate the convergence of high-frequency, a series of works propose different approaches: A Multi-scale DNN (MscaleDNN) method, originally proposed in [19] and completed in [20] uses a scaling down operation to convert higher frequency spectrum to a low frequency one before the learning is carried out with a small-sized DNN.. several works project data into a high dimensional space with a set of sinusoids [23], [24], which is similar to the design in MscaleDNN in [19], [20]; [25] revise a normal activation function $\\sigma (wx+b)$ by $\\sigma (e^{w}(x-b))$ , which can be more sensitive to the weight; [36] use meta-learning to obtain a good initialization for fast and effective image restoration; [16] explicitly impose high frequencies with higher priority in the loss function; [15], [37] design different types of activation functions." ], [ "Notations", "In the following, we consider the regression problem of fitting a target function $f^{*}\\in C_c(^d)$ .", "Clearly, $f^{*}\\in L^{2}(^d)$ .", "Specifically, we use a DNN, $h(,(t))$ with a parameter set $(t)$ , to fit the training dataset $\\lbrace (_{i},y_{i})\\rbrace _{i=1}^{n}$ of $n$ sample points, where $_{i}\\in ^d$ , $y_{i}=f^{*}(_{i})$ for each $i$ .", "For the convenience of notation, we denote $=(_{1},\\ldots ,_{n})^,$ =(y1,...,yn).", "It has been shown in [35], [38] that, if the number of neurons in each hidden layer is sufficiently large, then ${(t)-(0)}\\ll 1$ for any $t\\ge 0$ .", "In such cases, the the following function $h_{\\mathrm {lin}}(,)=h(,_{0})+\\nabla _{}h\\left(,_0\\right)\\cdot (-_{0}),$ is a very good approximation of DNN output $h(,(t))$ with $(0)=_{0}$ .", "Note that, we have the following requirement for $h$ which is easily satisfied for common DNNs: for any $\\in ^{m}$ , there exists a weak derivative of $h(\\cdot ,_0)$ with respect to $$ satisfying $\\nabla _{}h(\\cdot ,_0)\\in L^{2}(^d)$ .", "A two-layer neural network is $h(,(t))=\\sum _{j=1}^{m} a_j \\sigma (_j\\cdot + b_j),$ where $\\sigma $ is the activation function, $_j$ is the input weight, $a_j$ is the output weight, $b_j$ is the bias term.", "In this work, for any function $g$ defined on $^d$ , we use the following convention of the Fourier transform and its inverse: $[g]()=\\int _{^d}g()^{-2\\pi I^},\\quad g()=\\int _{^d}[g]()^{2\\pi I^}.$" ], [ "Fourier-domain Variational Problem for Supervised Learning", "To study the decaying rate limit w.r.t.", "frequency in DNN training, we propose a Fourier-domain variational problem for supervised learning, in which frequency bias can be imposed by weight term.", "To show the motivation and the rationality of the variational problem, we first introduce a linear frequency principle." ], [ "Motivation: Linear Frequency Principle", "In the large width limit, it is reasonable [35], [38] to assume a linear condition, i.e., $h(,)=h_{\\mathrm {lin}}(,)$ .", "Based on the linear condition, [8], [9] derived a Linear F-Principle (LFP) dynamics to effectively study the training dynamics of a two-layer NN with the mean square loss in the large width limit.", "Up to a multiplicative constant in the time scale, the gradient descent dynamics of a sufficiently wide two-layer NN is approximated by $\\partial _{t}[u](,t)=-\\,(\\gamma ())^{2}[u_{\\rho }](),$ where $u(,t)=h(,(t))-f^{*}()$ , $u_{\\rho }(,t)=u(,t)\\rho ()$ , $\\rho ()=\\frac{1}{n}\\sum _{i=1}^n\\delta (-_i)$ , accounting for the real case of a finite training dataset $\\lbrace (_i,y_i)\\rbrace _{i=1}^n$ , and $\\gamma ()$ depends on the initialization and frequency.", "For ReLU activation function, $(\\gamma ())^{2}=_{a(0), r(0)}\\left[\\frac{r(0)^{3}}{16 \\pi ^{4}{}^{d+3}}+\\frac{a(0)^{2} r(0)}{4 \\pi ^{2}{}^{d+1}}\\right],$ where $r(0)={(0)}$ and the two-layer NN parameters at initial $a(0)$ and $(0)$ are random variables with certain given distribution.", "For tanh activation function, $\\gamma ()$ exponentially decays w.r.t.", "frequency as shown in [9].", "The solution of the LFP model (REF ) is equivalent to that of the following optimization problem in a proper hypothesis space $F_\\gamma $ , $\\min _{h-h_{\\mathrm {ini}}\\in F_{\\gamma }}\\int _{^d}(\\gamma ())^{-2}{[h]()-[h_\\mathrm {ini}]()}^{2}{},$ subject to constraints $h(_{i})=y_{i}$ for $i=1,\\ldots ,n$ .", "The weight $(\\gamma ())^{-2}$ grows as the frequency $$ increases, which means that a large penalty is imposed on the high frequency part of $h()-h_{\\mathrm {ini}}()$ .", "As we can see, a random non-zero initial output of DNN leads to a specific type of generalization error.", "To eliminate this error, we use DNNs with an antisymmetrical initialization (ASI) trick  [39], which guarantees $h_{\\mathrm {ini}}()=0$ .", "Then the final output $h()$ is dominated by low frequency, and the DNN model possesses a good generalization." ], [ "Fourier-domain Variational Formulation", "Inspired by the variational formulation of LFP model, we propose a new continuum model for the supervised learning, which includes DNNs with gradient flow learning.", "This is a variational problem with a parameter $\\alpha >0$ to be determined later: $& \\min _{h\\in } Q_{\\alpha }[h] =\\int _{^d}\\langle \\rangle ^\\alpha {[h]()}^{2}{}, \\\\& \\mathrm {s.t.", "}\\quad h(_i)=y_i,\\quad i=1,\\cdots ,n,$ where $\\langle \\rangle =(1+{}^2)^{\\frac{1}{2}}$ is the “Japanese bracket” of $$ and $=\\lbrace h(x)|\\int _{^d}\\langle \\rangle ^\\alpha {[h]()}^{2}{}<\\infty \\rbrace $ .", "According to the equivalent theorem in [9], $-\\alpha $ is the decaying rate w.r.t.", "frequency in the gradient flow dynamics in (REF ).", "In this work, we study how the property of the solution in the variational problem depends on $\\alpha $ .", "Note that in the spatial domain, the evaluation on $n$ known data points is meaningless in the sense of $L^2$ functions.", "Therefore, we consider the problem in the frequency domain and define a linear operator $_{}:L^1(^d)\\cap L^2(^d)\\rightarrow ^n$ for the given sample set $$ to transform the original constraints into the ones in the Fourier domain: $_{}\\phi ^*=$ .", "More precisely, we define for $\\phi \\in L^1(^d)\\cap L^2(^d)$ $_{}\\phi :=\\left(\\int _{^d}\\phi ()^{2\\pi I\\cdot _{1}}{},\\cdots ,\\int _{^d}\\phi ()^{2\\pi I\\cdot _{n}}{}\\right)^$ The admissible function class reads as $_{,}=\\lbrace \\phi \\in L^1(^d)\\cap L^2(^d)\\mid _{}\\phi =\\rbrace .$ Notice that ${^{-1}[\\phi ]}_{H^{\\frac{\\alpha }{2}}}=\\left(\\int _{^d}\\langle \\rangle ^\\alpha {\\phi ()}^{2}{}\\right)^{\\frac{1}{2}}$ is a Sobolev norm, which characterizes the regularity of the final output function $h()=^{-1}[{\\phi }]()$ .", "The larger the exponent $\\alpha $ is, the better the regularity becomes.", "For example, when $d=1$ and $\\alpha =2$ , by Parseval's theorem, ${u}_{H^{1}}^2=\\int _{}(1+{\\xi }^2){[u](\\xi )}^{2}{\\xi }=\\int _{}u^2+\\frac{1}{4\\pi ^2}{\\nabla u}^2x.$ Accordingly, the Fourier-domain variational problem reads as a standard variational problem in spatial domain.", "This is true for any quadratic Fourier-domain variational problem, but of course our Fourier-domain variational formulation is not necessarily being quadratic.", "The details for general cases (non-quadratic ones) are left to future work.", "For the quadratic setting with exponent $\\alpha $ , i.e., Problem (REF ), it is roughly equivalent to the following spatial-domain variational problem: $\\min \\int _{^d}(u^2+{\\nabla ^{\\frac{\\alpha }{2}}u}^2)x.$ This is clear for integer $\\alpha /2$ , while fractional derivatives are required for non-integer $\\alpha /2$ .", "Back to our problem, after the above transformation, our goal is transformed into studying the following Fourier-domain variational problem, Problem 1 Find a minimizer $\\phi ^*$ in $_{,}$ such that $\\phi ^*\\in \\arg \\min _{\\phi \\in _{,}} {^{-1}[\\phi ]}_{H^{\\frac{\\alpha }{2}}}^2.$ We remark that the operator $_{}$ is the inverse Fourier transform with evaluations on sample points $$ .", "Actually, the linear operator $_{}$ projects a function defined on $^d$ to a function defined on 0-dimensional manifold $$ .", "Just like the (linear) trace operator $T$ in a Sobolev space projects a function defined on $d$ -dimensional manifold into a function defined on $(d-1)$ -dimensional boundary manifold.", "Note that the only function space over the 0-dimensional manifold $$ is the $n$ -dimensional vector space $^n$ , where $n$ is the number of data points, while any Sobolev (or Besov) space over $d$ -dimensional manifold ($d\\ge 1$ ) is an infinite dimensional vector space." ], [ "The Critical Decaying Rate", "In this section, we consider a critical exponent for $\\alpha $ , which leads to the existence/non-existence dichotomy to Problem REF .", "We first prove that there is no solution to the Problem REF in subcritical case $\\alpha < d$ , and for $\\alpha >d$ the optimal function is a continuous and nontrivial solution (See proof in Appendix.).", "Therefore, we conclude that to obtain a non-trivial interpolation among training data for supervised learning, such as DNN fitting, the decaying rate of high-frequency information can not be too fast, i.e., there exists a upper limit of the decaying rate w.r.t.", "frequency." ], [ "Subcritical Case: $\\alpha <d$", "In order to prove the nonexistence of the solution to the Problem REF in $\\alpha <d$ case, at first we need to find a class of functions that make the norm tend to zero.", "Let $\\psi _{\\sigma }()=(2\\pi )^{\\frac{d}{2}}\\sigma ^d ^{-2\\pi ^2\\sigma ^2{}^2}$ , then by direct calculation, we have $^{-1}[\\psi _{\\sigma }]()=^{-\\frac{{}^2}{2\\sigma ^2}}$ .", "For $\\alpha <d$ the following proposition shows that the norm ${^{-1}[\\psi _\\sigma ]}_{H^{\\frac{\\alpha }{2}}}^2$ can be sufficiently small as $\\sigma \\rightarrow 0$ .", "Proposition 1 (critical exponent) For any input dimension $d$ , we have $\\lim _{\\sigma \\rightarrow 0}{^{-1}[\\psi _\\sigma ]}_{H^{\\frac{\\alpha }{2}}}^2={\\left\\lbrace \\begin{array}{ll}0, & \\alpha <d, \\\\C_d, & \\alpha =d, \\\\\\infty , & \\alpha >d.\\end{array}\\right.", "}$ Here the constant $C_d=\\frac{1}{2}(d-1)!", "(2\\pi )^{-d}\\frac{2\\pi ^{d/2}}{\\Gamma \\left(d/2\\right)}$ only depends on the dimension $d$ .", "Remark 1 The function $^{-1}[\\psi ]$ can be any function in the Schwartz space, not necessarily Gaussian.", "Proposition REF still holds with (possibly) different $C_d$ .", "For every small $\\sigma $ , we can use $n$ rapidly decreasing functions $^{-1}[\\psi _{\\sigma }](-_{i})$ to construct the solution $^{-1}[\\phi _{\\sigma }]()$ of the supervised learning problem.", "However, according to Proposition REF , when the parameter $\\sigma $ tends to 0, the limit is the zero function in the sense of $L^2(^d)$ .", "Therefore we have the following theorem: Theorem 1 (non-existence) Suppose that $\\ne $ .", "For $\\alpha <d$ , there is no function $\\phi ^*\\in _{,}$ satisfying $\\phi ^*\\in \\arg \\min _{\\phi \\in _{,}}{^{-1}[\\phi ]}_{H^{\\frac{\\alpha }{2}}}^2.$ In other words, there is no solution to the Problem REF ." ], [ "Supercritical Case: $\\alpha >d$", "We then provide a theorem to establish the existence of the minimizer for Problem REF in the case of $\\alpha >d$ .", "Theorem 2 (existence) For $\\alpha >d$ , there exists $\\phi ^*\\in _{,}$ satisfying $\\phi ^*\\in \\arg \\min _{\\phi \\in _{,}}{^{-1}[\\phi ]}_{H^{\\frac{\\alpha }{2}}}^2.$ In other words, there exists a solution to the Problem REF .", "Remark 2 Note that, according to the Sobolev embedding theorem [40], [41], the minimizer in Theorem REF has smoothness index no less than $[\\frac{\\alpha -d}{2}]$ ." ], [ "Numerical Results", "In this section, we illustrate our results by solving Fourier-domain variational problems numerically.", "We use uniform mesh in frequency domain with mesh size $\\Delta \\xi $ and band limit $M\\Delta \\xi $ .", "In this discrete setting, the considered space becomes $^{(2M)^d}$ .", "We emphasize that the numerical solution with this setup always exists even for the subcritical case which corresponds to the non-existence theorem.", "However, as we will show later, the numerical solution is trivial in nature when $\\alpha <d$ ." ], [ "Special Case: One Data Point in One Dimension", "To simplify the problem, we start with a single point $X=0\\in $ with the label $Y=2$ .", "Denote $\\phi _j = \\phi (\\xi _j)$ for $j\\in $ .", "We also assume that the function $\\phi $ is an even function.", "Then according to the definition of $_{}$ , we have the following problem: Example 1 (Problem REF with a particular discretization) $& \\min _{\\phi \\in ^M} \\sum _{j=1}^M(1+{j}^2\\Delta \\xi ^2)^{\\frac{\\alpha }{2}}{\\phi _j}^{2}, \\\\& \\mathrm {s.t.", "}\\quad \\sum _{j=1}^M\\phi _j\\Delta \\xi = 1,$ Fig.", "REF shows that for this special case with a large $M$ , $h(x)$ is not an trivial function in $\\alpha >d$ case and degenerates to a trivial function in $\\alpha <d$ case.", "Figure: Fitting the function h(x)h(x) shown in equation () with different exponent α\\alpha 's.", "Here we take M=10 6 M=10^6, Δξ=0.01\\Delta \\xi =0.01, λ=1\\lambda =1 and different α\\alpha and observe that h(x)h(x) is not an trivial function in α>d\\alpha >d case and degenerates to a trivial function in α<d\\alpha <d case." ], [ "General Case: $n$ Points in {{formula:d563d3dc-2beb-497b-b854-37127aa631c3}} Dimension", "Assume that we have $n$ data points $_1,_2,\\ldots ,_n\\in ^d$ and each data point has $d$ components: $_i=\\left(x_{i1},x_{i2},\\ldots ,x_{id}\\right)^{equation*}and denote the corresponding label as \\left(y_1,y_2,\\ldots ,y_n\\right)^.", "For the sake of simplicity, we denote the vector (j_1,j_2,\\cdots ,j_d)^ by _{j_1\\ldots j_d}.", "Then our problem becomes\\begin{exam}[Problem \\ref {prob..VariationalPointCloud} with general discretization]{\\begin{@align}{1}{-1}& \\min _{\\phi \\in ^{(2M)^d}} \\sum _{j_1,\\ldots ,j_d=-M}^M(1+{_{j_1\\ldots j_d}}^2\\Delta \\xi ^2)^{\\frac{\\alpha }{2}}{\\phi _{j_1\\ldots j_d}}^{2}, \\\\& \\mathrm {s.t.", "}\\quad \\sum _{j_1,\\ldots ,j_d=-M}^M\\phi _{j_1\\ldots j_d}^{2\\pi I\\Delta \\xi _{j_1\\ldots j_d}^_k}=y_k, \\ \\ k=1,2,\\ldots ,d\\end{@align}}\\end{exam}$ In Fig.REF , we set $\\alpha =10$ in both cases to ensure $\\alpha >d$ and change the band limit $M$ .", "We observe that as $M$ increases, the fitting curve converges to a non-trivial curve.", "In Fig.REF , we set $M=1000$ in 1-dimensional case and $M=100$ in 2-dimensional case.", "By changing exponent $\\alpha $ , we can see in all cases, the fitting curves are non-trivial when $\\alpha >d$ , but degenerate when $\\alpha <d$ .", "Figure: 20 points in 2 dimensionFigure: 20 points in 2 dimension" ], [ "Conclusion ", "To understand the limit of the frequency bias in DNNs, we propose a Fourier-domain variational formulation and establish the sufficient and necessary conditions for the well-posedness of the Fourier-domain variational problem, followed by numerical demonstration.", "Our work suggests that there is a upper limit of the decaying rate w.r.t.", "frequency, i.e., high frequency cannot converge too fast, in order to obtain a nontrivial solution in DNN training, thus, pointing out the intrinsic high-frequency curse.", "For two-layer infinite-width neural networks, existing works have shown their solutions are equivalent to the solutions of particular Fourier-domain variational problems [9].", "However, for general non-linear DNNs, this equivalence is only qualitative.", "In addition, our Fourier-domain variational formulation provides a novel viewpoint for modelling machine learning problem, that is, imposing more constraints, e.g., higher regularity, on the model rather than the data (always isolated points in practice) can give us the well-posedness as dimension of the problem increases.", "This is different from the modelling in physics and traditional point cloud problems, in which the model is independent of dimension in general.", "Our work suggests a potential approach of algorithm design by considering a dimension-dependent model for data modelling.", "This work is sponsored by the National Key R&D Program of China Grant No.", "2019YFA0709503 (Z.", "X.", "), the Shanghai Sailing Program, the Natural Science Foundation of Shanghai Grant No.", "20ZR1429000 (Z.", "X.", "), the National Natural Science Foundation of China Grant No.", "62002221 (Z.", "X.", "), Shanghai Municipal of Science and Technology Project Grant No.", "20JC1419500 (Y.Z.", "), Shanghai Municipal of Science and Technology Major Project No.", "2021SHZDZX0102, and the HPC of School of Mathematical Sciences and the Student Innovation Center at Shanghai Jiao Tong University." ], [ "Checklist", " For all authors... Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?", "Did you describe the limitations of your work?", "See Section Did you discuss any potential negative societal impacts of your work?", "Have you read the ethics review guidelines and ensured that your paper conforms to them?", "If you are including theoretical results... Did you state the full set of assumptions of all theoretical results?", "Did you include complete proofs of all theoretical results?", "See Appendix.", "If you ran experiments... Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?", "See supplemental material.", "Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?", "See Section .", "Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?", "Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?", "The computation is done on personal computer.", "If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...", "If your work uses existing assets, did you cite the creators?", "Did you mention the license of the assets?", "Did you include any new assets either in the supplemental material or as a URL?", "Did you discuss whether and how consent was obtained from people whose data you're using/curating?", "Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?", "If you used crowdsourcing or conducted research with human subjects... Did you include the full text of instructions given to participants and screenshots, if applicable?", "Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?", "Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?" ], [ "Lemma ", "Lemma 1 Let the function $\\psi _{\\sigma }()=(2\\pi )^{\\frac{d}{2}}\\sigma ^d ^{-2\\pi ^2\\sigma ^2{}^2}$ , $\\in ^d$ .", "We have $\\lim _{\\sigma \\rightarrow 0}\\int _{^d}{}^{\\alpha }{\\psi _{\\sigma }()}^{2}{}={\\left\\lbrace \\begin{array}{ll}0, & \\alpha <d, \\\\C_d, & \\alpha =d, \\\\\\infty , & \\alpha >d.\\end{array}\\right.", "}$ Here the constant $C_d=\\frac{1}{2}(d-1)!", "(2\\pi )^{-d}\\frac{2\\pi ^{d/2}}{\\Gamma \\left(d/2\\right)}$ only depends on the dimension $d$ .", "In fact, $\\lim \\limits _{\\sigma \\rightarrow 0} \\int _{^d}{}^\\alpha |\\psi _\\sigma ()|^2&=\\lim \\limits _{\\sigma \\rightarrow 0} \\int _{^d}{}^\\alpha (2\\pi )^d\\sigma ^{2d}^{-4\\pi ^2\\sigma ^2{}^2}\\\\&=\\lim \\limits _{\\sigma \\rightarrow 0} (2\\pi )^d \\sigma ^{d-\\alpha } \\int _{^d} {\\sigma }^\\alpha ^{-4\\pi ^2{\\sigma }^2}{(\\sigma )}\\\\&=\\lim \\limits _{\\sigma \\rightarrow 0} (2\\pi )^d \\sigma ^{d-\\alpha } \\int _0^\\infty r^{\\alpha +d-1} ^{-4\\pi ^2r^2}r\\cdot \\omega _d,$ where $\\omega _d=\\frac{2\\pi ^{\\frac{d}{2}}}{\\Gamma \\left(\\frac{d}{2}\\right)}$ is the surface area of a unit $(d-1)$ -sphere.", "Notice that $\\int _0^\\infty r^{\\alpha +d-1} ^{-4\\pi ^2r^2}r&=\\int _0^1 r^{\\alpha +d-1} ^{-4\\pi ^2r^2}r + \\int _1^\\infty r^{\\alpha +d-1} ^{-4\\pi ^2r^2}r\\\\&\\le \\int _0^\\infty ^{-4\\pi ^2r^2}r + \\int _0^\\infty r^{[\\alpha ]+d} ^{-4\\pi ^2r^2}r\\\\&=\\frac{1}{8\\pi ^{\\frac{3}{2}}} + \\int _0^\\infty r^{[\\alpha ]+d} ^{-4\\pi ^2r^2}r$ and $\\int _0^\\infty r^{[\\alpha ]+d} ^{-4\\pi ^2r^2}r={\\left\\lbrace \\begin{array}{ll}\\frac{1}{2}\\left(\\frac{[\\alpha ]+d-1}{2}\\right)!", "(2\\pi )^{-([\\alpha ]+d+1)}, & [\\alpha ]+d\\mathrm {\\ is\\ odd},\\\\\\frac{\\sqrt{\\pi }}{2}(2\\pi )^{-([\\alpha ]+d+1)}(\\frac{1}{2})^{\\frac{[\\alpha ]+d}{2}}([\\alpha ]+d-1)!", "!, & [\\alpha ]+d\\mathrm {\\ is\\ even}.\\end{array}\\right.", "}$ Therefore, in both cases, the integral $\\int _0^\\infty r^{\\alpha +d-1} ^{-4\\pi ^2r^2}r$ is finite.", "Then we have $\\lim \\limits _{\\sigma \\rightarrow 0} \\int _{^d}{}^\\alpha |\\psi _\\sigma ()|^2&=\\lim \\limits _{\\sigma \\rightarrow 0} (2\\pi )^d \\sigma ^{d-\\alpha } \\int _0^\\infty r^{\\alpha +d-1} ^{-4\\pi ^2r^2}r\\cdot \\omega _d\\\\&={\\left\\lbrace \\begin{array}{ll}0, & \\alpha <d, \\\\\\infty , & \\alpha >d.\\end{array}\\right.", "}$ When $\\alpha =d$ , it follows that $\\int _0^\\infty r^{\\alpha +d-1} ^{-4\\pi ^2r^2}r=\\frac{1}{2}(2\\pi )^{-2d}(d-1)!.$ Therefore $\\lim \\limits _{\\sigma \\rightarrow 0} \\int _{^d}{}^\\alpha |\\psi _\\sigma (\\xi )|^2\\xi =\\frac{1}{2}(d-1)!", "(2\\pi )^{-d}\\frac{2\\pi ^{\\frac{d}{2}}}{\\Gamma \\left(\\frac{d}{2}\\right)},$ which completes the proof." ], [ "Proof of Proposition ", "Similar to the proof of Lemma REF , we have $\\lim \\limits _{\\sigma \\rightarrow 0} {^{-1}[\\psi _\\sigma ]}_{H^{\\frac{\\alpha }{2}}}^2&=\\lim \\limits _{\\sigma \\rightarrow 0} (2\\pi )^d \\sigma ^{d-\\alpha } \\int _{^d} (\\sigma ^2+{\\sigma }^2)^{\\frac{\\alpha }{2}} ^{-4\\pi ^2{\\sigma }^2}(\\sigma )\\\\&=\\lim \\limits _{\\sigma \\rightarrow 0} (2\\pi )^d \\sigma ^{d-\\alpha } \\int _0^\\infty r^{d-1}(\\sigma ^2+r^2)^{\\frac{\\alpha }{2}} ^{-4\\pi ^2r^2}r\\cdot \\omega _d.\\\\$ For $\\sigma <1$ , the following integrals are bounded from below and above, respectively: $\\int _0^\\infty r^{d-1}(\\sigma ^2+r^2)^{\\frac{\\alpha }{2}} ^{-4\\pi ^2r^2}r \\ge \\int _0^\\infty r^{\\alpha +d-1} ^{-4\\pi ^2r^2}r = C_1 >0,$ and $\\int _0^\\infty r^{d-1}(\\sigma ^2+r^2)^{\\frac{\\alpha }{2}} ^{-4\\pi ^2r^2}r &\\le \\int _0^1 r^{d-1}(1+r^2)^{\\frac{\\alpha }{2}} ^{-4\\pi ^2r^2}r +\\int _1^\\infty r^{d-1}((2r)^2)^{\\frac{\\alpha }{2}} ^{-4\\pi ^2r^2}r \\\\&\\le \\int _0^1 r^{d-1}(1+r^2)^{\\frac{\\alpha }{2}} ^{-4\\pi ^2r^2}r +2^\\alpha \\int _0^\\infty r^{\\alpha +d-1} ^{-4\\pi ^2r^2}r\\\\&= C_2 < \\infty ,$ where $C_1=\\int _0^\\infty r^{\\alpha +d-1} ^{-4\\pi ^2r^2}r$ and $C_2=\\int _0^1 r^{d-1}(1+r^2)^{\\frac{\\alpha }{2}} ^{-4\\pi ^2r^2}r +2^\\alpha \\int _0^\\infty r^{\\alpha +d-1} ^{-4\\pi ^2r^2}r$ .", "Therefore, we obtain the results for the subcritical ($\\alpha <d$ ) and supercritical ($\\alpha >d$ ) cases $\\lim \\limits _{\\sigma \\rightarrow 0} {^{-1}[\\psi _\\sigma ]}_{H^{\\frac{\\alpha }{2}}}^2&=\\lim \\limits _{\\sigma \\rightarrow 0} (2\\pi )^d \\sigma ^{d-\\alpha } \\int _0^\\infty r^{d-1}(\\sigma ^2+r^2)^{\\frac{\\alpha }{2}} ^{-4\\pi ^2r^2}r\\cdot \\omega _d\\\\&={\\left\\lbrace \\begin{array}{ll}0, & \\alpha <d, \\\\\\infty , & \\alpha >d.\\end{array}\\right.", "}$ For the critical case $\\alpha =d$ , we have $&\\lim \\limits _{\\sigma \\rightarrow 0} {^{-1}[\\psi _\\sigma ]}_{H^{\\frac{\\alpha }{2}}}^2\\\\&=\\lim \\limits _{\\sigma \\rightarrow 0} (2\\pi )^d \\int _0^\\infty r^{d-1}(\\sigma ^2+r^2)^{\\frac{\\alpha }{2}} ^{-4\\pi ^2r^2}r\\cdot \\omega _d\\\\&=\\lim \\limits _{\\sigma \\rightarrow 0} (2\\pi )^d \\int _0^\\infty r^{2d-1} ^{-4\\pi ^2r^2}r\\cdot \\omega _d + \\lim \\limits _{\\sigma \\rightarrow 0}\\left[\\frac{\\alpha }{2}(2\\pi )^d \\sigma ^2\\int _0^\\infty r^{2d-3}^{-4\\pi ^2r^2}r\\cdot \\omega _d+o(\\sigma ^2)\\right]\\\\&=\\lim \\limits _{\\sigma \\rightarrow 0} (2\\pi )^d \\int _0^\\infty r^{2d-1} ^{-4\\pi ^2r^2}r\\cdot \\omega _d\\\\&=\\frac{1}{2}(d-1)!", "(2\\pi )^{-d}\\frac{2\\pi ^{\\frac{d}{2}}}{\\Gamma \\left(\\frac{d}{2}\\right)}.$ Therefore the proposition holds." ], [ "Proof of Theorem ", "Given $=(_{1},\\ldots ,_{n})^ and $ =(y1,...,yn), let $=\\left(\\exp (-\\frac{{_j-_i}^2}{2\\sigma ^2})\\right)_{n\\times n}$ be an $n\\times n$ matrix.", "For sufficiently small $\\sigma $ , the matrix $$ is diagonally dominant, and hence invertible.", "So the linear system $^{(\\sigma )}=$ has a solution $^{(\\sigma )}=\\left(g^{(\\sigma )}_1,g^{(\\sigma )}_2,\\cdots ,g^{(\\sigma )}_n\\right)^.", "Let\\begin{equation*}\\phi _{\\sigma }()=\\sum _{i}g^{(\\sigma )}_{i}^{-2\\pi I^_i}\\psi _{\\sigma }(),\\end{equation*}where $ ()=(2)d2d-2222$ satisfying $ -1[]()=-222$.", "Thus\\begin{equation*}^{-1}[\\phi _{\\sigma }]()=\\sum _{i}g^{(\\sigma )}_{i}^{-1}[\\psi _{\\sigma }](-_{i})=\\sum _{i}g^{(\\sigma )}_{i}^{-\\frac{{-_{i}}^2}{2\\sigma ^2}}.\\end{equation*}In particular, for all $ i=1,2,,n$\\begin{equation*}^{-1}[\\phi _{\\sigma }](_i)=\\sum _{j}g^{(\\sigma )}_{j}^{-\\frac{{_i-_{j}}^2}{2\\sigma ^2}}=(^{(\\sigma )})_i=y_i.\\end{equation*}Therefore, $ ,$ for sufficiently small $ >0$.$ According to the above discussion, we can construct a sequence $\\lbrace \\phi _{\\frac{1}{m}}\\rbrace _{m=M}^\\infty \\subset _{,}$ , where $M$ is a sufficiently large positive integer to make the matrix $$ invertible.", "As Proposition REF shows, $\\lim _{m\\rightarrow +\\infty }{^{-1}[\\phi _{\\frac{1}{m}}]}_{H^{\\frac{\\alpha }{2}}}^2=0.$ Now, suppose that there exists a solution to the Problem REF , denoted as $\\phi ^*\\in _{,}$ .", "By definition, ${^{-1}[\\phi ^*]}_{H^{\\frac{\\alpha }{2}}}^2\\le \\min _{\\phi \\in _{,}} {^{-1}[\\phi ]}_{H^{\\frac{\\alpha }{2}}}^2\\le \\lim _{m\\rightarrow +\\infty }{^{-1}[\\phi _{\\frac{1}{m}}]}_{H^{\\frac{\\alpha }{2}}}^2=0.$ Therefore, $\\phi ^*()\\equiv 0$ and $_{}\\phi ^*=$ , which contradicts to the restrictive condition $_{}\\phi ^*=$ for the situation that $\\ne $ .", "The proof is completed." ], [ "Proof of Theorem ", "1.", "We introduce a distance for functions $\\phi ,\\psi \\in L^2(^d)$ : $(\\phi ,\\psi )={^{-1}[\\phi ]-^{-1}[\\psi ]}_{H^{\\frac{\\alpha }{2}}}.$ Under the topology induced by this distance, the closure of the admissible function class $_{,}$ reads as $\\overline{_{,}}:=\\overline{\\lbrace \\phi \\in L^1(^d)\\cap L^2(^d)\\mid _{}\\phi =\\rbrace }^{\\mathrm {dist}(\\cdot ,\\cdot )}.$ 2.", "We will consider an auxiliary minimization problem: to find $\\phi ^*$ such that $\\phi ^*\\in \\arg \\min _{\\phi \\in \\overline{_{,}}}{^{-1}[\\phi ]}_{H^{\\frac{\\alpha }{2}}}.$ Let $m:=\\inf _{\\phi \\in \\overline{_{,}}}{^{-1}[\\phi ]}_{H^{\\frac{\\alpha }{2}}}$ .", "According to the proof of Proposition REF and Theorem REF , for a small enough $\\sigma >0$ , the inverse Fourier transform of function $\\phi _{\\sigma }()=\\sum _{i}g^{(\\sigma )}_{i}^{-2\\pi I^_i}\\psi _{\\sigma }()$ has finite Sobolev norm ${^{-1}[\\phi _\\sigma ]}_{H^{\\frac{\\alpha }{2}}}<\\infty $ , where $\\psi _{\\sigma }()$ satisfies $^{-1}[\\psi _{\\sigma }]()=^{-\\frac{{}^2}{2\\sigma ^2}}$ , $=\\left(\\exp (-\\frac{{_j-_i}^2}{2\\sigma ^2})\\right)_{n\\times n}$ and $^{(\\sigma )}=\\left(g^{(\\sigma )}_1,g^{(\\sigma )}_2,\\cdots ,g^{(\\sigma )}_n\\right)^^{-1}$ .", "Thus $m<+\\infty $ .", "3.", "Choose a minimizing sequence $\\lbrace \\bar{\\phi }_k\\rbrace _{k=1}^\\infty \\subset \\overline{_{,}}$ such that $\\lim _{k\\rightarrow \\infty } {^{-1}[\\bar{\\phi }_k]}_{H^{\\frac{\\alpha }{2}}} =m.$ By definition of the closure, there exists a function $\\phi _k\\in _{,}$ for each $k$ such that ${^{-1}[\\bar{\\phi }_k]-^{-1}[\\phi _k]}_{H^{\\frac{\\alpha }{2}}}\\le \\frac{1}{k}.$ Therefore $\\lbrace \\phi _k\\rbrace _{k=1}^\\infty \\subset _{,}$ is also a minimizing sequence, i.e., $\\lim _{k\\rightarrow \\infty } {^{-1}[\\phi _k]}_{H^{\\frac{\\alpha }{2}}} =m.$ Then $\\lbrace ^{-1}[\\phi _k]\\rbrace _{k=1}^\\infty $ is bounded in the Sobolev space $H^{\\frac{\\alpha }{2}}(^d)$ .", "Hence there exist a weakly convergent subsequence $\\lbrace ^{-1}[\\phi _{n_k}]\\rbrace _{k=1}^\\infty $ and a function $^{-1}[\\phi ^*]\\in H^{\\frac{\\alpha }{2}}(^d)$ such that $^{-1}[\\phi _{n_k}]\\rightharpoonup ^{-1}[\\phi ^*] \\quad \\text{in } H^{\\frac{\\alpha }{2}}(^d)\\text{ as}\\ k\\rightarrow \\infty .$ Note that $m=\\inf _{\\phi \\in \\overline{_{,}}}{^{-1}[\\phi ]}_{H^{\\frac{\\alpha }{2}}}\\le {^{-1}[\\phi ^*]}_{H^{\\frac{\\alpha }{2}}}\\le \\liminf _{\\phi _{n_k}}{^{-1}[\\phi _{n_k}]}_{H^{\\frac{\\alpha }{2}}}=m,$ where we have used the lower semi-continuity of the Sobolev norm of $H^{\\frac{\\alpha }{2}}(^d)$ in the third inequality.", "Hence ${^{-1}[\\phi ^*]}_{H^{\\frac{\\alpha }{2}}}=m$ .", "4.", "We further establish the strong convergence that $^{-1}[\\phi _{n_k}]-^{-1}[\\phi ^*]\\rightarrow 0$ in $H^{\\frac{\\alpha }{2}}(^d)$ as $k\\rightarrow \\infty $ .", "In fact, since $^{-1}[\\phi _{n_k}]\\rightharpoonup ^{-1}[\\phi ^*] \\ \\text{in } H^{\\frac{\\alpha }{2}}(^d)\\text{ as}\\ k\\rightarrow \\infty $ and $\\lim _{k\\rightarrow \\infty }{^{-1}[\\phi _{n_k}]}_{H^{\\frac{\\alpha }{2}}}=m={^{-1}[\\phi ^*]}_{H^{\\frac{\\alpha }{2}}}$ , we have $&\\lim _{k\\rightarrow \\infty }{^{-1}[\\phi _{n_k}]-^{-1}[\\phi ^*]}_{H^{\\frac{\\alpha }{2}}}^2=\\lim _{k\\rightarrow \\infty } \\langle ^{-1}[\\phi _{n_k}]-^{-1}[\\phi ^*],^{-1}[\\phi _{n_k}]-^{-1}[\\phi ^*]\\rangle \\\\&=\\lim _{k\\rightarrow \\infty } \\langle ^{-1}[\\phi _{n_k}],^{-1}[\\phi _{n_k}]\\rangle + \\langle ^{-1}[\\phi ^*],^{-1}[\\phi ^*]\\rangle - \\langle ^{-1}[\\phi _{n_k}],^{-1}[\\phi ^*]\\rangle - \\langle ^{-1}[\\phi ^*],^{-1}[\\phi _{n_k}]\\rangle \\\\&=m^2+m^2- \\lim _{k\\rightarrow \\infty }\\left(\\langle ^{-1}[\\phi _{n_k}],^{-1}[\\phi ^*]\\rangle + \\langle ^{-1}[\\phi ^*],^{-1}[\\phi _{n_k}]\\rangle \\right)\\\\&=m^2+m^2-\\langle ^{-1}[\\phi ^*],^{-1}[\\phi ^*]\\rangle -\\langle ^{-1}[\\phi ^*],^{-1}[\\phi ^*]\\rangle =0.$ Here $\\langle \\cdot ,\\cdot \\rangle $ is the inner product of the Hilbert space $H^{\\frac{\\alpha }{2}}$ .", "5.", "We have $\\phi ^*\\in L^1(^d)$ because $\\int _{^d}{\\phi ^*()}=\\int _{^d}\\frac{\\langle \\rangle ^{\\frac{\\alpha }{2}}{\\phi ^*()}}{\\langle \\rangle ^{\\frac{\\alpha }{2}}}\\le {^{-1}[\\phi ^*]}_{H^{\\frac{\\alpha }{2}}} \\left(\\int _{^d}\\frac{1}{\\langle \\rangle ^\\alpha }\\right)^{\\frac{1}{2}}=Cm<+\\infty ,$ where $C:=\\left(\\int _{^d}\\frac{1}{\\langle \\rangle ^\\alpha }\\right)^{\\frac{1}{2}}<+\\infty $ .", "Hence $\\phi ^*\\in L^1(^d)\\cap L^2(^d)$ and $_{}\\phi ^*$ is well-defined.", "6.", "Recall that $_{}\\phi _{n_k}=$ .", "We have ${-_{}\\phi ^*}&=\\lim _{k\\rightarrow +\\infty }{_{}\\phi _{n_k}-_{}\\phi ^*}\\\\&=\\lim _{k\\rightarrow +\\infty }{\\int _{^d}(\\phi _{n_k}-\\phi ^*)^{2\\pi I}}\\\\&=\\lim _{k\\rightarrow +\\infty }{\\int _{^d}\\frac{\\langle \\rangle ^{\\frac{\\alpha }{2}}(\\phi _{n_k}-\\phi ^*)}{\\langle \\rangle ^{\\frac{\\alpha }{2}}}^{2\\pi I} }\\\\&\\le \\lim _{k\\rightarrow +\\infty }{^{-1}[\\phi _{n_k}]-^{-1}[\\phi ^*]}_{H^{\\frac{\\alpha }{2}}} \\left(\\int _{^d}\\frac{{^{2\\pi I}}^2}{\\langle \\rangle ^\\alpha }\\right)^{\\frac{1}{2}}\\\\&=C\\lim _{k\\rightarrow +\\infty }{^{-1}[\\phi _{n_k}]-^{-1}[\\phi ^*]}_{H^{\\frac{\\alpha }{2}}}=0.$ Hence $_{}\\phi ^*=$ and $\\phi ^*\\in _{,}$ .", "7.", "Note that $m=\\inf _{\\phi \\in \\overline{_{,}}}{^{-1}[\\phi ]}_{H^{\\frac{\\alpha }{2}}}\\le \\inf _{\\phi \\in _{,}}{^{-1}[\\phi ]}_{H^{\\frac{\\alpha }{2}}}\\le {^{-1}[\\phi ^*]}_{H^{\\frac{\\alpha }{2}}}=m.$ This implies that $\\inf _{\\phi \\in _{,}}{^{-1}[\\phi ]}_{H^{\\frac{\\alpha }{2}}}=m$ and $\\phi ^*\\in \\arg \\min _{\\phi \\in _{,}}{^{-1}[\\phi ]}_{H^{\\frac{\\alpha }{2}}}$ , which completes the proof." ], [ "Special Case: One Data Point in One Dimension", "To simplify the problem, we start with a single point $X=0\\in $ with the label $Y=2$ .", "Denote $\\phi _j = \\phi (\\xi _j)$ for $j\\in $ .", "We also assume that the function $\\phi $ is an even function.", "Then according to the definition of $_{}$ , we have the following problem: Example 2 (Problem REF with a particular discretization) $& \\min _{\\phi \\in ^M} \\sum _{j=1}^M(1+{j}^2\\Delta \\xi ^2)^{\\frac{\\alpha }{2}}{\\phi _j}^{2}, \\\\& \\mathrm {s.t.", "}\\quad \\sum _{j=1}^M\\phi _j\\Delta \\xi = 1,$ where we further assume $\\phi _0 = \\phi (0) = 0$ .", "If we denote $= {(\\phi _1, \\phi _2, \\ldots , \\phi _M)}^{, b = \\frac{1}{\\Delta \\xi }, = (1, 1, \\ldots , 1)\\in ^M and\\begin{equation*}= \\sqrt{\\lambda }\\begin{pmatrix}(1+1^2\\Delta \\xi ^2)^{\\frac{\\alpha }{4}} & & & \\\\& (1+2^2\\Delta \\xi ^2)^{\\frac{\\alpha }{4}} & & \\\\& & \\ddots & \\\\& & & (1+M^2\\Delta \\xi ^2)^{\\frac{\\alpha }{4}}\\end{pmatrix}.\\end{equation*}In fact this is a standard Tikhonov regularization~\\cite {1977Solutions} also known as ridge regression problem with the Lagrange multiplier \\lambda .", "The corresponding ridge regression problem is,\\begin{equation}\\min _{}{{- b}_2^2 + {}_2^2},\\end{equation}where we put \\lambda in the optimization term {}_2^2, instead of the constraint term {- b}_2^2.", "This problem admits an explicit and unique solution~\\cite {1977Solutions},\\begin{equation}= {(^{+ ^{)}^{-1}^{ b.", "}Here we need to point out that the above method is also applicable to the case that the matrix is not diagonal.", "}Back to our problem, in order to obtain the explicit expression for the optimal we need the following relation between the solution of the ridge regression and the singular-value decomposition (SVD).", "}By denoting \\tilde{} = and\\begin{equation*}\\tilde{} = ^{-1}= \\frac{1}{\\sqrt{\\lambda }}\\left( (1+1^2\\Delta \\xi ^2)^{\\frac{\\alpha }{4}}, (1+2^2\\Delta \\xi ^2)^{\\frac{\\alpha }{4}}, \\ldots , (1+M^2\\Delta \\xi ^2)^{\\frac{\\alpha }{4}} \\right),\\end{equation*}where is the diagonal matrix, the optimal solution~(\\ref {ridge_solution}) can be written as\\begin{equation*}= {(^{)}^{-1}{\\left( \\tilde{}^{\\tilde{} +}^{-1}^{-1}^{ b= {(^{)}^{-1}{\\left( \\tilde{}^{\\tilde{} +}^{-1}\\tilde{}^{ b= {(^{)}^{-1}\\tilde{},}where \\tilde{}={\\left( \\tilde{}^{\\tilde{} +}^{-1}\\tilde{}^{ b is the solution of ridge regression with \\tilde{} and \\tilde{}.", "In order to obtain the explicit expression for \\tilde{} we need the following relation between the solution of the ridge regression and the singular-value decomposition (SVD).", "}\\begin{lem} If \\tilde{}= , then this least-squares solution can be solved using SVD.", "Given the singular value decomposition\\begin{equation*}\\tilde{}= ^{,}with singular values \\sigma _i, the Tikhonov regularized solution can be expressed aspects\\begin{equation*}\\tilde{}= ^{ b,}where has diagonal values\\begin{equation*}D_{ii} = \\frac{\\sigma _i}{\\sigma _i^2 + 1},\\end{equation*}and is zero elsewhere.\\end{equation*}\\begin{proof} In fact, \\tilde{}={(\\tilde{}^{\\tilde{}+ \\tilde{}^{\\tilde{})}^{-1}\\tilde{}^{ b=(^+1)^{-1}^^^b̰\\\\=^{ b, which completes the proof.", "}}}Since \\tilde{}\\tilde{}^{ = \\dfrac{1}{\\lambda }\\sum _{j = 1}^M (1+j^2\\Delta \\xi ^2)^{-\\frac{\\alpha }{2}}, we have \\tilde{} = U \\Sigma ^{ with\\begin{equation*}U = 1, \\quad \\Sigma = \\frac{1}{\\sqrt{\\lambda }}{\\left( \\sum _{j = 1}^M (1+j^2\\Delta \\xi ^2)^{-\\frac{\\alpha }{2}} \\right)}^{\\frac{1}{2}} := {Z}/{\\sqrt{\\lambda }},\\end{equation*}\\begin{equation*}= {\\left( (1+1^2\\Delta \\xi ^2)^{-\\frac{\\alpha }{2}}/Z, (1+2^2\\Delta \\xi ^2)^{-\\frac{\\alpha }{2}}/Z, \\ldots , (1+M^2\\Delta \\xi ^2)^{-\\frac{\\alpha }{2}}/Z \\right)}^{.", "}Then we get the diagonal value\\begin{equation*}D = \\frac{{Z}/{\\sqrt{\\lambda }}}{{Z^2}/{\\lambda } + 1}.\\end{equation*}Therefore, by Lemma ~\\ref {ridge_svd}\\begin{equation*}\\tilde{} = D U b = \\dfrac{{1}/{\\sqrt{\\lambda }}}{{Z^2}/{\\lambda } + 1}{\\left( (1+1^2\\Delta \\xi ^2)^{-\\frac{\\alpha }{2}},(1+2^2\\Delta \\xi ^2)^{-\\frac{\\alpha }{2}}, \\ldots , (1+M^2\\Delta \\xi ^2)^{-\\frac{\\alpha }{2}} \\right)}^{ b.", "}Finally, for the original optimal solution{\\begin{@align*}{1}{-1}= {(^{)}^{-1}\\tilde{} = \\frac{1}{(Z^2 + \\lambda )\\Delta \\xi }{\\left( (1+1^2\\Delta \\xi ^2)^{-\\frac{\\alpha }{2}}, (1+2^2\\Delta \\xi ^2)^{-\\frac{\\alpha }{2}}, \\ldots , (1+M^2\\Delta \\xi ^2)^{-\\frac{\\alpha }{2}} \\right)}^{,}}which means\\begin{equation*}\\phi _j = \\frac{(1+j^2\\Delta \\xi ^2)^{-\\frac{\\alpha }{2}}}{(Z^2 + \\lambda )\\Delta \\xi }.\\end{equation*}To derive the function in x space, say h(x) then{\\begin{@align}{1}{-1}h(x) & = \\frac{1}{(Z^2 + \\lambda )} \\sum _{j=-M}^{M} (1+j^2\\Delta \\xi ^2)^{-\\frac{\\alpha }{2}} ^{2\\pi Ij x} \\nonumber \\\\& = \\frac{2}{(Z^2 + \\lambda )} \\sum _{j=1}^{M} (1+j^2\\Delta \\xi ^2)^{-\\frac{\\alpha }{2}} \\cos (2\\pi j x).\\end{@align}}\\end{@align*}Fig.~\\ref {fig:hx_is_nontrivial} shows that for this special case with a large M, h(x) is not an trivial function in \\alpha >d case and degenerates to a trivial function in \\alpha <d case.", "}\\end{equation*}\\end{equation*}}}\\subsubsection {General Case: n Points in d Dimension}Assume that we have n data points _1,_2,\\ldots ,_n\\in ^d and each data point has d components:\\begin{equation*}_i=\\left(x_{i1},x_{i2},\\ldots ,x_{id}\\right)^{equation*}and denote the corresponding label as \\left(y_1,y_2,\\ldots ,y_n\\right)^.", "For the sake of simplicity, we denote the vector (j_1,j_2,\\cdots ,j_d)^ by _{j_1\\ldots j_d}.", "Then our problem becomes\\begin{exam}[Problem \\ref {prob..VariationalPointCloud} with general discretization]{\\begin{@align}{1}{-1}& \\min _{\\phi \\in ^{(2M)^d}} \\sum _{j_1,\\ldots ,j_d=-M}^M(1+{_{j_1\\ldots j_d}}^2\\Delta \\xi ^2)^{\\frac{\\alpha }{2}}{\\phi _{j_1\\ldots j_d}}^{2}, \\\\& \\mathrm {s.t.", "}\\quad \\sum _{j_1,\\ldots ,j_d=-M}^M\\phi _{j_1\\ldots j_d}^{2\\pi I\\Delta \\xi _{j_1\\ldots j_d}^_k}=y_k, \\ \\ k=1,2,\\ldots ,d\\end{@align}}\\end{exam}\\end{equation*}The calculation of this example can be completed by the method analogous to the one used in subsection~\\ref {case1}.", "Let\\begin{equation}_j=\\left(^{2\\pi I\\Delta \\xi _{-M-M\\ldots -M}^_j},\\ldots , ^{2\\pi I\\Delta \\xi _{j_1j_2\\ldots j_d}^_j},\\ldots ,^{2\\pi I\\Delta \\xi _{MM\\ldots M}^_j}\\right)^\\ j=1,2,\\ldots ,n,\\end{equation}}\\end{proof}\\begin{equation}=\\left(_1,_2,\\ldots ,_n \\right)^^{n\\times (2M)^d},\\quad =\\left(y_1,y_2,\\ldots ,y_n\\right)^^{n\\times 1},\\end{equation}\\begin{equation}=\\lambda \\begin{pmatrix}\\ddots & & \\\\& (1+{_{j_1j_2\\ldots j_d}}^2\\Delta \\xi ^2)^{\\frac{\\alpha }{4}} & \\\\& & \\ddots \\end{pmatrix}\\in ^{(2M)^d\\times (2M)^d}.\\end{equation}We just need to solve the following equation:\\begin{equation}= {(^{+ ^{)}^{-1}^{ b.", "}Then we can get the output function h(x) by using inverse Fourier transform:\\begin{equation}h() = \\sum _{j_1,\\ldots ,j_d=-M}^M\\phi _{j_1\\ldots j_d}^{2\\pi I\\Delta \\xi _{j_1\\ldots j_d}\\cdot }\\end{equation}Since the size of the matrix is too large, it is difficult to solve by an explicit calculation.", "Thus we choose special n, d and M and show that h(x) is not a trivial solution (non-zero function).", "}In our experiment, we set the hyper-parameter M,\\alpha ,\\lambda ,\\Delta \\xi in advance.", "We set \\lambda =0,5,\\Delta \\xi =0.1 in 1-dimensional case and \\lambda =0.2,\\Delta \\xi =0.1 in 2-dimensional case.", "We select two data points \\lbrace (-0.5,0.9),(0.5,0.9)\\rbrace as the given points in 1-dimensional case and four points as given points in 2-dimensional case whose second coordinates are 0.5 so that it is convenient to observe the phenomenon.", "At first, we use formula (\\ref {Aj}), (\\ref {A}) and (\\ref {Gamma}) to calculate matrix , and vector .", "Then from the equation (\\ref {phi}) we can deduce vector .", "The final output function h() is obtained by inverse discrete Fourier transform (\\ref {h}).", "}In Fig.\\ref {fig:diff_M}, we set \\alpha =10 in both cases to ensure \\alpha >d and change the band limit M. We observe that as M increases, the fitting curve converges to a non-trivial curve.", "In Fig.\\ref {fig:diff_alpha}, we set M=1000 in 1-dimensional case and M=100 in 2-dimensional case.", "By changing exponent \\alpha , we can see in all cases, the fitting curves are non-trivial when \\alpha >d, but degenerate when \\alpha <d.\\end{equation}\\end{equation*}\\end{lem}\\right.}}\\right.}}}\\right.", "}}\\end{equation*}\\end{equation}}$" ] ]
2105.11675
[ [ "On the fluid-structure interaction of a flexible cantilever cylinder at\n low Reynolds numbers" ], [ "Abstract We present a numerical study to investigate the fluid-structure interaction of a flexible circular cantilever cylinder in a uniform cross-flow.", "We employ a fully-coupled fluid-structure solver based on the three-dimensional Navier-Stokes equations and the Euler-Bernoulli beam theory.", "We examine the dynamics of the cylinder for a wide range of reduced velocities ($U^*$), mass ratios ($m^*$), and Reynolds numbers ($Re$).", "Of particular interest is to explore the possibility of flow-induced vibrations in a slender cantilever cylinder of aspect ratio $AR=100$ at laminar subcritical $Re$ regime (i.e., no periodic vortex shedding).", "We assess the extent to which such a flexible cylindrical beam can sustain flow-induced vibrations and characterize the contribution of the beam's flexibility to the stability of the wake at low $Re$.", "We show that when certain conditions are satisfied, the flexible cantilever cylinder undergoes sustained large-amplitude vibrations.", "The frequency of the oscillations is found to match the frequency of the periodic fluid forces for a particular range of system parameters.", "In this range, the frequency of the transverse vibrations is shown to match the first-mode natural frequency of the cylinder, indicating the existence of the lock-in phenomenon.", "The range of the lock-in regime is shown to have a strong dependence on $Re$ and $m^*$.", "We discover that unlike the steady wake behind a stationary rigid cylinder, the wake of a low mass ratio flexible cantilever cylinder could lose its stability in the lock-in regime at Reynolds numbers as low as $Re=22$.", "A combined VIV-galloping type instability is shown to be the possible cause of the wake instability at this $Re$ regime.", "These findings attempt to generalize our understanding of the flow-induced vibrations in flexible cantilever structures and can have a profound impact on the development of novel flow-measurement sensors." ], [ "Introduction", "Flow-induced vibrations (FIVs) have significant consequences and are essential to predict in numerous fields, such as marine/offshore, civil, biomedical, and aerospace engineering.", "Considerable research has been done in recent decades to characterize the underlying mechanism and explore the practical aspects of flow-induced vibrations in a wide range of domains, including vibration control [1], [2], [3], energy harvesting [4], [5], [6], [7], [8], and sensing [9], [10].", "The phenomenon of flow-induced vibration in bluff bodies has received special attention in the literature due to complex vortex dynamics and nonlinear physics involved in the interaction of a bluff body and fluid flow.", "In this regard, the flow-induced vibration of an elastically-mounted rigid cylinder has served as a prototypical model for both experimental and numerical studies [11].", "It has been shown that asymmetric vortex shedding from the wake of an elastically-mounted rigid cylinder exerts unsteady transverse loads that could lead to sustained large-amplitude vibrations called vortex-induced vibrations (VIVs) [12].", "VIVs are characterized by a frequency match between the frequency of the periodic vortex shedding and the vibration frequency of the cylinder [12], [13].", "When the natural frequency of the cylinder is close to the vortex shedding frequency, the VIV phenomenon results in a complex evolution of the shedding frequency, which deviates from the Strouhal frequency of its stationary counterpart.", "In this frequency regime, the vortex formation locks on to the natural frequency of the structure, which in turn creates a strong coupling between the cylinder and fluid flow [12].", "Several studies have shown that the peak vibration amplitude of an elastically-mounted rigid cylinder, with only one degree-of-freedom in the transverse direction, is approximately $O(D)$  [14], where $D$ denotes the cylinder diameter.", "The magnitude of the peak vibration amplitude is known to be a function of fluid and structural parameters, such as Reynolds number and mass-damping ratio [15], and has been shown to have a slightly higher value for a two-degree-of-freedom cylinder [16].", "Comprehensive reviews regarding the VIV of elastically-mounted rigid cylinders could be found in Refs.", "[14], [17], [18], [19], [15].", "More recently, studies have focused on the dynamic response of flexible slender structures at high Reynolds numbers [20], [21], [22] to give new physical insight into the phenomenon of vortex-induced vibrations.", "Due to the complex interaction of nonlinear wake dynamics with numerous flexible modes, the VIV modeling and prediction poses serious challenges for long flexible structures.", "For example, studies on thin risers have found that ocean currents excite several vibration modes and frequencies along the span of a riser during VIVs [23], [24], [25].", "In a short-term perspective, VIV effects can lead to drag amplification and large dynamic bending stresses.", "These large-amplitude vibrations lead to fatigue failure in slender structures and marine risers in the long term if not controlled properly [26].", "An Experimental study on the VIV of a flexible cantilever cylinder in the laminar flow regime has found some distinct differences between the dynamics of a flexible cantilever cylinder and a flexible riser in the VIV regime.", "A flexible pinned-pinned beam, such as a marine riser, has been shown to vibrate at monotonically increasing frequencies with each eigenmode gradually growing in modal weight as the reduced velocity is increased [27].", "However, for a flexible cantilever cylinder, although higher modes are observed at higher reduced velocities, the cylinder has been shown to oscillate with only one vibration eigenmode during VIVs [28].", "In line with the works done in the field of vortex-induced vibrations, in our current work, we examine the dynamic response of a flexible cantilever cylinder at low Reynolds numbers to give new insight into the topic of flow-induced vibrations in flexible slender structures.", "Our interest in studying the dynamics of a flexible cantilever cylinder at low Reynolds numbers originates from the intriguing problem of sensing through whiskers in some mammals, such as rats and seals.", "Experimental studies on the mechanical response of isolated rat vibrissae (whiskers) to low-speed airflow have revealed that air currents of magnitude $0.5$ to $5.6$ m/s, typically found in natural environments, generate significant vibrissal motion that carries information about the direction and magnitude of the airflow  [29], [30].", "More interestingly, behavioral experiments have shown that rats use the information from their whiskers to localize airflow sources [31].", "To characterize the geometry of a rat's tapered conical whisker, two parameters, namely arc length $S$ and base diameter $D_\\mathrm {b}$ , are defined.", "Typically, in a rat's mystacial pad, the ratio of a whisker's arc length $S$ to base diameter $D_\\mathrm {b}$ is between $100<S/D_\\mathrm {b}<400$ and the wind speeds translate into Reynolds numbers $<50$ , based on $D_\\mathrm {b}$ of the whisker [29].", "In addition to rats, harbor seals, for example, have been shown to use their highly sensitive undulated whiskers to sense hydrodynamic information of water flows and detect fluid structures without auditory or visual cues [32].", "An experimental study on a model of a seal whisker has shown that a whisker in the wake of a stationary rigid cylinder undergoes large-amplitude oscillations, with the frequency of the oscillations being close to the shedding frequency of the upstream wake [33].", "These oscillations are known to help seals detect upstream wakes and estimate the size and shape of the wake-generating body [33].", "Although considerable research has been done to understand the problem of sensing through whiskers, there have been limited studies on the mechanism behind the dynamics of whiskers in fluid flow.", "In particular, the oscillatory response of a whisker at laminar subcritical Reynolds numbers, i.e., $Re < Re_\\mathrm {cr}\\approx 45$  [34], [35], is not well understood.", "In our current work, we investigate fluid-structure interaction of a flexible cantilever cylinder, as a simplified model of a whisker, to help answer two specific questions: (i) can we observe sustained vibrations in the flexible cantilever cylinder at subcritical $Re$ with laminar wake flow, and (ii) what is the relationship between the cylinder dynamics and stability of the wake at this $Re$ regime?", "Understanding the underlying fluid-structure dynamics of a flexible cantilever cylinder, inspired by the dynamics of whiskers, is of vital importance for developing novel flow-measurement sensors [9] and brings us one step closer towards a complete mapping of the sensing properties of whiskers.", "The key non-dimensional parameters involved in the fluid-structure interaction of the flexible cantilever cylinder are mass ratio $m^\\mathrm {*}$ , Reynolds number $Re$ , and reduced velocity $U^\\mathrm {*}$ defined as m* = 4 mD2f,       Re = fU0Df,       U* = U0fnD, where $m$ is the mass per unit length of the cylinder, $D$ is the cylinder diameter, $\\rho ^\\mathrm {f}$ and $\\mu ^\\mathrm {f}$ are the density and dynamic viscosity of the fluid, respectively, $U_0$ is the magnitude of the uniform flow velocity, and $f_\\mathrm {n}$ is the natural frequency of the first mode of vibration.", "The non-dimensional parameters studied in our current work are within $20\\le Re\\le 40$ , $U^*\\in [2,19]$ and $1\\le m^*\\le 1000$ , which cover a practical range of values in the laminar subcritical $Re$ regime.", "A schematic of the flexible cantilever cylinder is given in Fig.", "REF .", "The cylinder is connected to a fixed support at $z=0$ .", "Figure: Schematic of a flexible cantilever cylinder of diameter DD and length LL interacting with a uniform flow of velocity U 0 U_0.The Young's modulus and second moment of area of the cylinder are denoted by $E$ and $I$ , respectively.", "As shown in Fig.", "REF , due to fluid forces acting on the cylinder, it initially deforms in the streamwise direction.", "Also, depending on the system parameters, the cylinder could exhibit an unsteady dynamic response, resulting in periodic vortex shedding patterns in the wake region.", "A systematic analysis of the dynamic response of the cylinder is conducted in our current work, where we use high-fidelity numerical simulations to examine the fluid-structure interaction of the cylinder for a broad range of system parameters.", "The content of the paper is structured as follows.", "The governing equations for modeling the cylinder dynamics and the coupling strategy between the fluid and structural solvers are discussed in Section .", "In addition, we provide the results for the grid convergence study at the end of this section.", "In section  we cover the results of our study and discuss the dynamic response characteristics of the cylinder in detail.", "Finally, we finish the paper with a conclusion in section ." ], [ " Numerical Methodology", "This section presents a three-dimensional numerical framework for studying the fluid-structure interaction of the flexible cantilever cylinder with an incompressible viscous flow.", "To model the coupled dynamics of the cylinder, we employ a three-dimensional computational domain as shown in Fig.", "REF .", "The cylinder is placed at an offset distance of $15D$ and $45D$ from the inflow and outflow surfaces, respectively.", "Fixed structural support is imposed at one end of the cylinder ($z=0$ ), and the no-slip boundary condition is applied at the fluid-structure interface $\\Gamma ^\\mathrm {fs}$ .", "The size of the computational domain is $60D\\times 30D\\times L$ where a uniform velocity of ${u}^\\mathrm {f} = (U_0,0,0)$ with a magnitude of $U_0$ in the x-direction is given at the inflow surface.", "The slip boundary condition is applied to the side surfaces $\\Gamma ^{\\mathrm {f}}_{side-1}$ and $\\Gamma ^{\\mathrm {f}}_{side-2}$ , and the traction-free boundary condition, given by $ {\\sigma }^{\\mathrm {f}}.", "{n}^\\mathrm {f} = 0$ , is specified at the outflow surface.", "Figure: Schematic of the computational domain with details of the boundary conditions.In the following, we discuss the governing equations for modeling the dynamics of the flexible cantilever cylinder and present the strategy implemented to couple the fluid and structural solvers." ], [ "Governing equations", "We consider the three-dimensional incompressible Navier-Stokes equations coupled with the Euler-Bernoulli beam theory to examine the coupled dynamics of the flexible cantilever cylinder.", "We formulate the governing equation for the Euler-Bernoulli beam in a Lagrangian reference frame and take a body-fitted moving boundary approach based on the arbitrary Lagrangian-Eulerian (ALE) description [36] to formulate the unsteady Navier-Stokes equations for the viscous incompressible fluid.", "The body-fitted treatment of the fluid-structure interface through the ALE description of the flow field provides accurate modeling of the boundary layer over the deformable surface of the structure.", "The unsteady Navier-Stokes equations for a viscous incompressible fluid flow in an arbitrary Lagrangian-Eulerian reference frame on the fluid domain $\\Omega ^\\mathrm {f}(t)$ are fuft|xf + f(uf - um)uf = f + bf on f(t), uf = 0 on f(t), where ${u}^\\mathrm {f} = {u}^\\mathrm {f}({x}^\\mathrm {f},t)$ and ${u}^\\mathrm {m}={u}^\\mathrm {m}({x}^\\mathrm {f},t)$ denote the fluid and mesh velocities defined for each spatial point ${x}^\\mathrm {f} \\in \\Omega ^\\mathrm {f}(t)$ respectively, ${b}^\\mathrm {f}$ is the body force applied to the fluid and ${\\sigma }^\\mathrm {f}$ is the Cauchy stress tensor for a Newtonian fluid, given as f = -pI + f( uf + (uf)T), where $p$ denotes the fluid pressure, and $\\mu ^\\mathrm {f}$ is the dynamic viscosity of the fluid.", "The first term in Eq.", "(REF ) represents the partial derivative of ${u}^\\mathrm {f}$ with respect to time while the ALE referential coordinate $\\hat{x}^\\mathrm {f}$ is kept fixed.", "The fluid forcing acting on the beam's surface is calculated by integrating the surface traction at the first layer of the elements located on the fluid-structure interface.", "The instantaneous coefficients of lift and drag forces are quantified as CL = 112fU02DLfs (fn)ny d, CD = 112fU02DLfs (fn)nx d, where ${n}_\\mathrm {x}$ and ${n}_\\mathrm {y}$ are the Cartesian components of the unit outward normal vector ${n}$ .", "In the next section, we present the equation for modeling the structural dynamics of the flexible cantilever cylinder using the Euler-Bernoulli beam theory." ], [ "Euler-Bernoulli beam theory for a flexible structure", "We consider the flexible cantilever cylinder as a slender structure with relatively small lateral motions.", "Therefore, the Euler Bernoulli beam theory can be applied to model its dynamic response.", "Let $\\Omega ^\\mathrm {s}$ be the structural domain consisting of structure coordinates ${x}^\\mathrm {s} = (x,y,z)$ .", "We solve the transverse displacements ${w}^\\mathrm {s}(z,t)$ using the Euler-Bernoulli beam equation excited by the distributed unsteady fluid force per unit length ${f}^\\mathrm {s}$ .", "The motion of the flexible cantilever cylinder is governed by the fluid forces and involves integrating pressure and shear stress effects on the cylinder surface.", "Neglecting the damping and shear effects, we take the equation of motion for the flexible cantilever cylinder as: m2 ws(z,t)t2 + EI4 ws(z,t)z4 = fs(z,t), where $m=\\rho ^\\mathrm {s} A$ is the mass per unit length of the cylinder, with $A$ being the cross-sectional area of the cylinder.", "Under the cantilever (clamped-free) configuration, the boundary conditions at the clamped end of the cylinder are given as: ws(z,t)|z=0 = 0,      ws(z,t)z|z=0 = 0.", "To solve Eq.", "(REF ), we consider a mode superposition approach for the dynamic response of the cylinder.", "The $\\mathrm {n}^\\mathrm {th}$ mode natural frequency of the flexible cantilever cylinder is given by fn = n22L2 EIm+ma , where $\\mathrm {n}$ is the mode number, $m_\\mathrm {a}$ is the added mass of the fluid per unit length defined as $m_\\mathrm {a} = \\pi D^{2}\\rho ^\\mathrm {f}/4$ and $\\lambda _\\mathrm {n}$ is the dimensionless frequency parameter for the $\\mathrm {n}^\\mathrm {th}$ mode of vibration.", "The $\\lambda _\\mathrm {n}$ values are given in Table REF .", "The modal parameters are based on the values reported in Ref.", "[37] for flexible cantilever beams of constant cross section.", "Table: Modal parameters for flexible cantilever beams of constant cross section .The cylinder motion is solved using simple linear vibration analysis.", "The displacements from the mean position of the cylinder are assumed to be small and characterized based on the normal modes of the vibration found using an eigenvalue analysis.", "The mode shapes of the cantilever cylinder are taken as the sums of sine, cosine, sinh, and cosh functions of $\\lambda _\\mathrm {n} z/L$ written as: $ S^\\mathrm {n}\\left( z \\right) &=& \\cosh \\left( \\frac{\\lambda _\\mathrm {n}z}{L} \\right) - \\cos \\left( \\frac{\\lambda _\\mathrm {n} z}{L} \\right) - \\sigma _\\mathrm {n} \\sinh \\left( \\frac{\\lambda _\\mathrm {n} z}{L} \\right) \\nonumber \\\\& &+ \\sigma _\\mathrm {n} \\sin \\left( \\frac{\\lambda _\\mathrm {n} z}{L} \\right),$ where $S^\\mathrm {n}$ denotes the mode shape associated with the $\\mathrm {n}^\\mathrm {th}$ mode of vibration and $\\sigma _\\mathrm {n}$ is the non-dimensional parameter dependent on the mode number (see Table REF for $\\sigma _\\mathrm {n}$ values)." ], [ "Treatment of the fluid-structure interface", "We need to satisfy the continuity of velocity and traction at the fluid-structure interface.", "Let $\\Gamma ^\\mathrm {fs} = \\partial \\Omega ^\\mathrm {f}(0) \\cap \\partial \\Omega ^\\mathrm {s}$ be the fluid-structure interface at $t=0$ and $\\Gamma ^\\mathrm {fs}(t) = {\\varphi }^\\mathrm {s}(\\Gamma ^\\mathrm {fs},t)$ be the interface at time $t$ .", "The required conditions to be satisfied are as follows: uf(s(xs0,t),t) = us(xs0,t), s(,t) f(xf,t)n d(xf) + ts d= 0, where ${\\varphi }^\\mathrm {s}$ denotes the position vector that maps the initial position ${x}^\\mathrm {s}_0$ of the flexible cantilever cylinder to its position at time $t$ , i.e., ${\\varphi }^\\mathrm {s}({x}^\\mathrm {s},t) = {x}^\\mathrm {s}_0 + {w}^\\mathrm {s}({x}^\\mathrm {s},t)$ , ${t}^\\mathrm {s}$ is the fluid traction vector relating to the fluid forcing as ${f}^\\mathrm {s}(z,t) =\\int _{\\Gamma ^\\mathrm {fs}} {t}^\\mathrm {s} \\mathrm {d}\\Gamma $ , and ${u}^\\mathrm {s}$ is the velocity of the structure at time $t$ given by ${u}^\\mathrm {s} = \\partial {\\varphi }^\\mathrm {s}/\\partial t$ .", "Here, ${n}$ is the outer normal to the fluid-structure interface, $\\gamma $ is any part of the interface $\\Gamma ^\\mathrm {fs}$ in the reference configuration, $\\mathrm {d\\Gamma }$ is the differential surface area and ${\\varphi }^\\mathrm {s}(\\gamma ,t)$ is the corresponding fluid part at time $t$ .", "The above conditions are satisfied such that the fluid velocity is exactly equal to the velocity of the structure at the fluid-structure interface.", "To couple the fluid and structure equations, we use a nonlinear partitioned iterative approach based on the nonlinear iterative force correction (NIFC) scheme described in [38], [39].", "At each time step, the fluid traction applied to the surface of the cylinder is projected onto the eigenvectors to find the values of the generalized modal forces.", "The projected modal forces are then used to determine the modal amplitudes and displacements for the next time step.", "To account for the changes in the cylinder geometry, we explicitly control the motion of each mesh node while satisfying the kinematic consistency of the discretized interface.", "The movement of the internal finite element nodes is chosen such that the mesh quality does not deteriorate as the displacements of the cylinder become large.", "For this purpose, we assume the fluid mesh to represent a hyperelastic solid model.", "In addition, a standard Lagrangian finite element technique is used to adapt the mesh to the new geometry of the domain." ], [ "Grid convergence study", "The coupled dynamics of the flexible cantilever cylinder is examined through a numerical framework that has been verified and validated extensively for FSI problems in an earlier study [24].", "Thus, we proceed with the grid convergence study here.", "We discretize the computational domain into unstructured hexahedral finite element grids with a boundary layer mesh around the flexible cantilever cylinder.", "We start with a relatively coarse grid denoted by M1 and successively increase the number of elements by approximately a factor of 2 to achieve the M2 and M3 meshes.", "An isometric view of the discretized domain and a $z$ -plane slice of the unstructured grid for the M2 mesh is given in Fig.", "REF .", "For the grid convergence study, we have examined the dynamics of the cylinder at $Re = 40$ , $m^\\mathrm {*} = 1$ , and $U^\\mathrm {*} = 11$ .", "Grid convergence errors are calculated by taking the finest mesh, M3, as the reference case.", "The values of the frequency ratio ($f_\\mathrm {y}/f_\\mathrm {n}$ ), mean streamwise deformation ($\\overline{A_\\mathrm {x}}/D$ ), root-mean-square (rms) of the dimensionless transverse vibration amplitude ($A_\\mathrm {y}^{rms}/D$ ), and the force coefficients ($\\overline{C_\\mathrm {D}}$ and $C_\\mathrm {L}^{rms}$ ) are given in Table REF .", "According to Table REF , the relative errors using the M2 mesh are less than $2\\%$ ; therefore, the M2 mesh is chosen as the suitable grid for our present study.", "The results mentioned in Table REF are for a computational domain with 16 layers in the spanwise direction.", "After doing an independent grid convergence study on the number of spanwise layers, ranging from 8 to 64, we found that 16 layers are adequate to capture the essential three-dimensional features of the fluid-structure system.", "Figure: Computational finite element grid for the M2 mesh: (a) isometric view of the discretized computational domain; (b) representative z-plane slice of the unstructured grid with a closeup view of the boundary layer mesh.Table: Grid convergence study results for the flexible cantilever cylinder interacting with a uniform cross-flow at Re=40Re = 40, m * =1m^\\mathrm {*} = 1, and U * =11U^\\mathrm {*} = 11." ], [ " Results and discussion", "In this section, we present our results for the fluid-structure interaction of the flexible cantilever cylinder for $20\\le Re\\le 40$ , $U^*\\in [2,19]$ and $1\\le m^*\\le 1000$ .", "In addition, we discuss the wake dynamics for the range of studied parameters.", "Finally, we relate our findings to real-world observations regarding the oscillatory motion of whiskers in laminar fluid flow." ], [ " Response characteristics", "We first present the response characteristics of the flexible cantilever cylinder at $m^*=1$ for $20\\le Re\\le 40$ and $U^*\\in [2,19]$ .", "The root-mean-square (rms) values of the dimensionless transverse vibration amplitude $A_\\mathrm {y}^{rms}/D$ at the tip of the cylinder ($z/L=1$ ) is given in Fig.", "REF .", "We find that at $Re=20$ , the cylinder remains in its steady deflected position, i.e., $A_\\mathrm {y}^{rms} = 0$ , within the range of studied $U^*$ .", "This steady response is also observed at $Re=22$ for $U^*\\le 6$ and at $24 \\le Re\\le 40$ for $U^*\\le 5$ (see Fig.", "REF ).", "However, there is a particular range of $U^*$ within which the cylinder is shown to undergo sustained vibrations for $22\\le Re\\le 40$ .", "We observe that the peak of the transverse vibration amplitude in this range is within $U^*\\in [7,8]$ .", "As shown in Fig.", "REF , at $Re=22$ , the peak of the $A_\\mathrm {y}^{rms}/D$ is at $U^*=8$ with a magnitude of approximately $0.18$ .", "However, at higher $Re$ , the peak of the vibration amplitude shifts to $U^*=7$ , where the oscillations are shown to grow in magnitude as $Re$ is increased.", "For instance, the peak of the $A_\\mathrm {y}^{rms}/D$ has a magnitude of approximately $0.26$ at $Re=24$ , whereas at $Re=40$ , the maximum $A_\\mathrm {y}^{rms}/D$ is approximately equal to $0.49$ .", "We find that at $Re=40$ , the cylinder experiences sustained oscillations for reduced velocities between $U^*\\in [6,19]$ ; however, for lower $Re$ , the oscillations are present for a narrower range of $U^*$ .", "A more broadband oscillatory response with respect to $U^*$ at higher $Re$ is due to larger inertial fluid forces that overcome the viscous damping.", "Figure: Root-mean-square (rms) value of the dimensionless transverse vibration amplitude A y rms /DA_\\mathrm {y}^{rms}/D at z/L=1z/L=1 as a function of U * U^* at m * =1m^*=1 for 20≤Re≤4020\\le Re \\le 40.Fig.", "REF Ay-Cl demonstrates the time histories of the transverse vibration amplitude calculated from the mean deformed position of the cylinder $(A_\\mathrm {y}-\\overline{A_\\mathrm {y}})/{D}$ at $z/L=1$ and lift coefficient $C_\\mathrm {L}$ at $Re = 40$ , $m^* = 1$ , and $U^* = 7$ .", "We observe that the transverse vibrations at the tip of the cylinder are in-phase with the variations of the lift coefficient.", "In addition, we show that the peak of the dimensionless transverse vibration frequency ($f_{\\mathrm {y}}/f_\\mathrm {n}$ ) matches the peak of the dimensionless lift coefficient frequency ($f_{\\mathrm {C_L}}/f_\\mathrm {n}$ ) in the frequency domain at $f_{\\mathrm {y}}/f_\\mathrm {n}=f_{\\mathrm {C_L}}/f_\\mathrm {n}=1$ (see Fig.", "REF FFT-CLAy).", "This frequency match indicates that the lock-in phenomenon is driving the oscillations at $U^*=7$ .", "Figure: (a) Variations of the dimensionless transverse vibration amplitude calculated from the mean deformed position of the cylinder (A y -A y ¯)/D(A_\\mathrm {y}-\\overline{A_\\mathrm {y}})/{D}, probed at z/L=1z/L=1, and lift coefficient C L C_\\mathrm {L} in time domain; (b) power spectra of the (A y -A y ¯)/D(A_\\mathrm {y}-\\overline{A_\\mathrm {y}})/{D} and C L C_\\mathrm {L} in frequency domain.", "The results are gathered in the time window tU 0 /D∈[200,300]tU_{0}/D\\in [200, 300] at Re=40Re = 40, m * =1m^* = 1, and U * =7U^* = 7.To specify the range of the lock-in regime, we have provided the variations of the dimensionless transverse vibration frequency $f_{\\mathrm {y}}/f_\\mathrm {n}$ and lift coefficient frequency $f_{\\mathrm {C_L}}/f_\\mathrm {n}$ at $m^*=1$ for $Re=22, 30$ and 40 with respect to $U^*$ in Fig.", "REF .", "We show that at $Re=22$ , $f_{\\mathrm {y}}/f_\\mathrm {n}$ and $f_{\\mathrm {C_L}}/f_\\mathrm {n}$ are close to unity for $U^*\\in [7,9]$ and zero elsewhere.", "However, for $Re=30$ and 40, the lock-in regime is found to extend to a broader range of reduced velocities within $U^*\\in [6,13]$ .", "Figure: Variations of the dimensionless transverse vibration frequency f y /f n f_\\mathrm {y}/f_\\mathrm {n}, probed at z/L=1z/L=1, and lift coefficient frequency f C L/f n f_\\mathrm {C_L}/f_\\mathrm {n} with respect to U * U^*.", "The results are gathered at m * =1m^*=1 for Re=22,30Re = 22,30 and 40.An isometric view of the cylinder undergoing large-amplitude oscillations in the lock-in regime at $Re = 40$ , $m^* = 1$ , and $U^* = 7$ is illustrated in Fig.", "REF  trajectory.", "A figure-eight type motion trajectory is observed across the cylinder length.", "These trajectories are shown to grow in magnitude by moving towards the tip of the cylinder.", "The scalograms of the dynamic response of the cylinder in the streamwise and transverse directions are given in Figs.", "REF  Dx-scalogram and Dy-scalogram, respectively.", "We show that the cylinder exhibits a standing wave response, with oscillations being in the first mode of vibration in both the streamwise and transverse directions.", "Based on the scalograms of the cylinder response, the dimensionless frequency of the streamwise oscillations ($f_{\\mathrm {x}}/f_\\mathrm {n}$ ) is found to be twice that of the dimensionless frequency of the transverse vibrations ($f_{\\mathrm {y}}/f_\\mathrm {n}$ ), i.e., $f_{\\mathrm {x}}/f_\\mathrm {n}=2f_{\\mathrm {y}}/f_\\mathrm {n}~\\approx 2$ .", "This type of oscillatory response has been previously observed in two-degrees-of-freedom elastically mounted rigid cylinders undergoing VIVs [40].", "Figure: (a) Motion trajectory of the flexible cantilever cylinder (illustrated by black lines); the red filled dots represent the mean position of the cylinder nodes and the red line corresponds to the cylinder's steady deflected position.", "(b) Scalogram of the vibrations in the streamwise direction.", "(c) Scalogram of the vibrations in the transverse direction.", "The results are gathered at Re=40Re = 40, m * =1m^* = 1, and U * =7U^* = 7 in the time window tU 0 /D∈[200,300]tU_{0}/D\\in [200, 300].The motion trajectory of the tip of the cylinder at $Re = 40$ , and $m^* = 1$ is given in Fig.", "REF for $U^*\\in [6, 11]$ .", "The figure-eight shape of the motion trajectories is associated with the frequency ratio of $f_{\\mathrm {x}}/f_\\mathrm {y}\\approx 2$ in the lock-in regime.", "Figure: Motion trajectory of the flexible cantilever cylinder at z/L=1z/L=1, Re=40Re=40, and m * =1m^*=1 for U * ∈[6,11]U^*\\in [6, 11]." ], [ "Wake dynamics during lock-in", "Here, we examine the wake dynamics in the lock-in regime for the flexible cantilever cylinder at laminar subcritical $Re$ .", "A comparison between the wake of a stationary rigid cylinder at $Re=40$ , and the wake of the flexible cantilever cylinder at $z/L=0.5$ , $Re=40$ , $m^*=1$ , and $U^* = 7$ is given in Fig.", "REF .", "Figure: Comparison of the z-vorticity contour for the (a) stationary rigid cylinder and (b) flexible cantilever cylinder at z/L=0.5z/L=0.5, Re=40Re=40, m * =1m^*=1, U * =7U^* = 7, and tU 0 /D=200tU_{0}/D=200.We show that the wake of the stationary rigid cylinder is steady and symmetric with respect to the wake centerline at $Re = 40$ ; however, for the flexible cantilever cylinder, the wake is unstable at the same $Re$ .", "To illustrate, we have examined the z-vorticity ($\\omega _\\mathrm {z}$ ) contours at different cross-sections of the flexible cantilever cylinder.", "Figure: Isometric view of the spanwise z-vorticity contours at various cross-sections of the flexible cantilever cylinder, with z-plane slices of the contours shown in the right-hand side for z/L=1z/L=1, 0.50.5, and 0.As shown in Fig.", "REF , the wake of the cylinder is steady at $z/L=0$ where it is connected to fixed support; however, by approaching the tip of the cylinder, the flow starts to become unstable, and periodic vortex-shedding patterns are observed downstream.", "This finding suggests a connection between the cylinder motion and wake stability at laminar subcritical $Re$ .", "To examine this conjecture, we have provided the z-vorticity iso-surfaces of the three-dimensional wake structures at $Re=30$ and $m^*=1$ in Fig.", "REF for $U^*=3,6$ and 15.", "Figure: Wake structures visualized by the normalized z-vorticity iso-surfaces (ω z D/U 0 =-0.224,0.224\\omega _{z}D/U_{0} = -0.224,0.224) for the flexible cantilever cylinder at Re=30Re=30, and m * =1m^*=1.", "Red [blue] indicates regions of positive [negative] vortices.At the given $m^*$ and $Re$ , $U^*=3,6$ and 15 represent the pre-lock-in, lock-in, and post-lock-in regimes, respectively.", "We find that the flow field in the wake of the flexible cantilever cylinder is steady at $U^*=3$ (pre lock-in) and $U^*=15$ (post lock-in); however, an unsteady wake is observed at $U^*=6$ (see Fig.", "REF ).", "The phase diagram of the wake stability as a function of $Re$ and $U^*$ at $m^*=1$ is given in Fig.", "REF .", "Figure: Phase diagram of the wake stability as a function of ReRe and U * U^* at m * =1m^* = 1.", "Here, ∘\\circ denotes a steady wake, while ** represents an unsteady wake behind the flexible cantilever cylinder.We find that at $Re=20$ , the flow field is steady for all $U^*$ values; however, as $Re$ is increased, the wake becomes unsteady for a particular range of reduced velocities.", "As shown in Fig.", "REF , the flow field in the wake of the flexible cantilever cylinder is unstable at $Re=22$ for $U^*\\in [7,9]$ .", "The range of the wake unsteadiness is shown to become wider at higher $Re$ .", "For example, this range is between $U^*\\in [6,13]$ at $Re=30$ and increases to $U^*\\in [6,19]$ at $Re=40$ .", "An important point to note here is that there is a critical $U^*\\in [6,7]$ that marks the initiation of the wake unsteadiness for $22\\le Re\\le 40$ .", "This critical $U^*$ also marks the lower bound of the lock-in regime, as shown in the results of Section REF .", "Thus, we can infer that the range of the wake unsteadiness is closely correlated with the range of the lock-in regime at laminar subcritical $Re$ .", "We have provided the wake structures around the flexible cantilever cylinder at $Re=40$ and $m^*=1$ for $5\\le U^*\\le 14$ in Fig.", "REF .", "Figure: Wake structures visualized by the normalized z-vorticity iso-surfaces (ω z D/U 0 =-0.224,0.224\\omega _{z}D/U_{0} = -0.224,0.224) for the flexible cantilever cylinder at Re=40Re=40, and m * =1m^*=1.", "Red [blue] indicates regions of positive [negative] vortices.At $U^*=5$ , which represents the pre-lock-in regime, a steady wake flow is observed behind the cylinder; however, for $U^*\\ge 6$ , the wake is shown to become unstable, with two alternate vortices being shed from the cylinder wake in each cycle.", "In addition, the wake structures close to the fixed end of the cylinder are found to be steady, regardless of $U^*$ .", "As shown in Fig.", "REF , although an unsteady wake is observed for $U^*\\ge 6$ at distances between $z/L\\in [0.2,1]$ from the fixed end of the cylinder, the wake is shown to be steady for $z/L\\in [0,0.2]$ .", "Thus, we can deduce that the three-dimensional flow phenomena do not contribute to the wake stability at this $Re$ regime.", "For the flow around an isolated cylinder, the wake has been shown to first become three-dimensional at $Re\\approx 200$  [41].", "With these findings, we can deduce that the wake of a flexible cantilever cylinder could become unsteady at laminar subcritical $Re$ , provided that two essential requirements are met: (i) the flow needs to have sufficiently large inertia to overcome the viscous damping and (ii) the system parameters need to be in the lock-in range to sustain the unsteadiness in the wake.", "In the next section, we discuss the relationship between the cylinder motion and stability of the wake in detail." ], [ "Relationship between the cylinder dynamics and wake unsteadiness", "Here, we pinpoint the relationship between the cylinder motion and stability of the wake at laminar subcritical $Re$ .", "We recommend a combined VIV-galloping type instability as the possible cause of the wake unsteadiness for $Re<Re_\\mathrm {cr}$ .", "Galloping is a velocity-dependent and damping-controlled fluid-structure instability, which is generally observed in geometrically asymmetric structures [42].", "Although the flow field around an asymmetric structure is uniform in magnitude and direction, cross-flow oscillations of the asymmetric body alter the magnitude and direction of the incident flow with respect to the body coordinate system.", "This change, in turn, alters the fluid forces acting on the body and could trigger the galloping instability.", "A deviation from symmetric cross-section in transmission lines due to ice formation [43] or in marine cables due to marine organisms [44] are some examples of the galloping instability in engineering structures.", "Galloping is known to cause large-amplitude sustained oscillations in flexible or elastically-mounted structures [42].", "In contrast to vortex-induced vibrations, galloping instability is induced by a relative body motion rather than the unsteady fluctuations of the flow field; hence it can occur even for steady attached flows.", "When the transverse force acting on a flexible or elastically-mounted body increases in the direction of motion, it adds movement to the body, and the body will displace further until the opposing stiffness or damping overcomes the movements, or the transverse force decreases when the movement is increased.", "For a flexible cantilever cylinder interacting with fluid flow, the body is free to deform in the streamwise and transverse directions.", "Although displacements in the streamwise direction do not contribute to the stability of the wake [45], [46], relative movements in the transverse direction break the wake symmetry, altering the fluid forces acting on the cylinder.", "This symmetry breakdown, in turn, induces a galloping-type instability by creating negative damping in the combined fluid-structure system.", "The low-speed galloping-type instability, together with the frequency lock-in, is most arguably the mechanism that leads to sustained unsteadiness in the wake at laminar subcritical $Re$ .", "To better understand the relationship between the cylinder motion and stability of the wake at laminar subcritical $Re$ , we have provided the z-vorticity contours at the mid-section of the cylinder at $Re=40$ , $m^*=1$ , and $U^*=7$ in Fig.", "REF .", "We show that the wake region behind the cylinder is steady and symmetric at $tU_{0}/D=60$ ; however, for $tU_{0}/D\\in [65,75]$ , relative motion of the cylinder cross-section in the transverse direction, makes the wake lose its stability and become asymmetric.", "This symmetry breakdown, in turn, exerts a transverse load that further increases the cylinder motion.", "Finally, due to the coupling between the unsteady wake and the cylinder movements, large-amplitude transverse vibrations are observed for $tU_{0}/D\\in [80,85]$ .", "Figure: Contours of z-vorticity at the cross section of the cylinder at z/L=0.5z/L=0.5, Re=40Re=40, m * =1m^*=1, and U * =7U^* = 7 in the time window tU 0 /D∈[60,85]tU_{0}/D\\in [60, 85].In the next section, we investigate the effect of mass ratio $m^*$ on the dynamics of the flexible cantilever cylinder and further examine the wake structures in the lock-in regime." ], [ "Effect of mass ratio", "We first investigate the effect of mass ratio on the dynamic response of the flexible cantilever cylinder at $Re=40$ for $U^*\\in [2, 19]$ .", "We examine the response of the cylinder at four different mass ratios, namely $m^* = 1, 10, 100$ and 1000.", "The results for the rms value of the dimensionless transverse vibration amplitude $A_\\mathrm {y}^{rms}/D$ with respect to $U^*$ are given in Fig.", "REF .", "For all the studied mass ratios, we find that the cylinder stays at its steady deflected position for $U^*\\le 5$ .", "This steady response is present for the whole range of $U^*$ at $m^*=1000$ .", "However, a discrete change in the dynamic response of the cylinder is observed for higher $U^*$ values at $m^* = 1, 10$ and 100.", "We observe a sudden jump in the amplitude response of the cylinder at $U^*=6, 7$ and 8 at mass ratios $m^*=1, 10$ and 100, respectively.", "As shown in Fig.", "REF , the peak of the transverse vibration amplitude is at $U^*=7$ for $m^*=1$ and 10, and at $U^*=8$ for $m^*=100$ .", "The magnitude of the maximum $A_\\mathrm {y}^{rms}/D$ is shown to be approximately $0.49, 0.47$ and $0.39$ at $m^*=1, 10$ and 100, respectively.", "By further increasing the $U^*$ , a gradual decrease in the amplitude of the transverse vibrations is observed at $m^*=1$ ; however, for $m^*=10$ and 100, there is a sharp decrease in the amplitude of the transverse vibrations for $U^*>8$ .", "A steady response is observed for $U^*\\ge 11$ at $m^*=10$ and for $U^*\\ge 10$ at $m^*=100$ .", "Fig.", "REF shows the frequency response of the fluid-structure system in terms of the dimensionless transverse vibration frequency $f_\\mathrm {y}/f_\\mathrm {n}$ and the dimensionless lift coefficient frequency $f_\\mathrm {C_L}/f_\\mathrm {n}$ at $Re=40$ for $m^*=1,10$ and 100 with respect to $U^*$ .", "We show that for all three mass ratios, there is a frequency match between the frequency of the transverse vibrations $f_\\mathrm {y}$ , frequency of the lift coefficient $f_\\mathrm {C_L}$ , and the first-mode natural frequency of the cylinder $f_\\mathrm {n}$ for a specific range of $U^*$ .", "This range is within $U^*\\in [8,9]$ at $m^*=100$ and within $U^*\\in [7,10]$ at $m^*=10$ .", "At $m^*=1$ , the lock-in regime begins at $U^*=6$ , however, the beam is shown to oscillate in frequencies higher than its first-mode natural frequency at larger $U^*$ values.", "Figure: Root-mean-square value of the dimensionless transverse vibration amplitude A y rms /DA_\\mathrm {y}^{rms}/D at z/L=1z/L=1 as a function of U * U^* at Re=40Re = 40 for m * =1,10,100m^*=1, 10, 100 and 1000.Figure: Variations of the dimensionless transverse vibration frequency f y /f n f_\\mathrm {y}/f_\\mathrm {n}, probed at z/L=1z/L=1, and lift coefficient frequency f C L/f n f_\\mathrm {C_L}/f_\\mathrm {n} with respect to U * U^*.", "The results are gathered at Re=40Re=40 for m * =1,10m^* = 1, 10 and 100.Based on our findings, we observe that by increasing $m^*$ , the range of the lock-in regime becomes narrower.", "This behavior is because of stronger inertial coupling and added mass effects at lower mass ratios.", "It should be noted that for Reynolds numbers beyond $Re_\\mathrm {cr}\\approx 45$ , interactions between the unsteady wake and the cylinder motion could lead to sustained vibrations for mass ratios of $O(100-1000)$ , which are not examined in our current work.", "A qualitative representation of the cylinder motion trajectory at $z/L=1$ and $Re=40$ with respect to $U^*$ is given in Fig.", "REF for $m^*=1, 10,$ and 100.", "We show that as $m^*$ is increased, the motion trajectory of the cylinder in the lock-in regime shifts from a figure-eight type response at $m^*=1$ to a dominated motion in the transverse direction at $m^*=100$ .", "To examine the wake structures in the lock-in regime at different mass ratios, we have provided the z-vorticity iso-surfaces around the cylinder at $Re=40$ and $U^*=8$ for $m^*=1,10$ and 100 in Fig.", "REF .", "We observe that for all three $m^*$ , two alternate vortices are shed from the cylinder wake in each cycle.", "This finding suggests that at a fixed $Re$ , the vortex shedding patterns in the wake of the flexible cantilever cylinder are independent of the mass ratio $m^*$ .", "Figure: A representation of the motion trajectory of the flexible cantilever cylinder with respect to U * U^* at z/L=1z/L=1 and Re=40Re=40 for m * =1,10m^*=1, 10 and 100.", "The filled dot (.)", "represents a steady response.Figure: Wake structures visualized by the normalized z-vorticity iso-surfaces (ω z D/U 0 =-0.224,0.224\\omega _{z}D/U_{0} = -0.224,0.224) for the flexible cantilever cylinder at Re=40Re=40, and U * =8U^*=8.", "Red [blue] indicates regions of positive [negative] vortices.Finally, we connect our findings to the dynamic response of rat and seal whiskers in fluid flow.", "The presented results in this section for $m^*=1$ and 100 are qualitative representatives of a seal and a rat whisker in fluid flow, respectively.", "As mentioned in Section , the interaction of a rat's whisker with low-speed airflow occurs at $Re<50$ .", "Based on our results for the flexible cantilever cylinder, a rat's whisker at laminar subcritical $Re$ is expected to experience a VIV-galloping type instability in the lock-in regime.", "In addition, flutter instability at reduced velocities of $O(100)$ could appear in a rat's whisker, which requires further investigations.", "For the case of a seal whisker in water flow, we characterize the oscillations as a VIV-dominant mechanism that occurs due to interactions between the whisker and unsteady wake at $Re\\approx 1000$ .", "Our results for the dynamic response of the flexible cantilever cylinder at $m^*=1$ could be used to interpret the response of a seal whisker in low-speed laminar water flows.", "Based on the available data in the literature regarding the amplitude response of an elastically mounted rigid cylinder at laminar $Re$  [15], we can deduce that the amplitude response of the flexible cantilever cylinder at $Re\\approx 1000$ would be slightly higher than the values presented in our current work.", "In addition, the vibration amplitude of a seal whisker is predicted to be significantly lower than the vibration amplitude of the flexible cantilever cylinder under similar conditions.", "Based on the experiments by Refs [33], [47], lower amplitudes in a seal whisker, compared to the flexible cantilever cylinder, are due to the seal whisker's undulated geometry that helps reduce the fluid forces during VIVs.", "In this paper, we have investigated the fluid-structure interaction of a flexible cantilever cylinder at laminar subcritical $Re$ .", "Through numerical simulations, we assessed the dynamic response of the cylinder as a function of reduced velocity $U^*$ , for Reynolds numbers between $20\\le Re\\le 40$ and mass ratios between $1\\le m^*\\le 1000$ .", "We found that for $Re=20$ , the flexible cantilever cylinder remains in its steady deflected position for the whole range of studied $U^*$ and $m^*$ .", "However, for $22\\le Re\\le 40$ , we observed that the cylinder experiences sustained oscillations when certain conditions are satisfied.", "We showed that the frequency of the transverse vibrations matches the frequency of the periodic lift force during the oscillations.", "Also, these two frequencies were found to be approximately equal to the first-mode natural frequency of the cylinder for a particular range of $U^*$ .", "This specific range, known as the lock-in regime, was shown to strongly depend on the Reynolds number $Re$ and mass ratio $m^*$ ; we found that at laminar subcritical $Re$ , the range of the lock-in regime decreases by increasing $m^*$ , whereas this range was shown to increase by increasing $Re$ .", "Finally, we identified two requirements for the wake unsteadiness at laminar subcritical $Re$ : (i) the flow needs to have sufficiently large inertia to overcome the viscous damping, and (ii) the system parameters need to be in the lock-in range.", "When these two conditions are satisfied, the cylinder experiences a combined VIV-galloping type instability.", "This instability was shown to break the symmetry of the wake and lead to sustained large-amplitude vibrations and unsteadiness in the wake at laminar subcritical $Re$ .", "We anticipate that the presented systematic analysis can help improve our understanding of the lock-in mechanism in flexible cantilever structures.", "Further research is required towards a parametric investigation of the dynamic response of the cylinder at high reduced velocities of $O(100)$ , where potential flutter-type instabilities could be present.", "In addition, the effects of structural nonlinearities should be accounted from a practical viewpoint for different flow incidence with a broader range of $Re$ and $U^*$ to fully understand the dynamic instabilities of the coupled system.", "The authors would like to acknowledge the Natural Sciences and Engineering Research Council of Canada (NSERC) for the funding.", "This research was enabled in part through computational resources and services provided by WestGrid (https://westgrid.ca/), Compute Canada (https://computecanada.ca/), and the Advanced Research Computing facility at the University of British Columbia (https://arc.ubc.ca/)." ] ]
2105.11663
[ [ "Exact dimension of Furstenberg measures" ], [ "Abstract For a probability measure $\\mu$ on SL d (R), we consider the Furstenberg stationary measure on the space of flags.", "Under general non-degeneracy conditions, if $\\mu$ is discrete and if g log g d$\\mu$(g) < +$\\infty$, then the measure $\\nu$ is exact-dimensional." ], [ "Main results", "Let $\\mu $ be a probability measure on the group $G= SL_d(\\mathbb {R})$ of $d\\times d$ of matrices with determinant 1.", "We assume in all the paper that $\\mu $ is strongly irreducible (i.e.", "there is no finite union of subspaces of $\\mathbb {R}^d$ that is invariant by the natural linear action of $\\mu $ -a.e.", "matrix) and that the semi-group generated by the support of $\\mu $ is a Zariski dense subgroup in $SL_d(\\mathbb {R}).$ Then, let $\\mathcal {F}$ be the space of complete flags in $\\mathbb {R}^d$ , $ f \\in \\mathcal {F}\\iff f = U_0 \\subset U_1 \\subset \\ldots \\subset U_{d-1} \\subset U_d,$ where $ U_i$ is a vector space of dimension $i$ in $\\mathbb {R}^d, U_0 = \\lbrace 0\\rbrace , U_d = \\mathbb {R}^d.$ $SL_d(\\mathbb {R}) $ acts naturally on $\\mathcal {F}$ and (see e.g.", "[5], Proposition 4.7) there is a unique stationary probability measure $\\nu $ on $\\mathcal {F}$ (i.e.", "satisfying $\\int g_\\ast \\nu \\, d\\mu (g) = \\nu $ ).", "The measure $\\nu $ is called the Furstenberg measure on $\\mathcal {F}$ associated to $\\mu $ .", "We write $\\mathcal {M},$ or $ \\mathcal {M}(G),$ for the space of such probability measures on $ G$ such that $\\int _G \\log \\Vert g\\Vert \\, d\\mu (g) <+\\infty .", "$ Let $Q$ be a partition of $\\lbrace 0,1, \\ldots , d\\rbrace $ into intervals, $Q = \\lbrace q_0 = 0 < q_1 < \\ldots < q_k = d\\rbrace .$ The space of partial flags $\\mathcal {F}_Q$ is the space of increasing sequences of vector subspaces of $\\mathbb {R}^d$ , $ \\lbrace 0\\rbrace = U_0 \\subset U_1 \\subset \\ldots \\subset U_{k-1} \\subset U_k = \\mathbb {R}^d, $ with $ \\dim U_i = q_i $ for $i = 0, 1, \\ldots , k $ .", "The group $G$ acts naturally on $\\mathcal {F}_Q$ .", "A measure $ \\nu $ on $\\mathcal {F}_Q$ is called stationary if it satisfies $ \\int g_\\ast \\nu \\, d\\mu (g) \\; = \\; \\nu .$ For $ \\mu \\in \\mathcal {M}$ , there is a unique stationary probability measure $\\nu _Q$ on $\\mathcal {F}_Q .$ We are interested in geometric properties of the measures $\\nu _Q$ and in particular in their dimension.", "The spaces $\\mathcal {F}_Q$ are endowed with the natural metric invariant under all orthogonal rotations.", "Let $(X,\\rho ) $ be a metric space, $ \\nu $ a measure on $X.$ The lower dimension $ \\underline{\\delta }$ and the upper dimension $ \\overline{\\delta }$ of $(X, \\rho , \\nu )$ are defined by $ \\underline{\\delta }\\, = \\, \\underset{\\nu }{{\\textrm {ess.inf}}} \\, \\liminf _{r\\rightarrow 0} \\frac{\\log \\nu (B(x,r))}{\\log r} , \\quad \\overline{\\delta }\\, = \\, \\underset{\\nu }{{\\textrm {ess.sup}}} \\,\\limsup _{r\\rightarrow 0} \\frac{\\log \\nu (B(x,r))}{\\log r}.$ A measure $\\nu $ on $X$ is called exact-dimensional with dimension $\\delta $ if $\\underline{\\delta }= \\overline{\\delta }= \\delta .$ If the space $(X, \\rho ) $ is bilipschitz equivalent to an Euclidean $ \\mathbb {R}^n, n\\ge 1,$ and $\\nu $ is exact-dimensional of dimension $\\delta $ , then $\\delta $ is the smallest Hausdorff dimension of sets of positive $\\nu $ -measure (see e.g.", "[55], Prop 2.1).", "Our main result is Theorem 1.1 Let $\\mu \\in \\mathcal {M}$ be a discrete probability measure on $G:= SL_d(\\mathbb {R})$ .", "Let $Q$ be a partition of $\\lbrace 0,1, \\ldots , d\\rbrace $ into intervals.", "Then the unique stationary probability measure $\\nu _Q$ on the space $\\mathcal {F}_Q$ of partial flags is exact-dimensional.", "In particular, if $Q_1$ is the partition into singletons, then $\\mathcal {F}_{Q_1}$ is the space $\\mathcal {F}$ of complete flags and the Furstenberg measure $\\nu $ is exact-dimensional.", "In the case when $d = 2,$ $\\mathcal {F}$ is the space of lines in $\\mathbb {R}^2$ , theorem REF holds for a general measure $ \\mu \\in \\mathcal {M}$ (Hochman and Solomyak [29]).", "Theorem REF has been recently proven by A. Rapaport ([47]) for the partition $ Q = \\lbrace 1\\rbrace , \\lbrace 2, \\ldots , d\\rbrace $ in the case when the measure $\\mu $ is finitely supported.", "The proof of [47] extends to the partition $ Q = \\lbrace 1, \\ldots , d-1 \\rbrace , \\lbrace d\\rbrace $ .", "Question 1.2 Is theorem REF true for a general probability measure $ \\mu \\in \\mathcal {M}$ ?", "Let $ \\mu \\in \\mathcal {M}$ .", "Then, there are $d$ distinct Lyapunov exponents (they are distinct by [25] and [22]) $ \\chi _1 > \\chi _2 >\\ldots > \\chi _d, \\; {\\textrm {with }} \\chi _1 + \\ldots + \\chi _d = 0 ,$ such that for $f \\in \\mathcal {F}, f= \\lbrace \\lbrace 0\\rbrace \\subset U_1(f) \\subset \\ldots \\subset U_{d-1}(f) \\subset \\mathbb {R}^d \\rbrace ,$ $ j = 1\\ldots , d-1, $ $ \\sum _{i\\le j} \\chi _i \\; =\\; \\int \\log |\\text{det}_{U_j (f)} (g) | \\, d\\mu (g) d\\nu (f) ,$ where, for any subspace $U$ in $\\mathbb {R}^d$ , $|\\text{det}_U (g)|$ is the Jacobian of the linear mapping from $U$ to $gU$ , both endowed with the Euclidean metric.", "Let $Q$ be a partition of $\\lbrace 0,1, \\ldots , d\\rbrace $ into intervals; we define the Furstenberg entropy as the nonnegative number $ h(\\mathcal {F}_Q, \\mu , \\nu _Q ) $ given by $ h(\\mathcal {F}_Q, \\mu , \\nu _Q ) := \\int _G \\int _{\\mathcal {F}_Q} \\log \\frac{dg_\\ast \\nu _Q}{d\\nu _Q} (x)\\, \\frac{dg_\\ast \\nu _Q}{d\\nu _Q} (x) \\, d\\nu _Q (x) d\\mu (g) ,$ with the usual convention that $ 0 \\log 0 = 0.$ Remark 1.3 It follows from theorem REF below and from its proof, that for $\\mu \\in \\mathcal {M}, \\, \\mu $ -a.e.", "$g \\in G$ , the measure $ g_\\ast \\nu $ is absolutely continuous with respect to the measure $\\nu $ (see Corollary REF ).", "Let $\\chi _1> \\ldots > \\chi _d$ be the Lyapunov exponents of $(G, \\mu )$ and for $ i = 1, \\ldots d,$ set $\\ell _Q(i) $ for the index such that $ q_{\\ell _Q(i) -1} < i \\le q_{\\ell _Q(i)}.$ Then, Theorem 1.4 With the above notations, for any partition $Q$ of $\\lbrace 0,1, \\ldots , d\\rbrace $ into intervals, we have $ h(\\mathcal {F}_Q, \\mu , \\nu _Q ) \\; \\le \\; \\sum _{i,j: \\ell _Q(i) < \\ell _Q(j) } \\chi _i - \\chi _j .$ If there is equality in (REF ), then the measure $ \\nu _Q $ is exact dimensional with dimension $ \\dim \\mathcal {F}_Q$ .", "The inequality (REF ) is proven in Section .", "See the discussion after theorem REF for the equality case.", "In the case when $d = 2,$ $\\chi _1 >0 >\\chi _2 = - \\chi _1,$ and the dimension $\\delta $ is given by $ \\delta = \\frac{h(\\mathcal {F}, \\mu , \\nu )}{\\chi _1-\\chi _2} $ ([29]).", "In particular, theorem REF holds in that case (see [38] for a direct proof).", "A similar formula holds in many cases where the action of a group on its boundary is conformal (see e.g.", "[53] for, among other cases, the one of a discrete group of isometries of the hyperbolic space $\\mathbb {H}^n $ ).", "In the nonconformal cases he considers, [47] also gives a relation between the dimension, the exponents and partial entropies and relation (REF ) follows.", "Clearly, the upper dimension of $\\nu _Q $ is at most $\\dim \\mathcal {F}_Q; $ the other natural estimate from above of the dimension comes from the estimate of the Furstenberg entropy by the random walk entropy, as follows.", "Assume the distribution of $\\mu $ has finite Shannon entropy $ H(\\mu )$ .", "Then the limit $ h_{{\\textrm {RW}}} (\\mu ) : = \\lim \\limits _{n \\rightarrow \\infty }\\frac{1}{n} H(\\mu ^{(n)}) $ exists.", "For any partition $Q$ of $\\lbrace 0, 1, \\ldots d\\rbrace $ into intervals, we have ([12] and see [34] for background on the random walk entropy $ h_{{\\textrm {RW}}} (\\mu ) $ and applications) $ h(\\mathcal {F}_Q, \\mu , \\nu _Q ) \\; \\le \\; h_{{\\textrm {RW}}} (\\mu ) .", "$ Assume furthermore that $ \\mu \\in \\mathcal {M}$ .", "The dimension in Theorem REF is given by a formula involving exponents and some partial entropies (see (REF )).", "Together with relation (REF ), this implies an a priori bound on the dimension that we describe now.", "Let $Q$ be as above and denote $ \\lbrace 0 < \\lambda _1 \\le \\lambda _2 \\le \\ldots \\le \\lambda _N \\rbrace $ the differences of exponents $ \\chi _i - \\chi _j $ for all $(i,j)$ such that $ \\ell _Q(i) < \\ell _Q(j).$ We then define the continuous, piecewise affine function $ D_{\\mathcal {F}_Q, \\mu } $ on the interval $ [ 0,N= \\dim \\mathcal {F}_Q]$ as: $ D_{\\mathcal {F}_Q, \\mu } (0) := h_{{\\textrm {RW}}} (\\mu )\\quad {\\textrm {and}}\\quad D^{\\prime }_{\\mathcal {F}_Q, \\mu } (s) = - \\lambda _t\\; {\\textrm {for}} \\; s \\in (t-1,t), \\; t = 1, \\ldots , N. $ Following Kaplan-Yorke ([35]) and Douady-Oesterlé ([13]), the Lyapunov dimension $ \\dim _{{\\textrm {LY}}}(\\mathcal {F}_Q, \\mu )$ is defined by $ \\dim _{{\\textrm {LY}}}(\\mathcal {F}_Q, \\mu ) := {\\left\\lbrace \\begin{array}{ll}N & \\text{if } D_{\\mathcal {F}_Q, \\mu } (N ) \\ge 0,\\\\\\delta \\; \\text{such that } D_{\\mathcal {F}_Q, \\mu } (\\delta ) =0 & \\text{otherwise}.", "\\end{array}\\right.", "}$ Theorem 1.5 Let $ \\mu \\in \\mathcal {M}$ be discrete and have finite entropy.", "Then, the exact dimension $\\delta _Q$ of $ \\nu _Q$ satisfies $ \\delta _Q \\; \\le \\; \\dim _{{\\textrm {LY}}}(\\mathcal {F}_Q, \\mu ).$ In case of equality in (REF ) with $ \\delta < N $ , we have $ h(\\mathcal {F}_Q, \\mu _Q, \\nu _Q) \\; = \\; h_{{\\textrm {RW}}} (\\mu ).$ We discuss the proof of theorem REF in Section REF .", "For any parttion $ Q$ , the equality $ h(\\mathcal {F}_Q, \\mu , \\nu _Q) \\; = \\; h_{{\\textrm {RW}}} (\\mu )$ is important since it means that the flag space $\\mathcal {F}_Q$ is the Poisson boundary of the random walk (see [34]).", "In the case of $ Q= \\lbrace 1\\rbrace , \\lbrace 2, \\ldots , d\\rbrace $ ,(REF ) is Corollary 1.7 in [47].", "As discussed in the introduction of [47], in view of the the results in [32], [7], [26], [27], [29], [6], [3], [28], [54], [8], we may ask the following Question 1.6 Is there a Diophantine-type condition on the support of $ \\mu $ that ensures equality in (REF )?", "We can prove relation (REF ) in some examples: in section , we discuss the terms and the proof of the following Theorem 1.7 Let $ \\rho $ be a Hitchin representation of a hyperbolic cocompact surface group $ \\Gamma $ in $PSL_d(\\mathbb {R})$ and $ \\mu $ be an adapted probability measure on $ \\Gamma $ such that $ \\sum _g | g| \\, d\\mu (g) < + \\infty ,$ where $ |\\cdot | $ is some word metric on $\\Gamma $ .", "Consider the random walk on $PSL_d(\\mathbb {R})$ directed by the probability $ \\rho _\\ast (\\mu ) $ and $ \\nu $ the stationary measure on the space $ \\mathcal {F}$ of flags.", "Then, $\\nu $ is exact-dimensional and $ \\delta (\\nu ) \\; = \\; \\dim _{{\\textrm {LY}}}(\\mathcal {F}, \\mu ) .$ The main new feature of our paper is already present for finitely supported measures in the case $ d=3.$ The space $\\mathcal {F}_{0<1<3}$ is the space $\\mathcal {L}$ of lines in $\\mathbb {R}^3$ , the space $\\mathcal {F}_{0<2<3}$ is the space $\\mathcal {P}$ of planes in $\\mathbb {R}^3$ .", "The stationary measure $\\nu $ on $\\mathcal {F}$ projects on the stationary measures $\\nu _\\mathcal {L}$ and $\\nu _\\mathcal {P}$ and the fibers of the projections are one-dimensional.", "We know by [47] that $\\nu _\\mathcal {L}$ and $\\nu _\\mathcal {P}$ are exact-dimensional.", "Moreover, we know by [41] that the conditional measures on the fibers are exact-dimensional and [41] has formulae for the almost everywhere constant dimensions.", "This is not enough information to be able to conclude that the measure $\\nu $ is exact-dimensional and to compute its dimension.", "Indeed, in the setting of Theorem REF in dimension 3, as soon as the Hitchin representation is not Fuchsian, there exists a probability measure $ \\mu _0 \\in \\mathcal {M}(\\Gamma ) $ for which the dimensions do not add up for the projection from $( \\mathcal {F}, \\nu _0 ) $ to $ \\mathcal {P}$ .", "This is the consequence of two properties: for a Hitchin representation of a cocompact surface group, if $ \\chi _2 >0 $ , then the projection from $( \\mathcal {F}, \\nu _0 ) $ to $ \\mathcal {P}$ fails to preserve dimension (see proposition REF ).", "On the other hand, we observe that if the Hitchin representation is not Fuchsian, then one can find $ \\mu _0 \\in \\mathcal {M}(\\Gamma ) $ such that $ \\chi _2 >0 $ , see theorem REF .", "In the next subsection, we recall this phenomenon of dimension conservation and explain what is the third codimension one projection of $\\mathcal {F}$ that we consider and for which we will prove dimension conservation.", "We then introduce the corresponding formalism in higher dimensions." ], [ "Dimension conservation", "Let $(X, \\nu ), (X^{\\prime }, \\nu ^{\\prime }) $ be standard probability spaces, $ \\pi :(X,\\nu ) \\rightarrow (X^{\\prime }, \\nu ^{\\prime }) $ a measure preserving mapping.", "Recall that a disintegration of the measure $\\nu $ with respect to $\\pi $ is a measurable family of probability measures $ x^{\\prime } \\mapsto \\nu ^{x^{\\prime }} $ ( or $ x^{\\prime } \\mapsto \\nu ^{x^{\\prime }}_{X^{\\prime }}, x^{\\prime } \\mapsto \\nu ^{x^{\\prime }}_\\pi $ ) of probability measures on $X$ such that $ \\nu ^{x^{\\prime }} \\pi ^{-1} (x^{\\prime }) = 1 \\quad {\\textrm {and }} \\quad \\nu = \\int _{X^{\\prime }} \\nu ^{x^{\\prime }} \\, d\\nu ^{\\prime } (x^{\\prime }).$ Two families of disintegrations of the measure $\\nu $ with respect to $\\pi $ coincide $ \\nu ^{\\prime }$ -a.e..", "Assume now that $(X, d ), (X^{\\prime }, d^{\\prime } ) $ are separable metric spaces and that $\\pi : (X,d)\\rightarrow (X^{\\prime },d^{\\prime })$ is a Lipschitz mapping.", "Let $\\nu $ be a probability measure on $X$ .", "We say that the projection $\\pi $ is dimension conserving for $\\nu $ if the measure $\\nu $ is exact-dimensional with dimension $\\delta $ , the measure $\\pi _\\ast \\nu $ is exact-dimensional with dimension $\\delta ^{\\prime } $ , for $\\pi _\\ast \\nu $ -a.e.", "$ x^{\\prime } \\in X^{\\prime },$ the disintegration $\\nu ^{x^{\\prime }} $ is exact-dimensional on $ \\pi ^{-1} (x^{\\prime }) $ with dimension $\\delta - \\delta ^{\\prime }$ .", "The definition is adapted from Furstenberg ([19], [20]).", "Dimension conservation occurs often in the presence of iterations or randomness.", "Classical examples are the results à la Marstrand and Mattila for projections of measures along almost every direction in $\\mathbb {R}^d$ ([31]), see [30] and the more recent [15] and [51] for surveys.", "On the other hand, it is easy to construct examples (for instance the graphs of the Brownian trajectories or the graph of the Weierstraß function (see [50])) where 1 and 2 hold, but the conditional measures are Dirac measures with dimension 0 whereas $ \\delta > \\delta ^{\\prime }.$ See also [46] for an example in a context close to ours.", "We will now describe, in dimension 3, a third projection defined on $\\mathcal {F}$ with one-dimensional fibers for which we will be able to prove dimension conservation.", "We say that two flags $f= \\lbrace 0\\rbrace \\subset U_1 \\subset U_2 \\subset \\mathbb {R}^3 $ and $f^{\\prime }= \\lbrace 0\\rbrace \\subset U^{\\prime }_1 \\subset U^{\\prime }_2 \\subset \\mathbb {R}^3 $ are in general position if $ U_1 \\oplus U^{\\prime }_2 = U_2 \\oplus U^{\\prime }_1 = \\mathbb {R}^3.$ Set $\\mathcal {F}^{(2)} $ for the set of pairs of flags in general position.", "The mapping $ F: \\mathcal {F}^{(2)} \\rightarrow \\mathcal {P}\\times \\mathcal {L}\\times \\mathcal {F}$ defined by $ F(f,f^{\\prime }) \\; := \\; ( U_1 \\oplus U^{\\prime }_1, U_2 \\cap U^{\\prime }_2, f^{\\prime } ) $ has one-dimensional fibers.", "Indeed, fixing $U_1$ in $U_1 \\oplus U^{\\prime }_1$ determines $ U_2 = U_1 \\oplus (U_2 \\cap U^{\\prime }_2) ;$ alternatively, choosing $U_2 \\supset U_2 \\cap U^{\\prime }_2 $ determines $ U_1 = (U_1 \\oplus U^{\\prime }_1) \\cap U_2.$ Let $\\mu ^{\\prime }$ be the image of $\\mu $ under the mapping $g \\mapsto g^{-1}.$ The measure $\\mu ^{\\prime }$ admits a unique stationary probability measure $\\nu ^{\\prime }$ for the action of $G$ on $\\mathcal {F}$ .", "From our results will follow that, for $\\nu ^{\\prime }$ -a.e.", "$f^{\\prime }$ , all projections in the following sequence are dimension conserving for $ \\nu \\times \\delta _{f^{\\prime }} $ $(f,f^{\\prime }) \\mapsto ( U_1 \\oplus U^{\\prime }_1, U_2 \\cap U^{\\prime }_2, f^{\\prime } ) \\mapsto ( U_1 \\oplus U^{\\prime }_1, f^{\\prime } ) \\mapsto f^{\\prime } &{\\textrm { if }}& \\chi _2 \\ge 0,\\\\(f,f^{\\prime }) \\mapsto ( U_1 \\oplus U^{\\prime }_1, U_2 \\cap U^{\\prime }_2, f^{\\prime } ) \\mapsto ( U_2 \\cap U^{\\prime }_2, f^{\\prime } ) \\mapsto f^{\\prime } &{\\textrm { if }}& \\chi _2 \\le 0.$ In order to prove theorem REF on $ \\mathcal {F}$ when $ d =3 $ , we are reduced to three projections with one dimensional fibers, for which we want to prove exact dimensionality of the conditional measures on the fibers and dimension conservation.", "Moreover, we have arranged so that the exponents of the dynamics on the fibers are nondecreasing: $\\chi _3 -\\chi _1 \\; < \\; \\chi _3 - \\chi _2 \\; \\le \\; \\chi _2 - \\chi _1 \\; <0 &{\\textrm { if }}& \\chi _2 \\ge 0,\\\\\\chi _3 -\\chi _1 \\; < \\; \\chi _2 - \\chi _1 \\; \\le \\; \\chi _3 - \\chi _2 \\; <0 &{\\textrm { if }}& \\chi _2 \\le 0.$ Therefore, we can apply the strategy of [16] and [47] (following [42], [17], [4]), and work one exponent at a time.", "We first generalize the above picture to higher dimensions." ], [ "Topologies, configuration spaces and entropy", "A topology $T$ on the set $\\lbrace 1,\\ldots , d\\rbrace $ will be called admissible if $\\lbrace i,i+1,\\ldots , d\\rbrace \\in T$ for all $i$ .", "Given an admissible topology $T$ we denote by $T(i)$ the atom of $i$ i.e.", "the smallest set in $T$ containing $i$ .", "Notice that any topology $T$ is determined by listing its atoms $T(1),T(2),\\ldots , T(d)$ .", "And $T$ is admissible if, and only if, $T(i) \\subset \\lbrace i,i+1,\\ldots , d\\rbrace $ for all $i$ .", "Recall that a topology $T$ is finer than another $T^{\\prime }$ (equivalently, $T^{\\prime }$ is coarser than $T$ ), denoted $T \\prec T^{\\prime }$ , if $T\\supset T^{\\prime }$ .", "The coarsest admissible topology $T_0$ is (defined by the list of atoms) $\\lbrace 1,\\ldots , d\\rbrace \\lbrace 2,\\ldots , d\\rbrace \\ldots \\lbrace d\\rbrace ,$ the finest admissible topology $T_1$ is $\\lbrace 1\\rbrace \\ldots \\lbrace d\\rbrace $ .", "Figure: Admissible topologies for d=3d = 3, an arrow indicates a topology one step coarser than another, filtered topologies are indicated in gray.We say an admissible topology $T$ is one step finer than an admissible topology $T^{\\prime }$ (equivalently $T^{\\prime }$ is one step coarser than $T$ ), denoted $T \\overset{1}{\\prec } T^{\\prime }$ , if there exists a unique $i \\in \\lbrace 1,\\ldots , d\\rbrace $ such that $T(i) \\ne T^{\\prime }(i)$ and furthermore $T^{\\prime }(i) \\setminus T(i)$ is a singleton.", "Let $j$ be so that $\\lbrace j \\rbrace = T^{\\prime }(i) \\setminus T(i)$ .", "Then, $j >i $ and $ T(i) \\setminus \\lbrace i\\rbrace \\subset T(i) \\subset T(i) \\cup \\lbrace j\\rbrace = T^{\\prime }(i).$ We associate to a pair $T \\overset{1}{\\prec } T^{\\prime }$ its exponent $\\chi _{T,T^{\\prime }} $ $ \\chi _{T,T^{\\prime }} \\; = \\; \\chi _i - \\chi _j .", "$ An admissible topology is called filtered if it is generated by the coarsest admissible topology and some subset of $\\left\\lbrace \\lbrace 1\\rbrace , \\lbrace 1,2\\rbrace , \\ldots , \\lbrace 1,2,\\ldots , d\\rbrace \\right\\rbrace $ .", "Let $Q$ be a partition of $\\lbrace 0,1, \\ldots , d\\rbrace $ into intervals, $Q = \\lbrace q_0 = 0 < q_1 < \\ldots < q_k = d\\rbrace .$ We associate to it the filtered topology $T_Q$ generated by $T_0$ and the sets $ \\lbrace 1,2,\\ldots , q_j\\rbrace , j = 1, \\ldots ,k.$ There are exactly $2^{d-1}$ filtered topologies and they are all obtained that way.", "The topologies $T_1$ and $T_0$ are filtered and correspond to respectively the space of complete flags and the one-point flag space of the trivial partition $Q_0 = \\lbrace 1,2,\\ldots , d\\rbrace $ .", "Figure: Admissible topologies for d=4d = 4, an arrow indicates a topology one step coarser than another, filtered topologies are indicated in gray.Figures 1 and 2 represent the graphs of the one-step relations between admissible topologies in dimensions 3 and 4 respectively.", "In dimension 3, the new fibration correspond to the nonfiltered topology that is one-step finer than $T_1$ .", "One sees the two ways of further descending on the graph according to the sign of $ \\chi _2$ .", "In dimension 4, there are 40 admissible topologies and 92 one-step arrows.", "The non-trivial filtered topologies correspond to the spaces of partial flags with only one level missing or to the Grassmannians of lines, planes or three-dimensional spaces.", "This correspondence is extended to all admissible topologies by constructing the configuration spaces as follows.", "Given an admissible topology $T$ we define the configuration space $\\mathcal {X}_T$ to be the space of sequences $x = (x_I)_{I \\in T}$ indexed on $T$ where $x_I$ is a $|I|$ -dimensional subspace of $\\mathbb {R}^d$ for each $I \\in T$ , $x_{I \\cup J} = x_I + x_J$ for all $I,J \\in T$ , and $x_{I \\cap J} = x_I \\cap x_J$ for all $I,J \\in T$ .", "The configuration space $\\mathcal {X}_{T_1} $ is identified with the space of $d$ independent lines in $\\mathbb {R}^d$ ; $\\mathcal {X}_{T_0}$ is the space $\\mathcal {F}$ of complete flags.", "For $Q$ be a partition of $\\lbrace 0,1, \\ldots , d\\rbrace $ into intervals, the configuration space $\\mathcal {X}_{T_Q} $ is identified with the space of partial flags $ \\mathcal {F}_Q$ .", "If $T$ is finer than $T^{\\prime }$ there is a natural projection mapping $\\pi _{T,T^{\\prime }}:\\mathcal {X}_T \\rightarrow \\mathcal {X}_{T^{\\prime }}$ .", "In particular, all configuration spaces project onto $\\mathcal {X}_{T_0}$ .", "We say that two flags $f= \\lbrace 0\\rbrace \\subset U_1 \\subset \\ldots \\subset \\mathbb {R}^d $ and $f^{\\prime }= \\lbrace 0\\rbrace \\subset U^{\\prime }_1 \\subset \\ldots \\subset \\mathbb {R}^d $ are in general position if for all $j, 0 <j<d$ , $ U_j \\oplus U^{\\prime }_{d-j} = \\mathbb {R}^d.$ Set $\\mathcal {F}^{(2)} $ for the set of pairs of flags in general position.", "Given an admissible topology $T$ , we associate to $ (f,f^{\\prime }) \\in \\mathcal {F}^{(2)} $ the configuration $F_T(f,f^{\\prime }) \\in \\mathcal {X}_T $ given, for $ I \\in T$ , by $ [F_T(f,f^{\\prime })]_I \\; = \\oplus _{i\\in I} \\left( U_i \\cap U^{\\prime }_{d-i+1} \\right).$ Observe that if $T$ is finer than $T^{\\prime }$ , then $ F_{T^{\\prime }} = \\pi _{T,T^{\\prime }} \\circ F_T.$ Set, for an admissible topology $T$ and a fixed $ f^{\\prime } \\in \\mathcal {F}, \\, \\mathcal {X}_T^{f^{\\prime }} $ for the set of $F_T(f,f^{\\prime })$ for all $f$ such that $(f,f^{\\prime }) \\in \\mathcal {F}^{(2)}.$ In particular, we identify $\\mathcal {X}_{T_0}^{f^{\\prime }} $ with $\\lbrace f^{\\prime }\\rbrace $ .", "Let $\\mu ^{\\prime }$ be the image of $\\mu $ under the mapping $g \\mapsto g^{-1}.$ The measure $\\mu ^{\\prime }$ admits a unique stationary probability measure $\\nu ^{\\prime }$ for the action of $G$ on $\\mathcal {F}$ .", "Endow $\\mathcal {F}^{(2)} $ with the measure $ \\nu \\otimes \\nu ^{\\prime } $ and $\\mathcal {X}_T $ with the measure $ (\\mathcal {F}_T)_\\ast (\\nu \\otimes \\nu ^{\\prime } ).", "$ Write $ f^{\\prime } \\mapsto \\nu _T^{f^{\\prime }} $ for the $\\nu ^{\\prime }$ -a.e.", "defined family of disintegrations of $ (\\mathcal {F}_T)_\\ast (\\nu \\otimes \\nu ^{\\prime } ) $ with respect to the projection on the second coordinate in $\\mathcal {F}^{(2)} .$ By definition, for $\\nu ^{\\prime }$ -a.e.", "$ f^{\\prime }$ , $ \\nu ^{f^{\\prime }}_T $ is a probability measure supported by $\\mathcal {X}^{f^{\\prime }}_T $ and so that $ (\\mathcal {F}_T)_\\ast (\\nu \\otimes \\nu ^{\\prime } )= \\int _\\mathcal {F}\\nu _T^{f^{\\prime }} \\, d\\nu ^{\\prime }(f^{\\prime }).$ We define the entropy $\\kappa _{T} $ by $ \\kappa _{T} \\; := \\;\\int \\log \\frac{ dg_\\ast \\nu _T^{g^{-1} f^{\\prime }} }{ d\\nu _T^{f^{\\prime }} }(y)\\, dg_\\ast \\nu _T^{g^{-1} f^{\\prime }}(y) d\\nu ^{\\prime }(f^{\\prime })d\\mu (g) .$ We will see in Section that this integral makes sense and can be seen as a conditional mutual entropy $ H(gF_T, F_T| f^{\\prime }) $ .", "In particular, $\\kappa _{T_0} =0 $ .", "If $T$ is filtered and associated to the partition $Q$ , then the mapping of $\\mathcal {X}_{T_Q}$ onto $\\mathcal {F}_Q$ is a bilipschitz homeomorphism when restricted to each fiber $ \\mathcal {X}_{T_Q}^{f^{\\prime }}$ , and identifies $\\nu _T^{f^{\\prime }}$ with $\\nu _Q$ .", "Therefore, $ \\kappa _T \\; = \\; \\int _{G \\times \\mathcal {F}_Q} \\log \\frac{dg_\\ast \\nu _Q}{d\\nu _Q} (x)\\, dg_\\ast \\nu _Q (x) d\\mu (g) \\; =\\; h (\\mathcal {F}_Q, \\mu , \\nu _Q).$ Assume that the admissible topology $T$ is finer than the admissible topology $T^{\\prime }$ .", "Then, clearly, for $\\nu ^{\\prime }$ -a.e.", "$ f^{\\prime } \\in \\mathcal {F}$ , $( \\pi _{T,T^{\\prime }} )_\\ast \\nu _T^{f^{\\prime }} \\; = \\; \\nu _{T^{\\prime }}^{f^{\\prime }} $ and we set $ (\\nu _{T,T^{\\prime }}^{x^{\\prime }} , x^{\\prime } \\in \\mathcal {X}_{T^{\\prime }}^{f^{\\prime }} )$ for a family of disintegrations of the measure $ \\nu _T^{f^{\\prime }} $ with respect to $ \\pi _{T,T^{\\prime }} .$ The entropy difference $\\kappa _{T,T^{\\prime }} \\; := \\; \\kappa _T -\\kappa _{T^{\\prime }} $ can be seen as a conditional mutual entropy $ H (gF_T, F_T|F_{T^{\\prime }}, f^{\\prime }) $ and can be expressed in terms of the measures $\\nu _{T,T^{\\prime }}^{x^{\\prime }} $ (see below section ).", "A key step in our proof is Theorem 1.8 Fix $ \\mu \\in \\mathcal {M}.$ Assume $T$ and $T^{\\prime }$ are admissible topologies, with $T$ one step finer than $T^{\\prime }$ .", "With the above notations, for almost every $ x^{\\prime } $ , the measure $ \\nu _{T,T^{\\prime }}^{x^{\\prime }} $ is exact-dimensional with dimension $ \\gamma _{T,T^{\\prime }} $ given by $ \\gamma _{T,T^{\\prime }} \\; = \\; \\frac{\\kappa _{T,T^{\\prime }}}{\\chi _{T,T^{\\prime }}} .", "$ The proof of theorem REF is given in section .", "In dimension $ d= 2$ , there is only one pair $ T_1, T_0 $ , the measure $ \\nu _{T_1,T_0}^{f^{\\prime }} $ is the constant measure $\\nu $ and theorem REF comes from [29].", "In higher dimensions, if $ T = T_1 $ and $T^{\\prime }$ has one atom of the form $\\lbrace i, i+1\\rbrace $ , then theorem REF is theorem 2 from [41].", "The general scheme of the proof is the same: we find a one-dimensional parameterization of the support of the measure $ \\nu _{T,T^{\\prime }}^{x^{\\prime }}$ that is adapted to the $G$ -action, and then use a telescoping argument and the Maker ergodic theorem.", "It follows from the one-dimensional parameterization that if $T \\overset{1}{\\prec } T^{\\prime },$ then $ \\gamma _{T,T^{\\prime }} \\le 1$ (see lemma REF below) and this leads to a general estimate of $\\kappa _{T,T^{\\prime }} $ in terms of the differences of exponents associated to $\\pi _{T,T^{\\prime }} $ (see proposition REF .2 below).", "Inequality (REF ) in theorem REF corresponds to the particular case of $ \\kappa _{T_Q,T_0}.$ We provide a direct proof of (REF ) first, since we use it to ensure that the entropy $ \\kappa _T$ defined by equation (REF ) is finite.", "Since for all $T \\overset{1}{\\prec } T^{\\prime }, \\gamma _{T,T^{\\prime }} \\le 1,$ if we have equality in (REF ), all corresponding $\\gamma _{T,T^{\\prime }} $ are 1 and the equality case in theorem REF follows from theorem REF .1.", "In the next section, we reduce the proofs of our results to entropy/dimension statements related to the fine structure of the configuration spaces.", "A more detailed organization of the remaining proofs is given at the end of the next section." ], [ "Proof of theorem ", "Observe that for $T$ and $T^{\\prime }$ admissible topologies, $ T\\prec T^{\\prime } $ if, and only if, for all $ i = 1,\\ldots , d-1, T(i) \\subset T^{\\prime }(i), $ where $T(i)$ is the atom of $T$ containing $\\lbrace i\\rbrace $ .", "We denote $D_{T,T^{\\prime }} $ the set of pairs $(i , j)$ such that $ j \\in T^{\\prime }(i) \\setminus T(i).", "$ The numbers $\\chi _i - \\chi _j, (i,j) \\in D_{T,T^{\\prime }}$ are called the exponents of the pair $ T \\prec T^{\\prime }$ .", "Observe that $\\lbrace i\\rbrace \\subset T(i), T^{\\prime }(i) \\subset \\lbrace i, i+1, \\ldots d\\rbrace .", "$ Therefore, for $(i,j ) \\in D_{T,T^{\\prime }}, \\; i<j $ and the exponents $ \\chi _i - \\chi _j , (i,j) \\in D_{T,T^{\\prime }}$ are positive.", "Set $ N_{T, T^{\\prime }} := \\# D_{T,T^{\\prime }} = \\sum _{i=1}^d \\# (T^{\\prime }(i) \\setminus T(i)).$ Proposition 2.1 Let $T \\prec T^{\\prime }$ be a pair of admissible topologies, $ N := N_{T,T^{\\prime }} .$ 1.", "Then, there exists a sequence $T^0 =T^{\\prime },T^1,\\ldots ,T^N = T$ such that $T^{t}$ is one step finer than $T^{t-1}$ for $t=1,\\ldots ,N$ , and $\\chi _{T^1,T^0} \\le \\chi _{T^2,T^1} \\le \\ldots \\le \\chi _{T^{N},T^{N-1}}.$ 2.", "Moreover, $ \\overline{\\delta }_{T,T^{\\prime }} \\le N_{T,T^{\\prime }} $ and $ \\kappa _{T,T^{\\prime }} \\le \\sum _{ (i,j) \\in D_{T,T^{\\prime }} } (\\chi _i - \\chi _j),$ where $ \\overline{\\delta }_{T,T^{\\prime }} $ is the essentially constant (in $x^{\\prime }$ ) value of the upper dimension of the measure $ \\nu _{T,T^{\\prime }}^{x^{\\prime }} .$ Proposition REF .1 is proven in Section REF .", "Proposition REF .2 is proven at the end of Section after a more precise description of the metric spaces $ (\\pi _{T,T^{\\prime }})^{-1} (x^{\\prime }) $ for $ x^{\\prime } \\in \\mathcal {X}_{T^{\\prime }} .$ For any pair $T \\prec T^{\\prime }$ of admissible topologies, we henceforth choose and fix a sequence given by proposition REF .1.", "The entropy $ \\kappa _{T,T^{\\prime }} $ is the sum of the entropy differences $ \\kappa _{T^t, T^{t-1}} .$ Theorem REF yields a Ledrappier-Young formula for the entropy $ \\kappa _{T,T^{\\prime }} $ $ \\kappa _{T,T^{\\prime }} \\; = \\; \\sum _{t=1}^{N_{T,T^{\\prime }}} \\kappa _{T^t, T^{t-1}} \\;= \\; \\sum _{t=1}^{N_{T,T^{\\prime }}} \\chi _{T^t,T^{t-1}} \\, \\gamma _{T^t, T^{t-1}}\\, .$ The precise form of our main result is the dimension counterpart of (REF ): Theorem 2.2 Let $ \\mu \\in \\mathcal {M}$ and $T \\prec T^{\\prime }$ be a pair of admissible topologies.", "With the previous notations, for $\\nu ^{\\prime }$ -a.e.", "$f^{\\prime }\\in \\mathcal {F}$ , $ \\nu _{T}^{f^{\\prime }} $ -a.e $x^{\\prime } \\in \\mathcal {X}_{T^{\\prime }}^{f^{\\prime }} $ , the lower dimension $ \\underline{\\delta }_{T , T^{\\prime }}$ of the measure $ \\nu _{T,T^{\\prime }}^{x^{\\prime }} $ is at least $\\sum \\limits _{t=1}^{N_{T,T^{\\prime }}} \\gamma _{T^t, T^{t-1}};$ moreover, in the case when $ \\mu $ is discrete, the measure $ \\nu _{T,T^{\\prime }}^{x^{\\prime }} $ is exact-dimensional, with dimension $\\delta _{T , T^{\\prime }}$ given by $ \\delta _{T ,T^{\\prime }}\\; = \\; \\sum _{t=1}^{N_{T,T^{\\prime }}} \\gamma _{T^t, T^{t-1}}\\,.$ In the case when $T^{\\prime } = T_0,$ we can identify the measures $ \\nu _{T^t}^{f^{\\prime }} $ and $ \\nu _{T^t,T_0}^{ f^{\\prime }} $ on $\\mathcal {X}^{f^{\\prime }}_{T^t}$ and we obtain, setting $ D_T := D_{T, T_0} = \\lbrace (i,j), i<j , j \\notin T(i)\\rbrace , N_T := \\# D_T$ and, for $ (i,j) \\in D_T, \\gamma ^T_{i,j} $ for $\\gamma _{T^t,T^{t-1}}$ associated by proposition REF .1 to $(i,j)$ .", "Corollary 2.3 Assume $ \\mu $ is a discrete measure in $\\mathcal {M}$ and let $T$ be an admissible topology.", "With the previous notations, we have $ \\kappa _T \\;= \\; \\sum _{(i,j) \\in D_T} \\gamma ^T_{i,j} (\\chi _i -\\chi _j) $ and, for $\\nu ^{\\prime }$ -a.e.", "$f^{\\prime }\\in \\mathcal {F}$ , the measure $ \\nu _{T}^{f^{\\prime }} $ is exact-dimensional, with dimension $\\delta _{T}$ given by $ \\delta _{T}\\; = \\; \\delta _{T,T_0} \\; =\\; \\sum _{(i,j) \\in D_T} \\gamma ^T_{i,j} .$ In particular, if $T$ is a filtered admissible topology associated to a partition $Q$ , then $\\mathcal {X}_T$ is identified with $\\mathcal {F}_Q$ , the measure $ \\nu _{T}^{f^{\\prime }} $ is then identified with the measure $\\nu _Q$ for almost every $f^{\\prime }$ .", "If, moreover, the measure $ \\mu $ is discrete, theorem REF follows: for a discrete $ \\mu \\in \\mathcal {M}$ , the Furstenberg stationary measure $\\nu _Q$ on the space $\\mathcal {F}_Q$ of $Q$ -flags is exact-dimensional with dimension $\\delta _Q.$ There are numbers $ \\gamma ^Q_{i,j} := \\gamma _{i,j}^{T_Q} $ such that Comparing with theorem REF , the content of (REF ) is that the numbers $ \\gamma ^Q_{i,j} $ are the dimensions of certain conditional measures on specific 1-dimensional leaves in $ \\mathcal {F}_Q$ .", "$ \\delta _Q= \\sum _{i,j: \\ell _Q(i) < \\ell _Q(j) } \\gamma ^Q_{i,j} , \\quad \\quad h(\\mathcal {F}_Q,\\mu , \\nu _Q) = \\sum _{i,j: \\ell _Q(i) < \\ell _Q(j) } \\gamma ^Q_{i,j} (\\chi _i - \\chi _j).", "$ We endow $ \\mathcal {X}^{f^{\\prime }}_{T} $ with a smooth metric.", "The following limits are constants for $\\nu ^{\\prime }$ -a.e.", "$f^{\\prime } \\in \\mathcal {F}$ , $\\nu ^{f^{\\prime }}_{T^t} $ -a.e.", "$x^{\\prime } \\in \\mathcal {X}^{f^{\\prime }}_{T^t} $ , $\\nu ^{x^{\\prime }}_{T,T^t} $ -a.e.", "$y^{\\prime } \\in \\mathcal {X}^{f^{\\prime }}_{T} $ : $ \\underline{\\delta }^t := \\liminf _{r\\rightarrow 0} \\frac{\\log \\nu _{T, T^t}^{x^{\\prime }}( B (y^{\\prime } ,r))}{\\log r}, \\;\\; \\overline{\\delta }^t := \\limsup _{r\\rightarrow 0} \\frac{\\log \\nu _{T, T^t}^{x^{\\prime }} ( B (y^{\\prime } ,r))}{\\log r}.", "$ With this notation, lower and upper dimensions of the measure $ \\nu _{T,T^{\\prime }}^{x^{\\prime }} $ are respectively $\\underline{\\delta }^0 $ and $\\overline{\\delta }^0.$ For the first part of theorem REF , we have that for almost every $(f^{\\prime }, x^{\\prime }, y^{\\prime } )$ , the following relation (REF ) holds for all $ t = N_{T,T^{\\prime }}, N_{T,T^{\\prime }} -1, \\ldots , 1 $ $ \\underline{\\delta }^{t-1} \\;\\ge \\; \\underline{\\delta }^{t} + \\gamma _{T^{t}, T^{t-1}}.", "$ Indeed, since $T \\overset{1}{\\prec } T^{N_{T,T^{\\prime }}-1}$ , (REF ) holds for $ t = N_{T,T^{\\prime }} $ (with $\\underline{\\delta }^{N_{T,T^{\\prime }} } = 0 $ ) by theorem REF .", "By [42] lemma 11.3.1, (REF ) for $ t < N_{T,T^{\\prime }} $ follows from theorem REF as well.", "Theorem REF .1 follows by summing the relations (REF ) for $ t = N_{T,T^{\\prime }}, N_{T,T^{\\prime }} -1, \\ldots , 1 .$ For the second part of theorem REF , it remains to prove Theorem 2.4 Assume that the measure $\\mu \\in \\mathcal {M}$ is discrete.", "With the above notations, for almost every $(f^{\\prime }, x^{\\prime }, y^{\\prime } )$ , for all $ t = N_{T,T^{\\prime }}, N_{T,T^{\\prime }} -1, \\ldots , 1, $ $ \\overline{\\delta }^{t-1} \\le \\overline{\\delta }^{t} + \\gamma _{T^{t}, T^{t-1}}.", "$ Summing the relations (REF ) for $ t = N_{T,T^{\\prime }}, N_{T,T^{\\prime }} -1, \\ldots , 1 $ (with $\\overline{\\delta }^{N_{T,T^{\\prime }} } = 0 $ ) gives $ \\overline{\\delta }_{T ,T^{\\prime }} = \\overline{\\delta }^0 \\le \\sum _{t=1}^{N_{T,T^{\\prime }}} \\gamma _{T^t, T^{t-1}}.$ Comparing with theorem REF .1 gives the result.", "We prove theorem REF in Section .", "The analog of theorem REF in [42] is a counting argument (see section (10.2)) that uses partitions with finite entropy in the underlying space.", "This is not possible here.", "By working on the trajectory space of the underlying process, [16] and [47] perform this counting procedure in the case when the measure $\\mu $ has finite support.", "We follow the same scheme under the hypothesis that $ \\mu $ is discrete." ], [ "Properties of the partial dimensions", "The spaces $ (\\pi _{T^t, T^{t-1}})^{-1} (x)$ and the measures $ \\nu ^{x}_{T^t,T^{t-1}} $ for a.e.", "$x$ depend only on the arrow $ T^t \\overset{1}{\\prec } T^{t-1} $ and not on its environment $ T \\prec T^{\\prime }$ such that $T \\prec T^t \\overset{1}{\\prec } T^{t-1} \\prec T^{\\prime }$ .", "Indeed, the dimension $ \\gamma _{T^t,T^{t-1}}$ in formula (REF ) does not depend on the environment $ T \\prec T^{\\prime }$ .", "But still, there might be several arrows corresponding to the same pair $ (i,j), 0< i <j \\le d.$ We can write Lemma 2.5 Assume the diagram of projections $\\begin{matrix} T & \\longrightarrow & S \\\\\\downarrow 1 & & \\downarrow 1 & \\\\T^{\\prime } & \\longrightarrow & S^{\\prime }\\end{matrix}$ commutes and $i,j$ are such that $T(i) = T^{\\prime }(i) \\setminus \\lbrace j\\rbrace $ and $S(i) = S^{\\prime }(i) \\setminus \\lbrace j \\rbrace $ .", "Then, $ \\gamma _{T,T^{\\prime }} \\le \\gamma _{S,S^{\\prime }} .$ Lemma REF is proven in section .", "The example in section shows that, in general, $ \\gamma _{T,T^{\\prime }}< \\gamma _{S,S^{\\prime }} .$ Let $i,j$ satisfy $ 0< i <j \\le d$ and let $T_{i,j} $ be the topology defined by $ T_{i,j} (k) = \\lbrace k\\rbrace \\;\\; {\\textrm {if}} \\;\\; k\\ne i, \\; T_{i,j}(i) = \\lbrace i, j\\rbrace .$ The topology $T_{i,j} $ is admissible and one step coarser than $T_1$ .", "Corollary 2.6 Let $ T \\prec T^{\\prime \\prime }$ be admissible topologies.", "For $ T^t \\overset{1}{\\prec } T^{t-1} $ in the decomposition of $ T \\prec T^{\\prime \\prime }$ and $(i,j)$ such that $ T^{t-1}(i) = T^{t}(i )\\cup \\lbrace j\\rbrace $ , we have $ \\gamma _{T_1, T_{i,j} } \\le \\gamma _{T^t, T^{t-1}}.$ Apply lemma REF .to the commutative part of the diagram $\\begin{matrix}& & & T &\\\\& & & \\downarrow &\\\\& T_1 &\\longrightarrow & T^t &\\\\& \\downarrow 1 & & \\downarrow 1& \\\\& T_{i,j} &\\longrightarrow & T^{t-1} &\\longrightarrow T^{\\prime \\prime }.\\end{matrix}$ In the case when there are several pairs $\\lbrace (i_1,j_1), \\ldots , (i_k, j_k) \\rbrace $ with the same difference $ \\chi _i - \\chi _j $ , there are different decompositions given by proposition REF .1.", "If the measure $ \\mu $ is discrete, we can apply theorem REF .2 to each decomposition and get a common formula by grouping together the terms corresponding to the same difference $ \\chi _i - \\chi _j $ .", "Namely, set $ \\delta _{T^{t+k}, T^{t}} := \\sum \\limits _{\\ell =1}\\limits ^k \\gamma _{T^{t +\\ell }, T^{t +\\ell -1}} .$ By theorem REF .2 applied to $ T^{t+k} \\prec T^{t} ,$ this number $ \\delta _{T^{t+k}, T^{t}}$ is the exact dimension of the measures $ \\nu _{T^{t+k},T^t} $ and thus is independent of the order of the decomposition.", "We also obtain that $ \\kappa _{T^{t+k}, T^{t}} =(\\chi _i - \\chi _j ) \\delta _{T^{t+k}, T^{t}}.$" ], [ "Lyapunov Dimension", "Theorem REF follows from the previous results.", "Indeed, assume $ \\mu \\in \\mathcal {M}$ is discrete and has finite entropy and let $Q$ be a partition of $\\lbrace 0,1, \\ldots , d\\rbrace $ into intervals.", "The measure $ \\nu _Q$ is exact-dimensional with dimension $\\delta _Q$ .", "By equation (REF ), there are numbers $ \\gamma ^Q_{i,j} $ such that $ \\delta _Q= \\sum _{i,j: \\ell _Q(i) < \\ell _Q(j) } \\gamma ^Q_{i,j} .$ Moreover, $h(\\mathcal {F}_Q,\\mu , \\nu _Q) = \\sum _{i,j: \\ell _Q(i) < \\ell _Q(j) } \\gamma ^Q_{i,j} (\\chi _i - \\chi _j).$ We ordered $ \\lbrace 0 < \\lambda _1 \\le \\lambda _2 \\le \\ldots \\le \\lambda _N \\rbrace $ the $ \\lambda _k := \\chi _{i_k} - \\chi _{j_k} $ for all $ (i_k,j_k)$ such that $ \\ell _Q(i_k) < \\ell _Q(j_k) $ .", "We can define another continuous, piecewise affine Lyapunov function $ \\underline{D}_{\\mathcal {F}_Q, \\mu , \\nu _Q } $ on the interval $ [ 0, \\delta _Q ]$ such that $ \\underline{D}_{\\mathcal {F}_Q, \\mu , \\nu _Q } (0) = h(\\mathcal {F}_Q, \\mu , \\nu _Q)$ and the slope of $ \\underline{D}_{\\mathcal {F}_Q, \\mu , \\nu _Q } (t) $ on the successive intervals of length $ \\gamma ^Q_{i_k,j_k}$ is $ - \\lambda _k.$ By (REF ), $ \\underline{D}_{\\mathcal {F}_Q, \\mu , \\nu _Q } (\\delta _Q) = 0.$ On the other hand, using (REF ) and the fact that for all $(i,j), \\; \\gamma ^Q_{i,j} \\le 1 $ (Proposition REF .2), we see that, for all $ t \\in [0, \\delta _Q],$ $ \\underline{D}_{\\mathcal {F}_Q, \\mu , \\nu _Q } (t) \\; \\le \\; D_{\\mathcal {F}_Q, \\mu } (t).$ In particular, $ D_{\\mathcal {F}_Q, \\mu } (\\delta _Q) \\ge \\underline{D}_{\\mathcal {F}_Q, \\mu , \\nu _Q } (\\delta _Q) = 0.$ This shows (REF ).", "In case of equality with $\\delta _Q <N $ , we have necessarily $ h(\\mathcal {F}_Q, \\mu , \\nu _Q ) \\; = \\; h_{{\\textrm {RW}}} (\\mu )$ ." ], [ "Content of the paper", "Given the previous discussion, we still have to prove theorems REF , REF , REF , REF and REF , proposition REF and lemma REF .", "In section , we show proposition REF and discuss the geometry of the spaces $ \\mathcal {X}_{T}^{f^{\\prime }} $ .", "In particular, lemma REF describes the one-dimensional structure of $\\pi _{T,T^{\\prime }}^{-1}(x^{\\prime })$ for $ x^{\\prime } \\in \\mathcal {X}_{T^{\\prime }}^{f^{\\prime }} $ and $T \\overset{1}{\\prec } T^{\\prime }$ .", "We also discuss the multidimensional structure of the configuration spaces (proposition REF ).", "We recall in section the underlying trajectory space of the associated random walk, and the applications of Oseledets theorem to random walks of matrices.", "In section , we prove theorem REF .", "In section , we recall the notion of mutual information of random variables and its properties.", "We prove theorem REF and lemma REF in section and theorem REF in section .", "The central arguments in sections , and have a long history in ergodic theory.", "A short comment about background heads each of the corresponding sections.", "In section , we discuss the case of Hitchin representations of cocompact surface groups, exhibit examples of non-conservation of dimension and prove theorems REF and REF ." ], [ "Topologies and exponents", "Let us order the indices in $ D_{T, T^{\\prime }} $ in such a way that $\\chi _{i_1} - \\chi _{j_1} \\le \\chi _{i_2} - \\chi _{j_2} \\le \\cdots \\le \\chi _{i_{N_{T, T^{\\prime }} }} - \\chi _{j_{N_{T, T^{\\prime }} }}.$ Beginning with $T^0 = T^{\\prime }$ define the sequence inductively so that for each $t = 1,2,\\ldots , N_{T, T^{\\prime }} $ the topology $T^{t}$ is generated by $T^{t-1}$ together with the set $T^{t-1}(i_{t}) \\setminus \\lbrace j_{t}\\rbrace $ .", "Since $T^t \\prec T^{t-1}$ , it follows by induction that $T^t$ is admissible.", "We have that $T^t \\prec T^{t-1}$ ; we claim that $T^t$ is one step finer than $T^{t-1}$ .", "Indeed, if this is not the case, then there exists $j \\in T^{t-1}(i)$ such that $i_t < j < j_t$ and $T^{t}(j) \\setminus T^{t-1}(j) = \\lbrace j_t\\rbrace $ .", "Since $\\chi _j - \\chi _{j_t} < \\chi _{i_t} - \\chi _{j_t}$ it must be the case that $j_t \\in T(j)$ , otherwise $j_t$ would have been removed from $T^{t-1}(j)$ at some previous step.", "Similarly, since $\\chi _{i_t} - \\chi _j < \\chi _{i_t} - \\chi _{j_t}$ we obtain that $j \\in T(i_t)$ .", "However, these two facts imply that $j_t \\in T(i_t)$ which is a contradiction.", "It follows that $T^t$ is one step finer than $T^{t-1}$ as claimed.", "We finally claim that $T \\prec T^t$ .", "To see this, observe that $T^t$ is generated by $\\lbrace A(i) \\rbrace ,$ for $ i = 1, \\ldots ,d ,$ where $A(i) = T^{t-1}(i)$ if $i \\ne i_t$ and $A(i_t) = T^{t-1}(i_t) \\setminus \\lbrace j_t\\rbrace $ .", "Inductively, if $i \\ne i_t$ , one has $A(i) = T^{t-1}(i) \\supset T(i)$ .", "By construction $j_t \\notin T(i_t)$ and therefore $A(i_t) \\supset T(i_t)$ as well.", "Hence, $T \\prec T^t$ as claimed.", "We also record here the following fact about admissible topologies which will be used several times.", "Proposition 3.1 Let $T \\overset{1}{\\prec } T^{\\prime }$ be admissible topologies and $i,j$ be such that $T(i) = T^{\\prime }(i) \\setminus \\lbrace j\\rbrace $ .", "Then both $T^{\\prime }(i) \\setminus \\lbrace i\\rbrace $ and $T^{\\prime }(i) \\setminus \\lbrace i,j\\rbrace $ are in $T^{\\prime }$ .", "Since $T^{\\prime }$ is admissible we have $T^{\\prime }(i) \\subset \\lbrace i,i+1,\\ldots ,d\\rbrace ,$ and $\\lbrace i+1,i+2,\\ldots ,d\\rbrace \\in T^{\\prime }$ .", "Therefore, $T^{\\prime }(i) \\setminus \\lbrace i\\rbrace = T^{\\prime }(i) \\cap \\lbrace i+1,\\ldots ,d\\rbrace $ is in $T^{\\prime }$ as claimed.", "Repeating the argument for $T$ we obtain that $T^{\\prime }(i) \\setminus \\lbrace i,j\\rbrace = T(i) \\setminus \\lbrace i\\rbrace $ is in $T$ .", "Combining this with the fact that $T \\overset{1}{\\prec } T^{\\prime }$ we obtain $T^{\\prime }(i) \\setminus \\lbrace i,j\\rbrace = \\bigcup \\limits _{k \\in T^{\\prime }(i) \\setminus \\lbrace i,j\\rbrace }T(k) = \\bigcup \\limits _{k \\in T^{\\prime }(i) \\setminus \\lbrace i,j\\rbrace }T^{\\prime }(k),$ so that $T^{\\prime }(i) \\setminus \\lbrace i,j\\rbrace $ is in $T^{\\prime }$ as claimed." ], [ "Distances on configuration spaces", "Let $\\mathcal {G}_i$ be the Grassmannian manifold of $i$ -dimensional subspaces of $\\mathbb {R}^d$ .", "We fix on each $\\mathcal {G}_i$ a Riemannian metric which is invariant under the action of orthogonal transformations with the additional property that if $S,S^{\\prime } \\in \\mathcal {G}_i$ are such that $S + S^{\\prime }$ has dimension $i+1$ then the Riemannian distance satisfies $\\operatorname{dist}(S,S^{\\prime }) = \\angle (\\pi (S),\\pi (S^{\\prime })),$ where $\\pi :S+S^{\\prime } \\rightarrow (S+S^{\\prime })/(S\\cap S^{\\prime })$ is the projection onto the quotient space $(S+S^{\\prime })/(S \\cap S^{\\prime })$ , which is endowed with the inner product inherited from $\\mathbb {R}^d$ .", "From the definition it follows that when $\\dim (S+S^{\\prime }) = i+1$ one has $\\operatorname{dist}(S+W,S^{\\prime }+W) \\le \\operatorname{dist}(S,S^{\\prime }),$ for all subspaces $W$ such that $\\dim (S+W) = \\dim (S^{\\prime }+W)$ .", "Given an admissible topology $T$ we define the distance on the configuration space $\\mathcal {X}_{T}$ so that $\\operatorname{dist}((x_I)_{I \\in T}, (x_I^{\\prime })_{I \\in T}) = \\sum \\limits _{I \\in T}\\operatorname{dist}(x_I,x_I^{\\prime }).$ Proposition 3.2 Let $ T\\prec T^{\\prime } $ be admissible topologies and $ x^{\\prime } \\in \\mathcal {X}_{T^{\\prime }} $ .", "Then, $(\\pi _{T,T^{\\prime }} )^{-1} (x^{\\prime }) $ , endowed with the metric $\\operatorname{dist}$ , is locally bilipschitz homeomorphic to the Euclidean space $\\mathbb {R}^{N_{T,T^{\\prime }}}.$ By proposition REF .1, we take a sequence $T^0 =T^{\\prime },T^1,\\ldots ,T^N = T$ such that, $T^{t}$ is one step finer than $T^{t-1}$ for $t=1,\\ldots ,N$ .", "We prove by increasing induction on $t$ that $(\\pi _{T^t,T^{\\prime }} )^{-1} (x^{\\prime }) $ , endowed with the metric $\\operatorname{dist}$ , is locally bilipschitz homeomorphic to the Euclidean space $\\mathbb {R}^t.$ This is trivially true for $ t= 0$ .", "So, we assume for $t>0$ , that for any $ y^{\\prime } \\in (\\pi _{T^{t-1},T^{\\prime }} )^{-1} (x^{\\prime }) ,$ there is a neighborhood of $y^{\\prime }$ in $(\\pi _{T^{t-1},T^{\\prime }} )^{-1} (x^{\\prime }) $ which is bilipschitz homeomorphic to the Euclidean space $\\mathbb {R}^{t-1}.$ The fibers of the projections $ \\pi _{T^t, T^{t-1}}$ form a $C^\\infty $ foliation of $ (\\pi _{T^{t},T^{\\prime }} )^{-1} (x^{\\prime }) .", "$ We have to verify that the induced metric by $\\operatorname{dist}$ on the fibers is bilipschitz equivalent to the Euclidean one dimensional metric, uniformly in the neighborhood of $ y^{\\prime }$ .", "Let $(i, j)$ such that $T^t(i) = T^{t-1} (i) \\setminus \\lbrace j\\rbrace $ , we define a distance on each fiber of the projection $\\pi _{T^t,T^{t-1}}$ by setting $\\operatorname{dist}_{T^t,T^{t-1}}^{x^{\\prime }}(x_1,x_2) = \\operatorname{dist}((x_1)_{T(i)},(x_2)_{T(i)}),$ for each $x^{\\prime } \\in \\mathcal {X}_{T^{t-1}}$ and $x_1,x_2 \\in \\pi ^{-1}_{T^t,T^{t-1}}(x^{\\prime })$ .", "These distances are all Lipschitz equivalent on the fibers: Lemma 3.3 For all admissible topologies $T \\overset{1}{\\prec } T^{\\prime }$ , all $x^{\\prime } \\in \\mathcal {X}_{T^{\\prime }}$ and all $x_1,x_2 \\in \\pi _{T,T^{\\prime }}^{-1}(x^{\\prime })$ , one has $\\operatorname{dist}_{T,T^{\\prime }}^{x^{\\prime }}(x_1,x_2) \\le \\operatorname{dist}(x_1,x_2) \\le 2^d\\operatorname{dist}_{T,T^{\\prime }}^{x^{\\prime }}(x_1,x_2).$ The inequality $\\operatorname{dist}_{T,T^{\\prime }}^{x^{\\prime }}(x_1,x_2) \\le \\operatorname{dist}(x_1,x_2)$ is immediate from the definitions.", "For the second inequality we assume $x_1 \\ne x_2$ .", "Let $i,j$ be such that $T(i) = T^{\\prime }(i) \\setminus \\lbrace j\\rbrace $ .", "Since $(x_1)_{T(i)}$ and $(x_2)_{T(i)}$ are distinct codimension one subspaces of $x^{\\prime }_{T^{\\prime }(i)}$ their sum is $x^{\\prime }_{T^{\\prime }(i)}$ , and therefore $\\operatorname{dist}_{T,T^{\\prime }}^{x^{\\prime }}(x_1,x_2) = \\operatorname{dist}((x_1)_{T(i)},(x_2)_{T(i)}) \\ge \\operatorname{dist}((x_1)_{T(i)} + W,(x_2)_{T(i)}+W),$ for all subspaces $W$ containing neither $(x_1)_{T(i)}$ nor $(x_2)_{T(i)}$ .", "For each $I \\in T \\setminus T^{\\prime }$ one has $i \\in I$ and therefore $T(i) \\subset I$ .", "Noticing that $J := \\bigcup \\limits _{k \\in I \\setminus \\lbrace i\\rbrace }T(k) = \\bigcup \\limits _{k \\in I \\setminus \\lbrace i\\rbrace }T^{\\prime }(k)$ belongs to $T^{\\prime }$ we obtain $\\operatorname{dist}(x_1,x_2) &= \\sum \\limits _{I \\in T \\setminus T^{\\prime }}\\operatorname{dist}((x_1)_I,(x_2)_I)\\\\ &= \\sum \\limits _{I \\in T \\setminus T^{\\prime }}\\operatorname{dist}((x_1)_{T(i)} + x^{\\prime }_J, (x_2)_{T(i)} + x^{\\prime }_J) \\le 2^d \\operatorname{dist}_{T,T^{\\prime }}^{x^{\\prime }}(x_1,x_2).$ Consider $ T, T^{\\prime }$ admissible topologies with $T \\overset{1}{\\prec } T^{\\prime }$ , $ (i<j) $ such that $ T^{\\prime }(i )= T(i) \\cup \\lbrace j\\rbrace .$ Given $x \\in \\mathcal {X}_T$ , we use the metric (REF ) to define a bilipschitz homeomorphism $\\varphi _x:(-\\pi /2,\\pi /2) \\rightarrow \\pi _{T,T^{\\prime }}^{-1}(x^{\\prime })$ where $x^{\\prime } = \\pi _{T,T^{\\prime }}(x)$ .", "For this purpose let $V = x^{\\prime }_{T^{\\prime }(i)}/x^{\\prime }_{T^{\\prime }(i) \\setminus \\lbrace i,j\\rbrace }$ endowed with the inner product inherited from the ambient space $\\mathbb {R}^d$ (i.e.", "the inner product between two classes is calculated by taking representatives perpendicular to $x^{\\prime }_{T^{\\prime }(i) \\setminus \\lbrace i,j\\rbrace }$ ).", "One has that $(\\pi _{T,T^{\\prime }})^{-1}(x^{\\prime })$ consists in configurations $z$ with $z_{T^{\\prime }(k)} = x^{\\prime }_{T^{\\prime }(k)}$ for all $k \\ne i$ while $z_{T(i)}$ is a codimension one subspace of $x^{\\prime }_{T^{\\prime }(i)}$ which contains $x^{\\prime }_{T^{\\prime }(i) \\setminus \\lbrace i,j\\rbrace }$ and is distinct from $x^{\\prime }_{T^{\\prime }(i) \\setminus \\lbrace i\\rbrace }$ .", "It follows that $(\\pi _{T,T^{\\prime }})^{-1}(x^{\\prime })$ endowed with $dist^{x^{\\prime }}_{T,T^{\\prime }}$ is isometric to the space of one dimensional subspaces of $V$ minus the projection of $x^{\\prime }_{T^{\\prime }(i) \\setminus \\lbrace i\\rbrace }$ with the angle distance.", "Let $X,Y$ be a pair of unit vectors in $V$ such that $X$ has a representative in $x_{T(i)}$ , $Y$ has a representative in $x^{\\prime }_{T^{\\prime }(i)\\setminus \\lbrace i\\rbrace }$ and such that $\\cos \\angle (X,Y) >0 $ .", "Define $\\varphi _x :(-\\pi /2, + \\pi /2) \\rightarrow \\pi _{T,T^{\\prime }}^{-1} (x^{\\prime }) $ by associating to $u \\in (-\\pi /2, +\\pi /2) $ the configuration where the corresponding one dimensional subspace of $V$ contains a vector of the form $\\cos u X + \\sin u Y.$ Set $ \\theta : = \\angle (X,Y).", "$ Lemma 3.4 In the above context $\\varphi _x$ is a bilipschitz homeomorphism.", "Moreover, $ |\\tan \\frac{\\theta }{2} | \\, < \\, {\\textrm { Lip }} (\\varphi _x) \\, < \\, \\frac{1}{ |\\tan \\frac{\\theta }{2} |}.", "$ Let $e_1 = (1,0), e_2 = (0,1)$ be the standard basis of $\\mathbb {R}^2$ and $\\mathcal {G}_1(\\mathbb {R}^2)$ be the space of one dimensional subspaces with the angle distance.", "Let $L_2$ be the subspace generated by $e_2$ .", "The mapping $f:(-\\pi /2,\\pi /2) \\rightarrow \\mathcal {G}_1(\\mathbb {R}^2) \\setminus \\lbrace L_2\\rbrace $ where $f(u)$ is the subspace generated by $\\cos (u)e_1 + \\sin (u)e_2$ is an isometry.", "By [41], if $\\theta \\in (0,\\pi /2]$ then the mapping $g_{\\theta }:\\mathcal {G}_1(\\mathbb {R}^2) \\setminus \\lbrace L_2\\rbrace $ induced by the matrix $A = \\begin{pmatrix}\\sin (\\theta ) & 0\\\\ \\cos (\\theta ) &1\\end{pmatrix}$ has derivative $|dg_{\\theta }(L)| = \\frac{|\\text{det}(A)|}{\\Vert A_{\\vert L}\\Vert ^2}$ .", "This implies for the composition one has $|dg_{\\theta }\\circ f(u)| = \\frac{|\\sin (\\theta )|}{\\cos (u)^2\\sin (\\theta )^2 + (\\cos (u)\\cos (\\theta ) + \\sin (u))^2} = \\frac{|\\sin (\\theta )|}{1 + \\sin (2u)\\cos (\\theta )}.$ Letting $\\theta = \\operatorname{dist}(x_{T(i)},x^{\\prime }_{T^{\\prime }(i) \\setminus \\lbrace i\\rbrace })$ the fiber $\\pi _{T,T^{\\prime }}^{-1}(x^{\\prime })$ endowed with $\\operatorname{dist}_{T,T^{\\prime }}^{x^{\\prime }}$ is isometric to the projective space $\\mathcal {G}_1(\\mathbb {R}^2) \\setminus \\lbrace L_2 \\rbrace $ where $x$ is identified with the subspace generated by $\\sin (\\theta )e_1 + \\cos (\\theta )e_2$ .", "Under this identification $\\varphi _x$ corresponds to the composition $g_{\\theta }\\circ f$ , so we have obtained $|d\\varphi _x(u)| = \\frac{|\\sin (\\theta )|}{1 + \\sin (2u)\\cos (\\theta )}.$ To finish the proof of proposition REF , we still have to show that the homeomorphism $ \\varphi _z , z \\in (\\pi _{T^t,T^{\\prime }} )^{-1} (x^{\\prime }) $ depends continuously on $ z$ as a bilipschitz homeomorphism.", "Given the above construction, this is true since we can locally choose in a continuous way parameterizations of the spaces $V_z = [\\pi _{T^t,T^{t-1}} (z)]_{T^{t-1} (i)}/ [\\pi _{T^t,T^{t-1}} (z)]_{T^{t-1} (i)\\setminus \\lbrace i, j\\rbrace } $ and in these spaces vectors $ X_z, Y_z $ such that $X_z $ has a representative in $ z_{T^t (i)} ,$ $Y_z $ has a representative in $ [\\pi _{T^t,T^{t-1}} (z)]_{T^{t-1} (i)\\setminus \\lbrace i\\rbrace } $ and $ \\cos \\angle (X_z, Y_z) >0 .$ For $T \\prec T^{\\prime }$ admissible topologies, it follows from proposition REF that $ \\overline{\\delta }_{T,T^{\\prime }} \\le N_{T,T^{\\prime }} .$ Also since $T^t \\overset{1}{\\prec } T^{t-1} $ , we have $ \\overline{\\gamma }_{T^t,T^{t-1}} \\le 1.$ By formula (REF ), we have indeed $ \\kappa _{T,T^{\\prime }} \\le \\sum _{ (i,j) \\in D_{T,T^{\\prime }} } (\\chi _i - \\chi _j).$ Observe that relation (REF ) depends on theorem REF , which will be proven in section ." ], [ "One-dimensional coordinates", "Let $i<j $ and consider all arrows $ T, T^{\\prime }$ of admissible topologies with $T \\overset{1}{\\prec } T^{\\prime }$ , such that $ T^{\\prime }(i )= T(i) \\cup \\lbrace j\\rbrace .$ Recall that $T_{i,j} $ is the topology defined by $ T_{i,j} (k) = \\lbrace k\\rbrace \\;\\; {\\textrm {if}} \\;\\; k\\ne i, \\; T_{i,j}(i) = \\lbrace i, j\\rbrace .$ The topology $T_{i,j} $ is admissible and one step coarser than $T_1$ .", "Lemma 3.5 Let $T \\overset{1}{\\prec } T^{\\prime }$ be admissible topologies and $i,j$ be such that $T(i) = T^{\\prime }(i) \\setminus \\lbrace j\\rbrace $ .", "For all $ y^{\\prime },x^{\\prime }$ such that $ \\pi _{ T_{i,j},T^{\\prime }} y^{\\prime } = x^{\\prime },$ $ \\pi _{T_1, T} $ defines a bilipschitz homeomorphism between $(\\pi _{T,T^{\\prime }})^{-1} (x^{\\prime }) $ and $ ( \\pi _{T_1, T_{i,j}})^{-1} (y^{\\prime }).$ The Lipschitz constants depend only on $y^{\\prime }$ .", "Consider $y_1,y_2 \\in \\mathcal {X}_{T_1}$ , distinct and such that $\\pi _{T,T_{i,j}}(y_1) = \\pi _{T,T_{i,j}}(y_2) = y^{\\prime }$ .", "Set $x_1 = \\pi _{T_1,T}(y_1)$ and $x_2 = \\pi _{T_1,T}(y_2)$ and notice that $\\pi _{T,T^{\\prime }}(x_1) = \\pi _{T,T^{\\prime }}(x_2) = \\pi _{T_{i,j},T^{\\prime }}(y^{\\prime }) = x^{\\prime }$ .", "Since $\\pi _{T_1,T}$ consists of forgetting some subspaces of each sequence $(x_I)_{I \\in T_1}$ it is 1-Lipschitz, and in particular 1-Lipschitz as a mapping between $\\pi _{T_1,T_{i,j}}^{-1}(y^{\\prime })$ and $\\pi _{T,T^{\\prime }}^{-1}(x^{\\prime })$ .", "To prove that the inverse is also Lipschitz let $S_1 = (y_1)_{\\lbrace i \\rbrace }, S_2 = (y_2)_{\\lbrace i \\rbrace }$ .", "Notice that $S_1,S_2$ are distinct one dimensional subspaces of the two dimensional subspace $y^{\\prime }_{\\lbrace i,j\\rbrace }$ .", "Therefore $S_1 + S_2 = y^{\\prime }_{\\lbrace i,j\\rbrace }$ and $\\operatorname{dist}(S_1,S_2) = \\angle (S_1,S_2)$ .", "For each $I \\in T_{1} \\setminus T_{i,j}$ notice that $i \\in I$ and $j \\notin I$ .", "Setting $W_I = y^{\\prime }_{I \\setminus \\lbrace i\\rbrace }$ we have $(y_1)_I = W_I + S_1$ and $(y_2)_I = W_I + S_2$ from which it follows that $\\operatorname{dist}((y_1)_{I}, (y_2)_{I}) = \\angle (\\pi _{W_I^\\perp }(S_1),\\pi _{W_I^\\perp }(S_2)) \\le \\angle (S_1,S_2),$ where $\\pi _{W^\\perp }:\\mathbb {R}^d \\rightarrow W^{\\perp }$ is the orthogonal projection onto $W^{\\perp }$ .", "By Lemma REF , $\\operatorname{dist}(y_1,y_2) = \\sum \\limits _{I \\in T_1 \\setminus T_{i,j}}\\operatorname{dist}((y_1)_{I}, (y_2)_{I}) \\le 2^d \\angle (S_1,S_2).$ Because $y^{\\prime }$ is a configuration, the minimum over $I \\in T_1 \\setminus T_{i,j}$ of the angle between $W_I$ and $S_1+S_2 = y^{\\prime }_{\\lbrace i,j\\rbrace }$ is positive.", "It follows that there exists $c > 0$ which depends only on $y^{\\prime }$ such that $\\operatorname{dist}((y_1)_{I}, (y_2)_{I}) = \\angle (\\pi _{W_I^\\perp }(S_1),\\pi _{W_I^\\perp }(S_2)) \\ge c\\angle (S_1,S_2),$ for all $I \\in T_{1} \\setminus T_{i,j}$ .", "Notice that, since $ T_1 \\setminus T_{i,j} = \\lbrace I : I \\ni i {\\textrm { and }} I \\lnot \\ni j\\rbrace , $ $T \\setminus T^{\\prime } \\subset T_1 \\setminus T_{i,j}$ , and therefore $\\operatorname{dist}(x_1,x_2) = \\sum \\limits _{I \\in T \\setminus T^{\\prime }}\\angle (\\pi _{W_I^\\perp }(S_1),\\pi _{W_I^\\perp }(S_2)) \\ge 2^{-d}c\\operatorname{dist}(y_1,y_2).$ It follows that $\\pi _{T_1,T}$ is a bilipschitz homeomorphism between $\\pi _{T_1,T_{i,j}}^{-1}(y^{\\prime })$ and $\\pi _{T,T^{\\prime }}^{-1}(x^{\\prime })$ , as claimed.", "For $x \\in \\mathcal {X}_T ,$ $T \\overset{1}{\\prec } T^{\\prime }$ admissible topologies, lemma REF yields a Lipschitz homeomorphism between $\\pi _{T,T^{\\prime }}^{-1}(x^{\\prime })$ where $x^{\\prime } = \\pi _{T,T^{\\prime }}(x)$ and $(-\\frac{\\pi }{2},\\frac{\\pi }{2})$ and the Lipschitz constant depends on $x$ .", "Combining with lemma REF yields Corollary 3.6 Assume the diagram of projections $\\begin{matrix} T & \\longrightarrow & S \\\\\\downarrow 1 & & \\downarrow 1 & \\\\T^{\\prime } & \\longrightarrow & S^{\\prime }\\end{matrix}$ commutes and $i,j$ are such that $T(i) = T^{\\prime }(i) \\setminus \\lbrace j\\rbrace $ and $S(i) = S^{\\prime }(i) \\setminus \\lbrace j \\rbrace $ .", "Then, for $ x \\in \\mathcal {X}_T,$ there is a bilipschitz homeomorphism between $ (\\pi _{T,T^{\\prime }})^{-1} (\\pi _{T,T^{\\prime }}) (x) $ and $ (\\pi _{S,S^{\\prime }})^{-1} (\\pi _{T,S^{\\prime }}) (x) $ .", "The Lipschitz constant depends only on $ x$ ." ], [ "Oseledets multiplicative ergodic theorem", "We review in this section some applications of Oseledets multiplicative ergodic theorem to random walks on matrices.", "Let $\\mu $ be a probability measure on the group $G= SL_d(\\mathbb {R})$ of $d\\times d$ matrices with determinant 1.", "Recall that we assume in all the paper that $\\mu $ is strongly irreducible and that the semi-group generated by the support of $\\mu $ is a Zariski dense subgroup in $SL_d(\\mathbb {R}).$ Let $(\\Omega , m ) = (G^\\mathbb {Z}, \\mu ^\\mathbb {Z})$ be the probability space of independent trials of elements of $G$ with distribution $\\mu $ , $\\sigma $ the shift transformation on $\\Omega $ .", "Let $\\omega \\in \\Omega $ be the sequence $ (g_n) _{n \\in \\mathbb {Z}} $ .", "We denote $g_n$ the mapping $\\omega \\mapsto g_n(\\omega ) $ that associates to $ \\omega \\in \\Omega $ its coordinate $ g_n \\in G$ .", "In particular $ g_0(\\omega )$ defines a cocycle with values in $SL(d,\\mathbb {R})$ over the ergodic system $(\\Omega , m; \\sigma ).$ The multiplicative ergodic theorem gives Theorem 4.1 ([43]) With the above notations, assume that $\\int \\log \\Vert g\\Vert \\, d\\mu (g) <+\\infty .$ Then, there exist $d$ numbers $\\chi _1 >\\chi _2 > \\ldots >\\chi _d $ with $\\sum _i \\chi _i = 0 $ and, for $m$ -a.e.", "$\\omega \\in \\Omega ,$ a direct decomposition of $\\mathbb {R}^d$ into $d$ one-dimensional spaces $ \\mathbb {R}^d \\; = \\; E_1 (\\omega ) \\oplus E_2 (\\omega ) \\oplus \\ldots \\oplus E_d (\\omega ) \\;\\; \\; \\; {\\textrm {such that }} $ a vector $v \\ne 0 $ belongs to $E_i(\\omega ) $ if, and only if, $ \\lim \\limits _{n \\rightarrow +\\infty } \\frac{1}{n} \\log |g_{n-1}(\\omega ) \\circ \\ldots \\circ g_0(\\omega ) \\,v| = \\lim \\limits _{n \\rightarrow -\\infty } \\frac{1}{n} \\log |(g_{n}(\\omega ))^{-1} \\circ \\ldots \\circ (g_{-1}(\\omega ))^{-1}\\, v| \\; = \\; \\chi _i$ for $ m$ -a.e.", "$\\omega $ , all $i$ , $ \\lim \\limits _{n \\rightarrow \\pm \\infty } \\frac{1}{|n|} \\log | \\sin \\angle (E_i(\\sigma ^n \\omega ), \\sum _{j \\ne i} E_j(\\sigma ^n \\omega ) ) | \\; = \\; 0.$ The numbers $\\chi _i, i = 1, \\ldots , d$ are called the Lyapunov exponents of the cocycle $g_0$They are the same as in the Introduction, see the comment after Theorem REF below..", "Under our hypotheses, they indeed are distinct numbers ([22] and [25]) and the $E_i$ are one-dimensional.", "The directions $E_i (\\omega ) $ are defined $m$ -a.e.", "and are called Oseledets directions.", "For two distinct dimensional subspaces $E,E^{\\prime } \\subset \\mathbb {R}^d$ , $|\\sin \\angle (E,E^{\\prime })| $ is defined by $|\\sin \\angle (E,E^{\\prime })| = \\inf \\limits _{v \\ne 0 \\in E, v^{\\prime } \\ne 0 \\in E^{\\prime }} \\frac{ |v\\wedge v^{\\prime }|}{|v| |v^{\\prime }|}.", "$ The set $\\Omega _{reg}$ of points in $\\Omega $ such that properties 1 and 2 of Oseledets theorem hold is called the set of regular points.", "The set $\\Omega _{reg}$ is $\\sigma $ -invariant, measurable and has full measure in $\\Omega $ .", "Observe that by characterization 1, for all $i$ , the mapping $\\omega \\mapsto E_i(\\omega )$ is measurable on $\\Omega _{reg}$ and we have $ E_i (\\sigma \\omega )\\; = \\; g_0 E_i (\\omega ).$ Let $i = 1, \\ldots , d.$ We write $U_i(\\omega ) := \\oplus _{j=1}^i E_i(\\omega ) $ for the unstable spaces, $U^{\\prime }_i(\\omega ) := \\oplus _{j=d-i+1}^d E_i(\\omega ) $ for the stable spaces.cf.", "notations of the Introduction.", "Clearly, for $ \\omega \\in \\Omega _{reg}$ , the flags $E_-(\\omega )$ given by $\\lbrace 0\\rbrace = U_0 \\subset U_{1}(\\omega ) \\subset \\ldots \\subset U_d = \\mathbb {R}^d $ and $ E_+ (\\omega ) $ given by $ \\lbrace 0\\rbrace = U^{\\prime }_0 \\subset U^{\\prime }_{1}(\\omega ) \\subset \\ldots \\subset U^{\\prime }_{d-1} (\\omega ) \\subset U^{\\prime }_d = \\mathbb {R}^d $ are in general position.", "An important classical observation is the following: Proposition 4.2 For all $i,$ the mappings $\\omega \\mapsto U_i (\\omega )$ are measurable with respect to the $\\sigma $ -algebra generated by $(g_n)_{n \\le -1}$ ; for all $i^{\\prime },$ the mappings $\\omega \\mapsto U^{\\prime }_{i^{\\prime }} (\\omega )$ are measurable with respect to the $\\sigma $ -algebra generated by $(g_n)_{n \\ge 0}.$ In particular, $E_-$ and $E_+$ are independent.", "It suffices to prove that for any $i$ , $U_i$ depends only on $\\lbrace g_n\\rbrace _{n \\le -1}$ .", "We claim that, for $\\omega \\in \\Omega _{reg}, $ $U_i = \\lbrace v : \\limsup _{n \\rightarrow +\\infty } \\frac{1}{n} \\log |g_{-n}^{-1} \\circ \\ldots \\circ g_{-1}^{-1}\\, v| \\le - \\chi _i \\rbrace .$ This shows that $U_i$ is completely determined when one knows $\\lbrace g_n\\rbrace _{n \\le -1}$ .", "To prove the claim, observe that $\\lbrace v : \\limsup _{n \\rightarrow +\\infty } \\frac{1}{n} \\log |g_{-n}^{-1} \\circ \\ldots \\circ g_{-1}^{-1}\\, v| \\le - \\chi _i \\rbrace $ is a vector space that contains $E_j(\\omega ) , j\\le i$ by definition.", "It is exactly $U_i (\\omega )$ since any vector that has a nonzero component in one of the $E_\\ell (\\omega ), \\ell >i$ satisfies $\\limsup _{n \\rightarrow +\\infty } \\frac{1}{n} \\log |g_{-n}^{-1} \\circ \\ldots \\circ g_{-1}^{-1}\\, v| \\ge - \\chi _\\ell > -\\chi _i.$ One verifies in the same way that $ U^{\\prime }_{i^{\\prime }}(\\omega ) = \\lbrace v : \\limsup _{n \\rightarrow +\\infty } \\frac{1}{n} \\log |g_{n} \\circ \\ldots \\circ g_0\\, v| \\le \\chi _{i^{\\prime }}\\rbrace $ .", "Let $Q$ be a partition of $\\lbrace 0,1, \\ldots , d\\rbrace $ into intervals, $Q = \\lbrace q_0 = 0 <q_1 < \\ldots < q_k = d\\rbrace .$ Write $U_Q(\\omega ) \\in \\mathcal {F}_Q$ for the $Q$ -flag $ U_Q (\\omega ) := \\lbrace 0\\rbrace = U_0 \\subset U_{q_1}(\\omega ) \\subset \\ldots \\subset U_{q_{k-1}} (\\omega ) \\subset U_d = \\mathbb {R}^d.", "$ The set $\\mathcal {G}_i$ of $i$ -dimensional subspaces is identified with $\\mathcal {F}_{\\lbrace 0<i<d\\rbrace }.$ In particular, since $g_0$ is independent of $U_i(\\omega ), U_Q (\\omega ) $ and the distribution of $g_0$ is $\\mu $ , the distribution of $U_i$ (resp.", "$U_Q$ ) is a measure on $\\mathcal {G}_i $ (resp.", "$\\mathcal {F}_Q$ ) which is stationary under the action of $(G, \\mu ) $ .", "By uniqueness, the distribution of $U_i(\\omega ) $ is $\\nu _{\\lbrace 0<i<d\\rbrace },$ the distribution of $U_Q(\\omega ) $ is $\\nu _Q.$ Similarly, the distribution of $U^{\\prime }_{i}$ is the stationary measure $\\nu ^{\\prime }_{\\lbrace 0<i<d\\rbrace }$ for the action of $\\mu ^{\\prime }$ on $\\mathcal {G}_i .$ Let $\\Omega _+ := (g_n) _{n \\ge 0} $ (respectively $ \\Omega _- := (g_n) _{n \\le -1} $ ) be the space of one-sided sequences of elements of $G$ , $m_+$ (respectively $m_-$ ) the product measure with $g_k$ of distribution $\\mu $ for all $k \\ge 0$ (respectively for all $k<0$ ), $\\sigma $ the shift transformation.", "Then, by proposition REF , $ E_- $ (respectively $E_+$ ) can be seen as a mapping from $ \\Omega _-$ (respectively $ \\Omega _+$ ) into $\\mathcal {F}$ .", "By Oseledets theorem REF , for almost every $\\omega $ , the pair $ E(\\omega ) := (E_- (\\omega _-), E_+(\\omega _+))$ belongs to $\\mathcal {F}^{(2)}$ .", "We recall in our notations the key Furstenberg result Theorem 4.3 ([18]) Let $x \\in \\mathcal {F}, x = \\lbrace \\lbrace 0\\rbrace \\subset U_1(x) \\subset \\ldots \\subset U_{d-1}(x) \\subset \\mathbb {R}^d \\rbrace .$ For $m$ -a.e.", "$\\omega \\in \\Omega _+$ , all $j, j=1, \\ldots , d,$ $ \\lim \\limits _{n \\rightarrow +\\infty } \\frac{1}{n} \\log |\\text{det}_{U_j(x)} (g_{n-1} \\circ \\ldots \\circ g_0) | \\; = \\; \\sum _{i\\le j} \\chi _i .$ Under our hypotheses, the distributions of all exterior products $\\wedge _{i=1}^j g$ satisfy the conditions of Theorem 8.5 in [18].", "Observe that $ \\log |\\text{det}_{U_j(x)} (g_{n-1} \\circ \\ldots \\circ g_0) | &=& \\sum _{k=1}^{n-1} \\log |\\text{det}_{g_{k-1} \\circ \\ldots \\circ g_0 U_j(x) } (g_k)|\\\\&=& \\sum _{k=1}^{n-1} \\log |\\text{det}_{U_j (\\sigma ^k (\\omega _+, x))}(g_0(\\sigma ^k \\omega ))| .$ The last line converges to $ \\int \\log |\\text{det}_{U_j (f)} (g) | \\, d\\mu (g) d\\nu (f)$ by the ergodic theorem, so that we indeed have $ \\sum _{i\\le j} \\chi _i \\; = \\; \\int \\log |\\text{det}_{U_j (f)} (g) | \\, d\\mu (g) d\\nu (f).$ We used these relations in the Introduction to define the exponents $\\chi _j, j = 1,\\ldots , d,$ (cf.", "equation (REF )).", "Since the distribution of $ E_-(\\omega ) $ is $ \\nu $ and $ g_0 $ is independent of $ E_-(\\omega ) $ this can also be written, all $j, j=1, \\ldots , d,$ $ \\sum _{i\\le j} \\chi _i \\; = \\; \\int \\limits _{\\Omega }\\log \\left(|\\text{det}_{U_j(\\omega )}(g_0(\\omega ))|\\right) dm(\\omega ).$ Using property 2 in Oseledets multiplicative ergodic theorem REF , we get, for any subset $I \\subset \\lbrace 1,\\ldots ,d\\rbrace $ , setting $V_I(\\omega ) = \\bigoplus \\limits _{k \\in I}E_k(\\omega )$ , $ \\int \\limits _{\\Omega }\\log \\left(|\\text{det}_{V_I(\\omega )}(g_0(\\omega ))|\\right) dm(\\omega ) = \\sum \\limits _{k \\in I}\\chi _k.$" ], [ "Approximation of partial Oseledets configurations", "Given an admissible topology $T$ we define $E_T(\\omega ) = F_T(E_-(\\omega ),E_+(\\omega ))$ where $F_T$ is defined in section REF .", "Let $T \\prec T^{\\prime }$ be admissible topologies.", "We will extend the configuration $E_{T^{\\prime }}(\\omega )$ to a configuration $\\widehat{E}(\\omega )$ defined on $T$ by applying a deterministic function to $E_{T^{\\prime }}(\\omega )$ .", "For this purpose, for each $i = 1,\\ldots ,d$ , we let $\\widehat{E}_i(\\omega )$ be the one dimensional subspace of $E_{T^{\\prime }}(\\omega )_{T^{\\prime }(i)}$ which is perpendicular to $E_{T^{\\prime }}(\\omega )_{T^{\\prime }(i) \\setminus \\lbrace i\\rbrace }$ .", "The configuration $\\widehat{E}(\\omega )$ is defined by letting $\\widehat{E}(\\omega )_I = \\sum \\limits _{i \\in I}\\widehat{E}_i(\\omega )$ for all $I \\in T$ .", "Proposition 4.4 For $m$ -a.e.", "$\\omega $ , and all $I \\in T^{\\prime }$ one has $\\widehat{E}(\\omega )_I = E_{T^{\\prime }}(\\omega )_I$ .", "It suffices to verify that $\\widehat{E}(\\omega )_{T^{\\prime }(i)} = E_{T^{\\prime }}(\\omega )_{T^{\\prime }(i)}$ for $i = 1,\\ldots ,d$ .", "When $i = d,$ this is trivial since $E_{T^{\\prime }}(\\omega )_{T^{\\prime }(d)} = E_d(\\omega )$ and therefore $\\widehat{E}(\\omega )_{T^{\\prime }(d)} = E_d(\\omega )$ as well.", "Suppose that the claim is true for $i+1,\\ldots ,d$ , then $\\widehat{E}(\\omega )_{T^{\\prime }(i)\\setminus \\lbrace i\\rbrace } = \\sum \\limits _{j \\in T^{\\prime }(i)\\setminus \\lbrace i\\rbrace }\\widehat{E}(\\omega )_{T^{\\prime }(j)} = \\sum \\limits _{j \\in T^{\\prime }(i)\\setminus \\lbrace i\\rbrace }E(\\omega )_{T^{\\prime }(j)} = E(\\omega )_{T^{\\prime }(i) \\setminus \\lbrace i\\rbrace }.$ It follows that $\\widehat{E}_i(\\omega )$ is complementary to the codimension one subspace $\\widehat{E}(\\omega )_{T^{\\prime }(i) \\setminus \\lbrace i\\rbrace }$ within $E(\\omega )_{T^{\\prime }(i)}$ .", "Taking the sum one obtains $\\widehat{E}(\\omega )_{T^{\\prime }(i)} = E_{T^{\\prime }}(\\omega )_{T^{\\prime }(i)}$ , as claimed.", "We now show that using the extension above one may approximate $E_T(\\omega )$ using only $E_{T^{\\prime }}(\\omega )$ and $g_{-1}(\\omega ),\\ldots ,g_{-n}(\\omega )$ up to an error which is exponentially small as $n \\rightarrow +\\infty $ .", "Lemma 4.5 For $m$ -a.e.", "$\\omega $ when $n \\rightarrow +\\infty $ one has $\\operatorname{dist}(g_{-1}(\\omega )\\cdots g_{-n}(\\omega )\\widehat{E}(\\sigma ^{-n}(\\omega ))_T,E_{T}(\\omega )) \\le \\exp (-\\chi n + o(n)),$ where $\\chi = \\min \\limits _{(i,j) \\in D_{T,T^{\\prime }}}\\chi _i - \\chi _j$ .", "To simplify calculations for each $i = 1,\\ldots ,d$ let $w_i(\\omega )$ be a unit vector generating $E_i(\\omega )$ , and $v_i(\\omega )$ be a unit vector generating $\\widehat{E}_i(\\omega )$ .", "Write $v_i(\\omega ) = \\sum \\limits _{j \\in T^{\\prime }(i)}a_{i,j}(\\omega )w_j(\\omega ).$ Since $v_i(\\sigma ^{-n}\\omega )$ is a unit vector $|a_{i,j}(\\sigma ^{-n}\\omega )| \\le 1$ for all $j$ , all $n$ , and $m$ -a.e.", "$\\omega $ .", "Furthermore, since the angle between $E_i(\\sigma ^{-n}\\omega )$ and $\\sum \\limits _{j \\in T^{\\prime }(i) \\setminus \\lbrace i \\rbrace }E_j(\\sigma ^{-n}\\omega )$ is at least $e^{-o(n)}$ , we obtain $|a_{i,i}(\\sigma ^{-n}\\omega )| \\ge e^{-o(n)}$ when $n \\rightarrow +\\infty $ for $m$ -a.e.", "$\\omega $ .", "It suffices to show that $\\operatorname{dist}(g_{-1}(\\omega )\\cdots g_{-n}(\\omega )\\widehat{E}(\\sigma ^{-n}(\\omega ))_{T(i)},E_{T}(\\omega )_{T(i)}) \\le \\exp (-\\chi n + o(n)),$ when $n \\rightarrow +\\infty $ for all $i = 1,\\ldots ,d$ .", "The claim is trivially true when $i = d$ .", "Suppose that the claim is true for $i+1,\\ldots ,d$ , so in particular one has that $\\operatorname{dist}(g_{-1}(\\omega )\\cdots g_{-n}(\\omega )\\widehat{E}(\\sigma ^{-n}(\\omega ))_{J},E_{T}(\\omega )_{J}) \\le \\exp (-\\chi n + o(n)),$ as $n \\rightarrow +\\infty $ where $J = T(i) \\setminus \\lbrace i\\rbrace $ .", "Let $z_n(\\omega )$ be a unit vector in the intersection of $\\widehat{E}(\\sigma ^{-n}(\\omega ))_{T(i)}$ with the subspace $W_i(\\sigma ^{-n}(\\omega )),$ where $ W_i (\\omega ) $ is the space generated by $\\lbrace w_j (\\omega ) , j = i $ and $ j \\in T^{\\prime }(i) \\setminus T(i) \\rbrace .", "$ We have $ e^{-o(n)} \\le |<z_n , w_i (\\sigma ^{-n} (\\omega ) )> | $ and $ \\Vert z_n \\Vert \\le 1.$ If we write $z_n(\\omega ) = b_{n,i}(\\omega )w_i(\\sigma ^{-n}\\omega ) + \\sum \\limits _{j \\in T^{\\prime }(i) \\setminus T(i)}b_{n,j}(\\omega )w_j(\\sigma ^{-n}(\\omega ),$ We have $|b_{n,i}(\\omega )| \\ge e^{-o(n)}$ while $|b_{n,j}(\\omega | \\le 1$ for all $j$ .", "To conclude notice that $g_{-1}(\\omega )\\cdots g_{-n}(\\omega )z_n(\\omega ) = e^{\\chi _i n + o(n)}b_{n,i}(\\omega )w_i(\\omega ) + \\sum \\limits _{j \\in T^{\\prime }(i) \\setminus T(i)}e^{\\chi _j n+o(n)}b_{n,j}(\\omega )w_j(\\omega ).$ It follows that the angle between $E_i(\\omega )$ and the subspace $L_n(\\omega )$ generated by $g_{-1}(\\omega )\\cdots g_{-n}(\\omega )z_n(\\omega )$ is at most $e^{-\\chi ^{\\prime } n + o(n)}$ where $\\chi ^{\\prime } = \\min \\limits _{j \\in T^{\\prime }(i) \\setminus T(i)} \\chi _i - \\chi _j \\ge \\chi $ .", "Since $g_{-1}(\\omega )\\cdots g_{-n}(\\omega )\\widehat{E}(\\omega )_{T(i)} = L_n(\\omega ) + g_{-1}(\\omega )\\cdots g_{-n}(\\omega )\\widehat{E}(\\omega )_{J},$ the claim follows.", "Assume in the above discussion that $T \\overset{1}{\\prec } T^{\\prime }$ and that $ i<j $ is such that $T^{\\prime }(i) = T(i) \\cup \\lbrace j \\rbrace $ .", "Set $ x^{\\prime } = E_{T^{\\prime }} (\\omega ) $ .", "Then the space $ W_i (\\omega ) $ , generated by $ E_i(\\omega ), E_j(\\omega ) $ is a representative of the vector space $V = x^{\\prime }_{T^{\\prime }(i)}/x^{\\prime }_{T^{\\prime }(i) \\setminus \\lbrace i,j\\rbrace }$ discussed in Lemma REF .", "Let $\\pi \\:= \\pi _{T,T^{\\prime }} $ be the projection from $\\mathcal {X}_{T}$ to $\\mathcal {X}_{T^{\\prime }}$ and consider the coordinates $\\varphi _{x(\\omega )}$ given by lemma REF on $\\pi ^{-1}(E_{T^{\\prime }}(\\omega )),$ setting $x (\\omega )= E_{T}(\\omega )$ and $x^{\\prime } (\\omega )= E_{T^{\\prime }}(\\omega )$ .", "The distance $\\varphi _{x(\\omega )}$ on $W_i(\\omega )$ is equivalent to the metric $ \\operatorname{dist}_{T,T^{\\prime }}^{x^{\\prime }(\\omega )}$ (see (REF )) on $ \\pi ^{-1}(x^{\\prime }(\\omega ))$ .", "Lemma 4.6 Let $T \\overset{1}{\\prec } T^{\\prime }$ be admissible topologies and $ i<j $ such that $T^{\\prime }(i) = T(i) \\cup \\lbrace j \\rbrace $ .", "Fix $ \\beta >0 $ and let $ x_n \\in \\pi ^{-1}(x^{\\prime }(\\sigma ^{-n}\\omega ))$ satisfy $\\varphi _{x(\\sigma ^{-n} \\omega )} (x_n)) \\le \\beta $ .", "Then for $m$ -a.e.", "$ \\omega $ , as $ n \\rightarrow \\infty ,$ one has $ \\operatorname{dist}_{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )} (g_{-1}(\\omega ) \\circ \\ldots \\circ g_{-n}(\\omega ) (x_n) , {E_{T}( \\omega )} ) \\; \\le \\; \\exp (- \\chi _{T,T^{\\prime }} n +o(n) ) .", "$ By theorem REF .2, the distance from $ E_T (\\sigma ^{-n} \\omega ) $ to $ {E_{T^{\\prime }}(\\sigma ^{-n} \\omega )}$ is at least $ \\exp (-o(n)) .$ Therefore, $ \\operatorname{dist}_{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma ^{-n}(\\omega )} ( (x_n) , {E_{T^{\\prime }}(\\sigma ^{-n} \\omega )} ) \\ge \\exp (-o(n)) $ as well.", "Following the proof of lemma REF , we have that the point $ x_n $ satisfies $\\operatorname{dist}_{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )} (g_{-1}(\\omega ) \\circ \\ldots \\circ g_{-n}(\\omega ) (x_n) , {E_{T}( \\omega )} ) \\; \\le \\; \\exp (- \\chi _{T,T^{\\prime }} n +o(n) ) .$ The lemma follows." ], [ "Proof of theorem ", "Theorem REF states that some entropy is estimated from above by exponents.", "For random walks on matrices, this is a fundamental observation of Furstenberg ([18]).", "Theorem REF and its proof are one more variant of the original proof: one shows equality for a big family of random walks on the same group and one approximates using this family.", "The exponents are continuous and the entropy has some upper semi-continuous properties.", "This should be sufficient for the astute reader, but we will give a detailed proof anyway.", "In particular, it gives some a priori quasi invariance of stationary measures (see Corollary REF )." ], [ "Mollification of $\\mu $", "For each $n=1,2,3,\\ldots $ fix a probability $\\lambda _n$ with a smooth positive density with respect to Haar measure on the orthogonal group of $\\mathbb {R}^d$ , in such a way that $\\lim \\limits _{n \\rightarrow +\\infty }\\lambda _n = \\delta _{Id}$ where $\\delta _{Id}$ is the point mass at the identity.", "Let $\\mu _n = \\lambda _n * \\mu $ so one has, for all continuous $h:G \\rightarrow \\mathbb {R}$ $ \\int \\limits _{G} h(g) d\\mu _n(g) = \\int \\limits _{G} \\int h(rg) d\\lambda _n(r) d\\mu (g).$ Let $\\eta $ be the orthogonally invariant probability on $\\mathcal {F}$ .", "Lemma 5.1 For each $n$ there is a unique $\\mu _n$ -stationary probability $\\nu _n$ on $\\mathcal {F}$ .", "Furthermore, $\\nu _n$ has a continuous positive density with respect to $\\eta $ .", "For any $\\mu _n$ -stationary probability we have $\\nu _n = \\mu _n * \\nu _n = (\\lambda _n * \\mu ) * \\nu _n = \\lambda _n * (\\mu * \\nu _n).$ Since, for any probability $m$ on $\\mathcal {F}$ , the convolution $\\lambda _n*m$ has a continuous positive density with respect to $\\eta $ it follows that any $\\mu _n$ -stationary probability has this property.", "However, any two distinct extremal $\\mu _n$ -stationary probabilities must be mutually singular.", "This implies that $\\nu _n$ is unique as claimed.", "Endow $ \\mathcal {M} (G \\times \\mathcal {F}) $ with the topology of convergence over continuous function $ \\varphi $ on $(G\\times \\mathcal {F})$ with $|\\varphi (g,f) |\\le C \\log \\Vert g\\Vert $ for some constant $C$ .", "Lemma 5.2 One has $\\lim \\limits _{n \\rightarrow +\\infty }(\\mu _n \\times \\nu _n) = \\mu \\times \\nu $ in $ \\mathcal {M}(G\\times \\mathcal {F}).$ We have $ \\lim \\limits _{n \\rightarrow \\infty } \\mu _n = \\mu $ and, since any weak*-limit of $\\nu _n $ is $ \\mu $ -stationary, $ \\lim \\limits _{n \\rightarrow \\infty }( \\mu _n \\times \\nu _n )= \\mu \\times \\nu .", "$ A priori, we have convergence against only continuous functions with compact support on $ G\\times \\mathcal {F}$ .", "We want to ensure that there is also convergence in $ \\mathcal {M}(G\\times \\mathcal {F}).$ By Skorohod's representation theorem, there exists a probability space $(P,\\mathcal {E},\\mathbb {P})$ and random elements on this space such that $\\lim \\limits _{n \\rightarrow +\\infty }(A_n,F_n) = (A,F)$ almost surely, where $(A,F)$ has distribution $\\mu \\times \\nu $ and $(A_n,F_n)$ has distribution $\\mu _n \\times \\nu _n$ for each $n$ .", "Let $ \\varphi $ be a continuous function $ \\varphi $ on $(G\\times \\mathcal {F})$ with $|\\varphi (g,f) |\\le C \\log \\Vert g\\Vert $ for some constant $C$ .", "With these auxiliary random elements we calculate (denoting integration with respect to $\\mathbb {P}$ by $\\mathbb {E}\\left(\\right)$ as usual) $\\lim \\limits _{n \\rightarrow +\\infty }\\int \\varphi (g,f) d\\mu _n (g) d\\nu _n (f) = \\lim \\limits _{n \\rightarrow +\\infty }\\mathbb {E}\\left(\\left(\\varphi (A_n,F_n)\\right)\\right).$ Observe that, since left composition with an orthogonal transformation does not change the norm a linear mapping, the random variables $\\left(|\\varphi (A_n,F_n)|\\right)_{n= 1,2,\\ldots }$ are uniformly integrable.", "It follows that $ \\lim \\limits _{n \\rightarrow +\\infty }\\mathbb {E}\\left(\\varphi (A_n,F_n)\\right) \\; = \\; \\mathbb {E}\\left(\\varphi (A,F)\\right).", "$ Lemma 5.3 For each $n$ and each $i = 1,\\ldots ,d$ let $ \\chi _{i,n}$ be such that $ \\sum _{j\\le i } \\chi _{j,n} := \\int \\limits _{G} \\int \\limits _{\\mathcal {F}} \\log \\left(|\\text{det}_{U_i(f)}(g)|\\right) d\\nu _n(f)d\\mu _n(g)$ .", "Then one has $\\lim \\limits _{n \\rightarrow +\\infty }\\chi _{i,n} = \\chi _{i}$ for each $i = 1,\\ldots ,d$ .", "We may apply the preceding lemma since $ |\\text{det}_{U_i(f)}(g)| \\le d!", "|\\log g|.", "$" ], [ "Quasi-independence and mutual information", "Let $ (\\Omega , \\mathbb {P})$ be a probability space.", "Given $X:\\Omega \\rightarrow \\mathcal {X}$ and $Y:\\Omega \\rightarrow \\mathcal {Y}$ taking values in Polish spaces, we say $ X $ and $ Y$ are quasi-independent if the distribution $ \\mathbb {P}_{X,Y} $ of $X,Y$ is absolutely continuous with respect to the product $ \\mathbb {P}_X\\otimes \\mathbb {P}_Y $ of the distributions of $X$ and $Y$ .", "We write $ f(x,y) $ or $ f_\\mathbb {P}(x,y) $ for the Radon-Nikodym derivative $ \\frac{d\\mathbb {P}_{X,Y} }{d(\\mathbb {P}_X\\otimes \\mathbb {P}_Y)}.$ In that case, the disintegration of the measure $ \\mathbb {P}_{X,Y} $ with respect to the projections over $ X $ and $ Y$ is respectively $ \\mathbb {P}^x = f(x,y) \\mathbb {P}_Y $ and $ \\mathbb {P}^y = f(x,y) \\mathbb {P}_X $ .", "For $ X $ and $ Y$ as above, define the mutual information between $X$ and $Y$ as $I(X,Y) = I_{\\mathbb {P}}(X,Y) = \\sup \\sum \\limits _{A \\in \\mathcal {A}}\\log \\left(\\frac{\\mathbb {P}_{(X,Y)}(A)}{(\\mathbb {P}_{X}\\times \\mathbb {P}_{Y})(A)}\\right)\\mathbb {P}_{(X,Y)}(A),$ where the supremum is over finite Borel partitions $\\mathcal {A}$ of $\\mathcal {X}\\times \\mathcal {Y}$By convention, $ \\log \\left(\\frac{\\mathbb {P}_{(X,Y)}(A)}{(\\mathbb {P}_{X}\\times \\mathbb {P}_{Y})(A)}\\right)\\mathbb {P}_{(X,Y)}(A) =0 $ if $ \\mathbb {P}_{X,Y} (A) = 0,$ the sum is $ +\\infty $ if there is one $ A \\in \\mathcal {A}$ such that $ \\mathbb {P}_{X,Y} (A) \\ne 0$ and $ (\\mathbb {P}_{X}\\times \\mathbb {P}_{Y})(A) =0.$.", "Directly from the definition one sees that $I(X,Y) = I(Y,X)$ .", "By Jensen inequality $0 \\le I(X,Y) \\le +\\infty $ with equality to 0 if and only if $X$ and $Y$ are independent.", "If $X$ takes countably many values and has finite entropy $H(X)$ in the sense of [49] one has $I(X,Y) \\le H(X)$ .", "It was shown in [14] that $I(X,Y)$ is the supremum over any sequence of partitions which generate the Borel $\\sigma $ -algebra in $\\mathcal {X}\\times \\mathcal {Y}$ .", "It was shown in [21] and [44] that if $I(X,Y) < +\\infty $ then $X$ and $Y$ are quasi-independent and $ I(X,Y) \\; = \\; \\int \\log f(x,y)\\, d\\mathbb {P}_{X,Y} (x,y) \\; =\\; \\int f(x,y)\\log f(x,y) \\,d\\mathbb {P}_X(x)d\\mathbb {P}_Y(y) .$ Let $Q$ be a partition of $\\lbrace 0,1,\\ldots d\\rbrace $ into intervals and $\\pi :\\mathcal {F}\\rightarrow \\mathcal {F}_Q$ the projection from $\\mathcal {F}$ into the corresponding space of partial flags.", "Recall that $\\nu _Q = (\\pi _Q)_*(\\nu )$ and set in what follows $\\nu _{Q,n} = (\\pi _Q)_*\\nu _n$ .", "We define the probability $\\mathbb {P}$ on $G \\times \\mathcal {F}_Q$ so that $ \\int h(g,f) d\\mathbb {P}(g,f) = \\int \\limits _{G} \\int \\limits _{\\mathcal {F}_Q} h(g,gf) d\\nu _Q(f) d\\mu (g)$ and the mutual information $I $ between the coordinate projections on $G \\times \\mathcal {F}_{Q}$ with respect to $\\mathbb {P}$ .", "Inequality (REF ) will follow directly from Proposition 5.4 With the above notations, $ I \\le \\sum \\limits _{\\ell _Q(i) < \\ell _Q(j)}\\chi _i - \\chi _j.$ Indeed, by proposition REF , the variables $G$ and $ \\mathcal {F}_Q$ are quasi-independent under $\\mathbb {P}$ , with density $ \\frac{d g_\\ast \\nu _Q}{d \\nu _Q} (f) $ .", "In particular, $ h(\\mathcal {F}_Q,\\mu ,\\nu _Q) = \\int \\log \\left(\\frac{dg_*\\nu _{Q}}{d\\nu _{Q}}(f)\\right) dg_*\\nu _Q(f)d\\mu (g) = I \\le \\sum \\limits _{\\ell _Q(i) < \\ell _Q(j)}\\chi _i - \\chi _j,$ which is the statement of (REF ).", "We analogously define the probability $\\mathbb {P}_n$ on $G \\times \\mathcal {F}_Q$ so that $ \\int h(g,f) d\\mathbb {P}_n (g,f) = \\int \\limits _{G} \\int \\limits _{\\mathcal {F}_Q} h(g,gf) d\\nu _{Q,n}(f) d\\mu _n(g)$ and the mutual information $I _n$ between the coordinate projections on $G \\times \\mathcal {F}_{Q}$ with respect to $\\mathbb {P}_n$ .", "Lemma 5.5 In the above context, $I \\le \\liminf \\limits _{n \\rightarrow +\\infty }I_n$ .", "By Dobrushin theorem the supremum may be taken over partitions whose sets belong to any generating set for the Borel $\\sigma $ -algebra.", "Therefore we may consider only partitions into sets satisfying $\\lim \\limits _{n \\rightarrow +\\infty }\\mathbb {P}_n(A) = \\mathbb {P}(A)$ .", "The inequality follows immediately.", "Lemma 5.6 For each $n$ one has $I_n = \\sum \\limits _{\\ell _Q(i) < \\ell _Q(j)}\\chi _{i,n} - \\chi _{j,n}$ .", "Let $\\varphi _n$ be the density of $\\nu _{Q,n}$ with respect to the rotationally invariant probability $\\eta _{Q}$ on $\\mathcal {F}_Q$ .", "By the Gelfand-Yaglom-Perez theorem one has $I_n &= \\int \\limits _{G \\times \\mathcal {F}_Q} \\log \\left(\\frac{dg_*\\nu _{Q,n}}{d\\nu _{Q,n}}(f)\\right)d\\mathbb {P}_n(g,f)\\\\ &= \\int \\limits _{G}\\int \\limits _{\\mathcal {F}_Q}\\log \\left(\\frac{dg_*\\nu _{Q,n}}{d\\nu _{Q,n}}(gf)\\right)d\\nu _{Q,n}(f)d\\mu _n(g)\\\\ &= \\int \\limits _{G}\\int \\limits _{\\mathcal {F}_Q}\\log \\left(\\frac{\\varphi _n(f)}{\\varphi _n(gf)}\\frac{dg_*\\eta _{Q}}{d\\eta _{Q}}(gf)\\right)d\\nu _{Q,n}(f)d\\mu _n(g)$ By $\\mu _n$ -stationarity of $\\nu _{Q,n}$ the integrals involving $\\varphi _n$ cancel, and one obtains $I_n &= \\int \\limits _{G} \\int \\limits _{\\mathcal {F}_Q} \\log \\left(\\frac{dg_*\\eta _{Q}}{d\\eta _{Q}}(gf)\\right)d\\nu _{Q,n}(f)d\\mu _n(g).$ We have the following explicit formula for $\\frac{dg_*\\eta _{Q}}{d\\eta _Q}(gf)$ Proposition 5.7 For $Q = \\lbrace q_0 = 0 < q_1 < \\cdots < q_k = d\\rbrace $ and $\\eta = \\eta _Q$ the rotation invariant measure then $\\frac{dg\\eta }{d\\eta }(gx) = \\frac{|\\text{det}_{U_{q_1}(x)} (g)|^{q_2} |\\text{det}_{U_{q_2}(x)} (g)|^{q_3-q_1} \\cdots |\\text{det}_{U_{q_{k-1}}(x)} (g)|^{d-q_{k-2}}}{|\\text{det}(g)|^{q_{k-1}}}.$ In particular on the space of full flags one has $\\frac{dg\\eta }{d\\eta }(gx) = \\frac{|\\text{det}_{U_{1}(x)} (g)|^{2} |\\text{det}_{U_{2}(x)} (g)|^{2} \\cdots |\\text{det}_{U_{k-1}(x)} (g)|^{2}}{|{\\text{det}(g)|^{d-1}}}.$ Proposition REF is proven in the next subsection.", "Given proposition REF , we may write $I_n &= \\int \\limits _{G} \\int \\limits _{\\mathcal {F}_Q} \\log \\left(\\frac{|\\text{det}_{U_{q_1}(f)}(g)|^{q_2}|\\text{det}_{U_{q_2}(f)}(g)|^{q_3-q_1} \\ldots |\\text{det}_{U_{q_{k-1}}(f)}(g)|^{d-q_{k-2}}}{|\\text{det}(g)|^{q_{k-1}}}\\right)d\\nu _{Q,n}(f)d\\mu _n(g)\\\\ &= \\left(\\sum \\limits _{j = 1}^{k-1} (q_{j+1}-q_{j-1})\\sum \\limits _{i = 1}^{q_j} \\chi _{i,n}\\right) - q_{k-1}\\sum \\limits _{i = 1}^{d} \\chi _{i,n}\\\\ &= \\sum \\limits _{\\ell _{Q}(i) < \\ell _{Q}(j)}\\chi _{i,n} - \\chi _{j,n},$ where the last equality follows by direct computation.", "Using lemma REF , proposition REF follows.", "Corollary 5.8 Let $ \\mu \\in \\mathcal {M}(G), \\nu $ the stationary measure on $ \\mathcal {F}$ .", "Then, $ h( \\mathcal {F}, \\mu ,\\nu ) \\; \\le \\; \\sum _{0 < i<j \\le d} \\chi _i - \\chi _j \\; < \\; + \\infty .$ In particular, for $ \\mu $ -a.e.", "$g \\in G , g_\\ast \\nu $ is absolutely continuous with respect to $ \\nu $ and the function $ \\log \\frac{dg_\\ast \\nu }{d\\nu } $ is integrable with respect to $ g_\\ast \\nu .$" ], [ "Proof of proposition ", "Lemma 5.9 Let $Q = \\lbrace 0 < i < d\\rbrace $ and $\\eta = \\eta _Q$ be the unique rotationally invariant probability on $\\mathcal {F}_Q$ the Grasmannian manifold of $i$ -dimensional subspaces of $\\mathbb {R}^d$ .", "Then $\\frac{dg\\eta }{d\\eta }(gx) = \\frac{|\\text{det}_x (g)|^d}{|\\text{det}(g) |^i},$ for all $g \\in \\operatorname{GL}_d(\\mathbb {R})$ .", "Let $\\pi _x:\\mathbb {R}^d \\rightarrow x$ be the orthogonal projection onto $x$ and $\\pi _{\\mathbb {R}^d/x}:\\mathbb {R}^d \\rightarrow \\mathbb {R}^d/x$ the canonical projection.", "The quotient space $\\mathbb {R}^d/x$ is endowed with the inner product such that the projection is an isometry when restricted to the orthogonal complement of $x$ .", "Since $\\eta $ is invariant under orthogonal transformations we may compose $g$ with such transformations on both sides.", "In particular we may assume that $g(x) = x$ and therefore $g$ induces a transformation on $\\mathbb {R}^d/x$ which we denote by $g$ as well.", "We may further assume that: There is an orthogonal basis $v_1,\\ldots ,v_i$ of $x$ and positive eigenvalues $\\sigma _1,\\ldots ,\\sigma _i > 0$ such that $gv_k = \\sigma _k v_k$ for $k = 1,\\ldots ,i$ .", "There is an orthogonal basis $w_1,\\ldots ,w_{d-i}$ of $\\mathbb {R}^d/x$ and positive eigenvalues $\\mu _1,\\ldots ,\\mu _{d-i}> 0$ such that $gw_k = \\mu _k w_k$ for $k = 1,\\ldots ,d-i$ .", "Notice that $\\sigma _1\\cdots \\sigma _i = |\\text{det}_x (g)|$ while $\\mu _1\\cdots \\mu _{d-i} = |\\text{det}(g) |/|\\text{det}_x (g)|$ .", "A local parametrization of $\\mathcal {F}_Q$ around $x$ is given by identifying each linear mapping $\\varphi :x \\rightarrow \\mathbb {R}^d/x$ with the subspace $x_\\varphi $ such that $v \\in x_\\varphi $ if and only if $\\varphi (\\pi _x(v)) = \\pi _{\\mathbb {R}^d/x}(v)$ .", "Identify each $\\varphi $ with its matrix $\\left(a_{lk}\\right)_{k,l}$ where $\\varphi (v_k) = \\sum \\limits _{l}a_{lk} w_l$ .", "With this identification, we define a volume form $\\omega $ on $\\mathcal {F}_Q$ such that in any coordinates constructed as above one has $\\omega (0) = \\pm \\bigwedge _{k,l} da_{lk}$ .", "Since $\\omega $ is orthogonally invariant it must define a volume on $\\mathcal {F}_Q$ which is a constant multiple of $\\eta $ .", "In this chart the action of $g$ on $\\mathcal {F}_Q$ maps $\\varphi $ to $g_{\\vert \\mathbb {R}^d/x}\\circ \\varphi \\circ g^{-1}_{\\vert x}$ so that $g_{\\vert \\mathbb {R}^d/x}\\circ \\varphi \\circ g^{-1}_{\\vert x}(v_k) = g\\varphi (\\sigma _k^{-1}v_k) = \\sum \\limits _{l}a_{lk}\\frac{\\mu _{l}}{\\sigma _k}w_l.$ It follows that the pull-back under $g$ of $\\omega $ satisfies $g^*\\omega (0) = \\prod \\limits _{k,l}\\frac{\\mu _l}{\\sigma _k} \\omega (0) = \\frac{|\\text{det}g|^i}{|\\text{det}_x g|^d}\\omega (0)$ , which implies the desired claim.", "Corollary 5.10 If $Q$ is obtained by a splitting an interval $k+1 < k+m$ of $Q^{\\prime }$ into $k+1 < k+i < k+m$ then $\\frac{dg\\eta _{Q,Q^{\\prime }}^x}{d\\eta _{Q,Q^{\\prime }}^{gx}}(gx) = \\frac{|\\text{det}_{U_{k+i}(x)} (g)|^{m}}{|\\text{det}_{U_{k}(x)} (g)|^{m-i}|\\text{det}_{U_{k+m}(x)} (g)|^i},$ for all $g \\in \\operatorname{GL}_d(\\mathbb {R})$ and all $x \\in \\mathcal {F}_Q$ .", "The fiber of the projection from $\\mathcal {F}_Q$ to $\\mathcal {F}_{Q^{\\prime }}$ which contains $x$ is naturally identified with the the $i$ -dimensional Grasmannian of $U_{k+m}(x)/U_k(x)$ .", "The Jacobian of $g$ as a mapping from $U_{k+m}(x)/U_k(x)$ to its image is $|\\text{det}_{U_{k+m}(x)} (g)|/|\\text{det}_{U_k(x)} (g)|$ .", "The Jacobian of the restriction of $g$ to the subspace of $U_{k+m}(x)/U_k(x)$ represented by $U_{k+i}$ is $|\\text{det}_{U_{k+i}} (g)|/|\\text{det}_{U_{k}} (g)|$ .", "The result follows replacing these values in the previous lemma.", "Proposition REF follows from the previous corollary by splitting $\\lbrace 0 < d\\rbrace $ successively into $\\lbrace 0 < q_{k-1} < d\\rbrace $ , $\\lbrace 0 < q_{k-2} < q_{k-1} < d\\rbrace $ , etc." ], [ "Conditional mutual information", "Given $X:\\Omega \\rightarrow \\mathcal {X} , Y:\\Omega \\rightarrow \\mathcal {Y}$ and $Z:\\Omega \\rightarrow \\mathcal {Z} $ taking values in polish spaces, one may define for $\\mathbb {P}_\\mathbb {Z}$ -a.e.", "$z \\in \\mathcal {Z} $ the disintegrations $ \\mathbb {P}^z_{X\\times Y} $ with respect to the projections on $\\mathcal {Z}$ and the mutual information $ I_{\\mathbb {P}^z}(X,Y)$ of $X$ and $Y$ given $Z$ using these conditional distributions $\\mathbb {P}^{z}$ of $(X,Y)$ given $Z$ .", "Define the conditional mutual information $ I(X,Y\\vert Z)$ of $ X $ and $Y$ given $Z$ by $ I(X,Y \\vert Z) \\; := \\; I_{\\mathbb {P}^z} (X,Y) $ and the conditional mutual entropy $ H(X,Y\\vert Z)$ of $ X $ and $Y$ given $Z$ by $ H(X,Y \\vert Z) \\; := \\; \\int I(X,Y\\vert Z) \\, d\\mathbb {P}_Z(z) .$ Observe that $X$ and $Y$ are conditionally independent given $Z$ if, and only if, $ I(X,Y \\vert Z) = 0 {\\textrm { almost everywhere}}, $ if, and only if, $ H(X,Y \\vert Z) = 0 .$ Observe also that, by definition, $ I(X,Y \\vert Z) = I(Y,X \\vert Z) $ .", "If $W,X,Y,Z$ are measurable functions from $\\Omega $ into Polish spaces then one has the following chain rule $ H(W,(X,Y)\\vert Z) &=&H(W,X\\vert Y,Z) + H(W,Y\\vert Z).$ Let $ \\mathbb {P}$ be the measure on $ G \\times \\mathcal {F}$ introduced in section REF : for any positive measurable function $h $ on $ G \\times \\mathcal {F}$ , $ \\int h(g,f) \\,d\\mathbb {P}(g, f ) \\; = \\; \\int h(g, gf) \\, d\\mu (g) d\\nu (f).$ We showed that $ h(\\mathcal {F}, \\mu ,\\nu ) = I_\\mathbb {P}(G, \\mathcal {F}) < +\\infty .$ Lemma 6.1 The measure $\\mathbb {P}$ is the distribution of the variables $ (g_{-1}, E_- ) $ on $ (\\Omega , m).$ In particular, $ g_{-1}(\\omega ) $ and $ E_-(\\omega ) $ are quasi-independent.", "We indeed have, by invariance of $m,$ for any positive $h $ on $ G \\times \\mathcal {F}$ $ \\int h(g_{-1}(\\omega ) , E_- (\\omega )) \\, dm(\\omega ) &=& \\int h(g_{-1}(\\sigma \\omega ), E_- (\\sigma \\omega )) \\, dm(\\omega )\\\\ &= & \\int h(g_{0}(\\omega ), g_0(\\omega )E_- (\\omega )) \\, dm(\\omega ) = \\int h \\, d\\mathbb {P}.$ Using proposition REF , we may write, for $m_+$ -a.e.", "$ \\omega _+$ : $ h(\\mathcal {F}, \\mu ,\\nu ) \\;=\\; I (g_{-1}, E_-) \\;= \\; I(g_{-1}, E_- \\vert E_+) \\;= \\; I (g_{-1}, E_-\\vert g_0, g_1, \\ldots ).$ Let $T$ be an admissible topology.", "Recall that we defined the entropy $\\kappa _{T} $ by equation (REF ) $ \\kappa _{T} \\; := \\; \\int \\log \\frac{ dg_\\ast \\nu _T^{g^{-1} f^{\\prime }} }{ d\\nu _T^{f^{\\prime }} }(g,y)\\, dg_\\ast \\nu _T^{g^{-1} f^{\\prime }}(y) d\\nu ^{\\prime }(f^{\\prime }) d\\mu (g) .$ For $m$ -a.e.", "$ \\omega \\in \\Omega $ write $ E_T (\\omega ) \\in \\mathcal {X}_T^{E_+(\\omega )} $ for $ E_T(\\omega ) := F_T (E_-(\\omega ), E_+ (\\omega ) ) .$ The next three propositions give other useful expressions for $\\kappa _T $ .", "Proposition 6.2 With the above notations, we have $ \\kappa _T \\; = \\; H(g_{-1} , E_T\\vert E_+ ) \\; <\\; +\\infty .", "$ Observe first that the RHS of (REF ) is finite because $ H (g_{-1} , E_T \\vert E_+ ) \\le H (g_{-1} , E_-\\vert E_+) < +\\infty .", "$ The distribution of $ g_{-1} (\\omega ) $ given $E_+ (\\omega )$ is $\\mu $ and the distribution of $E_T (\\omega )$ given $E_+ (\\omega )$ is $ \\nu _T^{ E_+(\\omega ) } $ .", "Remains to compute the joint distribution of $ (g_{-1}(\\omega ),E_T(\\omega ) )$ given $E_+ (\\omega )$ .", "We claim that it projects to $\\mu $ with conditional measures given by $g_{-1}(\\omega )_*\\nu _T^{g_{-1}(\\omega )^{-1}E_+(\\omega )}$ .", "It follows that, for $m_+$ -a.e.", "$ E_+ (\\omega _+),$ $ I (g_{-1}(\\omega ),E_T (\\omega ) \\vert E_+(\\omega ) ) = \\int \\log \\frac{ dg_\\ast \\nu _T^{g^{-1} E_+(\\omega )} }{ d\\nu _T^{E_+(\\omega ) } }(g,y)\\, dg_\\ast \\nu _T^{g^{-1} E_+(\\omega )} (y) d\\mu (g).$ By integrating in $E_+ (\\omega ) $ , i.e.", "in $f^{\\prime }$ with respect to the measure $ \\nu ^{\\prime },$ we find that $ H (g_{-1} , E_T\\vert E_+ ) $ is given by $ H (g_{-1} , E_T\\vert E_+ ) \\; = \\; \\int \\log \\frac{ dg_\\ast \\nu _T^{g^{-1} f^{\\prime }} }{ d\\nu _T^{f^{\\prime }} }(g,y)\\, d\\mu (g) dg_\\ast \\nu _T^{g^{-1} f^{\\prime }}(y) d\\nu ^{\\prime }(f^{\\prime }), $ which is the formula defining $\\kappa _T $ in relation (REF ).", "We prove the claim: the distribution of $(g,f,f^{\\prime }) = ( g_{-1}, E_-, E_+) (\\omega ) $ is given by $ d\\mu (g) dg_\\ast \\nu (f) d\\nu ^{\\prime } (f^{\\prime }) $ and the distribution of $ ( g_{-1}(\\omega ), E_T(\\omega ))= ( g_{-1}(\\omega ), F_T (E_-(\\omega ), E_+(\\omega ))) $ is given by $ d\\mu (g) (F_T)_\\ast \\left[dg_\\ast \\nu (f) d\\nu ^{\\prime } (f^{\\prime })\\right].", "$ Remains to compute $ (F_T)_\\ast \\left[dg_\\ast \\nu (f) d\\nu ^{\\prime } (f^{\\prime })\\right]$ .", "We have $ (F_T)_\\ast \\left[dg_\\ast \\nu (f) d\\nu ^{\\prime } (f^{\\prime })\\right] &=& (F_T\\circ g)_\\ast \\left[d\\nu (f) d(g^{-1})_\\ast \\nu ^{\\prime } (f^{\\prime })\\right] \\\\ &=&(g\\circ F_T)_\\ast \\left[\\frac{d(g^{-1})_\\ast \\nu ^{\\prime } }{d\\nu ^{\\prime }}(f^{\\prime }) d\\nu (f) d\\nu ^{\\prime } (f^{\\prime })\\right] ,$ where we used that $ F_T \\circ g (x,x^{\\prime }) := F_T(gx, gx^{\\prime }) = g \\circ F_T (x,x^{\\prime })$ and that, by Theorem REF , $h(\\mathcal {F}, \\mu ^{\\prime } , \\nu ^{\\prime }) < +\\infty $ , which guarantees that $ \\frac{d(g^{-1})_\\ast \\nu ^{\\prime } }{d\\nu ^{\\prime }}(f^{\\prime }) $ exists for $ \\mu \\times \\nu ^{\\prime }$ -a.e.", "$(g,f^{\\prime })$ .", "We then take into account that $F_T $ is the identity on the second coordinate, and that $ (F_T)_\\ast \\left[\\nu \\times \\nu ^{\\prime } \\right] \\; = \\; \\int \\nu _T^{f^{\\prime }} \\, d\\nu (f^{\\prime })$ to obtain $ (g\\circ F_T)_\\ast \\left[\\frac{d(g^{-1})_\\ast \\nu ^{\\prime } }{d\\nu ^{\\prime }}(f^{\\prime }) d\\nu (f) d\\nu ^{\\prime } (f^{\\prime })\\right]&=& g_\\ast \\left[ \\frac{d(g^{-1})_\\ast \\nu ^{\\prime } }{d\\nu ^{\\prime }}(f^{\\prime }) d\\nu _T^{f^{\\prime }} (x) d\\nu ^{\\prime } (f^{\\prime })\\right] \\\\ &=& g_\\ast \\left[ d\\nu _T^{f^{\\prime }} (x) d(g^{-1})_\\ast \\nu ^{\\prime } (f^{\\prime })\\right]\\\\&=& dg_\\ast \\nu _T^{g^{-1} f^{\\prime }} d\\nu ^{\\prime }(f^{\\prime }).", "$ It follows that the distribution of $(g,x) = g_{-1}(\\omega ),E_T(\\omega ) $ given $f^{\\prime } = E_+ (\\omega )$ is indeed $ d\\mu (g) dg_\\ast \\nu _T^{g^{-1} f^{\\prime }}(x) $ .", "Proposition 6.3 We also have $ \\kappa _T \\;=\\; H (g_{-1} , E_T (\\omega ) \\vert ( E_+(\\omega ) , g_0, g_1, \\ldots ) )\\;= \\; H (g_{-1} , E_T (\\omega ) \\vert \\omega _+) .$ By proposition REF , the conditional measures on $E_{T_1}(\\omega ) $ with respect to $E_+ (\\omega ) $ and $E_+ (\\omega ), g_0, g_1, \\ldots $ coincide with $ \\nu _{T_1}^{E_+(\\omega )}.$ We have, for all admissible $T$ , $E_T(\\omega ) = \\pi _{T_1,T} E_{T_1} (\\omega ), $ so that the conditional measures on $E_{T}(\\omega ) $ with respect to $E_+ (\\omega ) $ and $E_+ (\\omega ), g_0, g_1, \\ldots $ coincide with $(\\pi _{T_1,T} )_\\ast \\nu _{T_1}^{E_+(\\omega )}= \\nu _{T}^{E_+(\\omega )}.$ The projection $ \\omega \\in \\Omega \\mapsto E_T(\\omega ) \\in \\mathcal {X}_T^{E_+(\\omega _+)} $ admits disintegrations that we denote $ m_T^{x}.$ We still denote by $ m_T^{x}$ the projection of $ m_T^{x}$ to $\\Omega _-$ .", "For instance, $m_{T_0}^{f^{\\prime }} = m_- $ for $\\nu ^{\\prime }$ -a.e.", "$f^{\\prime }$ , $ m_{T_1}^{f} $ is the distribution of $ \\omega _- $ given the unstable flag $f$ .", "By Proposition REF and (REF ), the variables $ g_{-1} $ and $ E_T $ are quasi-independent (Indeed, since $ E_+ $ is $ E_T$ measurable, $ I(g_{-1}, E_T) = I(g_{-1}, E_+) + H (g_{-1} , E_T \\vert E_+ ) $ equals $ \\kappa _T$ and thus is finite).", "It follows that the density $ f $ of the measure $ m_T^{E_T(\\omega )}$ restricted to $ g_{-1}(\\omega ) $ with respect to $ \\mu $ is given by $f(\\omega )= \\frac{ d(g_{-1} (\\omega ))_\\ast \\nu _T^{(g_{-1}(\\omega ))^{-1} E_+(\\omega )} }{ d\\nu _T^{E_+(\\omega ) } }(E_T(\\omega )) .$ This yields another formula for $ \\kappa _T :$ $ \\kappa _T \\; = \\; \\int \\left(\\log \\frac{dm_T^{E_T(\\omega )}}{d\\mu } (g_{-1}(\\omega )) \\right) \\, dm(\\omega )\\; =\\; \\int \\log f(\\omega )\\, dm(\\omega )$ Proposition 6.4 We have, for $m$ -a.e.", "$\\omega $ , $\\kappa _T = \\lim \\limits _{n \\rightarrow \\infty } \\frac{1}{n} \\log \\frac{dm_T^{E_T(\\omega )}}{d\\otimes _1^n \\mu } (g_{-1}(\\omega ), \\ldots , g_{-n}(\\omega )).$ We claim that the ratio $f^{(n)}(\\omega ) := \\frac{dm_T^{E_T(\\omega )}}{d\\otimes _1^n \\mu } (g_{-1}(\\omega ), \\ldots , g_{-n}(\\omega ) )$ is given by $ f ^{(n)} (\\omega )= \\prod _{j=0}^{n-1} f(\\sigma ^{-j} \\omega ).$ Then $\\frac{1}{n} \\log f^{(n)} (\\omega )$ is an ergodic average of the function $ \\log f(\\omega ).$ By the Birkhoff ergodic theorem this average converges to $ \\int \\log f (\\omega ) \\, dm(\\omega ) = \\kappa _T$ (the function $\\log f $ is integrable by Corollary REF ).", "We prove the claim: by the law of composition of conditional probabilities, the ratio $ f^{(n)}(\\omega )/ f^{(n-1)}(\\omega ) $ is the density with respect to $ \\mu $ of the conditional measures of $m$ relative to $ (g_{-1}(\\omega ), \\ldots , g_{-n+1}(\\omega ), E_T(\\omega ) ) $ restricted to $ g_{-n}(\\omega ).$ But the $\\sigma $ -algebra generated by $ (g_{-1}(\\omega ), \\ldots , g_{-n+1}(\\omega ), E_T(\\omega ) ) $ is the same as the $\\sigma $ -algebra generated by $ (E_T(\\sigma ^{-n+1}\\omega ), g_0(\\sigma ^{-n+1}\\omega ), \\ldots , g_{n-2}(\\sigma ^{-n+1}\\omega )).$ By proposition REF and stationarity, this is the density of the measure $ m_T^{E_T(\\sigma ^{-n+1}\\omega )}$ restricted to $ g_{-n }(\\omega ) = g_{-1}(\\sigma ^{-n+1} \\omega )$ , that is $f(\\sigma ^{-n+1}\\omega ).$" ], [ "Entropy difference", "For any pair of admissible topologies $T \\prec T^{\\prime }$ we defined $\\kappa _{T,T^{\\prime }} = \\kappa _T - \\kappa _{T^{\\prime }}$ .", "By relation (REF ), $\\kappa _{T,T^{\\prime }} <+\\infty $ and if $T \\prec T^{\\prime } \\prec T^{\\prime \\prime }$ one has $\\kappa _{T,T^{\\prime \\prime }} = \\kappa _{T,T^{\\prime }} + \\kappa _{T^{\\prime },T^{\\prime \\prime }}.$ By the chain rule for conditional mutual entropy, relation(REF ) and proposition REF , we have $\\kappa _{T,T^{\\prime }} = H(g_{-1},E_T\\vert E_{T^{\\prime }}) = H(g_{-1},E_T\\vert (E_{T^{\\prime }},g_0,g_1,\\ldots ))= H(g_{-1},E_T\\vert (E_{T^{\\prime }},\\omega _+)).$ Recall that in the introduction we defined, for $\\nu ^{\\prime }$ -a.e.", "$f^{\\prime } \\in \\mathcal {F},$ $ \\nu _{T^{\\prime }}^{f^{\\prime }} $ -a.e.", "$ x^{\\prime } \\in \\mathcal {X}_{T^{\\prime }}^{f^{\\prime }} , $ $ \\nu _{T,T^{\\prime }}^{x^{\\prime }} $ as a family of disintegrations of the measure $ \\nu _T^{f^{\\prime }} $ with respect to $ \\pi _{T,T^{\\prime }} .$ Then, $ \\frac{ dg_\\ast \\nu _T^{f^{\\prime }} }{ d\\nu _T^{gf^{\\prime }} }(y) = \\frac{ dg_\\ast \\nu _{TT^{\\prime }}^{ x^{\\prime }} }{ d\\nu _{T,T^{\\prime }}^{gx^{\\prime }} }(y) \\frac{ dg_\\ast \\nu _{T^{\\prime }}^{ f^{\\prime }} }{ d\\nu _{T^{\\prime }}^{gf^{\\prime }} }(x^{\\prime }) .$ Proposition 6.5 If $T \\prec T^{\\prime }$ are admissible topologies then $\\kappa _{T,T^{\\prime }} = \\int \\limits _{\\Omega }\\log \\left(\\frac{dg_0(\\omega )\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}}{d\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma (\\omega ))}}(E_T(\\sigma (\\omega ))\\right) dm(\\omega ).$ We write $\\kappa _T $ as $ \\kappa _{T} \\; = \\; \\mathbb {E} \\left[I (g_{-1},E_T \\vert E_+ ) (\\omega ) \\right].$ Reporting the explicit expression for $ I (g_{-1}(\\omega ),E_T (\\omega ) \\vert E_+(\\omega ) ),$ we have $ \\kappa _T &=& \\mathbb {E}\\left[ \\log \\frac{d (g_{-1} (\\omega )) _\\ast \\nu _T^{(g_{-1}(\\omega ) )^{-1}E_+(\\omega )} }{d\\nu _T^{E_+(\\omega )}} (E_T(\\omega ))\\right]\\\\ &=& \\mathbb {E}\\left[\\log \\frac{ d(g_{0} (\\omega )) _\\ast \\nu _T^{E_+(\\omega )} }{d\\nu _T^{E_+(\\sigma \\omega )}} (E_T(\\sigma \\omega ))\\right],$ where the second line follows by $\\sigma $ -invariance.", "The proposition follows by making the difference $ \\kappa _{T,T^{\\prime }} = \\kappa _T -\\kappa _{T^{\\prime }} $ and applying (REF ).", "Fix $ T \\prec T^{\\prime } .$ For a.e.", "$ x \\in (\\pi _{T,T^{\\prime }})^{-1} (E_{T^{\\prime }} (\\omega ) ), $ set $f_\\omega (x) := \\frac{dg_0(\\omega )\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}}{d\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma (\\omega ))}}(x).$ From proposition REF , follows $ \\kappa _{T,T^{\\prime }} \\; =\\; \\mathbb {E} \\left[ \\log f_\\omega (E_T(\\omega )) \\right].$ Recall formula (REF ) for $\\kappa _T $ and $\\kappa _{T^{\\prime }} .$ With the same notations, we have $ \\kappa _{T,T^{\\prime }}\\; = \\; \\int \\left(\\log \\frac{dm_T^{E_T(\\omega )}}{dm_{T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}}(\\omega )) \\right) \\, dm(\\omega )$ and the following corollary of Proposition REF Corollary 6.6 Assume $T \\prec T^{\\prime }$ are admissible topologies.", "Then, we have, for $m$ -a.e.", "$\\omega $ , $\\kappa _{T,T^{\\prime }} = \\lim \\limits _{n \\rightarrow \\infty } \\frac{1}{n} \\log \\frac{dm_T^{E_T(\\omega )}}{dm_{T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}} (g_{-1}(\\omega ), \\ldots , g_{-n}(\\omega )).$" ], [ "Zero entropy difference", "Proposition 6.7 If $\\kappa _{T,T^{\\prime }} = 0$ then $\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}$ is the point mass at $E_{T}(\\omega )$ for $m$ -a.e.", "$\\omega \\in \\Omega $ .", "In particular, it is exact dimensional with dimension 0.", "By proposition REF , we may assume that $ T \\overset{1}{\\prec } T^{\\prime }$ and that $ i<j$ are such that $ T^{\\prime }(i) = T(i) \\cup \\lbrace j\\rbrace .$ By proposition REF , we have $\\kappa _{T,T^{\\prime }} = \\int \\limits _{\\Omega }\\log \\left(\\frac{dg_0(\\omega )\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}}{d\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma (\\omega ))}}(E_T(\\sigma (\\omega ))\\right) dm(\\omega ).$ Therefore, by Jensen inequality, if $\\kappa _{T,T^{\\prime }} = 0$ then for $m$ -almost every $\\omega $ we have $g_{0}(\\omega )\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}}(\\omega ) = \\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma (\\omega ))}$ .", "Since $m$ is $\\sigma $ -invariant, we obtain that $m$ -almost everywhere one has $\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )} = g_{-1}(\\omega )\\ldots g_{-n}(\\omega )\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma ^{-n}(\\omega ))}$ for all $n \\ge 1$ .", "Observe that, since $E_+$ and $E_-$ are in general position, one has $E_{T^{\\prime }}(\\omega )_{T^{\\prime }(i) \\setminus \\lbrace i\\rbrace } \\ne E_{T}(\\omega )_{T(i)}$ for $m$ -almost every $\\omega \\in \\Omega $ , and therefore $\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}\\left(\\lbrace E_{T^{\\prime }}(\\omega )_{T^{\\prime }(i) \\setminus \\lbrace i\\rbrace }\\rbrace \\right) = 0$ .", "Using the coordinate $ \\varphi _{x(\\omega )} $ (see lemma REF ), we have $ \\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )} [\\varphi _{x(\\omega )}((-\\pi /2, \\pi /2))] = 1.$ By Poincaré recurrence, for almost every $ \\omega \\in \\Omega _- $ , there exists an infinite sequence $n_k, k\\in \\mathbb {N}$ and $ \\alpha >0$ such that $ \\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma ^{-n_k}(\\omega ))} $ converges to $\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}$ as $ k \\rightarrow \\infty $ and $\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}\\left( (\\varphi _{x(\\sigma ^{-n_k} (\\omega ))} ) ((-\\alpha , \\alpha ))\\right) > \\alpha .$ Hence, for those $ \\omega $ for which lemma REF hold, $ g_{-1}(\\omega )\\ldots g_{-n_k}(\\omega )$ sends any neighborhood of $ E_{T}(\\sigma ^{-n_k}(\\omega )) $ to a small neighborhood of $ E_{T}(\\omega ) $ .", "This is possible only if $ \\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )} $ is concentrated at $ E_{T}(\\omega ) $ ." ], [ "Proof of theorem ", "Theorem REF is an almost everywhere statement.", "It uses a telescoping argument mixed with weak type (1,1) techniques as in the proof of Shannon-McMillan-Breiman theorem for finite entropy partitions.", "The proof follows [29] and [41].", "In this section, we assume that $ T \\overset{1}{\\prec } T^{\\prime }$ and that $ i<j$ are such that $ T^{\\prime }(i) = T(i) \\cup \\lbrace j\\rbrace .$ Let $ \\chi := \\chi _{T,T^{\\prime }}, \\kappa := \\kappa _{T,T^{\\prime }}.", "$ we want to show that for $ \\nu ^{\\prime } $ -a.e.", "$ f^{\\prime } \\in \\mathcal {F},$ $ \\nu _{T^{\\prime }}^{f^{\\prime }} $ -a.e.", "$ x^{\\prime } \\in \\mathcal {X}_{T^{\\prime }}^{f^{\\prime }}, $ the conditional measure $\\nu _{T,T^{\\prime }}^{x^{\\prime }} $ is exact-dimensional with dimension $ \\kappa / \\chi .$" ], [ "Length of stationary neighborhoods", "Let $\\pi \\:= \\pi _{T,T^{\\prime }} $ be the projection from $\\mathcal {X}_{T}$ to $\\mathcal {X}_{T^{\\prime }}$ and consider the coordinates given by lemma REF on $\\pi ^{-1}(E_{T^{\\prime }}(\\omega )),$ setting $x = E_{T}(\\omega )$ and $x^{\\prime } = E_{T^{\\prime }}(\\omega )$ .", "For each $\\alpha , 0 < \\alpha < \\pi /2$ let $N^{\\alpha }(\\omega )$ (respectively $N^{\\alpha ,+}(\\omega ), N^{\\alpha ,-}(\\omega )$ ) the set $ \\varphi ((-\\alpha , \\alpha )) $ (respectively $\\varphi ( [0, \\alpha )), \\varphi ((-\\alpha , 0] )$ .", "Recall that we denoted by $ \\eta $ the rotation invariant probability measure on $\\pi ^{-1}(E_{T^{\\prime }}(\\omega )).$ The measure $ \\varphi _\\ast du $ has a bounded continuous density with respect to $ \\eta $ (the bound depends on $\\omega $ ).", "From lemma REF follows Lemma 7.1 For all $\\alpha , 0 < \\alpha <\\pi /2$ , for $m$ -a.e.", "$\\omega \\in \\Omega $ one has $\\eta \\left(g_{-1}(\\omega )\\ldots g_{-n}(\\omega )N^{\\alpha }(\\sigma ^{-n}(\\omega ))\\right) = e^{-\\chi n + o(n)}\\text{ as }n \\rightarrow +\\infty ,$ and the same holds for $N^{\\alpha ,+}$ and $N^{\\alpha ,-}$ .", "In the sequel, we set, for each $\\alpha , 0 < \\alpha < \\pi /2$ and for each $n \\ge 1$ , $N_n^\\alpha (\\omega ) $ for the interval in $ \\pi ^{-1}(E_{T^{\\prime }}( \\omega )) $ given by $N_n^{\\alpha }(\\omega )= (g_{-1}(\\omega )\\ldots g_{-n}(\\omega ))(N^{\\alpha } (\\sigma ^{-n}(\\omega )).$ $ N_n^{\\alpha ,+} (\\omega ), N_n^{\\alpha ,-} (\\omega ) $ are defined similarly, $ N_n^{\\alpha ,\\pm } (\\omega )$ is a choice of one of the three intervals." ], [ "Probability of stationary neighborhoods", "Theorem REF will follow by comparing lemma REF and the following Proposition 7.2 For all $\\alpha , 0 < \\alpha <\\pi /2$ , for $m$ -a.e.", "$\\omega \\in \\Omega $ , one has $\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}(N_n^{\\alpha ,\\pm }(\\omega )) = e^{-\\kappa n + o(n)}\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma ^{-n}(\\omega ))}(N^{\\alpha ,\\pm ^{\\prime } }(\\sigma ^{-n}(\\omega )))\\text{ as }n \\rightarrow +\\infty .$ In this formula, the sign $\\pm ^{\\prime }$ is not necessarily the same as $ \\pm .$ Recall that we set $f_\\omega (x) = \\frac{dg_0(\\omega )\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}}{d\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma (\\omega ))}}(x)$ , and that by (REF ), $ \\kappa \\; =\\; \\int \\log f_\\omega (E_T(\\omega )) dm (\\omega ) \\; =\\; \\mathbb {E} \\left[ \\int _{\\pi ^{-1}(E_{T^{\\prime }}(\\omega ))}\\log f_\\omega (x)\\, d\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}(x) \\right] .$ Since $ \\kappa < +\\infty ,$ for $m$ -a.e.", "$ \\omega $ , $ \\int _{\\pi ^{-1}(E_{T^{\\prime }}(\\omega ))} \\log f_\\omega (x)\\, d\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}(x) < +\\infty .$ Set for each $\\alpha , 0 < \\alpha < \\pi /2$ , for each $n \\ge 1$ , $f_{n}^{\\alpha , \\pm }(\\omega ) = \\frac{\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}(N_n^{\\alpha , \\pm }(\\omega ))}{\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma (\\omega ))}(g_0(\\omega ) N_n^{\\alpha , \\pm }(\\omega ))} = \\frac{\\int \\limits _{g_0(\\omega )N_n^{\\alpha , \\pm }(\\omega )}f_{\\omega }(x)d\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma (\\omega ))}(x)}{\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma (\\omega ))}(g_0(\\omega ) N_n^{\\alpha , \\pm }(\\omega ))}.$ Lemma 7.3 For all $\\alpha , 0 < \\alpha <\\pi /2$ , for $m$ -a.e.", "$\\omega \\in \\Omega $ one has $\\lim \\limits _{n \\rightarrow +\\infty }f_n^{\\alpha ,\\pm }(\\omega ) = f_{\\omega }(E_{T}(\\sigma (\\omega )))$ .", "Furthermore $\\int \\sup \\limits _{n \\ge 1}|\\log (f_n^{\\alpha ,\\pm }(\\omega ))| \\,dm(\\omega ) < +\\infty $ .", "By lemma REF the intervals $N_n^{\\alpha ,\\pm }(\\omega )$ intersect to $E_{T}(\\omega )$ when $n \\rightarrow +\\infty $ for $m$ -a.e.", "$\\omega \\in \\Omega $ .", "By the Lebesgue differentiation theorem this implies $\\lim \\limits _{n \\rightarrow +\\infty }f_n^{\\alpha ,\\pm }(\\omega ) = f_{\\omega }(g_0(\\omega )E_{T}(\\omega )) = f_{\\omega }(E_{T}(\\sigma (\\omega ))),$ for $m$ -a.e.", "$\\omega \\in \\Omega $ as claimed.", "For the second claim, let $M f_{\\omega }$ be the maximal function defined by $M f_{\\omega }(x) = \\sup \\limits _{N \\supset \\lbrace x\\rbrace } \\frac{1}{\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma (\\omega ))}(N)}\\int \\limits _{N} f_{\\omega }(x)\\,d\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma (\\omega ))}(x),$ where the supremum is over intervals $N$ starting or finishing at $x$ in $\\pi ^{-1}(E_{T^{\\prime }}(\\sigma (\\omega ))).$ Since $ \\pi ^{-1}(x^{\\prime })$ is one-dimensional, for all $x^{\\prime } \\in \\mathcal {X}_{T^{\\prime }}$ , the maximal operator is of weak type (1,1) and we have (see [41], [41] for details) $ \\int \\log \\left(M f_{\\omega }(E_{T}(\\sigma (\\omega )))\\right)\\, dm(\\omega ) < +\\infty .$ It remains to show that $\\inf \\limits _{n}\\log (f_n^{\\alpha ,\\pm }(\\omega ))$ is $m$ -integrable.", "See [41] for the argument in a very similar setting.", "From Birkhoff ergodic theorem we have $\\kappa &= \\lim \\limits _{n \\rightarrow +\\infty } \\frac{1}{n}\\sum \\limits _{k = 1}^{n}\\log \\left(\\frac{dg_{-k}\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma ^{-k}(\\omega ))}}{d\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma ^{-k+1}(\\omega ))}}(E_{T}(\\sigma ^{-k+1}(\\omega )))\\right)\\\\ &= \\lim \\limits _{n \\rightarrow +\\infty } \\frac{1}{n}\\sum \\limits _{k = 1}^{n}\\log \\left(f_{\\sigma ^{-k}(\\omega )}(E_{T}(\\sigma (\\sigma ^{-k}(\\omega ))))\\right)$ To conclude, using Lemma REF and Maker theorem (see e.g.", "[47]) we obtain $\\kappa &= \\lim \\limits _{n \\rightarrow +\\infty } \\frac{1}{n}\\sum \\limits _{k = 1}^{n}\\log \\left(f_{n-k}^{\\alpha , \\pm }(\\sigma ^{-k}(\\omega ))\\right)\\\\ &= \\lim \\limits _{n \\rightarrow +\\infty } \\frac{1}{n}\\sum \\limits _{k = 1}^{n}\\log \\left(\\frac{\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma ^{-k}(\\omega ))}(N_{n-k}^{\\alpha , \\pm }(\\sigma ^{-k}(\\omega )))}{\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma ^{-k+1}(\\omega ))}(g_{-k}(\\omega ) N_{n-k}^{\\alpha , \\pm }(\\sigma ^{-k}(\\omega )))}\\right)\\\\ &= \\lim \\limits _{n \\rightarrow +\\infty } \\frac{1}{n}\\sum \\limits _{k = 1}^{n}\\log \\left(\\frac{\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma ^{-k}(\\omega ))}(g_{-k-1}(\\omega ) \\ldots g_{-n}(\\omega ) N^{\\alpha , \\pm }(\\sigma ^{-n}(\\omega )) )}{\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma ^{-k+1}(\\omega ))}(g_{-k}(\\omega ) \\ldots g_{-n}(\\omega ) N^{\\alpha , \\pm }(\\sigma ^{-n}(\\omega )))}\\right)\\\\ &= \\lim \\limits _{n \\rightarrow +\\infty } \\frac{1}{n}\\log \\left(\\frac{\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma ^{-n}(\\omega ))}(N^{\\alpha , \\pm }(\\sigma ^{-n}(\\omega )))}{\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}(N_n^{\\alpha , \\pm }(\\omega ))}\\right).$" ], [ "Exact dimensionality", "We finish the proof of theorem REF .", "Firstly, if $ \\kappa =0, $ by proposition REF , the measure $ \\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )} $ is exact-dimensional with dimension 0.", "Indeed, since $ \\chi >0 $ and $\\kappa =0 $ , we have $ 0= \\kappa /\\chi .$ So we may assume $ \\kappa >0.$ By proposition REF , for $m$ -a.e.", "$ \\omega $ , all $\\alpha , 0 < \\alpha < \\pi /2$ , $n$ going to infinity, $ \\frac{1}{n} \\log \\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}(N_n^{\\alpha ,\\pm }(\\omega )) \\; =\\; - \\kappa + o(1) + \\frac{1}{n} \\log \\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma ^{-n} \\omega )}(N^{\\alpha ,\\pm ^{\\prime } }(\\sigma ^{-n} \\omega )) .$ On the other hand, we have, by lemma REF , for $m$ -a.e.", "$ \\omega $ , all $\\alpha , 0 < \\alpha < \\pi /2$ , $n$ going to infinity, $ B_{\\pi ^{-1} (E_{T^{\\prime }}(\\omega ))}^\\pm (E_T(\\omega ), e^{-n\\chi + o(n)}) \\subset N_n^{\\alpha , \\pm ^{\\prime } } (\\omega ) \\subset B_{\\pi ^{-1} (E_{T^{\\prime }}(\\omega ))}^\\pm (E_T(\\omega ), e^{-n\\chi + o(n)}) ,$ where $ B_{\\pi ^{-1} (E_{T^{\\prime }}(\\omega ))}^\\pm (x,r) $ is either of the two intervals of size $r$ based on $x$ .", "It follows that for $m$ -a.e.", "$ \\omega $ , $\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}$ -a.e.", "$x$ , $\\liminf \\limits _{r \\rightarrow 0}\\frac{\\log (\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega ))}(B(x,r))}{\\log (r)} \\ge \\frac{\\kappa }{\\chi }.$ We cannot estimate directly $ \\limsup \\limits _{r \\rightarrow 0}\\frac{\\log (\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega ))}(B(x,r))}{\\log (r)} $ in the same way, because we do not know a priori that $ \\liminf \\limits _{n\\rightarrow \\infty }\\frac{1}{n} \\log (\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\sigma ^{-n}\\omega ))}(N^{\\alpha , \\pm } (\\sigma ^{-n}\\omega )) =0.$ The observation is that to compute $\\limsup \\limits _{r \\rightarrow 0}\\frac{\\log (\\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega ))}(B(x,r))}{\\log (r)} , $ it suffices to consider values of $-\\log r$ in $ \\mathbb {N}$ with a positive density.", "We claim that we can take $ \\alpha $ large enough that one of $ \\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}(N^{\\alpha , + } (\\omega ))$ and $ \\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}(N^{\\alpha , - } (\\omega ))$ is at least some $c >0$ on a set $ \\Omega ^{\\prime } \\subset \\Omega $ of positive measure.", "This finishes the proof, because, by Birkhoff ergodic theorem, for $m$ -a.e.", "$\\omega $ the sequence $ n_k $ such that $ \\sigma ^{-n_k} \\omega \\in \\Omega ^{\\prime } $ has positive density.", "Remains to prove the claim.", "We prove it by contradiction: if it is not true, $ \\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}(N^{\\alpha } (\\omega )) = 0 $ $ m$ -a.e.", "for all $\\alpha $ .", "But then, the measure $ \\nu _{T,T^{\\prime }}^{E_{T^{\\prime }}(\\omega )}$ is concentrated on $ E_{T^{\\prime }}(\\omega )_{T^{\\prime }(i) \\setminus \\lbrace i\\rbrace } $ , a contradiction with the fact that $E_+$ and $E_-$ are in general position." ], [ "Proof of lemma ", "Assume the diagram of projections $\\begin{matrix} T & \\longrightarrow & S \\\\\\downarrow 1 & & \\downarrow 1 & \\\\T^{\\prime } & \\longrightarrow & S^{\\prime }\\end{matrix}$ commutes and $i,j$ are such that $T(i) = T^{\\prime }(i) \\setminus \\lbrace j\\rbrace $ and $S(i) = S^{\\prime }(i) \\setminus \\lbrace j \\rbrace $ .", "Then, by Corollary REF , for $x^{\\prime } \\in \\mathcal {X}_{S^{\\prime }} $ and $ y^{\\prime } \\in (\\pi _{T^{\\prime },S^{\\prime }})^{-1} (x^{\\prime }) $ there is a bilipschitz homeomorphism between $ (\\pi _{S,S^{\\prime }})^{-1} (x^{\\prime }) $ and $ (\\pi _{T,T^{\\prime }})^{-1} (y^{\\prime }) $ .", "The measure $ \\nu ^{x^{\\prime }}_{S,S^{\\prime }} $ is the average over $ y^{\\prime }$ of the measures $ (\\pi _{T,S} )_\\ast \\nu ^{y^{\\prime }}_{T,T^{\\prime }} $ , the average being taken under $ d\\nu ^{x^{\\prime }}_{T^{\\prime },S^{\\prime }}(y^{\\prime }) .$ Lemma REF then follows from [42] Lemma 11.3.2.", "$ \\Box $" ], [ "Proof of Theorem ", "In this section, we assume that the measure $ \\mu $ is discrete and prove Theorem REF .", "The general idea of the proof is that entropy conservation implies dimension conservation.", "In [42], dynamical balls are disjoint ellipsoids with exponentially big eccentricities, not suitable for dimension estimates.", "But they behave very well for entropy estimates, even when considering their slices by invariant foliations.", "If one takes the slices in increasing order of size, this forces dimension conservation.", "For IFS or stationary measures, the dynamical balls are not disjoint any more.", "The idea of [16] is to look at the dynamical balls and the slicing at the level of the invertible dynamics on $\\Omega $ .", "Since the measure $\\mu $ is discrete, working at the level of the space $\\Omega $ is possible here as well." ], [ "Setup", "Recall that we consider $ T \\prec T^{\\prime } $ and the decomposition $T^0 ,T^1,\\ldots ,T^{N_{T,T^{\\prime }}} $ , where $T^0 = T^{\\prime }$ and $T^{N_{T,T^{\\prime }}} = T$ of proposition REF with $ T^t \\overset{1}{\\prec } T^{t-1}$ for $ t = 1, \\ldots {N_{T,T^{\\prime }}}$ .", "Recall that for each admissible topology $S$ we set $E_S(\\omega ) = F_{S}(E(\\omega ))$ .", "We will use $x$ to denote a point in $\\mathcal {X}_{T}$ and always set $x_t = \\pi _{T,T^t}(x)$ and $x_{t-1} = \\pi _{T,T^{t-1}}(x)$ .", "We fix $t$ and set $\\chi = \\chi _{T^t,T^{t-1}}$ , $\\kappa = \\kappa _{T^t,T^{t-1}}$ and $\\delta = \\overline{\\delta ^{t}}$ .", "Then, $\\gamma _{T^t,T^{t-1}} = \\kappa /\\chi $ .", "We are interested in comparing for $\\nu _{\\omega } = \\nu _{T,T_0}^{E_+(\\omega )}$ -almost every $x$ , the upper dimension at $x$ of $\\nu _{T, T^t}^{x_t}$ and $\\nu _{T, T^{t-1}}^{x_{t-1}}$ .", "For this purpose we fix $\\varepsilon > 0$ and set $r_n = \\exp (-n(\\chi - \\varepsilon ))$ for all $n$ in what follows." ], [ "Approximating configurations", "Lemma 8.1 There exist functions $f_n$ such that, for $m$ -a.e.", "$ \\omega ,$ setting $E_{T,n}(\\omega ) = f_n(E_{T^{t-1}}(\\omega ),g_{-1}(\\omega ),\\ldots ,g_{-n}(\\omega ))$ there exists $n(\\omega )$ such that for all $n \\ge n(\\omega )$ , $E_{T,n}(\\omega ) \\in B(E_{T}(\\omega ),r_n)$ .", "The construction of Proposition REF guarantees that $\\chi = \\min \\left\\lbrace \\chi _{T^s,T^{s-1}}: s = t,t+1,\\ldots , N_{T,T^{\\prime }}\\right\\rbrace .$ Therefore the claim follows directly from lemma REF .", "Corollary 8.2 For $m$ -a.e.", "$ \\omega $ , $\\nu _{\\omega }$ -almost every $x$ and $m_{T^t}^{x_t}$ -almost every $(h_n)_{n \\le -1}$ there exists $N(\\omega , x_t,(h_n)_{n \\le -1})$ such that, for all $n \\ge N(\\omega , x_t,(h_n)_{n \\le -1})$ .", "$f_n(x_{t-1},h_{-1},\\ldots ,h_{-n}) \\in B\\left(\\lim \\limits _{k \\rightarrow +\\infty }f_k(x_{t-1},h_{-1},\\ldots ,h_{-k}), r_n\\right).$" ], [ "Approximating conditional probabilities", "Let $T$ be an admissible topology.", "By definition, $E_T(\\omega ) \\in \\mathcal {X}_T^{E_+(\\omega _+)} $ and, if $T \\prec T^{\\prime }, \\; E_{T^{\\prime }}(\\omega ) = \\pi _{T,T^{\\prime }} (E_T(\\omega )).$ We introduced in section REF the disintegrations of the projection $ \\omega \\in \\Omega \\mapsto E_T(\\omega ) \\in \\mathcal {X}_T^{E_+(\\omega _+)} $ and their restrictions $ m_T^{E_T(\\omega )}$ to $\\Omega _-$ .", "With our notations, we have, for $ \\nu ^{\\prime } $ -a.e.", "$f^{\\prime }, \\nu _{T^t}^{f^{\\prime }} $ -a.e.", "$ z \\in \\mathcal {X}_{T^{t-1}}, $ $ m_{T^{t-1}}^z \\; = \\; \\int _{\\mathcal {X}_{T^t, T^{t-1}}^z} m_{T^{t}}^y \\, d\\nu _{T^t, T^{t-1}}^z(y).", "$ Given $x \\in \\mathcal {X}_T$ , let $A_n(x_{t-1})$ be the set of sequences $(h_{n})_{n \\le -1}$ such that, for all $ m \\ge n,$ $f_m(x_{t-1},h_{-1},\\ldots ,h_{-m}) \\in B\\left(\\lim \\limits _{k \\rightarrow +\\infty }f_k(x_{t-1},h_{-1},\\ldots ,h_{-k}), r_m\\right)$ .", "We let $[h_{-n},\\ldots ,h_{-1}]$ denote the set of sequences $(h_n^{\\prime })_{n \\le -1}$ such that $h_{-n}=h_{-n}^{\\prime },\\ldots ,h_{-1}=h_{-1}^{\\prime }$ .", "Lemma 8.3 For $ m $ -a.e.", "$ \\omega $ , $\\nu _{\\omega }$ -almost every $x$ and $m_{T^{t}}^{x_t}$ -almost every $(h_n)_{n \\le -1}$ , there exists $N(\\omega , x_t,(h_n)_{n \\le -1})$ such that, for all $n \\ge N(\\omega , x,(h_n)_{n \\le -1})$ , $m_{T^{t}}^{x_t}([h_{-n},\\ldots ,h_{-1}] \\cap A_n(x_{t-1})) \\le \\exp (n(\\kappa + \\varepsilon ))m_{T^{t-1}}^{x_{t-1}}([h_{-n},\\ldots ,h_{-1}] \\cap A_n(x_{t-1})).$ For $m$ -a.e.", "$ \\omega $ , consider the measure $\\mathbb {P}_\\omega $ on the space $\\mathcal {X}_T \\times \\lbrace (h_k)_{k \\le -1} \\rbrace $ which projects to $\\nu _{\\omega }$ on $\\mathcal {X}_T$ and whose disintegrations on the fibers projecting to $x \\in \\mathcal {X}_T$ are given by $x \\mapsto m_{T^{t}}^{x_t}$ .", "By corollary REF , for $m$ -a.e.", "$ \\omega $ , $\\nu _{\\omega }$ -almost every $x$ , $A_n(x_{t-1})$ is increasing with $n$ and $\\cup _n A_n(x_{t-1})= \\Omega _-$ up ot a set of $m_{T^{t}}^{x_t}$ -measure 0.", "By the martingale convergence theorem, the conditional expectation with respect to $\\mathbb {P}_\\omega $ of the indicator of the event $\\lbrace (h_k)_{k \\le -1} \\in A_n(x_{t-1})\\rbrace $ with respect to the $\\sigma $ -algebras generated by $x_{t},h_{-1},\\ldots ,h_{-n}$ (respectively $x_{t-1},h_{-1},\\ldots ,h_{-n}$ ) converge to 1.", "The first conditional expectation is given at $m$ -a.e.", "$ \\omega $ , $\\nu _{\\omega }$ -almost every $x$ and $m_{T^t}^{x_t}$ -almost every $(h_n)_{n \\le -1}$ by $\\frac{m_{T^{t}}^{x_t}([h_{-n},\\ldots ,h_{-1}] \\cap A_n(x_{t-1}))}{m_{T^{t}}^{x_t}([h_{-n},\\ldots ,h_{-1}])} .$ The second one by $ \\frac{\\int \\limits _{\\mathcal {X}_{T^t, T^{t-1}}^{x_{t-1}}}m_{T^{t}}^{y}([h_{-n},\\ldots ,h_{-1}] \\cap A_n(x_{t-1}))\\,d\\nu _{T^t,T^{t-1}}^{x_{t-1}}(y)}{\\int \\limits _{\\mathcal {X}_{T^t,T^{t-1}}^{x_{t-1}}} m_{T^{t}}^{y}([h_{-n},\\ldots ,h_{-1}] )\\,d\\nu _{T^t,T^{t-1}}^{x_{t-1}}(y) }.$ Using (REF ), this last expression is $ \\frac{m_{T^{t-1}}^{x_{t-1}}([h_{-n},\\ldots ,h_{-1}] \\cap A_n(x_{t-1}))}{m_{T^{t-1}}^{x_{t-1}}([h_{-n},\\ldots ,h_{-1}])} $ Thus, at $m$ -a.e.", "$ \\omega $ , $\\nu _{\\omega }$ -almost every $x$ and $m_{T^t}^{x_t}$ -almost every $(h_n)_{n \\le -1}$ , $\\lim \\limits _{n \\rightarrow +\\infty }\\frac{m_{T^{t}}^{x_t}([h_{-n},\\ldots ,h_{-1}] \\cap A_n(x_{t-1}))}{m_{T^{t}}^{x_t}([h_{-n},\\ldots ,h_{-1}])} = \\lim \\limits _{n \\rightarrow +\\infty }\\frac{m_{T^{t-1}}^{x_{t-1}}([h_{-n},\\ldots ,h_{-1}] \\cap A_n(x_{t-1}))}{m_{T^{t-1}}^{x_{t-1}}([h_{-n},\\ldots ,h_{-1}])} = 1.$ We have shown (in corollary REF ) that for $ m$ -a.e.", "$\\omega $ , $\\nu _\\omega $ -almost every $x$ and $m_{T^{t}}^{x_t}$ -almost every $(h_n)_{n \\le -1}$ one has $\\lim \\limits _{n \\rightarrow +\\infty }\\frac{1}{n}\\log \\left(\\frac{m_{T^t}^{x_t}([h_{-n},\\ldots ,h_{-1}])}{m_{T^{t-1}}^{x_{t-1}}([h_{-n},\\ldots ,h_{-1}])}\\right) = \\kappa .$ Combining these three statements we obtain that for $m$ -a.e.", "$ \\omega $ , $\\nu _\\omega $ -almost every $x$ and $m_{T^{t}}^{x_t}$ -almost every $(h_n)_{n \\le -1}$ one has $\\lim \\limits _{n \\rightarrow +\\infty }\\frac{1}{n}\\log \\left(\\frac{m_{T^{t}}^{x_t}([h_{-n},\\ldots ,h_{-1}] \\cap A_n(x_{t-1}))}{m_{T^{t-1}}^{x_{t-1}}([h_{-n},\\ldots ,h_{-1}] \\cap A_n(x_{t-1}))}\\right) = \\kappa ,$ from which the desired result follows immediately." ], [ "Lebesgue Density", "Given $x \\in \\mathcal {X}_T$ let $B_n(x_{t})$ denote the set of sequences $(h_n)_{n \\le -1}$ such that $m_{T^{t}}^{x_t}([h_{-n},\\ldots ,h_{-1}] \\cap A_n(x_{t-1})) \\le \\exp (n(\\kappa + \\varepsilon ))m_{T^{t-1}}^{x_{t-1}}([h_{-n},\\ldots ,h_{-1}] \\cap A_n(x_{t-1})).$ Lemma 8.4 For $m $ -a.e.", "$ \\omega $ and $\\nu _{\\omega }$ almost every $x$ there exists $N(\\omega ,x)$ such that, for all $n \\ge N(\\omega ,x )$ , $m_{T^t}^{x_t}\\left(\\left\\lbrace \\lim \\limits _{k \\rightarrow +\\infty }f_k(x_{t-1},h_{-1},\\ldots ,h_{-k}) \\in B(x,r_n)\\right\\rbrace \\cap A_n(x_{t-1}) \\cap B_n(x_t)\\right) \\ge \\exp (-n(\\delta + \\varepsilon )\\chi ).$ The probability on the left-hand side in the statement is $\\int \\limits _{B(x,r_n)}m_{T}^{y}(A_n(x_{t-1}) \\cap B_n(x_t))d\\nu _{T,T^t}^{x_{t}}(y) = \\int \\limits _{B(x,r_n)}m_{T}^{y}(A_n(y_{t-1}) \\cap B_n(y_t))d\\nu _{T,T^t}^{x_{t}}(y).$ Let $C_n(x) = \\bigcap \\limits _{m \\ge n}B_n(x_t)$ so that $C_n(x)$ is increasing with $n$ .", "We have $\\int \\limits _{B(x,r_n)}m_{T}^{y}(A_n(y_{t-1}) \\cap B_n(y_{t}))d\\nu _{T,T^t}^{x_t}(y) \\ge \\int \\limits _{B(x,r_n)}m_{T}^{y}(A_n(y_{t-1}) \\cap C_n(y_t))d\\nu _{T,T^t}^{x_t}(y).$ By lemma REF , for $ m $ -a.e.", "$ \\omega $ , $\\nu _\\omega $ -a.e.", "$x$ , the function $y \\mapsto m_{T}^{y}(A_n(y_{t-1}) \\cap C_n(y_t))$ increases to 1 at $\\nu _{T,T^{t}}^{x_t}$ -a.e.", "$y$ .", "Applying the Lebesgue differentiation theorem (justified since configuration spaces are bilipschitz homeomorphic to Euclidean spaces) we obtain a set $L(\\omega , x_t )$ of $\\nu _{T,T^t}^{x_t}$ -full measure such that for all $z$ in this set $\\lim \\limits _{n \\rightarrow +\\infty }\\frac{1}{\\nu _{T,T^t}^{x_t}(B(z,r_n))}\\int \\limits _{B(z,r_n)}m_{T}^{y}(A_n(y_{t-1}) \\cap C_n(y_t))d\\nu _{T,T^t}^{x_t}(y) = 1.$ For $m$ -a.e.", "$ \\omega $ , $\\nu _\\omega $ -a.e.", "$x$ belongs to $ L(\\omega , x_t)$ .", "Moreover, by hypothesis, for $m $ -a.e.", "$ \\omega $ and $\\nu _\\omega $ -a.e.", "$x$ , there exists $N(\\omega , x)$ such that for $n \\ge N(\\omega ,x) $ , $\\nu _{T,T^t}^{x_t}(B(x,r_n)) \\ge \\exp (-n(\\delta +\\varepsilon )\\chi )$ .", "The result follows." ], [ "Proof of the theorem", "We now complete the proof of Theorem 2.4.", "We begin with Lemma REF and observe that if $\\lim \\limits _{k \\rightarrow +\\infty }f_k(x_{t-1},h_{-1},\\ldots ,h_{-k}) \\in B(x,r_n)$ and $(h_n)_{n \\le -1} \\in A_n(x_{t-1}),$ then in fact $f_n(x_{t-1},h_{-1},\\ldots ,h_{-n}) \\in B(x,2r_n)$ .", "This implies that for $m $ -a.e.", "$ \\omega $ , $\\nu _{\\omega }$ -a.e.", "$x$ and all $n \\ge N(\\omega ,x)$ given by Lemma REF one has $\\exp (-n(\\delta +\\varepsilon )\\chi ) &\\le m_{T^t}^{x_t}\\left(\\left\\lbrace \\lim \\limits _{k \\rightarrow +\\infty }f_k(x_{t-1},h_{-1},\\ldots ,h_{-k}) \\in B(x,r_n)\\right\\rbrace \\cap A_n(x_{t-1}) \\cap B_n(x_t)\\right)\\\\ &\\le m_{T^t}^{x_t}\\left(\\left\\lbrace f_n(x_{t-1},h_{-1},\\ldots ,h_{-n}) \\in B(x,2r_n)\\right\\rbrace \\cap A_n(x_{t-1}) \\cap B_n(x_t)\\right).$ Notice that both $D_n(x_{t-1}) : = \\lbrace (h_n)_{n \\le -1}: f_n(x_{t-1},h_{-1},\\ldots ,h_{-n}) \\in B(x,2r_n)\\rbrace $ and $B_n(x_t)$ are a union of cylinders $[h_{-n},\\ldots ,h_{-1}]$ .", "Therefore from the second line above and the definition of $B_n(x_t)$ we obtain $\\exp (-n(\\delta +\\varepsilon )\\chi ) &\\le \\sum \\limits _{[h_{-n},\\ldots ,h_{-1}] \\subset B_n(x_t) \\cap D_n(x_{t-1})}m_{T^t}^{x_t}([h_{-n},\\ldots ,h_{-1}] \\cap A_n(x_{t-1}))\\\\ &\\le \\exp (n(\\kappa +\\varepsilon ))\\sum \\limits _{[h_{-n},\\ldots ,h_{-1}] \\subset D_n(x_{t-1})}m_{T^{t-1}}^{x_{t-1}}([h_{-n},\\ldots ,h_{-1}] \\cap A_n(x_{t-1})).$ For $m $ -a.e.", "$ \\omega $ , $\\nu _{\\omega }$ -a.e.", "$x$ and all $n \\ge N(\\omega ,x)$ , we have shown that $\\exp (-n(\\delta +\\varepsilon )\\chi )\\exp (-n(\\kappa +\\varepsilon )) \\le m_{T^{t-1}}^{x_{t-1}}\\left(\\left\\lbrace f_n(x_{t-1},h_{-1},\\ldots ,h_{-n}) \\in B(x,2r_n)\\right\\rbrace \\cap A_n(x_{t-1})\\right).$ Since whenever $(h_n)_{n \\le -1} \\in A_n(x_{t-1})$ we have that $f_n(x_{t-1},h_{-1},\\ldots ,h_{-n})$ and $\\lim \\limits _{k \\rightarrow +\\infty }f_k(x_{t-1},h_{-1},\\ldots ,h_{-k})$ are at distance at most $r_n$ , this implies that for $m $ -a.e.", "$ \\omega $ , $\\nu _{\\omega }$ -a.e.", "$x$ and all $n \\ge N(\\omega ,x)$ , $\\exp (-n(\\delta +\\varepsilon )\\chi )\\exp (-n(\\kappa +\\varepsilon )) &\\le m_{T^{t-1}}^{x_{t-1}}\\left(\\left\\lbrace (h_k)_{k \\le -1}: \\lim \\limits _{k \\rightarrow +\\infty }f_k(x_{t-1},h_{-1},\\ldots ,h_{-n}) \\in B(x,3r_n)\\right\\rbrace \\right)\\\\ &= \\nu _{T,T^{t-1}}^{x_{t-1}}(B(x,3r_n)).$ Since this holds for all $\\varepsilon > 0$ it follows that $\\overline{\\delta ^{t-1}} \\le \\delta + \\frac{\\kappa }{\\chi } = \\overline{\\delta ^t} + \\gamma _{T^t,T^{t-1}} $ as claimed." ], [ "Application to Hitchin representations of compact surface groups", "In this section, we first illustrate by an example the discussion of section 1.2.", "Take $ \\mu $ discrete in $\\mathcal {M}( SL_3(\\mathbb {R}) )$ .", "Consider the nine arrows of Figure 1 describing projections from $ \\mathcal {X}^{f^{\\prime }}_T $ to $ \\mathcal {X}^{f^{\\prime }}_{T^{\\prime }} $ , where $T \\overset{1}{\\prec } T^{\\prime }$ , in dimension 3.", "By Theorem REF , six of these projections are dimension conserving.", "There is a natural family of examples of random walks on $ SL_3(\\mathbb {R}) $ for which two of the other projections are not dimension conserving as soon as the middle exponent $ \\chi _2 $ is not 0.", "Namely, these are the random walks on images of a surface group by a Hitchin representation in $ SL_3(\\mathbb {R}) .$ We present these examples and then extend the discussion to Hitchin representations in $ PSL_d(\\mathbb {R}) $ , for all $ d\\ge 3.$" ], [ "Hitchin component in dimension 3", "Consider a closed surface $ \\Sigma $ of genus at least two and the group $ \\Gamma := \\pi _1 (\\Sigma ).$ A representation $ \\rho : \\Gamma \\rightarrow PSL_2(\\mathbb {R}) $ is called Fuchsian if it is discrete and cocompact.", "A representation $ \\rho : \\Gamma \\rightarrow SL_3(\\mathbb {R}) $ is also called Fuchsian if it is the composition of a Fuchsian representation and the canonical irreducible representation of $ PSL_2(\\mathbb {R}) $ into $ SL _3(\\mathbb {R}) $ .", "It is called Hitchin if it can be obtained by a deformation of a Fuchsian representation.", "Hitchin representations have been studied from many points of view, we only list the properties we are going to use.", "We shall describe points in $ \\mathcal {F}$ as pairs $ (\\zeta , \\overline{\\zeta })$ , where $ \\zeta $ in a point in the projective plane $ \\mathbb {R}\\mathbb {P}^2$ and $ \\overline{\\zeta }$ a line in $ \\mathbb {R}\\mathbb {P}^2 $ containing $ \\zeta .$ The pairs $ (\\zeta , \\overline{\\zeta }), (\\eta , \\overline{\\eta }) $ are in general position if, and only if, $ \\zeta \\notin \\overline{\\eta }, \\eta \\notin \\overline{\\zeta }.$ By classical results of Koszul [33], Goldman [23] and Choi-Goldman [9], if the representation $ \\rho $ is Hitchin, then there exists a $ C^1$ convex subset $\\Delta \\subset \\mathbb {R}\\mathbb {P}^2$ invariant under $\\rho (\\Gamma )$ and a Hölder continuous mapping $( \\xi ,\\overline{\\xi }): \\mathbb {S}^1 \\rightarrow \\mathcal {F}$ such that – for $ s \\ne t \\in \\mathbb {S}^1, \\; (\\xi , \\overline{\\xi }) (s) $ and $ (\\xi , \\overline{\\xi })(t)$ are in general position, – $ \\xi (\\mathbb {S}^1) $ is the boundary $ \\partial \\Delta $ , – for $t \\in \\mathbb {S}^1, \\, $ $ \\overline{\\xi }(t) $ is the tangent direction to $ \\partial \\Delta $ at $ \\xi (t)$ and – the set $ \\Lambda := (\\xi , \\overline{\\xi }^{\\prime }) (\\mathbb {S}^1) $ is an invariant set for the action of $ \\rho (\\Gamma ) $ on $ \\mathcal {F}$ .", "The set $\\Lambda $ consists in the tangent elements to $ \\partial \\Delta $ .", "So, the convex $ \\Delta $ admits a cocompact group of projective mappings, i.e.", "it is divisible.", "A classical result of Benzécri is that the boundary is of class $C^2$ if, and only if, $\\Delta $ is an ellipse, if, and only if, the representation is conjugated to a Fuchsian representation.", "In that case, $ \\chi _2 = 0$ and all the dimension questions reduce to the $PSL_2(\\mathbb {R}) $ case.", "Therefore, we may assume that representation $ \\rho $ is Hitchin but not a Fuchsian representation.", "Such representations were studied in detail by Y. Benoist.", "In particular, he showed that – the boundary $ \\partial \\Delta $ is $ C^{1+ \\beta } $ for some $ \\beta >0 $ , but not $ C^{1+ abs.cont.}", "$ ([2]), – the group $ \\rho (\\Gamma ) $ is Zariski dense in $SL_3(\\mathbb {R}) $ and its action on $\\mathbb {R}^3$ is strongly irreducible ([1]).", "Let $\\mu \\in \\mathcal {M}(\\Gamma ) $ ; consider the random walk $ (\\rho (\\Gamma ), \\rho _\\ast (\\mu )) $ and the stationary measures $ \\nu , \\nu ^{\\prime }$ on $ \\mathcal {F}$ .", "The measures $ \\nu $ and $ \\nu ^{\\prime }$ are supported on $ \\Lambda .$ For $m $ -a.e.", "$ \\omega \\in \\Omega ,$ there are $ (\\xi _+, \\overline{\\xi }_+)(\\omega )$ and $(\\xi _-, \\overline{\\xi }_-) (\\omega ) $ distinct points in $ \\Lambda $ that are the supports of the limit measures of $ \\left(g_{-1}(\\omega ) \\ldots g_{-n}(\\omega )\\right)_\\ast \\nu $ and, respectively, $ \\left(g_{0}(\\omega )^{-1} \\ldots g_{n-1}(\\omega )^{-1}\\right)_\\ast \\nu ^{\\prime } $ as $ n \\rightarrow +\\infty $ .", "The distribution of $(\\xi _+, \\overline{\\xi }_+)(\\omega ) $ is $ \\nu $ , the distribution of $ (\\xi _-, \\overline{\\xi }_-)(\\omega ) $ is $ \\nu ^{\\prime } $ .", "The point $ \\xi _+(\\omega ) $ is the direction of the expanding $ E_1(\\omega ) $ , the point $ \\xi _-(\\omega ) $ is the direction of the contracting $ E_3(\\omega ) $ .", "The central direction $ E_2(\\omega ) $ is obtained as $ \\overline{\\xi }_+(\\omega ) \\cap \\overline{\\xi }_-(\\omega ) .$ Proposition 9.1 Let $ \\rho $ be a Hitchin representation of $ \\Gamma $ and $ \\mu $ be an adapted probability measure on $ \\Gamma $ such that $ \\sum _g | g| \\, d\\mu (g) < + \\infty ,$ where $ |\\cdot | $ is some word metric on $\\Gamma $ .", "Consider the random walk on $ SL_3(\\mathbb {R})$ directed by the probability $ \\rho _\\ast (\\mu ) $ , $ \\chi _1 > \\chi _2 > \\chi _3 $ its Lyapunov exponents.", "Let $ \\mathcal {F}, \\mathcal {L}, \\mathcal {P}$ be the spaces of flags, lines and planes in $\\mathbb {R}^3$ , $ \\nu , \\nu _\\mathcal {L}, \\nu _\\mathcal {P}$ the respective stationary measure and $ \\delta , \\delta _\\mathcal {L}, \\delta _\\mathcal {P}$ their dimensions.", "Assume $ \\chi _2 >0 $ .", "Then, $ \\delta _\\mathcal {P}< \\delta _\\mathcal {L}= \\delta $ .", "Moreover, the projections $ \\nu \\rightarrow \\nu _\\mathcal {P}$ are not dimension conserving.", "Assume $ \\chi _2 <0 $ .", "Then, $ \\delta _\\mathcal {L}< \\delta _\\mathcal {P}=\\delta $ .", "Moreover, the projections $ \\nu \\rightarrow \\nu _\\mathcal {L}$ are not dimension conserving.", "Assume $\\chi _2 =0 .$ Then, $ \\delta = \\delta _\\mathcal {L}= \\delta _\\mathcal {P}.$ All the natural projections are dimension conserving.", "By the above discussion, our results apply in this setting.", "We can consider – The distribution $ \\nu $ of $f = (\\xi _+, \\overline{\\xi }^{\\prime }_+ )(\\omega ) \\in \\mathcal {F}$ .", "It has entropy $ h $ and dimension $ \\delta .$ – The distribution $ \\nu _\\mathcal {L}$ of $ \\xi _+(\\omega ) $ .", "It has entropy $ h _\\mathcal {L}$ and dimension $ \\delta _\\mathcal {L}.$ Observe that, once one knows $ \\xi (t) \\in \\partial \\Delta , \\, \\overline{\\xi }(t) $ is the tangent direction to $ \\partial \\Delta $ at $ \\xi (t) $ , so it is uniquely determined.", "In other words, the projection from $ \\nu $ to $ \\nu _\\mathcal {L}$ is a.e.", "one-to-one.", "By [41] $ h_\\mathcal {L}= h $ , but, a priori, there is no dimension conservation and we only get $ \\delta \\ge \\delta _\\mathcal {L}.$ – The distribution $ \\nu _\\mathcal {P}$ of $ \\overline{\\xi }_+(\\omega )$ .", "It has entropy $ h _\\mathcal {P}$ and dimension $ \\delta _\\mathcal {P}.$ Observe that, similarly, once one knows $ \\overline{\\xi }(t)$ is a tangent direction to $ \\partial \\Delta $ at some point, then this point is $ \\xi (t) $ , so it is uniquely determined.", "In other words, the projection from $ \\nu $ to $ \\nu _\\mathcal {P}$ is a.e.", "one-to-one.", "By [41] again, $ h_\\mathcal {P}= h $ , but, a priori, there is no dimension conservation and we only get $ \\delta \\ge \\delta _\\mathcal {P}.$ We choose $ f^{\\prime } = (\\eta ,\\overline{\\eta }) \\in \\mathcal {F}$ .", "Write $\\mathcal {L}\\mathcal {P}:= \\mathcal {X}^{f^{\\prime }}_{\\lbrace 1,3\\rbrace , \\lbrace 2\\rbrace , \\lbrace 3\\rbrace }, \\mathcal {L}^{\\prime } := \\mathcal {X}^{f^{\\prime }}_{\\lbrace 1,3\\rbrace , \\lbrace 2,3\\rbrace , \\lbrace 3\\rbrace }$ and $ \\mathcal {P}^{\\prime } := \\mathcal {X}^{f^{\\prime }}_{\\lbrace 1,2,3\\rbrace , \\lbrace 2\\rbrace , \\lbrace 3\\rbrace } .$ For $ \\nu ^{\\prime } $ -a.e.", "$ f^{\\prime } \\in \\mathcal {F}$ , write $ \\nu _{\\mathcal {L}\\mathcal {P}}^{f^{\\prime }}, \\nu _{\\mathcal {L}^{\\prime }}^{f^{\\prime }} , \\nu _{\\mathcal {P}^{\\prime }} ^{f^{\\prime }}$ for the corresponding conditional measures.", "We can also consider the distribution $ \\nu _{\\mathcal {L}\\mathcal {P}}^{f^{\\prime }} $ of the couple made of the point $\\overline{\\xi }_+ \\cap \\overline{\\eta }$ and the line $ (\\eta , \\xi _+) $ given $ f^{\\prime } = (\\eta , \\overline{\\eta }).$ It has entropy $ h _{\\mathcal {L}\\mathcal {P}}$ and dimension $ \\delta _{\\mathcal {L}\\mathcal {P}}.$ Again, this determines $ ( \\xi , \\overline{\\xi }) $ by intersection with $ \\partial \\Delta $ and the dimension on fibers of the projection from $ \\mathcal {F}$ to $ \\mathcal {X}^{f^{\\prime }}_{\\mathcal {L}\\mathcal {P}} $ is 0.", "But now, by theorem REF , there is dimension conservation, so $ h = h_{\\mathcal {L}\\mathcal {P}} $ and $ \\delta = \\delta _{\\mathcal {L}\\mathcal {P}}.$ Assume $ \\chi _2 \\ge 0.$ We project both $ \\nu _\\mathcal {L}$ and $ \\nu _{\\mathcal {L}\\mathcal {P}}^{f^{\\prime }} $ to the space $ \\mathcal {L}^{\\prime }$ of lines going through $ \\eta $ by associating the line going through $ \\xi $ and $\\eta $ in the first case and by forgetting $ \\overline{\\xi }\\cap \\overline{\\eta }$ in the second case.", "The projection and the image measure $ \\nu _{\\mathcal {L}^{\\prime }}^{f^{\\prime }} $ depend on $ f^{\\prime }$ .", "For $ \\nu ^{\\prime }$ -a.e.", "$ f^{\\prime }$ , we have entropy $h_{\\mathcal {L}^{\\prime }}$ and dimension $\\delta _{\\mathcal {L}^{\\prime }}$ on $ \\mathcal {L}^{\\prime }.$ Both projections have almost everywhere trivial fibers: intersecting the line $ (\\eta , \\xi )$ with $ \\partial \\Delta $ determines everything.", "Therefore, $ h = h_\\mathcal {L}= h_{\\mathcal {L}\\mathcal {P}} = h_{\\mathcal {L}^{\\prime }}.$ Moreover, by theorem REF , both projections have dimension conservation (observe that $ \\chi _{\\mathcal {L},\\mathcal {L}^{\\prime }} = \\chi _1 - \\chi _3 \\ge \\chi _{\\mathcal {L}^{\\prime }} = \\chi _1 - \\chi _2$ ).", "So we obtain $ \\delta = \\delta _\\mathcal {L}= \\delta _{\\mathcal {L}\\mathcal {P}} = \\delta _{\\mathcal {L}^{\\prime }} = \\frac{h}{\\chi _1 -\\chi _2} .$ Remain to understand the projections of both $ \\nu _\\mathcal {P}$ and $ \\nu _{\\mathcal {L}\\mathcal {P}}^{f^{\\prime }}$ on the space $ \\mathcal {P}^{\\prime }$ of points of $ \\overline{\\eta }$ .", "The projection and the image measure $ \\nu _{\\mathcal {P}^{\\prime }}^{f^{\\prime }} $ depend on $ f^{\\prime }$ .", "For $ \\nu ^{\\prime }$ -a.e.", "$ f^{\\prime }$ , write $h_{\\mathcal {P}^{\\prime }} $ and $\\delta _{\\mathcal {P}^{\\prime }} $ for the entropy and the dimension of $ \\nu _{\\mathcal {P}^{\\prime }}^{f^{\\prime }} $ .", "Once more, knowing the point $E_2 $ in $ \\overline{\\eta }$ determines the rest by drawing the unique other tangent to $ \\partial \\Delta $ going through $E_2$ .", "So, $ h_\\mathcal {P}=h_{\\mathcal {P}^{\\prime }} = h_{\\mathcal {L}\\mathcal {P}} $ (both $ h_\\mathcal {P}$ and $ h_{\\mathcal {L}\\mathcal {P}} $ are $h$ by the above discussion) and all the entropies are the same $h$ .", "Moreover, since $ \\chi _{\\mathcal {P},\\mathcal {P}^{\\prime }} = \\chi _1 - \\chi _3 \\ge \\chi _{\\mathcal {P}^{\\prime }} = \\chi _2 - \\chi _3,$ $ \\delta _\\mathcal {P}= \\delta _{\\mathcal {P}^{\\prime }} = \\frac{h}{\\chi _2 - \\chi _3}.", "$ If $ \\chi _2 = 0 $ , then $ \\chi _1 - \\chi _2 = \\chi _2 - \\chi _3 , \\delta _{\\mathcal {L}^{\\prime }} = \\delta _{\\mathcal {P}^{\\prime }} ,$ all the dimensions coincide and there is dimension conservation at all the projections of Figure 1.", "If $ \\chi _2 > 0, $ then $ \\chi _2 - \\chi _3 > \\chi _1 - \\chi _2$ and $ \\delta _{\\mathcal {P}^{\\prime }} < \\delta _{\\mathcal {L}^{\\prime }} $ .", "So $ \\delta _\\mathcal {P}< \\delta $ and the projection from $ \\nu $ to $\\nu _\\mathcal {P}$ is not dimension conserving.", "In the same way, $ \\delta _{\\mathcal {P}^{\\prime }} < \\delta _{\\mathcal {L}\\mathcal {P}} $ and, for $ \\nu ^{\\prime }$ -a.e.", "$ f^{\\prime }$ , the projection from $ \\nu _{\\mathcal {L}\\mathcal {P}}^{f^{\\prime }}$ to $ \\nu _{\\mathcal {P}^{\\prime }}^{f^{\\prime }} $ is not dimension conserving either.", "In the case when $ \\chi _2 \\le 0$ , the discussion is the same, exchanging the role of points and lines and of $ \\chi _2 - \\chi _3 $ and $ \\chi _1 - \\chi _2$ .", "By the above proof, we have $ h = \\delta (\\nu ) \\min \\lbrace ( \\chi _1 - \\chi _2), ( \\chi _2 - \\chi _3 )\\rbrace , $ independently of the sign of $\\chi _2 .$ Observe that, since $\\rho (\\Gamma ) $ is discrete in $SL_3(\\mathbb {R}), \\;h = h_{{\\textrm {RW}}} (\\mu ) $ ([39]).", "Theorem REF follows when $ d = 3$ ." ], [ "Rigidity of Hitchin representations", "In this section, we prove that the hypotheses of proposition REF are satisfied for some probability measure in $ \\mathcal {M}(\\rho (\\gamma )) $ if the representation $ \\rho $ is Hitchin but not Fuchsian, namely that one can find such a measure with $ \\chi _2 \\ne 0$ .", "We have the Theorem 9.2 Let $ \\rho $ be a Hitchin representation of a cocompact surface group in $ SL_3(\\mathbb {R}) $ such that for all probability measures in $ \\mathcal {M}(\\rho (\\Gamma )), \\, \\chi _2 \\le 0 .$ Then, the representation $ \\rho $ is Fuchsian.", "Such variational characterizations of Fuchsian representations among Hitchin components have been proven by M. Crampon ([11]) and R. Potrie and A. Sambarino ([45]) in greater generality.", "Theorem REF is a variant of their results adapted to the dimension 3.", "By proposition REF , our hypothesis is that for all $\\mu \\in \\mathcal {M}(\\Gamma ),$ the dimensions $ \\delta _\\mathcal {L}, \\delta _\\mathcal {P}$ of the stationary measures on the spaces of lines and planes satisfy $ \\delta _\\mathcal {L}\\; \\le \\; \\delta _\\mathcal {P}.$ We are going to use thermodynamical formalism for the geodesic flow on $ \\rho (\\Gamma ) \\setminus H \\Delta ,$ where $ H\\Delta $ is the homogeneous tangent bundle to $ \\Delta $ and a construction of [10] to obtain, for any Hitchin representation $ \\rho $ , some $ \\mu \\in \\mathcal {M}(\\rho (\\Gamma )) $ such that $ \\delta _\\mathcal {L}= 1 .$ Since $ \\nu _\\mathcal {P}$ is also supported on a $ C^1 $ circle, (REF ) implies that $ \\delta _\\mathcal {P}= 1 $ as well.", "Using dynamics of the geodesic flow and [2], section 6, this will imply that the representation is Fuchsian.", "Recall that all matrices $ \\rho (\\gamma ), \\gamma \\in \\Gamma , \\rho (\\gamma ) \\ne Id, $ have three distinct real eigenvalues with absolute values $ e^{\\ell _1(\\gamma )} > e^{\\ell _2(\\gamma )} > e^{\\ell _3(\\gamma )} $ ([36]).", "Let $ \\varphi $ be the linear functional on $ \\Sigma := \\lbrace ( \\ell _1, \\ell _2, \\ell _3 ) \\in \\mathbb {R}^3: \\ell _1 + \\ell _2+ \\ell _3 = 0 \\rbrace $ defined by $ \\varphi := \\ell _1 - \\ell _2 .$ Recall that the geodesic flow on $ \\rho (\\Gamma ) \\setminus H\\Delta $ is an Anosov flow.", "There exists a Hölder continuous function $f$ on $ H\\Delta $ such that for any $\\gamma \\in \\Gamma , \\gamma \\ne Id,$ $ \\ell _1(\\gamma ) - \\ell _2 (\\gamma ) \\; = \\; \\int _{\\sigma _\\gamma } f, $ where $ \\sigma _\\gamma $ is the periodic orbit associated to $ \\gamma $ (see [45], sections 2 and 7).", "Moreover, for any ergodic invariant measure $m$ for the geodesic flow, $ \\int f \\, dm $ is the positive Lyapunov exponent of the geodesic flow for $m$ ([2], Lemma 6.5).", "In particular, the equilibrium measure $ m_0$ for $ f $ is absolutely continuous along unstable manifolds.", "Fix a point $ o \\in \\Delta .$ Then, the Gibbs-Patterson-Sullivan construction (see e.g.", "[40]) yields an equivariant family of measures $ \\nu _0 $ at the boundary such that for all $ \\gamma \\in \\Gamma , \\frac{d(\\rho (\\gamma ))_\\ast \\nu _0}{d\\nu _0} (\\xi ) $ is a Hölder continuous function and with the property that, if a set $A$ of points in $ \\partial \\Delta $ is $\\nu _0$ -negligible, then the set of geodesics with end in $A $ is $m_0$ -negligible.", "By the absolute continuity of the stable foliation, this implies that $ \\nu _0 $ is absolutely continuous on $ \\partial \\Delta .$ Recall that the limit set $ \\Lambda $ of $ \\rho (\\Gamma )$ projects one-to-one in $ \\partial \\Delta .$ Denote by $ \\nu $ the lift of the measure $ \\nu _0 $ to $ \\Lambda $ .", "Next step consists in finding a random walk in $ \\mathcal {M}(\\rho (\\Gamma ) )$ such that $ \\nu $ is the stationary measure on $\\mathcal {F}$ or equivalently such that $\\nu _0 $ is the stationary measure for the action on $ \\partial \\Delta $ .", "Lemma 9.3 Let $\\Gamma $ be a co-compact group of isometries of $\\mathbb {H}^2$ , $ \\rho $ a Hitchin non-Fuchsian representation of $\\Gamma $ in $SL_3(\\mathbb {R}) $ , $\\Delta $ the open convex proper subset of $ \\mathbb {R}\\mathbb {P}^2 $ invariant under $ \\rho (\\Gamma ), \\, \\nu _0$ be a finite measure on $ \\partial \\Delta $ such that for all $ \\gamma \\in \\Gamma , \\frac{d(\\rho (\\gamma ))_\\ast \\nu _0}{d\\nu _0} (\\xi ) $ is a Hölder continuous function.", "Then there exists a probability measure $ \\mu _0 \\in \\mathcal {M}(\\Gamma ) $ such that $ \\nu _0$ is $ \\rho _\\ast (\\mu _0)$ -stationary.", "We can apply [10], theorem 1.1, to the action of $ \\Gamma $ on the hyperbolic plane with the measure $\\nu _{\\partial \\Delta } = (\\xi ^{-1})_\\ast \\nu _0 .$ Since the mapping $ \\xi $ is Hölder continuous and $ \\Gamma $ -equivariant, the measure $ \\nu _{\\partial \\Delta } $ has Hölder continuous Radon-Nikodym derivatives under the action of $\\Gamma $ as well.", "Let $ \\mu _0 $ be the measure given by [10] Theorem 1.1 and such that $\\nu _{\\partial \\Delta } $ is the stationary measure under $ \\mu $ .", "The measure $ \\mu $ has whole support on $ \\Gamma $ and satisfies $ \\sum _g \\mu _0 (g) d(o, go ) < + \\infty $ ([10], page 488).", "It does indeed belong to $ \\mathcal {M}(\\Gamma ).$ To summarize, the measure $ \\mu := \\rho _\\ast \\mu _0$ on $\\rho (\\Gamma ) $ has the property that $ \\nu _\\mathcal {L}$ is absolutely continuous and is the measure at infinity of the SRB measure of the geodesic flow on $ H\\Delta .$ Moreover, by (REF ), $ \\delta _\\mathcal {P}= 1.$ Consider the dual representation $ \\rho ^\\ast (\\gamma ) = (\\rho (\\gamma )^t)^{-1} $ and the measure $ \\mu ^\\ast := (\\rho ^\\ast )_\\ast \\mu _0 $ .", "We have the same entropy $ h = h_{{\\textrm {RW}}} (\\mu ) = h_{{\\textrm {RW}}} (\\mu ^\\ast ) $ and the opposite exponents.", "Therefore the dimension of the stationary measure $ \\nu ^\\ast _\\mathcal {L}$ is $ \\delta _\\mathcal {P}=1.$ By the variational principle again, $ \\nu ^\\ast _\\mathcal {L}$ is absolutely continuous.", "By [2] Proposition 6.2, the representation is Fuchsian.", "Using the dual representation $ \\rho ^\\ast $ , we also have Corollary 9.4 Let $ \\rho $ be a Hitchin representation of a cocompact surface group in $ SL_3(\\mathbb {R}) $ such that for all probability measures in $ \\mathcal {M}(\\rho (\\Gamma )), \\, \\chi _2 \\ge 0 .$ Then, the representation $ \\rho $ is Fuchsian." ], [ "Hitchin components in higher dimensions", "Consider the surface group $ \\Gamma .$ As before, a representation $ \\rho : \\Gamma \\rightarrow PSL_d(\\mathbb {R}) $ is called Fuchsian if it is the composition of a Fuchsian representation and the canonical irreducible representation of $ PSL_2(\\mathbb {R}) $ into $ PSL _d(\\mathbb {R}) $ .", "It is called Hitchin if it can be obtained by a deformation of a Fuchsian representation.", "Geometric properties of Hitchin representations have been studied, notably by F. Labourie (see [36], [37] for history, background, the properties we use below and much more).", "Let $ \\rho :\\gamma \\rightarrow PSL_d(\\mathbb {R}) $ be a Hitchin representation and denote again by $ \\rho (\\Gamma ) $ a lift of the representation to $ SL_d(\\mathbb {R}) $ .", "By [24], proposition 14, the action of $ \\rho (\\Gamma )$ on $ \\mathbb {R}^d$ is strongly irreducible.", "By [36], theorem 1.5, the matrix $ \\rho (\\gamma ) $ , for $ \\gamma $ non-trivial has all eigenvalues real and distinct.", "In particular, for $ \\gamma $ non trivial, there is a unique attracting fixed point $ \\gamma ^+ $ for the action of $\\rho ( \\gamma )$ on $ \\mathcal {F}$ .", "By definition, the limit set $\\Lambda $ is the closure of the set of all $ \\gamma ^+, \\gamma \\ne Id \\in \\Gamma .$ Moreover, the projection from the limit set $ \\Lambda $ to $ \\mathbb {R}\\mathbb {P}^{d-1} $ is one-to-one ([36] theorem 4.1).", "Let $ \\mu $ be an adapted probability measure on $ \\Gamma $ such that $ \\sum _\\gamma |g| \\mu (\\gamma ) < +\\infty $ for some word metric $|\\cdot | $ on $ \\Gamma $ .", "The measure $ \\rho _\\ast (\\mu ) $ does not necessarily belong to $\\mathcal {M}(SL_d (\\mathbb {R}]))$ (see [24] and [48] for the description of the possible Zariski closures of $ \\rho (\\Gamma )$ ), but the action is proximal on all exterior products and we can apply [25] to the random walk $ (\\rho (\\Gamma ) , \\rho _\\ast (\\mu ) ):$ the exponents $ \\chi _1 > \\ldots >\\chi _d $ are distinct and satisfy $ \\sum _i \\chi _i = 0 $ .", "Since the action on $\\mathbb {R}^d$ is strongly irreducible, there is a unique stationary measure $ \\nu $ on $ \\mathbb {R}\\mathbb {P}^{d-1} $ .", "That measure has a unique lift to $ \\Lambda $ and therefore, there is a unique stationary measure on $ \\mathcal {F}$ .", "For the same reason, there is a unique stationary measure for $\\mu ^{\\prime }$ on $ \\mathcal {F}$ .", "All our discussion and theorem REF apply, the measure $ \\nu $ is exact dimensional with dimension $\\delta $ and entropy $ h := h(\\mathcal {F}, \\mu , \\nu ) $ .", "Since $ \\rho (\\Gamma ) $ is discrete ([36], theorem 1.5), we have $ h = h_{{\\textrm {RW}}} (\\mu ) $ ([39]).", "Proposition 9.5 Let $ \\lambda := \\inf _{i<j} (\\chi _i -\\chi _j).$ Then $ \\delta = h/ \\lambda .$ More generally, let $T \\ne T_0$ be an admissible topology, $ \\kappa _T, \\delta _T$ as defined in (REF ) and corollary REF .", "Let $ \\lambda _T: = \\inf _{i<j, j \\notin T(i)} (\\chi _i -\\chi _j).$ Then, $ \\kappa _T= h $ and $ \\delta _T= h/ \\lambda _T.$ In particular, theorem REF follows in all dimensions.", "Another consequence is that, as in dimension 3, comparing $ \\lambda $ and $ \\lambda _T $ is enough to decide whether the projection from $ \\mathcal {F}$ to $ \\mathcal {X}_T^{f^{\\prime }} $ is dimension conserving for $ \\nu ^{\\prime }$ -a.e.", "$ f^{\\prime }.$ Corollary 9.6 Let $T $ be an admissible topology such that $ T_1 \\overset{1}{\\prec } T $ .", "Then $ \\delta _T = \\delta $ unless there is a unique $ i $ with $ \\lambda = \\chi _i - \\chi _{i+1} $ and furthermore $ T = \\lbrace 1\\rbrace , \\lbrace 2\\rbrace , \\ldots , \\lbrace i, i+1\\rbrace , \\ldots , \\lbrace d\\rbrace .$ The measure $\\nu $ is supported by the limit set $ \\Lambda \\subset \\mathcal {F}.$ Labourie showed that $ \\Lambda $ is a hyperconvex Frenet curve with Property (H).", "Namely: There is a Hölder continuous $\\Gamma $ -equivariant mapping $ \\xi : \\mathbb {S}^1 \\rightarrow \\Lambda ,$ $ \\xi (t) = \\lbrace 0\\rbrace \\subset \\xi _1 (t) \\subset \\ldots \\subset \\xi _i (t) \\subset \\ldots \\subset \\xi _d (t) = \\mathbb {R}^d .", "$ For any distinct points $ t_1 \\ldots , t _\\ell $ integers $ d_1, \\ldots , d_\\ell $ with $ p := \\sum _{j=1}^\\ell d_j ,$ the following sum is direct $ \\xi _{d_1, \\ldots , d_\\ell }(t_1, \\ldots , t_\\ell ) \\; : = \\; \\xi _{d_1} (t_1) \\oplus \\ldots \\oplus \\xi _{d_\\ell } (t_\\ell ) $ and, if the distinct $ t_1 \\ldots , t _\\ell $ all converge to $ x$ , then $ \\xi _{d_1, \\ldots , d_\\ell }(t_1, \\ldots , t_\\ell ) $ converge to $ \\xi _p (x) $ .", "for any triple of distinct points $ (s,t,t^{\\prime })$ , any integer $i, 0< i < d$ , $ \\xi _i (s) \\oplus (\\xi _i (t) \\cap \\xi _{d-i+1} (t^{\\prime }) )\\oplus \\xi _{d-i -1} (t^{\\prime }) \\; = \\; \\mathbb {R}^d.$ Property (2) defines a hyperconvex Frenet curve ([36], theorem 1.4) and property (3) is relation (6) in [36] theorem 4.1.", "(Property (3) is called Property (H) in [36] section 7.1.4.)", "Let $T^1$ be an admissible topology such that $ T^1 \\overset{1}{\\prec } T_0 $ .", "We claim that there is a unique integer $ i , 0< i< d,$ such that $ T^1(k) = \\lbrace k, k+1, \\ldots , d \\rbrace {\\textrm { for }} k \\ne i, \\quad T^1(i) = \\lbrace i, i+2, \\ldots , d \\rbrace .$ Indeed, by definition, there is $ i , 0< i< d,$ such that $ T^1(k) = T_0(k) $ for $ k \\ne i $ and $ j >i $ such that $T^1(i) = T_0(i) \\setminus \\lbrace j\\rbrace $ .", "By Proposition REF , $ T^1(i) \\setminus \\lbrace i,j \\rbrace \\in T_0, $ and this is possible only if $ j = i+1.$ Fix $ t^{\\prime } \\in \\mathbb {S}^1$ and set $f^{\\prime } := \\xi (t^{\\prime }) .$ By lemma REF , the set $ \\mathcal {X}_{T^1}^{f^{\\prime }} $ is bilipschitz homeomorphic to an open interval.", "We associate to $ t \\in \\mathbb {S}^1, t \\ne t^{\\prime },$ a configuration $ \\Psi (t ) \\in \\mathcal {X}_{T^1}^{f^{\\prime }} $ by setting: $ \\Psi (t) _{{T^1}(k )} &=& \\xi _{d-k+1} (t^{\\prime }) \\; {\\textrm { for }} \\; k \\ne i ,\\\\\\Psi (t) _{{T^1}(i)} &=& \\left(\\xi _i(t) \\cap \\xi _{d-i+1}(t^{\\prime }) \\right) \\oplus \\xi _{d-i-1 }(t^{\\prime }) .$ Lemma 9.7 The mapping $ \\Psi $ is an homeomorphism between $ \\mathbb {S}^1 \\setminus \\lbrace t^{\\prime }\\rbrace $ and its image in $ \\mathcal {X}_{T^1}^{f^{\\prime }}.$ By the hyperconvexity property (2), dim$\\left(\\xi _i(t) \\cap \\xi _{d-i+1}(t^{\\prime }) \\right) =1$ and that space is disjoint from $ \\xi _{d-i-1} (t^{\\prime }) $ ; therefore the mapping $ \\Psi $ is continuous.", "Since $ \\Psi $ is a mapping between two open intervals, it suffices to show that $ \\Psi $ is one-to-one.", "Assume by contradiction that there is $ s\\ne t,t^{\\prime } $ such that $ \\left(\\xi _i(s) \\cap \\xi _{d-i+1}(t^{\\prime }) \\right) \\oplus \\xi _{d-i -1}(t^{\\prime }) = \\left(\\xi _i(t) \\cap \\xi _{d-i+1}(t^{\\prime }) \\right) \\oplus \\xi _{d-i -1}(t^{\\prime }) .$ By property (3), $ \\xi _i (s) $ should be in direct sum with $ \\left(\\xi _i(s) \\cap \\xi _{d-i+1}(t^{\\prime }) \\right) \\oplus \\xi _{d-i -1}(t^{\\prime }) $ , which is possible only if $ \\xi _i(s) \\cap \\xi _{d-i+1}(t^{\\prime }) = \\lbrace 0\\rbrace .", "$ This contradicts hyperconvexity.", "By lemma REF , for any $ x^{\\prime } \\in \\mathcal {X}_{T^1}^{f^{\\prime }}$ , there at most one point $ s \\in \\mathbb {S}^1 $ such that $ \\pi _{T_1, {T^1}} ( \\xi (s) ) = x^{\\prime }, $ i.e.", "$ (\\pi _{T_1, {T^1}})^{-1} (x^{\\prime }) $ is at most one point.", "So, $ \\kappa _{T_1, {T^1} } = 0 .$ By theorem REF , we have $ \\delta _{T^1} =h /\\chi _{{T^1}, T_0} .$ For a general admissible topology $ T,$ we apply proposition REF and obtain a topology $ T^1 $ such that $ T\\prec T^1 \\overset{1}{\\prec } T_0 $ and $ \\chi _{T^1, T_0} = \\lambda _T.", "$ Since $ T_1 \\prec T \\prec T^1$ , $ \\kappa _{T, T^1} = 0$ and $ \\kappa _T = h $ .", "By (REF ) and corollary REF , $ \\delta _T $ is the same as $ \\delta _{T^1} = h / \\lambda _T.$" ] ]
2105.11712
[ [ "Single boson exchange representation of the functional renormalization\n group for strongly interacting many-electron systems" ], [ "Abstract We present a reformulation of the functional renormalization group (fRG) for many-electron systems, which relies on the recently introduced single boson exchange (SBE) representation of the parquet equations [Phys.", "Rev.", "B 100, 155149 (2019)].", "The latter exploits a diagrammatic decomposition, which classifies the contributions to the full scattering amplitude in terms of their reducibility with respect to cutting one interaction line, naturally distinguishing the processes mediated by the exchange of a single boson in the different channels.", "We apply this idea to the fRG by splitting the one-loop fRG flow equations for the vertex function into SBE contributions and a residual four-point fermionic vertex.", "Similarly as in the case of parquet solvers, recasting the fRG algorithm in the SBE representation offers both computational and interpretative advantages: the SBE decomposition not only significantly reduces the numerical effort of treating the high-frequency asymptotics of the flowing vertices, but it also allows for a clear physical identification of the collective degrees of freedom at play.", "We illustrate the advantages of an SBE formulation of fRG-based schemes, by computing through the merger of dynamical mean-field theory and fRG the susceptibilities and the Yukawa couplings of the two-dimensional Hubbard model from weak to strong coupling, for which we also present an intuitive physical explanation of the results.", "The SBE formulation of the one-loop flow equations paves a promising route for future multiboson and multiloop extensions of fRG-based algorithms." ], [ "Introduction", "A major challenge for the theory of many-electron systems is to correctly describe the competing microscopical processes occurring on very different length and time scales.", "This task becomes particularly hard in the nonperturbative regime of intermediate to strong interactions, where several fascinating phenomena of condensed matter physics take place.", "The recently developed [1], [2], [3] merger of the dynamical mean field theory (DMFT) [4], [5] and the functional renormalization group (fRG) [6], [7], [8], [9], [10], coined as DMF2RG [1], can be regarded as one of the diagrammatic extensions [11] of DMFT designed to be applied in the most challenging parameter regimes of quantum many-body Hamiltonians.", "By combining the unbiased [9] diagrammatic structure of the fRG to the intrinsic nonperturbative content [12] of DMFT, the DMF2RG offers a particularly promising route to tackle crucial nonperturbative features of the many-electron physics.", "However, the impact of DMF2RG calculations will eventually depend on the numerical performance of its actual implementations, where the evident bottleneck is posed by the treatment of the two-particle vertex functions [13], [14], due to the large number of momentum and frequency variables needed for their definition.", "The single boson exchange (SBE) decomposition, recently introduced [15] to rationalize the treatment of parquet-type diagrams [16], [17], [18], [19], [20], [13], [21], [22], [23], leads to a significant reduction of the computational effort [24], [25].", "Due to the qualitative similarity of the diagrammatic structure in parquet- and fRG-based approximations, it seems quite natural to exploit similar ideas also in an fRG context, recasting the fRG flow equations within the SBE formalism.", "On a general level, we recall that applying the SBE formalism corresponds to recasting the two-particle diagrams in terms of their reducibility/irreducibility with respect to the cutting of an interaction line.", "Due to the two-particle nature of the electronic (Coulomb) interaction, the SBE classification shares important qualitative features with the parquet formalism as, for example, the high-frequency asymptotic properties [13], [26], [14] of the corresponding irreducibile diagrams.", "At the same time, the SBE classification of diagrams circumvents important problems (such as the multiple divergences of the irreducible vertices [27], [28], [29], [30], [31], [32], [33], [34], [12]) which affect parquet-based approaches in the nonperturbative regime.", "The specific application of the SBE formalism to the fRG presented here relies on the partial bosonization of the vertex function [35], [36], [37], similar to the channel decomposition [38], [39], [40], [41], [42], [3] already adopted in the context of fRG and parquet solvers [43], [24] (for recent developments in this direction see also Refs.", "[44], [45], [46], [47]).", "In addition to the screened interaction, a fermion-boson Yukawa coupling  [48], [49] (or Hedin vertex [50]) is determined from the vertex asymptotics, similarly to the construction of the kernel functions describing the high-frequency asymptotics, see Ref. [14].", "Their relation allows to recover the flow equations of the screened interaction and Yukawa coupling in the SBE representation.", "We note that the obtained structure is apparently the same as the one reported in Ref.", "[51], where instead of the high-frequency limit the zero frequency value has been used.", "Several bosonization procedures have been already developed for the weak coupling fRG, by applying, for instance, the Hubbard-Stratonovich transformation on the bare action [52], [53], [54], [55], [56], or by means of the dynamical bosonization [57], [36], [58].", "Instead, the description of the DMF2RG vertex in terms of exchanged bosons is a highly nontrivial task.", "Indeed, the complex frequency structure of the initial DMFT vertex function [13] prevents a straightforward application of the Gaussian integration in the Hubbard-Stratonovic procedure or of the dynamical bosonization.", "In this perspective, the SBE decomposition offers a relatively simple way to circumvent this problem, expressing the DMFT vertex in terms of bosonic propagators and Yukawa couplings, which can then be used as initial conditions of a mixed boson-fermion flow.", "In this respect, it is worth stressing that the fermion-boson approach [59], [60] has proven to be useful also in the context of diagrammatic extensions of DMFT [61] not based on the fRG formalisms.", "After presenting the detailed derivation of the SBE representation of fRG and DMF2RG, we will illustrate its main features by selected applications of the approach to the 2D Hubbard model See Ref.", "[109] for a recent overview of computational results for the 2D Hubbard model.", "both at half filling from weak to strong coupling, as well as finite doping.", "In the latter case, we exploit the transparent nature of the SBE formulation to investigate the physical mechanisms responsible for the enhanced $d$ -wave pairing fluctuations found in the DMF2RG calculations.", "The paper is organized as follows: In Sec.", "we introduce the Hubbard model and present the SBE decomposition together with its implementation in both the fRG and DMF2RG flow.", "In Secs.", "and we discuss the results for the susceptibilities and Yukawa couplings at half filling and out of it, providing also a comparison with the fermionic formalism.", "In order to identify the mechanisms responsible for strong $d$ -wave pairing correlations, we perform a diagnostics [63], [28], [64], [65], [66] of the corresponding fluctuations.", "We finally conclude with a summary and an outlook in Sec. .", "We consider the single-band Hubbard model in two dimensions, $\\begin{split}\\mathcal {H}=\\sum _{i\\ne j,\\sigma }t_{ij}c^\\dagger _{i\\sigma }c_{j\\sigma }+U\\sum _i n_{i\\uparrow }n_{i\\downarrow }-\\mu \\sum _{i,\\sigma } n_{i\\sigma },\\end{split}$ where $c_{i\\sigma }$ ($c^{\\dagger }_{i\\sigma }$ ) annihilates (creates) an electron with spin $\\sigma $ at the lattice site $i$ ($n_{i\\sigma }=c^{\\dagger }_{i\\sigma }c_{i\\sigma }$ ), $t_{ij}=-t$ is the hopping between nearest-neighbor sites, $t_{ij}=-t^{\\prime }$ the hopping between next-nearest-neighbor sites (the Fourier transform of $t_{ij}$ gives the bare dispersion $\\epsilon _\\mathbf {k}$ ), and $U$ the on-site Coulomb interaction.", "The filling is fixed by adjusting the chemical potential $\\mu $ .", "In the following we use $t\\equiv 1$ as the energy unit." ], [ "SBE decomposition", "In this section we present the SBE decomposition as introduced in Ref. [15].", "All diagrams contributing to the two-particle vertex can be divided into $U$ -reducible and $U$ -irreducible, depending on whether the removal of a bare interaction vertex cuts the diagram into two disconnected parts or not, respectively.", "Moreover, among the $U$ -reducible diagrams, we can identify three different channels, depending on how the fermionic legs are connected to the removed interaction $U$ .", "In particular, we find $U$ -particle-particle ($U$ -$pp$ ), $U$ -particle-hole ($U$ -$ph$ ) and $U$ -particle-hole-crossed ($U$ -$\\overline{ph}$ ) reducible diagrams [15].", "This classification of diagrams is alternative, and not equivalent, to the more common one, based on the notion of two-particle reducibility (for the relation to the latter we refer to the Appendix ).", "Indeed, there are some diagrams which are two-particle reducible in a given channel, but $U$ -irreducible.", "These diagrams are the so-called box diagrams (see Fig.", "REF ).", "Figure: Representative diagrams of the UU irreducibility.", "Here we choose the particle-particle (pppp) channel as an example.", "(a) two-particle- and UU-reducible diagram in the pppp channel.", "Since this diagram has two interaction vertices at its both ends, it contributes to the screened interaction.", "(b) Two-particle- and UU-reducible diagram, contributing to the Yukawa coupling, as there is only one bare interaction vertex which can be removed to disconnect the diagram.", "(c) Diagram which is two-particle reducible but UU-irreducible, therefore contributing to the SBE rest function.In general, a diagram which is $U$ -reducible in a given channel is also two-particle reducible in the same channel, but not vice versa, the only exception being the diagram composed by a single interaction vertex.", "Within the SBE decomposition, it is quite natural to interpret the collection of all $U$ -reducible diagrams in a given channel as an effective interaction between two fermions, mediated by the exchange of a boson, representing a collective fluctuation.", "As mentioned in Refs.", "[15], [67], [24] and illustrated explicitly in the following, the SBE decomposition not only significantly reduces the numerical complexity of the vertex functions, but it also allows for a clear physical identification of the collective degrees of freedom arising in a strongly correlated electron system.", "In a system with the U(1)-charge and SU(2)-spin symmetries, along with translational invariance, the two-particle vertex can be expressed as [13]: $\\begin{split}V_{\\sigma _1\\sigma _2\\sigma _3\\sigma _4}(k_1,k_2,k_3,k_4) &=V(k_1,k_2,k_3)\\delta _{\\sigma _1\\sigma _3}\\delta _{\\sigma _2\\sigma _4}\\\\&-V(k_2,k_1,k_3)\\delta _{\\sigma _1\\sigma _4}\\delta _{\\sigma _2\\sigma _3},\\end{split}$ where $\\sigma _i$ represents the spin quantum number, and $k_i=(\\mathbf {k}_i,\\nu _i)$ is a collective variable including the crystal momentum and a fermionic Matsubara frequency.", "The fourth fermionic variable, $k_4$ , on which the vertex depends, is fixed by momentum and energy conservation.", "According to the SBE decomposition, we represent the function $V=V_{\\uparrow \\downarrow \\uparrow \\downarrow }$ as $\\begin{split}V(k_1,k_2,k_3) &= \\Lambda _\\text{$U$irr}(k_1,k_2,k_3) - 2U+ \\mathcal {S}_{k_1 k_3}(k_1+k_2)\\\\&+ \\mathcal {M}_{k_1 k_3}(k_2-k_3) +\\frac{1}{2} \\mathcal {M}_{k_1 k_4}(k_3-k_1)\\\\& +\\frac{1}{2} \\mathcal {C}_{k_1 k_4}(k_3-k_1),\\end{split}$ where $k_4=k_1+k_2-k_3$ , $\\mathcal {S}$ , $(\\mathcal {M}+\\mathcal {C})/2$ , and $\\mathcal {M}$ represent the sum of all $U$ -$pp$ , $U$ -$ph$ , and $U$ -$\\overline{ph}$ reducible diagrams, respectively, the function $\\Lambda _\\text{$U$irr}$ accounts for all fully $U$ -irreducible diagrams, and a term $2U$ has been subtracted in order to avoid double counting of the bare interaction, which is already included in the $U$ -reducible channels.", "The functions $\\mathcal {S}$ , $\\mathcal {M}$ , and $\\mathcal {C}$ , corresponding to the pairing, charge and magnetic channel respectively, depend on two fermionic variables, and a bosonic one, indicated in brackets in Eq.", "(REF ).", "Since each of them is $U$ -reducible in a given channel, their dependencies on the fermionic arguments can be factorized.", "We can therefore express them as Mkk'(q)hmk(q) Dm(q) hmk'(q), Ckk'(q)hck(q) Dc(q) hck'(q), Skk'(q)hsk(q) Ds(q) hsk'(q), where we name $h$ (${h}$ ) as left-sided (right-sided) Yukawa coupling and $D$ as screened interaction, which plays the role of an effective bosonic propagator.", "The right-sided Yukawa couplings ${h}$ can be related to their respective left-sided ones, $h$ through the relations hXk(q) = hXk+q(-q),      X=m,c, hsk(q) = hsk(q).", "By considering the symmetries V(k1,k2,k3) = V(k2,k1,k1+k2-k3), V(k1,k2,k3) = V(k3,k1+k2-k3,k1), where Eq.", "(REF ) corresponds to the simultaneous exchange of the ingoing and outgoing variables, and Eq.", "(REF ) to the simultaneous exchange of the two ingoing variables with the two outcoming ones, one can easily prove that $h^X_{k+q}(-q)=h^X_k(q)$ , with $X=m,c$ .", "Therefore the relation $h_k^X(q)=\\bar{h}_k^X(q)$ holds for all channels.", "For this reason, from now on we label by $h^X$ both the left- and the right-sided Yukawa couplings.", "It is worthwhile to stress that the equivalence $h=\\bar{h}$ holds because of the choice of notation (REF ), different choices may lead to to more complicated relations between $h$ and $\\bar{h}$ .", "The screened interactions $D$ are related to the physical susceptibilities [15], [67], that is Dm(q) = U + U2 m(q), Dc(q) = U - U2 c(q), Ds(q) = U - U2 s(q), where $\\chi ^m$ , $\\chi ^c$ , and $\\chi ^s$ are the magnetic, charge, and pairing susceptibilities of the system, respectively.", "The Yukawa couplings are connected to the so-called three-legged correlators via hmk(q) = 1-sgn ck+q,ck, m-q 1PI1+Um(q), hck(q) = 1-ck+q,ck, -q 1PI1-Uc(q), hsk(q) = 1+cq-k,ck, q 1PI1-Us(q), where the symbol $\\langle \\cdots \\rangle _\\text{1PI}$ indicates the connected average with amputated external propagators, and the fermionic bilinears are defined as mq = ksgn  ck+q,ck,, q = kck+q,ck,, q = k cq-k,ck,.", "Here, and from now on, the symbol $\\int _k=T\\sum _\\nu \\int _\\mathrm {B.Z.", "}\\frac{d^2\\mathbf {k}}{(2\\pi )^2}$ denotes a sum over fermionic Matsubara frequencies and a momentum integration over the Brillouin zone.", "Notice that in Eq.", "(REF ) the division by a term $1\\pm U\\chi $ is necessary to avoid double counting as it removes from the three-legged correlators all those $U$ -reducible diagrams which are already included in the screened interaction $D$ .", "It is interesting to consider the asymptotic high frequency behavior of the Yukawa couplings and bosonic propagators: DX(q,) = U, hX(k,)(q) = hXk(q,) = 1.", "The limits of large bosonic frequency $\\Omega $ are quite trivial to prove, as in this case both the susceptibilities and the three-legged correlators are zero.", "More interesting is the large fermionic frequency limit for the Yukawa coupling.", "In this limit, one can show by diagrammatic arguments [14] that the three-legged correlator approaches $\\pm U \\chi $ (the sign depending on the channel), which makes the Yukawa coupling approaching 1." ], [ "Flow equations", "In this section we derive the one-loop ($1\\ell $ ) fRG flow equations within the SBE decomposition.", "The dependencies of the various functions on the RG scale $\\Lambda $ are implicit to keep the notation lighter.", "We start by focusing on the flow equations within the channel decomposition of Husemann and Salmhofer [39], [51], who divided the different contributions to the vertex according to the notion of two-particle reducibility.", "The flow equation for a given two-particle reducible channel $\\phi $ reads $\\begin{split}\\partial _\\Lambda \\phi ^X_{kk^{\\prime }}(q) = \\int _p L^X_{kp}(q) \\left[\\widetilde{\\partial }_\\Lambda \\Pi ^X_p(q) \\right] L^X_{pk^{\\prime }}(q), \\end{split}$ with $X=m,c$ or $s$ .", "The bubbles are defined as mk(q) = -G(k)G(k+q), ck(q) = G(k)G(k+q), sk(q) = -G(k)G(q-k), with $G(k)=[(\\Theta ^\\Lambda (k)/(i\\nu -\\epsilon _\\mathbf {k}+\\mu ))^{-1}-\\Sigma (k)]^{-1}$ the propagator, and $\\Sigma $ the self-energy.", "The symbol $\\widetilde{\\partial }_\\Lambda $ denotes a derivative only on the explicit RG scale dependence of the propagator introduced by the cutoff function $\\Theta ^\\Lambda (k)$ .", "The $L$ -functions are given by Lmkk'(q) = V(k',k,k+q), Lckk'(q) = 2V(k,k',k+q) - V(k',k,k+q), Lskk'(q) = V(k,q-k,k'), with $V$ defined in Eqs.", "(REF ) and (REF ).", "Each two-particle reducible channel consists of two contributions, one which is also $U$ -reducible and one which is not.", "We can therefore express $\\phi $ as $\\phi ^X_{kk^{\\prime }}(q) = h^X_{k}(q)\\,D^X(q)\\,h^X_{k^{\\prime }}(q) + \\mathcal {R}^X_{kk^{\\prime }}(q),$ where we identify the $U$ -irreducible contribution $\\mathcal {R}^X$ as rest function.", "It obeys the asymptotic relations [14] $\\begin{split}\\lim _{\\nu \\rightarrow \\infty } \\mathcal {R}^X_{(\\mathbf {k},\\nu ),k^{\\prime }}(q) =&\\lim _{\\nu ^{\\prime }\\rightarrow \\infty } \\mathcal {R}^X_{k,(\\mathbf {k}^{\\prime },\\nu ^{\\prime })}(q) \\\\= &\\lim _{\\Omega \\rightarrow \\infty } \\mathcal {R}^X_{kk^{\\prime }}(\\mathbf {q},\\Omega ) = 0.", "\\end{split}$ With the help of Eqs.", "(REF ) and (REF ), we further notice that $\\lim _{\\nu ^{\\prime }\\rightarrow \\infty }L^X_{k,(\\mathbf {k},\\nu ^{\\prime })}(q) = h^X_k(q) D^X(q).$ By inserting the limits (REF ) and (REF ) into Eq.", "(REF ), we hence obtain the flow equations for the screened interactions, Yukawa couplings and rest functions: DX(q) = [DX(q)]2p hXp(q) [Xp(q) ] hXp(q), hXk(q) = p LXkp(q) [Xp(q) ] hXp(q), RXkk'(q) = p LXkp(q) [Xp(q) ] LXpk'(q), where $\\mathcal {L}^X_{kp}(q)=L^X_{kp}(q)-h^X_{k}(q)\\,D^X(q)\\,h^X_{p}(q)$ .", "Alternatively, the above equations can be derived through the explicit introduction of three bosonic fields via as many Hubbard-Stratonovich transformations (see Appendix ).", "We note that a similar decomposition has been used in Ref.", "[51], where the flow equations for the screened interactions and Yukawa couplings exhibit the same structure.", "However, instead of using the vertex asymptotics, they are determined by its value at the lowest Matsubara frequency.", "By applying the same line of reasoning, one can generalize the flow Eq.", "(REF ), derived within the $1\\ell $ truncation, to the multiloop extension introduced in Refs.", "[68], [69], [42], [44].", "In the following we discuss its application to the conventional fRG as well as its merger with the DMFT in the DMF2RG." ], [ "fRG", "Within the conventional fRG, the fully $U$ -irreducible vertex function $\\Lambda _{U_\\mathrm {irr}}$ corresponds to the sum of the rest functions of the three channels, and hence does not include any contributions from diagrams that are fully two-particle irreducible." ], [ "Cutoff scheme", "In order to run a fRG flow, we regularize the bare Green's function by making use of the so-called $\\Omega $ -cutoff [39], [51], that is $G_0^\\Lambda (\\mathbf {k},\\nu ) = \\frac{\\Theta ^\\Lambda (\\nu )}{i\\nu +\\mu -\\epsilon _\\mathbf {k}},$ where the cutoff function is given by $\\Theta ^\\Lambda (\\nu ) = \\frac{\\nu ^2}{\\nu ^2+\\Lambda ^2}.$" ], [ "Initial conditions", "The fRG initial conditions at $\\Lambda =\\Lambda _\\text{ini}$ read as DXini(q) = U, hXini,k(q) = 1, RXini,kk'(q) = 0, that, by comparing with Eq.", "(REF ), is equivalent to imposing $V_{\\text{ini}}=U$ ." ], [ "DMF", "The DMF2RG [1], [3] differs from the fRG by the initial conditions (and the cutoff scheme used), see Fig.", "REF .", "To determine the initial conditions in this case, we apply the decomposition in Eq.", "(REF ) also to the DMFT vertex.", "The local bosonic propagators and Yukawa couplings can be computed directly from the Anderson impurity model (AIM), from Eqs.", "(REF )-(REF ), while the fully $U$ -irreducible vertex can be extracted by subtraction from Eq.", "(REF ).", "Figure: Central idea of DMF2RG: Starting from an effective AIM, nonlocal correlation effects are gradually included by the flow." ], [ "Cutoff scheme", "When introducing a scale dependence on the bare fermionic propagator through a cutoff, one has to keep in mind that the regularized bare propagator has to smoothly interpolate between the bare propagator of the self-consistent AIM and the bare lattice one, specifically: G0,ini(k,)=G0,AIM()=1i+-hyb(), G0,fin(k,)= G0(k,)=1i+-k, where $\\Delta _{\\text{hyb}}(\\nu )$ is the hybridization function of the AIM.", "Furthermore, we require the DMFT solution for the local propagator to be conserved at each step of the flow, that is [3] $\\int _\\text{B.Z.", "}\\frac{d^2\\mathbf {k}}{(2\\pi )^2}G(\\mathbf {k},\\nu )\\Big \\vert _{\\Sigma =\\Sigma _{\\rm dmft}} = \\mathcal {G}_{\\text{AIM}}(\\nu ),$ with $\\mathcal {G}_{\\text{AIM}}(\\nu )=[\\mathcal {G}_{0,\\text{AIM}}^{-1}(\\nu )-\\Sigma _{\\rm dmft}(\\nu )]^{-1}$ the full Green's function of the AIM, and $\\Sigma _\\mathrm {dmft}$ the DMFT self-energy.", "We therefore make the following choice for the regularized Green's function $\\begin{split}G^\\Lambda _0(\\mathbf {k},\\nu ) &= \\Theta ^\\Lambda (\\nu )G_0(\\mathbf {k},\\nu ) + \\Xi ^\\Lambda (\\nu )\\mathcal {G}_{0,\\text{AIM}}(\\nu ),\\end{split}$ where we choose the function $\\Theta ^\\Lambda (\\nu )$ to be a smooth frequency cutoff, namely the same as in Eq.", "(REF ), which imposes the boundary values for the RG scale $\\Lambda _\\mathrm {ini}=+\\infty $ , and $\\Lambda _\\mathrm {fin}=0$ .", "The function $\\Xi ^\\Lambda (\\nu )$ is determined by the DMFT self-consistency relation (REF ).", "With the above definitions it is straightforward to check that the boundary conditions (REF ) are consistently fulfilled.", "This choice for the DMF2RG cutoff (REF ) has a direct intuitive physical meaning: at a given scale $\\Lambda $ , indeed, the fermionic modes with energies $|\\nu |\\gtrsim \\Lambda $ are nonlocal and their contributions on top of the DMFT solution have been included, while the ones with energies $|\\nu |\\lesssim \\Lambda $ still belong to the AIM and do not yet contribute to the generation of nonlocal correlations." ], [ "Initial conditions", "The DMF2RG employs the DMFT solution as a correlated starting point for the RG flow.", "The initial conditions for the screened interactions and Yukawa couplings therefore read DXini(q) = DXloc(), hXini,k(q) = hXloc,(), where $D_\\text{loc}$ and $h_\\text{loc}$ are the bosonic propagator and Yukawa couplings of the self-consistent AIM.", "Concerning the rest functions, we set their initial value to zero, so that they will represent only the nonlocal contributions, the local ones being already included in the $U$ -irreducible vertex function of the AIM, $\\Lambda _{U\\text{irr}}^{\\text{loc}}$ .", "The flow equations for the screened interactions, Yukawa couplings, and rest functions are the same as in the plain fRG." ], [ "Form factor decomposition", "In order to simplify the set of flow equation described above, throughout this work we project the Yukawa coupling dependence onto the secondary spatial momentum $\\mathbf {k}$ onto $s$ -wave form factors, $f^s_\\mathbf {k}\\equiv 1$ , that is we approximate $h^X_{k}(q) \\sim h^X_\\nu (q) = \\int _\\mathbf {k}f^s_\\mathbf {k}\\, h^X_{(\\mathbf {k},\\nu )}(q),$ with $\\int _\\mathbf {k}$ a shorthand for $\\int _\\mathrm {B.Z.", "}\\frac{d^2\\mathbf {k}}{(2\\pi )^2}$ .", "Within this approximation, Eq.", "(REF ) becomes DX(q) = [ DX(q)]2 ThX(q)[X(q)] hX(q), hX(q) = T' LX'(q)[X'(q)] hX'(q), RX'(q) = T LX(q)[X(q)] LX'(q), where X(q) = k(fsk)2 X(k,)(q), L'X(q) = kk' fskfsk' LX(k,),(k',')(q).", "We finally note that if the residual vertex develops a strong momentum dependence, considering a limited number of form factors is not justified a priori and has to be verified." ], [ "$d$ -wave pairing channel", "In the doped regime we include $d$ -wave pairing fluctuations into our parametrization of the two-particle vertex by $\\begin{split}V(k_1,k_2,k_3) &= \\Lambda _\\text{$U$irr}^{\\mathrm {loc}}(\\nu _1,\\nu _2,\\nu _3) - 2U\\\\&+ \\mathcal {S}_{\\nu _1 \\nu _3}(k_1+k_2) - \\mathcal {D}_{k_1 k_3}(k_1+k_2)\\\\&+ \\mathcal {M}_{\\nu _1 \\nu _3}(k_2-k_3) +\\frac{1}{2} \\mathcal {M}_{\\nu _1 \\nu _4}(k_3-k_1)\\\\&+\\frac{1}{2} \\mathcal {C}_{\\nu _1 \\nu _4}(k_3-k_1),\\end{split}$ where $\\mathcal {M}_{\\nu \\nu ^{\\prime }}(q)$ , $\\mathcal {C}_{\\nu \\nu ^{\\prime }}(q)$ , and $\\mathcal {S}_{\\nu \\nu ^{\\prime }}(q)$ are the the $s$ -wave projections of the couplings of Eq.", "(REF ), and, in order to deal with a positive quantity, we have chosen $\\mathcal {D}$ with a minus sign in front of it.", "The function $\\Lambda _\\text{$U$irr}^{\\mathrm {loc}}(\\nu _1,\\nu _2,\\nu _3)$ represents the sum of $U$ -irreducible diagrams at the DMFT level and therefore does not flow, that is in the doped regime we neglect the flow of rest functions in the magnetic, charge, and $s$ -wave pairing channels.", "The $d$ -wave pairing channel $\\mathcal {D}_{kk^{\\prime }}(q)$ is given by $\\mathcal {D}_{kk^{\\prime }}(q) = \\mathcal {D}_{\\nu \\nu ^{\\prime }}(q) f^d_{\\mathbf {k}-\\mathbf {q}/2} f^d_{\\mathbf {k}^\\prime -\\mathbf {q}/2},$ with the $d$ -wave form factor $f^d_\\mathbf {k}= \\cos k_x - \\cos k_y$ .", "In essence, the function $\\mathcal {D}_{\\nu \\nu ^{\\prime }}(q)$ represents the sum of all diagrams that are reducible in the two-particle-$pp$ channel and at the same time exhibit a $d$ -wave symmetry in the dependency on secondary momenta $\\mathbf {k}$ and $\\mathbf {k}^\\prime $ .", "Due to the locality of the bare Hubbard interaction $U$ , all the above mentioned diagrams are $U$ -irreducible, that is the $d$ -wave pairing channel will consist exclusively of its rest function.", "Its flow equation reads $\\partial _\\Lambda \\mathcal {D}_{\\nu \\nu ^{\\prime }}(q) = T\\sum _{\\nu ^{\\prime \\prime }} L^d_{\\nu \\nu ^{\\prime \\prime }}(q) \\left[\\widetilde{\\partial }_\\Lambda \\Pi ^d_{\\nu ^{\\prime \\prime }}(q)\\right] L^d_{\\nu ^{\\prime \\prime }\\nu ^{\\prime }}(q),$ with $\\Pi ^d_\\nu (q) = \\int _\\mathbf {k}\\left(f^d_\\mathbf {k}\\right)^2 G(k)G(q-k),$ and $L^d_{\\nu \\nu ^{\\prime }}(q) = \\int _\\mathbf {k}\\int _{\\mathbf {k}^{\\prime }} f^d_\\mathbf {k}f^d_{\\mathbf {k}^{\\prime }} V(k,q-k,k^{\\prime }),$ where $k=(\\mathbf {k},\\nu )$ , and $k^{\\prime }=(\\mathbf {k}^{\\prime },\\nu ^{\\prime })$ .", "This corresponds to restrict ourselves to the pure $d$ -wave first harmonic contribution.", "The flow equations for the Yukawa couplings and bosonic propagators of the magnetic, charge, and $s$ -wave pairing channels are the same as in the previous section, with the difference that the feedback of the $d$ -wave pairing channel on them is included through Eq.", "(REF )." ], [ "Results at half filling", "To illustrate the potential of our SBE implementation of the DMF2RG scheme (see Appendix  for the details on the numerical implementation), we first consider the testbed case of half filling.", "The expected physical behavior, encoded in the response functions of the different channels, is quite clear in this situation, due to the underlying particle-hole symmetry of the problem." ], [ "Weak-coupling fRG and DMF", "We start by comparing different methods in the weak-coupling regime, where also conventional perturbative approximations can be applied as a benchmark.", "In particular, for the calculations of physical susceptibilities presented in Fig.", "REF , we have exploited: the random phase approximation (RPA), the conventional $1\\ell $ fRG, the multiloop extension which amounts to the summation of the parquet diagrams, the DMFT, and the DMF2RG.", "In order to allow for a quantitative comparison of all the approaches considered on an equal footing, these specific weak-coupling fRG and DMF2RG calculations have been performed including the feedback of the self-energy in the RG flow, while the flow of $\\mathcal {R}^X$ was consistently neglected in all channels.", "In the upper panels of Fig.", "REF , we show the screened interactions $D^X(\\mathbf {q},\\Omega )$ for zero transfer frequency $\\Omega =0$ and a specific momentum transfer.", "The lower panels show the corresponding susceptibilities that are connected with the screened interactions via Eq.", "(REF ).", "As expected, for the lowest values of $U$ , the different calculations display a very good agreement, which is gradually reduced for larger values of the coupling.", "Consistent to the physics expected in the half-filled Hubbard model, all methods outline predominant antiferromagnetic (AF) correlations for the coupling values considered, though they differ on the quantitative level.", "In particular, the RPA yields, for all values of $U$ , the largest magnetic effects.", "These get suppressed, to different amounts, in the results of the other approaches, because they all capture, differently from RPA, the competition with the other channels.", "In particular, a sizable suppression of the AF susceptibility for $U > t$ is clearly observed, within fRG, already at the level of the $1\\ell $ calculations.", "Not surprisingly, the suppression due to the channel interference becomes stronger in the PA results, reflecting the inclusion of the corresponding fluctuation effects at all loop orders.", "Due to the fully two-particle irreducible diagrams which are not included in the PA, it appears to systematically underestimate $\\chi _m$ as compared to numerically exact determinant quantum Monte Carlo data [44], [70] for larger values of $U$ .", "At the given temperature $T=0.2t$ , suppression effects are observed also in the DMFT data.", "The reduction of the magnetic fluctuations with respect to the $1\\ell $ fRG is ascribable to self-energy effects, which reduce more quickly the electronic coherence in DMFT than in fRG.", "This is consistent with the observation that DMF2RG results closely resemble the ones of DMFT in this parameter regime.", "Indeed, the nonlocal correlations included at the $1\\ell $ level on top of the DMFT results, induce, as expected, a further reduction of the magnetic correlations, but their quantitative impact remains marginal for $U<3t$ .", "We note that nonlocal multiloop corrections, which recover the PA result, have a stronger impact on the suppression than the local DMFT ones at $1\\ell $ level.", "We turn now to discuss the other sectors, which are secondary in the physics of the half-filled model.", "The corresponding results are reported in the central and right panels of Fig.", "REF , showing that the correlation effects beyond RPA are overall weaker and significantly less dependent by the particular approach chosen than in the magnetic sector.", "In particular, the slight suppression of the uniform charge and $s$ -wave static susceptibilities induced by the interplay with their complementary channels get reflected in a slight increase of corresponding screened interaction with respect to the RPA values.", "We note that including the flow of the fermionic rest functions $\\mathcal {R}^X$ leads only to negligible corrections of the results shown in Fig.", "REF , see Appendix .", "In the weak-coupling regime, the screened interactions and susceptibilities obtained without $\\mathcal {R}^X$ correctly describe the physical behavior, justifying the application of the SBE approximation." ], [ "Intermediate- to strong-coupling DMF", "After this preliminary comparison, we move towards the more challenging parameter regime of intermediate to strong couplings.", "Hence, from now on, we will focus on DMF2RG results for the physical susceptibilities of the different sectors, whereas, for simplicity, we turn off the flow of the nonlocal self-energy corrections.", "Especially in the strong-coupling regime, this simplification is not expected [71], [72], [11], [3] to affect the final results for the response functions in a significant way.", "We start our analysis of the strong-coupling regime, by presenting in Fig.", "REF our DMF2RG results for the whole momentum dependence of the static susceptibilities (at zero bosonic frequency) in the magnetic and in the charge sectors for the highest coupling considered $U=16t$ , which is significantly beyond the critical interaction value of the Mott-Hubbard metal to insulator transition of DMFT ($U_{\\rm MIT}(T\\!=\\!0) \\sim 12 t$ ), and for a temperature slightly above the DMFT Néel temperature (see leftmost panel of Fig.", "REF for the precise location in the DMFT phase-diagram).", "Note that at half filling particle-hole symmetry implies $\\chi ^s(\\bf {q},\\omega )=\\chi ^c(\\bf {q}+\\bf {Q})$ for the pairing susceptibility, with $\\bf {Q}=(\\pi ,\\pi )$ .", "The magnetic susceptibility exhibits a very pronounced peak around momentum $\\bf {q}=(\\pi ,\\pi )$ , a hint of strong AF fluctuations.", "This indicates that, similarly as for weak-coupling data, the nonlocal correlations captured by the $1\\ell $ DMF2RG do not suppress the ordering temperature of DMFT in a significant way.", "At the same time, the static response in the charge (and pairing) sector bears very clear hallmarks of the strong-coupling physics.", "Except for a residual momentum modulation, these response functions appear massively suppressed (note the different order of magnitude of the scales in the two panels), reflecting the almost insulating nature of the Mott-Hubbard physics at finite $T$ .", "Figure: Magnetic and charge susceptibilities in the BZ for transfer frequency, at n=1n=1 and for U=16tU=16t (t ' =0t^{\\prime }=0), T=0.286tT=0.286t.Following the analysis of Ref.", "[12], it is instructive to relate the results presented so far to the behavior of the corresponding generalized susceptibilities, which describe the underlying fermionic scattering processes.", "We recall that, in general This rigorously holds for the exact solutions of the problem and/or for approximations based on a definite subset of diagrams, such as RPA, PA, DMFT.", "At the level of a truncated (1$\\ell $ ) fRG/DMF2RG, deviations between the susceptibilities computed via the flow and those computed by summing the internal frequencies (post-processing) may occur [68], [42]., the susceptibilities $\\chi ^X(q)$ describing the physical response of the system can be obtained by the generalized ones $\\chi ^X_{\\nu \\nu ^{\\prime }}(q)$ by summing over all the fermionic Matsubara frequencies $\\nu , \\nu ^{\\prime }$ .", "In our notation, the explicit definition reads, also referred to as \"post-processing\": m'(q) = m(q)' + m(q) Lm'(q) m'(q), c'(q) = -c(q)' - c(q) Lc'(q) c'(q), where $L^X_{\\nu \\nu ^{\\prime }}(q) = h^X_\\nu (q) D^X(q) {h}^X_{\\nu ^{\\prime }}(q) + \\mathcal {L}_{\\nu \\nu ^{\\prime }}^X(q).$ From a numerical side, we note that a larg number of Matsubara frequencies are required for the internal summation in order to account correctly for the high-frequency asymptotics.", "A restricted frequency summation yields unphysical negative values in the charge response function due to the formation of the local moment [12].", "Figure: Generalized charge and magnetic susceptibilities as obtained from the self-consistent AIM (top) and the DMF2RG (bottom), at half filling and U=4tU=4t (t ' =0t^{\\prime }=0), T=0.25tT=0.25t.Figure: Same as Fig.", "for U=16tU=16t and T=0.286tT=0.286t.For illustrative reasons, we here explicitly discuss the $\\Omega =0$ case of the generalized charge and magnetic susceptibilities for two rather different interaction values, namely $U=4t$ and $U=16t$ , which correspond, in DMFT, to a Fermi-liquid paramagnetic metallic (PM) and Mott-Hubbard paramagnetic insulating (PI) regime, respectively.", "The corresponding data are shown in Figs.", "REF and REF , where we report the on-site Precisely, these are generalized susceptibilities of the auxiliary AIM associated to the corresponding self-consistent DMFT solution.", "We recall that in the limit of $d\\rightarrow \\infty $ the physical susceptibilities computed via the double Matsubara summation exactly coincide with the local (momentum-summed) susceptibilities of the lattice.", "generalized magnetic (left panels) and charge (right panels) DMFT solutions used as input of our DMF2RG calculations in the upper panels and the corresponding DMF2RG results for the uniform ($\\bf q=0$ ) generalized susceptibility in the lower ones.", "As discussed in Ref.", "[12], the frequency dependence of generalized charge susceptibilities represents a sensitive compass for identifying the underlying physics in fundamental many electron models, since it directly describes the impact of correlations on the electronic mobility and unveils its link to the changes of magnetic response.", "A first glance to the data of Figs.", "REF -REF shows that the qualitative modifications in the frequency structures of the generalized susceptibilities occurs indeed in the charge sector.", "In particular, sign changes in relevant frequency structures of the generalized charge susceptibility take place from weak ($U=4t$ ) to strong coupling ($U=16t$ ), reflecting important differences of the dominating physical mechanisms at play in the two regimes.", "For $U=4t$ shown in Fig.", "REF , the leading frequency dependence in the generalized charge susceptibilities of both the AIM (DMFT) and DMF2RG results appears in the main diagonal structure.", "This assumes positive values, larger at low frequencies $\\nu =\\nu ^{\\prime }$ and slowly decaying for larger $\\nu =\\nu ^{\\prime }$ values.", "This feature is a typical hallmark of the metallic physics in the perturbative regime, as it arises from the bubble term contribution $\\chi ^0_{\\nu \\nu ^{\\prime }}=-\\beta \\Pi ^c_{\\nu \\nu ^{\\prime }}(\\Omega =0)= -\\beta G(\\nu ) G(\\nu ^{\\prime }) \\delta _{\\nu \\nu ^{\\prime }} >0 $ , built upon a metallic $G(\\nu )$ .", "The role of vertex corrections, for $U=4t$ , appears merely quantitative, yielding an overall slight suppression (enhancement) of all entries of $\\chi ^{c}_{\\nu \\nu ^{\\prime }}$ ($\\chi ^{m}_{\\nu \\nu ^{\\prime }}$ ), responsible for the emergence of small negative (positive) off-diagonal contributions (faint bluish/reddish color for small $\\nu \\ne \\nu ^{\\prime }$ in the right/left upper panels of Fig.", "REF ).", "The moderate size of the vertex corrections is highlighted by the comparison of the two channels, whose results are both dominated by a positive diagonal frequency structure.", "Due to the proximity to an AF instability, the inclusion of nonlocal correlations in DMF2RG strongly affects the generalized susceptibility of the predominant magnetic channel, which gets significantly enhanced with respect to the corresponding AIM data.", "At the same time, the impact of nonlocal correlations appears marginal in the charge sector.", "The data computed at $U=16t$ (Fig.", "REF ) display relevant differences, which are induced by significant vertex correction effects.", "In the magnetic channels these drive an overall enhancement of the generalized susceptibility (diffuse reddish zone clearly visible in the left panels), which gradually overcomes the residual diagonal contribution of the bubble term.", "Such generalized enhancement of $\\chi ^{m}_{\\nu \\nu ^{\\prime }}$ encodes the typical Curie behavior of the local magnetic response in this regime.", "At the same time, the corresponding vertex corrections in the charge sector strongly suppress the physical response, by flipping the sign of the diagonal entries of $\\chi ^c_{\\nu \\nu ^{\\prime }}$ up to an energy scale of order $U$ .", "Specifically, the largely negative low-frequency diagonal structure visible in the right panels of Fig.", "REF is directly responsible [28], [12] for freezing the local density fluctuations in the Mott PI regime For $U=16t$ , the negative values in the main diagonal are counterbalanced by the positive contributions at high frequencies, giving rise to an overall positive charge response.. Physically, it can be interpreted [12] as the charge counterpart of the local moment formation in the magnetic sector.", "We note that the strong differentiation of the charge and magnetic response, which is a typical hallmark the Mott PM phase, has a clear nonperturbative origin: The negative diagonal structure emerging at low-frequency is associated to negative eigenvalues of $\\chi ^c_{\\nu \\nu ^{\\prime }}$ .", "This implies that sign flips with respect to the (positive) eigenvalues of the weak-coupling regime must occur by increasing $U$ .", "By any vanishing eigenvalue, however, the matrix $\\chi ^c_{\\nu \\nu ^{\\prime }}$ becomes singular leading to multiple divergences [27], [76], [28], [30], [33], [32], [34] of the irreducible vertex function and to the corresponding [31] problem of the multivaluedness [77], [78], [79], [80], [81] of the Luttinger-Ward functionals, which have been extensively discussed in the recent literature.", "As these singularities cannot be captured by any perturbative approach (including truncated fRG and PA) [12], for a proper description of the intermediate to strong-coupling regime it is pivotal to include the associated nonperturbative physics (such as local moment formation and its Kondo screening [12]) through a DMFT starting point, emphasizing the necessity of a DMF2RG treatment.", "In this specific case, the DMF2RG data (lower panels of Fig.", "REF ) display a further increase of the generalized magnetic susceptibility at ${\\bf q=} (\\pi , \\pi )$ , due to the proximity to the AF instability in the phase diagram, as well as a further suppression of the generalized charge susceptibility at ${\\bf q}=0$ with respect to the AIM results, consistent with the incompressible nature of the Mott insulating ground state.", "All calculations presented above have been performed neglecting the flow of the fermionic rest functions $\\mathcal {R}^X$ , consistently within the SBE framework.", "While this approximation allows for a significant reduction of the numerical effort, it is important to verify to what extent its application is justified in the different coupling regimes.", "We briefly recall here that, when fully considering the flow of the rest functions $\\mathcal {R}^X$ , the SBE-based implementation becomes formally equivalent to the one used in the conventional fRG based on the 1PI vertex function, see Ref.", "[3] for the specific case of DMF2RG [1].", "We first analyze the frequency dependence of the (previously neglected) $\\mathcal {R}^X$ functions, evaluated for the same bosonic variables ($\\mathbf {q},\\Omega $ ) of the susceptibilities shown in the preceding subsection, at $U=4t$ and $U=16t$ .", "The corresponding results are reported in Fig.", "REF .", "This highlights a key feature of the rest function, namely its characteristic decay to zero at large frequencies.", "Qualitatively, this reflects a general feature of the SBE diagrammatics [15]: the high-frequency asymptotics of the two-particle vertex functions is fully captured by the effective interactions, $D^X$ , and by the Yukawa couplings, $h^X$ .", "From a more quantitative perspective, the high-frequency decay appears particularly pronounced at large couplings: In contrast to $U=4t$ , for $U=16t$ the rest function exhibits a significant frequency dependence only for the lowest Matsubara frequencies, which, at this temperature, represents the leading frequency dependence of the whole 1PI vertex function.", "Despite the large values of the rest functions at strong coupling, the single-boson contributions $h^X D^X h^X$ are of the same order or even larger over an infinitely broad frequency range.", "Furthermore, we checked that the $\\mathcal {R}^X$ have a marginal feedback effect on the flow of the Yukawa couplings due to cancellations with the fermionic bubbles in the insulating phase.", "Figure: Rest functions magnetic, charge and pairing channels (from top to bottom) at U=4tU=4t, T=0.25tT=0.25t (left), and U=16tU=16t, T=0.286tT=0.286t (right), for n=1n=1 (and t ' =0t^{\\prime }=0).After examining the high-frequency decay properties of $\\mathcal {R}^X$ , which highlight the overall convenience of an SBE-based formalism, we turn now to the analysis of the specific impact that neglecting them has on the final results of our DMF2RG calculations.", "In Fig.", "REF , we report the susceptibilities of the predominant magnetic channels, computed with and without the inclusion of $\\mathcal {R}^X$ , for three different values of $U$ depicted on the phase diagram in the inset.", "The parameter set chosen is particularly relevant due to its proximity to the AF instability of the DMFT calculations, and the expected relevant contribution of large magnetic fluctuations.", "We note immediately that at half filling the magnetic susceptibilities obtained without $\\mathcal {R}^X$ correctly describe the physical properties expected in the whole parameter region considered, including the strong-coupling Mott-insulating regime at $U=16 t$ .", "Furthermore, the corrections to the magnetic response induced by the inclusion of $\\mathcal {R}^X$ in our DMF2RG calculations appear to be overall rather marginal (see insets of Fig.", "REF ) and further decreasing at larger interaction values.", "Figure: Left panel: Néel temperature as obtained from DMFT (left panel).", "Other panels: Magnetic susceptibility at zero bosonic frequencyalong the selected BZ path (shown as dashed line in Fig.", "), with and without the flow of the rest function ℛ X \\mathcal {R}^X, for n=1n=1 (t ' =0t^{\\prime }=0) and different values of UU (second, third, and forth panel).", "The different panels correspond to the crosses in the first panel, showing their location with respect to the Néel temperature.As a consequence of this, the inclusion of the rest function has also a minor impact on the determination of the temperature of the AF instability (Néel temperature $T_{\\rm N}$ ), which can be finite in all approaches where the Mermin-Wagner theorem is violated.", "In Fig.", "REF the inverse magnetic susceptibility is shown as a function of the temperature, with and without the inclusion of $\\mathcal {R}^X$ .", "Both data-sets clearly display a linear mean-field like critical behavior, resulting in $T_{\\rm N}=0.4042t$ and $T_{\\rm N}=0.3986t$ respectively.", "These values, both slightly lower than the corresponding DMFT results, are quite close.", "Hence, the overall effect of neglecting $\\mathcal {R}^X$ appears to be marginal, even quantitatively, within the $1\\ell $ DMF2RG scheme, in particular for the dominant magnetic channel.", "Not surprisingly, the corrections induced by $\\mathcal {R}^X$ are even lower in the other channels: charge and pairing susceptibilities determined with and without the rest function display almost no difference (not shown).", "Figure: Néel temperature T N T_{\\rm N} at U=8tU=8t." ], [ "Yukawa couplings", "In this subsection, we focus on the second main ingredient of the SBE formalism, that is the (fermion-boson) Yukawa couplings, s. also Refs.", "[82], [67], [47], [83].", "The corresponding results for the magnetic and the charge sector are shown in Fig.", "REF , while the $s$ -wave pairing one can be obtained from the relation $h^s_\\nu (\\mathbf {q},\\Omega )=h^c_\\nu (\\mathbf {q}+\\mathbf {Q},\\Omega )$ , with $\\mathbf {Q}=(\\pi ,\\pi )$ , valid when the particle-hole symmetry is realized.", "At weak coupling ($U=4t$ ) we observe for both channels a moderate dependence of the static ($\\Omega \\!=\\!0$ ) Yukawa couplings on the fermionic frequency $\\nu $ .", "This results in an overall slight suppression of the asymptotic value of the coupling ($h^X\\sim 1$ ) and encodes the metallic screening effects [84], [85] occurring in the proximity of the Fermi level.", "The inclusion of nonlocal effects in DMF2RG induces mild corrections with respect to the DMFT (AIM) results (black dashed line).", "In particular, it can be noted that, in both sectors, the fluctuations associated to a transfer momentum of $\\bf {q}=(\\pi ,\\pi )$ tend to magnify the frequency dependence displayed by the local DMFT (AIM) calculations, while the uniform ones ${\\bf q}=(0,0)$ have the opposite effect.", "At strong-coupling, the situation appears essentially reversed.", "The DMFT (AIM) calculations performed at $U=16t$ show a much stronger dependence of the couplings on the fermionic Matsubara frequency and a dycothomic behavior in the two channels – a typical hallmark [12], [86] of large vertex corrections.", "In particular, the magnetic Yukawa coupling gets significantly enhanced at low-frequencies with respect to the $h^m=1$ value, whereas the largest value is found in DMF2RG for the AF ordering wavevector.", "The charge channel, instead, exhibits a strong suppression, that even makes $h^c$ negative for the smallest frequencies.", "Here, the strongest suppression, according to our DMF2RG results occurs for $\\mathbf {q}=(0,0)$ , consistent to a vanishing isothermal compressibility in the Mott-Hubbard ground state [5], [86], [87].", "Physically, these opposite behaviors should be regarded as the nonperturbative fingerprints [12] of the local moment in the Yukawa couplings, since its formation enhances the system's response to a magnetic perturbation and, at the same time, freezes the fluctuations of the electronic density." ], [ "Results at finite doping", "To showcase the validity of our method in a physically more relevant parameter regime, we have also performed DMF2RG calculations for the non particle-hole symmetric case.", "In particular, we consider an intermediate-coupling of $U=8t$ and a fairly low temperature $T=0.044t$ , doping the system to the filling $n=0.82$ , and frustrating the magnetic correlations with a next to nearest neighbor hopping $t^\\prime =-0.2t$ .", "As the Mermin-Wagner theorem is not fulfilled at the level of the $1\\ell $ truncation of the DMF2RG [1], [68], [42], [3], we observe a magnetic instability in the considered parameter region.", "Hence, in order to highlight the relation between superconductivity and magnetism, we stop the flow before reaching the final scale, as customarily done in the fRG treatment of pseudocritical transitions.", "In particular, the divergence of the magnetic propagator has been prevented, by stopping the flow when the magnetic propagator $D^m(q)$ exceeds a value of $8\\times 10^3 t$ , which, for $U=8t$ , corresponds to a susceptibility of $\\sim 120/t$ , resulting in a stopping scale $\\Lambda _\\mathrm {cr} \\simeq 0.067 t$ .", "To account for $d$ -wave pairing correlations, we have included the flow of the $d$ -wave pairing channel (consisting exclusively of its rest function), as explained in Sec.", "REF .", "We neglect the rest functions of the magnetic, charge and $s$ -wave pairing channels since these appear to yield only minor corrections, together with nonlocal corrections to the self-energy.", "Figure: Magnetic, charge, ss- and dd-wave pairing susceptibilities (from left to right) at zero bosonic frequency, as functions of the momentum 𝐪\\mathbf {q}, forn=0.82n=0.82, U=8tU=8t, t ' =-0.2tt^\\prime =-0.2t, and T=0.044tT=0.044t, determined at the stopping scale Λ cr ≃0.067t\\Lambda _\\mathrm {cr} \\simeq 0.067t." ], [ "Susceptibilities", "Here, we will first discuss the results obtained for the magnetic $\\chi ^m$ , charge $\\chi ^c$ , and $d$ -wave pairing $\\chi ^d$ susceptibilities computed at the stopping scale $\\Lambda _\\mathrm {cr}$ .", "While the first two can be directly extracted from the respective propagators, see Eqs.", "(REF ) and (REF ), the $d$ -wave pairing one is computed through $\\chi ^d(q) = T \\sum _\\nu \\Pi ^d_\\nu (q) + T^2\\sum _{\\nu \\nu ^{\\prime }} \\Pi ^d_\\nu (q) L^d_{\\nu \\nu ^{\\prime }}(q) \\Pi ^d_{\\nu ^{\\prime }}(q),$ where we have neglected terms of the type $T^2\\sum _{\\nu \\nu ^{\\prime }} \\Pi ^{sd}_\\nu (q) L^s_{\\nu \\nu ^{\\prime }}(q) \\Pi ^{sd}_{\\nu ^{\\prime }}(q)$ , with $\\Pi ^{sd}_\\nu (q)$ the mixed $s$ -$d$ -wave pairing bubble, as it is nonzero only for $q\\ne 0$ and generally rather small S. Heinzelmann, private communication..", "In Fig.", "REF , we show the static susceptibilities in the whole Brillouin zone.", "Specifically, the magnetic susceptibility is close to a divergence at momenta $(\\pi , \\pi \\pm 2\\pi \\eta )$ (as well as, by symmetry, at $(\\pi \\pm 2\\pi \\eta ,\\pi )$ ) with $\\eta \\simeq 0.08$ , which indicates the tendency towards an incommensurate magnetic instability [9], [89], [90], [91].", "We recall that magnetic long-range orders with incommensurate wave vectors have been found in several mean-field studies [92], [93], [94], [95] of the Hubbard model at finite doping.", "Similar conclusions have been also drawn when including fluctuations beyond the static mean-field, for example, by means of expansions in the hole-density [96], [97], [98], [99], or exploiting extensions of DMFT [100], [101].", "DMFT calculations suggest that the ordering wave vector is related to the Fermi surface geometry not only at weak, but also at strong coupling [102].", "Differently, the charge susceptibility exhibits a rather weak dependence on $\\mathbf {q}$ , that is, only moderate deviations are observed from the local description of DMFT in this sector.", "We expect this feature not to depend on the fact that we have a small finite $\\Lambda _{\\textrm {cr}}$ .", "At a closer inspection, anyway, one notes two peaks located at $(\\pi ,0)$ and $(0,\\pi )$ .", "These signal the presence of mild charge-stripes correlations.", "Eventually, we focus on the $d$ -wave pairing susceptibility.", "Although its absolute values are not excessively large, it presents a pronounced $\\mathbf {q}$ -dependence with a well-defined peak at $\\mathbf {q}=\\mathbf {0}$ .", "Hence, one can reasonably expect that, if we were able to continue the flow below $\\Lambda _\\mathrm {cr}$ , $\\chi ^d$ would further increase, possibly even diverging at some finite scale.", "This heuristic expectation is supported by the analysis presented in the following sections." ], [ "Yukawa couplings", "In this section, we analyze the real part of the magnetic, charge, and $s$ -wave pairing Yukawa couplings at the stopping scale $\\Lambda _\\mathrm {cr}$ shown in Fig.", "REF .", "The magnetic Yukawa coupling exhibits a behavior that qualitatively resembles the one in the weak-coupling regime of the half-filled case (compare with upper left panel of Fig.", "REF ) despite the relatively large value of $U=8t$ .", "This happens because the local moment and, thus, its fingerprints in the Yukawa coupling, gets weakened by hole-doping: its low-frequency part is suppressed with respect to the high-frequency ($h^m=1$ ) value, due to the electronic screening.", "Finally, the $s$ -wave pairing Yukawa coupling appears overall suppressed, in spite a relatively small upturn at the lowest frequencies.", "Not surprisingly, it displays a rather marginal momentum dependence and, thus, small deviations from the AIM result.", "Figure: Yukawa couplings in the magnetic, charge, and ss-wave pairing channel (from left to right) at zero bosonic frequency, as a function of the fermionic Matsubara frequency ν\\nu ,for the same parameters as in Fig.", "and for various choices of the spatial momentum 𝐪\\mathbf {q}." ], [ "Diagnostics of the correlations", "The aim of this section is to thoroughly inspect all terms contributing to the sizable enhancement of the susceptibility $\\chi ^d$ , which we briefly discussed above.", "In the spirit of the post-processing procedures [28], [63], [64], [65] recently applied[15], [103], [64], [65], [66] to several quantum many-body approaches to the Hubbard model, we decompose Eqs.", "(REF ) and (REF ), in order to distinguish the different contributions to the $d$ -wave susceptibility.", "In particular, by combining Eq.", "(REF ) with (REF ), we get $\\begin{split}L^d_{\\nu \\nu ^{\\prime }}(\\mathbf {q},\\Omega ) &= \\frac{1}{2}\\mathcal {M}^d_{\\nu ,\\Omega -\\nu }(\\nu ^{\\prime }-\\nu ) + \\mathcal {M}^d_{\\nu ,\\Omega -\\nu }(\\Omega -\\nu -\\nu ^{\\prime })\\\\&+\\frac{1}{2}\\mathcal {C}^d_{\\nu ,\\Omega -\\nu }(\\nu ^{\\prime }-\\nu )+\\mathcal {D}_{\\nu \\nu ^{\\prime }}(\\mathbf {q},\\Omega ),\\end{split}$ where we have introduced the functions $X^d_{\\nu \\nu ^{\\prime }}(\\Omega ) = -\\int _\\mathbf {q}\\frac{\\cos q_x + \\cos q_y}{2} X_{\\nu \\nu ^{\\prime }}(\\mathbf {q},\\Omega ),$ with $X=\\mathcal {M}$ or $\\mathcal {C}$ , depending only on frequencies.", "We can therefore split the function $L^d$ in three distinct contributions: Ld(m)'() = 12Md,-('-) + Md,-(--'), Ld(c)'() = 12Cd,-('-) , and $\\mathcal {D}_{\\nu \\nu ^{\\prime }}(q)$ .", "Figure: Different diagrammatic contributions to the dd-wave pairing susceptibility.", "The solid lines represent fermionic propagators, the wavy ones the magnetic channel ℳ\\mathcal {M}, the dashed ones the charge channel 𝒞\\mathcal {C}, and the gray circles the dd-wave form factors f 𝐤 d f_\\mathbf {k}^d.", "(a) Bare bubble term.", "(b-c) Maki-Thompson contributions with one exchanged magnetic and charge boson respectively.", "(d) N≥2N\\ge 2 boson contributions: due to the inaccuracy of the 1ℓ1\\ell truncation, the equivalence of χ d(N b ) \\chi ^{d (N_\\mathrm {b})} with these diagrams is not exact.", "Notice that we have not drawn the Yukawa couplings, even though they display a nontrivial structure.By inserting this decomposition into the expression for the susceptibility (REF ), we obtain $\\chi ^d(q) \\!", "= \\!", "\\chi ^{d(0)}(q) \\!", "+\\!", "\\chi ^{d(m)}(q) \\!", "+ \\!", "\\chi ^{d(c)}(q) \\!+ \\!", "\\chi ^{d(N_\\mathrm {b})}(q)$ with d(0)(q) = Td(q), d(m)(q) = T2'd(q) Ld(m)'() d'(q), d(c)(q) = T2'd(q) Ld(c)'() d'(q), d(Nb)(q) = T2'd(q) D'(q) d'(q), Here, $\\chi ^{d(0)}$ is the bare bubble term, while $\\chi ^{d(m)}$ and $\\chi ^{d(c)}$ represent the Maki-Thompson contributions to the susceptibility, as diagrammatically shown in Fig.", "REF .", "Finally, since the $d$ -wave pairing channel entails all diagrams which are two-particle ($pp$ ) reducible but $U$ -irreducible, the remaining $\\chi ^{d(N_\\mathrm {b})}$ term can be interpreted, as the sum of $N$ -boson processes.", "The physics encoded in the latter processes will be analyzed separately in the final part of this section.", "Here, we only remark that this representation becomes exact only in the multiloop extension.", "Before commenting on the results, we note that, from an fRG perspective, the AF susceptibility gradually evolves from the beginning of the flow [89], [39], [85], [3], while the superconducting $d$ -wave susceptibility emerges only in proximity of the critical scale [90], [104].", "Figure: Contributions to the dd-wave static susceptibility for n=0.82n=0.82, U=8tU=8t, t ' =-0.2tt^\\prime =-0.2t, and T=0.044tT=0.044t along a path in the BZ: χ d(m) \\chi ^{d(m)} and χ d(c) \\chi ^{d(c)} indicate the Maki-Thompson term with the insertion of a magnetic and a charge bosonic line respectively, and χ d(N b ) \\chi ^{d(N_\\mathrm {b})} the NN-boson processes, with N≥2N\\ge 2.", "In addition, the bare dd-wave pairing bubble χ d(0) \\chi ^{d(0)} as well as the total susceptibility χ d \\chi ^d are shown.In Fig.", "REF , we plot the different contributions to the $d$ -wave static susceptibility defined above, as obtained at the stopping scale $\\Lambda _\\mathrm {cr}$ .", "We observe that in the considered parameter region the magnetic Maki-Thompson processes yield the most relevant contribution.", "Moreover, we find a fairly large contribution of the multiboson terms.", "Indeed, we expect that, by approaching a $d$ -wave pairing instability, the multiboson term would dominate over the Maki-Thompson ones, eventually driving the divergence of the associated susceptibility at the thermodynamic instability.", "More in general, if one were able to separate the $N<\\bar{N}$ ($\\chi ^{d(N_\\mathrm {b}<\\bar{N})}$ ) from the $N\\ge \\bar{N}$ ($\\chi ^{d(N_\\mathrm {b}\\ge \\bar{N})}$ ) boson processes in the susceptibility, arbitrarily close to the critical point one would detect $\\chi ^{d(N_\\mathrm {b}\\ge \\bar{N})}(\\mathbf {q}\\simeq \\mathbf {0},0) > \\chi ^{d(N_\\mathrm {b}<\\bar{N})}(\\mathbf {q}\\simeq \\mathbf {0},0),$ for every finite $\\bar{N}$ .", "This happens because the divergence of the susceptibility is due to a term $(1-\\lambda _d)^{-1}$ , with $\\lambda _d$ , the maximum eigenvalue of the matrix product between the $d$ -wave bubble and the two-particle irreducible vertex in the $d$ -wave pairing channel, approaching 1.", "Since the term $(1-\\lambda _d)^{-1}$ results from a resummation of infinite order diagrams, it can only be encoded in the $N\\ge \\bar{N}$ term, with the $N<\\bar{N}$ one scaling as $\\bar{N}\\lambda _d^{\\bar{N}}\\sim \\bar{N}$ close to the instability.", "In general, a measure of the maximum $\\bar{N}$ for which Eq.", "(REF ) is fulfilled, might be exploited to quantify the actual proximity to a thermodynamic ($d$ -wave pairing) instability.", "As anticipated above, we eventually draw our attention on the last term in Eq.", "(REF ) identified by the presence of $\\mathcal {D}_{\\nu \\nu ^{\\prime }}(q)$ , which entails all the reducible scattering processes in the $d$ -wave pairing channel.", "Due to the $d$ -wave symmetry in momentum space, $\\mathcal {D}_{\\nu \\nu ^{\\prime }}(q)$ arises exclusively from scattering events involving the exchange of two or more bosons.", "A refined inspection of these scattering processes is then possible via a corresponding decomposition of $\\mathcal {D}_{\\nu \\nu ^{\\prime }}(q)$ .", "According to Eqs.", "(REF ) and (REF ), we can classify the different contributions to $\\mathcal {D}_{\\nu \\nu ^{\\prime }}(q)$ directly from the corresponding flow equation: Dmm'(q) = T Ld(m)(q)[d(q)] Ld(m)'(q), Dcc'(q) = T Ld(c)(q)[d(q)] Ld(c)'(q), Dmc'(q) = T Ld(m)(q)[d(q)] Ld(c)'(q) +T Ld(c)(q)[d(q)] Ld(m)'(q), DNb3'(q) = D'(q) - Dmm'(q) - Dcc'(q) - Dmc'(q).", "Figure: Diagrammatic representation of the two-boson contributions to the flow equation for 𝒟 νν ' (q)\\mathcal {D}_{\\nu \\nu ^{\\prime }}(q), defined in Eqs. ()-().", "The symbols are the same as in Fig.", ", where the \"ticks\" on the fermionic lines represent the single scale derivatives ∂ ˜ Λ \\widetilde{\\partial }_\\Lambda .", "Note that for each of the diagrams there is another contribution with the tick on the lower fermionic line.The diagrammatic representation of the terms $\\mathcal {D}^{mm}$ , $\\mathcal {D}^{cc}$ , and $\\mathcal {D}^{mc}$ is shown in Fig.", "REF .", "These two-boson processes can be associated to the so-called Aslamazov-Larkin diagrams.", "As one might expect, a closer inspection of Eqs.", "(REF )-(REF ) shows that they are not fully reconstructed during the flow because the functions $L^{d(m)}$ , and $L^{d(c)}$ (as well as the nonlocal self-energy if included) also depend on the fRG scale.", "This feature is a typical artifact of the $1\\ell $ truncation, which can be fully resolved in the framework of future multiloop extensions of the approach.", "With this caveat, it is nonetheless possible to interpret $\\mathcal {D}^{mm}$ , $\\mathcal {D}^{cc}$ , and $\\mathcal {D}^{mc}$ as the two-boson contributions to the $d$ -wave pairing channel, and the remainder $\\mathcal {D}^{N_\\mathrm {b}\\ge 3}$ in terms of higher order processes in the number of exchanged bosons.", "Figure: Different contributions to the dd-wave pairing channel for the same parameters as in Fig.", ", as obtained from Eq.", "() at zero bosonic frequency and spatial momentum and evaluated at ν=ν ' =ν 0 ≡πT\\nu =\\nu ^\\prime =\\nu _0\\equiv \\pi T (s. also Fig.", "), as a function of the fRG scale Λ\\Lambda .In Fig.", "REF we plot the various terms contributing to $\\mathcal {D}_{\\nu \\nu ^{\\prime }}(q)$ as a function of the fRG scale $\\Lambda $ .", "We observe that the multiboson ($N_b \\ge 3$ ) term develops at significantly lower scales as compared to the two-boson ones.", "At the same time, it increases considerably by approaching the stopping scale.", "This behavior is consistent to the following general consideration: If the system is close enough (in temperature, doping, or other parameters) to a thermodynamic instability, at some point $\\mathcal {D}^{N_\\mathrm {b}\\ge 3}$ will overtake the other terms in Eq.", "(REF ), and eventually diverge at the transition itself.", "In fact, similar as discussed for the susceptibility, sufficiently close to the transition the $N\\ge {N}$ term is going to exceed the sum of the $N<\\bar{N}$ terms for every finite $\\bar{N}$ .", "Consistent with these general consideration, in the framework of our $1\\ell $ DMF2RG, we indeed observe that the multiboson term $\\mathcal {D}^{N_\\mathrm {b}\\ge 3}$ becomes dominating at a finite scale just before the end of the flow.", "Hence, for the selected parameter choice, an important precondition for the onset of a $d$ -wave pairing instability has been already realized.", "This represents a promising hint that a true superconducting transition may be unveiled at lower temperatures by means of higher loop-order calculations.", "Figure: Different contributions to the dd-wave pairing channel at the stopping scale Λ cr \\Lambda _\\mathrm {cr}, for zero bosonic frequency and spatial momentum and the same parameters as in Fig. .", "Aslamazov-Larkin diagram (see text) with two exchanged magnetic bosons (top left), two exchanged charge bosons (top right), one charge and one magnetic boson (bottom right), and total dd-wave pairing channel (bottom left) total dd-wave pairing channel.Note that the scale in ν\\nu ,ν ' \\nu ^\\prime is differentin order to better resolve the frequency structures.In Fig.", "REF we show the frequency structure of the two-boson contributions to the $d$ -wave pairing channel as well as $\\mathcal {D}$ itself, at $q=(\\mathbf {0},0)$ as functions of the fermionic Matsubara frequencies $\\nu $ , $\\nu ^{\\prime }$ .", "Being associated only to $U$ -irreducible diagrams, the $d$ -wave pairing channel is a rapidly decaying function of the Matsubara frequencies.", "It exhibits a structure centered around the frequencies $\\nu =\\pm \\nu _0$ , $\\nu ^{\\prime }=\\pm \\nu _0$ ($\\nu _0=\\pi T$ ), where it assumes fairly large values.", "About 50% of this structure is generated by two magnetic boson processes and (most of) the rest by multiboson ones, as the $\\mathcal {D}^{cc}$ and $\\mathcal {D}^{mc}$ terms play a very marginal role in the formation of $d$ -wave pairing correlations.", "It appears hence natural to conclude that among all multiboson terms, the one consisting only of multiple magnetic boson processes will have the largest weight.", "This observation suggests that $d$ -wave pairing correlations in the normal phase of the Hubbard model are mostly generated by (incommensurate or not) AF fluctuations, consistent with previous findings in fRG [90], [39], [85], [3] and DMF2RG [1], [3], as well as in recent numerical analyses [63], [105], [65]." ], [ "Remarks on the bosonization of the d-wave pairing channel", "In view of further reductions of the numerical complexity in future applications, it might be helpful to describe also the $d$ -wave pairing channel $\\mathcal {D}_{\\nu \\nu ^{\\prime }}(q)$ in terms of single boson processes.", "Since the bare interaction $U$ has $s$ -wave symmetry, the diagrammatic argument of the SBE decomposition does not hold in this case.", "Hence, a proper decomposition of the $d$ -wave pairing channel that factorizes the dependence on the fermionic frequencies is needed.", "Inspired by earlier fRG works where the Yukawa coupling has been calculated from the channel functions at given values of the fermionic frequencies [51], we can define the $d$ -wave screened interaction and boson-fermion coupling as Dd(q) = D0, 0(q)+D0, -0(q)2, hd(q) = D, 0(q)+D, -0(q)2Dd(q), where $\\bar{\\nu }_0$ is a fixed fermionic frequency (eventually $q$ -dependent).", "We notice that in Eq.", "(REF ) the symmetrization over $\\pm \\bar{\\nu }_0$ is necessary to guarantee the correct symmetries for $D^d$ and $h^d$ .", "In this way the $d$ -wave pairing channel can be expressed in a similar form as the other channels: $\\mathcal {D}_{\\nu \\nu ^{\\prime }}(q) = h^d_\\nu (q)\\,D^d(q)\\,h^d_{\\nu ^{\\prime }}(q) + \\mathcal {R}^d_{\\nu \\nu ^{\\prime }}(q).$ Setting $\\bar{\\nu }_0=\\infty $ in Eq.", "(REF ), as for the $s$ -wave channels, would lead to the conditions $h^d=D^d=0$ and $\\mathcal {D}=\\mathcal {R}^d$ , for which an effective decomposition of $d$ -wave pairing channel would not be possible.", "Hence, another choice of $\\bar{\\nu }_0$ is necessary, e.g., $\\bar{\\nu }_0=\\pm \\pi T$ .", "The resulting flow equation for $D^d$ and $h^d$ can be then extracted by applying Eq.", "(REF ) to the flow of the interaction $\\mathcal {D}$ in Eq.", "(REF ).", "A key point to keep in mind when choosing $\\bar{\\nu }_0$ is that, once a $d$ -wave pairing instability occurs, the divergence of the effective interaction must be reabsorbed into the bosonic propagator, while the Yukawa coupling and the rest function should remain finite, similarly to what happens in the other competing channels." ], [ "Conclusions", "We have applied the recently introduced SBE representation to the fRG and DMF2RG, which relies on a diagrammatic decomposition in contributions mediated by the exchange of a single boson in the different channels.", "Specifically, the ($1\\ell $ ) flow equations for the two-particle vertex are recast into SBE contributions and a residual four-point fermion vertex.", "The SBE-based formulation leads to a substantial reduction of the numerical effort, since the corresponding rest function is significantly localized in frequency space, especially in the strong-coupling regime.", "This justifies the approximation to significantly restrict the total number of frequencies taken into account in the RG flow or even to fully neglect the rest function.", "The reduced numerical effort facilitates the applicability of the fRG and DMF2RG to the most interesting regime of intermediate to strong correlations and/or low $T$ .", "The advantage of this implementation is well illustrated by hands of our DMF2RG calculations of the 2D Hubbard model performed up to very strong interaction ($U = 16$ t) at and out of half-filling.", "Moreover, the SBE decomposition naturally allows for a clear physical identification of the relevant degrees of freedoms.", "As pertinent example, we exploited this specific feature of our approach to diagnose the tendency towards a $d$ -wave superconducting instability of the doped Hubbard model in terms of magnetic and charge driven processes.", "The derivation of the SBE-based RG flow within the $1\\ell $ truncation represents the natural starting point for future multiloop extensions [68], [69], [42], [44], as well as for the systematic inclusion of multiboson contributions.", "Within these methodological extensions, fRG- and DMF2RG-based computation schemes can be brought to a quantitative level for all coupling strengths.", "Physically, we expect that nonlocal correlation effects, associated to higher loop order processes, might become progressively more pronounced, especially in the most challenging low-temperature regime.", "Finally, the present formalism offers the possibility to explicitly introduce bosonic fields and study the flow of a mixed boson-fermion system.", "This would allow, for example, to study the effect of bosonic fluctuations on top of mean-field solutions [106], [91], [107], [108] below the (pseudo) critical scale, where symmetry breaking occurs." ], [ "Acknowledgments", "The authors thank P. Chalupa, J. Hauck, C. Honerkamp, F. Krien, F. Kugler, W. Metzner, and T. Schäfer for valuable discussions and W. Metzner for a careful reading of the manuscript.", "We acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) through Project No.", "AN 815/6-1 and from the Austrian Science Fund (FWF) through Project No.", "I 2794-N35." ], [ "Comparison with conventional fermionic formalism", "We here discuss the relation of the SBE formalism with the channel asymptotics introduced in Ref.", "[14] (see also Ref. [83]).", "Both approaches rely on a similar classification of the diagrams contributing to the two-particle reducible contribution to the vertex function.", "For this reason, it is possible to connect the screened interactions $D^X$ to the so-called $\\mathcal {K}^{(1)X}$ functions via $D^X(q)=U + (\\mathrm {sgn}X) \\mathcal {K}^{(1)X}(q),$ where $\\mathrm {sgn}\\,m=1$ , and $\\mathrm {sgn}\\,c=\\mathrm {sgn}\\,s=-1$ .", "$\\mathcal {K}^{(1)X}$ is defined as  [14]: $\\mathcal {K}^{(1)X}(q)=\\lim _{\\nu ,\\nu ^{\\prime }\\rightarrow \\infty } \\phi ^X_{(\\mathbf {k},\\nu ),(\\mathbf {k}^{\\prime },\\nu ^{\\prime })}(q),$ with $\\phi ^X$ the sum of all two-particle reducible diagrams in the $m$ , $c$ , or $s$ channel, as in Eq.", "(REF ).", "Similarly, the Yukawa coupling is related to the $\\mathcal {K}^{(2)X}$ asymptotic function by $h^X_k(q) = 1 + \\frac{\\mathcal {K}^{(2)X}_k(q)}{D^X(q)},$ with $\\mathcal {K}^{(2)X}_k(q)=\\lim _{\\nu ^{\\prime }\\rightarrow \\infty } \\phi ^X_{k,(\\mathbf {k}^{\\prime },\\nu ^{\\prime })}(q) - \\mathcal {K}^{(1)X}(q).$ At this stage the SBE decomposition seems to offer no substantial computational gain as compared to the channel asymptotics, the only exception being in the vicinity of a critical point, where in the asymptotic formalism both $\\mathcal {K}^{(1)X}$ and $\\mathcal {K}^{(2)X}$ acquire large values, while in the SBE one this occurs only for $D^X$ , the Yukawa coupling being always finite.", "However, the most important difference is visible in the rest functions.", "Indeed, the SBE and the asymptotic rest functions are related via (see also Ref.", "[108]) $\\mathcal {R}^{\\mathrm {SBE},X}_{kk^{\\prime }}(q)=\\mathcal {R}^{\\mathrm {asym},X}_{kk^{\\prime }}(q)-[h^X_k(q)-1]D^X(q)[h^X_{k^{\\prime }}(q)-1],$ with $\\mathcal {R}^{\\mathrm {asym},X}_{kk^{\\prime }}(q) = \\phi _{kk^{\\prime }}(q) - \\mathcal {K}^{(1)X}_{kk^{\\prime }}(q) - \\mathcal {K}^{(2)X}_{k}(q) -\\mathcal {K}^{(2)X}_{k^{\\prime }}(q).$ From Eq.", "(REF ) we notice two important facts: First, in the asymptotics formalism some rest functions can diverge at a given critical point as they contain $D^X$ .", "Therefore, in this regime it is not safe to neglect them.", "Second, $\\mathcal {R}^{\\mathrm {SBE},X}$ contains less diagrams than $\\mathcal {R}^{\\mathrm {asym}X}$ , as the latter still includes some $U$ reducible contribution.", "Therefore, an approximation which neglects $\\mathcal {R}^{\\mathrm {SBE},X}$ can be always justified as the selection of a well-defined class of diagrams.", "For completeness we report the fRG and DMF2RG results of Fig.", "REF obtained by including the flow of the rest functions $\\mathcal {R}^X$ , where the SBE-based implementation is equivalent to the fRG and DMF2RG respectively.", "In Fig.", "REF we show the impact of the inclusion/neglection of the rest function in the magnetic screened interaction, which displays the largest difference, for both conventional fRG and DMF2RG.", "We notice that, as discussed in the main text for stronger coupling, sizable differences between the two approaches (with and without $\\mathcal {R}^X$ ) arise only in the vicinity of the Néel transition.", "We then conclude that, away from the AF transition point, the tiny differences fully justify the approximation to neglect the rest function, while close to the critical transition the inclusion/neglection of the rest functions will result in a (generally small) change of the values for the critical temperature or coupling." ], [ "Alternative derivation of the flow equations", "In this Appendix we derive the flow equations presented in the main text as obtained by an alternative derivation based on an explicit introduction of the order parameter fluctuation field via the Hubbard-Stratonovich transformation (HST).", "In particular, after having split the bare Hubbard interaction into three equal terms ($Un_{\\uparrow }n_{\\downarrow }=3Un_{\\uparrow }n_{\\downarrow } - 2 Un_{\\uparrow }n_{\\downarrow }$ ), we apply three different HST on the first three terms, one on each physical channel.", "In formulas $\\begin{split}\\mathcal {Z}_\\text{Hubbard}&=\\int \\mathcal {D}\\left(\\psi ,{\\psi }\\right)e^{-\\mathcal {S}_\\text{Hubbard}\\left[\\psi ,{\\psi }\\right]}\\\\&=\\int \\mathcal {D}\\Phi \\mathcal {D}\\left(\\psi ,{\\psi }\\right)e^{-\\mathcal {S}_\\text{bos}\\left[\\psi ,{\\psi },\\Phi \\right]},\\end{split}$ where $\\Phi =(\\phi _c,{\\phi }_m,\\phi _p,\\phi _p^*)$ collects all bosonic fields, $\\begin{split}\\mathcal {S}_\\text{Hubbard}\\left[\\psi ,{\\psi }\\right] &= -\\int _{k,\\sigma } {\\psi }_{k,\\sigma } \\left(i\\nu +\\mu -\\epsilon _\\mathbf {k}\\right) \\psi _{k,\\sigma } \\\\&+ U \\int _0^\\beta d\\tau \\sum _i n_{\\uparrow ,i}(\\tau ) n_{\\downarrow ,i}(\\tau ),\\end{split}$ and $\\begin{split}\\mathcal {S}_\\text{bos}&\\left[\\psi ,{\\psi },\\Phi \\right] =-\\int _{k,\\sigma } {\\psi }_{k,\\sigma } \\left(i\\nu +\\mu -\\epsilon _k\\right) \\psi _{k,\\sigma }\\\\&+\\frac{1}{2}\\int _{q}\\phi _c(-q) \\frac{1}{U} \\phi _c(q)+\\frac{1}{2}\\int _{q}{\\phi }_m(-q) \\cdot \\frac{1}{U} {\\phi }_m(q)\\\\&+\\int _{q}\\phi ^*_p(q)\\frac{1}{U} \\phi _p(q)+\\int _{k,q,\\sigma }\\phi _c(q) \\,{\\psi }_{k+\\frac{q}{2},\\sigma }\\psi _{k-\\frac{q}{2},\\sigma }\\\\&+\\int _{k,q,\\sigma ,\\sigma ^{\\prime }}{\\phi }_m(q)\\cdot \\,{\\psi }_{k+\\frac{q}{2},\\sigma }{\\tau }_{\\sigma \\sigma ^{\\prime }}\\psi _{k-\\frac{q}{2},\\sigma ^{\\prime }}\\\\&+\\int _{k,q}\\left[\\phi _p(q) \\,{\\psi }_{\\frac{q}{2}+k,\\uparrow }{\\psi }_{\\frac{q}{2}-k,\\downarrow }+\\phi ^*_p(q) \\,\\psi _{\\frac{q}{2}-k,\\downarrow }\\psi _{\\frac{q}{2}+k,\\uparrow }\\right]\\\\ &-2U \\int _0^\\beta d\\tau \\sum _i n_{\\uparrow ,i}(\\tau ) n_{\\downarrow ,i}(\\tau ).\\end{split}$ The remaining (not bosonized) $-2U$ term in $\\mathcal {S}_\\text{bos}$ , avoids double counting of the bare interaction.", "To derive the functional flow equations, a regulator term is added to the definition of the generating functionals by replacing the bare action as $\\begin{split}Z^\\Lambda \\left[\\eta ,{\\eta },J\\right]&=\\int \\mathcal {D}\\Phi \\int \\mathcal {D}\\left(\\psi ,{\\psi }\\right)\\\\&\\times e^{-S^\\Lambda _\\text{bos}\\left[\\psi ,{\\psi },\\Phi \\right]+\\left({\\psi },\\eta \\right)+\\left({\\eta },\\psi \\right) + \\Phi J},\\end{split}$ where $S^\\Lambda _\\text{bos}\\left[\\psi ,{\\psi },\\Phi \\right]=S_\\text{bos}\\left[\\psi ,{\\psi },\\Phi \\right]+\\int _{k,\\sigma } {\\psi }_{k,\\sigma }\\,R^\\Lambda (k)\\,\\psi _{k,\\sigma }.$ Its value at some initial scale $\\Lambda _\\text{ini}$ depends on the formalism used.", "For instance, in the plain fRG we impose $R^{\\Lambda \\rightarrow \\Lambda _\\text{ini}}(k)\\rightarrow \\infty $ , such that, from the saddle point equation of the functional integral, the initial conditions are determines by the bare ones.", "Differently, in the DMF2RG, the cutoff must fulfill $R^{\\Lambda _{\\text{ini}}}(k) = \\epsilon _{{k}} - \\Delta _\\text{AIM}\\left(\\nu \\right),$ so that the rescaled action at $\\Lambda _\\text{ini}$ is $\\mathcal {S}^{\\Lambda _\\text{ini}}\\left[\\psi ,{\\psi },\\Phi \\right] =\\mathcal {S}_\\text{AIM}\\left[\\psi ,{\\psi },\\Phi \\right],$ where $\\mathcal {S}_\\text{AIM}\\left[\\psi ,{\\psi }\\right]$ is the action of the self-consistent AIM.", "Here the same procedure involving HST for the Hubbard interaction at the impurity has been performed.", "In this way, the initial condition for the resulting effective action reads $\\Gamma ^{\\Lambda _\\text{ini}}\\left[\\psi ,{\\psi },\\Phi \\right]=\\Gamma _\\text{AIM}\\left[\\psi ,{\\psi },\\Phi \\right],$ where $\\Gamma _\\text{AIM}\\left[\\psi ,{\\psi },\\Phi \\right]$ is the effective action of the self-consistent AIM.", "By expanding it in terms of 1PI functions, one recovers the initial conditions given in the main text, where the screened interactions $D^X$ and the Yukawa couplings $h^X$ play the role of bosonic propagators and fermion-boson interactions respectively.", "Note that we do not explicitly introduce a regulator for bosonic fluctuations.", "Indeed, the bosonic path integral at this stage has to be performed in some way to recover the fermionic formalism.", "For this scope, a second scale $\\Lambda _b$ can be introduced, responsible for the integration over $\\Phi $ and which is associated to a second cutoff function $R_b^{\\Lambda _b}$ regularizing the bare bosonic propagator.", "In this context, the bosonic flow is shown to be ineffective when integrated before the fermionic flow integration [54]." ], [ "Numerical implementation", "In this Appendix we discuss a few numerical details.", "Regarding the DMFT calculation, we solve the AIM with the exact diagonalization (ED) by discretizing the bath into 4 sites.", "The local magnetic, charge and pairing screened interactions $D^X$ and Yukawa couplings $h^X$ are computed by Lehmann representation as in Ref.", "[42] in a relatively large frequency range as well as the two-particle Green's function needed to extract the $U$ -irreducible vertex $\\Lambda _{U\\mathrm {irr}}^\\mathrm {loc}$ .", "Regarding the flow equations of $D^X$ and $h^X$ , we use a frequency domain ranging from 30 to 64 positive values for both the bosonic and the fermionic Matsubara frequencies, depending on the convergence of the calculation.", "The treatment of the frequency asymptotics here simplifies since the screened interactions $D^X$ and Yukawa couplings $h^X$ tend to 1 and $U$ , respectively, at large frequencies, see Eqs.", "(REF ) or (REF ).", "Regarding the Matsubara summation, in general we select a range from 100 to 256 positive frequencies, depending on the physical regime.", "Contrary to the fRG, in the DMF$^2$ RG the single-scale propagator decays faster at large frequency, simplifying the convergence in the Matsubara summation.", "Moreover, it is noteworthy to mention that, as stated in Sec.", "REF , the computation of the charge susceptibility in the Mott phase requires a larger number of frequencies in the Matsubara summation, which must be then extended, despite the relatively high temperature range, to 1000 to recover its physical value [12].", "While part of the momentum dependence is projected onto form-factors as explained in the main text, the transfer momentum dependence has been patched similarly to Ref.", "[107], retaining 38 patches in the reduced Brillouin zone $\\mathcal {B}_\\mathrm {red}=\\lbrace (k_x,k_y): 0\\le k_y\\le k_x \\le \\pi \\rbrace $ .", "Finally, the flow equations have been solved using the adaptive Runge-Kutta Cash-Karp 54 method and the momentum integration over the Brillouin zone is carried out via an adaptive cubature technique." ] ]
2105.11749
[ [ "Fredholm modules over categories, Connes periodicity and classes in\n cyclic cohomology" ], [ "Abstract We replace a ring with a small $\\mathbb C$-linear category $\\mathcal{C}$, seen as a ring with several objects in the sense of Mitchell.", "We introduce Fredholm modules over this category and construct a Chern character taking values in the cyclic cohomology of $\\mathcal C$.", "We show that this categorified Chern character is homotopy invariant and is well-behaved with respect to the periodicity operator in cyclic cohomology.", "For this, we also obtain a description of cocycles and coboundaries in the cyclic cohomology of $\\mathcal C$ (and more generally, in the Hopf-cyclic cohomology of a Hopf module category) by means of DG-semicategories equipped with a trace on endomorphism spaces." ], [ "Introduction", "In his celebrated work , Connes extended differential calculus beyond the framework of manifolds to include noncommutative spaces such as that of leaves of a foliation or the orbit space of the action of a group on a manifold.", "For this, he began by considering Fredholm modules over an algebra $A$ which could in general be noncommutative.", "When $A$ is commutative, such as the space of smooth functions on a manifold $M$ , examples of Fredholm modules over $A$ may be obtained by considering elliptic operators on $M$ .", "More generally, by considering Schatten classes inside the collection of bounded operators on a Hilbert space, Connes studied the notion of $p$ -summable Fredholm modules over $A$ in .", "The Fredholm modules over $A$ lead to Chern characters taking values in the cyclic cohomology of $A$ .", "Moreover, these cohomology classes are related by means of Connes' periodicity operator.", "In this paper, we study Fredholm modules over linear categories, along with their Chern characters taking values in cyclic cohomology.", "Our idea is to have a counterpart of the algebraic notion of modules over a category, a subject which has been highly developed in the literature (see, for instance, , , , , , ).", "A small preadditive category is treated as a ring with several objects, following an idea first advanced by Mitchell .", "We note that there is also a well-developed study of spaces in algebraic geometry over categories (see, for instance, , , ).", "It is also important to mention here the work of Baez with the category of Hilbert spaces as well as the recent work of Henriques , Henriques and Penneys with fusion categories with potential applications to physics.", "Let $\\mathcal {C}$ be a small linear category.", "We consider pairs $(H,\\mathcal {F})$ , where $H$ is a functor $H:\\mathcal {C}\\longrightarrow SHilb_{\\mathbb {Z}_2}$ taking values in $\\mathbb {Z}_2$ -graded separable Hilbert spaces and $\\mathcal {F}=\\lbrace \\mathcal {F}_X:H(X)\\longrightarrow H(X)\\rbrace _{X\\in Ob(\\mathcal {C})}$ is a family of bounded and involutive linear operators each of degree 1.", "When the elements of $\\mathcal {F}$ satisfy certain commutator conditions with respect to the operators $\\lbrace H(f)\\rbrace _{f\\in Mor(\\mathcal {C})}$ , we say that the pair $(H,\\mathcal {F})$ is a Fredholm module over the category $\\mathcal {C}$ .", "Following the methods of Connes , we construct Chern characters of these Fredholm modules taking values in the cyclic cohomology of $\\mathcal {C}$ and study how they are related by means of the periodicity operator.", "We hope this is the first step towards a larger program which mixes together the techniques in categorical algebra with those in differential geometry.", "The paper consists of two parts.", "In the first part, we study cyclic cohomology.", "We work more generally with a small linear category $\\mathcal {D}_H$ whose morphism spaces carry a well-behaved action of a Hopf algebra $H$ .", "In other words, $\\mathcal {D}_H$ is a small Hopf-module category (or $H$ -category) in the sense of Cibils and Solotar .", "We recall that in , , , Connes and Moscovici introduced Hopf-cyclic cohomology as a generalization of Lie algebra cohomology adapted to noncommutative geometry.", "For an $H$ -category $\\mathcal {D}_H$ , we describe the cocycles and coboundaries that determine its Hopf cyclic cohomology groups by extending Connes' original construction of cyclic cohomology from and in terms of cycles and closed graded traces on differential graded algebras.", "An important role in our paper is played by “semicategories,” which are categories that may not contain identity maps.", "This notion, introduced by Mitchell , is precisely what we need in order to categorify non-unital algebras.", "We work with the Hopf cyclic cohomology groups $HC^\\bullet _H(\\mathcal {D}_H,M)$ having coefficients in $M$ , where $M$ is a stable anti-Yetter Drinfeld module in the sense of .", "Let $k$ be a field.", "After collecting some preliminaries in Section 2, we begin in Section 3 by considering the universal differential graded Hopf module semicategory (or DGH-semicategory) associated to the $H$ -category $\\mathcal {D}_H$ .", "For a DGH-semicategory $(\\mathcal {S}_H,\\hat{\\partial }_H)$ and $n\\ge 0$ , we let an $n$ -dimensional closed graded $(H,M)$ -trace on $\\mathcal {S}_H$ be a collection of maps $ \\hat{{T}}^H:=\\lbrace \\hat{{T}}_X^H:M \\otimes Hom^n_{\\mathcal {S}_H}(X,X) \\longrightarrow k\\rbrace _{X \\in Ob(\\mathcal {S}_H)}$ satisfying certain conditions (see Definition REF ).", "A cycle over $\\mathcal {D}_H$ consists of a triple $(\\mathcal {S}_H,\\hat{\\partial }_H,\\hat{T}^H)$ along with an $H$ -linear semifunctor $\\rho :\\mathcal {D}_H\\longrightarrow \\mathcal {S}_H^0$ .", "In Theorem REF , we provide a description of the cocycles $Z^\\bullet _H(\\mathcal {D}_H,M)$ in Hopf cyclic cohomology in terms of characters of cycles over $\\mathcal {D}_H$ .", "This result is an $H$ -linear categorical version of Connes' .", "It also follows from Theorem REF that there is a one-one correspondence between $Z^\\bullet _H(\\mathcal {D}_H,M)$ and the collection of $n$ -dimensional closed graded $(H,M)$ -traces on the universal DGH-semicategory $\\Omega (\\mathcal {D}_H)$ associated to $\\mathcal {D}_H$ .", "In Sections 4 and 5, we provide a description of the space $B^\\bullet _H(\\mathcal {D}_H,M)$ of coboundaries.", "Throughout, we take $k=\\mathbb {C}$ .", "We consider families $\\eta $ of automorphisms $\\eta = \\lbrace \\eta (X)\\in Aut_{\\mathcal {D}_H}(X)\\rbrace _{X\\in Ob(\\mathcal {D}_H)}$ such that $h(\\eta (X))=\\epsilon (h)\\eta (X)$ for all $h\\in H$ and $X\\in Ob(\\mathcal {D}_H)$ .", "We show that these families form a group, which we denote by $\\mathbb {U}_H(\\mathcal {D}_H)$ .", "Further, we show that the inner automorphism of $\\mathcal {D}_H$ induced by conjugating with an element $\\eta \\in \\mathbb {U}_H(\\mathcal {D}_H)$ induces the identity functor on $HC^\\bullet _H(\\mathcal {D}_H,M)$ .", "Using this, we obtain in Proposition REF a set of sufficient conditions for the Hopf cyclic cohomology of an $H$ -category to be zero.", "We say that a cycle $(\\mathcal {S}_H,\\hat{\\partial }_H,\\hat{T}^H)$ is vanishing if $\\mathcal {S}_H^0$ is an $H$ -category and $\\mathcal {S}_H^0$ satisfies the assumptions in Proposition REF .", "We describe the elements of $B^\\bullet _H(\\mathcal {D}_H,M)$ in Theorem REF as the characters of vanishing cycles over $\\mathcal {D}_H$ .", "Finally, in Theorem REF , we use categorified cycles and vanishing cycles to construct a product in Hopf-cyclic cohomologies $ HC^p_H(\\mathcal {D}_H,M) \\otimes HC^q_H(\\mathcal {D}^{\\prime }_H,M^{\\prime }) \\longrightarrow HC^{p+q}_H(\\mathcal {D}_H \\otimes \\mathcal {D}^{\\prime }_H,M \\square _H M^{\\prime })\\qquad p,q\\ge 0$ where $\\mathcal {D}_H$ and $\\mathcal {D}^{\\prime }_H$ are $H$ -linear categories and $M$ and $M^{\\prime }$ are stable anti-Yetter Drinfeld modules over $H$ satisfying certain conditions.", "In the second part of the paper, we study Fredholm modules and Chern classes.", "For this, we assume $H=\\mathbb {C}=M$ and consider a small $\\mathbb {C}$ -linear category $\\mathcal {C}$ .", "Let $p\\ge 1$ be an integer.", "We will say that a pair $(H,\\mathcal {F})$ over $\\mathcal {C}$ as in (REF ) is a $p$ -summable Fredholm module if it satisfies $[\\mathcal {F},f]:= \\left(\\mathcal {F}_Y \\circ {H}(f) - {H}(f) \\circ \\mathcal {F}_X \\right) \\in \\mathcal {B}^p\\left({H}(X),{H}(Y)\\right)$ for any morphism $f:X\\longrightarrow Y$ in $\\mathcal {C}$ (see Definition REF ).", "Here, $\\mathcal {B}^p\\left({H}(X),{H}(Y)\\right)$ is the $p$ -th Schatten class inside the space of bounded linear operators from $H(X)$ to $H(Y)$ .", "We mention here that in this paper, we will consider only even Fredholm modules.", "We hope to tackle the case of odd Fredholm modules over linear categories in a future paper .", "Let $H^\\bullet _\\lambda (\\mathcal {C}):=HC^\\bullet _{\\mathbb {C}}(\\mathcal {C},\\mathbb {C})$ denote the cyclic cohomology groups of $\\mathcal {C}$ .", "Corresponding to a $p$ -summable Fredholm module $(H,\\mathcal {F})$ and any $2m\\ge p-1$ , we construct a DG-semicategory $(\\Omega _{(H,\\mathcal {F})}\\mathcal {C},\\partial ^{\\prime })$ along with a closed graded trace $\\hat{Tr}_s=\\lbrace Tr_s:Hom^{2m}_{\\Omega _{(H,\\mathcal {F})}}(X,X) \\longrightarrow \\mathbb {C}\\rbrace _{X\\in Ob(\\mathcal {C})}$ of dimension $2m$ .", "Let $CN_\\bullet (\\mathcal {C})$ denote the cyclic nerve of $\\mathcal {C}$ and $CN^\\bullet (\\mathcal {C})$ its linear dual.", "By taking the character of the cycle $(\\Omega _{(H,\\mathcal {F})}\\mathcal {C},\\partial ^{\\prime },\\hat{Tr}_s)$ over $\\mathcal {C}$ , we obtain $\\phi ^{2m}\\in CN^{2m}(\\mathcal {C})$ which is given by (see Theorem REF ) $\\phi ^{2m}(f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{2m}):= Tr_s\\left(H({f}^0)[\\mathcal {F},f^1][\\mathcal {F},f^2] \\ldots [\\mathcal {F},f^{2m}]\\right)$ for any $f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{2m} \\in CN_{2m}(\\mathcal {C})$ .", "Then, $\\phi ^{2m}$ lies in the space $Z^{2m}_\\lambda (\\mathcal {C})$ of cocycles for the cyclic cohomology of $\\mathcal {C}$ .", "The Chern character $ch^{2m}(H,\\mathcal {F})$ of the Fredholm module $(H,\\mathcal {F})$ will be the class of $\\phi ^{2m}$ in the cyclic cohomology $H^{2m}_\\lambda (\\mathcal {C})$ of $\\mathcal {C}$ .", "We relate the Chern characters by means of the periodicity operator in Section 7.", "We know that the action of the periodicity operator $S:H^\\bullet _\\lambda (\\mathcal {C})\\longrightarrow H^{\\bullet +2}_\\lambda (\\mathcal {C})$ is given by taking the product as in (REF ) with a certain class in the cohomology $H^2_{\\lambda }(\\mathbb {C})$ .", "If $(H,\\mathcal {F})$ is a $p$ -summable Fredholm module over $\\mathcal {C}$ and $2m\\ge p-1$ , we show in Theorem REF that $S(\\phi ^{2m})=-(m+1)\\phi ^{2m+2} \\qquad \\text{in}~ H^{2m+2}_\\lambda (\\mathcal {C})$ Finally, in Section 8, we describe the homotopy invariance of the Chern character.", "For this, we consider a family $\\lbrace (\\rho _t,\\mathcal {F}_t)\\rbrace _{t\\in [0,1]}$ of $p$ -summable Fredholm modules $\\lbrace \\rho _t:\\mathcal {C} \\longrightarrow SHilb_{\\mathbb {Z}_2}\\rbrace _{t \\in [0,1]}\\qquad \\mathcal {F}_t(X):\\rho _t(X)\\longrightarrow \\rho _t(X)$ each having the same underlying Hilbert space and satisfying some conditions.", "Then, if the $\\rho _t$ and $\\mathcal {F}_t$ vary in a strongly continuous manner with respect to $t\\in [0,1]$ , we show in Theorem REF that the $(p+2)$ -dimensional character $ch^{p+2}(H_t,\\mathcal {F}_t)\\in H^{p+2}_\\lambda (\\mathcal {C})$ is independent of $t\\in [0,1]$ .", "Notations: Throughout the paper, $H$ is a Hopf algebra over the field $k$ of characteristic zero, with comultiplication $\\Delta $ , counit $\\varepsilon $ and bijective antipode $S$ .", "We will use Sweedler's notation for the coproduct $\\Delta (h)= h_1 \\otimes h_2$ and for a left $H$ -coaction $\\rho :M \\longrightarrow H \\otimes M$ , $\\rho (m)= m_{(-1)} \\otimes m_{(0)}$ (with the summation sign suppressed).", "The small cyclic category of Connes will be denoted by $\\Lambda $ .", "The Hochschild differential will always be denoted by $b$ and the modified Hochschild differential (with the last face operator missing) will be denoted by $b^{\\prime }$ .", "On any cocyclic module $C$ , we will denote by $\\tau _n$ the unsigned cyclic operator on $C^n(C)$ and by $\\lambda _n$ the signed cyclic operator $(-1)^n\\tau _n$ on $C^n(C)$ .", "The complex computing cyclic cohomology of $C$ will be denoted by $C^\\bullet _\\lambda (C) := Ker(1-\\lambda )$ .", "Accordingly, the cyclic cocycles and cyclic coboundaries will be denoted by $Z^\\bullet _\\lambda (C)$ and $B^\\bullet _\\lambda (C)$ respectively.", "Acknowledgements: We are grateful to Gadadhar Misra for several useful discussions." ], [ "Preliminaries on $H$ -categories and Hopf-cyclic cohomology", "A small Hopf module category may be treated as a “Hopf algebra with several objects.” In this section, we will collect some preliminaries on Hopf module categories and on Hopf cyclic cohomology.", "We note that the Hopf cyclic cohomology introduced by Connes and Moscovici (, , ) has been developed extensively by a number of authors (see, for instance, , , , , , , , , , , ).", "Definition 2.1 (see Cibils and Solotar ) Let $H$ be a Hopf algebra over a field $k$ .", "A $k$ -linear category $\\mathcal {D}_H$ is said to be a left $H$ -module category if (i) $Hom_{\\mathcal {D}_H}(X,Y)$ is a left $H$ -module for all $X,Y \\in Ob(\\mathcal {D}_H)$ (ii) $h(\\text{id}_X)=\\varepsilon (h)\\text{id}_X$ for all $X \\in Ob(\\mathcal {D}_H)$ and $h \\in H$ (iii) the composition map is a morphism of $H$ -modules, i.e., $h(gf)=(h_1g)(h_2f)$ for any $h \\in H$ , $f \\in Hom_{\\mathcal {D}_H}(X,Y)$ and $g \\in Hom_{\\mathcal {D}_H}(Y,Z)$ .", "A small left $H$ -module category will be called a left $H$ -category.", "We will denote by $Cat_H$ the category of all left $H$ -categories with $H$ -linear functors between them.", "For more on Hopf-module categories, we refer the reader, for instance, to , , , .", "Let $\\mathcal {D}_H$ be a left $H$ -category.", "We set $CN_n(\\mathcal {D}_H):=\\bigoplus Hom_{\\mathcal {D}_H}(X_1,X_0) \\otimes Hom_{\\mathcal {D}_H}(X_2,X_1) \\otimes \\ldots \\otimes Hom_{\\mathcal {D}_H}(X_0,X_n)$ where the direct sum runs over all $(X_0,X_1,\\ldots ,X_n) \\in Ob(\\mathcal {D}_H)^{n+1}$ .", "Lemma 2.2 Let $M$ be a right $H$ -module.", "For each $n\\ge 0$ , $M \\otimes CN_n(\\mathcal {D}_H)$ is a right $H$ -module with action determined by $(m \\otimes f^0 \\otimes \\ldots \\otimes f^{n})h:= mh_1 \\otimes S(h_2)(f^0 \\otimes \\ldots \\otimes f^{n})$ for any $m \\in M$ , $f^0 \\otimes \\ldots \\otimes f^{n} \\in CN_n(\\mathcal {D}_H)$ and $h \\in H$ .", "We now recall the notion of a stable anti-Yetter-Drinfeld module (SAYD) module from .", "Definition 2.3 Let $H$ be a Hopf algebra with a bijective antipode $S$ .", "A $k$ -vector space $M$ is said to be a right-left anti-Yetter-Drinfeld module over $H$ if $M$ is a right $H$ -module and a left $H$ -comodule such that $\\rho (mh)=(mh)_{(-1)} \\otimes (mh)_{(0)}= S(h_3)m_{(-1)}h_1 \\otimes m_{(0)}h_2$ for all $m \\in M$ and $h \\in H$ , where $\\rho : M \\longrightarrow H \\otimes M,~ m \\mapsto m_{(-1)} \\otimes m_{(0)}$ is the coaction.", "Moreover, $M$ is said to be stable if $m_{(0)}m_{(-1)}=m$ .", "We now take the Hopf-cyclic cohomology $HC^\\bullet _H(\\mathcal {D}_H,M)$ of an $H$ -category $\\mathcal {D}_H$ with coefficients in a SAYD module $M$ (see also ).", "This generalizes the construction of the Hopf-cyclic cohomology for $H$ -module algebras with coefficients in an SAYD module (see and also ).", "For each $n \\ge 0$ , we set $\\begin{array}{ll}C^n(\\mathcal {D}_H,M):=Hom_k(M \\otimes CN_n(\\mathcal {D}_H),k) \\qquad \\qquad C^n_H(\\mathcal {D}_H,M):=Hom_H(M \\otimes CN_n(\\mathcal {D}_H),k)\\end{array}$ where $k$ is considered as a right $H$ -module via the counit.", "It is clear from the definition that an element in $C^n_H(\\mathcal {D}_H,M)$ is a $k$ -linear map $\\phi :M \\otimes CN_n(\\mathcal {D}_H) \\longrightarrow k$ satisfying $\\phi \\left(mh_1 \\otimes S(h_2)(f^0 \\otimes \\ldots \\otimes f^{n})\\right)=\\varepsilon (h)\\phi (m \\otimes f^0 \\otimes \\ldots \\otimes f^{n})$ We recall that a (co)simplicial module is said to be para-(co)cyclic if all the relations for a (co)cyclic module are satisfied except $\\tau _n^{n+1}=id$ (see, for instance ).", "The following may be verified directly.", "Proposition 2.4 Let $\\mathcal {D}_H$ be a left $H$ -category and let $M$ be a right-left SAYD module over $H$ .", "Then, (1) we have a para-cocyclic module $C^\\bullet (\\mathcal {D}_H,M):=\\lbrace C^n(\\mathcal {D}_H,M)\\rbrace _{n \\ge 0}$ with the following structure maps $(\\delta _i\\phi )(m \\otimes f^0 \\otimes \\ldots \\otimes f^{n})&= {\\left\\lbrace \\begin{array}{ll}\\phi (m \\otimes f^0 \\otimes \\ldots \\otimes f^if^{i+1} \\otimes \\ldots \\otimes f^{n}) \\quad ~~~~~~~~~~~~~~~~ 0 \\le i \\le n-1\\\\\\phi \\big (m_{(0)} \\otimes \\big (S^{-1}(m_{(-1)})f^n\\big )f^0 \\otimes \\ldots \\otimes f^{n-1}\\big ) \\quad ~~~~~~~~~~~~i=n\\\\\\end{array}\\right.", "}\\\\\\vspace{5.69046pt}(\\sigma _i\\psi )(m \\otimes f^0 \\otimes \\ldots \\otimes f^{n})&= {\\left\\lbrace \\begin{array}{ll} \\psi (m \\otimes f^0 \\otimes \\ldots \\otimes f^i \\otimes id_{X_{i+1}}\\otimes f^{i+1} \\otimes \\ldots \\otimes f^{n}) \\quad ~~~~~ 0 \\le i \\le n-1\\\\\\psi (m \\otimes f^0 \\otimes \\ldots \\otimes f^{n} \\otimes id_{X_{0}}) \\quad ~~~~~~~~~~~~~~~~~~~~~~~~~~ i=n\\\\\\end{array}\\right.", "}\\\\\\vspace{5.69046pt}(\\tau _n\\varphi )(m \\otimes f^0 \\otimes \\ldots \\otimes f^{n})&=\\varphi \\big (m_{(0)} \\otimes S^{-1}(m_{(-1)})f^n \\otimes f^0 \\otimes \\ldots \\otimes f^{n-1}\\big )$ for any $\\phi \\in C^{n-1}(\\mathcal {D}_H,M)$ , $\\psi \\in C^{n+1}(\\mathcal {D}_H,M)$ , $\\varphi \\in C^{n}(\\mathcal {D}_H,M)$ , $m \\in M$ and $ f^0 \\otimes \\ldots \\otimes f^n \\in Hom_{\\mathcal {D}_H}(X_1,X_0) \\otimes Hom_{\\mathcal {D}_H}(X_2,X_1) \\otimes \\ldots \\otimes Hom_{\\mathcal {D}_H}(X_0,X_n)$ .", "(2) by restricting to right $H$ -linear morphisms $C^n_H(\\mathcal {D}_H,M)=Hom_H(M \\otimes CN_n(\\mathcal {D}_H),k),$ we obtain a cocyclic module $C^\\bullet _H(\\mathcal {D}_H,M):=\\lbrace C^n_H(\\mathcal {D}_H,M)\\rbrace _{n \\ge 0}$ .", "The cohomology of the cocyclic module $C^\\bullet _H(\\mathcal {D}_H,M)$ is referred to as the Hopf-cyclic cohomology of the $H$ -category $\\mathcal {D}_H$ with coefficients in the SAYD module $M$ .", "The corresponding cohomology groups are denoted by $HC_H^\\bullet (\\mathcal {D}_H,M)$ .", "Remark 2.5 As $k$ contains $\\mathbb {Q}$ , we recall that the cohomology of a cocyclic module ${C}$ can be expresed alternatively as the cohomology of the following complex (see, for instance ): ${\\begin{matrix}C^0_\\lambda ({C}) &\\xrightarrow{}& \\dots &\\xrightarrow{}& C^n_\\lambda ({C}) &\\xrightarrow{}& C^{n+1}_\\lambda ({C})&\\xrightarrow{}&\\dots \\end{matrix}}$ where $C^n_\\lambda ({C})=Ker(1-\\lambda )\\subseteq C^n({C})$ , $b=\\sum _{i=0}^{n+1} (-1)^i \\delta _i$ and $\\lambda =(-1)^n \\tau _n$ .", "In particular, an element $\\phi \\in C^n_H(\\mathcal {D}_H,M)$ is a cyclic cocycle if and only if $b(\\phi )=0 \\quad \\text{and} \\quad (1-\\lambda )(\\phi )=0$ In this paper, the cocycles and coboundaries of a cocyclic module will always refer to this complex.", "Proposition 2.6 Let $\\mathcal {D}_H$ be a left $H$ -category and let $M$ be a right-left SAYD module.", "Then: (1) We obtain a para-cyclic module $C_\\bullet (\\mathcal {D}_H,M):=\\lbrace C_n(\\mathcal {D}_H,M):=M \\otimes CN_n(\\mathcal {D}_H)\\rbrace _{n \\ge 0}$ with the following structure maps $d_i(m \\otimes f^0 \\otimes \\ldots \\otimes f^{n})&= {\\left\\lbrace \\begin{array}{ll}m \\otimes f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^if^{i+1} \\otimes \\ldots \\otimes f^{n} \\quad ~~~~~~~~~~~~~~~~ 0 \\le i \\le n-1\\\\m_{(0)} \\otimes \\big (S^{-1}(m_{(-1)})f^n\\big )f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{n-1} \\quad ~~~~~~~~~~~~i=n\\\\\\end{array}\\right.", "}\\\\\\vspace{5.69046pt}s_i(m \\otimes f^0 \\otimes \\ldots \\otimes f^{n})&= {\\left\\lbrace \\begin{array}{ll} m \\otimes f^0 \\otimes f^1 \\otimes \\ldots f^i \\otimes id_{X_{i+1}}\\otimes f^{i+1} \\otimes \\ldots \\otimes f^{n} \\quad ~~~~~ 0 \\le i \\le n-1\\\\m \\otimes f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{n} \\otimes id_{X_{0}} \\quad ~~~~~~~~~~~~~~~~~~~~~~~~~~ i=n\\\\\\end{array}\\right.", "}\\\\\\vspace{5.69046pt}t_n(m \\otimes f^0 \\otimes \\ldots \\otimes f^{n})&=m_{(0)} \\otimes S^{-1}(m_{(-1)})f^n \\otimes f^0 \\otimes \\ldots \\otimes f^{n-1}$ for any $m \\in M$ and $ f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^n \\in Hom_{\\mathcal {D}_H}(X_1,X_0) \\otimes Hom_{\\mathcal {D}_H}(X_2,X_1) \\otimes \\ldots \\otimes Hom_{\\mathcal {D}_H}(X_0,X_n)$ .", "(2) By passing to the tensor product over $H$ , we obtain a cyclic module $C_\\bullet ^H(\\mathcal {D}_H,M):=\\lbrace C_n^H(\\mathcal {D}_H,M)=M \\otimes _H CN_n(\\mathcal {D}_H)\\rbrace _{n \\ge 0}$ .", "The cyclic homology groups corresponding to the cyclic module $C_\\bullet ^H(\\mathcal {D}_H,M)$ will be denoted by $HC^H_\\bullet (\\mathcal {D}_H,M)$ ." ], [ "Traces, cocycles and DGH-semicategories", "We continue with $\\mathcal {D}_H$ being a left $H$ -category and $M$ a right-left SAYD module over $H$ .", "Our purpose is to develop a formalism analogous to that of Connes in order to interpret the cocycles $Z^\\bullet _H(\\mathcal {D}_H,M)$ of the complex $C^\\bullet _H(\\mathcal {D}_H,M)$ and its coboundaries $B^\\bullet _H(\\mathcal {D}_H,M)$ as characters of differential graded semicategories.", "In this section, we will describe $Z^\\bullet _H(\\mathcal {D}_H,M)$ , for which we will need the framework of DG-semicategories.", "Let us first recall the notion of a semicategory introduced by Mitchell in (for more on semicategories, see, for instance, ).", "Definition 3.1 (see ) A semicategory $\\mathcal {C}$ consists of a collection $Ob(\\mathcal {C})$ of objects together with a set of morphisms $Hom_{\\mathcal {C}}(X,Y)$ for each $X,Y \\in Ob(\\mathcal {C})$ and an associative composition.", "A semifunctor $F:\\mathcal {C} \\longrightarrow \\mathcal {C}^{\\prime }$ between semicategories assigns an object $F(X) \\in Ob(\\mathcal {C}^{\\prime })$ to each $X \\in Ob(\\mathcal {C})$ and a morphism $F(f) \\in Hom_{\\mathcal {C}^{\\prime }}(F(X),F(Y))$ to each $f \\in Hom_\\mathcal {C}(X,Y)$ and preserves composition.", "A left $H$ -semicategory is a small $k$ -linear semicategory $\\mathcal {S}_H$ such that (i) $Hom_{\\mathcal {S}_H}(X,Y)$ is a left $H$ -module for all $X,Y \\in Ob(\\mathcal {S}_H)$ (ii) $h(gf)=(h_1g)(h_2f)$ for any $h \\in H$ , $f \\in Hom_{\\mathcal {S}_H}(X,Y)$ and $g \\in Hom_{\\mathcal {S}_H}(Y,Z)$ .", "It is clear that any ordinary category may be treated as a semicategory.", "Conversely, to any $k$ -semicategory $\\mathcal {C}$ , we can associate an ordinary $k$ -category $\\tilde{\\mathcal {C}}$ by setting $Ob(\\tilde{\\mathcal {C}})=Ob(\\mathcal {C})$ and adjoining unit morphisms as follows: $Hom_{\\tilde{\\mathcal {C}}}(X,Y):&=\\left\\lbrace \\begin{array}{ll} Hom_\\mathcal {C}(X,X) \\bigoplus k & \\mbox{if $X=Y$}\\\\Hom_\\mathcal {C}(X,Y) & \\mbox{if $X \\ne Y$} \\\\ \\end{array}\\right.$ A morphism in $Hom_{\\tilde{\\mathcal {C}}}(X,Y)$ will be denoted by $\\tilde{f}=f+\\mu $ , where $f \\in Hom_{\\mathcal {C}}(X,Y)$ and $\\mu \\in k$ .", "It is understood that $\\mu =0$ whenever $X \\ne Y$ .", "Any semifunctor $F:\\mathcal {C} \\longrightarrow \\mathcal {D}$ where $\\mathcal {D}$ is an ordinary category may be extended to an ordinary functor $\\tilde{F}:\\tilde{\\mathcal {C}} \\longrightarrow \\mathcal {D}$ .", "If $\\mathcal {S}_H$ is a left $H$ -semicategory, we note that $\\tilde{\\mathcal {S}}_H$ is a left $H$ -category in the sense of Definition REF .", "Definition 3.2 A differential graded semicategory (DG-semicategory) $(\\mathcal {S},\\hat{\\partial })$ is a $k$ -linear semicategory $\\mathcal {S}$ such that (i) $Hom^\\bullet _\\mathcal {S}(X,Y)=\\big (Hom^n_\\mathcal {S}(X,Y),\\hat{\\partial }^n_{XY}\\big )_{n \\ge 0}$ is a cochain complex of $k$ -spaces for each $X,Y \\in Ob(\\mathcal {S})$ .", "(ii) the composition map $Hom^\\bullet _\\mathcal {S}(Y,Z) \\otimes Hom^\\bullet _\\mathcal {S}(X,Y) \\longrightarrow Hom^\\bullet _\\mathcal {S}(X,Z)$ is a morphism of complexes.", "Equivalently, we have $\\hat{\\partial }^n_{XZ}(gf)=\\hat{\\partial }^{n-r}_{YZ}(g)f+(-1)^{n-r}g\\hat{\\partial }^{r}_{XY}(f) $ for any $f \\in Hom_\\mathcal {S}(X,Y)^r$ and $g \\in Hom_\\mathcal {S}(Y,Z)^{n-r}$ .", "Whenever the meaning is clear from context, we will drop the subscript and simply write $\\hat{\\partial }^\\bullet $ for the differential on any $Hom^\\bullet _{\\mathcal {S}}(X,Y)$ .", "A small DG-semicategory may be treated as a differential graded (but not necessarily unital) $k$ -algebra with several objects.", "The DG-semicategories may be treated in a manner similar to DG-categories (see, for instance, , ).", "For instance, there is an obvious notion of DG-semifunctor between DG-semicategories.", "We also note that if $\\mathcal {S}$ is a DG-semicategory, the morphisms in degree 0 determine a semicategory $\\mathcal {S}^0$ .", "We now construct a “universal DG-semicategory” associated to a given $k$ -linear semicategory, similar to the construction of the universal differential graded algebra associated to a (not necessarily unital) $k$ -algebra (see, for instance, ).", "Let $\\Omega \\mathcal {C}$ be the semicategory with $Ob(\\Omega \\mathcal {C}):=Ob(\\mathcal {C})$ and $Hom_{\\Omega \\mathcal {C}}(X,Y)=\\bigoplus \\limits _{n \\ge 0}Hom^n_{\\Omega \\mathcal {C}}(X,Y)$ , where $ Hom^n_{\\Omega \\mathcal {C}}(X,Y):=\\left\\lbrace \\begin{array}{ll}Hom_{\\mathcal {C}}(X,Y) & \\mbox{if $n=0$} \\\\\\\\ \\underset{(X_1,...,X_n)\\in Ob(\\mathcal {C})^n}{\\mbox{\\Large $\\bigoplus $}}Hom_{\\tilde{\\mathcal {C}}}(X_1,Y)\\otimes Hom_{\\mathcal {C}}(X_2,X_1)\\otimes \\dots \\otimes Hom_{\\mathcal {C}}(X,X_n) & \\mbox{if $n\\ge 1$} \\\\\\end{array} \\right.$ Here the sum runs over the ordered tuples $(X_1,...,X_n)\\in Ob(\\mathcal {C})^n$ .", "In particular, $(\\Omega \\mathcal {C})^0={\\mathcal {C}}$ .", "For $n\\ge 1$ , an element of the form $\\tilde{f}^0\\otimes f^1\\otimes ...\\otimes f^n$ in $Hom^n_{\\Omega \\mathcal {C}}(X,Y)$ will be denoted by $\\tilde{f}^0df^1 \\ldots df^n=(f^0+\\mu ) df^1\\dots df^n$ and said to be homogeneous of degree $n$ .", "By abuse of notation, we will continue to use $\\tilde{f}^0df^1 \\ldots df^n=(f^0+\\mu ) df^1\\dots df^n$ to denote an element of $Hom^n_{\\Omega \\mathcal {C}}(X,Y)$ even when $n=0$ .", "In that case, it will be understood that $\\mu =0$ .", "The composition in $\\Omega \\mathcal {C}$ is determined by $f^0\\circ df^1\\circ \\dots \\circ df^n= f^0df^1\\dots df^n \\qquad (df^0)\\circ f^1=d(f^0f^1)-f^0(df^1) \\qquad df^1\\circ \\dots \\circ df^n= df^1\\dots df^n$ In particular, it follows that $\\begin{array}{l}((f^0+\\mu )df^1...df^i)\\cdot ((g^0+\\mu ^{\\prime })dg^1...dg^j)\\\\= (f^0+\\mu )\\left(df^1....df^{i-1}d(f^ig^0)dg^1...dg^j+\\underset{l=1}{\\overset{i-1}{\\sum }}(-1)^{i-l}df^1...d(f^lf^{l+1})...df^idg^0dg^1...dg^j\\right)\\\\\\quad + (-1)^i(f^0+\\mu )f^1df^2...df^idg^0dg^1...dg^j+\\mu ^{\\prime } (f^0+\\mu )df^1...df^idg^1...dg^j\\\\\\end{array}$ For each $X,Y \\in Ob(\\Omega \\mathcal {C})$ , the differential $\\partial ^n_{XY}:Hom^n_{\\Omega \\mathcal {C}}(X,Y) \\longrightarrow Hom^{n+1}_{\\Omega \\mathcal {C}}(X,Y)$ is determined by setting $\\partial ^n_{XY}((f^0+\\mu )df^1 \\ldots df^n):=df^0df^1 \\ldots df^n$ It follows from definition that $\\partial ^{n+1}_{XY}\\circ \\partial ^n_{XY}=0$ .", "Therefore, $Hom^\\bullet _{\\Omega \\mathcal {C}}(X,Y):=\\big (Hom^n_{\\Omega \\mathcal {C}}(X,Y),\\partial ^n_{XY}\\big )_{n \\ge 0}$ is a cochain complex for each $X, Y \\in Ob(\\Omega \\mathcal {C})$ .", "It may also be verified that the composition in $\\Omega \\mathcal {C}$ is a morphism of complexes.", "Thus, $\\Omega \\mathcal {C}$ is a DG-semicategory.", "Proposition 3.3 Let $\\mathcal {C}$ be a small $k$ -linear semicategory.", "Then, the associated DG-semicategory $(\\Omega \\mathcal {C},\\partial )$ is universal in the following sense: given (i) any DG-semicategory $(\\mathcal {S},\\hat{\\partial })$ and (ii) a $k$ -linear semifunctor $\\rho :\\mathcal {C} \\longrightarrow \\mathcal {S}^0$ , there exists a unique DG-semifunctor $\\hat{\\rho }:(\\Omega \\mathcal {C},\\partial ) \\longrightarrow (\\mathcal {S},\\hat{\\partial })$ such that the restriction of $\\hat{\\rho }$ to the semicategory $\\mathcal {C}$ is identical to $\\rho : \\mathcal {C} \\longrightarrow \\mathcal {S}^0$ .", "We extend $\\rho $ to obtain a DG-semifunctor $\\hat{\\rho }:(\\Omega \\mathcal {C},\\partial ) \\longrightarrow (\\mathcal {S},\\hat{\\partial })$ as follows: $\\begin{array}{c}\\hat{\\rho }(X):=\\rho (X)\\\\\\hat{\\rho }((f^0+\\mu )df^1\\ldots df^n):=\\rho (f^0)\\circ \\hat{\\partial }^0(\\rho (f^1))\\circ \\ldots \\circ \\hat{\\partial }^0(\\rho (f^n))+\\mu \\hat{\\partial }^0(\\rho (f^1)) \\circ \\ldots \\circ \\hat{\\partial }^0(\\rho (f^n))\\end{array}$ for all $X \\in Ob(\\Omega \\mathcal {C})=Ob(\\mathcal {C})$ and $(f^0+\\mu )df^1\\ldots df^n \\in Hom^n_{\\Omega \\mathcal {C}}(X,Y)$ , $n\\ge 1$ .", "Since each $\\rho (f^i)$ is a morphism of degree 0 in $\\mathcal {S}$ , it follows from () and (REF ) that $ \\hat{\\rho }(((f^0+\\mu )df^1...df^n)\\circ ((f^{n+1}+\\mu ^{\\prime })df^{n+2}... df^m))=\\hat{\\rho }((f^0+\\mu )df^1...df^n)\\circ \\hat{\\rho }((f^{n+1}+\\mu ^{\\prime })df^{n+2}... df^m)$ It is also clear by construction that $\\hat{\\rho }|_{\\mathcal {C}}=\\rho $ .", "Moreover, we have $\\begin{array}{ll}\\hat{\\partial }^n\\left(\\hat{\\rho }((f^0+\\mu )df^1\\ldots df^n)\\right)&= \\hat{\\partial }^n\\left(\\rho (f^0)\\hat{\\partial }^0(\\rho (f^1)) \\ldots \\hat{\\partial }^0(\\rho (f^n))\\right)+\\mu \\hat{\\partial }^n\\left( \\hat{\\partial }^0(\\rho (f^1)) \\ldots \\hat{\\partial }^0(\\rho (f^n))\\right)\\\\&= {\\hat{\\partial }}^0(\\rho (f^0)) {\\hat{\\partial }}^0(\\rho (f^1)) \\ldots {\\hat{\\partial }}^0(\\rho (f^n))+\\rho (f^0) {\\hat{\\partial }}^n\\left({ \\hat{\\partial }}^0(\\rho (f^1)) \\ldots {\\hat{\\partial }}^0(\\rho (f^n))\\right)\\\\&= \\hat{\\partial }^0(\\rho (f^0)) \\hat{\\partial }^0(\\rho (f^1)) \\ldots \\hat{\\partial }^0(\\rho (f^n))=\\hat{\\rho }\\left(\\partial ^n((f^0+\\mu )df^1\\ldots df^n)\\right)\\end{array}$ The uniqueness of $\\hat{\\rho }$ is also clear from (REF ) and (REF ).", "Definition 3.4 A left DGH-semicategory is a left $H$ -semicategory $\\mathcal {S}_H$ equipped with a DG-semicategory $(\\mathcal {S}_H,\\hat{\\partial }_H)$ structure such that for all $n\\ge 0$ : (a) $Hom^n_{\\mathcal {S}_H}(X,Y)$ is a left $H$ -module for $X,Y \\in Ob(\\mathcal {S}_H)$ .", "(b) $\\hat{\\partial }^n_{H}:Hom^n_{\\mathcal {S}_H}(X,Y) \\longrightarrow Hom^{n+1}_{\\mathcal {S}_H}(X,Y)$ is $H$ -linear for $X,Y \\in Ob(\\mathcal {S}_H)$ .", "We can similarly define the notion of a DGH-semifunctor between DGH-semicategories.", "If $(\\mathcal {S}_H,\\hat{\\partial }_H)$ is a left DGH-semicategory, we note that $\\mathcal {S}_H^0$ is a left $H$ -semicategory.", "Proposition 3.5 Let $\\mathcal {D}_H$ be a left $H$ -category.", "Then, the universal DG-semicategory $(\\Omega (\\mathcal {D}_H),\\partial _H)$ associated to $\\mathcal {D}_H$ is a left DGH-semicategory with the $H$ -action determined by $h \\cdot \\left((f^0+\\mu )df^1 \\ldots df^n\\right):=(h_1f^0+\\mu \\varepsilon (h_1))d(h_2f^1) \\ldots d(h_{n+1}f^n)$ for all $h \\in H$ and $(f^0+\\mu )df^1 \\ldots df^n\\in Hom_{\\Omega (\\mathcal {D}_H)}(X,Y)$ .", "This is immediate from the definitions in (REF ) and (REF ).", "Definition 3.6 Let $(\\mathcal {S}_H,\\hat{\\partial }_H)$ be a left DGH-semicategory and $M$ be a right-left SAYD module over $H$ .", "A closed graded $(H,M)$ -trace of dimension $n$ on $\\mathcal {S}_H$ is a collection of $k$ -linear maps $\\hat{{T}}^H:=\\lbrace \\hat{{T}}_X^H:M \\otimes Hom^n_{\\mathcal {S}_H}(X,X) \\longrightarrow k\\rbrace _{X \\in Ob(\\mathcal {S}_H)}$ such that $&\\hat{{T}}_X^H\\big (mh_1 \\otimes S(h_2)f\\big )=\\varepsilon (h)\\hat{{T}}_X^H(m \\otimes f)\\\\&\\hat{{T}}_X^H\\big (m \\otimes \\hat{\\partial }_{H}^{n-1}(f^{\\prime })\\big )=0 \\\\&\\hat{{T}}_X^H\\big (m \\otimes g^{\\prime }g)=(-1)^{ij}~\\hat{{T}}_Y^{H}\\big (m_{(0)} \\otimes \\left(S^{-1}(m_{(-1)})g\\right)g^{\\prime }\\big )$ for all $h \\in H$ , $m \\in M$ , $f \\in Hom^n_{\\mathcal {S}_H}(X,X)$ , $f^{\\prime } \\in Hom^{n-1}_{\\mathcal {S}_H}(X,X)$ , $g \\in Hom^i_{\\mathcal {S}_H}(X,Y)$ , $g^{\\prime } \\in Hom^j_{\\mathcal {S}_H}(Y,X)$ and $i+j=n$ .", "Definition 3.7 An $n$ -dimensional $\\mathcal {S}_H$ -cycle with coefficients in a SAYD module $M$ is a triple $(\\mathcal {S}_H,\\hat{\\partial }_H,\\hat{T}^H)$ such that (i) $(\\mathcal {S}_H,\\hat{\\partial }_H)$ is a left DGH-semicategory.", "(ii) $\\hat{T}^H$ is a closed graded $(H,M)$ -trace of dimension $n$ on $\\mathcal {S}_H$ .", "Let $\\mathcal {D}_H$ be a left $H$ -category.", "By an $n$ -dimensional cycle over $\\mathcal {D}_H$ , we mean a tuple $(\\mathcal {S}_H,\\hat{\\partial }_H, \\hat{T}^H,\\rho )$ such that (i) $(\\mathcal {S}_H,\\hat{\\partial }_H, \\hat{T}^H)$ is an $n$ -dimensional $\\mathcal {S}_H$ -cycle with coefficients in a SAYD module $M$ .", "(ii) $\\rho :\\mathcal {D}_H \\longrightarrow \\mathcal {S}_H^0$ is an $H$ -linear semifunctor.", "We fix a left $H$ -category $\\mathcal {D}_H$ .", "Given an $n$ -dimensional cycle $(\\mathcal {S}_H,\\hat{\\partial }_H, \\hat{T}^H,\\rho )$ over $\\mathcal {D}_H$ , we define its character $\\phi \\in C^n_H(\\mathcal {D}_H,M)$ by setting $\\phi :M\\otimes CN_n(\\mathcal {D}_H)\\longrightarrow k\\qquad \\phi (m \\otimes f^0 \\otimes \\ldots \\otimes f^n):=\\hat{T}^H_{X_0}\\big (m \\otimes \\rho (f^0)\\hat{\\partial }^0_H\\left(\\rho (f^1)\\right) \\ldots \\hat{\\partial }^0_H\\left(\\rho (f^n)\\right)\\big )$ for $m \\in M$ and $f^0 \\otimes \\ldots \\otimes f^n \\in Hom_{\\mathcal {D}_H}(X_1,X_0) \\otimes Hom_{\\mathcal {D}_H}(X_2,X_1) \\otimes \\ldots \\otimes Hom_{\\mathcal {D}_H}(X_0,X_n)$ .", "We will often suppress the semifunctor $\\rho $ and refer to $\\phi $ simply as the character of the $n$ -dimensional cycle $(\\mathcal {S}_H,\\hat{\\partial }_H, \\hat{T}^H)$ .", "We now have a characterization of the space $Z^n_H(\\mathcal {D}_H,M)$ of $n$ -cocycles in the Hopf-cyclic cohomology of the category $\\mathcal {D}_H$ with coefficients in the SAYD module $M$ .", "Theorem 3.8 Let $\\mathcal {D}_H$ be a left $H$ -category and $M$ be a right-left SAYD module over $H$ .", "Let $\\phi \\in C^n_H(\\mathcal {D}_H,M)$ .", "Then, the following conditions are equivalent: (1) $\\phi $ is the character of an $n$ -dimensional cycle over $\\mathcal {D}_H$ , i.e., there is an $n$ -dimensional cycle $(\\mathcal {S}_H,\\hat{\\partial }_H, \\hat{T}^H)$ with coefficients in $M$ and an $H$ -linear semifunctor $\\rho :\\mathcal {D}_H \\longrightarrow \\mathcal {S}_H^0$ such that $\\begin{array}{ll}\\phi (m \\otimes f^0 \\otimes \\ldots \\otimes f^n)&=\\hat{T}^H_{X_0}((id_M \\otimes \\hat{\\rho })(m \\otimes f^0df^1\\ldots df^n))\\vspace{1.4457pt}\\\\&=\\hat{T}^H_{X_0}\\big (m \\otimes \\rho (f^0)\\hat{\\partial }_H^0\\left(\\rho (f^1)\\right) \\ldots \\hat{\\partial }_H^0\\left(\\rho (f^n)\\right)\\big )\\\\\\end{array}$ for any $m \\in M$ and $f^0 \\otimes \\ldots \\otimes f^n \\in Hom_{\\mathcal {D}_H}(X_1,X_0) \\otimes Hom_{\\mathcal {D}_H}(X_2,X_1) \\otimes \\ldots \\otimes Hom_{\\mathcal {D}_H}(X_0,X_n)$ .", "(2) There exists a closed graded $(H,M)$ -trace ${T}^H$ of dimension $n$ on $\\left(\\Omega (\\mathcal {D}_H),\\partial _H\\right)$ such that $\\phi (m \\otimes f^0 \\otimes \\ldots \\otimes f^n)={T}^H_{X_0}(m \\otimes f^0df^1 \\ldots df^n)$ for any $m \\in M$ and $f^0 \\otimes \\ldots \\otimes f^n \\in Hom_{\\mathcal {D}_H}(X_1,X_0) \\otimes Hom_{\\mathcal {D}_H}(X_2,X_1) \\otimes \\ldots \\otimes Hom_{\\mathcal {D}_H}(X_0,X_n)$ .", "(3) $\\phi \\in Z^n_H(\\mathcal {D}_H,M)$ .", "(1) $\\Rightarrow $ (2).", "By the universal property of $\\Omega (\\mathcal {D}_H)$ , the $H$ -linear semifunctor $\\rho :\\mathcal {D}_H \\longrightarrow \\mathcal {S}_H^0$ can be extended to a DGH-semifunctor $\\hat{\\rho }:\\Omega (\\mathcal {D}_H) \\longrightarrow \\mathcal {S}_H$ as in (REF ).", "We define a collection ${T}^H:=\\lbrace {T}^H_X:M \\otimes Hom^n_{\\Omega (\\mathcal {D}_H)}(X,X) \\longrightarrow k\\rbrace _{X \\in Ob(\\Omega (\\mathcal {D}_H))}$ of $k$ -linear maps given by ${T}_X^H(m \\otimes (f^0+\\mu )df^1 \\ldots df^n):=\\hat{T}^H_{X}\\big (m \\otimes \\hat{\\rho }((f^0+\\mu )df^1 \\ldots df^n)\\big )$ for any $m \\in M$ and $f^0 \\otimes \\ldots \\otimes f^n \\in Hom_{\\mathcal {D}_H}(X_1,X) \\otimes Hom_{\\mathcal {D}_H}(X_2,X_1) \\otimes \\ldots \\otimes Hom_{\\mathcal {D}_H}(X,X_n)$ .", "In particular, it follows from (REF ) that $\\phi (m \\otimes f^0 \\otimes \\ldots \\otimes f^n)=\\hat{T}^H_{X}\\big (m \\otimes \\rho (f^0)\\hat{\\partial }_H^0\\left(\\rho (f^1)\\right) \\ldots \\hat{\\partial }_H^0\\left(\\rho (f^n)\\right)\\big )={T}_X^H(m \\otimes f^0df^1 \\ldots df^n)$ It may be verified that the collection ${T}^H$ is an $n$ -dimensional closed graded $(H,M)$ -trace on $\\Omega (\\mathcal {D}_H)$ .", "(2) $\\Rightarrow $ (1).", "Suppose that we have a closed graded $(H,M)$ -trace ${T}^H$ of dimension $n$ on $\\Omega (\\mathcal {D}_H)$ satisfying (REF ).", "Then, the triple $(\\Omega (\\mathcal {D}_H), \\partial _H, {T}^H)$ forms an $n$ -dimensional cycle over $\\mathcal {D}_H$ with coefficients in $M$ .", "Further, by observing that $\\partial ^0_{H}(f)=df$ for any $f \\in Hom_{\\mathcal {D}_H}(X,Y)$ , we get (REF ).", "(1) $\\Rightarrow $ (3).", "Let $(\\mathcal {S}_H,\\hat{\\partial }_H, \\hat{T}^H)$ be an $n$ -dimensional cycle over $\\mathcal {D}_H$ with coefficients in $M$ and $\\rho :\\mathcal {D}_H \\longrightarrow \\mathcal {S}^0_H$ be an $H$ -linear semifunctor satisfying $\\phi (m \\otimes f^0 \\otimes \\ldots \\otimes f^n)=\\hat{T}^H_{X_0}\\big (m \\otimes \\rho (f^0)\\hat{\\partial }_H^0\\left(\\rho (f^1)\\right) \\ldots \\hat{\\partial }_H^0\\left(\\rho (f^n)\\right)\\big )$ for any $m \\in M$ and $f^0 \\otimes \\ldots \\otimes f^n \\in Hom_{\\mathcal {D}_H}(X_1,X_0) \\otimes Hom_{\\mathcal {D}_H}(X_2,X_1) \\otimes \\ldots \\otimes Hom_{\\mathcal {D}_H}(X_0,X_n)$ .", "For simplicity of notation, we will drop the functor $\\rho $ .", "To show that $\\phi $ is an $n$ -cocycle, it suffices to check that (see (REF )) $b(\\phi )=0\\quad \\text{and} \\quad (1-\\lambda )(\\phi )=0$ where $b=\\sum \\limits _{i=0}^{n+1}(-1)^i\\delta _i$ and $\\lambda =(-1)^n\\tau _n$ .", "For any $p^0 \\otimes \\ldots \\otimes p^{n+1} \\in Hom_{\\mathcal {D}_H}(X_1,X_0) \\otimes Hom_{\\mathcal {D}_H}(X_2,X_1) \\otimes \\ldots \\otimes Hom_{\\mathcal {D}_H}(X_0,X_{n+1})$ , we have $&\\sum \\limits _{i=0}^{n+1}(-1)^i\\delta _i(\\phi )(m \\otimes p^0 \\otimes \\ldots \\otimes p^{n+1})\\\\&= \\sum \\limits _{i=0}^{n}(-1)^i\\phi (m \\otimes p^0 \\otimes \\ldots \\otimes p^ip^{i+1} \\otimes \\ldots \\otimes p^{n+1})~ + (-1)^{n+1}\\phi \\big (m_{(0)} \\otimes \\big (S^{-1}(m_{(-1)})p^{n+1}\\big )p^0 \\otimes p^1 \\otimes \\ldots \\otimes p^{n}\\big )\\\\&= \\hat{T}^H_{X_0}\\big (m \\otimes p^0p^1\\hat{\\partial }_H^0(p^2) \\ldots \\hat{\\partial }_H^0(p^{n+1})\\big ) ~+ \\sum \\limits _{i=1}^{n}(-1)^i \\hat{T}^H_{X_0}\\big (m \\otimes p^0\\hat{\\partial }_H^0(p^1) \\ldots \\hat{\\partial }_H^0(p^ip^{i+1})\\ldots \\hat{\\partial }_H^0(p^{n+1})\\big )~ +\\\\& \\quad (-1)^{n+1}\\hat{T}^H_{X_{n+1}}\\big (m_{(0)} \\otimes \\big (S^{-1}(m_{(-1)})p^{n+1}\\big )p^0 \\hat{\\partial }_H^0(p^1)\\ldots \\otimes \\hat{\\partial }_H^0(p^n)\\big )$ Now using the equality $\\hat{\\partial }_H^0(fg)=\\hat{\\partial }_H^0(f)g+f\\hat{\\partial }_H^0(g)$ for any $f$ and $g$ of degree 0, we have $&\\big (p^0\\hat{\\partial }_H^0(p^1) \\ldots \\hat{\\partial }_H^0(p^n)\\big )p^{n+1}\\\\&=\\sum \\limits _{i=1}^n (-1)^{n-i} p^0\\hat{\\partial }_H^0(p^1) \\ldots \\hat{\\partial }_H^0(p^ip^{i+1}) \\ldots \\hat{\\partial }_H^0(p^{n+1}) + (-1)^n p^0p^1\\hat{\\partial }_H^0(p^2) \\ldots \\hat{\\partial }_H^0(p^{n+1})$ Thus, using the condition in (), we obtain $&\\sum \\limits _{i=0}^{n+1}(-1)^i\\delta _i(\\phi )(m \\otimes p^0 \\otimes \\ldots \\otimes p^{n+1})\\\\&= (-1)^n \\hat{T}^H_{X_0}\\big (m \\otimes \\big ( p^0\\hat{\\partial }_H^0(p^1) \\ldots \\hat{\\partial }_H^0(p^n)\\big )p^{n+1}\\big ) + (-1)^{n+1}\\hat{T}^H_{X_{n+1}}\\big (m_{(0)} \\otimes \\big (S^{-1}(m_{(-1)})p^{n+1}\\big )p^0 \\hat{\\partial }_H^0(p^1)\\ldots \\hat{\\partial }_H^0(p^n)\\big )=0$ Next, using (), (), and the $H$ -linearity of $\\hat{\\partial }_H$ , we have $&\\big (\\left(1-(-1)^n\\tau _n\\right)\\phi \\big )(m \\otimes f^0 \\otimes \\ldots \\otimes f^{n})\\\\&=\\phi (m \\otimes f^0 \\otimes \\ldots \\otimes f^{n})- (-1)^n\\phi \\big (m_{(0)} \\otimes S^{-1}(m_{(-1)})f^n \\otimes f^0 \\otimes \\ldots \\otimes f^{n-1}\\big )\\\\&=\\hat{T}^H_{X_0}(m \\otimes f^0\\hat{\\partial }_H^0(f^1) \\ldots \\hat{\\partial }_H^0(f^n))-(-1)^n \\hat{T}^H_{X_n}\\big (m_{(0)} \\otimes \\big (S^{-1}(m_{(-1)})f^n\\big )\\hat{\\partial }_H^0(f^0)\\hat{\\partial }_H^0(f^1) \\ldots \\hat{\\partial }_H^0(f^{n-1})\\big )\\\\&= (-1)^{n-1} \\hat{T}^H_{X_n}\\big (m_{(0)} \\otimes \\big (S^{-1}(m_{(-1)})\\hat{\\partial }_H^0(f^n)\\big )f^0\\hat{\\partial }_H^0(f^1) \\ldots \\hat{\\partial }^0_H(f^{n-1}) \\big )+\\\\& \\quad (-1)^{n-1} \\hat{T}^H_{X_n}\\big (m_{(0)} \\otimes \\big (S^{-1}(m_{(-1)})f^n\\big )\\hat{\\partial }^0_H(f^0) \\hat{\\partial }_H^0(f^1)\\ldots \\hat{\\partial }^0_H(f^{n-1})\\big )\\\\&= (-1)^{n-1} \\hat{T}^H_{X_n}\\big (m_{(0)} \\otimes \\hat{\\partial }_H^{n-1}\\big ((S^{-1}(m_{(-1)})f^n)f^0\\hat{\\partial }_H^0(f^1) \\ldots \\hat{\\partial }_H^0(f^{n-1})\\big )\\big )=0$ (3) $\\Rightarrow $ (2).", "Let $\\phi \\in Z^n_H(\\mathcal {D}_H,M)$ .", "For each $X \\in Ob(\\Omega (\\mathcal {D}_H))$ , we define an $H$ -linear map ${T}^H_X:M \\otimes Hom^n_{\\Omega (\\mathcal {D}_H)}(X,X) \\longrightarrow k$ given by ${T}^H_X(m \\otimes (f^0+\\mu )df^1\\ldots df^{n}):= \\phi (m \\otimes f^0 \\otimes \\ldots \\otimes f^{n})$ for $f^0 \\otimes \\ldots \\otimes f^{n} \\in Hom_{\\mathcal {D}_H}(X_1,X) \\otimes Hom_{\\mathcal {D}_H}(X_2,X_1) \\otimes \\ldots \\otimes Hom_{\\mathcal {D}_H}(X,X_{n})$ .", "We now verify that the collection $\\lbrace {T}^n_X:M \\otimes Hom^n_{\\Omega (\\mathcal {D}_H)}(X,X) \\longrightarrow k\\rbrace _{X \\in Ob(\\Omega (\\mathcal {D}_H))}$ is a closed graded $(H,M)$ -trace on $(\\Omega (\\mathcal {D}_H),\\partial _H)$ .", "For any $(p^0+\\mu )dp^1\\ldots dp^{n-1} \\in Hom^{n-1}_{\\Omega (\\mathcal {D}_H)}(X,X)$ , we have ${T}_X^{H}\\big (m \\otimes \\partial _H^{n-1}((p^0+\\mu )dp^1\\ldots dp^{n-1})\\big )&={T}_X^{H}\\big (m \\otimes 1dp^0dp^1\\ldots dp^{n-1}\\big )=\\phi (m \\otimes 0 \\otimes p^0 \\otimes \\ldots \\otimes p^{n-1})=0$ This proves the condition in ().", "Using (REF ), it is also clear that $\\lbrace {T}^n_X:M \\otimes Hom^n_{\\Omega (\\mathcal {D}_H)}(X,X) \\longrightarrow k\\rbrace _{X \\in Ob(\\Omega (\\mathcal {D}_H))}$ satisfies condition (REF ).", "Finally, for any $g^{\\prime }=(g^0+\\mu ^{\\prime })dg^1\\ldots dg^{r} \\in Hom^r_{\\Omega (\\mathcal {D}_H)}(Y,X)$ and $g=(g^{r+1}+\\mu )dg^{r+2}\\ldots dg^{n+1} \\in Hom^{n-r}_{\\Omega (\\mathcal {D}_H)}(X,Y)$ , we have $&{T}_X^{H}\\big (m \\otimes g^{\\prime }g\\big )\\\\&=\\sum \\limits _{j=1}^r (-1)^{r-j}~ {T}_X^{H}\\big (m \\otimes (g^0+\\mu ^{\\prime })dg^1 \\ldots d(g^jg^{j+1}) \\ldots dg^{n+1}\\big ) + (-1)^r~ {T}_X^{H}\\big (m \\otimes (g^0+\\mu ^{\\prime })g^1dg^2 \\ldots dg^{n+1}\\big )\\\\&\\textrm { }+{T}_X^{H}\\big (m \\otimes \\mu (g^0+\\mu ^{\\prime })dg^1 \\ldots dg^rdg^{r+2}\\ldots dg^{n+1}\\big ) \\\\&= \\sum \\limits _{j=1}^r (-1)^{r-j} \\phi (m \\otimes g^0 \\otimes \\ldots \\otimes g^jg^{j+1} \\otimes \\ldots \\otimes g^{n+1}) + (-1)^r~ \\phi (m \\otimes g^0g^1 \\otimes g^2 \\otimes \\ldots \\otimes g^{n+1})\\\\& \\textrm { }+ (-1)^r~ \\mu ^{\\prime }\\phi (m \\otimes g^1 \\otimes g^2 \\otimes \\ldots \\otimes g^{n+1})+\\mu \\phi (m \\otimes g^0\\otimes g^1 \\otimes ...\\otimes g^r\\otimes g^{r+2} \\otimes \\ldots \\otimes g^{n+1})\\\\&= \\sum \\limits _{j=0}^r (-1)^{r+j} \\phi (m \\otimes g^0 \\otimes \\ldots \\otimes g^jg^{j+1} \\otimes \\ldots \\otimes g^{n+1})+ (-1)^r~ \\mu ^{\\prime }\\phi (m \\otimes g^1 \\otimes g^2 \\otimes \\ldots \\otimes g^{n+1})\\\\ &\\textrm { }+\\mu \\phi (m \\otimes g^0\\otimes g^1 \\otimes ...\\otimes g^r\\otimes g^{r+2} \\otimes \\ldots \\otimes g^{n+1})$ On the other hand, we have $\\begin{array}{ll}&(-1)^{r(n-r)}~{T}_Y^{H}\\Big (m_{(0)} \\otimes \\big (S^{-1}(m_{(-1)})g\\big )g^{\\prime }\\Big )\\\\&=(-1)^{r(n-r)}~{T}_Y^{H}\\Big (m_{(0)} \\otimes \\left([S^{-1}\\left((m_{(-1)})_{n-r+1}\\right)(g^{r+1}+\\mu )][d\\left(S^{-1}\\left((m_{(-1)})_{n-r}\\right)g^{r+2}\\right)] \\ldots [d\\left(S^{-1}\\left((m_{(-1)})_{1}\\right)g^{n+1}\\right)]\\right)\\circ \\\\&\\qquad ((g^0+\\mu ^{\\prime })dg^1\\ldots dg^{r}) \\Big )\\\\&=(-1)^{r(n-r)} \\sum \\limits _{j=r+2}^{n} (-1)^{n-j+1} ~{T}_Y^{H}\\Big (m_{(0)} \\otimes [S^{-1}\\left((m_{(-1)})_{n-r}\\right)(g^{r+1}+\\mu )]\\ldots d\\big [\\big (S^{-1}((m_{(-1)})_{n-j+1})(g^{j}g^{j+1})\\big ]\\ldots dg^r)+\\\\&\\quad (-1)^{r(n-r)}~{T}_Y^{H}\\Big (m_{(0)} \\otimes [S^{-1}\\left((m_{(-1)})_{n-r+1}\\right)(g^{r+1}+\\mu )]\\ldots d[\\big (S^{-1}((m_{(-1)})_1)g^{n+1}\\big )g^{0}] \\ldots dg^r\\Big )+\\\\& \\quad (-1)^{r(n-r)} (-1)^{n-r} {T}_Y^{H}\\Big (m_{(0)} \\otimes \\left([S^{-1}\\left((m_{(-1)})_{n-r}\\right)((g^{r+1}+\\mu )g^{r+2})]\\right) \\ldots [d\\left(S^{-1}\\left((m_{(-1)})_{1}\\right)g^{n+1}\\right)](dg^0dg^1\\ldots dg^{r}) \\Big )\\\\&\\textrm { }+(-1)^{r(n-r)}\\mu ^{\\prime }{T}_Y^{H}\\Big (m_{(0)} \\otimes \\left.", "[S^{-1}\\left((m_{(-1)})_{n-r+1}\\right)(g^{r+1}+\\mu )][d\\left(S^{-1}\\left((m_{(-1)})_{n-r}\\right)g^{r+2}\\right)] \\ldots [d\\left(S^{-1}\\left((m_{(-1)})_{1}\\right)g^{n+1}\\right)]\\right.\\\\&\\qquad dg^1\\ldots dg^{r} \\Big )\\\\&=(-1)^{r(n-r)} \\sum \\limits _{j=r+2}^{n}(-1)^{n-j+1} ~\\phi \\Big (m_{(0)} \\otimes S^{-1}\\left((m_{(-1)})_{n-r}\\right)g^{r+1} \\otimes \\ldots \\otimes \\big (S^{-1}((m_{(-1)})_{n-j+1})(g^{j}g^{j+1}) \\otimes \\ldots \\otimes g^r)+\\\\&\\quad (-1)^{r(n-r)}~\\phi \\Big (m_{(0)} \\otimes S^{-1}\\left((m_{(-1)})_{n-r+1}\\right)g^{r+1} \\otimes \\ldots \\otimes \\big (S^{-1}((m_{(-1)})_1)g^{n+1}\\big )g^{0} \\otimes \\ldots \\otimes g^r\\Big )+\\\\& \\quad (-1)^{r(n-r)} (-1)^{n-r} \\phi \\Big (m_{(0)} \\otimes S^{-1}\\left((m_{(-1)})_{n-r}\\right)(g^{r+1}g^{r+2}) \\otimes \\ldots \\otimes \\left(S^{-1}\\left((m_{(-1)})_{1}\\right)g^{n+1}\\right) \\otimes g^0 \\otimes g^1 \\otimes \\ldots \\otimes g^{r} \\Big )\\\\& \\quad (-1)^{r(n-r)} (-1)^{n-r} \\mu \\phi \\Big (m_{(0)} \\otimes S^{-1}\\left((m_{(-1)})_{n-r}\\right)g^{r+2} \\otimes \\ldots \\otimes \\left(S^{-1}\\left((m_{(-1)})_{1}\\right)g^{n+1}\\right) \\otimes g^0 \\otimes g^1 \\otimes \\ldots \\otimes g^{r} \\Big )\\\\&\\textrm { }+(-1)^{r(n-r)}\\mu ^{\\prime }\\phi \\Big (m_{(0)} \\otimes \\left.", "S^{-1}\\left((m_{(-1)})_{n-r+1}\\right)g^{r+1}\\otimes S^{-1}\\left((m_{(-1)})_{n-r}\\right)g^{r+2}\\otimes \\ldots \\otimes (S^{-1}\\left((m_{(-1)})_{1}\\right)g^{n+1}\\otimes \\right.", "g^1\\otimes \\ldots \\otimes g^{r} \\Big ) \\\\\\end{array}$ Using repeatedly the fact that $\\phi =(-1)^n\\tau _n\\phi $ , we get $&(-1)^{r(n-r)}~T_Y^{n}\\Big (m_{(0)} \\otimes \\big (S^{-1}(m_{(-1)})g\\big )g^{\\prime }\\Big )\\\\&=-\\sum \\limits _{j=r+1}^{n} (-1)^{r+j} \\phi (m \\otimes g^0 \\otimes \\ldots \\otimes g^jg^{j+1} \\otimes \\ldots \\otimes g^{n+1})-(-1)^{n+r+1}\\phi \\Big (m_{(0)} \\otimes \\big (S^{-1}(m_{(-1)})g^{n+1}\\big )g^0 \\otimes g^1 \\otimes \\ldots \\otimes g^n\\Big )\\\\&\\textrm { } + (-1)^r~ \\mu ^{\\prime }\\phi (m \\otimes g^1 \\otimes g^2 \\otimes \\ldots \\otimes g^{n+1}) +\\mu \\phi (m \\otimes g^0\\otimes g^1 \\otimes ...\\otimes g^r\\otimes g^{r+2} \\otimes \\ldots \\otimes g^{n+1})$ The condition () now follows using the fact that $b(\\phi )=0$ .", "This proves the result.", "Remark 3.9 From the statement and proof of Theorem REF , it is clear that there is a one to one correspondence between $n$ -dimensional closed graded $(H,M)$ -traces on $\\Omega (\\mathcal {D}_H)$ and $Z^n_H(\\mathcal {D}_H,M)$ ." ], [ "Linearization by matrices and Hopf-cyclic cohomology", "In the previous section, we described the spaces $Z^\\bullet _H(\\mathcal {D}_H,M)$ .", "The next aim is to find a characterization of $B^\\bullet _H(\\mathcal {D}_H,M)$ which will be done in several steps.", "For this, we will show in this section that the Hopf-cyclic cohomology of an $H$ -category $\\mathcal {D}_H$ is the same as that of its linearization $\\mathcal {D}_H \\otimes M_r(k)$ by the algebra of $r\\times r$ -matrices.", "We observe that $\\mathcal {D}_H \\otimes M_r(k)$ is also a left $H$ -category.", "We denote by $\\overline{Cat}_H$ the category whose objects are left $H$ -categories and whose morphisms are $H$ -linear semifunctors.", "We denote by $Vect_k$ the category of all $k$ -vector spaces and by $H\\text{-}Mod$ the category of all left $H$ -modules.", "Let $Hom_H(-,k):H\\text{-}Mod \\longrightarrow Vect_k$ be the functor that takes $N \\mapsto Hom_H(N,k)$ .", "We fix $r\\ge 1$ .", "For $1\\le i,j\\le r$ and $\\alpha \\in k$ , we let $E_{ij}(\\alpha )$ denote the elementary matrix in $M_r(k)$ having $\\alpha $ at $(i,j)$ -th position and 0 everywhere else.", "We will often use $E_{ij}$ for $E_{ij}(1)$ .", "For each $1\\le p\\le r$ , we have an inclusion $inc_p:\\mathcal {D}_H \\longrightarrow \\mathcal {D}_H \\otimes M_r(k)$ in $\\overline{Cat}_H$ which fixes the objects and $inc_p(f)=f\\otimes E_{pp}=f \\otimes E_{pp}(1)$ for any morphism $f \\in \\mathcal {D}_H$ .", "For any right-left SAYD-module $M$ , the inclusion $inc_p:\\mathcal {D}_H \\longrightarrow \\mathcal {D}_H \\otimes M_r(k)$ induces an inclusion map $({inc}_p,M):M \\otimes CN_n(\\mathcal {D}_H) \\longrightarrow M \\otimes CN_n\\left(\\mathcal {D}_H \\otimes M_r(k)\\right) $ which takes $m \\otimes f^0 \\otimes \\ldots \\otimes f^n \\mapsto m \\otimes (f^0 \\otimes E_{pp}) \\otimes \\ldots \\otimes (f^n \\otimes E_{pp})$ .", "This induces a morphism of Hochschild complexes $C_\\bullet (inc_p,M)^{hoc}:C_\\bullet (\\mathcal {D}_H,M)^{hoc} \\longrightarrow C_\\bullet \\left(\\mathcal {D}_H \\otimes M_r(k),M\\right)^{hoc}$ .", "Applying the functor $Hom_H(-,k)$ , we obtain morphisms of Hochschild complexes $C^{\\bullet }_H(inc_1,M)^{hoc}:C^{\\bullet }_H\\left(\\mathcal {D}_H \\otimes M_r(k),M\\right)^{hoc} \\longrightarrow C^{\\bullet }_H(\\mathcal {D}_H,M)^{hoc}$ .", "We also have the induced morphism of double complexes computing cyclic homology $C_{\\bullet \\bullet }(inc_p,M)^{cy}:C_{\\bullet \\bullet }(\\mathcal {D}_H,M)^{cy} \\longrightarrow C_{\\bullet \\bullet }\\left(\\mathcal {D}_H \\otimes M_r(k),M\\right)^{cy}$ .", "Applying the functor $Hom_H(-,k)$ , we obtain a morphism of double complexes computing cyclic cohomology $C^{\\bullet \\bullet }_H(inc_1,M)^{cy}:C^{\\bullet \\bullet }_H(\\mathcal {D}_H\\otimes M_r(k),M)^{cy} \\longrightarrow C_H^{\\bullet \\bullet }\\left(\\mathcal {D}_H ,M\\right)^{cy}$ .", "For each $n \\ge 0$ , there is an $H$ -linear trace map $tr^M:M \\otimes CN_n\\left(\\mathcal {D}_H \\otimes M_r(k)\\right) \\longrightarrow M \\otimes CN_n(\\mathcal {D}_H) $ given by $tr^M\\left(m \\otimes (f^0 \\otimes B^0) \\otimes \\ldots \\otimes (f^n \\otimes B^n)\\right):=(m \\otimes f^0 \\otimes \\ldots \\otimes f^n)\\text{trace}(B^0\\ldots B^n)$ for any $m \\in M$ and $(f^0 \\otimes B^0) \\otimes \\ldots \\otimes (f^n \\otimes B^n) \\in CN_n\\left(\\mathcal {D}_H \\otimes M_r(k)\\right)$ .", "It may be verified easily that the trace map as in (REF ) defines a morphism $C_\\bullet (tr^M):C_\\bullet \\left(\\mathcal {D}_H \\otimes M_r(k),M\\right) \\longrightarrow C_\\bullet (\\mathcal {D}_H,M)$ of para-cyclic modules.", "In particular, we have an induced morphism between underlying Hochschild complexes $C_\\bullet ({tr^M})^{hoc}:C_\\bullet \\left(\\mathcal {D}_H \\otimes M_r(k),M\\right)^{hoc} \\longrightarrow C_\\bullet (\\mathcal {D}_H,M)^{hoc}$ Proposition 4.1 The maps $C_\\bullet ({inc}_1,M)^{hoc}$ and $C_\\bullet (tr^M)^{hoc}$ are homotopy inverses of each other.", "It may be easily verified that $C_\\bullet (tr^M)^{hoc} \\circ C_\\bullet (inc_1,M)^{hoc}=id$ .", "To show that $C_\\bullet (inc_1,M)^{hoc} \\circ C_\\bullet (tr^M)^{hoc} \\sim id$ , we define $k$ -linear maps $\\lbrace \\hbar _i: C_{n}\\left(\\mathcal {D}_H \\otimes M_r(k),M\\right) \\longrightarrow C_{n+1}\\left(\\mathcal {D}_H \\otimes M_r(k),M\\right)\\rbrace _{0\\le i\\le n}$ by setting: $\\begin{array}{ll}\\hbar _i\\left(m \\otimes (f^0 \\otimes B^0) \\otimes \\ldots \\otimes (f^n \\otimes B^n)\\right):=& m \\otimes \\sum _{1 \\le j,k,l,\\ldots ,p,q\\le r} (f^0 \\otimes E_{j1}(B^0_{jk})) \\otimes (f^1 \\otimes E_{11}(B^1_{kl})) \\otimes \\ldots \\\\ & \\quad \\otimes (f^i \\otimes E_{11}(B^i_{pq})) \\otimes (id_{X_{i+1}} \\otimes E_{1q}(1)) \\otimes (f^{i+1} \\otimes B^{i+1}) \\otimes \\ldots \\\\& \\quad \\ldots \\otimes (f^n \\otimes B^n)\\end{array}$ for $0\\le i<n$ and $\\begin{array}{ll}\\hbar _n\\left(m \\otimes (f^0 \\otimes B^0) \\otimes \\ldots \\otimes (f^n \\otimes B^n)\\right):= & m \\otimes \\sum _{1 \\le j,k,m,\\ldots ,p,q \\le r} (f^0 \\otimes E_{j1}(B^0_{jk})) \\otimes (f^1 \\otimes E_{11}(B^1_{km})) \\otimes \\ldots \\\\ & \\quad \\ldots \\otimes (f^n \\otimes E_{11}(B^n_{pq})) \\otimes (id_{X_0} \\otimes E_{1q}(1))\\end{array}$ We now verify that $\\hbar ^n:=\\sum _{i=0}^n (-1)^i\\hbar _i$ is a pre-simplicial homotopy (see, for instance, ) between $C_\\bullet (inc_1,M)^{hoc} \\circ C_\\bullet (tr^M)^{hoc}$ and $id_{C_\\bullet \\left(\\mathcal {D}_H \\otimes M_r(k),M\\right)}$ .", "For this, we need to verify the following identities: $\\begin{array}{lll}d_i\\hbar _{i^{\\prime }}=\\hbar _{i^{\\prime }-1}d_i & \\text{for}~ i<i^{\\prime }\\\\d_i\\hbar _i=d_i\\hbar _{i-1} & \\text{for}~ 0< i \\le n\\\\d_i\\hbar _{i^{\\prime }}= \\hbar _{i^{\\prime }}d_{i-1} & \\text{for}~ i>i^{\\prime }+1\\\\d_0\\hbar _0= id_{C_\\bullet \\left(\\mathcal {D}_H \\otimes M_r(k),M\\right)^{hoc}}& \\text{and}~ d_{n+1}\\hbar _n=C_\\bullet (inc_1,M)^{hoc} \\circ C_\\bullet (tr^M)^{hoc}\\end{array}$ where $d_i:C_{n+1}\\left(\\mathcal {D}_H \\otimes M_r(k),M\\right) \\longrightarrow C_{n}\\left(\\mathcal {D}_H \\otimes M_r(k),M\\right)$ , $0 \\le i \\le n+1$ are the face maps.", "We only verify the last one in (REF ) because the others follow similarly.", "Using the fact that $E_{1q}(1)E_{j1}(B_{jk})=0$ unless $q=j$ , we have $\\begin{array}{ll}&d_{n+1}\\hbar _n\\left(m \\otimes (f^0 \\otimes B^0) \\otimes \\ldots \\otimes (f^n \\otimes B^n)\\right)\\\\& \\quad =d_{n+1}\\big (m \\otimes \\sum _{1 \\le j,k,l,\\ldots ,p,q \\le r} (f^0 \\otimes E_{j1}(B^0_{jk})) \\otimes (f^1 \\otimes E_{11}(B^1_{kl})) \\otimes \\ldots \\otimes (f^n \\otimes E_{11}(B^n_{pq})) \\otimes (id_{X_{0}} \\otimes E_{1q}(1))\\big )\\\\& \\quad =m_{(0)} \\otimes \\sum _{1 \\le j,k,l,\\ldots ,p,q \\le r} \\left(S^{-1}(m_{(-1)})(id_{X_{0}} \\otimes E_{1q}(1))\\right)(f^0 \\otimes E_{j1}(B^0_{jk}))\\otimes (f^1 \\otimes E_{11}(B^1_{kl})) \\otimes \\ldots \\\\&\\qquad \\ldots \\otimes (f^n \\otimes E_{11}(B^n_{pq}))\\\\& \\quad = m \\otimes \\sum _{1 \\le j,k,l,\\ldots ,p,q \\le r}\\left(f^0 \\otimes E_{1q}(1)E_{j1}(B^0_{jk})\\right) \\otimes (f^1 \\otimes E_{11}(B^1_{kl})) \\otimes \\ldots \\otimes (f^n \\otimes E_{11}(B^n_{pq}))\\\\& \\quad = m \\otimes \\sum _{1 \\le j,k,l,\\ldots ,p\\le r}\\left(f^0 \\otimes E_{1j}(1)E_{j1}(B^0_{jk})\\right) \\otimes (f^1 \\otimes E_{11}(B^1_{kl})) \\otimes \\ldots \\otimes (f^n \\otimes E_{11}(B^n_{pj}))\\\\& \\quad = m \\otimes \\sum _{1 \\le j,k,l,\\ldots ,p\\le r}(f^0 \\otimes E_{11}(B^0_{jk})) \\otimes (f^1 \\otimes E_{11}(B^1_{kl})) \\otimes \\ldots \\otimes (f^n \\otimes E_{11}(B^n_{pj}))\\\\& \\quad = \\left(m \\otimes (f^0 \\otimes E_{11}) \\otimes \\ldots \\otimes (f^n \\otimes E_{11})\\right) \\sum _{1 \\le j,k,l,\\ldots ,p\\le r}(B^0_{jk}B^1_{kl}\\ldots B^n_{pj})\\\\& \\quad = \\left(m \\otimes (f^0 \\otimes E_{11}) \\otimes \\ldots \\otimes (f^n \\otimes E_{11})\\right) \\sum _{1 \\le j \\le r}(B^0B^1 \\ldots B^n)_{jj}\\\\& \\quad = \\left(m \\otimes (f^0 \\otimes E_{11}) \\otimes \\ldots \\otimes (f^n \\otimes E_{11})\\right)trace(B^0B^1 \\ldots B^n)\\\\& \\quad = \\left(C_\\bullet (inc_1,M)^{hoc} \\circ C_\\bullet (tr^M)^{hoc}\\right)\\left(m \\otimes (f^0 \\otimes B^0) \\otimes \\ldots \\otimes (f^n \\otimes B^n)\\right)\\end{array}$ This proves the result.", "Proposition 4.2 Let $\\mathcal {D}_H$ be a left $H$ -category and $M$ be a right-left SAYD module.", "Then, (1) The morphisms $\\begin{array}{ll}HC^{\\bullet }_H(inc_1,M)^{hoc}:HC^{\\bullet }_H\\left(\\mathcal {D}_H \\otimes M_r(k),M\\right)^{hoc} \\longrightarrow HC^{\\bullet }_H(\\mathcal {D}_H,M)^{hoc}\\\\HC^{\\bullet }_H(tr^M)^{hoc}:HC^{\\bullet }_H(\\mathcal {D}_H,M)^{hoc} \\longrightarrow HC^{\\bullet }_H\\left(\\mathcal {D}_H \\otimes M_r(k),M\\right)^{hoc}\\end{array}$ induced by $C^{\\bullet }_H(inc_1,M)^{hoc}$ and $C^{\\bullet }_H(tr^M)^{hoc}$ are mutually inverse isomorphisms of Hochschild cohomologies.", "(2) We have isomorphisms $ (10,0)*{HC^\\bullet _H(\\mathcal {D}_H,M)}; (40,3)*{HC^{\\bullet }_H(tr^M)}; (40,-3.3)*{HC^{\\bullet }_H(inc_1,M)}; {(25,1)*{}; (60,1)*{}}; {(60,-1)*{};(25,-1)*{}};\\quad HC^\\bullet _H\\left(\\mathcal {D}_H \\otimes M_r(k),M\\right)$ (1) By Proposition REF , we know that $C_\\bullet (tr^M)^{hoc} \\circ C_\\bullet (inc_1,M)^{hoc}=id_{C_\\bullet (\\mathcal {D}_H,M)^{hoc}}$ and $C_\\bullet (inc_1,M)^{hoc} \\circ C_\\bullet (tr^M)^{hoc} \\sim id_{C_\\bullet \\left(\\mathcal {D}_H \\otimes M_r(k),M\\right)^{hoc}}$ .", "Thus, applying the functor $Hom_H(-,k)$ , we obtain $C^{\\bullet }_H(inc_1,M)^{hoc} \\circ C^{\\bullet }_H(tr^M)^{hoc}=id_{C^{\\bullet }_H(\\mathcal {D}_H,M)^{hoc}} \\qquad C^{\\bullet }_H(tr^M)^{hoc} \\circ C^{\\bullet }_H(inc_1,M)^{hoc} \\sim id_{C^{\\bullet }_H(\\mathcal {D}_H \\otimes M_r(k),M)^{hoc}}$ Therefore, $C^{\\bullet }_H(inc_1,M)^{hoc}$ and $C^{\\bullet }_H(tr^M)^{hoc}$ are homotopy inverses of each other.", "(2) This follows immediately from (1) and the Hochschild to cyclic spectral sequence.", "Corollary 4.3 For an $n$ -cocycle $\\phi \\in Z^n_H(\\mathcal {D}_H,M)$ , the $n$ -cocycle $\\tilde{\\phi }=Hom_H(tr^M,k)(\\phi ) =\\phi \\circ tr^M\\in Z^n_H(\\mathcal {D}_H \\otimes M_r(k),M)$ may be described as follows $\\tilde{\\phi }\\left(m \\otimes (f^0 \\otimes B^0) \\otimes \\ldots \\otimes (f^n \\otimes B^n)\\right)=\\phi (m \\otimes f^0 \\otimes \\ldots \\otimes f^n)trace(B^0\\ldots B^n)$" ], [ "Vanishing cycles on an $H$ -category and coboundaries", "From now onwards, we will always assume that $k=\\mathbb {C}$ .", "In this section, we will describe the spaces $B^\\bullet _H(\\mathcal {D}_H,M)$ .", "We will then use the formalism of categorified cycles and vanishing cycles developed in this paper to obtain a product on Hopf cyclic cohomologies of $H$ -categories.", "We begin by recalling the notion of an inner automorphism of a category.", "Definition 5.1 [see , p 24] Let $\\mathcal {D}_H$ be a left $H$ -category.", "An automorphism $\\Phi \\in Hom_{Cat_H}(\\mathcal {D}_H,\\mathcal {D}_H)$ is said to be inner if $\\Phi $ is isomorphic to the identity functor $id_{\\mathcal {D}_H}$ .", "In particular, there exist isomorphisms $\\lbrace \\eta (X):X\\longrightarrow \\Phi (X)\\rbrace _{X\\in Ob(\\mathcal {D}_H)}$ such that $\\Phi (f)=\\eta (Y)\\circ f \\circ (\\eta (X))^{-1}$ for any $f \\in Hom_{\\mathcal {D}_H}(X,Y)$ .", "We now set $\\mathbb {G}(\\mathcal {D}_H):=\\underset{X\\in Ob(\\mathcal {D}_H)}{\\prod }\\textrm { }Aut_{\\mathcal {D}_H}(X)$ By definition, an element $\\eta \\in \\mathbb {G}(\\mathcal {D}_H)$ corresponds to a family of automorphisms $\\lbrace \\eta (X):X\\longrightarrow X\\rbrace _{X\\in Ob(\\mathcal {D}_H)}$ .", "We now set $\\mathbb {U}_H(\\mathcal {D}_H):=\\lbrace \\mbox{$\\eta \\in \\mathbb {G}(\\mathcal {D}_H)$ $\\vert $ $h(\\eta (X))=\\varepsilon (h)\\eta (X)$ for every $h\\in H$ and $X\\in Ob(\\mathcal {D}_H)$}\\rbrace $ Lemma 5.2 $\\mathbb {U}_H(\\mathcal {D}_H)$ is a subgroup of $\\mathbb {G}(\\mathcal {D}_H)$ .", "The element $\\mathbf {e}=\\underset{X\\in Ob(\\mathcal {D}_H)}{\\prod } id_X$ is the identity of the group $\\mathbb {G}(\\mathcal {D}_H)$ .", "By definition of an $H$ -category, we know that $h\\cdot id_X=\\varepsilon (h)\\cdot id_X$ for each $X \\in Ob(\\mathcal {D}_H)$ and $h \\in H$ .", "Thus, $\\mathbf {e} \\in \\mathbb {U}_H(\\mathcal {D}_H)$ .", "Now, suppose that $\\eta , \\eta ^{\\prime } \\in \\mathbb {U}_H(\\mathcal {D}_H)$ .", "Then, for each $X \\in Ob(\\mathcal {D}_H)$ and $h \\in H$ , $h\\left((\\eta \\circ \\eta ^{\\prime })(X)\\right)=h(\\eta (X) \\circ \\eta ^{\\prime }(X))=(h_1\\eta (X)) \\circ (h_2\\eta ^{\\prime }(X))=(\\varepsilon (h_1)\\eta (X)) \\circ (\\varepsilon (h_2)\\eta ^{\\prime }(X))=\\varepsilon (h)(\\eta (X) \\circ \\eta ^{\\prime }(X))$ Hence, $\\eta \\circ \\eta ^{\\prime } \\in \\mathbb {U}_H(\\mathcal {D}_H)$ .", "Also, $\\eta ^{-1} \\in \\mathbb {G}(\\mathcal {D}_H)$ corresponds to a family of morphisms $\\lbrace \\eta ^{-1}(X):=\\eta (X)^{-1}:X \\longrightarrow X\\rbrace _{X \\in Ob(\\mathcal {D}_H)}$ .", "Then, for each $h \\in H$ and $X \\in Ob(\\mathcal {D}_H)$ , $\\varepsilon (h)id_X=h(\\eta (X) \\circ \\eta ^{-1}(X))=(\\varepsilon (h_1)\\eta (X)) \\circ (h_2 \\eta ^{-1}(X))=\\eta (X) \\circ (h \\eta ^{-1}(X))$ which gives $\\varepsilon (h)\\eta ^{-1}(X)=h \\eta ^{-1}(X)$ .", "Therefore, $\\eta ^{-1} \\in \\mathbb {U}_H(\\mathcal {D}_H)$ .", "Lemma 5.3 Let $\\mathcal {D}_H$ be a left $H$ -category and let $\\eta \\in \\mathbb {U}_H(\\mathcal {D}_H)$ .", "(1) Consider $\\Phi _\\eta :\\mathcal {D}_H\\longrightarrow \\mathcal {D}_H$ defined by $\\Phi _\\eta (X)=X\\qquad \\Phi _\\eta (f):=\\eta (Y)\\circ f\\circ \\eta (X)^{-1}$ for every $X\\in Ob(\\mathcal {D}_H)$ and $f\\in Hom_{\\mathcal {D}_H}(X,Y)$ .", "Then, $\\Phi _\\eta : \\mathcal {D}_H\\longrightarrow \\mathcal {D}_H$ is an inner automorphism of $\\mathcal {D}_H$ .", "(2) Consider $\\tilde{\\Phi }_\\eta : \\mathcal {D}_H\\otimes M_2(k)\\longrightarrow \\mathcal {D}_H\\otimes M_2(k)$ defined by $\\tilde{\\Phi }_\\eta (X)=X \\qquad \\tilde{\\Phi }_\\eta (f\\otimes B)=(id_Y\\otimes E_{11}+ \\eta (Y)\\otimes E_{22})\\circ (f\\otimes B)\\circ (id_X\\otimes E_{11}+ \\eta (X)^{-1}\\otimes E_{22})$ for every $X\\in Ob(\\mathcal {D}_H\\otimes M_2(k))=Ob(\\mathcal {D}_H)$ and $f\\otimes B \\in Hom_{\\mathcal {D}_H\\otimes M_2(k)}(X,Y)$ .", "Then, $\\tilde{\\Phi }_\\eta : \\mathcal {D}_H\\otimes M_2(k)\\longrightarrow \\mathcal {D}_H\\otimes M_2(k)$ is an inner automorphism.", "(1) Using the fact that $\\eta , \\eta ^{-1} \\in \\mathbb {U}_H(\\mathcal {D}_H)$ , we have $\\begin{array}{ll}h(\\Phi _\\eta (f))=(h_1\\eta (Y)) \\circ (h_2f) \\circ (h_3\\eta (X)^{-1})&=(\\varepsilon (h_1)\\eta (Y)) \\circ (h_2f) \\circ (\\varepsilon (h_3)\\eta (X)^{-1})\\\\&=\\eta (Y) \\circ (h_1f) \\circ (\\varepsilon (h_2)\\eta (X)^{-1})\\\\&=\\eta (Y)\\circ (hf)\\circ \\eta (X)^{-1}\\end{array}$ for any $h \\in H$ and $f\\in Hom_{\\mathcal {D}_H}(X,Y)$ .", "By Definition REF , we now see that $\\Phi _\\eta $ is an inner automorphism.", "(2) Setting $\\tilde{\\eta }(X):X \\longrightarrow X$ in $\\mathcal {D}_H\\otimes M_2(k)$ as $\\tilde{\\eta }(X)=id_X\\otimes E_{11}+ \\eta (X)\\otimes E_{22}$ , we see that $\\begin{array}{ll}\\tilde{\\Phi }_\\eta (f\\otimes B)&=(id_Y\\otimes E_{11}+ \\eta (Y)\\otimes E_{22})\\circ (f\\otimes B)\\circ (id_X\\otimes E_{11}+ \\eta (X)^{-1}\\otimes E_{22})\\\\&=\\tilde{\\eta }(Y) \\circ (f\\otimes B)\\circ \\tilde{\\eta }(X)^{-1}\\end{array}$ for any $f\\otimes B \\in Hom_{\\mathcal {D}_H\\otimes M_2(k)}(X,Y)$ .", "Considering the $H$ -action on the category $\\mathcal {D}_H\\otimes M_2(k)$ , we have $\\begin{array}{ll}h\\left(\\tilde{\\Phi }_\\eta ((f \\otimes B)\\right)&=h_1(id_Y\\otimes E_{11}+ \\eta (Y)\\otimes E_{22})\\circ h_2(f\\otimes B)\\circ h_3(id_X\\otimes E_{11}+ \\eta (X)^{-1}\\otimes E_{22})\\\\&=(h_1id_Y\\otimes E_{11}+ h_1\\eta (Y)\\otimes E_{22})\\circ h_2(f\\otimes B)\\circ (h_3id_X\\otimes E_{11}+ h_3\\eta (X)^{-1}\\otimes E_{22})\\\\&=\\varepsilon (h_1)(id_Y\\otimes E_{11}+ \\eta (Y)\\otimes E_{22})\\circ h_2(f\\otimes B)\\circ \\varepsilon (h_3)(id_X\\otimes E_{11}+ \\eta (X)^{-1}\\otimes E_{22})\\\\&=(id_Y\\otimes E_{11}+ \\eta (Y)\\otimes E_{22})\\circ h(f\\otimes B)\\circ (id_X\\otimes E_{11}+ \\eta (X)^{-1}\\otimes E_{22})\\\\&=\\tilde{\\Phi }_\\eta (h(f \\otimes B))\\end{array}$ for any $h \\in H$ and $f\\otimes B \\in Hom_{\\mathcal {D}_H\\otimes M_2(k)}(X,Y)$ .", "By Definition REF , we now see that $\\Phi _{\\tilde{\\eta }}$ is an inner automorphism.", "For any $\\eta \\in \\mathbb {U}_H(\\mathcal {D}_H)$ , we will always denote by $\\Phi _\\eta $ and $\\tilde{\\Phi }_\\eta $ the inner automorphisms defined in Lemma REF .", "Lemma 5.4 Let $M$ be a right-left SAYD module over $H$ .", "Then, (1) A semifunctor $\\alpha \\in Hom_{\\overline{Cat}_H}(\\mathcal {D}_H,\\mathcal {D}_H^{\\prime })$ induces a morphism (for all $n\\ge 0$ ) $C^n_H(\\alpha ,M):C^n_H(\\mathcal {D}_H^{\\prime },M)=Hom_H(M \\otimes CN_n(\\mathcal {D}_H^{\\prime }),k)\\longrightarrow C^n_H(\\mathcal {D}_H,M)=Hom_H(M \\otimes CN_n(\\mathcal {D}_H),k)$ determined by $C^{n}_H(\\alpha ,M)(\\phi )(m \\otimes f^0 \\otimes \\ldots \\otimes f^n)=\\phi \\left(m \\otimes \\alpha (f^0) \\otimes \\ldots \\otimes \\alpha (f^n)\\right)$ for any $\\phi \\in C^n_H(\\mathcal {D}_H^{\\prime },M)$ , $m \\in M$ and $f^0 \\otimes \\ldots \\otimes f^n \\in CN_n(\\mathcal {D}_H)$ .", "This leads to a morphism $C^{\\bullet \\bullet }_H(\\alpha ,M)^{cy}:C^{\\bullet \\bullet }_H(\\mathcal {D}_H^{\\prime },M)^{cy} \\longrightarrow C^{\\bullet \\bullet }_H(\\mathcal {D}_H,M)^{cy}$ of double complexes and induces a functor $HC^\\bullet _H(-,M):\\overline{Cat}_H^{op} \\longrightarrow Vect_k$ .", "(2) Let $\\eta \\in \\mathbb {U}_H(\\mathcal {D}_H)$ .", "Then, $\\Phi _\\eta $ induces the identity map on $HC^\\bullet _H(\\mathcal {D}_H,M)$ .", "(1) Since $\\phi $ and $\\alpha $ are $H$ -linear, the morphisms $C^n_H(\\alpha ,M)$ are well-defined and well behaved with respect to the maps appearing in the Hochschild and cyclic complexes.", "The result follows.", "(2) Let $\\eta \\in \\mathbb {U}_H(\\mathcal {D}_H)$ and $\\Phi _\\eta \\in Hom_{Cat_H}(\\mathcal {D}_H,\\mathcal {D}_H)$ be the corresponding inner automorphism.", "By Proposition REF , the maps $HC^\\bullet _H(inc_1,M)$ and $HC^\\bullet _H(tr^M)$ are mutually inverse isomorphisms of Hopf-cyclic cohomology groups.", "Thus, we have $HC^\\bullet _H(inc_2,M) \\circ \\left(HC^\\bullet _H(inc_1,M)\\right)^{-1} =HC^\\bullet _H(inc_2,M) \\circ HC^\\bullet _H(tr^M)= HC^\\bullet _H\\left(tr^M \\circ (inc_2,M)\\right)=id$ Further, we have the following commutative diagram in the category $\\overline{Cat}_H$ : ${\\begin{matrix}\\mathcal {D}_H &\\xrightarrow{}& \\mathcal {D}_H \\otimes M_2(k) &\\xleftarrow{}& \\mathcal {D}_H\\\\{\\scriptstyle id_{\\mathcal {D}_H}}\\downarrow \\mathbox{mphantom}{\\scriptstyle id_{\\mathcal {D}_H}}&& \\mathbox{mphantom}{\\scriptstyle \\tilde{\\Phi }_\\eta }\\downarrow {\\scriptstyle \\tilde{\\Phi }_\\eta }&& \\mathbox{mphantom}{\\scriptstyle \\Phi _\\eta }\\downarrow {\\scriptstyle \\Phi _\\eta }&&\\\\\\mathcal {D}_H &\\xrightarrow{}&\\mathcal {D}_H \\otimes M_2(k)&\\xleftarrow{}& \\mathcal {D}_H\\\\\\end{matrix}}$ Thus, by applying the functor $HC^\\bullet _H(-,M)$ to the commutative diagram (REF ) and using (REF ), we obtain $HC^\\bullet _H(\\Phi _\\eta ,M)&= \\left(HC^\\bullet _H(inc_2,M)\\right)\\circ HC^\\bullet _H(inc_1,M)^{-1} \\circ HC^\\bullet _H(id_{\\mathcal {D}_H},M)\\circ \\left(HC^\\bullet _H(inc_1,M)\\right) \\circ HC^\\bullet _H(inc_2,M)^{-1}\\\\&=id_{HC^\\bullet _H(\\mathcal {D}_H,M)}$ Proposition 5.5 Let $\\mathcal {D}_H$ be a left $H$ -category.", "Suppose that there is a semifunctor $\\upsilon \\in Hom_{\\overline{Cat}_H}(\\mathcal {D}_H,\\mathcal {D}_H)$ and an $\\eta \\in \\mathbb {U}_H\\left(\\mathcal {D}_H \\otimes M_2(k)\\right)$ such that (1) $\\upsilon (X)=X \\qquad \\forall X \\in Ob(\\mathcal {D}_H)$ (2) $\\Phi _\\eta (f \\otimes E_{11}+\\upsilon (f) \\otimes E_{22})=\\upsilon (f) \\otimes E_{22}$ for all $f \\in Hom_{\\mathcal {D}_H}(X,Y)$ and $X,Y \\in Ob(\\mathcal {D}_H)$ .", "Then, $HC^\\bullet _H(\\mathcal {D}_H,M)=0$ .", "Let $\\alpha , \\alpha ^{\\prime } \\in Hom_{\\overline{Cat}_H}\\left(\\mathcal {D}_H, \\mathcal {D}_H \\otimes M_2(k)\\right)$ be the semifunctors defined by $\\begin{array}{c}\\alpha (X):=X \\qquad \\alpha (f):=f \\otimes E_{11} + \\upsilon (f) \\otimes E_{22}\\\\\\alpha ^{\\prime }(X):=X \\qquad \\alpha ^{\\prime }(f):=\\upsilon (f) \\otimes E_{22}\\end{array}$ for all $X \\in Ob(\\mathcal {D}_H)$ and $f \\in Hom_ {\\mathcal {D}_H}(X,Y)$ .", "Then, by assumption, $\\alpha ^{\\prime }=\\Phi _\\eta \\circ \\alpha $ .", "Therefore, applying the functor $HC^\\bullet _H(-,M)$ and using Lemma REF (2), we get $HC^\\bullet _H(\\alpha ^{\\prime },M)=HC^\\bullet _H(\\alpha ,M) \\circ HC^\\bullet _H(\\Phi _\\eta ,M)=HC^\\bullet _H(\\alpha ,M):HC^\\bullet _H(\\mathcal {D}_H\\otimes M_2(k),M)\\longrightarrow HC^\\bullet _H(\\mathcal {D}_H,M)$ Let $\\phi \\in Z^n_H(\\mathcal {D}_H,M)$ and $\\tilde{\\phi }=Hom_H(tr^M,k)(\\phi ) =\\phi \\circ tr^M\\in Z^n_H(\\mathcal {D}_H \\otimes M_2(k),M)$ as in Corollary REF .", "Let $[\\tilde{\\phi }]$ denote the cohomology class of $\\tilde{\\phi }$ .", "Then, by (REF ), we have $HC^\\bullet _H(\\alpha ,M)([\\tilde{\\phi }])=HC^\\bullet _H(\\alpha ^{\\prime },M)([\\tilde{\\phi }])$ , i.e., $\\tilde{\\phi }\\circ (id_M\\otimes CN_n(\\alpha )) + B^n_H(\\mathcal {D}_H,M)=\\tilde{\\phi }\\circ (id_M\\otimes CN_n(\\alpha ^{\\prime })) + B^n_H(\\mathcal {D}_H,M)$ so that $\\tilde{\\phi }\\circ (id_M\\otimes CN_n(\\alpha )) - \\tilde{\\phi }\\circ (id_M\\otimes CN_n(\\alpha ^{\\prime })) \\in B^n_H(\\mathcal {D}_H,M)$ .", "Applying the definition of $\\tilde{\\phi }$ , we now have $\\begin{array}{ll}&(\\tilde{\\phi }\\circ (id_M\\otimes CN_n(\\alpha ))) (m \\otimes f^0 \\otimes \\ldots \\otimes f^n)\\\\&\\quad =\\tilde{\\phi }\\left(m \\otimes \\alpha (f^0) \\otimes \\ldots \\otimes \\alpha (f^n)\\right)\\\\& \\quad = \\tilde{\\phi }\\left(m \\otimes (f^0 \\otimes E_{11} + \\upsilon (f^0) \\otimes E_{22}) \\otimes \\ldots \\otimes (f^n \\otimes E_{11} + \\upsilon (f^n) \\otimes E_{22})\\right)\\\\& \\quad = \\phi (m \\otimes f^0 \\otimes \\ldots \\otimes f^n)+\\phi (m \\otimes \\upsilon (f^0) \\otimes \\ldots \\otimes \\upsilon (f^n))\\end{array}$ Similarly, $(\\tilde{\\phi }\\circ (id_M\\otimes CN_n(\\alpha ^{\\prime }))) (m \\otimes f^0 \\otimes \\ldots \\otimes f^n)=\\phi (m \\otimes \\upsilon (f^0) \\otimes \\ldots \\otimes \\upsilon (f^n))$ .", "Thus, $\\phi =\\tilde{\\phi }\\circ (id_M\\otimes CN_n(\\alpha )) - \\tilde{\\phi }\\circ (id_M\\otimes CN_n(\\alpha ^{\\prime }))\\in B^n_H(\\mathcal {D}_H,M)$ .", "This proves the result.", "Definition 5.6 Let $(\\mathcal {S}_H,\\hat{\\partial }_H, \\hat{{T}}^H)$ be an $n$ -dimensional $\\mathcal {S}_H$ -cycle with coefficients in a SAYD module $M$ over $H$ (see, Definition REF ).", "Then, we say that the cycle $(\\mathcal {S}_H,\\hat{\\partial }_H, \\hat{{T}}^H)$ is vanishing if $\\mathcal {S}^0_H$ is a left $H$ -category and $\\mathcal {S}^0_H$ satisfies the assumptions in Proposition REF .", "We now recall from the algebra $\\mathbf {C}$ of infinite matrices $(a_{ij})_{i,j\\in \\mathbb {N}}$ with entries from $\\mathbb {C}$ satisfying the following conditions (see also ) (i) the set $\\lbrace \\mbox{$a_{ij}$ $\\vert $ $i,j\\in \\mathbb {N}$}\\rbrace $ is finite, (ii) the number of non-zero entries in each row or each column is bounded.", "Identifying $M_2(\\mathbf {C})=\\mathbf {C} \\otimes M_2(\\mathbb {C})$ , we recall the following result from : Lemma 5.7 There exists an algebra homomorphism $\\omega :\\mathbf {C} \\longrightarrow \\mathbf {C}$ and an invertible element $\\tilde{U} \\in M_2(\\mathbf {C})$ such that the corresponding inner automorphism $\\Xi :M_2(\\mathbf {C}) \\longrightarrow M_2(\\mathbf {C})$ satisfies $\\Xi (B \\otimes E_{11}+\\omega (B) \\otimes E_{22})=\\omega (B) \\otimes E_{22} \\qquad \\forall B \\in \\mathbf {C}$ Then, $HC^\\bullet (\\mathbf {C})=0$ .", "Remark 5.8 We note that the condition in (REF ) ensures that $\\omega (\\mathbf {1})\\ne \\mathbf {1}$ , where $\\mathbf {1}$ is the unit element of $\\mathbf {C}$ .", "For any $k$ -algebra $\\mathcal {A}$ , we may define a $k$ -linear category $\\mathcal {A} \\otimes \\mathcal {D}_H$ by setting $Ob(\\mathcal {A} \\otimes \\mathcal {D}_H)=Ob(\\mathcal {D}_H)$ and $Hom_{\\mathcal {A} \\otimes \\mathcal {D}_H}(X,Y)=\\mathcal {A} \\otimes Hom_{\\mathcal {D}_H}(X,Y)$ .", "The category $\\mathcal {A} \\otimes \\mathcal {D}_H$ is a left $H$ -category via the action $h(a \\otimes f):=a \\otimes hf$ for any $h \\in H$ , $a \\otimes f \\in \\mathcal {A} \\otimes Hom_{\\mathcal {D}_H}(X,Y)$ .", "Lemma 5.9 We have $HC^\\bullet _H(\\mathbf {C} \\otimes \\mathcal {D}_H,M)=0$ .", "We will verify that the category $\\mathbf {C} \\otimes \\mathcal {D}_H$ satisfies the assumptions of Proposition REF .", "Let $\\omega $ and $\\tilde{U}$ be as in Lemma REF .", "We now define $\\upsilon :\\mathbf {C} \\otimes \\mathcal {D}_H \\longrightarrow \\mathbf {C} \\otimes \\mathcal {D}_H$ given by $\\upsilon (X):=X \\qquad \\upsilon (B \\otimes f):=\\omega (B) \\otimes f$ for any $X \\in Ob(\\mathbf {C} \\otimes \\mathcal {D}_H)$ and $B \\otimes f \\in Hom_{\\mathbf {C} \\otimes \\mathcal {D}_H}(X,Y)$ .", "Since $\\omega :\\mathbf {C} \\longrightarrow \\mathbf {C}$ is an algebra homomorphism, it follows that $\\upsilon $ is a semifunctor.", "By the definition of the $H$ -action on $\\mathbf {C} \\otimes \\mathcal {D}_H$ , it is also clear that $\\upsilon $ is $H$ -linear.", "Using the identification $\\mathbf {C} \\otimes \\mathcal {D}_H \\otimes M_2(\\mathbb {C}) = M_2(\\mathbf {C}) \\otimes \\mathcal {D}_H$ , we now define an element $\\eta \\in \\mathbb {G}(\\mathbf {C} \\otimes \\mathcal {D}_H \\otimes M_2(\\mathbb {C}))=\\mathbb {G}( M_2(\\mathbf {C}) \\otimes \\mathcal {D}_H)$ given by the family of morphims $\\lbrace \\eta (X):=\\tilde{U} \\otimes id_X \\in Hom_{M_2(\\mathbf {C}) \\otimes \\mathcal {D}_H}(X,X)=M_2(\\mathbf {C}) \\otimes Hom_{\\mathcal {D}_H}(X,X)\\rbrace _{X \\in Ob(\\mathcal {D}_H)}$ Since $\\tilde{U}$ is a unit in $M_2(\\mathbf {C})$ , it follows that each $\\eta (X)$ in (REF ) is an automorphism.", "Since $H$ acts trivially on $M_2(\\mathbf {C})$ , we see that $\\eta \\in \\mathbb {U}_H(\\mathbf {C} \\otimes \\mathcal {D}_H \\otimes M_2(\\mathbb {C}))$ .", "Moreover, for any $\\tilde{B} \\otimes f \\in Hom_{M_2(\\mathbf {C}) \\otimes \\mathcal {D}_H}(X,Y)=M_2(\\mathbf {C}) \\otimes Hom_{\\mathcal {D}_H}(X,Y)$ , we have $\\Phi _\\eta (\\tilde{B} \\otimes f)=\\eta (Y) \\circ (\\tilde{B} \\otimes f) \\circ \\eta (X)^{-1}=(\\tilde{U} \\otimes id_Y) \\circ (\\tilde{B} \\otimes f) \\circ (\\tilde{U}^{-1} \\otimes id_X)=\\tilde{U}\\tilde{B}\\tilde{U}^{-1} \\otimes f=\\Xi (\\tilde{B}) \\otimes f$ Therefore, for any $B \\otimes f \\in \\mathbf {C} \\otimes Hom_{\\mathcal {D}_H}(X,Y)$ , we have $\\begin{array}{ll}\\Phi _\\eta ((B \\otimes f) \\otimes E_{11} + \\upsilon (B \\otimes f) \\otimes E_{22})&=\\Phi _\\eta (B \\otimes f \\otimes E_{11} + \\omega (B) \\otimes f \\otimes E_{22})\\\\&= \\Phi _\\eta (B \\otimes E_{11} \\otimes f + \\omega (B) \\otimes E_{22} \\otimes f)\\\\&= \\Xi (B \\otimes E_{11} + \\omega (B) \\otimes E_{22}) \\otimes f\\\\&=\\omega (B) \\otimes E_{22} \\otimes f=\\upsilon (B \\otimes f) \\otimes E_{22}\\end{array}$ This proves the result.", "We are now ready to describe elements in the space $B^n_H(\\mathcal {D}_H,M)$ .", "Theorem 5.10 An element $\\phi \\in C^n_H(\\mathcal {D}_H,M)$ is a coboundary iff $\\phi $ is the character of an $n$ -dimensional vanishing $\\mathcal {S}_H$ -cycle $(\\mathcal {S}_H,\\hat{\\partial }_H, \\hat{{T}}^H,\\rho )$ over $\\mathcal {D}_H$ .", "Let $\\phi $ be the character of an $n$ -dimensional vanishing $\\mathcal {S}_H$ -cycle $(\\mathcal {S}_H,\\hat{\\partial }_H, \\hat{{T}}^H, \\rho )$ .", "By definition, $\\hat{{T}}^H$ is an $n$ -dimensional closed graded $(H,M)$ -trace on the $H$ -semicategory $\\mathcal {S}_H$ and that $\\mathcal {S}_H^0$ is an ordinary $H$ -category.", "We now define $\\psi \\in C^n_H(\\mathcal {S}_H^0,M)$ by setting $\\psi (m \\otimes g^0 \\otimes \\ldots \\otimes g^n):=\\hat{{T}}^H_{X_0}\\big (m \\otimes g^0\\hat{\\partial }_H^0(g^1) \\ldots \\hat{\\partial }_H^0(g^n)\\big )$ for $m \\in M$ and $g^0 \\otimes \\ldots \\otimes g^n \\in Hom_{\\mathcal {S}^0_H}(X_1,X_0) \\otimes Hom_{\\mathcal {S}^0_H}(X_2,X_1) \\otimes \\ldots \\otimes Hom_{\\mathcal {S}^0_H}(X_0,X_n)$ .", "Then, by the implication (1) $\\Rightarrow $ (3) in Theorem REF , we have that $\\psi \\in Z^n_H(\\mathcal {S}_H^0,M)$ .", "Since $HC^n_H(\\mathcal {S}_H^0,M)=0$ , we have that $\\psi =b\\psi ^{\\prime }$ for some $\\psi ^{\\prime } \\in C^{n-1}_H(\\mathcal {S}_H^0,M)$ .", "By Lemma REF , the semifunctor $\\rho \\in Hom_{\\overline{Cat}_H}(\\mathcal {D}_H,\\mathcal {S}_H^0)$ induces a map $C^{n-1}_H(\\rho ,M):C^{n-1}_H(\\mathcal {S}_H^0,M)\\longrightarrow C_H^{n-1}(\\mathcal {D}_H,M)$ .", "Setting $\\psi ^{\\prime \\prime }:=C^{n-1}_H(\\rho ,M)(\\psi ^{\\prime })$ , we have $\\left(\\psi ^{\\prime \\prime }\\right)(m \\otimes p^0 \\otimes \\ldots \\otimes p^{n-1})=\\psi ^{\\prime }\\left(m \\otimes \\rho (p^0) \\otimes \\ldots \\otimes \\rho (p^{n-1})\\right)$ for any $m \\in M$ and $p^0 \\otimes \\ldots \\otimes p^{n-1} \\in CN_{n-1}(\\mathcal {D}_H)$ .", "Therefore, $\\begin{array}{ll}\\phi (m \\otimes f^0 \\otimes \\ldots \\otimes f^n)&=\\hat{{T}}^H_{X_0}\\big (m \\otimes \\rho (f^0)\\hat{\\partial }_H^0\\left(\\rho (f^1)\\right) \\ldots \\hat{\\partial }_H^0\\left(\\rho (f^n)\\right)\\big )=\\psi \\left(m \\otimes \\rho (f^0) \\otimes \\ldots \\otimes \\rho (f^n)\\right)\\\\&= (b\\psi ^{\\prime })\\left(m \\otimes \\rho (f^0) \\otimes \\ldots \\otimes \\rho (f^n)\\right)=(b\\psi ^{\\prime \\prime })(m \\otimes f^0 \\otimes \\ldots \\otimes f^n)\\\\\\end{array}$ for any $m \\in M$ and $f^0 \\otimes \\ldots \\otimes f^n \\in Hom_{\\mathcal {D}_H}(X_1,X_0) \\otimes Hom_{\\mathcal {D}_H}(X_2,X_1) \\otimes \\ldots \\otimes Hom_{\\mathcal {D}_H}(X_0,X_n)$ .", "Thus, $\\phi \\in B^n_H(\\mathcal {D}_H,M)$ .", "Conversely, suppose that $\\phi \\in B^n_H(\\mathcal {D}_H,M)$ .", "Then, $\\phi =b\\psi $ for some $\\psi \\in C^{n-1}_H(\\mathcal {D}_H,M)$ .", "We now extend $\\psi $ to get an element $\\psi ^{\\prime } \\in C^{n-1}_H(\\mathbf {C} \\otimes \\mathcal {D}_H,M)$ as follows: $\\psi ^{\\prime }\\left(m \\otimes (B^0 \\otimes f^0) \\otimes \\ldots \\otimes (B^{n-1} \\otimes f^{n-1})\\right)=\\psi (m \\otimes B^0_{11}f^0 \\otimes \\ldots \\otimes B^{n-1}_{11}f^{n-1})$ We now set $\\phi ^{\\prime }=b\\psi ^{\\prime } \\in Z^n_H(\\mathbf {C} \\otimes \\mathcal {D}_H,M)$ .", "We now consider the $H$ -linear semifunctor $\\rho :\\mathcal {D}_H \\longrightarrow \\mathbf {C} \\otimes \\mathcal {D}_H$ which fixes objects and takes any morphism $f$ to $\\mathbf {1} \\otimes f$ .", "Then, we have $\\begin{array}{ll}\\left(C^n_H(\\rho ,M)(\\phi ^{\\prime })\\right)(m \\otimes f^0 \\otimes \\ldots \\otimes f^n)&=\\phi ^{\\prime }\\left(m \\otimes \\rho (f^0) \\otimes \\ldots \\otimes \\rho (f^n)\\right)=(b\\psi ^{\\prime })\\left(m \\otimes \\rho (f^0) \\otimes \\ldots \\otimes \\rho (f^n)\\right)\\\\&=(b\\psi )(m \\otimes f^0 \\otimes \\ldots \\otimes f^n)=\\phi (m \\otimes f^0 \\otimes \\ldots \\otimes f^n)\\\\\\end{array}$ Since $\\phi ^{\\prime }\\in Z^n_H(\\mathbf {C} \\otimes \\mathcal {D}_H,M)$ , the implication (3) $\\Rightarrow $ (2) in Theorem REF gives us a closed graded $(H,M)$ -trace ${T}^H$ of dimension $n$ on the DGH-semicategory $\\left(\\Omega (\\mathbf {C} \\otimes \\mathcal {D}_H),\\partial _H\\right)$ such that ${T}^H_{X_0}\\left(m \\otimes \\rho (f^0)\\partial _H^0\\left(\\rho (f^1)\\right) \\ldots \\partial _H^0\\left(\\rho (f^n)\\right)\\right)=\\phi ^{\\prime }\\left(m \\otimes \\rho (f^0) \\otimes \\ldots \\otimes \\rho (f^n)\\right)=\\phi (m \\otimes f^0 \\otimes \\ldots \\otimes f^n)$ Since $\\left(\\Omega \\left(\\mathbf {C} \\otimes \\mathcal {D}_H\\right)\\right)^0=\\mathbf {C}\\otimes \\mathcal {D}_H$ is a left $H$ -category, we see that $\\phi $ is the character associated to the cycle $\\left(\\Omega \\left(\\mathbf {C} \\otimes \\mathcal {D}_H\\right),\\partial _H,{T}^H, {\\rho } \\right)$ over $\\mathcal {D}_H$ .", "From the proof of Lemma REF , we know that $\\mathbf {C} \\otimes \\mathcal {D}_H$ satisfies the assumptions in Proposition REF .", "Hence, $\\left(\\Omega \\left(\\mathbf {C} \\otimes \\mathcal {D}_H\\right),\\partial _H,{T}^H, \\rho \\right)$ is a vanishing cycle over $\\mathcal {D}_H$ .", "From this, the result follows.", "For the remaining part of this section, we shall suppose that $H$ is cocommutative.", "If $\\mathcal {D}_H$ , $\\mathcal {D}_H^{\\prime }$ are left $H$ -categories, we observe that $\\mathcal {D}_H \\otimes \\mathcal {D}_H^{\\prime }$ then becomes a left $H$ -category under the diagonal action of $H$ .", "Let $M$ , $M^{\\prime }$ be left $H$ -comodules equipped respectively with coactions $\\rho _M:M\\longrightarrow H\\otimes M$ and $\\rho _{M^{\\prime }}:M^{\\prime }\\longrightarrow H\\otimes M^{\\prime }$ .", "Since $H$ is cocommutative, $M$ may be treated as a right $H$ -comodule and we can form the cotensor product $M \\square _H M^{\\prime }$ defined by the kernel ${\\begin{matrix}M \\square _H M^{\\prime } :=Ker\\left(M \\otimes M^{\\prime }\\right.", "&\\xrightarrow{}& \\left.", "M \\otimes H \\otimes M^{\\prime }\\right)\\\\\\end{matrix}}$ in $Vect_k$ .", "It follows by that the map $\\rho _M \\otimes id_{M^{\\prime }}$ gives $M \\square _H M^{\\prime }$ a left $H$ -comodule structure.", "We also note that $M\\otimes M^{\\prime }$ carries a right $H$ -module structure via the diagonal action.", "Lemma 5.11 Let $H$ be a cocommutative Hopf algebra and $M$ , $M^{\\prime }$ be right-left SAYD modules over $H$ such that $M \\square _H M^{\\prime }$ is a right $H$ -submodule of $M \\otimes M^{\\prime }$ .", "Then, $M \\square _H M^{\\prime }$ is also an SAYD module over $H$ .", "For any $m \\otimes m^{\\prime } \\in M \\square _H M^{\\prime }$ , we have $&\\left((m \\otimes m^{\\prime })h\\right)_{(-1)} \\otimes \\left((m \\otimes m^{\\prime })h\\right)_{(0)}=(mh_1 \\otimes m^{\\prime }h_2)_{(-1)} \\otimes (mh_1 \\otimes m^{\\prime }h_2)_{(0)}= (mh_1)_{(-1)} \\otimes (mh_1)_{(0)} \\otimes m^{\\prime }h_2\\\\&=S(h_{13})m_{(-1)}h_{11} \\otimes m_{(0)}h_{12} \\otimes m^{\\prime }h_2=S(h_{3})m_{(-1)}h_{1} \\otimes m_{(0)}h_{2} \\otimes m^{\\prime }h_4$ On the other hand, we have $&S(h_3)(m \\otimes m^{\\prime })_{(-1)}h_1 \\otimes (m \\otimes m^{\\prime })_{(0)}h_2=S(h_3)m_{(-1)}h_1 \\otimes (m_{(0)} \\otimes m^{\\prime })h_2=S(h_3)m_{(-1)}h_1 \\otimes m_{(0)}h_{21} \\otimes m^{\\prime }h_{22}\\\\&=S(h_4)m_{(-1)}h_1 \\otimes m_{(0)}h_{2} \\otimes m^{\\prime }h_{3}$ Since $H$ is cocommutative, we see that the two expressions are the same.", "This proves that $M \\square _H M^{\\prime }$ is an anti-Yetter-Drinfeld module.", "We now check that it is also stable.", "Using the cocommutativity of $H$ and the stability of $M$ , $M^{\\prime }$ , we have $(m \\otimes m^{\\prime })_{(0)}(m \\otimes m^{\\prime })_{(-1)}=m_0m_1\\otimes m^{\\prime }m_2= m_{00}m_{01}\\otimes m^{\\prime }m_1=m_0\\otimes m^{\\prime }m_1=m\\otimes m^{\\prime }_0m^{\\prime }_{-1}=m\\otimes m^{\\prime }$ for any $m \\otimes m^{\\prime } \\in M \\square _H M^{\\prime }$ .", "Let $(\\mathcal {S}_H,\\hat{\\partial }_H)$ and $(\\mathcal {S}^{\\prime }_H,\\hat{\\partial }^{\\prime }_H)$ be DGH-semicategories.", "Then, their tensor product $\\mathcal {S}_H \\otimes \\mathcal {S}^{\\prime }_H$ is the DG-semicategory defined by setting $Ob( \\mathcal {S}_H \\otimes \\mathcal {S}^{\\prime }_H)=Ob(\\mathcal {S}_H) \\times Ob(\\mathcal {S}^{\\prime }_H)$ and $Hom_{\\mathcal {S}_H \\otimes \\mathcal {S}^{\\prime }_H}^n\\left((X,X^{\\prime }),(Y,Y^{\\prime })\\right)=\\bigoplus \\limits _{i+j=n}Hom^i_{\\mathcal {S}_H}(X,Y) \\otimes _k Hom^j_{\\mathcal {S}^{\\prime }_H}(X^{\\prime },Y^{\\prime })$ The composition in $ \\mathcal {S}_H \\otimes \\mathcal {S}^{\\prime }_H$ is given by the rule: $(g \\otimes g^{\\prime })\\circ (f \\otimes f^{\\prime })=(-1)^{deg(g^{\\prime }) deg(f)} (gf \\otimes g^{\\prime }f^{\\prime })$ for homogeneous $f:X \\longrightarrow Y$ , $g: Y \\longrightarrow Z$ in $\\mathcal {S}_H$ and $f^{\\prime }:X^{\\prime } \\longrightarrow Y^{\\prime }$ , $g^{\\prime }: Y^{\\prime } \\longrightarrow Z^{\\prime }$ in $\\mathcal {S}^{\\prime }_H$ .", "The differential $(\\hat{\\partial }_H\\otimes \\hat{\\partial }^{\\prime }_H)^n :Hom_{\\mathcal {S}_H \\otimes \\mathcal {S}^{\\prime }_H}^n\\left((X,X^{\\prime }),(Y,Y^{\\prime })\\right) \\longrightarrow Hom_{\\mathcal {S}_H \\otimes \\mathcal {S}^{\\prime }_H}^{n+1}\\left((X,X^{\\prime }),(Y,Y^{\\prime })\\right)$ is determined by $(\\hat{\\partial }_H\\otimes \\hat{\\partial }^{\\prime }_H)^n(f_i \\otimes g_j)=\\hat{\\partial }^i_{H}(f_i) \\otimes g_j + (-1)^i f_i \\otimes {\\hat{\\partial ^{\\prime }}}^{j}_{H}(g_j)$ for any $f_i \\in Hom^i_{\\mathcal {S}_H}(X,Y)$ and $g_j \\in Hom^j_{\\mathcal {S}^{\\prime }_H}(X^{\\prime },Y^{\\prime })$ such that $i+j=n$ .", "Clearly, $(\\mathcal {S}_H \\otimes \\mathcal {S}^{\\prime }_H)^0=\\mathcal {S}^0_H \\otimes \\mathcal {S}^{\\prime 0}_H$ .", "Theorem 5.12 Let $H$ be a cocommutative Hopf algebra and $M$ , $M^{\\prime }$ be right-left SAYD modules over $H$ such that $M \\square _H M^{\\prime }$ is a right $H$ -submodule of $M \\otimes M^{\\prime }$ .", "Let $\\mathcal {D}_H$ , $\\mathcal {D}_H^{\\prime }$ be left $H$ -categories.", "Then, we have a pairing $HC^p_H(\\mathcal {D}_H,M) \\otimes HC^q_H(\\mathcal {D}^{\\prime }_H,M^{\\prime }) \\longrightarrow HC^{p+q}_H(\\mathcal {D}_H \\otimes \\mathcal {D}^{\\prime }_H,M \\square _H M^{\\prime })$ for $p$ , $q\\ge 0$ .", "Let $\\phi \\in Z^p_H(\\mathcal {D}_H,M)$ and $\\phi ^{\\prime } \\in Z^q_H(\\mathcal {D}^{\\prime }_H,M)$ .", "We may express $\\phi $ and $\\phi ^{\\prime }$ respectively as the characters of $p$ and $q$ -dimensional cycles $(\\mathcal {S}_H,\\hat{\\partial }_H, \\hat{T}^H,\\rho )$ and $(\\mathcal {S}^{\\prime }_H,\\hat{\\partial }^{\\prime }_H,\\hat{T}^{\\prime H},\\rho ^{\\prime })$ over $\\mathcal {D}_H$ and $\\mathcal {D}^{\\prime }_H$ with coefficients in $M$ and $M^{\\prime }$ respectively.", "We now consider the collection $\\hat{T^H} \\# \\hat{T}^{\\prime H}:=\\lbrace {(\\hat{T}^H \\# \\hat{T}^{\\prime H})}_{(X,X^{\\prime })}:M \\square _H M^{\\prime } \\otimes Hom^{p+q}_{\\mathcal {S}_H\\otimes \\mathcal {S}^{\\prime }_H}\\left((X,X^{\\prime }),(X,X^{\\prime })\\right) \\longrightarrow \\mathbb {C}\\rbrace _{(X,X^{\\prime }) \\in Ob\\left(\\mathcal {S}_H\\otimes \\mathcal {S}^{\\prime }_H\\right)}$ of $\\mathbb {C}$ -linear maps defined by ${(\\hat{T}^H \\# \\hat{T}^{\\prime H})}_{(X,X^{\\prime })}(m \\otimes m^{\\prime } \\otimes f \\otimes f^{\\prime }):=\\hat{T}^H_X(m \\otimes f_p)\\hat{T}^{\\prime H}_{X^{\\prime }}(m^{\\prime } \\otimes f^{\\prime }_q)$ for any $m \\otimes m^{\\prime } \\in M \\square _H M^{\\prime }$ and $f \\otimes f^{\\prime }= (f_i \\otimes f^{\\prime }_j)_{i+j=p+q} \\in Hom^{p+q}_{\\mathcal {S}_H \\otimes \\mathcal {S}^{\\prime }_H}\\left((X,X^{\\prime }),(X,X^{\\prime })\\right)$ .", "We will now prove that $\\hat{T}^H \\# \\hat{T}^{\\prime H}$ is a $(p+q)$ -dimensional closed graded trace on the DGH-semicategory $\\mathcal {S}_H\\otimes \\mathcal {S}^{\\prime }_H$ with coefficients in $M \\square _H M^{\\prime }$ .", "For any $m \\otimes m^{\\prime } \\in M \\square _H M^{\\prime }$ and $g \\otimes g^{\\prime }= (g_i \\otimes g^{\\prime }_j)_{i+j=p+q-1} \\in Hom^{p+q-1}_{\\mathcal {S}_H \\otimes \\mathcal {S}^{\\prime }_H}\\left((X,X^{\\prime }),(X,Y)\\right)$ , we have $\\begin{array}{ll}&(\\hat{T}^H \\# \\hat{T}^{\\prime H})_{(X,X^{\\prime })}\\left(m \\otimes m^{\\prime } \\otimes (\\hat{\\partial }_H \\otimes \\hat{\\partial }^{\\prime }_H)^{p+q-1}(g \\otimes g^{\\prime })\\right)\\\\&=\\underset{i+j=p+q-1}{\\sum }(\\hat{T}^H \\# \\hat{T}^{\\prime H})_{(X,X^{\\prime })}\\left(m \\otimes m^{\\prime } \\otimes \\hat{\\partial }^i_{H}(g_i) \\otimes g^{\\prime }_j + (-1)^{i} m \\otimes m^{\\prime } \\otimes g_i \\otimes \\hat{\\partial ^{\\prime }}^j_H(g^{\\prime }_j) \\right)\\\\&=\\hat{T}^H_X(m \\otimes \\hat{\\partial }^{p-1}_{H}(g_{p-1}))\\hat{T}^{\\prime H}_{X^{\\prime }}(m^{\\prime } \\otimes g^{\\prime }_q)+(-1)^{p}\\hat{T}^H_X(m \\otimes g_p)\\hat{T}^{\\prime H}_{X^{\\prime }}(m^{\\prime } \\otimes \\hat{\\partial ^{\\prime }}^{q-1}_{H}(g^{\\prime }_{q-1}))=0\\end{array}$ This proves the condition in ().", "Next for any homogeneous $f:X \\longrightarrow Y$ , $g: Y \\longrightarrow X$ in $\\mathcal {S}_H$ and $f^{\\prime }:X^{\\prime } \\longrightarrow Y^{\\prime }$ , $g^{\\prime }: Y^{\\prime } \\longrightarrow X^{\\prime }$ in $\\mathcal {S}^{\\prime }_H$ , we have $\\begin{array}{ll}&(\\hat{T}^H \\# \\hat{T}^{\\prime H})_{(X,X^{\\prime })}\\left(m \\otimes m^{\\prime } \\otimes (g \\otimes g^{\\prime })(f \\otimes f^{\\prime })\\right)\\\\&=(-1)^{deg(g^{\\prime })deg(f)}(\\hat{T}^H \\# \\hat{T}^{\\prime H})_{(X,X^{\\prime })}(m \\otimes m^{\\prime } \\otimes gf \\otimes g^{\\prime }f^{\\prime })\\\\&=(-1)^{deg(g^{\\prime })deg(f)}\\hat{T}^H_X(m \\otimes (gf)_p)\\hat{T}^{\\prime H}_{X^{\\prime }}(m^{\\prime } \\otimes (g^{\\prime }f^{\\prime })_q)\\\\&=(-1)^{deg(g^{\\prime })deg(f)}(-1)^{deg(g)deg(f)}(-1)^{deg(g^{\\prime })deg(f^{\\prime })}\\hat{T}^H_Y(m \\otimes (fg)_p)\\hat{T}^{\\prime H}_{Y^{\\prime }}(m^{\\prime } \\otimes (f^{\\prime }g^{\\prime })_q)\\\\&=(-1)^{deg(g^{\\prime })deg(f)}(-1)^{deg(g)deg(f)}(-1)^{deg(g^{\\prime })deg(f^{\\prime })}(-1)^{deg(g)deg(f^{\\prime })}(\\hat{T}^H \\#\\hat{T}^{\\prime H})_{(Y,Y^{\\prime })}\\left(m \\otimes m^{\\prime } \\otimes (f \\otimes f^{\\prime })(g \\otimes g^{\\prime })\\right)\\\\&=(-1)^{deg(g \\otimes g^{\\prime })deg(f \\otimes f^{\\prime })}(\\hat{T}^H \\# \\hat{T}^{\\prime H})_{(Y,Y^{\\prime })}\\left(m \\otimes m^{\\prime } \\otimes (f \\otimes f^{\\prime })(g \\otimes g^{\\prime })\\right)\\end{array}$ This proves the condition in ().", "We may similarly verify the condition in (REF ).", "Thus, we get a $(p+q)$ -dimensional cycle $\\left(\\mathcal {S}_H \\otimes \\mathcal {S}^{\\prime }_H, \\hat{\\partial }_H\\otimes \\hat{\\partial }^{\\prime }_H, \\hat{T}^H\\#\\hat{T}^{\\prime H}, \\rho \\otimes \\rho ^{\\prime }\\right)$ with coefficients in $M \\square _H M^{\\prime }$ over the category $\\mathcal {D}_H \\otimes \\mathcal {D}^{\\prime }_H$ .", "Then, the character of this cycle, denoted by $\\phi \\# \\phi ^{\\prime } \\in Z^{p+q}_H(\\mathcal {D}_H \\otimes \\mathcal {D}^{\\prime }_H, M \\square _H M^{\\prime })$ , gives a well defined map $\\gamma :Z^p_H(\\mathcal {D}_H,M) \\otimes Z^q_H(\\mathcal {D}^{\\prime }_H,M^{\\prime }) \\longrightarrow Z^{p+q}_H(\\mathcal {D}_H \\otimes \\mathcal {D}^{\\prime }_H, M \\square _H M^{\\prime })$ .", "We now verify that the map $\\gamma $ restricts to a pairing $\\begin{array}{ll}B^p_H(\\mathcal {D}_H,M) \\otimes Z^q_H(\\mathcal {D}^{\\prime }_H,M^{\\prime }) \\longrightarrow B^{p+q}_H(\\mathcal {D}_H \\otimes \\mathcal {D}^{\\prime }_H, M \\square _H M^{\\prime })\\end{array}$ For this, we let $\\phi \\in Z^p_H(\\mathcal {D}_H,M)$ be the character of a $p$ -dimensional vanishing cycle $\\left(\\mathcal {S}_H,\\hat{\\partial }_H,\\hat{T}^H, \\rho \\right)$ over $\\mathcal {D}_H$ .", "In particular, it follows from Definition REF that $\\mathcal {S}^0_H$ is an ordinary left $H$ -category.", "From the implication (1) $\\Rightarrow $ (2) in Theorem REF , it follows that we might as well take $\\mathcal {S}^{\\prime 0}_H$ to be an ordinary left $H$ -category.", "In fact, we could assume that $\\mathcal {S}^{\\prime }_H=\\Omega \\mathcal {D}^{\\prime }_H$ .", "Then, $\\mathcal {S}^0_H\\otimes \\mathcal {S}^{\\prime 0}_H$ is an ordinary left $H$ -category.", "It suffices to show that the tuple $\\left(\\mathcal {S}_H \\otimes \\mathcal {S}^{\\prime }_H, {\\hat{\\partial }}_H \\otimes {\\hat{\\partial }^{\\prime }}_H, \\hat{T}^H\\#\\hat{T}^{\\prime H}, \\rho \\otimes \\rho ^{\\prime }\\right)$ is a vanishing cycle.", "Since $\\left(\\mathcal {S}_H,{\\hat{\\partial }}_H,\\hat{T}^H\\right)$ is a vanishing cycle, we have an $H$ -linear semifunctor $\\upsilon : \\mathcal {S}^0_H \\longrightarrow \\mathcal {S}^0_H$ and an $\\eta \\in \\mathbb {U}(\\mathcal {S}^0_H \\otimes M_2(\\mathbb {C}))$ satisfying the conditions in Proposition REF .", "Extending $\\upsilon $ , we get the the $H$ -linear semifunctor $\\upsilon \\otimes id: \\mathcal {S}^0_H \\otimes \\mathcal {S}^{\\prime 0}_H \\longrightarrow \\mathcal {S}^0_H \\otimes \\mathcal {S}^{\\prime 0}_H$ .", "Identifying, $\\mathcal {S}^0_H \\otimes \\mathcal {S}^{\\prime 0}_H \\otimes M_2(\\mathbb {C}) \\cong \\mathcal {S}^0_H \\otimes M_2(\\mathbb {C}) \\otimes \\mathcal {S}^{\\prime 0}_H$ , we obtain $\\tilde{\\eta } \\in \\mathbb {U}(\\mathcal {S}^0_H \\otimes M_2(\\mathbb {C}) \\otimes \\mathcal {S}^{\\prime 0}_H)$ given by $\\lbrace \\tilde{\\eta }(X,X^{\\prime })=\\eta (X) \\otimes id_{X^{\\prime }} \\in Hom_{\\mathcal {S}^0_H \\otimes M_2(\\mathbb {C}) \\otimes \\mathcal {S}^{\\prime 0}_H}((X,X^{\\prime }),(X,X^{\\prime }))=Hom_{\\mathcal {S}^0_H \\otimes M_2(\\mathbb {C})}(X,X) \\otimes Hom_{\\mathcal {S}^{\\prime 0}_H}(X^{\\prime },X^{\\prime })\\rbrace $ It may also be easily verified that $\\Phi _{\\tilde{\\eta }}(f \\otimes f^{\\prime } \\otimes E_{11} + (\\upsilon \\otimes id)(f \\otimes f^{\\prime }) \\otimes E_{22})=(\\upsilon \\otimes id)(f \\otimes f^{\\prime }) \\otimes E_{22}$ Thus, we see that the category $(\\mathcal {S}_H \\otimes \\mathcal {S}^{\\prime }_H)^0=\\mathcal {S}^0_H \\otimes \\mathcal {S}^{\\prime 0}_H$ satisfies the conditions in Proposition REF .", "Therefore, the tuple $\\left(\\mathcal {S}_H \\otimes \\mathcal {S}^{\\prime }_H, {\\hat{\\partial }}_H \\otimes {\\hat{\\partial }^{\\prime }}_H, \\hat{T}^H\\#\\hat{T}^{\\prime H}, \\rho \\otimes \\rho ^{\\prime }\\right)$ is a vanishing cycle.", "This proves the result." ], [ "Characters of Fredholm modules over categories", "In the rest of this paper, we will study Fredholm modules and Chern characters.", "We fix a small $\\mathbb {C}$ -linear category $\\mathcal {C}$ .", "Our categorified Fredholm modules will consist of functors from $\\mathcal {C}$ taking values in separable Hilbert spaces.", "Let $SHilb$ be the category whose objects are separable Hilbert spaces and whose morphisms are bounded linear maps.", "Given separable Hilbert spaces $\\mathcal {H}_1$ and $\\mathcal {H}_2$ , let $\\mathcal {B}(\\mathcal {H}_1,\\mathcal {H}_2)$ denote the space of all bounded linear operators from $\\mathcal {H}_1$ to $\\mathcal {H}_2$ and $\\mathcal {B}^\\infty (\\mathcal {H}_1,\\mathcal {H}_2) \\subseteq \\mathcal {B}(\\mathcal {H}_1,\\mathcal {H}_2)$ be the space of all compact operators.", "For any bounded operator $T \\in \\mathcal {B}(\\mathcal {H}_1,\\mathcal {H}_2)$ , let $\\mu _n(T)$ denote the $n$ -th singular value of $T$ .", "In other words, $\\mu _n(T)$ is the $n$ -th (arranged in decreasing order) eigenvalue of the positive operator $|T|:=(T^*T)^{\\frac{1}{2}}$ .", "For $1 \\le p < \\infty $ , the $p$ -th Schatten class is defined to be the space $\\mathcal {B}^p(\\mathcal {H}_1,\\mathcal {H}_2):=\\lbrace T \\in \\mathcal {B}(\\mathcal {H}_1,\\mathcal {H}_2)~|~ \\sum \\mu _n(T)^p < \\infty \\rbrace $ Clearly, $\\mathcal {B}^p(\\mathcal {H}_1,\\mathcal {H}_2) \\subseteq \\mathcal {B}^q(\\mathcal {H}_1,\\mathcal {H}_2)$ for $p \\le q$ .", "For $p=1$ , the space $\\mathcal {B}^1(\\mathcal {H}_1,\\mathcal {H}_2)$ is the collection of all trace class operators from $\\mathcal {H}_1$ to $\\mathcal {H}_2$ .", "For $T \\in \\mathcal {B}^1(\\mathcal {H}_1,\\mathcal {H}_2)$ , we write $Tr(T):=\\sum \\mu _n(T)$ .", "It is well known that $Tr(T_1T_2)=Tr(T_2T_1) \\quad \\forall T_1 \\in \\mathcal {B}^{n_1}(\\mathcal {H}, \\mathcal {H}^{\\prime }),~ T_2 \\in \\mathcal {B}^{n_2}(\\mathcal {H}^{\\prime }, \\mathcal {H}), \\frac{1}{n_1}+\\frac{1}{n_2}=1$ We note that $\\mathcal {B}^p(\\mathcal {H}_1,\\mathcal {H}_2)$ is an “ideal\" in the following sense: consider the functor $\\begin{array}{c} \\mathcal {B}(-,-):SHilb^{op} \\otimes SHilb \\longrightarrow Vect_\\mathbb {C} \\qquad \\mathcal {B}(-,-)(\\mathcal {H}_1,\\mathcal {H}_2):=\\mathcal {B}(\\mathcal {H}_1,\\mathcal {H}_2)\\\\ \\mathcal {B}(-,-)(\\phi _1,\\phi _2):\\mathcal {B}(\\mathcal {H}_1, \\mathcal {H}_2) \\longrightarrow \\mathcal {B}(\\mathcal {H}_1^{\\prime },\\mathcal {H}_2^{\\prime })\\qquad T \\mapsto \\phi _2T\\phi _1\\\\\\end{array}$ taking values in the category $Vect_{\\mathbb {C}}$ of $\\mathbb {C}$ -vector spaces.", "Then, $\\mathcal {B}^p(-,-)$ is a subfunctor of $\\mathcal {B}(-,-)$ .", "In other words, for morphisms $\\phi _1:\\mathcal {H}_1^{\\prime }\\longrightarrow \\mathcal {H}_1$ , $\\phi _2:\\mathcal {H}_2\\longrightarrow \\mathcal {H}_2^{\\prime }$ and any $T \\in \\mathcal {B}^p(\\mathcal {H}_1,\\mathcal {H}_2)$ , we have $\\phi _2T\\phi _1 \\in \\mathcal {B}^p(\\mathcal {H}_1^{\\prime },\\mathcal {H}_2^{\\prime })$ .", "We fix the following convention for the commutator notation: Let ${H}:\\mathcal {C} \\longrightarrow SHilb$ be a functor and $\\mathcal {G}:=\\lbrace \\mathcal {G}_X:{H}(X) \\longrightarrow {H}(X)\\rbrace _{X \\in Ob(\\mathcal {C})}$ be a collection of bounded linear operators.", "Then, we set $[\\mathcal {G},-]:\\mathcal {B}(H(X),H(Y))\\longrightarrow \\mathcal {B}(H(X),H(Y)) \\qquad [\\mathcal {G},T]:= \\mathcal {G}_Y \\circ T - T\\circ \\mathcal {G}_X\\in \\mathcal {B}(H(X),H(Y))$ We now let $SHilb_{\\mathbb {Z}_2}$ be the category whose objects are $\\mathbb {Z}_2$ -graded separable Hilbert spaces and whose morphims are bounded linear maps.", "Let ${H}:\\mathcal {C} \\longrightarrow SHilb_{\\mathbb {Z}_2}$ be a functor and $\\mathcal {G}:=\\lbrace \\mathcal {G}_X:{H}(X) \\longrightarrow {H}(X)\\rbrace _{X \\in Ob(\\mathcal {C})}$ be a collection of bounded linear operators of the same degree $|\\mathcal {G}|$ .", "Then, we set $\\begin{array}{c}[\\mathcal {G},-]:\\mathcal {B}(H(X),H(Y))\\longrightarrow \\mathcal {B}(H(X),H(Y)) \\\\\\mbox{$[\\mathcal {G},T]:= \\mathcal {G}_Y \\circ T -(-1)^{|\\mathcal {G}||T|} T\\circ \\mathcal {G}_X\\in \\mathcal {B}(H(X),H(Y))$}\\\\\\end{array}$ for each $X$ , $Y\\in \\mathcal {C}$ .", "Definition 6.1 Let $\\mathcal {C}$ be a small $\\mathbb {C}$ -category and let $p \\in [1,\\infty )$ .", "We consider a pair $(H,\\mathcal {F})$ as follows.", "(1) A functor ${H}:\\mathcal {C} \\longrightarrow SHilb_{\\mathbb {Z}_2}$ such that $H(f):H(X)\\longrightarrow H(Y)$ is a linear operator of degree 0 for each $f \\in Hom_{\\mathcal {C}}(X,Y)$ .", "(2) A collection $\\mathcal {F}:=\\lbrace \\mathcal {F}_X:{H}(X) \\longrightarrow {H}(X)\\rbrace _{X \\in Ob(\\mathcal {C})}$ of bounded linear operators of degree 1 such that $\\mathcal {F}_X^2=id_{{H}(X)}$ for each $X \\in Ob(\\mathcal {C})$ .", "The pair $({H},\\mathcal {F})$ is said to be a $p$ -summable even Fredholm module over the category $\\mathcal {C}$ if every $f \\in Hom_{\\mathcal {C}}(X,Y)$ satisfies $[\\mathcal {F},f]:= \\left(\\mathcal {F}_Y \\circ {H}(f) - {H}(f) \\circ \\mathcal {F}_X \\right) \\in \\mathcal {B}^p\\left({H}(X),{H}(Y)\\right)$ Taking $H=\\mathbb {C}=M$ in Definition REF , we note that a closed graded trace of dimension $n$ on a DG-semicategory $(\\mathcal {S},\\hat{\\partial })$ is a collection of $\\mathbb {C}$ -linear maps $\\hat{T}:=\\lbrace \\hat{T}_X: Hom^n_\\mathcal {S}(X,X) \\longrightarrow \\mathbb {C}\\rbrace _{X \\in Ob(\\mathcal {S})}$ satisfying the following two conditions $\\hat{T}_X\\left(\\hat{\\partial }^{n-1}(f)\\right)=0\\qquad \\hat{T}_X(gg^{\\prime })=(-1)^{ij}~ \\hat{T}_Y(g^{\\prime }g)$ for all $f \\in Hom^{n-1}_{\\mathcal {S}}(X,X)$ , $g \\in Hom^i_{\\mathcal {S}}(Y,X)$ , $g^{\\prime } \\in Hom^j_{\\mathcal {S}}(X,Y)$ and $i+j=n$ .", "Accordingly, we will consider cycles $(\\mathcal {S},\\hat{\\partial },\\hat{T},\\rho )$ over $\\mathcal {C}$ by setting $H=\\mathbb {C}=M$ in Definition REF .", "Let $({H},\\mathcal {F})$ be a pair that satisfies conditions (1) and (2) in Definition REF .", "We define a graded-semicategory $\\Omega ^{\\prime }\\mathcal {C}=\\Omega _{(H,\\mathcal {F})}\\mathcal {C}$ as follows: we put $Ob(\\Omega ^{\\prime }\\mathcal {C}):=Ob(\\mathcal {C})$ and for any $X$ , $Y\\in \\mathcal {C}$ , $j\\ge 0$ , we set $Hom^j_{\\Omega ^{\\prime }{\\mathcal {C}}}(X,Y)$ to be the linear span in $\\mathcal {B}(H(X),H(Y))$ of the operators $H(\\tilde{f}^0)[\\mathcal {F},f^1][\\mathcal {F},f^2] \\ldots [\\mathcal {F},f^{j}]$ where $\\tilde{f}^0\\otimes f^1\\otimes ...\\otimes f^j$ is a homogeneous element of degree $j$ in $Hom_{\\Omega \\mathcal {C}}(X,Y)$ .", "Here, we write $H(\\tilde{f}^0)=H({f}^0)+\\mu \\cdot id$ , where $\\tilde{f}^0=f^0+\\mu $ .", "Using the fact that $[\\mathcal {F},f]H(f^{\\prime })=[\\mathcal {F},f\\circ f^{\\prime }]-H(f)[\\mathcal {F},f^{\\prime }]$ for composable morphisms $f$ , $f^{\\prime }$ in $\\mathcal {C}$ , we observe that $\\Omega ^{\\prime }\\mathcal {C}$ is closed under composition.", "We set $\\begin{array}{c}\\partial ^{\\prime }:=[\\mathcal {F},-]: \\mathcal {B}\\left({H}(X),{H}(Y)\\right) \\longrightarrow \\mathcal {B}\\left({H}(X),{H}(Y)\\right)\\\\\\partial ^{\\prime } T=[\\mathcal {F},T]=\\mathcal {F}_Y \\circ T -(-1)^{|T|} T\\circ \\mathcal {F}_X\\\\\\end{array}$ We now have the following Lemma.", "Lemma 6.2 Let $({H},\\mathcal {F})$ be a pair that satisfies conditions (1) and (2) in Definition REF .", "Then, (a) $(\\Omega ^{\\prime }\\mathcal {C},\\partial ^{\\prime })$ is a DG-semicategory and $\\Omega ^{\\prime 0}\\mathcal {C}$ is an ordinary category.", "(b) There is a canonical semifunctor $\\rho ^{\\prime }=\\rho _{H}:\\mathcal {C}\\longrightarrow \\Omega ^{\\prime 0}\\mathcal {C}$ which is identity on objects and takes any $f\\in Hom_{\\mathcal {C}}(X,Y)$ to $H(f)\\in \\mathcal {B}(H(X),H(Y))$ .", "This extends to a unique DG-semifunctor $\\hat{\\rho }^{\\prime }=\\hat{\\rho }_{H}:(\\Omega \\mathcal {C},\\partial )\\longrightarrow (\\Omega ^{\\prime }\\mathcal {C},\\partial ^{\\prime })$ such that the restriction of $\\hat{\\rho }^{\\prime }$ to $\\mathcal {C}$ is identical to $\\rho ^{\\prime }$ .", "(c) Suppose that $({H},\\mathcal {F})$ is a $p$ -summable Fredholm module.", "Choose $n\\ge p-1$ .", "Then, for $X$ , $Y\\in Ob(\\mathcal {C})$ and $k\\ge 0$ , we have $Hom^k_{\\Omega ^{\\prime }{\\mathcal {C}}}(X,Y)\\subseteq \\mathcal {B}^{(n+1)/k}(H(X),H(Y))$ .", "(a) Since each $\\mathcal {F}_X$ is a degree 1 operator and $\\mathcal {F}_Y[\\mathcal {F},f]=-[\\mathcal {F},f]\\mathcal {F}_X$ for any $f\\in Hom_{\\mathcal {C}}(X,Y)$ , we have $\\partial ^{\\prime }\\left(Hom^j_{\\Omega ^{\\prime } \\mathcal {C}}(X,Y)\\right) \\subseteq Hom^{j+1}_{\\Omega ^{\\prime } \\mathcal {C}}(X,Y)$ .", "We now check that $\\partial ^{\\prime 2}=0$ .", "For any homogeneous element $H(\\tilde{f}^0)[\\mathcal {F},f^1][\\mathcal {F},f^2] \\ldots [\\mathcal {F},f^{j}]$ of degree $j$ , we have $\\begin{array}{ll}\\partial ^{\\prime 2}\\left(H(\\tilde{f}^0)[\\mathcal {F},f^1][\\mathcal {F},f^2] \\ldots [\\mathcal {F},f^{j}]\\right)&=\\partial ^{\\prime }\\left(\\mathcal {F}_Y\\circ \\left(H(\\tilde{f}^0)[\\mathcal {F},f^1][\\mathcal {F},f^2] \\ldots [\\mathcal {F},f^{j}]\\right)\\right)\\\\&\\textrm { }-(-1)^j \\partial ^{\\prime } \\left(\\left(H(\\tilde{f}^0)[\\mathcal {F},f^1][\\mathcal {F},f^2] \\ldots [\\mathcal {F},f^{j}]\\right)\\circ \\mathcal {F}_X\\right)\\\\& =\\mathcal {F}_Y^2 \\circ H(\\tilde{f}^0)[\\mathcal {F},f^1][\\mathcal {F},f^2] \\ldots [\\mathcal {F},f^{j}]\\\\& \\quad -(-1)^{j+1} \\mathcal {F}_Y \\circ H(\\tilde{f}^0)[\\mathcal {F},f^1][\\mathcal {F},f^2] \\ldots [\\mathcal {F},f^{j}]\\circ \\mathcal {F}_X \\\\& \\quad - (-1)^j \\big ( \\mathcal {F}_Y \\circ H(\\tilde{f}^0)[\\mathcal {F},f^1][\\mathcal {F},f^2] \\ldots [\\mathcal {F},f^{j}]\\circ \\mathcal {F}_X \\\\&\\quad - (-1)^{j+1} H(\\tilde{f}^0)[\\mathcal {F},f^1][\\mathcal {F},f^2] \\ldots [\\mathcal {F},f^{j}]\\circ \\mathcal {F}_X^2 \\big )\\\\& =0\\end{array}$ The fact that $\\partial ^{\\prime }$ is compatible with composition follows by direct computation.", "It is also easy to see that $\\Omega ^{\\prime 0}\\mathcal {C}$ is an ordinary category.", "(b) This is immediate using the universal property in Proposition REF .", "(c) This is a consequence of Hölder's inequality and the condition (REF ) in Definition REF .", "For any $\\mathbb {Z}_2$ -graded Hilbert space $\\mathcal {H}$ , the grading operator on it will be denoted by $\\epsilon _{\\mathcal {H}}$ or simply $\\epsilon $ .", "For any $T \\in \\mathcal {B}(H(X), H(Y))$ such that $[\\mathcal {F},T] \\in \\mathcal {B}^1(H(X), H(Y))$ , we define $Tr_s(T):=\\frac{1}{2}~Tr\\left(\\epsilon \\mathcal {F}_Y[\\mathcal {F},T]\\right)=\\frac{1}{2}~Tr\\left(\\epsilon \\mathcal {F}_Y\\partial ^{\\prime }(T)\\right)=\\frac{1}{2}~ Tr\\left(\\epsilon \\mathcal {F}_Y(\\mathcal {F}_Y \\circ T - (-1)^{|T|}~ T \\circ \\mathcal {F}_X)\\right)$ Proposition 6.3 Let $({H},\\mathcal {F})$ be a $p$ -summable Fredholm module over $\\mathcal {C}$ .", "Take $ 2m \\ge p-1$ .", "Then, the collection $\\hat{Tr}_s=\\lbrace Tr_s:Hom^{2m}_{\\Omega ^{\\prime }\\mathcal {C}}(X,X) \\longrightarrow \\mathbb {C}\\rbrace _{X\\in Ob(\\mathcal {C})}$ defines a closed graded trace of dimension $2m$ on $(\\Omega ^{\\prime }\\mathcal {C},\\partial ^{\\prime })$ .", "From Lemma REF (a), it is clear that for any $T\\in Hom^{2m}_{\\Omega ^{\\prime }\\mathcal {C}}(X,X)$ , we have $[\\mathcal {F},T]\\in Hom^{2m+1}_{\\Omega ^{\\prime }\\mathcal {C}}(X,X)$ .", "Applying Lemma REF (c), it follows that $[\\mathcal {F},T]\\in \\mathcal {B}^1(H(X),H(X))$ .", "Accordingly, each of the maps $Tr_s:Hom^{2m}_{\\Omega ^{\\prime }\\mathcal {C}}(X,X) \\longrightarrow \\mathbb {C}$ is well-defined.", "For $T^{\\prime }\\in Hom^{2m-1}_{\\Omega ^{\\prime }\\mathcal {C}}(X,X)$ , we notice that $Tr_s(\\partial ^{\\prime }T^{\\prime })=\\frac{1}{2}~Tr\\left(\\epsilon \\mathcal {F}_X(\\partial ^{\\prime 2}T^{\\prime })\\right)=0$ We now consider $T_1\\in Hom^{i}_{\\Omega ^{\\prime }\\mathcal {C}}(X,Y)$ , $T_2\\in Hom^{j}_{\\Omega ^{\\prime }\\mathcal {C}}(Y,X)$ such that $i+j=2m$ .", "We notice that $\\begin{array}{c}\\epsilon \\mathcal {F}_Y\\partial ^{\\prime }(T_1)=\\partial ^{\\prime }(T_1)\\epsilon \\mathcal {F}_X \\qquad \\epsilon \\mathcal {F}_X\\partial ^{\\prime }(T_2)=\\partial ^{\\prime }(T_2)\\epsilon \\mathcal {F}_Y\\end{array}$ We note that $i\\equiv j \\mbox{(mod $2$)}$ .", "Using (REF ) and (REF ), we now have $\\begin{array}{ll}2\\cdot Tr_s(T_1T_2)=Tr\\left(\\epsilon \\mathcal {F}_Y\\partial ^{\\prime }(T_1T_2)\\right) & = Tr\\left(\\epsilon \\mathcal {F}_Y\\partial ^{\\prime }(T_1)T_2\\right)+(-1)^iTr\\left(\\epsilon \\mathcal {F}_YT_1\\partial ^{\\prime }(T_2)\\right)\\\\& = Tr\\left(\\partial ^{\\prime }(T_1)\\epsilon \\mathcal {F}_XT_2\\right)+(-1)^iTr\\left(\\partial ^{\\prime }(T_2)\\epsilon \\mathcal {F}_YT_1\\right)\\\\& =Tr\\left(\\epsilon \\mathcal {F}_XT_2\\partial ^{\\prime }(T_1)\\right)+(-1)^iTr\\left(\\epsilon \\mathcal {F}_X\\partial ^{\\prime }(T_2)T_1\\right)\\\\& = Tr\\left(\\epsilon \\mathcal {F}_XT_2\\partial ^{\\prime }(T_1)\\right)+(-1)^jTr\\left(\\epsilon \\mathcal {F}_X\\partial ^{\\prime }(T_2)T_1\\right)\\\\&= (-1)^{ij}2\\cdot Tr_s(T_2T_1)\\end{array}$ Theorem 6.4 Let $({H},\\mathcal {F})$ be a $p$ -summable Fredholm module over $\\mathcal {C}$ .", "Take $2m \\ge p-1$ .", "Then, the tuple $(\\Omega ^{\\prime }\\mathcal {C},\\partial ^{\\prime },\\hat{Tr}_s,\\rho ^{\\prime })$ defines a $2m$ -dimensional cycle over $\\mathcal {C}$ .", "Then, $\\phi ^{2m} \\in CN^{2m}(\\mathcal {C})=C^{2m}_{\\mathbb {C}}(\\mathcal {C},\\mathbb {C})=Hom(CN_{2m}(\\mathcal {C}),\\mathbb {C})$ defined by $\\phi ^{2m}(f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{2m}):= Tr_s\\left(H({f}^0)[\\mathcal {F},f^1][\\mathcal {F},f^2] \\ldots [\\mathcal {F},f^{2m}]\\right)$ for any $f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{2m} \\in Hom_{\\mathcal {C}}(X_1,X) \\otimes Hom_{\\mathcal {C}}(X_2,X_1) \\otimes \\ldots \\otimes Hom_{\\mathcal {C}}(X,X_{2m})$ is a cyclic cocycle over $\\mathcal {C}$ .", "It follows directly from Lemma REF and Proposition REF that $(\\Omega ^{\\prime }\\mathcal {C},\\partial ^{\\prime },\\hat{Tr}_s,\\rho ^{\\prime })$ is a $2m$ -dimensional cycle over $\\mathcal {C}$ .", "The rest follows by applying Theorem REF with $H=\\mathbb {C}=M$ .", "We will refer to $\\phi ^{2m}$ as the $2m$ -dimensional character associated with the $p$ -summable even Fredholm module $({H},\\mathcal {F})$ over the category $\\mathcal {C}$ .", "Remark 6.5 The appearance of only even cyclic cocycles in Theorem REF is due to the following fact from : if $T \\in \\mathcal {B}(H(X), H(X))$ is homogeneous of odd degree, then $Tr_s(T)=0$ ." ], [ "Periodicity of Chern character for Fredholm modules", "We continue with $\\mathcal {C}$ being a small $\\mathbb {C}$ -category.", "Taking $H=\\mathbb {C}=M$ , we denote the cyclic cohomology groups of $\\mathcal {C}$ by $H^\\bullet _\\lambda (\\mathcal {C}):=HC^\\bullet _{\\mathbb {C}}(\\mathcal {C},\\mathbb {C})$ .", "The cyclic complex corresponding to the cocyclic module $\\lbrace CN^n(\\mathcal {C})=Hom_{\\mathbb {C}}(CN_n(\\mathcal {C}),\\mathbb {C})\\rbrace _{n\\ge 0}$ as in (REF ) will be denoted by $C^\\bullet _\\lambda (\\mathcal {C})$ .", "The cocycles of this complex will be denoted by $Z^\\bullet _\\lambda (\\mathcal {C}):=Z^\\bullet _{\\mathbb {C}}(\\mathcal {C},\\mathbb {C})$ and the coboundaries by $B^\\bullet _\\lambda (\\mathcal {C}):=B^\\bullet _{\\mathbb {C}}(\\mathcal {C},\\mathbb {C})$ .", "Let $({H},\\mathcal {F})$ be a $p$ -summable Fredholm module over $\\mathcal {C}$ .", "We take $2m \\ge p-1$ .", "Let $\\phi ^{2m}$ be the $2m$ -dimensional character associated to the Fredholm module $({H},\\mathcal {F})$ .", "We denote by $ch^{2m}(H,\\mathcal {F})\\in H^{2m}_\\lambda (\\mathcal {C})$ the cohomology class of $\\phi ^{2m}$ .", "Since $\\mathcal {B}^p(H(X), H(Y)) \\subseteq \\mathcal {B}^q(H(X), H(Y))$ for any $p \\le q$ , the Fredholm module $({H},\\mathcal {F})$ is also $(p+2)$ -summable.", "Using Theorem REF , we then have the $(2m+2)$ -dimensional character $\\phi ^{2m+2}$ associated to $({H},\\mathcal {F})$ .", "We will show that the cyclic cocycles $\\phi ^{2m}$ and $\\phi ^{2m+2}$ are related to each other via the periodicity operator.", "If $\\mathcal {C}$ and $\\mathcal {C}^{\\prime }$ are small $\\mathbb {C}$ -categories, from the proof of Theorem REF it follows that there is a pairing on cyclic cocycles $Z^r_\\lambda (\\mathcal {C}) \\otimes Z^s_\\lambda (\\mathcal {C}^{\\prime }) \\longrightarrow Z^{r+s}_\\lambda (\\mathcal {C} \\otimes \\mathcal {C}^{\\prime }) \\qquad \\phi \\otimes \\phi ^{\\prime } \\mapsto \\phi \\# \\phi ^{\\prime }$ which descends to a pairing on cyclic cohomologies: $H^r_\\lambda (\\mathcal {C}) \\otimes H^s_\\lambda (\\mathcal {C}^{\\prime }) \\longrightarrow H^{r+s}_\\lambda (\\mathcal {C} \\otimes \\mathcal {C}^{\\prime })$ given by ${(\\hat{T}^\\phi \\# \\hat{T}^{\\phi ^{\\prime }})}_{(X,X^{\\prime })}(f \\otimes f^{\\prime }):=\\hat{T}^\\phi _X( f_r)\\hat{T}^{\\phi ^{\\prime }}_{X^{\\prime }}(f^{\\prime }_s)$ for any $f \\otimes f^{\\prime }= \\underset{i+j=r+s}{\\sum }(f_i \\otimes f^{\\prime }_j)\\in Hom^{r+s}_{\\mathcal {S} \\otimes \\mathcal {S}^{\\prime }}\\left((X,X^{\\prime }),(X,X^{\\prime })\\right)$ .", "Here $\\phi $ and $\\phi ^{\\prime }$ are expressed respectively as the characters of $r$ and $s$ -dimensional cycles $(\\mathcal {S},\\hat{\\partial }, \\hat{T}^\\phi ,\\rho )$ and $(\\mathcal {S}^{\\prime },\\hat{\\partial }^{\\prime },\\hat{T}^{\\phi ^{\\prime }},\\rho ^{\\prime })$ over $\\mathcal {C}$ and $\\mathcal {C}^{\\prime }$ .", "In particular, $\\phi \\# \\phi ^{\\prime }$ is the character of the $(r+s)$ -dimensional cycle $\\left(\\mathcal {S} \\otimes \\mathcal {S}^{\\prime }, \\hat{\\partial } \\otimes \\hat{\\partial ^{\\prime }}, \\hat{T}^\\phi \\# \\hat{T}^{\\phi ^{\\prime }}, \\rho \\otimes \\rho ^{\\prime } \\right)$ over $\\mathcal {C} \\otimes \\mathcal {C}^{\\prime }$ .", "For a morphism $f$ in $\\mathcal {C}$ , we will often suppress the functor $\\rho $ and write the morphism $\\rho (f)$ in $\\mathcal {S}^0$ simply as $f$ .", "Similarly, when there is no danger of confusion, we will often write the morphism $H(f)$ simply as $f$ .", "Now setting $\\mathcal {C}^{\\prime }=\\mathbb {C}$ (the category with one object) and considering the cyclic cocycle $\\psi \\in H^2_\\lambda (\\mathbb {C})$ determined by $\\psi (1,1,1)=1$ , we obtain the periodicity operator: $S:Z^r_\\lambda (\\mathcal {C}) \\longrightarrow Z^{r+2}_\\lambda (\\mathcal {C}) \\qquad S(\\phi ):=\\phi \\# \\psi $ for any $r \\ge 0$ and $\\phi \\in Z^r_\\lambda (\\mathcal {C})$ .", "Lemma 7.1 Let $\\phi \\in Z^r_\\lambda (\\mathcal {C})$ .", "For any $f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{r+2} \\in CN_{r+2}(\\mathcal {C})$ , we have $\\begin{array}{ll}(S(\\phi ))(f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{r+2})&=\\hat{T}^\\phi _X(f^0f^1f^2\\hat{\\partial }f^3 \\ldots \\hat{\\partial }f^{r+2})+\\hat{T}^\\phi _X(f^0\\hat{\\partial }f^1(f^2f^3) \\ldots \\hat{\\partial }f^{r+2})+ \\ldots \\\\&\\quad +\\hat{T}^\\phi _X(f^0\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^{i-1} (f^if^{i+1})\\hat{\\partial }f^{i+2} \\ldots \\hat{\\partial }f^{r+2}) + \\ldots \\\\& \\quad +\\hat{T}^\\phi _X(f^0\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^r (f^{r+1}f^{r+2}))\\end{array}$ We consider the 2-dimensional trace $\\hat{T}^\\psi $ on the DG-semicategory $(\\Omega \\mathbb {C},\\partial )$ such that $\\psi \\in Z^2_\\lambda (\\mathbb {C})$ is the character of the corresponding cycle over $\\mathbb {C}$ .", "We first observe that we have the following equalities in $\\Omega \\mathbb {C}$ : $\\partial 1= (\\partial 1)1+1 (\\partial 1), \\qquad 1(\\partial 1)1=0, \\qquad 1(\\partial 1)^2=(\\partial 1)^21$ We illustrate the proof for $r=2$ .", "The general case will follow similarly.", "By definition, we have $\\begin{array}{ll}&(S(\\phi ))(f^0 \\otimes f^1 \\otimes f^2 \\otimes f^3 \\otimes f^{4})\\\\& \\quad =(\\phi \\# \\psi )(f^0 \\otimes f^1 \\otimes f^2 \\otimes f^3 \\otimes f^{4})\\\\& \\quad ={(\\hat{T}^\\phi \\# \\hat{T}^\\psi )}\\left( (f^0 \\otimes 1)(\\hat{\\partial } \\otimes \\partial )(f^1 \\otimes 1) (\\hat{\\partial } \\otimes \\partial )(f^2 \\otimes 1) (\\hat{\\partial } \\otimes \\partial )(f^3 \\otimes 1) (\\hat{\\partial } \\otimes \\partial )(f^{4} \\otimes 1) \\right)\\\\& \\quad ={(\\hat{T}^\\phi \\# \\hat{T}^\\psi )}\\left( (f^0 \\otimes 1) (\\hat{\\partial }f^1 \\otimes 1 +f^1 \\otimes \\partial 1) (\\hat{\\partial }f^2 \\otimes 1 +f^2 \\otimes \\partial 1)(\\hat{\\partial }f^3 \\otimes 1 +f^3 \\otimes \\partial 1) (\\hat{\\partial }f^{4} \\otimes 1 +f^{4} \\otimes \\partial 1) \\right)\\\\& \\quad ={(\\hat{T}^\\phi \\# \\hat{T}^\\psi )} \\Big (f^0 \\hat{\\partial }f^1 \\hat{\\partial }f^2 \\hat{\\partial }f^3 \\hat{\\partial }f^4 \\otimes 1+f^0 \\hat{\\partial }f^1 \\hat{\\partial }f^2 \\hat{\\partial }f^3f^4 \\otimes 1 \\partial 1 + f^0\\hat{\\partial }f^1 \\hat{\\partial }f^2f^3f^4 \\otimes 1(\\partial 1)^2 +\\\\&\\qquad f^0 \\hat{\\partial }f^1f^2f^3 \\hat{\\partial }f^4 \\otimes 1(\\partial 1)^21 + f^0 \\hat{\\partial }f^1f^2f^3f^4 \\otimes 1(\\partial 1)^3 + f^0f^1f^2 \\hat{\\partial }f^3\\hat{\\partial }f^4 \\otimes 1 (\\partial 1)^2 \\\\&\\quad + f^0f^1f^2\\hat{\\partial }f^3 f^4 \\otimes 1(\\partial 1)^3 -f^0f^1f^2f^3\\hat{\\partial }f^4 \\otimes 1(\\partial 1)^31 + f^0f^1f^2f^3f^4 \\otimes 1(\\partial 1)^4\\Big )\\\\& \\quad =\\hat{T}^\\phi \\left(f^0 \\hat{\\partial }f^1 \\hat{\\partial }f^2f^3f^4\\right) \\hat{T}^\\psi \\left(1(\\partial 1)^2\\right) + \\hat{T}^\\phi \\left(f^0\\hat{\\partial }f^1f^2f^3\\hat{\\partial }f^4\\right) \\hat{T}^\\psi \\left(1(\\partial 1)^21\\right)\\\\& \\qquad + \\hat{T}^\\phi \\left(f^0f^1f^2\\hat{\\partial }f^3 \\hat{\\partial }f^4\\right) \\hat{T}^\\psi \\left(1(\\partial 1)^2\\right)\\\\& \\quad = \\hat{T}^\\phi \\left(f^0\\hat{\\partial }f^1 \\hat{\\partial }f^2f^3f^4\\right) + \\hat{T}^\\phi \\left(f^0 \\hat{\\partial }f^1f^2f^3\\hat{\\partial }f^4\\right)+ \\hat{T}^\\phi \\left(f^0f^1f^2\\hat{\\partial }f^3 \\hat{\\partial }f^4\\right)\\end{array}$ The last equality follows by using the fact that $\\hat{T}^\\psi \\left(1(\\partial 1)^2\\right)=\\psi (1,1,1)=1$ .", "Proposition 7.2 Let $\\phi $ be the character of an $r$ -dimensional cycle $(\\mathcal {S},\\hat{\\partial },\\hat{T}^{\\phi },\\rho )$ over $\\mathcal {C}$ .", "Then, $S(\\phi )$ is a coboundary.", "In particular, we have $S(\\phi )=b\\psi $ , where $\\psi \\in CN^{r+1}(\\mathcal {C})$ is given by $\\psi (f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{r+1})=\\sum \\limits _{j=1}^{r+1} (-1)^{j-1}~\\hat{T}^{\\phi }\\left(f^0\\hat{\\partial }f^1\\ldots \\hat{\\partial }f^{j-1} f^j \\hat{\\partial }f^{j+1} \\ldots \\hat{\\partial }f^{r+1}\\right)$ Again, we illustrate the case of $r=2$ .", "The general computation is similar.", "$\\begin{array}{ll}&(b\\psi )(f^0\\otimes f^1 \\otimes f^2 \\otimes f^3 \\otimes f^4)\\\\&=\\psi (f^0f^1 \\otimes f^2 \\otimes f^3 \\otimes f^4) - \\psi (f^0 \\otimes f^1f^2 \\otimes f^3 \\otimes f^4) + \\psi (f^0 \\otimes f^1 \\otimes f^2f^3 \\otimes f^4) - \\psi (f^0 \\otimes f^1 \\otimes f^2 \\otimes f^3f^4)\\\\&\\quad + \\psi (f^4f^0 \\otimes f^1 \\otimes f^2 \\otimes f^3)\\\\&=\\hat{T}^\\phi (f^0f^1f^2\\hat{\\partial }f^3\\hat{\\partial }f^4)-\\hat{T}^\\phi (f^0f^1\\hat{\\partial }f^2f^3\\hat{\\partial }f^4)+ \\hat{T}^\\phi (f^0f^1\\hat{\\partial }f^2\\hat{\\partial }f^3f^4) \\\\&\\quad -\\hat{T}^\\phi (f^0f^1f^2\\hat{\\partial }f^3\\hat{\\partial }f^4)+\\hat{T}^\\phi (f^0\\hat{\\partial }(f^1f^2)f^3\\hat{\\partial } f^4-\\hat{T}^\\phi (f^0\\hat{\\partial }(f^1f^2)\\hat{\\partial }f^3f^4)\\\\&\\quad +\\hat{T}^\\phi (f^0f^1\\hat{\\partial }(f^2f^3)\\hat{\\partial }f^4)-\\hat{T}^\\phi (f^0\\hat{\\partial }f^1f^2f^3\\hat{\\partial }f^4)+ \\hat{T}^\\phi (f^0\\hat{\\partial }f^1\\hat{\\partial }(f^2f^3)f^4)\\\\&\\quad -\\hat{T}^\\phi (f^0f^1\\hat{\\partial }f^2\\hat{\\partial }(f^3f^4))+ \\hat{T}^\\phi (f^0\\hat{\\partial }f^1f^2\\hat{\\partial }(f^3f^4))- \\hat{T}^\\phi (f^0\\hat{\\partial }f^1\\hat{\\partial }f^2f^3f^4)\\\\&\\quad +\\hat{T}^\\phi (f^4f^0f^1\\hat{\\partial }f^2\\hat{\\partial }f^3)- \\hat{T}^\\phi (f^4f^0\\hat{\\partial }f^1f^2\\hat{\\partial }f^3) + \\hat{T}^\\phi (f^4f^0\\hat{\\partial }f^1\\hat{\\partial }f^2f^3)\\\\&= \\hat{T}^\\phi (f^0f^1f^2\\hat{\\partial }f^3\\hat{\\partial }f^4) +\\hat{T}^\\phi (f^0\\hat{\\partial }f^1(f^2f^3)\\hat{\\partial }f^4) + \\hat{T}^\\phi (f^0\\hat{\\partial }f^1\\hat{\\partial }f^2f^3f^4) \\\\&= (S(\\phi ))(f^0\\otimes f^1 \\otimes f^2 \\otimes f^3 \\otimes f^4)\\end{array}$ Theorem 7.3 Let $\\mathcal {C}$ be a small $\\mathbb {C}$ -category and let $({H},\\mathcal {F})$ be a $p$ -summable even Fredholm module over $\\mathcal {C}$ .", "Take $ 2m \\ge p-1$ .", "Then, $S(\\phi ^{2m})=-(m+1)\\phi ^{2m+2} \\qquad \\text{in}~ H^{2m+2}_\\lambda (\\mathcal {C})$ We will show that $S(\\phi ^{2m})+(m+1)\\phi ^{2m+2}=b\\psi $ for some $\\psi \\in Z^{2m+1}_\\lambda (\\mathcal {C})$ .", "By Theorem REF , we know that $\\phi ^{2m}$ is the character of the $2m$ -dimensional cycle $(\\Omega ^{\\prime } \\mathcal {C}, \\partial ^{\\prime }, \\hat{Tr}_s,\\rho ^{\\prime })$ over the category $\\mathcal {C}$ .", "Applying Lemma REF and using the fact that $Tr_s(T)=0$ for any homogeneous $T$ of odd degree, we have $\\begin{array}{ll}(S(\\phi ^{2m}))(f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{2m+2}) &= \\sum \\limits _{j=0}^{2m+1} Tr_s\\left(f^0[\\mathcal {F},f^1]\\ldots [\\mathcal {F},f^{j-1}](f^jf^{j+1})[\\mathcal {F},f^{j+2}]\\ldots [\\mathcal {F},f^{2m+2}] \\right)\\end{array}$ Further, $\\begin{array}{ll}\\phi ^{2m+2}(f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{2m+2})= Tr_s\\left( f^0[\\mathcal {F},f^1]\\ldots \\ldots [\\mathcal {F},f^{2m+2}] \\right)\\end{array}$ so that $\\begin{array}{lr}&\\left(S(\\phi ^{2m})+(m+1)\\phi ^{2m+2}\\right)(f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{2m+2})\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\\\&= \\sum \\limits _{j=0}^{2m+1} Tr_s\\left( f^0[\\mathcal {F},f^1]\\ldots [\\mathcal {F},f^{j-1}](f^jf^{j+1})[\\mathcal {F},f^{j+2}]\\ldots [\\mathcal {F},f^{2m+2}] \\right)\\\\& +(m+1)Tr_s\\left(f^0[\\mathcal {F},f^1]\\ldots \\ldots [\\mathcal {F},f^{2m+2}] \\right)\\qquad \\qquad \\qquad \\qquad \\qquad \\end{array}$ We now consider $\\psi =\\sum \\limits _{j=0}^{2m+1} (-1)^{j-1} \\psi ^j$ , where $\\psi ^j(f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{2m+1})=Tr\\left(\\epsilon \\mathcal {F} f^j [\\mathcal {F},f^{j+1}]\\ldots [\\mathcal {F},f^{2m+1}][\\mathcal {F},f^0][\\mathcal {F},f^1]\\ldots [\\mathcal {F},f^{j-1}] \\right)$ Since $2m \\ge p-1$ and $(H,\\mathcal {F})$ is a $p$ -summable even Fredholm module over $\\mathcal {C}$ , it follows that the operator $\\epsilon \\mathcal {F} f^j [\\mathcal {F},f^{j+1}]\\ldots [\\mathcal {F},f^{2m+1}][\\mathcal {F},f^0][\\mathcal {F},f^1]\\ldots [\\mathcal {F},f^{j-1}]$ is trace class.", "We observe that $\\tau \\psi ^j=\\psi ^{j-1}$ for $1\\le j\\le 2m+1$ and $\\tau \\psi ^0=\\psi ^{2m+1}$ .", "It follows that $(1-\\lambda )(\\psi )=0$ .", "Hence, $\\psi \\in C^{2m+1}_\\lambda (\\mathcal {C})=Ker(1-\\lambda )$ .", "Using (REF ), we have $\\begin{array}{ll}&(b\\psi ^j)(f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{2m+2})\\\\& \\quad =\\sum \\limits _{i=0}^{2m+1} (-1)^i ~\\psi ^j(f^0 \\otimes \\ldots \\otimes f^if^{i+1} \\otimes \\ldots \\otimes f^{2m+2})+ \\psi ^j(f^{2m+2}f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{2m+1})\\\\& \\quad =Tr\\left(\\epsilon \\mathcal {F}f^{j+1}[\\mathcal {F},f^{j+2}]\\ldots [\\mathcal {F},f^{2m+2}]f^0[\\mathcal {F},f^1]\\ldots [\\mathcal {F},f^j]\\right)~+\\\\& \\qquad (-1)^{j-1}Tr\\left(\\epsilon \\mathcal {F}f^{j+1}[\\mathcal {F},f^{j+2}]\\ldots [\\mathcal {F},f^{2m+2}][\\mathcal {F},f^0][\\mathcal {F},f^1]\\ldots f^j \\right)+\\\\& \\qquad Tr\\left(\\epsilon \\mathcal {F}f^{j}[\\mathcal {F},f^{j+1}]\\ldots [\\mathcal {F},f^{2m+2}]f^0[\\mathcal {F},f^1]\\ldots [\\mathcal {F},f^{j-1}]\\right)\\end{array}$ We now set $\\beta ^j=[\\mathcal {F},f^{j+2}]\\ldots [\\mathcal {F},f^{2m+2}]f^0[\\mathcal {F},f^1]\\ldots [\\mathcal {F},f^{j-1}]$ .", "Then, we have $\\begin{array}{ll}[\\mathcal {F},\\beta ^j]=\\mathcal {F}\\beta ^j- (-1)^{2m}\\beta ^j \\mathcal {F}&=\\mathcal {F}[\\mathcal {F},f^{j+2}]\\ldots [\\mathcal {F},f^{2m+2}]f^0[\\mathcal {F},f^1]\\ldots [\\mathcal {F},f^{j-1}]-\\\\&\\qquad [\\mathcal {F},f^{j+2}]\\ldots [\\mathcal {F},f^{2m+2}]f^0[\\mathcal {F},f^1]\\ldots [\\mathcal {F},f^{j-1}]\\mathcal {F}\\\\&= (-1)^{j-1}[\\mathcal {F},f^{j+2}]\\ldots [\\mathcal {F},f^{2m+2}][\\mathcal {F},f^0][\\mathcal {F},f^1]\\ldots [\\mathcal {F},f^{j-1}]\\end{array}$ With $\\alpha ^j=f^j \\mathcal {F}f^{j+1}$ , we get $\\begin{array}{ll}&(-1)^{j-1}Tr\\left(\\epsilon \\mathcal {F}f^{j+1}[\\mathcal {F},f^{j+2}]\\ldots [\\mathcal {F},f^{2m+2}][\\mathcal {F},f^0][\\mathcal {F},f^1]\\ldots [\\mathcal {F},f^{j-1}]f^j \\right)\\\\& \\quad =Tr\\left(\\epsilon \\mathcal {F}f^{j+1}[\\mathcal {F},\\beta ^j]f^j\\right)=Tr_s\\left(\\alpha ^j[\\mathcal {F},\\beta ^j]\\right)=Tr_s([\\mathcal {F},\\alpha ^j]\\beta ^j)\\end{array}$ where we have used the fact that $Tr_s$ is a closed graded trace and $Tr_s(T)=Tr(\\epsilon T)$ for any operator that is trace class (see ).", "Thus, we have $\\begin{array}{ll}&(b\\psi ^j)(f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{2m+2})=-Tr_s\\left([\\mathcal {F},f^j] \\mathcal {F}f^{j+1}\\beta ^j\\right)+Tr_s([\\mathcal {F},\\alpha ^j]\\beta ^j)+Tr_s\\left( \\mathcal {F}f^j[\\mathcal {F},f^{j+1}]\\beta ^j\\right)\\end{array}$ Since $\\mathcal {F}[\\mathcal {F},f^jf^{j+1}]=\\mathcal {F}[\\mathcal {F},f^j]f^{j+1}+\\mathcal {F}f^j[\\mathcal {F},f^{j+1}]=-[\\mathcal {F},f^j]\\mathcal {F}f^{j+1}+ \\mathcal {F}f^j[\\mathcal {F},f^{j+1}],$ we obtain $\\begin{array}{ll}(b\\psi ^j)(f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{2m+2})&=Tr_s\\big (\\left(\\mathcal {F}[\\mathcal {F},f^jf^{j+1}] + [\\mathcal {F},\\alpha ^j]\\right)\\beta ^j\\Big )\\end{array}$ As $\\mathcal {F}[\\mathcal {F},f^jf^{j+1}] +[\\mathcal {F},\\alpha ^j]=\\mathcal {F}[\\mathcal {F},f^jf^{j+1}] +\\mathcal {F} \\alpha ^j + \\alpha ^j \\mathcal {F} =[\\mathcal {F},f^j][\\mathcal {F},f^{j+1}]+2f^jf^{j+1}$ we get $\\begin{array}{ll}(b\\psi )(f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{2m+2})&=\\sum \\limits _{j=0}^{2m+1} (-1)^{j-1} (b\\psi ^j) (f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{2m+2})\\\\&=\\sum \\limits _{j=0}^{2m+1}(-1)^{j-1} \\big ( 2~Tr_s\\left(f^jf^{j+1}\\beta ^j\\right) + Tr_s\\left([\\mathcal {F},f^j][\\mathcal {F},f^{j+1}]\\beta ^j\\right) \\big )\\\\&=\\sum \\limits _{j=0}^{2m+1} 2~Tr_s \\left(f^0[\\mathcal {F},f^1]\\ldots [\\mathcal {F},f^{j-1}](f^jf^{j+1})[\\mathcal {F},f^{j+2}]\\ldots [\\mathcal {F},f^{2m+2}] \\right)\\\\& \\quad +\\sum \\limits _{j=0}^{2m+1}Tr_s\\left(f^0[\\mathcal {F},f^1] \\ldots [\\mathcal {F},f^{2m+2}]\\right)\\\\&=\\sum \\limits _{j=0}^{2m+1}~2~Tr_s\\left(f^0[\\mathcal {F},f^1]\\ldots [\\mathcal {F},f^{j-1}](f^jf^{j+1})[\\mathcal {F},f^{j+2}]\\ldots [\\mathcal {F},f^{2m+2}] \\right)\\\\& \\quad +(2m+2)Tr\\left(f^0[\\mathcal {F},f^1] \\ldots [\\mathcal {F},f^{2m+2}]\\right)\\end{array}$ The result now follows by (REF )." ], [ "Homotopy invariance of the Chern character", "Let $SHilb_2$ be the full subcategory of $SHilb_{\\mathbb {Z}_2}$ whose objects are of the form $\\mathcal {D}=\\mathcal {H}\\oplus \\mathcal {H}$ , for some separable Hilbert space $\\mathcal {H}$ .", "If $\\mathcal {D}=\\mathcal {H}\\oplus \\mathcal {H}\\in SHilb_2$ , we denote by $F(\\mathcal {D})$ the morphism in $SHilb_2(\\mathcal {D},\\mathcal {D})= \\mathcal {B}(\\mathcal {H} \\oplus \\mathcal {H}, \\mathcal {H} \\oplus \\mathcal {H})$ given by the matrix $\\begin{pmatrix} 0 & 1\\\\ 1 & 0 \\\\ \\end{pmatrix}$ swapping the two copies of $\\mathcal {H}$ .", "Lemma 8.1 Let $\\mathcal {C}$ be a small $\\mathbb {C}$ -category and $\\lbrace {H}_t:\\mathcal {C} \\longrightarrow SHilb_2\\rbrace _{t \\in [0,1]}$ be a family of functors such that for each $X\\in Ob(\\mathcal {C})$ , we have $H_t(X)=H_{t^{\\prime }}(X)$ for all $t$ , $t^{\\prime }\\in [0,1]$ .", "We put $H(X):=H_t(X)$ for all $t \\in [0,1]$ .", "For each $f:X \\longrightarrow Y$ in $\\mathcal {C}$ , we assume that the function $ p_f:[0,1] \\longrightarrow SHilb_{\\mathbb {Z}_2}({H}_t(X),{H}_t(Y))\\qquad t \\mapsto {H}_t(f)$ is strongly $C^1$ .", "Then if $\\delta _t(f):=p_f^{\\prime }(t)$ , we have $\\delta _t(fg)={H}_t(f) \\circ \\delta _t(g) + \\delta _t(f) \\circ {H}_t(g)$ for composable morphisms $f$ , $g$ in $\\mathcal {C}$ .", "We have $\\begin{array}{ll}&\\delta _t(fg)-{H}_t(f) \\circ \\delta _t(g) - \\delta _t(f) \\circ {H}_t(g)\\\\& \\quad =p^{\\prime }_{fg}(t)-{H}_t(f) \\circ p^{\\prime }_g(t) -p^{\\prime }_f(t) \\circ {H}_t(g)\\\\& \\quad =\\lim \\limits _{s \\rightarrow 0} \\frac{1}{s} \\left(p_{fg}(t+s)-p_{fg}(t) - {H}_t(f)~ \\circ ~p_g(t+s)+ {H}_t(f)~ \\circ ~ p_g(t)- p_f(t+s)~ \\circ ~ {H}_t(g)+p_f(t)~ \\circ ~ {H}_t(g)\\right)\\\\& \\quad =\\lim \\limits _{s \\rightarrow 0} \\frac{1}{s}\\left({H}_{t+s}(fg) -{H}_{t}(fg) - {H}_t(f){H}_{t+s}(g) + {H}_t(f) {H}_{t}(g) - {H}_{t+s}(f) {H}_t(g)+{H}_{t}(f) {H}_t(g)\\right)\\\\& \\quad =\\lim \\limits _{s \\rightarrow 0} \\frac{1}{s}\\left({H}_{t+s}(f)-{H}_{t}(f)\\right)\\left({H}_{t+s}(g)-{H}_{t}(g)\\right)\\\\& \\quad =\\lim \\limits _{s \\rightarrow 0}\\frac{1}{s} \\left(p_f(t+s)-p_f(t)\\right) \\left(p_g(t+s)-p_g(t)\\right)\\\\& \\quad = p_f^{\\prime }(t) \\lim \\limits _{s \\rightarrow 0} \\left(p_g(t+s)-p_g(t)\\right)=0\\end{array}$ For each $n \\in \\mathbb {Z}_{\\ge 0}$ , we now define an operator $A:CN^{n}(\\mathcal {C}) \\longrightarrow CN^n(\\mathcal {C})$ given by $A:=1+\\lambda +\\lambda ^2+\\ldots +\\lambda ^n$ where $\\lambda $ is the (signed) cyclic operator.", "We observe that if $\\psi \\in C^n_\\lambda (\\mathcal {C})=Ker(1-\\lambda )$ , then $A\\psi =(n+1)\\psi $ .", "From the relation $(1-\\lambda )(1+2\\lambda +3\\lambda ^2+\\cdots +(n+1)\\lambda ^n)=A-(n+1)\\cdot 1$ it is immediate that $Ker(A) \\subseteq Im(1-\\lambda )$ .", "Let $B_0: CN^{n+1}(\\mathcal {C}) \\longrightarrow CN^n(\\mathcal {C})$ be the map defined as follows: $(B_0\\phi )(f^0 \\otimes \\ldots \\otimes f^n):=\\phi (id_{X_0} \\otimes f^0 \\otimes \\ldots \\otimes f^n)-(-1)^{n+1}\\phi (f^0 \\otimes \\ldots \\otimes f^n \\otimes id_{X_0})$ for any $f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{n} \\in Hom_{\\mathcal {C}}(X_1,X_0) \\otimes Hom_{\\mathcal {C}}(X_2,X_1) \\otimes \\ldots \\otimes Hom_{\\mathcal {C}}(X_0,X_{n})$ .", "We now set $B:=AB_0: CN^{n+1}(\\mathcal {C}) \\longrightarrow CN^n(\\mathcal {C})$ Lemma 8.2 We have (1) $bA=Ab^{\\prime }$ .", "(2) $bB+Bb=0$ .", "(1) This follows from the general fact that the dual $CN^\\bullet (\\mathcal {C})$ of the cyclic nerve of $\\mathcal {C}$ is a cocyclic module (see, for instance, ).", "(2) For any $f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{n} \\in Hom_{\\mathcal {C}}(X_1,X_0) \\otimes Hom_{\\mathcal {C}}(X_2,X_1) \\otimes \\ldots \\otimes Hom_{\\mathcal {C}}(X_0,X_{n})$ and $\\phi \\in CN^n{\\mathcal {C}}$ , we have $\\begin{array}{lll}&(B_0b\\phi )(f^0\\otimes \\ldots \\otimes f^{n})\\\\&\\quad =(b\\phi )(id_{X_0} \\otimes f^0 \\otimes \\ldots \\otimes f^n)-(-1)^{n+1}(b\\phi )(f^0 \\otimes \\ldots \\otimes f^n \\otimes id_{X_0})\\\\& \\quad = \\phi (f^0 \\otimes \\ldots \\otimes f^{n}) + \\sum \\limits _{i=0}^{n-1}(-1)^{i+1} \\phi (id_{X_0} \\otimes f^0 \\otimes \\ldots \\otimes f^if^{i+1} \\otimes \\ldots \\otimes f^{n}) +(-1)^{n+1} \\phi (f^n \\otimes f^0 \\otimes \\ldots \\otimes f^{n-1}) \\\\& \\qquad -(-1)^{n+1}\\left(\\sum \\limits _{i=0}^{n-1}(-1)^{i} \\phi (f^0 \\otimes \\ldots \\otimes f^if^{i+1} \\otimes \\ldots \\otimes f^{n} \\otimes id_{X_0}) \\right)\\end{array}$ On the other hand, $\\begin{array}{lll}&(b^{\\prime }B_0\\phi )(f^0\\otimes \\ldots \\otimes f^{n})\\\\& \\quad =\\sum \\limits _{i=0}^{n-1}(-1)^{i} \\phi (id_{X_0} \\otimes f^0 \\otimes \\ldots \\otimes f^if^{i+1} \\otimes \\ldots \\otimes f^{n})-(-1)^n\\sum \\limits _{i=0}^{n-1}(-1)^{i} \\phi (f^0 \\otimes \\ldots \\otimes f^if^{i+1} \\otimes \\ldots \\otimes f^{n} \\otimes id_{X_0})\\end{array}$ Thus, we obtain $(B_0b+b^{\\prime }B_0)(\\phi )(f^0\\otimes \\ldots \\otimes f^{n})=\\phi (f^0 \\otimes \\ldots \\otimes f^{n})+(-1)^{n+1} \\phi (f^n \\otimes f^0 \\otimes \\ldots \\otimes f^{n-1})$ Therefore, $(B_0b+b^{\\prime }B_0)(\\phi )=\\phi -\\lambda \\phi $ Now, by applying the operator $A$ to both sides of (REF ), we have $AB_0b+Ab^{\\prime }B_0=0$ The result now follows from part (1).", "Proposition 8.3 The image of the map $B:CN^{n+1}(\\mathcal {C}) \\longrightarrow CN^{n}(\\mathcal {C})$ is $C^n_\\lambda (\\mathcal {C})$ .", "Let $\\phi \\in C^n_\\lambda (\\mathcal {C})$ and let $R:=\\bigoplus \\limits _{X,Y \\in Ob(\\mathcal {C})}Hom(X,Y)$ .", "Then $A$ is an algebra with mutiplication given by composition wherever possible and 0 otherwise.", "We choose a linear map $\\eta : A \\longrightarrow \\mathbb {C}$ such that $\\begin{array}{ll}\\eta (f)=0 &\\qquad \\text{for}~ f \\in Hom_\\mathcal {C}(X,Y), ~X \\ne Y\\\\\\eta (id_X)=1 & \\qquad \\forall X \\in Ob(\\mathcal {C})\\end{array}$ We now define $\\psi \\in CN^{n+1}(\\mathcal {C})$ by setting $\\begin{array}{ll}\\psi (f^0\\otimes \\ldots \\otimes f^{n+1}):=&\\eta (f^0)\\phi (f^1\\otimes \\ldots \\otimes f^{n+1})+\\\\&\\quad (-1)^n \\left(\\phi \\left(f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{n}\\right)\\eta (f^{n+1}) -\\eta (f^0)\\phi \\left(id_{X_1} \\otimes f^1 \\otimes \\ldots \\otimes f^{n}\\right)\\eta (f^{n+1})\\right)\\end{array}$ for any $f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{n+1} \\in Hom_{\\mathcal {C}}(X_1,X_0) \\otimes Hom_{\\mathcal {C}}(X_2,X_1) \\otimes \\ldots \\otimes Hom_{\\mathcal {C}}(X_0,X_{n+1})$ .", "We observe that if the tuple $(f^1, \\ldots , f^{n+1})$ is not cyclically composable, i.e., $X_0 \\ne X_1$ , then the first term vanishes as $\\eta (f^0)=0$ .", "Similarly, if the tuple $(f^0, \\ldots , f^{n})$ is not cyclically composable, i.e., $X_{n+1} \\ne X_0$ , then the second term vanishes.", "For the last term, $\\eta (f^0)$ and $\\eta (f^{n+1})$ will be non zero only if $X_1=X_0$ and $X_0=X_{n+1}$ which means that $X_{n+1}=X_1$ and the tuple $(id_{X_1},f^1, \\ldots , f^{n})$ is cyclically composable.", "Then, for any $g^0 \\otimes g^1 \\otimes \\ldots \\otimes g^{n} \\in Hom_{\\mathcal {C}}(Y_1,Y_0) \\otimes Hom_{\\mathcal {C}}(Y_2,Y_1) \\otimes \\ldots \\otimes Hom_{\\mathcal {C}}(Y_0,Y_{n})$ , we have $\\begin{array}{ll}\\psi (id_{Y_0} \\otimes g^0 \\otimes \\ldots \\otimes g^{n})&=\\eta (id_{Y_0})\\phi (g^0\\otimes \\ldots \\otimes g^{n})+(-1)^n \\big (\\phi \\left(id_{Y_0} \\otimes g^0 \\otimes \\ldots \\otimes g^{n-1}\\right)\\eta (g^{n}) \\\\& \\quad -\\phi \\left(\\eta (id_{Y_0})id_{Y_0} \\otimes g^0 \\otimes \\ldots \\otimes g^{n-1}\\right)\\eta (g^{n})\\big )\\\\&=\\phi (g^0\\otimes \\ldots \\otimes g^{n})\\end{array}$ Also $\\begin{array}{ll}\\psi (g^0 \\otimes \\ldots \\otimes g^{n} \\otimes id_{Y_0})&=\\eta (g^0)\\phi (g^1\\otimes \\ldots \\otimes g^{n} \\otimes id_{Y_0})+(-1)^n \\big (\\phi \\left(g^0 \\otimes \\ldots \\otimes g^{n}\\right)\\eta (id_{Y_0})\\\\&\\quad -\\phi \\left(\\eta (g^0)id_{Y_1} \\otimes g^1 \\otimes \\ldots \\otimes g^{n}\\right)\\eta (id_{Y_0})\\big )\\\\& =(-1)^n\\phi \\left(g^0 \\otimes \\ldots \\otimes g^{n}\\right) \\\\\\end{array}$ where the second equality follows from the fact that $\\phi \\in C_\\lambda ^n(\\mathcal {C})$ and that $\\eta (g^0)=0$ whenever $Y_1\\ne Y_0$ .", "Thus, $\\begin{array}{ll}(B_0\\psi )(g^0 \\otimes \\ldots \\otimes g^n)&=\\psi (id_{Y_0} \\otimes g^0 \\otimes \\ldots \\otimes g^n)-(-1)^{n+1}\\psi (g^0 \\otimes \\ldots \\otimes g^n \\otimes id_{Y_0})\\\\&=2\\phi (g^0 \\otimes \\ldots \\otimes g^n)\\end{array}$ Since $\\phi \\in Ker(1-\\lambda )$ , we now have $B\\psi =2A\\phi =2(n+1)\\phi $ .", "Thus, $\\phi \\in Im(B)$ .", "Conversely, let $\\phi \\in Im(B)$ .", "Then, $\\phi =B\\psi $ for some $\\psi \\in CN^{n+1}(\\mathcal {C})$ .", "Using the fact that $(1-\\lambda )A=0$ , we have $\\begin{array}{c}(1-\\lambda )(\\phi )=(1-\\lambda )(B\\psi )=((1-\\lambda )AB_0)\\psi =0\\\\\\end{array}$ This proves the result.", "Proposition 8.4 Let $\\psi \\in CN^n(\\mathcal {C})$ be such that $b\\psi \\in C^{n+1}_\\lambda (\\mathcal {C})$ .", "Then, (1) $B\\psi \\in Z^{n-1}_\\lambda (\\mathcal {C})$ i.e., $b(B\\psi )=0$ and $(1-\\lambda )(B\\psi )=0$ .", "(2) $S(B\\psi )=n(n+1)b\\psi $ in $H^{n+1}_\\lambda (\\mathcal {C})$ .", "(1) We know that $(1-\\lambda )(B \\psi )=(1-\\lambda )(AB_0)(\\psi )=0$ .", "Further, for any $\\phi \\in Ker(1-\\lambda )$ , we have $B_0\\phi =0$ .", "Therefore, it follows that $bB\\psi =-Bb\\psi =-AB_0b\\psi =0$ .", "(2) We have to show that $SB\\psi -n(n+1)b\\psi =b\\zeta $ for some $\\zeta \\in C^{n}_\\lambda (\\mathcal {C})$ .", "We set $\\phi =B\\psi $ .", "Then, $\\phi $ is the character of an $(n-1)$ -dimensional cycle $(\\mathcal {S},\\hat{\\partial },\\hat{T},\\rho )$ over $\\mathcal {C}$ .", "By Proposition REF , we have $S\\phi =b\\psi ^{\\prime }$ , where $\\psi ^{\\prime } \\in CN^n(\\mathcal {C})$ is given by $\\psi ^{\\prime }(f^0 \\otimes \\ldots \\otimes f^{n})=\\sum \\limits _{j=1}^n (-1)^{j-1}~ \\hat{T}\\left(f^0\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^{j-1} f^{j} \\hat{\\partial }f^{j+1} \\ldots \\hat{\\partial }f^n\\right)$ Suppose we have $\\psi ^{\\prime \\prime } \\in CN^n(\\mathcal {C})$ such that $\\psi ^{\\prime \\prime }-\\psi \\in B^n(\\mathcal {C})$ and $\\zeta =\\psi ^{\\prime }-n(n+1)\\psi ^{\\prime \\prime } \\in C^{n}_\\lambda (\\mathcal {C})$ .", "This would give $b\\zeta =b\\psi ^{\\prime }-n(n+1)b \\psi ^{\\prime \\prime }=SB\\psi -n(n+1)b\\psi $ We set $\\theta :=B_0\\psi $ , $\\theta ^{\\prime }:=\\frac{1}{n}\\phi $ and $\\theta ^{\\prime \\prime }:=\\theta -\\theta ^{\\prime }\\in CN^{n-1}(\\mathcal {C})$ .", "Since $B\\psi \\in Z^{n-1}_\\lambda (\\mathcal {C})$ , we have $A\\theta ^{\\prime \\prime }=AB_0\\psi -\\frac{1}{n}A\\phi =B\\psi -\\frac{1}{n}AB\\psi =B\\psi -\\frac{1}{n}nB\\psi =0$ Since $Ker(A) \\subseteq Im(1-\\lambda )$ , we have $\\theta ^{\\prime \\prime }=(1-\\lambda )(\\psi _1)$ for some $\\psi _1 \\in CN^{n-1}(\\mathcal {C})$ .", "We take $\\psi ^{\\prime \\prime }=\\psi -b\\psi _1$ .", "We now show that $(1-\\lambda )(\\zeta )=0$ , i.e, $(1-\\lambda )(\\psi ^{\\prime })=n(n+1)(1-\\lambda )(\\psi ^{\\prime \\prime })$ where $\\zeta =\\psi ^{\\prime }-n(n+1)\\psi ^{\\prime \\prime }$ .", "We see that $(\\tau _n\\psi ^{\\prime })(f^0 \\otimes \\ldots \\otimes f^{n})=\\psi ^{\\prime }(f^n \\otimes f^0 \\otimes \\ldots \\otimes f^{n-1})=\\sum \\limits _{j=0}^{n-1} (-1)^{j}~ \\hat{T}\\left(\\hat{\\partial }f^0\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^{j-1} f^{j} \\hat{\\partial }f^{j+1} \\ldots \\hat{\\partial }f^{n-1}f^n\\right)$ where we have used the fact that $\\hat{T}$ is a graded trace.", "For $1\\le j\\le n-1$ , we now set $\\omega _j:=f^0(\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^{j-1}) f^{j} (\\hat{\\partial }f^{j+1} \\ldots \\hat{\\partial }f^{n-1})f^n$ Then, $\\begin{array}{ll}\\hat{\\partial }\\omega _j&=(\\hat{\\partial }f^0\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^{j-1}) f^{j} (\\hat{\\partial }f^{j+1} \\ldots \\hat{\\partial }f^{n-1})f^n+(-1)^{j-1}f^0(\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^{j-1} \\hat{\\partial }f^{j} \\hat{\\partial }f^{j+1} \\ldots \\hat{\\partial }f^{n-1})f^n +\\\\&\\quad (-1)^n f^0(\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^{j-1}) f^{j} (\\hat{\\partial }f^{j+1} \\ldots \\hat{\\partial }f^{n-1} \\hat{\\partial }f^n)\\end{array}$ Thus, $\\begin{array}{ll}0=\\hat{T}(\\hat{\\partial }\\omega _j)&=\\hat{T}\\left(\\hat{\\partial }f^0\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^{j-1} f^{j} \\hat{\\partial }f^{j+1} \\ldots \\hat{\\partial }f^{n-1}f^n\\right)+(-1)^{j-1}\\hat{T}\\left(f^0\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^{j-1} \\hat{\\partial }f^{j} \\hat{\\partial }f^{j+1} \\ldots \\hat{\\partial }f^{n-1}f^n\\right) +\\\\&\\quad (-1)^n \\hat{T}\\left(f^0\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^{j-1} f^{j} \\hat{\\partial }f^{j+1} \\ldots \\hat{\\partial }f^{n-1} \\hat{\\partial }f^n\\right)\\end{array}$ Therefore, $\\begin{array}{ll}&(1-\\lambda )(\\psi ^{\\prime })(f^0 \\otimes \\ldots \\otimes f^{n})\\\\&=-\\sum \\limits _{j=1}^n (-1)^{j}~ \\hat{T}\\left(f^0\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^{j-1} f^{j} \\hat{\\partial }f^{j+1} \\ldots \\hat{\\partial }f^n\\right)-(-1)^n \\sum \\limits _{j=0}^{n-1} (-1)^{j}~ \\hat{T}\\left(\\hat{\\partial }f^0\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^{j-1} f^{j} \\hat{\\partial }f^{j+1} \\ldots \\hat{\\partial }f^{n-1}f^n\\right)\\\\&=-\\big ((-1)^n\\hat{T}\\left(f^0\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^{n-1}f^n\\right)+(-1)^n \\hat{T}\\left(f^0\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^{n-1}f^n\\right) +\\\\&\\quad \\sum \\limits _{j=1}^{n-1} (-1)^{j}~\\left(\\hat{T}\\left(f^0\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^{j-1} f^{j} \\hat{\\partial }f^{j+1} \\ldots \\hat{\\partial }f^n\\right) +(-1)^n \\hat{T}\\left(\\hat{\\partial }f^0\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^{j-1} f^{j} \\hat{\\partial }f^{j+1} \\ldots \\hat{\\partial }f^{n-1}f^n\\right)\\right)\\big )\\\\&=(-1)^{n+1}~(n+1)\\hat{T}(f^nf^0\\hat{\\partial }f^1 \\ldots \\hat{\\partial }f^{n-1})=(-1)^{n+1} ~(n+1)\\phi (f^nf^0\\otimes f^1 \\otimes \\ldots \\otimes f^{n-1})\\end{array}$ Hence, $(1-\\lambda )(\\psi ^{\\prime })(f^0 \\otimes \\ldots \\otimes f^{n})=(-1)^{n+1} ~(n+1)\\phi (f^nf^0\\otimes f^1 \\otimes \\ldots \\otimes f^{n-1})$ On the other hand, using the definition of $\\psi ^{\\prime \\prime }$ and the fact that $(1-\\lambda )b=b^{\\prime }(1-\\lambda )$ , we have $(1-\\lambda )(\\psi ^{\\prime \\prime })=(1-\\lambda )(\\psi )-(1-\\lambda )(b\\psi _1)=(1-\\lambda )(\\psi )-b^{\\prime }(1-\\lambda )(\\psi _1)=(1-\\lambda )(\\psi )-b^{\\prime }\\theta ^{\\prime \\prime }$ Since $b\\psi \\in C^{n+1}_\\lambda (\\mathcal {C})$ , we have from (REF ) that $(1-\\lambda )(\\psi )=(B_0b+b^{\\prime }B_0)(\\psi )=b^{\\prime }B_0\\psi =b^{\\prime }\\theta =b^{\\prime }\\theta ^{\\prime }+b^{\\prime }\\theta ^{\\prime \\prime }$ .", "Hence, $(1-\\lambda )(\\psi ^{\\prime \\prime })=b^{\\prime }\\theta ^{\\prime }=\\frac{1}{n}~b^{\\prime }\\phi $ Since $\\phi =B\\psi \\in Z^{n-1}_\\lambda (\\mathcal {C})$ , $b\\phi =0$ and therefore $(1-\\lambda )(\\psi ^{\\prime \\prime })(f^0 \\otimes \\ldots \\otimes f^{n})=\\frac{1}{n}~(b^{\\prime }\\phi )(f^0 \\otimes \\ldots \\otimes f^{n})= \\frac{1}{n} ~(-1)^{n-1}\\phi (f^nf^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{n-1})$ The result now follows by comparing (REF ) and (REF ).", "Proposition 8.5 Let $\\mathcal {C}$ be a small $\\mathbb {C}$ -category and $\\lbrace {H}_t:\\mathcal {C} \\longrightarrow SHilb_2\\rbrace _{t \\in [0,1]}$ be a family of functors such that for each $X\\in Ob(\\mathcal {C})$ , we have $H_t(X)=H_{t^{\\prime }}(X)$ for all $t$ , $t^{\\prime }\\in [0,1]$ and $H_t(f)$ is of degree zero for each $f \\in Hom_\\mathcal {C}(X,Y)$ and $t\\in [0,1]$ .", "We put $H(X):=H_t(X)$ for all $t \\in [0,1]$ .", "Let $\\mathcal {F}$ be the family of operators $\\mathcal {F}=\\left\\lbrace (F(H(X))=\\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\\\\\end{pmatrix}\\right\\rbrace _{X\\in Ob(\\mathcal {C})}$ Let $p=2m$ be an even integer.", "We assume that (1) for each $f \\in Hom_\\mathcal {C}(X,Y)$ , the association $ t \\mapsto [\\mathcal {F},{H}_t(f)]$ is a continuous map $\\zeta _f:[0,1] \\longrightarrow \\mathcal {B}^p(H(X), H(Y)) \\qquad t \\mapsto [\\mathcal {F},{H}_t(f)]$ (2) for each $f \\in Hom_\\mathcal {C}(X,Y)$ , the association $p_f:[0,1] \\longrightarrow SHilb_{\\mathbb {Z}_2}({H}_t(X),{H}_t(Y))\\qquad t \\mapsto {H}_t(f)$ is piecewise strongly $C^1$ .", "Let $({H}_t,\\mathcal {F})$ be the corresponding $p$ -summable Fredholm modules over $\\mathcal {C}$ .", "Then, the class in $H^{p+2}_\\lambda (\\mathcal {C})$ of the $(p+2)$ -dimensional character of the Fredholm module $({H}_t,\\mathcal {F})$ is independent of $t$ .", "For any $t \\in [0,1]$ , let $\\phi _{t}$ be the $p$ -dimensional character of the Fredholm module $({H}_{t},\\mathcal {F})$ .", "We will show that $S(\\phi _{t_1})=S(\\phi _{t_2})$ for any $t_1, t_2 \\in [0,1]$ .", "By assumption, we know that there exists a finite set $R=\\lbrace 0= r_0\\le r_1 < ... <r_k\\le r_{k+1}=1\\rbrace \\subseteq [0,1]$ such that $p_f:[0,1] \\longrightarrow SHilb_{\\mathbb {Z}_2}({H}_t(X),{H}_t(Y))$ is continuously differentiable in each $[r_i,r_{i+1}]$ .", "By abuse of notation, we set for each $f \\in Hom_\\mathcal {C}(X,Y)$ : $\\delta _t(f):=p_f^{\\prime }(t)\\in SHilb_{\\mathbb {Z}_2}({H}_t(X),{H}_t(Y))$ Here, it is understood that if $t=r_i$ for some $1\\le i\\le k$ , we use the right hand derivative when $r_i$ is treated as a point of $[r_i,r_{i+1}]$ and the left hand derivative when $r_i$ is treated as a point of $[r_{i-1},r_i]$ .", "Using Lemma REF , we know that $\\delta _t(fg)={H}_t(f) \\circ \\delta _t(g) + \\delta _t(f) \\circ {H}_t(g)$ for any $t \\in [0,1]$ and for any pair of composable morphisms $f$ and $g$ in $\\mathcal {C}$ .", "For any $t \\in [0,1]$ and $1 \\le j \\le p+1$ , we set $\\psi _t^j(f^0 \\otimes \\ldots \\otimes f^{p+1}):=Tr\\left(\\epsilon {H}_t(f^0)[\\mathcal {F}, {H}_t(f^1)]\\ldots [\\mathcal {F}, {H}_t(f^{j-1})]\\delta _t(f^j)[\\mathcal {F}, {H}_t(f^{j+1})] \\ldots [\\mathcal {F}, {H}_t(f^{p+1})]\\right)$ Using the expression in (REF ) and the fact that $\\epsilon {H}(f)={H}(f)\\epsilon $ for any morphism $f \\in \\mathcal {C}$ , it may be easily verified that $b\\psi _t^j=0$ .", "For example, when $j=1$ , we have (suppressing the functor ${H}$ ) $\\begin{array}{ll}&(b\\psi _t^1)(f^0 \\otimes \\ldots \\otimes f^{p+2})\\\\& \\quad =\\sum \\limits _{i=0}^{p+1} \\psi _t^1(f^0 \\otimes \\ldots f^if^{i+1} \\otimes \\ldots \\otimes f^{p+2}) + \\psi _t^j(f^{p+2}f^0 \\otimes f^1 \\otimes \\ldots \\otimes f^{p+2})\\\\& \\quad = Tr\\left(\\epsilon f^0f^1\\delta _t(f^2)[\\mathcal {F},f^3]\\ldots [\\mathcal {F},f^{p+2}]\\right)-Tr\\left(\\epsilon f^0 \\delta _t(f^1f^2)[\\mathcal {F},f^3]\\ldots [\\mathcal {F},f^{p+2}]\\right)\\\\& \\qquad + Tr\\left(\\epsilon f^0 \\delta _t(f^1)[\\mathcal {F},f^2f^3]\\ldots [\\mathcal {F},f^{p+2}]\\right)-Tr\\left(\\epsilon f^0 \\delta _t(f^1)[\\mathcal {F},f^2][\\mathcal {F},f^3f^4]\\ldots [\\mathcal {F},f^{p+2}]\\right) + \\ldots \\\\& \\qquad \\ldots - Tr\\left(\\epsilon f^0 \\delta _t(f^1)[\\mathcal {F},f^2]\\ldots [\\mathcal {F},f^{p+1}f^{p+2}]\\right) + Tr\\left(\\epsilon f^{p+2}f^0 \\delta _t(f^1)[\\mathcal {F},f^2][\\mathcal {F},f^3f^4]\\ldots [\\mathcal {F},f^{p+1}]\\right) \\\\&\\quad =0\\end{array}$ We then define $\\begin{array}{ll}\\psi _t:=\\sum \\limits _{j=0}^{p+1} (-1)^{j-1} \\psi _t^j\\end{array}$ We have $b\\psi _t=0$ .", "For fixed $f$ , it follows from the compactness of $[0,1]$ and the assumptions (1) and (2) that the families $\\lbrace {H}_t(f) \\rbrace _{t \\in [0,1]}$ , $\\lbrace p_f(t)\\rbrace _{t\\in [0,1]}$ and $\\lbrace \\delta _t(f)\\rbrace _{t \\in [0,1]}$ are uniformly bounded.", "For the sake of simplicity, we assume that there is only a single point $r\\in R$ such that $t_1\\le r\\le t_2$ .", "Then, we form $\\psi \\in CN^{p+1}(\\mathcal {C})$ by setting $\\psi (f^0 \\otimes \\ldots \\otimes f^{p+1}):=\\int _{t_1}^{r}\\psi _t(f^0 \\otimes \\ldots \\otimes f^{p+1})dt+\\int _{r}^{t_2}\\psi _t(f^0 \\otimes \\ldots \\otimes f^{p+1})dt$ We now have $\\begin{array}{ll}&\\psi (id_{X_0} \\otimes f^0 \\otimes \\ldots \\otimes f^{p})\\\\&=\\int _{t_1}^{r}\\psi _t(id_{X_0} \\otimes f^0 \\otimes \\ldots \\otimes f^{p})dt+\\int _{r}^{t_2}\\psi _t(id_{X_0} \\otimes f^0 \\otimes \\ldots \\otimes f^{p})dt\\\\&=\\int _{t_1}^{r} \\big (\\sum \\limits _{j=0}^{p}(-1)^{j}Tr\\left(\\epsilon [\\mathcal {F}, {H}_t(f^0)]\\ldots [\\mathcal {F}, {H}_t(f^{j-1})]\\delta _t(f^j)[\\mathcal {F}, {H}_t(f^{j+1})] \\ldots [\\mathcal {F}, {H}_t(f^{p})]\\right)\\big )dt\\\\&\\textrm { }+\\int _{r}^{t_2} \\big (\\sum \\limits _{j=0}^{p}(-1)^{j}Tr\\left(\\epsilon [\\mathcal {F}, {H}_t(f^0)]\\ldots [\\mathcal {F}, {H}_t(f^{j-1})]\\delta _t(f^j)[\\mathcal {F}, {H}_t(f^{j+1})] \\ldots [\\mathcal {F}, {H}_t(f^{p})]\\right)\\big )dt\\\\\\end{array}$ Let $\\phi :[0,1] \\longrightarrow Z^p_\\lambda (\\mathcal {C})$ be the map given by $t \\mapsto \\phi _t$ .", "We now claim that $\\psi (id_{X_0} \\otimes f^0 \\otimes \\ldots \\otimes f^{p})=\\int _{t_1}^{r}\\phi ^{\\prime }(t)(f^0 \\otimes \\ldots \\otimes f^p) ~dt+\\int _{r}^{t_2}\\phi ^{\\prime }(t)(f^0 \\otimes \\ldots \\otimes f^p) ~dt$ Indeed, we have $\\begin{array}{ll}\\phi ^{\\prime }(t)(f^0 \\otimes \\ldots \\otimes f^p)&=\\lim \\limits _{s \\rightarrow 0}\\frac{1}{s}(\\phi _{t+s}-\\phi _t)(f^0 \\otimes \\ldots \\otimes f^p)\\\\&=\\lim \\limits _{s \\rightarrow 0}\\big (Tr\\big (\\epsilon \\frac{1}{s}\\left( {H}_{t+s}(f^0)-{H}_t(f^0)\\right) [\\mathcal {F}, {H}_{t+s}(f^{1})] \\ldots [\\mathcal {F}, {H}_{t+s}(f^{p})]\\big )\\\\& \\quad +Tr\\big (\\epsilon {H}_{t}(f^0) [\\mathcal {F},\\frac{1}{s}\\left({H}_{t+s}(f^1)-{H}_t(f^1)\\right)] [\\mathcal {F}, {H}_{t+s}(f^{2})] \\ldots [\\mathcal {F}, {H}_{t+s}(f^{p})]\\big )+ \\ldots \\\\& \\quad +Tr\\big (\\epsilon {H}_{t}(f^0)[\\mathcal {F}, {H}_{t}(f^1) ] \\ldots [\\mathcal {F},\\frac{1}{s}\\left({H}_{t+s}(f^p)-{H}_t(f^p)\\right)]\\big )\\big )\\end{array}$ By (1), we know that the association $t \\mapsto [\\mathcal {F},{H}_{t}(f)]$ is a continuous map for each morphism $f \\in \\mathcal {C}$ .", "Therefore, we have $\\begin{array}{ll}\\lim \\limits _{s \\rightarrow 0}\\big (Tr\\big (\\epsilon {H}_{t}(f^0)[\\mathcal {F}, {H}_{t}(f^1)]\\ldots [\\mathcal {F}, {H}_{t}(f^{j-1})] [\\mathcal {F},\\frac{1}{s}\\left({H}_{t+s}(f^{j})-{H}_t(f^j)\\right)] \\ldots [\\mathcal {F}, {H}_{t+s}(f^{p})]\\big )\\big )=\\\\=\\lim \\limits _{s \\rightarrow 0} (-1)^j \\big (Tr\\big (\\epsilon [\\mathcal {F},{H}_{t}(f^0)]\\ldots [\\mathcal {F}, {H}_{t}(f^{j-1})] \\frac{1}{s}\\left({H}_{t+s}(f^{j})-{H}_t(f^j)\\right)[\\mathcal {F}, {H}_{t+s}(f^{j+1})] \\ldots [\\mathcal {F}, {H}_{t+s}(f^{p})]\\big )\\big )\\\\= (-1)^j Tr\\big (\\epsilon [\\mathcal {F},{H}_{t}(f^0)][\\mathcal {F}, {H}_{t}(f^1)]\\ldots [\\mathcal {F}, {H}_{t}(f^{j-1})] \\delta _t(f^j) [\\mathcal {F}, {H}_{t}(f^{j+1})] \\ldots [\\mathcal {F}, {H}_{t}(f^{p})]\\big )\\end{array}$ From this, we obtain $\\begin{array}{ll}&\\int _{t_1}^{r}\\phi ^{\\prime }(t) (f^0 \\otimes \\ldots \\otimes f^p) ~dt+\\int _{r}^{t_2}\\phi ^{\\prime }(t) (f^0 \\otimes \\ldots \\otimes f^p)~dt\\\\&=\\int _{t_1}^{r} \\sum \\limits _{j=0}^p (-1)^j Tr\\big (\\epsilon [\\mathcal {F},{H}_{t}(f^0)][\\mathcal {F}, {H}_{t}(f^1)]\\ldots [\\mathcal {F}, {H}_{t}(f^{j-1})] \\delta _t(f^j)[\\mathcal {F}, {H}_{t}(f^{j+1})] \\ldots [\\mathcal {F}, {H}_{t}(f^{p})]\\big )~ dt\\\\&\\textrm { }+\\int _{r}^{t_2} \\sum \\limits _{j=0}^p (-1)^j Tr\\big (\\epsilon [\\mathcal {F},{H}_{t}(f^0)][\\mathcal {F}, {H}_{t}(f^1)]\\ldots [\\mathcal {F}, {H}_{t}(f^{j-1})] \\delta _t(f^j) [\\mathcal {F}, {H}_{t}(f^{j+1})] \\ldots [\\mathcal {F}, {H}_{t}(f^{p})]\\big )~ dt\\\\&=\\psi (id_{X_0} \\otimes f^0 \\otimes \\ldots \\otimes f^{p})\\end{array}$ Hence $\\begin{array}{ll}\\psi (id_{X_0} \\otimes f^0 \\otimes \\ldots \\otimes f^{p})&=\\phi _{t_2}(f^0 \\otimes \\ldots \\otimes f^{p})-\\phi _{r}(f^0 \\otimes \\ldots \\otimes f^{p})+\\phi _{r}(f^0 \\otimes \\ldots \\otimes f^{p})-\\phi _{t_1}(f^0 \\otimes \\ldots \\otimes f^{p})\\\\&=\\phi _{t_2}(f^0 \\otimes \\ldots \\otimes f^{p})-\\phi _{t_1}(f^0 \\otimes \\ldots \\otimes f^{p})\\\\\\end{array}$ Since $\\psi (f^0 \\otimes \\ldots \\otimes f^{p} \\otimes id_{X_0} )=0$ , we now have $\\begin{array}{ll}(B_0\\psi ) (f^0 \\otimes \\ldots \\otimes f^{p})&=\\psi (id_{X_0} \\otimes f^0 \\otimes \\ldots \\otimes f^{p})-\\psi (f^0 \\otimes \\ldots \\otimes f^{p} \\otimes id_{X_0} )\\\\&=(\\phi _{t_2}-\\phi _{t_1})(f^0 \\otimes \\ldots \\otimes f^{p})\\end{array}$ Since $b\\psi =0$ , using Proposition REF and the fact that $\\phi _{t_2}-\\phi _{t_1} \\in Ker(1-\\lambda )$ , we have $0=S(B\\psi )=S(AB_0\\psi )=(p+1)S(\\phi _{t_2}-\\phi _{t_1})$ This proves the result.", "Theorem 8.6 Let $\\mathcal {C}$ be a small $\\mathbb {C}$ -category and $\\lbrace \\rho _t:\\mathcal {C} \\longrightarrow SHilb_2\\rbrace _{t \\in [0,1]}$ be a family of functors such that for each $X\\in Ob(\\mathcal {C})$ , we have $\\rho _t(X)=\\rho _{t^{\\prime }}(X)$ for all $t$ , $t^{\\prime }\\in [0,1]$ .", "We put $\\rho (X):=\\rho _t(X)$ for all $t \\in [0,1]$ .", "Further, for each $t \\in [0,1]$ , let $\\mathcal {F}_t:=\\left\\lbrace \\mathcal {F}_t(X):=\\begin{pmatrix} 0 & \\mathcal {Q}_t(X) \\\\ \\mathcal {P}_t(X) & 0 \\\\\\end{pmatrix}:\\rho (X)\\longrightarrow \\rho (X) \\right\\rbrace _{X\\in Ob(\\mathcal {C})}$ with $\\mathcal {P}_t(X)=\\mathcal {Q}_t^{-1}(X)$ be such that $({\\rho }_t,\\mathcal {F}_t)$ is a $p$ -summable Fredholm module over the category $\\mathcal {C}$ .", "We set $\\rho (X)=\\rho ^{\\prime }(X)\\oplus \\rho ^{\\prime }(X)\\in SHilb_2$ .", "We further assume that for some even integer $p$ and for any $f \\in Hom_\\mathcal {C}(X,Y)$ , we have (1) $t \\mapsto {\\rho }_t^+(f)-\\mathcal {Q}_t{\\rho }_t^-(f)\\mathcal {P}_t$ is a continuous map from $[0,1]$ to $\\mathcal {B}^p({\\rho }^{\\prime }(X),{\\rho }^{\\prime }(Y))$ , where $\\rho _t^\\pm $ are the two components of the morphism $\\rho _t$ of degree zero.", "(2) $t \\mapsto {\\rho }_t^+(f)$ and $t \\mapsto \\mathcal {Q}_t{\\rho }_t^-(f)\\mathcal {P}_t$ are piecewise strongly $C^1$ maps from $[0,1]$ to $SHilb(\\rho ^{\\prime }(X),\\rho ^{\\prime }(Y))$ .", "Then, the $(p+2)$ -dimensional character $\\text{ch}^{p+2}({\\rho }_t,\\mathcal {F}_t) \\in H^{p+2}_\\lambda (\\mathcal {C})$ is independent of $t \\in [0,1]$ .", "For each $t \\in [0,1]$ , we set $\\mathcal {T}_t:=\\begin{pmatrix} 1 & 0 \\\\ 0 & \\mathcal {Q}_t \\\\\\end{pmatrix}$ .", "Then, $\\mathcal {T}_t^{-1}=\\begin{pmatrix} 1 & 0 \\\\ 0 & \\mathcal {P}_t \\\\\\end{pmatrix}$ and $\\mathcal {F}^{\\prime }_t:=\\mathcal {T}_t\\mathcal {F}_t\\mathcal {T}_t^{-1}=\\begin{pmatrix} 0 & 1\\\\ 1& 0 \\\\\\end{pmatrix}.$ For each $t \\in [0,1]$ , we also define a functor ${H}_t:\\mathcal {C} \\longrightarrow SHilb_{\\mathbb {Z}_2}$ given by ${H}_t(X):={\\rho }(X) \\qquad {H}_t(f):=\\mathcal {T}_t{\\rho }_t(f)\\mathcal {T}_t^{-1}$ Then, we have $[\\mathcal {F}^{\\prime }_t,{H}_t(f)]=\\begin{pmatrix} 0 & \\mathcal {Q}_t{\\rho _t}^-(f)\\mathcal {P}_t -{\\rho }_t^+(f)\\\\ {\\rho }_t^+(f)- \\mathcal {Q}_t{\\rho _t}^-(f)\\mathcal {P}_t & 0\\end{pmatrix}$ Therefore, using assumption (1), we see that the map $t \\mapsto [\\mathcal {F}^{\\prime },{H}_t(f)]$ from $[0,1]$ to $\\mathcal {B}^p({H}_t(X),{H}_t(Y))$ is continuous for each $f \\in Hom_\\mathcal {C}(X,Y)$ .", "Further, ${H}_t(f)=\\mathcal {T}_t{\\rho }_t(f)\\mathcal {T}_t^{-1}=\\begin{pmatrix} {\\rho }_t^+(f) & 0\\\\ 0 & \\mathcal {Q}_t{\\rho _t}^-(f)\\mathcal {P}_t\\end{pmatrix}$ Therefore, by applying assumption (2), we see that the map $t \\mapsto {H}_t(f)$ is piecewise strongly $C^1$ .", "Since trace is invariant under similarity, the result now follows using Proposition REF .", "Theorem 8.7 Let $\\mathcal {C}$ be a small $\\mathbb {C}$ -category and $\\lbrace \\rho _t:\\mathcal {C} \\longrightarrow SHilb_2\\rbrace _{t \\in [0,1]}$ be a family of functors such that for each $X\\in Ob(\\mathcal {C})$ , we have $\\rho _t(X)=\\rho _{t^{\\prime }}(X)$ for all $t$ , $t^{\\prime }\\in [0,1]$ .", "We put $\\rho (X):=\\rho _t(X)$ for all $t \\in [0,1]$ .", "Further, for each $t \\in [0,1]$ and $X\\in Ob(\\mathcal {C})$ , let $\\mathcal 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2105.11736
[ [ "Dynamics of vortex defect formation in two dimensional Coulomb crystals" ], [ "Abstract We study the non-equilibrium dynamics of two dimensional planar ion Coulomb crystals undergoing a structural buckling transition to a three plane configuration, driven by a reduction of the transverse confining frequency.", "This phase transition can be theoretically modeled using a mapping to a two dimensional Ginzburg-Landau theory with complex order parameter field.", "We demonstrate that finite rate quenches result in creation of stable topological vortices, which are localized point regions around which the phase of the order parameter field winds a multiple of 2{\\pi}.", "The density of the defects as a function of quench rate is investigated using molecular dynamics simulations and its scaling is shown to be consistent with Kibble-Zurek theory of defect formation.", "Following the quench, the annihilation of vortex and anti-vortex pairs results in the relaxation of defect density that follows a diffusive scaling with a logarithmic correction.", "This work highlights the potential for investigating complex non-equilibrium statistical physics of topological defects in an experimentally accessible ion trap setting." ], [ "Introduction", "Trapped ions is one of the most prominent quantum technologies.", "Doppler laser cooling can bring ions to low temperatures at which the ions crystallize forming regular structures, known as Coulomb crystals, whose shape is determined by the trapping parameters.", "Linear chain crystals in Paul traps have the simplest phonon spectrum and have been widely used in metrology [1], quantum computing [2], [3] and quantum simulations [4].", "In Penning traps, large two dimensional planar crystals can be readily created and these structures have also been used in metrology and quantum information processing applications [5].", "Beyond these simple Coulomb crystal geometries, there is great interest in exploring more complex structural phases and the transitions between them [6], [7], [8], [9], [10].", "The strong long-range Coulomb interactions between particles leads to highly non-linear non-trivial dynamics, whose investigation is of fundamental interest in the fields of nonlinear science, complex systems and solid-state physics; it is useful as a platform for studying complex non-linear and non-equilibrium dynamics in areas including the simulation of Klein-Gordon fields on a lattice [11], Kibble-Zurek mechanism of defect formation [12], [13], [14], dynamics of discrete solitons [15], [16], [17], dry friction [18], energy transport [19], [20] and synchronization [21].", "Coulomb crystals with more complexity also provide new lattice geometries for quantum simulations [22].", "In this paper, we investigate numerically the non-equilibrium structural phase transition from a quasi-two-dimensional 1-plane crystal to a 3-plane crystal.", "We focus on the Kibble-Zurek mechanism of the formation of topological defects and the subsequent coarsening dynamics of annihilation of defects and anti-defect pairs.", "Previous studies of KZ mechanism in ion crystal systems have focused on the linear to zigzag phase transition in a quasi-one-dimensional system [12], [13].", "This are symmetry breaking phase transitions and the resulting defects are either kinks [12], [13] if the $Z_2$ symmetry is broken or helical twists [14] if the $U(1)$ symmetry is broken.", "The 1-plane to 3-plane structural phase transition considered in this paper can be described by an XY 6-clock model with an intermediate Kosterlitz-Thouless phase.", "The defects are $U(1)$ point vortices whose physics is considerably richer [23].", "Previously, simulations of finite quenches in a two dimensional XY spin model have shown that the density of vortices is dictated by both the KZ mechanism and coarsening dynamics of annihilation of vortex/anti-vortex pairs [24].", "This observation was corroborated in an experimental study of colloids undergoing a phase transition via Kosterlitz– Thouless–Halperin–Nelson–Young (KTHNY) melting scenario [25].", "The molecular dynamics simulations of microscopic ion crystal dynamics presented in this paper are consistent with the dynamics observed in [24], [25].", "An observed deviation from the theoretical coarsening scaling may be attributed to the pinning effect of the emergent 6-clock potential.", "Our work introduces a new platform for investigating the non-equilibrium KT phase transition and more generally the collective dynamics of interacting vortices.", "The paper is organized as follows.", "Section presents the microscopic model, reviews its mapping to the Ginzburg-Landau field theory and the phase diagram of a 1-plane to 3-plane structural transition.", "Section uses molecular dynamics simulations to demonstrate the existence of topological defects in the 3-plane phase and presents an algorithm for determining their location.", "Section focuses on molecular dynamics investigation of finite rate quenches, where the KZ mechanism and coarsening dictate the evolution of the average number of defects in the system." ], [ "Ginzburg-Landau model of 1- to 3-plane transition", "Ion traps confine repulsively interacting ions in space either by rapidly varying oscillatory electric fields, as in the Paul traps, or a combination of static electric and magnetic fields, as in the Penning traps [26].", "The dynamics of ion Coulomb crystal can be often approximated with the so-called pondermotive approximation or pseudopotential theory, which replaces the time-varying trapping fields experienced by particles by an effective time-independent harmonic potential.", "The laser cooling reduces the temperature of the ions such that they can form regular crystal-like configurations, whose overall shape is determined by the trap parameters.", "We will consider a system of $N$ ions confined to a periodic cell in the $x$ -$y$ plane and by the harmonic confinement in the $z$ -direction.", "The potential energy is given by $V = \\frac{1}{2} m\\omega _z \\sum _j^N z_j^2 + \\mathcal {K} \\sum _{i<j}\\frac{1}{|\\textbf {r}_i-\\textbf {r}_j|} $ where $\\textbf {r}_j = (x_j,y_j,z_j)$ are the coordinates of the $j$ th ion, $\\mathcal {K}\\equiv q^2/4\\pi \\epsilon _0$ , $q$ is the charge of the ion, $\\epsilon _0$ is the vacuum permittivity, $m$ is the mass of the ion and $\\omega _z$ is the trapping frequency in the $z$ -direction.", "The periodic boundary conditions results in a homogeneous spacing in the ion crystal.", "In a real experimental system, the open boundary conditions and the harmonic confinement in the $x$ and $y$ direction would result in inhomogeneous spacing between the ions; the ions are closer together in the centre of the crystal and further apart near the edges.", "The system with periodic boundary condition can be viewed as an approximation to a central region of a large ion crystal, where the spacing is approximately homogeneous and the boundary effects can be neglected.", "Above a certain critical value of $\\omega _z=\\omega _z^{(c)}$ the lowest energy configuration is a planar triangular lattice.", "When the confining frequency is reduced to below $\\omega _z^{(c)}$ , the 1-planar crystal configuration undergoes a buckling structural transition into a 3-planes, all of which in triangular lattice geometry but with double the lattice spacing (see Figure REF ).", "This buckling instability has been predicted in an early theoretical work by Dubin [27] and has been observed experimentally in [28].", "Recently, Podolsky et.", "al.", "[23] derived a Ginzburg-Landau (GL) field theory for this transition thereby proving that it is in the universality class of a two dimensional XY model [23].", "The GL field theory is derived by Taylor expanding the non-linear Coulomb interaction term in equation (REF ) in displacements around the equilibrium lattice positions.", "In [23] it was shown that one must keep the terms up to the sixth order in the expansion to correctly capture the critical properties of the structural phase transition.", "The GL free energy density is give by $\\frac{f}{\\mathcal {K}}= \\frac{\\gamma }{2} |\\nabla \\psi |^2+ \\epsilon |\\psi |^2+u|\\psi |^4+v|\\psi |^6+\\frac{w}{2}\\left[\\psi ^6 + (\\psi ^*)^6\\right] $ where $\\epsilon =\\frac{1}{\\sqrt{3}}\\left(\\frac{m\\omega _z^2 a^2}{2\\mathcal {K}}-I_2\\right)$ , $u=3/\\sqrt{3}/4 I_4$ , $w=\\frac{5}{8\\sqrt{3}}I_6$ , $v=-\\frac{25}{4\\sqrt{3}}I_6$ , $I_2=6.683$ , $I_4=3.56$ , $\\gamma =0.223$ , $I_6=2.558$ and $a$ is the lattice spacing.", "The order parameter field at a lattice point with coordinates $(x_j,y_j$ , $\\psi (x_j,y_j)$ , is an implicit function of the transverse displacement $z_j = \\textrm {Re}\\left[\\psi e^{i\\textbf {K}\\cdot \\textbf {r}_j}\\right].", "$ Here $\\textbf {K}$ is the base vectors of the first Brillouin zone of the triangular lattice given by $\\textbf {K} = (4\\pi /3,0)$ , $\\textbf {r}_i = n_1 \\textbf {a}_1 + n_2\\textbf {a}_2$ , with $\\textbf {r}_{1,2} = (\\frac{1}{2},\\pm \\sqrt{3}/2)$ .", "The order parameters $\\psi $ is complex and can be expressed as $\\psi =|\\psi |e^{i\\theta }$ .", "Figure: Structural buckling transition between a 1-plane and 3-plane configuration.", "(a) The transverse displacement of the ions in a crystal at different values of ω z \\omega _z in the vicinity of the critical ω z (c) \\omega _z^{(c)}.", "(b) The triangular lattice structure of the 3-plane phase.Figure: (a) Single well potential of GL theory for ϵ>0\\epsilon >0 (b) Mexican hat potential for ϵ<0\\epsilon <0.The potential energy for the mean field configuration, which neglects the spatial fluctuations in the order parameter, $V(\\psi )/\\mathcal {K}=\\epsilon |\\psi |^2+u|\\psi |^4+v|\\psi |^6+\\frac{w}{2}\\left[\\psi ^6+(\\psi ^*)^6\\right]$ , is shown in Figure REF .", "For $\\epsilon >0$ the order parameter is zero, $\\psi =0$ , and the system is in the 1-plane phase.", "The potential is a single well and since the order parameter has no preferred direction, the 1-plane phase is disordered.", "For $\\epsilon <0$ the order parameter is non-zero and the system is in the 3-plane phase.", "The potential is a Mexican hat but with 6 equally spaced wells which correspond to the local order of the 6 degenerate lattice arrangements shown in Figure REF (b).", "This corresponds to the 6-clock phase which has the discrete $Z_2\\times Z_3$ symmetry.", "At higher energies the order parameter can easily overcome the energy barrier between the neighboring minima, and the symmetry changes to the broken $U(1)$ continuous symmetry.", "Two dimensional systems with the broken $U(1)$ symmetry in the order parameters support topological defect vortex configuration, which are localized regions where the field winds around the Mexican hat potential.", "The presence of these topological defects drastically alters the physics of the system leading to the existence of the KT phase [29], [30].", "Thus, near the critical point of the 1-plane to 3-plane structural phase transition the system can exist in 3 phases, disordered, KT and the 6-face clock phases, depending on the value of $\\epsilon $ and temperature $T$ .", "The phase diagram was derived in [23] and is sketched in Figure REF .", "The KT phase is characterized by a change in behavior in correlation length.", "For $T>T_{KT}$ , the system is disordered, the correlation length decays exponentially and there is a finite density of unbound vortices.", "For $T<T_{KT}$ , there is a quasi-long range order with a power law decays of the correlation length and vortices and anti-vortices form bound pairs.", "For $T<T_6$ , the system is in the 6-clock phase, which again exhibits a long-range order with exponentially decaying correlation length.", "Figure: Phase diagram of a structural 1-plane to 3-plane phase transition of a Coulomb crystals.", "The control parameter space determined by the temperature TT and coefficient ϵ\\epsilon which is a function of the ratio mω z 2 a 2 /𝒦m\\omega _z^2a^2/\\mathcal {K}.", "The line T KT T_{KT} indicates a transition between 1-plane phase and a 3-plane phase in quasi-long-range ordered KT phase.", "The line T 6 T_6 indicates a transition between the KT phase and the long range ordered 3-plane phase with the Z 6 Z_6 symmetry.Figure: a) Example of a configuration with topological defects in the 3-plane structural phase represented as color plot (top) or arrow plot (bottom).", "The observed defects are asters and vortices (topological charge, s=+1s=+1) or cross-hairs and anti-vortices (topological charge, s=-1s=-1) (b) Instantaneous configurations of a quasi-two-dimensional ion crystal with 9858 Be + \\textrm {Be}^{+} ions at a temperature of 20μ20 \\mu K and γ=8.96×10 -7 ps -1 \\gamma = 8.96 \\times 10^{-7} \\textrm {ps}^{-1} in the 3-plane structural phase at the end of a non-equilibrium quench from a disordered single plane phase at different rate τ Q -1 \\tau _Q^{-1}.", "The quench is implemented by reducing the transverse confining frequency ω z \\omega _z.", "The density of topological defects increases with increasing quench rate.", "Each color corresponds to a phase angle Θ\\Theta of the local order parameter ψ=|ψ|e iΘ \\psi =|\\psi | e^{i\\Theta }." ], [ "Topological defects in the 3-plane phase", "To verify the prediction of the existence of the topological defects, we have performed molecular dynamics simulations of ion Coulomb crystals confined in a box with periodic boundary conditions in the $x$ and $y$ direction.", "The topological defects are produced by quenching a system from a 1-plane disordered phase into the 3-plane phase.", "We use a Langevin thermostat to simulate the interaction of the ions with the cooling laser beam, which thermalized the ions.", "The equations of motion for the $j$ th ion are given by $m \\partial _{tt} x_j & = & - m\\gamma \\partial _t x_j -\\partial _{x_j} V_c + \\theta _{xj}(t) \\\\m \\partial _{tt} y_j & = & - m\\gamma \\partial _t y_j -\\partial _{y_j} V_c + \\theta _{yj}(t) \\\\m \\partial _{tt} z_j & = & -m\\omega (t)^2- m\\gamma \\partial _t z_j -\\partial _{z_j} V_c + \\theta _{zj}(t) ,$ where $m$ is the mass of the ion, $\\omega (t)$ is the transverse confining frequency, $V_c$ is the Coulomb interaction energy, $\\gamma $ is the damping coefficient.", "The force $(\\theta _{xj}, \\theta _{yj},\\theta _{zj})$ is is the stochastic thermal force satisfying $\\langle \\theta _{\\alpha ,j}(t) \\rangle = 0$ and $\\langle \\theta _{\\alpha ,j}(t) \\theta _{\\beta ,k}(t^{\\prime }) \\rangle = 2 m\\gamma k_B T \\delta _{\\alpha \\beta } \\delta _{jk} \\delta (t-t^{\\prime })$ , where $\\langle ...\\rangle $ denotes ensemble averaging.", "The integration of the equation of motion is performed using GPU accelerated OpenMM [31] framework, and Ewald sums are used to approximate the Coulomb interactions in the $x$ and $y$ directions.", "To determine the location of defects in a given ion crystal configuration, one must compute the local order parameter field $\\psi $ using the individual ion coordinates.", "Using equation (REF ) the order parameter in the 6-clock phase can be written as $z_i & = & \\textrm {Re}[\\Psi e^{\\textbf {K}\\cdot \\textbf {r}_i}] \\\\& = & |\\psi |\\cos \\left(\\frac{\\pi (2n_i+1)}{6}+\\delta \\Theta _i +\\textbf {K}\\cdot \\textbf {r}_i\\right),$ where $n_i\\in \\lbrace 1,...,6\\rbrace $ determines the clock state at the position of the $i$ th ion and $\\delta \\Theta _i$ is the fluctuation about this phase.", "Denoting the splitting between the planes as $h\\equiv \\textrm {max}_i(|\\langle z_i \\rangle |))$ , where $z_i$ is the $z$ -coordinate of an ion either in the + or - sublattice of the 3-plane structural phase, one finds that $h=|\\psi |\\cos (\\pi /6)\\langle \\cos (\\delta \\Theta )\\rangle $ and equation (REF ) can be written as $z_i = \\frac{h}{\\cos (\\pi /6)} \\cos \\left(\\delta \\Theta _i+\\frac{\\pi (2n_i+1)}{6}+ \\textbf {K}\\cdot \\textbf {r}_i\\right).", "$ The values of clock state at each point, $n_i$ , are determined by allocating a phase value to an arbitrary chosen patch of three adjacent ions and then assigning all other patches the best matching value relative to this chosen reference.", "After assigning $n_i$ , the correction term, $\\delta \\Theta _i$ , is obtained by solving numerically the non-linear equation (REF ) using gradient descent algorithm.", "Figure REF (a) shows a typical configuration with topological defects.", "The locations of the defects are determined by finding localized regions on the boundary of which the phase $\\Theta $ winds an integer multiple of $2\\pi $ .", "The topological charge $s$ of a defect is defined as a winding number along the contour $C$ encircling the defect i.e.", "$s = \\frac{1}{2\\pi } \\int _C \\partial _l \\Theta dl$ where $l$ is the position along the path of the chosen contour.", "In our simulations, we observe 4 types of point defects: asters and vortices with charge +1, and cross-hairs and anti-vortices with charge -1.", "The existence of such defects may be predicted on general homotopy theoretic arguments [32].", "Here, we have demonstrated that in 3-plane Coulomb crystal all four types of defects are energetically stable.", "The crystal configuration shown in the Figure REF (a) is in the 6-face clock phase rather than the KT phase, as there are clear boundary lines between the domains separating the regions with phase angles of the possible minimum energy configuration.", "Increasing the temperature in the system blurs the boundary lines until the system reaches a KT phase, where there is no energetically preferred phase angles $\\Theta $ .", "In the KT phase there is no long-range order in the system - the defects are mobile and oppositely charged defects tend to annihilate.", "The long range order established by entering the clock phase and the energy of the $Z_6$ domain walls, appears to have a stabilizing effect on the point defects, reducing significantly their mobility." ], [ "Kibble-Zurek mechanism of defect formation", "Quenching the system from a disordered phase into the ordered phase at different rates results in different defect densities as can be seen in Figure REF (b).", "The relationship between the number of defects and the quench rates across symmetry breaking phase transitions was initially investigated by Kibble in the context of cosmology [33] and Zurek in the context of condensed matter [34], in what became known as Kibble-Zurek (KZ) theory.", "Experimentally, the KZ mechanism for a 6-clock model has been investigated in the context of ferroelectric materials [35], [36].", "One should note, however, that in ferroelectrics the transition happens in three dimensions where there is no KT phenomena.", "Lets review the KZ mechanism as applied to continuous second order phase transitions.", "Consider approaching the critical point of a symmetry breaking second order phase transition.", "The correlation length $\\xi $ , defined by $\\langle \\psi (0,t)\\psi (r,t)\\rangle \\sim e^{-r/\\xi }$ , diverges as a power law of the control parameter $\\xi =\\xi _0/|\\epsilon |^\\mu $ , where $\\mu $ is the critical exponent.", "The system also slows down on the approach to the critical point i.e.", "the relaxation time, $\\tau $ , defined as $\\langle \\psi (0)\\psi (t)\\rangle \\sim e^{-t/\\tau }$ diverges as $\\tau = \\tau _0/|\\epsilon |^{\\mu }$ .", "KZ mechanism proposes that the correlation length freezes out when the relaxation time is equal to the time left until the crossing of the point i.e.", "the freeze out time $\\hat{t}$ is found by solving $\\tau (\\hat{t})= \\tau _Q$ .", "This “cross-over\" time $\\hat{t}$ marks a transition between the adiabatic dynamical regime, where the correlation lengths adjust to its equilibrium value, and impulsive regime, where the correlation length is fixed.", "For a linear quench $\\epsilon = t/\\tau _Q$ , one observes $\\hat{t} = (\\tau _0 \\tau ^{\\mu }_Q)^{1/(1+\\mu )}$ and the freeze-out correlation length is $\\hat{\\xi } = \\xi (\\hat{t}) = \\xi _0 (\\tau _q/\\tau _0)^{\\nu /(1+\\mu )}$ .", "In the two dimensional system, the number of defects $n$ is inversely proportional to the square of the correlation length scale in the system i.e.", "$n\\propto \\xi ^{-2}$ .", "Thus, the KZ prediction for the defect density in the end of the quench is $n_f = \\hat{\\xi }^{-2}\\sim (\\tau _q/\\tau _0)^{-2\\nu /(1+\\mu )}$ .", "For KT phase transition, the same arguments applies except the correlation length diverge exponentially at the critical point rather than algebraically i.e.", "$\\xi \\sim \\exp (a |\\epsilon |^{-\\mu }|)$ and $\\tau \\sim \\exp (b |\\epsilon |^{-\\nu }|)$ , where $\\epsilon $ .", "In this case, the equation for the freeze-out time, $\\hat{t}=\\tau _Q$ cannot be solved exactly, but can still be determined numerically [24], [25].", "An additional complication is that the generated vortex defects can be highly mobile and the process of annihilation of defects with opposite topological charges leads to the growth of correlation length in the system.", "Thus, the number of defects in the “impulsive\" regime is not fixed but continuously decreases as the defects annihilate.", "In the context of sudden quenches, this grows of correlation length is known as coarsening.", "In [37], [24], the authors propose that for a quench into the KT phase, one should account for both KZ mechanism and coarsening expressing the time evolution of correlation length as Figure: Evolution of the average number of defects following quenches from 1-plane disordered phase into a 3-lane 6 face clock phase at finite rate 1/τ Q 1/\\tau _Q.", "The simulated system contained 9858 Be + \\textrm {Be}^{+} ions at a temperature of 20μ20\\mu K and γ=8.96×10 -7 ps -1 \\gamma = 8.96 \\times 10^{-7} \\textrm {ps}^{-1} confined in a periodic box in the xyxy directions of size 1395.00×1376.981395.00\\times 1376.98 μ\\mu m. The starting and ending frequencies were set to ω z (i) =7.70 Mhz \\omega _z^{(i)}=7.70\\textrm { Mhz} and ω z (f) =7.42 MHz \\omega _z^{(f)}=7.42\\textrm { MHz}, and the critical frequency is ω z (i) =7.60 MHz \\omega _z^{(i)}=7.60\\textrm { MHz}.Figure: Late time evolution of the average number of defects for three quenches at different rates 1/τ Q 1/\\tau _Q.", "The slopes of the dashed lines is -1, indicating a nlnn∼t -1 n\\ln n \\sim t^{-1} coarsening scaling.Figure: Plot of the number of defects at the end of the quench, n f n_f as a function of quench rates.$\\xi (t)&= {\\left\\lbrace \\begin{array}{ll}\\xi _{eq}(t),& \\textrm {for }t <\\hat{t}\\\\\\hat{\\xi } + f(t),& \\textrm {for }t \\ge \\hat{t} \\end{array}\\right.", "}, $ where $f(t)$ is the function representing the growth of the correlation length due to coarsening.", "Equation (REF ) expresses the idea that before the KZ freeze-out time, $\\hat{t}$ , the correlation length adopts its thermal equilibrium value, and after crossing $\\hat{t}$ the correlation length is growing via coarsening.", "In a two dimensional system, which is quenched from disordered into ordered phase in the presence of linear damping, one expects the coarsening to proceed via diffusing law, where the correlation length grows as the square root of time $f(t) \\sim t^{1/2}$ [38].", "Several studies noted that the approach to diffusive law can be slow and that the coarsening is more accurately described by including a logarithmic correction, $f(t) \\sim \\left(t/\\ln t\\right)^{1/2}$ [39], [40].", "In [24], a good agreement between the analytic expression (REF ) and the numerical simulations of quenches in a two dimensional XY model is found.", "We have carried out molecular dynamics simulations to verify whether the dynamically crossing the 1-plane to 3-plane structural transition will result in the same defect scaling behavior.", "We simulate the dynamics of $N=9858$ $\\textrm {Be}^{+}$ ions at a temperature of $20\\mu \\textrm {K}$ and $\\gamma = 8.96 \\times 10^{-7} \\textrm {ps}^{-1}$ confined to a periodic box in an $xy$ directions and a harmonic potential in the $z$ directions by numerically integrating equation (REF )-().", "The system is first thermalized at a confinement frequency $\\omega _z^{(i)}$ sufficiently far from the critical frequency $\\omega _z^{(c)}$ such that the correlation length is small i.e.", "of the order of the lattice spacing.", "After that the confining frequency is decreased linearly at a rate, $\\omega _z = \\omega _z^{(i)}+t(\\omega _z^{(f)}-\\omega _z^{(i)})/\\tau _Q$ , such that the system undergoes a transition between a 1-plane and 3-plane structural phases at a rate $1/\\tau _Q$ .", "The defects are counted using the method presented in Section , and in order to obtain the ensemble averaged defect number $\\langle n(t) \\rangle $ the simulations are carried out $\\sim 140$ times for each $\\tau _Q$ .", "Figure REF shows the evolution of the defect density as function of time following phase transition at different quench rates.", "Several dynamical regimes can be seen in the figure.", "Initially, the system is in the 1-plane phase far from phase transition point, the correlation length is small and the density of defects is large.", "As $\\omega _z$ approaches the critical frequency, the defect density decreases adjusting to the new equilibrium values.", "This regime crosses over to rapid relaxation, where the density of vortices decreases significantly.", "According to the KZ theory, the crossover occurs at $t>\\hat{t}$ .", "The relaxation continues at a slower pace consistent with the power-law coarsening dynamics.", "Finally, the relaxation slows further as the system enters the clock phase and the domains are stabilized.", "In Figure REF we zoom in into the late time evolution at three different quench rates to verify whether the relaxation in the system is due to coarsening.", "The growth of the correlation length due to coarsening in the KT phase is predicted to follow $\\xi \\sim (t \\ln t)^{1/2}$ and consequently the number of defects follow a powerlaw with a logarithmic correction $n \\ln n \\sim t^{-1}$ [39].", "In Figure REF , one can see that there is a region where the powerlaw with the logarithmic correction is valid.", "However, this regime crosses over fairly quickly to a regime of slower relaxation, which we believe is either due to the stabilizing effect of entering into the clock phase or the pinning of the vortices by the discreteness of the lattice.", "Finally, Figure REF shows the number of defects at the end of the quench at $t=\\tau _Q$ as a function of $\\tau _Q$ .", "We observe a powerlaw scaling with an exponent of -0.80 i.e.", "$n(t=\\tau _Q) \\propto \\tau _Q^{-0.80}$ .", "While an accurate determination of the KZ scaling law involves numerical estimation of $\\hat{t}$ , previous simulation results for quenches of the two dimensional XY spin model found a powerlaw scaling of -0.72 at intermediate values of $\\tau _Q$ .", "The exponent of -0.80 observed in our simulations is consistent with these previous findings." ], [ "Conclusion", "In this paper we have studied the non-equilibrium dynamics of a planar Coulomb crystals undergoing a structural transition from a 1-planar to 3-planar configurations.", "The mapping to the Ginzburg-Landau theory reveals that this phase transition corresponds to a transition from a disordered paramagnetic phase to an ordered 6-clock phase with an intermediate KT phase.", "We used molecular dynamics simulations to confirm that the KT and the 6-clock phase support stable topological defect structures: vortices, anti-vortices, cross-hairs and asters.", "The density of defects depends on the quench rate of the structural transition as predicted by the Kibble-Zurek theory of defect formation.", "We have verified that the defect scaling law is consistent with KZ scaling previously observed in numerical simulation of two dimensional XY spin model.", "Moreover, we have observed signatures of coarsening due to defect/anti-defect annihilation, which follows a relaxation powerlaw with a logarithmic correction.", "Our work demonstrates that large planar ion Coulomb crystals can be used as model for studying non-equilibrium statistical physics of vortices in an effectively quasi-two-dimensional system.", "Such Coulomb crystals with open boundary conditions can be realized in Penning traps.", "While the focus of this paper was on the system with repulsive Coulomb interactions, similar transition is expected for a lattice with dipolar interactions [23].", "We hope that our work will stimulate research towards experimental study of the predicted topological defects and their rich collective dynamics." ] ]
2105.11680
[ [ "Simple But Effective Redundant Odometry for Autonomous Vehicles" ], [ "Abstract Robust and reliable ego-motion is a key component of most autonomous mobile systems.", "Many odometry estimation methods have been developed using different sensors such as cameras or LiDARs.", "In this work, we present a resilient approach that exploits the redundancy of multiple odometry algorithms using a 3D LiDAR scanner and a monocular camera to provide reliable state estimation for autonomous vehicles.", "Our system utilizes a stack of odometry algorithms that run in parallel.", "It chooses from them the most promising pose estimation considering sanity checks using dynamic and kinematic constraints of the vehicle as well as a score computed between the current LiDAR scan and a locally built point cloud map.", "In this way, our method can exploit the advantages of different existing ego-motion estimating approaches.", "We evaluate our method on the KITTI Odometry dataset.", "The experimental results suggest that our approach is resilient to failure cases and achieves an overall better performance than individual odometry methods employed by our system." ], [ "Introduction", "The ability to estimate the ego-motion, also called odometry, is a vital part of most autonomous mobile systems.", "Nowadays, autonomous vehicles are typically equipped with multiple sensors, such as cameras or LiDARs, as different sensing modalities have individual strengths.", "Therefore, a large number of algorithms [35], [14], [15] have been proposed exploiting both, visual and LiDAR information to explore the advantages of both and compensate for the drawbacks of the other.", "In most cases, researchers focus on achieving better odometry results by designing a potentially complex multi-sensor fusion algorithm.", "Even if different pose estimation systems' performances are remarkable within specific datasets, such algorithms may not always operate reliably under all conditions, setups, and application domains, as we will illustrate below in Sec.", "REF .", "Thus, failure detection and recovery steps are useful in real-world applications.", "We investigate in this paper a strategy that runs multiple odometry approaches in parallel and then selects the method that appears to be the best one.", "Thus, newly proposed individual methods can easily be added to our approach.", "Few studies are investigating redundant odometry methods running in parallel on autonomous mobile systems.", "One example is the work by Luo et al.", "[18], who propose to estimate odometry as an average of parallel odometry pipelines.", "This approach, however, is vulnerable since already one gross estimation error of one method will affect the overall performance.", "Santamaria-Navarro et al.", "[27] propose another solution, which runs several methods in parallel.", "It first sticks to one odometry method and only falls back to an alternative odometry pipeline if the initial method's divergence is detected.", "While this method is straightforward to realize, it does not utilize to its full extent the potential of all algorithms that run in parallel.", "Figure: Our redundant odometry system employs multiple odometry algorithms to provide proposals shown as differently colored dots for the each timestamp, exploiting LiDAR and camera data.", "Our approach uses sanity checks and a Chamfer distance-based score to choose the most promising pose estimation from all candidates.In this work, we investigate the possibility of building a redundant odometry system with sanity checks, and that can deal with failure cases of different odometry methods.", "As sketched in Fig.", "REF , instead of sticking to a specific odometry method, the idea is to run various odometry algorithms in parallel and choose the most promising one.", "We target to avoid explicit recovery behaviors for individual odometry pipelines in case of failures but propose to switch dynamically between estimation algorithms and independently of previous decisions.", "Furthermore, we target a computationally light-weight and flexible system.", "The main contribution of this paper is a redundant odometry system to provide a more resilient ego-motion estimation than the individual odometry methods that the system consist of.", "Instead of sticking to one specific odometry method, the proposed system runs multiple odometry algorithms in parallel and exploits LiDAR and camera data.", "It combines sanity checks exploiting dynamic and kinematic constraints of the vehicle and Chamfer distance-based scores between sensor data and a local map to select the most promising pose estimation among all the odometry candidates.", "We make three key claims: Our approach is able to (i) use redundant odometry pipelines to provide better odometry results, (ii) generally yields the best results compared to the individual methods used, and, (iii) avoids several failure cases in different situations.", "The paper and our experimental evaluation back up these claims.", "Furthermore, the source code of our approach is available at: https://github.com/PRBonn/MutiverseOdometry" ], [ "Related Work", "Odometry estimation is a widely investigated topic in robotics.", "It can be achieved using different types of sensors, including wheels [32], inertial sensing [13], cameras [28], thermal cameras [9], radars [7], LiDARs [26], sonars [1] as well as various combinations [11], [34], [35], [8], [3], [20].", "Although much work has been done, only a small amount of research has been devoted to making a robust and redundant system to deal with failures of individual ego-motion estimation approaches.", "Works show that redundancy can make a robotic system more robust.", "Rollinson et al.", "[25] show a robust estimation of the configuration of an articulated robot that uses a large number of redundant proprioceptive sensors (encoders, gyros, accelerometers) embedded in an unscented Kalman filter.", "They formulate the state estimation problem to leverages the redundancy in the proprioceptive information provided by the robot's joint angle encoders and inertial sensors.", "Additionally, they introduce a novel outlier detector that can identify corrupted measurements by utilizing the Mahalanobis distance matrix.", "Kim et al.", "[16] point out that redundant systems are important due to adversary attacks that corrupt measurement.", "They propose an observer for a nonlinear system to detect sensor attacks.", "The importance of a redundant odometry system has also been shown on Mars Exploration Rovers  [19].", "It successfully demonstrated a switching behavior between the wheel, inertia, and visual odometry, where visual odometry was used as compensation for any unforeseen slip encountered during a drive.", "Another redundant odometry system was proposed by Luo et al.", "[18] taking an average estimate over all odometry estimates that are currently in the active mode.", "Others use fallback solutions and execute homing behaviors [24], sampling-based motion proposals in a particle filter [30], or combinations of dense and feature-based methods [33].", "Carnevale et al.", "[6] propose a way to fuse the information coming from two typologies of redundant radio-frequency identification sensors with complementary characteristics using a Kalman filter framework.", "A switching criterion is embedded in the filtering approach where the observer uses uncertainties of the measurements and their average values to provide a switching behavior.", "Recently, Ebadi et al.", "[10] created an autonomous mapping and positioning system for the DARPA Subterranean Challenge.", "This work uses visual-inertial, LiDAR, and wheeled odometry as a front-end for a multi-robot mapping and positioning system to explore perceptually-degraded environments.", "They use the generalized iterative closest point (GICP) algorithm [29] to estimate scan-to-scan and scan-to-map matching that provides the relative transformation between the consecutive timestamps.", "The related approach to our work has recently been proposed by Santamaria-Navarro et al.", "[27] in the context of UAVs.", "Their system called heterogeneous redundant odometry (HeRO) evaluates the pose estimations from each pipeline by checking gaps and jumps within the transforms sequence.", "In this method, the different odometries run in parallel, and the potentially best estimation is used as the output of the system until the uncertainty of the estimation exceeds certain quality criteria.", "Then, the system falls back to another method while at the same time trying to reinitialize the former method.", "Unlike HeRO, which treats each odometry pipeline as a separate entity with a separated history of transformations and transformed point clouds, our method does not need any reinitialization procedure.", "It switches to an alternative method or fallbacks to a constant velocity model if all other methods fail.", "We focus on picking the best transform independently from the previous transforms and use a chain of transformation and transformed point clouds as a common history.", "HeRO assesses the best odometry based on the internal state of the odometry algorithms, while our method aims at finding a more objective metric that can choose fairly the best odometry at every timestamp.", "Based on HeRO, Palieri et al.", "[22] propose a health monitor system for the odometry estimation, which exploits LiDAR-based scan-to-scan and scan-to-map methods.", "While this method is similar to our approach, we do not need to manually specify the main odometry or submental ones.", "Our method chooses the most promising odometry by each candidate's performance scores at every timestamp without relying on any prior assumptions or knowledge about the odometry performance.", "Different from all the methods mentioned above, we propose an integrated visual-LiDAR odometry system that uses multiple odometry algorithms in parallel, a set of dynamics and kinematic sanity checks that filter out potential estimation failures, and a point cloud Chamfer distance-based criterion, which chooses the most promising pose estimation at each timestamp, independently from the history." ], [ "Our Approach", "We propose a redundant odometry system enhanced with sanity checks.", "These steps are visualized in Fig.", "REF .", "We use the modular design to keep our system flexible.", "There are four modules in our system.", "First, the input preprocessing module exploiting multiple sensing modalities as the input to the system (see Sec.", "REF ).", "Second, the odometry candidates module employing multiple different odometry algorithms to estimate the candidate transformations at each timestamp as odometry proposals (see Sec.", "REF ).", "Third, the sanity-check module using vehicle dynamics and kinematics properties to check for failure cases in each odometry proposal (see Sec.", "REF ).", "Fourth, the scoring module exploiting the Chamfer distance to score the odometry proposals and selects the final transformation from the most recent vehicle motion (see Sec.", "REF )." ], [ "Input Preprocessing Module", "In this work, we do not focus on choosing the best sensor or combination to improve odometry but provide a flexible system that allows us to use multiple sensing modalities and ego-motion algorithms in parallel to improve the robustness of the odometry.", "To this end, our system uses the data from a 3D LiDAR scanner and a monocular camera as an example to show that our system can exploit the redundancy of multiple sensing modalities to improve its robustness.", "Specifically, we use SIFT features [31] extract from images obtained by a monocular camera and voxelized point clouds from LiDAR scans as the input to different odometry methods.", "Such a redundancy design at the input level has the advantage that our approach is still functional even in case of sensor failures." ], [ "Stack of Odometry Pipelines", "Our approach combines multiple odometry methods into a redundant system, which is robust to failure cases and provides better odometry results.", "The odometry algorithms used in this paper are generalized iterative closest point (GICP) [29], point-to-plane iterative closest point (P2P-ICP) [26], normal distributions transform (NDT) [4], Color iterative closest point (ColorICP) [23], and Huang's method [14] for combining LiDAR and camera data.", "Moreover, we also employ the common constant velocity model (CVM), which directly uses the last timestamp estimation without using the sensor data.", "Our approach is not limited only to those methods and can be easily extended.", "P2P-ICP [26], [36] minimizes the sum of the squared distance between the point from the source points and the tangent plane at its corresponding point in the target point cloud for every point.", "GICP [29], [17] combines the Iterative Closest Point (ICP) and point-to-plane ICP into a single probabilistic framework.", "It models the locally planar surface structure from both scans and minimizes the objective function in a typical ICP-based framework.", "This procedure may be interpreted as a plane-to-plane approach embedded in the ICP framework.", "ColorICP [23], [36] optimizes jointly geometrical information (in the sense of point-to-plane ICP) together with RGB color information that is assigned to the LiDAR points which are inside the field of view of the camera(s).", "NDT [4], [17] uses a representation of the target point cloud to obtain a smooth surface described as a set of local probability density functions in the subdivided space as a grid of cells.", "A density function is computed for each cell based on the point distribution within the cell.", "To estimate the pose, NDT maximizes the likelihood that the current scan points lay on the reference scan surface.", "Huang's method [14] employs the 5-point algorithm [21] within a RANSAC loop that finds 5 of the 6 transformation parameters (rotation and translation vector up to scale) between the previous camera image and the current image with the SIFT descriptor.", "It then uses a grid search-based 1D ICP to estimate the scale.", "Besides the odometry information provided by different algorithms based on different sensing modalities, our system also employs the so-called constant velocity model (CVM), which takes the same transformation from the previous timestamp as an additional odometry estimation with the assumption that the acceleration of the car is zero.", "Although the constant velocity model is simple, it will always pass the sanity checks and make the system robust.", "Those odometry methods are run in parallel using the same input data and share the same odometry history.", "This means only the transformation that is finally selected by our approach at each timestamp will be stored and serves as the initial guess for all the odometry methods in the next timestamp.", "In this paper, we exploit six different odometry methods to exploit the redundancy of multiple odometries to improve overall performance and robustness.", "Note that we can easily add more methods if desired.", "Since our method can switch dynamically for each timestamp independently from previous decisions, multiple odometry methods can be easily integrated into our system, even during online processing." ], [ "Sanity Checks", "In robotics, we typically know the used hardware and the physical model of our robot or a typical car.", "Therefore, we use the dynamic and kinematic constraints of a car to check the odometry proposals provided by different algorithms, which we refer to as sanity checks.", "In this paper, we exploit two types of sanity checks, one based on the dynamic constraint, and the other based on the kinematic constraint.", "We refer to the maximum acceleration $a_{max}$ a typical car can reach for formulating the dynamic constraint.", "Our system first checks the acceleration resulting from all odometry proposals.", "If the estimated transformation exceeds the bound of the maximum possible acceleration, it will be rejected from further considerations.", "For the kinematic constraint, we use the Ackermann vehicle model [37], which puts constraints on the kinematic possibilities of the car movement.", "If using a differential drive model, one obviously must use the corresponding model.", "As shown in Fig.", "REF , given the forward velocity $v$ and the steering angle rate $\\delta $ , the car's trajectory follows the Ackermann model if there is no slip or any other highly dynamic conditions such as drifting or similar.", "It means that in real operations, the car's sideward movements are limited to a certain range.", "Applying this fact, our system uses the last pose estimation to calculate the forward velocity.", "Based on the forward velocity and the Ackermann model, the side velocity $v_\\text{s}$ is calculated based on the derivation [5], $v_{\\text{s}}(\\dot{\\beta }, v) = & f \\frac{\\frac{v}{f}+l(1-\\cos \\frac{\\dot{\\beta }}{f})}{\\sin \\frac{\\dot{\\beta }}{f}}(1-\\cos \\frac{\\dot{\\beta }}{f}) + l\\sin \\frac{\\dot{\\beta }}{f} ,$ where $v$ is a forward velocity of the vehicle, and $\\dot{\\beta }$ is the turning rate, $f = \\frac{1}{\\Delta \\tau }$ , where $\\Delta \\tau $ is the difference between timestamps of the current frame and previous frame, $l$ is the distance from the rear axis to the placement of the sensor.", "A proposal will then be rejected for future considerations if $|v_{s} - v^*| > v_{th},$ where $v^*$ is taken from the odometry proposals and $v_{th}$ is a maximum deviation from the side velocity that is acceptable.", "Note that even if all sensors or odometry algorithms based on sensor data fail to pass the sanity checks, the constant velocity model will always pass the sanity checks and makes the proposed method fairly robust to failures.", "Figure: Ackermann vehicle model.", "The variable ll is the distance between the front and rear axis, vv is the velocity of the rear axis center, δ\\delta is the steering angle, PP is the instantaneous center of rotation, and rr is the radius around PP.", "See  for details." ], [ "Scoring Candidate Transformations", "All the odometry proposals, which pass the sanity checks, will be evaluated based on the Chamfer distance metric [2] computed on the point cloud.", "Thus we need at least one sensor that generates point cloud data.", "We use a 3D LiDAR in our system.", "The metric on the point cloud is defined as follow, $D_{\\text{chamfer}}(M,N) = \\frac{1}{M} \\sum _{m \\in M}^{} d_N(m) ,$ where $M$  is a set of points in the source point cloud, $N$  is a set of points in the target point cloud, and $d_{N}$  is the minimum distance between point $m$ and its nearest neighbor point in the target point cloud $N$ given the transformation between point cloud $M$ and $N$ .", "In addition to the fact that point clouds after transformation should close to each other, we apply a small search radius $r_s$ for correspondence matching and delete the points for which no correspondence could be found within the radius.", "The reason to use the Chamfer distance as the criterion is that it is fast to calculate and independent from all the baseline odometry methods.", "Since each odometry candidate in our system estimates the transforms from the same source point cloud to the same target point cloud, this metric allows us to assess the candidate transformations estimated by different odometry methods.", "Our system applies this metric to all the success odometry proposals, and the lowest score is chosen as the final odometry result.", "$T^* = \\underset{T \\in {T_1,T_2,...}}{\\arg \\min } \\frac{1}{M} \\sum _{m \\in M}^{} d_N(m) ,$ The chosen transformation is stored in the odometry history and employed as the initial guess for the next iteration.", "We use the last $N_{\\text{map}}$ scans with the stored odometry to build a local point cloud map.", "Figure: Trajectories of different methods evaluated on the KITTI Dataset.", "For the sake of clarity, we only show the results of our method together with the best and the worst results of the baseline methods employed by our system.We present our experiments to showcase the capabilities of our method and to support our key claims that our approach is able to (i) use redundant odometry pipelines to provide better odometry results, (ii) generally yields the best results compared to the individual methods used, and, (iii) avoids several failure cases in different situations.", "For evaluation purposes, we use the KITTI Odometry Benchmark [12] which provides point clouds generated by a Velodyne HDL-64E 3D laser scanner (10 Hz, 64 laser beams, range: 100m) and color images from PointGrey Flea2 video cameras (10 Hz, resolution: $1382 \\times 512$ pixels).", "We tested our approach on an Intel Core i9-9900K CPU with 16 cores @ 3.60GHz with 32 GB RAM.", "For all the experiments, we use $l = 1.0\\,m $ , $a_{\\text{max}} = 6.0\\,m/s^2$ and $v_{th} = 0.8\\,m/s$ , the search radius $r_s = 0.5\\,m$ , and the number of scans used to build the local map is $N_{\\text{map}} = 10$ ." ], [ "Odometry Performance", "The experiments presented in this section are designed to show our approach's odometry performance and support the claim that using our redundant system yields better overall odometry results.", "To evaluate the odometry results, we use the KITTI odometry dataset sequences 00-10 and the KITTI odometry metric [12], which considers relative errors concerning translation and rotation averaged over different distances.", "For the qualitative results, we show the estimated trajectories of our method compared to those generated by the individual odometry algorithms employed by our redundant system, see in Fig.", "REF .", "For the sake of clarity, we only show the trajectories generated by our method as well as those of the best and the worst individual odometry.", "We can see that, in all sequences, our method always tracks the ground truth poses well no matter how bad the worst individual odometry candidate is, and there is no individual odometry baseline that can always generate good results.", "For the quantitative results, we show all evaluation results on all sequences in Tab.", "REF .", "In terms of translation error, our method is clearly superior in 10 out of 11 sequences to all other methods individually and in the other one sequence achieves a comparable performance to the best individual odometry algorithm.", "Overall, our method outperforms all baselines with the smallest average translation error of $1.11\\,\\%$ .", "In terms of rotation error, our method achieves the second-best average rotation error of $0.0045\\,\\text{deg}/\\text{m}$ .", "To further insight into how our system chooses different proposals, we illustrate when and how often different methods have been selected on sequence 00 in Fig.", "REF .", "We can see that our method switches between different proposals frequently according to the sanity checks and scoring results.", "Table: Results on KITTI Odometry" ], [ "Ablation Study on Odometry Candidates", "The second experiment shows the ablation study on different odometry combinations.", "We use the same KITTI odometry dataset sequence 00-10 and the same metric as in the first experiment to evaluate the odometry results of different odometry input combinations.", "As shown in Tab.", "REF , we can see that no matter what combination of the odometry candidate methods is provided, our system can always outperforms the individual odometry methods (see results in Tab.", "REF for the individual numbers per sequence)." ], [ "Robustness", "The third experiment illustrates our system's robustness and supports the claim that the proposed redundant odometry system is robust to different failure cases.", "We use four different cases to show that our system can successfully cover the failure cases where the individual candidate odometry methods fail, see Fig.", "REF .", "The failure cases refer to the situations where the odometry yields a clearly wrong estimate with respect to the ground truth poses.", "The failure cases can be caused by the dynamic objects, the lack of features of the environments, or the wrong data associations between different observations.", "In Fig.", "REF , we can see that in all cases when the individual entities fail, our system switches in between the odometry candidates at the right time, choosing the correct estimate and provide proper odometry results.", "Figure: Point-to-point ICP failure.Figure: Trajectory comparison.To gain further insight, we provide an analysis of the sanity checks.", "If we do not apply sanity checks, our KITTI 00-10 overall score drops from $1.11\\,\\%$ to $1.37\\,\\%$ in terms of the average translation error.", "The results of the acceleration sanity checks on KITTI sequence 01 is shown in Fig.", "REF .", "We can see in Fig.", "REF that there are some large accelerations that indicate the failures of the odometry estimations.", "After using sanity checks, see Fig.", "REF , our system provides smoother velocity estimates, see  Fig.", "REF , and provide better overall results, see Fig.", "REF ." ], [ "Conclusion", "In this paper, we presented a robust and resilient odometry system.", "Our method is simple, effective, easy to implement, and we provide the source code for further use.", "Our system exploits the redundancy of running multiple odometry pipelines, including multiple streams, in parallel.", "It uses the dynamic and kinematic constraints-based sanity checks and the Chamfer distance-based criteria to avoid failure cases and select the most promising odometry estimation among all proposals.", "For that, at least one sensor that generates point clouds, such as a LiDAR, is required.", "We evaluated our approach on the KITTI dataset and supported all claims made in this paper.", "The experiments suggest that the proposed system is resilient and robust to odometry failures and achieves better performance than all the baselines in terms of odometry accuracy.", "Despite these encouraging results, there is space for further improvements.", "In future work, we plan to use learning-based approaches to design better selection criteria for choosing the most promising odometry estimate." ] ]
2105.11783
[ [ "Safe Model-based Off-policy Reinforcement Learning for Eco-Driving in\n Connected and Automated Hybrid Electric Vehicles" ], [ "Abstract Connected and Automated Hybrid Electric Vehicles have the potential to reduce fuel consumption and travel time in real-world driving conditions.", "The eco-driving problem seeks to design optimal speed and power usage profiles based upon look-ahead information from connectivity and advanced mapping features.", "Recently, Deep Reinforcement Learning (DRL) has been applied to the eco-driving problem.", "While the previous studies synthesize simulators and model-free DRL to reduce online computation, this work proposes a Safe Off-policy Model-Based Reinforcement Learning algorithm for the eco-driving problem.", "The advantages over the existing literature are three-fold.", "First, the combination of off-policy learning and the use of a physics-based model improves the sample efficiency.", "Second, the training does not require any extrinsic rewarding mechanism for constraint satisfaction.", "Third, the feasibility of trajectory is guaranteed by using a safe set approximated by deep generative models.", "The performance of the proposed method is benchmarked against a baseline controller representing human drivers, a previously designed model-free DRL strategy, and the wait-and-see optimal solution.", "In simulation, the proposed algorithm leads to a policy with a higher average speed and a better fuel economy compared to the model-free agent.", "Compared to the baseline controller, the learned strategy reduces the fuel consumption by more than 21\\% while keeping the average speed comparable." ], [ "Introduction", "With the advancement in the vehicular connectivity and autonomy, Connected and Automated Vehicles (CAVs) have the potential to operate in a safer and more time- and fuel-efficient manner [1].", "With Vehicle-to-Vehicle (V2V) and Vehicle-to-Infrastructure (V2I) communication, the controller has access to real-time look-ahead information including the terrain, infrastructure and surrounding vehicles.", "Intuitively, with connectivity technologies, controllers can plan a speed profile that allows the ego vehicle to intelligently pass more signalized intersections in green phase with less changes in speed.", "This problem is formulated as the eco-driving problem, which aims to minimize weighted sum of the fuel consumption and the travel time between two designated locations by co-optimizing the speed trajectory and the powertrain control strategy [2], [3].", "The literature related to eco-driving focuses on different perspectives.", "Regarding powertrain configuration, the difference is whether the powertrain is equipped with a single power source [4], [3], [5], [6] or a hybrid electric architecture [7], [8], [9].", "The latter involves modeling multiple power sources and devising optimal control algorithms that can synergistically split the power demand in order to efficiently utilize the electric energy stored in battery.", "Maamria et al.", "[10] systematically compares the computational requirements and the optimality of different eco-driving formulations solved offline via Deterministic Dynamic Programming (DDP).", "Meanwhile, difference also exists in the complexity of the driving scenarios.", "Ozatay et al.", "[4] proposed a framework providing advisory speed profile using online optimization conducted on a cloud-based server without considering the real-time traffic light variability.", "Olin et al.", "[9] implemented a eco-driving framework to evaluate real-world fuel economy benefits from in-vehicle testing.", "As traffic lights are not explicitly considered in these studies, the eco-driving controller is required to be coupled with other decision-making agents, such as human drivers or rule-based controller [11].", "Other studies have explicitly modeled and considered Signal Phase and Timings (SPaTs).", "Jin et al.", "[3] formulated the problem as a Mixed Integer Linear Programming (MILP) for conventional vehicles with Internal Combustion Engine (ICE).", "Asadi et al.", "[12] used traffic simulation models and proposed to solve the problem considering probabilistic SPaT with DDP.", "Sun et al.", "[6] formulated the eco-driving problem as a distributionally robust stochastic optimization problem with collected real-world data.", "Guo et al.", "[8] proposed a bi-level control framework with a hybrid vehicle.", "In this work, the eco-driving problem of Connected and Automated Hybrid Electric Vehicles (CAHEVs) with the capability of passing traffic lights autonomously is studied.", "With the alternative electrical power source and the V2I communication to receive Signal Phase and Timing messages from signalized intersection, CAHEVs have the potential of significantly reducing the travel time and fuel consumption compared to conventional vehicles.", "Meanwhile, as the system becomes more complex and autonomous, the control logic becomes more sophisticated and computationally demanding.", "Specifically, the complexities come from three aspects.", "First, the system dynamics and constraints are nonlinear, non-smooth and non-convex.", "Second, the driving scenarios are probabilistic.", "Third, the design of vehicle velocity and battery $SoC$ profile requires long-term planning, and a short-sighted policy is subject to local optima.", "Recently, the use of Deep Reinforcement Learning (DRL) in the context of eco-driving has caught considerable attention.", "DRL provides a train-offline, execute-online methodology with which the policy is learned from the historical data or the interaction with simulated environments.", "Shi et al.", "[13] modeled the conventional vehicles with ICE as a simplified model and implemented Q-learning to minimize the $CO_2$ emission at signalized intersections.", "Li et al.", "[14] applies an actor-critic algorithm on the ecological ACC problem in car-following mode.", "Pozzi et al.", "[15] designed a velocity planner considering the signalized intersection and hybrid powertrain configuration with Deep Deterministic Policy Gradient (DDPG).", "Zhu et al.", "[16] formulates the eco-driving problem as a Partially Observable Markov Decision Process (POMDP) and approaches it with Proximal Policy Optimization (PPO).", "While the strategies with Model-Free Reinforcement Learning (MFRL) in these studies show improvements on the average fuel economy and reductions in onboard computation, the methodology has a fundamental drawback.", "To teach the agent to drive under complex driving scenarios while satisfying all the constraints from powertrain and traffic rules, a complex and ofter cumbersome rewarding/penalizing mechanism needs to be designed.", "Furthermore, under such setup, the agent learns to satisfy constraints by minimizing the expected cost.", "For scenarios that are rare yet catastrophic, the scale of the cost penalizing constraint violation needs to be significantly larger than the learning objective itself[13].", "As a result, such extrinsic rewarding mechanism increases the design period and deteriorates the final performance.", "On the other hand, Model Predictive Control (MPC) offers a framework to impose constraints, and the performance can be improved iteratively with a learned terminal cost function (also known as Approximate Dynamic Programming [17]).", "To be consistent with other works in literature, the technique is referred to as Model-based Reinforcement Learning (MBRL) in this work.", "Compared to MFRL methods, the advantages are two-fold.", "First, the method does not require any extrinsic reward to teach agents to satisfy constraints.", "Instead, the controller satisfies the short-term constraints by formulating an online constrained trajectory optimization problem.", "Second, the use of model significantly reduces the sample complexity of the learning algorithm [18].", "In [19], [20], on-policy MBRL algorithms with known dynamics are proposed for robotic applications.", "In [21], the system dynamics is learned as ensembles of neural network models, and the online optimization is solved via Cross Entropy Method (CEM).", "Thananjeyan et.al extended the MBRL algorithm from [21] and the concept of safe set from [22], and proposed the safe on-policy MBRL algorithm for iterative tasks in [23] and multi-start, multi-goal tasks in [24].", "In [23], [24], safe sets were approximated by kernel density function, and we extend the idea to a deep generative model for the high-dimensional safe set approximation.", "While the use of model increases the sample efficiency in the aforementioned studies, each transition collection becomes more computationally expensive due to the online trajectory optimization.", "To synthesize the historical data and reduce the overall training wall-time, we propose Safe Model-based Off-policy Reinforcement Learning (SMORL), a safety-critical model-based off-policy Q-learning algorithm for systems with known dynamics.", "To obtain the value function from Q function, an actor is explicitly trained as in DDPG [25] and Twin Delayed Deep Deterministic policy gradient algorithm (TD3)[26].", "As the behavior policy is directly obtained from trajectory optimization and impacted by the learned Q function implicitly, there is a mismatch between the state-action visitation of the target policy and the state-action distribution in the experience buffer collected by the behavior policy.", "The mismatch, known in offline reinforcement learning [27], causes accumulating bootstrapping error and deteriorates the performance.", "In this work, Batch Constrained Q-learning (BCQ) [28] is used to address the distributional mismatch.", "The contributions of the work are two-fold.", "First, an off-policy MBRL algorithm SMORL is proposed.", "The off-policy nature of the algorithm allows the use of experience replay [29] and increases the sample efficiency compared to other MBRL algorithm in literature.", "Second, the proposed algorithm is successfully applied to the eco-driving problem.", "Without any extrinsic reward design, the agent trained with SMORL shows dominating performance compared to the one previously trained with MFRL algorithm in [16].", "The remainder of the paper is organized as follows.", "Sec.", "presents the simulation environment and the eco-driving problem formulation.", "Sec.", "introduces the preliminaries of the mathematical concepts, and Sec.", "presents the main algorithm SMORL.", "Sec.", "explains the detailed implementation of SMORL on the eco-driving problem.", "Sec.", "shows the training details and benchmarks the performance.", "Finally, Sec.", "concludes the study and identifies the future research directions.", "As collecting data in the real-world driving data is expensive and potentially unsafe, a model of the environment is developed for training and validation purposes.", "The environment model, named EcoSim, consists of a Vehicle Dynamics and Powertrain (VD&PT) model and a microscopic traffic simulator.", "Fig.REF shows EcoSim and its interaction with the controller and the learning algorithm.", "The controller commands three control inputs, namely, the Internal Combustion Engine (ICE) torque, the electric motor torque and the mechanical brake torque.", "The component-level torques collectively determine the HEV powertrain dynamics, the longitudinal dynamics of the ego vehicle and its progress along the trip.", "As in [30], it is assumed that cellular communication technologies are available onboard, and the real-time SPaT informaiton of traffic lights in the geographic region of interest remains available during the trip.", "The DRL agent utilizes the SPaT from the upcoming traffic light while ignoring the SPaT from any other traffic light regardless of the availability.", "Specifically, the controller receives the distance to the upcoming traffic light, its current status and its SPaT program as part of the observation.", "Finally, a navigation application with Global Positioning System (GPS) is assumed to be available on the vehicle such that the locations of the origin and destination, the remaining distance and the speed limits along the entire trip are available at every point during the trip.", "Figure: The Structure of The Environment ModelA forward-looking dynamic powertrain model is developed for fuel economy evaluation and control strategy verification over real-world routes.", "In this work, a P0 mild-hybrid electric vehicle (mHEV) is considered, equipped with a 48V Belted Starter Generator (BSG) performing torque assist, regenerative braking and start-stop functions.", "The input/output relation among the modeled powertrain components is illustrated in Fig.", "REF .", "Figure: Block Diagram of 48V P0 Mild-Hybrid Drivetrain.The engine is modeled as low-frequency quasi-static nonlinear maps based on steady state engine test bench data provided by supplier.", "The map of instantaneous fuel consumption $\\dot{m}_{\\mathrm {fuel}}$ is a function of engine angular velocity $\\omega _{\\mathrm {eng}}$ and engine torque $T_{\\mathrm {eng}}$ , and the maps of torque limits $T_{\\mathrm {eng}}^{\\mathrm {min}}$ and $T_{\\mathrm {eng}}^{\\mathrm {max}}$ are functions of engine angular velocity $\\omega _{\\mathrm {eng}}$ .", "The battery $SoC$ and voltage $V_{\\mathrm {batt}}$ are governed by a zero-th order equivalent circuit model shown as follows: $I_t = \\frac{V_{\\mathrm {OC}}(SoC_t) - \\sqrt{V_{\\mathrm {OC}}^2(SoC_t) -4 R_0(SoC_t) P_{\\mathrm {bsg},t}}}{2R_0(SoC_t)} ,\\\\SoC_{t+1} = SoC_t -\\frac{\\Delta t}{C_{\\mathrm {nom}}} (I_t + I_{\\mathrm {aux}}),$ where $t$ is the discretized time index, and $\\Delta t$ is the time discretization that is set to be $1 s$ in the study.", "The power consumed by auxiliaries is modeled by a calibrated constant current bias $I_{\\mathrm {aux}}$ .", "The cell open circuit voltage $V_{\\mathrm {OC}}$ and internal resistance $R_0$ are maps of $SoC$ from a battery pack supplier.", "The vehicle dynamics model is based on the road-load equation: $\\begin{aligned}{v}_{\\mathrm {veh},t+1}=v_{\\mathrm {veh}, t} + \\Delta t & \\biggl ( \\frac{T_{\\mathrm {out},t}-T_{\\mathrm {brk},t}}{MR_\\mathrm {w}} - \\dfrac{C_\\mathrm {d}\\rho _\\mathrm {a} \\Omega _\\mathrm {f}v_{\\mathrm {veh},t}^2}{2M} \\\\& - g \\cos {\\alpha } C_\\mathrm {r}v_{\\mathrm {veh},t} - g\\sin {\\alpha } \\biggr ) \\end{aligned}$ Here, the four terms inside the bracket of the left hand side are associated with the forward propulsion force, the tire rolling resistance, the aerodynamic drag, and the road grade, respectively.", "$T_{\\mathrm {brk}}$ is the brake torque applied on wheel, $C_{\\mathrm {d}}$ is the aerodynamic drag coefficient, $\\rho _\\mathrm {a}$ is the air density, $A_{\\mathrm {f}}$ is the effective aerodynamic frontal area, $C_\\mathrm {r}$ is rolling resistance coefficient, and $\\alpha $ is the road grade.", "Besides the aforementioned models, which are directly associated with either the states or the objective in the eco-driving Optimal Control Problem (OCP) formulation, BSG, torque converter and transmission are also modeled in the study.", "The BSG is modeled as a quasi-static efficiency map to compute the electrical power output $P_{\\mathrm {bsg}}$ .", "A torque converter model is developed with the purpose of computing the losses during the traction and regeneration modes.", "The transmission model is based on a static gearbox, and its efficiency $\\eta _{\\mathrm {trans}}$ is scheduled as a nonlinear map of the gear number $n_\\mathrm {g}$ , the transmission input shaft torque $T_{\\mathrm {trans}}$ and the transmission input speed $\\omega _{\\mathrm {trans}}$ .", "The detailed mathematical models of these components can be found in [16].", "The forward vehicle model as shown in Fig.", "REF was calibrated and validated using experimental data from chassis dynamometer.", "Vehicle velocity, battery $SoC$ , gear number, engine speed and the fuel consumption were used to evaluate the model with the experimental data.", "Fig.", "REF shows the sample results from model verification over the FTP-75 regulatory drive cycle.", "Results indicate that the the vehicle velocity and $SoC$ are accurately predicted by the model.", "Some of the mismatches in the battery $SoC$ can be attributed to the assumptions made in the simplified battery model such as modeling electrical auxiliary loads as constant current bias.", "Further, the final value of the fuel consumption estimated by the model over the FTP-75 drive cycle is within 4% of the actual fuel consumption which verifies that the model can be used for energy and fuel prediction over real-world routes.", "Figure: Validation of Vehicle Velocity, SoCSoC and Fuel Consumed over FTP Cycle." ], [ "Traffic Model", "A large-scale microscopic traffic simulator is developed in an open source software Simulation of Urban Mobility (SUMO) [31] as part of the environment.", "In order to recreate realistic mixed urban and highway trips for training, the map of the city of Columbus, OH, US is downloaded from the online database OpenStreetMap [32].", "The map contains the length, shape, type and speed limit of the road segments and the detailed program of each traffic light at signalized intersections.", "Fig.", "REF highlights the area that is covered in the study.", "In the shaded area, 10,000 random passenger car trips, each of which the total distance is randomly distributed from 5 $km$ to 10 $km$ , are generated as the training set.", "Another 100 trips, indicated by the origins (red markers) and destinations (blue markers) in Fig.", "REF , are generated following the same distribution as the testing set.", "In addition, the departure time of each trip follows a geometric distribution with the success rate $p=0.01$ .", "The variation and the randomness of the trips used for training enhance the richness of the environment, which subsequently leads to a learned policy that is less subject to local minimum and agnostic to specific driving condition (better generalizability) [33].", "The interface between the traffic simulator and the VD&PT model is established via Traffic Control Interface (TraCI) as part of the SUMO package.", "At any given time, the vehicle speed is calculated based on VD&PT model in response to the component-level torque commands.", "Subsequently, the location of the ego vehicle and the connectivity information, including SPaT and GPS signals, are determined and updated via TraCI.", "In the eco-driving problem, the objective is to minimize the weighted sum of fuel consumption and travel time between two designated locations.", "The optimal control problem is mathematically formulated as follows: $\\min _{\\left\\lbrace u_t \\right\\rbrace _{t=1}^\\infty }\\: &\\mathbb {E} \\left[ \\sum _{t=1}^\\infty \\left[\\lambda \\dot{m}_{\\text{fuel},t} + (1-\\lambda ) \\right] \\Delta t \\cdot \\mathbb {I}\\left[s_t < s_{\\text{total}}\\right] \\right]\\\\\\text{where} \\:&u_t = \\left[T_{\\text{eng},t}, T_{\\text{bsg},t}, T_{\\text{brk},t}\\right]^T \\\\\\text{s.t.}", "\\: & SoC_{t+1} = f_{\\text{batt}}(v_{\\text{veh},t}, SoC_t, u_t) \\\\& v_{\\mathrm {veh},t+1} = f_{\\text{veh}}(v_{\\text{veh}, t}, SoC_t, u_t) \\\\& T_{\\mathrm {eng}}^{\\min }(\\omega _{\\mathrm {eng},t}) \\le T_{\\mathrm {eng},t} \\le T_{\\mathrm {eng}}^{\\max }(\\omega _{\\mathrm {eng},t}) \\\\& T_{\\mathrm {bsg}}^{\\min }(\\omega _{\\mathrm {bsg},t}) \\le T_{\\mathrm {bsg},t} \\le T_{\\mathrm {bsg}}^{\\max }(\\omega _{\\mathrm {bsg},t}) \\\\& I^{\\min } \\le I_t \\le I^{\\max } \\\\& SoC^{\\text{min}} \\le SoC_t \\le SoC^{\\text{max}} \\\\& SoC_T \\ge SoC^{\\mathrm {T}} \\\\& 0 \\le v_{\\mathrm {veh},t} \\le v_{\\mathrm {lim},t} \\\\& (t, s_t) \\notin \\mathcal {S}_{\\mathrm {red}}.", "$ Here, $\\dot{m}_{\\mathrm {fuel},t}$ is the instantaneous fuel consumption.", "$\\lambda $ is a normalized weight on the fuel consumption.", "$\\omega _{\\mathrm {eng}}$ and $\\omega _{\\mathrm {bsg}}$ are the engine and BSG angular velocities, respectively, and they are static functions of vehicle speed $v_{\\mathrm {veh}}$ and gear number $n_\\mathrm {g}$ .", "$f_{\\mathrm {batt}}$ and $f_{\\mathrm {veh}}$ are the battery and vehicle dynamics, respectively, introduced in Sec.", "REF .", "Eqn.", "() to () are the constraints imposed by the powertrain components.", "Eqn.", "() and Eqn.", "() are the constraints on the instantaneous battery $SoC$ and terminal $SoC$ for charge sustaining, respectively.", "Here, the subscript $T$ represents the time at which the vehicle reaches the destination.", "$SoC^\\mathrm {min}$ , $SoC^\\mathrm {max}$ and $SoC^\\mathrm {T}$ are commonly set to 30%, 80% and 50%.", "Eqn.", "() and () are the constraints imposed by traffic conditions.", "The set $\\mathcal {S}_{\\mathrm {red}}$ represents the set in which the traffic light at the certain location is in red phase, as indicated by the red lines in Fig.", "REF .", "The problem is formulated as a infinite horizon problem in which the stage cost becomes zero once the system reaches the goal set, i.e.", "the travelled distance $s_t$ is greater than or equal to the total distance of the trip $s_{\\mathrm {total}}$ while keeping the terminal $SoC_T$ greater than or equal to $SoC^{\\mathrm {T}}$ .", "In addition, anytime that the vehicle violates the traffic light constraints, i.e.", "Eqn.", "(), the trip is considered failure and the goal set is not reached.", "Figure: Feasible Set Imposed by Traffic LightsTo solve the aforementioned optimization formulation as an OCP, a MBRL algorithm is proposed, and the preliminaries of the algorithm are included in the next section." ], [ "Preliminaries on MBRL", "The nonlinear, stochastic, time-invariant system is considered in this work: $\\begin{aligned}x_{t+1} &= f(x_t,u_t,w_t)\\\\x_t &\\in \\mathcal {X} \\subseteq \\mathbb {R}^n, \\: t \\in \\mathbb {N}_{+}\\\\u_t &\\in \\mathcal {U}(x_t) \\subseteq \\mathbb {R}^m, \\: t \\in \\mathbb {N}_{+}\\\\w_t &\\in \\mathcal {W} \\subseteq \\mathbb {R}^p, \\: t \\in \\mathbb {N}_{+}.", "\\end{aligned}$ Here, $x_t$ , $u_t$ and $w_t$ are the state, control, and uncertainty at time $t$ .", "$\\mathcal {X}$ and $\\mathcal {U}$ are the feasible sets for states and inputs, respectively.", "The uncertainties are assumed to be independent and identically distributed (i.i.d.).", "Let $\\pi :\\mathcal {X} \\rightarrow \\mathcal {U}$ be an feasibile deterministic policy and $\\Pi $ be the set of all feasible deterministic policies.", "The objective of the OCP is to reach the goal set $\\mathcal {G}\\subseteq \\mathbb {R}^n$ while finding the optimal policy $\\pi ^*$ that minimizes the expectation of the discounted sum of the costs defined as follows: ${\\begin{array}{c}\\pi ^* = \\operatornamewithlimits{\\arg \\!\\min }_{\\pi \\in \\Pi }\\eta (\\pi ), \\; \\text{where} \\\\\\eta (\\pi ) = \\mathbb {E}_{w_t}\\left[ \\sum _{t=0}^\\infty \\gamma ^{t}c\\left(x_t,u_t\\right)\\right], \\\\\\text{where } u_t = \\pi (x_t).\\end{array}}$ Here, $\\gamma $ is the discount factor that prioritizes the immediate rewards and ensures the sum over the infinite horizon remains finite.", "As in [24], the following assumption is made regarding the cost function.", "Assumption 3.1 (Costs) The cost is zero for the states inside the goal set $\\mathcal {G}$ and positive for the states outside, i.e.", "$\\exists \\epsilon > 0$ such that $c(x,u)>\\epsilon \\mathbb {I}_{\\mathcal {G}^C(x)}$ where $\\mathbb {I}$ is the indicator function and $\\mathcal {G}^C$ is the complement of the goal set $\\mathcal {G}$ .", "As in[34], [22], [24], the following definitions are given.", "Definition 3.1 (Robust Control Invariant Set) A set $\\mathcal {C}\\subseteq \\mathcal {X}$ is said to be a robust control invariant set for the system Eqn.", "(REF ) if for all $x(t) \\in \\mathcal {C}$ , there exists a $u(t)\\in \\mathcal {U}$ such that $f(x(t),u(t), w(t))\\in \\mathcal {C}$ , for all $w(t)\\in \\mathcal {W}$ and $t\\in \\mathbb {N}_+$ .", "Definition 3.2 (Robust Successor Set $\\textit {Suc}(\\mathcal {S})$ ) For a given set $\\mathcal {S}$ , its robust successor set $\\text{Suc}(\\mathcal {S})$ is defined as $\\begin{aligned}\\text{Suc}(\\mathcal {S})=&\\left\\lbrace x^{\\prime }\\in \\mathbb {R}^n: \\exists x \\in \\mathcal {S}, \\exists w \\in \\mathcal {W} \\right.", "\\\\& \\quad \\quad \\quad \\quad \\: \\left.", "\\text{such that } x^{\\prime } = f(x, \\pi (x), w) \\right\\rbrace .\\end{aligned}$ Definition 3.3 (Robust Reachable Set $\\mathcal {R}_N(x_0^j)$ ) For a given initial state $x_0^j$ , the N-step robust reachable set $\\mathcal {R}_N(x_0^j)$ of the system defined in Eqn.", "(REF ) in a closed loop policy $\\pi $ at iteration $j$ is defined recursively as $\\begin{aligned}\\mathcal {R}_{i+1}^{\\pi }(x_0^j) &= \\text{Suc}(\\mathcal {R}_i^{\\pi }(x_0^j)) \\cap \\mathcal {X},\\\\\\mathcal {R}_0^\\pi (x_0^j) &= x_0^j,\\end{aligned}$ where $i = 0, 1, \\dots , N-1$ .", "Definition 3.4 (Safe Set) The safe set $\\mathcal {SS}^j$ contains the full evolution of the system at iteration j, $\\mathcal {SS}^j=\\left\\lbrace \\bigcup _{k=0}^\\infty \\mathcal {R}_k^{\\pi } (x_0^j) \\bigcup \\mathcal {G}\\right\\rbrace .", "$ As shown in [22], the exact form of the safe set in REF is a robust control invariant set.", "As calculating its exact form is intractable, especially for high dimensional nonlinear system, it is, in practice, approximated as $\\widetilde{\\mathcal {SS}}^j = \\bigcup _{k\\in \\mathcal {M}^j} x^k,$ where $x^k=\\lbrace x_t^k:t\\in \\mathbb {N}_+\\rbrace $ is the trajectory at iteration $k$ , and $\\mathcal {M}^j = \\lbrace k\\in [0,j): \\lim _{t\\rightarrow \\infty } x_t^k\\in \\mathcal {G}\\rbrace $ is the set of indices of which the trajectories were successfully driven to the goal.", "As the safe set in this work is constantly evolving during training, the iteration index $j$ will be neglected in the remaining work.", "For any policy $\\pi $ , the value function $V^\\pi :\\mathcal {X}\\rightarrow \\mathbb {R}$ , the Q function $Q^\\pi :\\mathcal {X}\\times \\mathcal {U} \\rightarrow \\mathbb {R}$ and the advantage function $A^\\pi :\\mathcal {X}\\times \\mathcal {U} \\rightarrow \\mathbb {R}$ are defined as follows: $V^{\\pi }(x_t) &= {\\left\\lbrace \\begin{array}{ll}\\mathbb {E}_{\\pi }\\left[\\displaystyle {\\sum _{i=t}^\\infty }\\gamma ^{i-t}c\\left(x_i,u_i\\right)|x_t\\right], & x_t \\in \\mathcal {SS}\\\\\\infty , & \\text{otherwise.}\\end{array}\\right.", "}\\\\Q^{\\pi }(x_t,u_t) &= {\\left\\lbrace \\begin{array}{ll} \\mathbb {E}_{\\pi }\\left[\\displaystyle {\\sum _{i=t}^\\infty }\\gamma ^{i-t}c\\left(x_i,u_i\\right)|x_t,u_t\\right], & \\begin{aligned}&x_t \\in \\mathcal {SS}\\\\ &u_t \\in \\mathcal {U} \\end{aligned} \\\\\\infty & \\text{otherwise.}\\end{array}\\right.", "}\\\\A^{\\pi }(x_t,u_t) &=Q^{\\pi }(x_t,u_t) - V^{\\pi }(x_t).$" ], [ "Proposed Method", "In this work, an off-policy model-based deep reinforcement learning algorithm with approximated safe set is proposed.", "At any given time $t$ during policy execution, the following trajectory optimization problem with a receding horizon of $H$ steps is solved: $\\begin{aligned}\\min _{\\left\\lbrace \\tilde{u}_k \\right\\rbrace _{k=t}^{t+H-1}} & \\mathbb {E}\\left[ \\sum _{k=t}^{t+H-1} \\gamma ^{k-t}c(\\tilde{x}_k,\\tilde{u}_k) + \\gamma ^H V^{\\pi } (\\tilde{x}_{k+H}) \\right] \\\\\\text{s.t.}", "\\:& \\tilde{x}_{k+1} = f(\\tilde{x}_k,\\tilde{u}_k,w_k)\\\\& \\tilde{x}_t = x_t \\\\& \\tilde{x}_k \\in \\mathcal {X}, \\: k = t, \\dots , t+H-1 \\\\& \\tilde{x}_{t+H} \\in \\widetilde{\\mathcal {SS}}\\\\& \\tilde{u}_k \\in \\mathcal {U}, \\: k = t, \\dots , t+H-1,\\end{aligned} $ where $\\tilde{x}$ and $\\tilde{u}$ are the variables for states and control actions in the predicted trajectory.", "Compared to the formulation in [19], the state $\\tilde{x}_k$ and action $\\tilde{u}_k$ are explicitly constrained to be within the feasible region in the receding horizon and the terminal state $\\tilde{x}_{t+H}$ to be within the safe set $\\widetilde{\\mathcal {SS}}$ .", "With the presence of uncertainties in the dynamic system, solving the exact form of the above stochastic optimization problem can be challenging.", "In [21], [23], [24], Cross Entropy Method (CEM) is used to solve the problem with unknown dynamics as a chance constraint problem.", "In Section REF , techniques will be discussed to simplify and solve the optimization in the eco-driving problem.", "As most of the model-based deep reinforcement learning methods with trajectory optimization in literature learns the value function as the terminal cost for the MPC [21], [19], [24], [20], the learning algorithm becomes on-policy.", "While the trajectory optimization increases the sample efficiency and helps exploration[19], solving the trajectory optimization problem makes each data sample more computationally expensive.", "As a result, the training wall time is not necessarily reduced.", "In this work, the off-policy Q-learning [35] is instead proposed.", "To use the learned Q function in trajectory optimization, the following equation needs to be solved: $\\min _{\\left\\lbrace \\tilde{u}_k \\right\\rbrace _{k=t}^{t+H}} & \\mathbb {E}\\left[ \\sum _{k=t}^{t+H-1} \\gamma ^{k-t}c(\\tilde{x}_k,\\tilde{u}_k) + \\gamma ^H Q^{\\pi }_\\theta (\\tilde{x}_{t+H},\\tilde{u}_{t+H}) \\right],$ where $Q_\\theta ^{\\pi }$ is the approximated $Q$ function parametrized by $\\theta $ .", "Compared to solving Eqn.", "(REF ), solving Eqn.", "(REF ) requires one extra computational step: $V^\\pi (\\tilde{x}_{t+H}) = \\min _{\\tilde{u}_{t+H}} Q_\\theta ^{\\pi } (\\tilde{x}_{t+H},\\tilde{u}_{t+H}).", "$ Depending on the dimension of the problem, solving Eqn.", "(REF ) can be computationally intractable, especially for online control.", "Several algorithms, e.g.", "DDPG, TD3 and dueling network [36] are proposed to obtain the value function from the Q function.", "In this work, the off-policy actor-critic algorithm TD3 is used since it reduces the overestimation and is shown to be more stable than DDPG.", "Specifically, with the sample $(x_j, u_j, c_j, x^{\\prime }_j)$ from the experience replay buffer $\\mathcal {D}$ [29], the target for the Q function during training is constructed as follows, $y_j = c_j + \\gamma \\max _{i=1,2}Q_{\\theta ^{\\prime }_i}(x^{\\prime }_j,u^{\\prime }_j),\\\\u^{\\prime }_j = \\pi _{\\phi ^{\\prime }}(x^{\\prime }_j) .$ Here, $Q_{\\theta _1}$ and $Q_{\\theta _2}$ are two independently trained critic networks.", "$Q_{\\theta ^{\\prime }_1}$ and $Q_{\\theta ^{\\prime }_2}$ are the corresponding target networks.", "$\\pi _{\\phi }$ and $\\pi _{\\phi ^{\\prime }}$ are the actor network and its target network, respectively.", "The critics are then updated following ${\\begin{array}{c}\\theta _i \\leftarrow \\theta _i - \\alpha \\nabla _{\\theta _i} \\left[\\dfrac{1}{N}\\sum _{j=1}^N\\left(y_j-Q_{\\theta _i}(x_j,u_j)\\right)^2 \\right], \\: i = 1, 2, \\\\\\left\\lbrace (x_j, u_j, c_j, x^{\\prime }_j) \\sim \\mathcal {D} \\right\\rbrace _{j=1}^N.\\end{array}}$ where $\\alpha $ is the learning rate, and $N$ is the batch size.", "In the off-policy learning algorithm used here, the behavior policy is the trajectory optimization where state and action constraints within the receding horizon are satisfied thanks to the constrained optimization formulation.", "However, the trained actor $\\pi _\\phi $ makes decisions solely based on the Q function.", "The resulting mismatch between the distribution of state-action pairs induced by the actor $\\pi _\\phi $ and that collected by the behavior policy results in extrapolation error leading to unstable training [27].", "As an example in the eco-driving problem, the trajectory optimization ensures the power is solely generated from ICE when the battery $SoC$ is at the lower limit $SoC^{\\mathrm {min}}$ .", "Accordingly, no state-action pair resembling low $SoC$ and high motor torque can be collected, which leads to an extrapolation in the Q function near the region.", "The error can eventually cause unstable training or inferior performance.", "To address the extrapolation error induced by the mismatch in distributions, Batch Constrained Q-learning (BCQ) [28] originally proposed for offline reinforcement learning is used.", "Here, a generative model, specifically a Variational Autoencoder (VAE) [37], $G_\\omega (x)$ is trained to resemble the state-action distribution in the experience replay buffer.", "The background on VAE and the training objective are covered in Appendix .", "Note that samples from the generative model $a^{\\prime } \\sim G_\\omega (x^{\\prime })$ should ideally match the distribution collected by the behavior policy.", "Instead of selecting action following Eqn.", "(REF ), the action is now selected as ${\\begin{array}{c}u^{\\prime }_j = \\operatornamewithlimits{\\arg \\!\\min }_{u_{j,k} + \\xi _\\phi (x_j, u_{j,k}, \\Phi )} \\left[\\max _{i=1,2} Q_{\\theta _i^{\\prime }}(x^{\\prime }_j,u_{j,k} + \\xi _\\phi (x_j, u_{j,k}, \\Phi ))\\right],\\\\\\left\\lbrace u_{j,k} \\sim G_\\omega (x^{\\prime }_j) \\right\\rbrace _{k=1}^n.\\end{array}}$ Here $n$ is the hyperparameter that is the number of actions sampled from the generative model.", "The action $u^{\\prime }$ used for the target value for the Q function is selected as the best among the $n$ sampled ones.", "Note that there is no longer an actor network mapping from state to action.", "Instead, to ensure the agent can learn on top of the actions sampled from the generative model imitating the behavior policy from the experience buffer, a perturbation network $\\xi _\\phi $ whose output is clipped between $[-\\Phi ,\\Phi ]$ is trained.", "The perturbation network $\\xi _{\\phi }$ is updated by deterministic policy gradient theorem from [38] as $\\phi \\leftarrow \\phi - \\alpha \\nabla _\\phi \\left[\\dfrac{1}{N} \\sum _{j=1}^N Q_{\\theta _1}(x_j, u_j + \\xi _\\phi (x_j, u_j, \\Phi ))\\right].", "$ To reduce the accumulating error from bootstrapping, all the target networks are updated with a slower rates as $\\theta ^{\\prime }_i &\\leftarrow \\tau \\theta _i + (1-\\tau ) \\theta ^{\\prime }_i, \\quad i = 1, 2, \\\\\\phi ^{\\prime } &\\leftarrow \\tau \\phi + (1-\\tau ) \\phi ^{\\prime },$ where $\\tau $ is a constant on the order of $10^{-3}$ to $10^{-1}$ .", "In [23], [24], the safe set is approximated by kernel density estimation, which typically works well only for problems in low dimensions.", "Here, we extend the approximation to high-dimensional setting by using deep generative models.", "Following the notion in [23], the safe set is approximated as $\\widetilde{\\mathcal {SS}} = \\lbrace x:p_\\psi (x) \\ge \\delta \\rbrace ,$ where $p_\\psi :\\mathcal {X}\\rightarrow [0,1]$ is the probability that a state is inside the safe set parametrized by $\\psi $ , and the constant $\\delta $ regulates how exploratory the controller is.", "Note that the generative model used for safe set approximation needs to model the probability explicitly and can be slow in sampling, whereas the generative model resembling the distribution of state-action pairs in the experience replay needs to be fast in sampling while the explicit probability is not required.", "Due to the aforementioned consideration, autoregressive model with Long Short-term Memory (LSTM) [39] is used.", "The description of the model as well as the training objective is included in Appendix .", "In Sec., the use of the autoregressive model in the application of eco-driving is motivated as the dimension of the problem can get large once the future conditions are sampled discretely.", "In summary, Safe Model-based Off-policy Reinforcement Learning (SMORL) is proposed.", "The algorithm builds on SAVED [23] and extends it to be an off-policy algorithm with the methods proposed in BCQ.", "The detailed step-by-step algorithm is included in Algorithm .", "[] Initialize Q-networks $Q_{\\theta _1}$ , $Q_{\\theta _2}$ independently, and duplicate target networks $Q_{\\theta ^{\\prime }_1}$ , $Q_{\\theta ^{\\prime }_2}$ .", "Initialize the perturbation network $\\xi _\\phi $ , its target network $\\xi _\\phi ^{\\prime }$ and VAE $G_\\omega =\\left\\lbrace E_{\\omega _1}, D_{\\omega _2} \\right\\rbrace $ .", "Initialize the experience replay buffer $\\mathcal {D}$ .", "Collect $N_0$ successfully executed trajectories with a baseline controller and initialize the safe set $\\widetilde{\\mathcal {SS}}$ .", "$n_{iter}\\in {1,\\dots ,N_{iter}}$ $j^{th}$ trajectory NOT finished Select control action $u_t$ by solving trajectory optimization in Eqn.", "(REF ).", "Sample mini-batch of $N$ transitions $(x, u, c, x^{\\prime })$ from $\\mathcal {D}$ .", "For each transition, sample $n$ actions $u^{\\prime }_j$ from $G_\\omega (x^{\\prime })$ and $n$ perturbations from $\\xi _\\phi (x^{\\prime },u^{\\prime },\\Phi )$ .", "Update the critic networks $Q_{\\theta _1}$ , the target networks $Q_{\\theta _2}$ following Eqn.", "(REF ) and $Q_{\\theta ^{\\prime }_1}$ , $Q_{\\theta ^{\\prime }_2}$ following Eqn.", "(REF ).", "Update perturbation network $\\xi _\\phi $ following Eqn.", "(REF ).", "Update VAE $G_\\omega $ by maximizing Eqn.", "(REF ).", "$x_T\\in \\mathcal {G}$ Push the trajectory $\\left\\lbrace (x_t,u_t,c_t,x_{t+1}) \\right\\rbrace ^T$ to $\\mathcal {D}$ .", "Update the safe set $\\widetilde{\\mathcal {SS}}$ with minibatchs sampled from $\\mathcal {D}$ following Eqn.", "(REF ).", "Safe Model-based Off-policy Reinforcement Learning (SMORL)" ], [ "Trajectory Optimization", "Specific to the eco-driving problem, the state vector $x_t$ is defined as a vector with 88 states.", "A description of the states are listed in Tab.", "REF .", "Here, the first seven elements of the state vector are the battery $SoC$ , the vehicle speed $v_{\\mathrm {veh}}$ , the current speed limit $v_{\\mathrm {lim}}$ , the next speed limit $v_{\\mathrm {lim}}^{\\prime }$ , the distance to the next speed limit $s_{\\mathrm {lim}}$ , the distance to the upcoming traffic light $s_{\\mathrm {tls}}$ and the total remaining distance $s_{\\mathrm {rem}}$ .", "The remaining 81 elements are the sampled upcoming traffic light status in the next 80 seconds $x_{\\mathrm {tfc}}$ .", "For example, if the upcoming traffic light has 20 seconds remaining for the current red phase and will remain in green for the rest of the 80 seconds, the first 21 elements of the sampled upcoming traffic light status are 0, and the rest are set to 1.", "Compared to the manually extracted feature representation in [16], the sampled representation reduces the discontinuity and results in a better performance.", "Table: The State and Action Spaces of the Eco-driving ProblemAs the vehicle considered in this study is assumed equipped with connected features, e.g.", "advanced mapping and V2I connectivity, and surrounding vehicles are not included in the study, it is assumed that the ego vehicle can deterministically predict the uncertainties from driving conditions within the receding horizon in this study.", "Sun et al.", "[6] suggests by formulating the problem as a chance constraint or a distributionally robust optimization problem, uncertainties in SPaT can be considered without additional computational load.", "In Eqn.", "(REF ), the receding horizon $H$ is in time domain.", "While it is easier to incorporate the time-based information such as SPaT received from V2I communication in time domain, an iterative dynamic look-ahead process is required to process any distance-based route feature, such as speed limits, grade, traffic light and stop sign locations.", "For example, the controller requires the speed limits as the constraints to generate speed trajectory while the speed limits can change based on the distance travelled by the speed trajectory.", "In this study, the value and Q functions are learned in time domain for the ease of integration with time-based traffic simulator, while the trajectory optimization is conducted in spatial domain.", "As SPaTs and speed limits do not depend on the decision made by the ego vehicle in distance domain, they are incorporated into the optimization problem as constraints, and only the vehicle speed, battery $SoC$ and the time at which the vehicle reaches the given distance are considered as the state in the trajectory optimization.", "Define the optimization state $z\\in \\mathcal {Z} \\subseteq \\mathbb {R}^3$ as $z_s=\\begin{bmatrix}v_{\\mathrm {veh},s}, SoC_s, t_s\\end{bmatrix}^T.$ Here, $s$ is the index in the discretized spatial domain with $\\Delta s = 10 m$ , and the dynamics of $z$ in time and spatial domains are converted following $\\dfrac{\\Delta z}{\\Delta s} = \\dfrac{\\Delta z}{\\Delta t}\\dfrac{\\Delta t}{\\Delta s} = \\dfrac{\\Delta z}{\\Delta t}\\dfrac{1}{v_{\\mathrm {veh}}}.$ As a result, the trajectory optimization is formulated as $\\min _{\\left\\lbrace \\tilde{u} \\right\\rbrace _{k=s_t}^{s_t+H_s-1}} \\: &\\sum _{k=s_t}^{s_t+H_s-1} \\gamma ^{t_k} c(\\tilde{z}_k,\\tilde{u}_k) + \\gamma ^{t_{H_s}} V^\\pi (\\mathcal {G}(x_t,z_{H_s}))\\\\\\text{where:} \\: & \\\\c(\\tilde{x}_k, \\tilde{u}_k) = &\\left( \\lambda \\dot{m}_{\\mathrm {fuel},k} + (1-\\lambda ) \\right) \\dfrac{\\Delta s}{v_{\\mathrm {veh},k}} \\cdot \\mathbb {I}\\left[s_k < s_{\\mathrm {total}}\\right]\\\\\\text{s.t.}", "\\: & SoC_{k+1} = f_{\\mathrm {batt},s}\\left(\\tilde{z}_k, \\tilde{u}_k\\right) \\\\& v_{\\mathrm {veh},k+1} = f_{\\mathrm {veh},s}\\left(\\tilde{z}_k, \\tilde{u}_k\\right)\\\\& T_{\\mathrm {eng}}^{\\min }(\\omega _{\\mathrm {eng},k}) \\le T_{\\mathrm {eng},k} \\le T_{\\mathrm {eng}}^{\\max }(\\omega _{\\mathrm {eng},k}) \\\\& T_{\\mathrm {bsg}}^{\\min }(\\omega _{\\mathrm {bsg},k}) \\le T_{\\mathrm {bsg},k} \\le T_{\\mathrm {bsg}}^{\\max }(\\omega _{\\mathrm {bsg},k}) \\\\& I^{\\min } \\le I_k \\le I^{\\max } \\\\& SoC^{\\text{min}} \\le SoC_k \\le SoC^{\\text{max}}\\\\& 0 \\le v_{\\mathrm {veh},k} \\le v_{\\mathrm {lim},k}\\\\& (t_k, s_k) \\notin \\mathcal {S}_{\\mathrm {red}}\\\\& \\mathcal {G}(x_t, z_{H_s}) \\in \\widetilde{\\mathcal {SS}}.$ Here, $s_t$ is the spatial index corresponding to the distance the ego vehicle has traveled at the time $t$ .", "$H_s=20$ is the prediction step in spatial domain, making the total prediction horizon 200 $m$ .", "$\\mathcal {G}: \\mathcal {X} \\times \\mathcal {Z} \\rightarrow \\mathcal {X}$ is the function that takes the full state $x_t$ and the terminal optimization state $z_{H_s}$ and determines the predicted terminal full state $\\tilde{x}_{t+t_{H_s}}$ .", "For example, suppose there are 15 seconds left in the current green phase and $t_{H_s}$ in the optimization state is 10 seconds, i.e.", "it takes 10 seconds for the ego vehicle to travel the future 200 $m$ , there will be 5 seconds left in the current green phase at the end of the prediction horizon.", "The trajectory optimization problem is solved by Deterministic Dynamic Programming (DDP) [40].", "The optimal deterministic policy $\\mu _k^*: \\mathcal {Z} \\rightarrow \\mathcal {U}$ , $k=1, 2, \\dots , H_{s}-1$ , along with the optimal cost-to-go function $\\mathcal {J}_k: \\mathcal {Z} \\rightarrow \\mathbb {R}$ , $k=1, 2, \\dots , H_s$ can be calculated through backward recursion as $\\mathcal {J}_{H_s}(z) = V^\\pi \\left(\\mathcal {G}(x_t,z)\\right) + \\mathcal {P}_N(z) , \\\\\\mathcal {F}_k(z,u) = c(z,u) + \\mathcal {P}_k(z) + \\mathcal {J}_{k+1}(f_k(x,u)),\\\\\\mu _k^*=\\operatornamewithlimits{\\arg \\!\\min }_{\\mu _k} \\mathcal {F}_k(z, \\mu _k(z)) ,\\\\\\mathcal {J}_{k}(z) = \\mathcal {F}_k(z, \\mu ^*_k(z)) .$ Here, $\\mathcal {F}: \\mathcal {Z}\\times \\mathcal {U}\\rightarrow \\mathbb {R}$ is the cost-to-go associated with a given immediate action and then the optimal policy.", "$\\mathcal {P}_k:\\mathcal {Z}\\rightarrow \\mathbb {R}$ and $\\mathcal {P}_N:\\mathcal {Z}\\rightarrow \\mathbb {R}$ are penalty functions introduced to ensure no constraint violation in the predicted trajectory.", "Solving Eqn.", "(REF ) is computationally intensive yet highly parallelizable.", "Considering onboard GPU is readily available nowadays on self-driving vehicles, a CUDA-based Parallel DDP (PDDP) solver in [41] is used in this work.", "In the cases where the stochasity within the prediction horizon cannot be ignored, other gradient-free optimization methods, such as CEM or random shooting method [42], can be used as the trajectory optimizer." ], [ "Q Learning", "Fig.", "REF shows the architecture of the neural network associated with the Q-learning.", "Upon receiving the state vector, the sampled traffic light status $w_{\\mathrm {tfc}}$ are fed to a pre-trained autoencoder with Multilayer Perceptron (MLP) of size $(81, 100, 5, 100, 81)$ for dimensionality reduction.", "The remaining states along with the actions are concatenated with the latent states from the encoder, and subsequently fed into another MLP of size $(200, 100, 50)$ to output the Q function for critic and perturbation for actor.", "The critic and the actor do not share parameters in this work.", "Figure: The Network Architecture of the Value and Q Functions.To accelerate the training and improve generalizability, the state of the vehicle is randomized for every 50 steps in simulation.", "When the domain randomization occurs, the battery $SoC$ and the vehicle velocity $v_{\\mathrm {veh},t}$ are sampled from uniform distributions $\\text{Uniform}(SoC^{\\mathrm {min}}, SoC^{\\mathrm {max}})$ and $\\text{Uniform}(0, v_{\\mathrm {lim},t})$ , respectively.", "To guarantee feasibility during the trip, the domain randomization is disabled $200 m$ within signalized intersections or $1000 m$ within the final destination." ], [ "Safe Set Approximation", "In the eco-driving problem, two types of constraints can induce feasibility issue, namely, the battery terminal $SoC$ constraint and the constraint imposed by traffic rules at signalized intersections.", "For the first case, the goal set is considered not reached when the vehicle is near the destination and it cannot sufficiently charge the battery back to $SoC^\\mathrm {T}$ in the remaining distance.", "For the second case, the trip is considered failed when the vehicle breaks traffic rules at signalized intersections, and infeasibility occurs when the vehicle speed is too high and there is not enough distance to brake to stop in front of a traffic light in red or stop sign.", "A conservative low speed controller that only uses ICE is used to collect the initial data for the experience replay buffer.", "During training, only samples from the trips that reach the goal set without violating any constraints are added to the experience replay buffer.", "In the eco-driving problem, the sampled traffic light $x_{\\mathrm {tfc}}$ is binary, while the other variables are continuous.", "As the PDDP solver also discretize the continuous state space, we consider the loss of accuracy with the same discretization is acceptable.", "The discretized states as in one-hot form in each dimension are fed into a LSTM network with 50 units sequentially as shown in Fig.", "REF .", "The outputs from LSTM are then masked according to the number of categorical classes in each dimension.", "Finally, the softmax operator ensures the outputs to be proper conditional probability distribution.", "As an alternative to LSTM, Causal 1D Convolutional Neural Networks (Causal Conv1D) [43] was also implemented as the network for the autoregressive model.", "The key difference is that the states in all dimensions can be fed into Causal Conv1D in parallel, whereas each dimension needs to be fed into LSTM sequentially.", "For applications with long sequence, Causal Conv1D can be more efficient and accurate [44].", "For the specific problem, Caucal Conv1D shows no noticeable advantage over LSTM in either accuracy or inference speed.", "As a result, LSTM is chosen as it has less hyperparameters.", "When the receding horizon ($200 m$ ) in this study is longer than the critical braking distance [16], the vehicle will never violate any constraints imposed by signalized intersections.", "Nevertheless, using the safe set to constrain the terminal state is still essential for the following reason.", "At the last step of the receding horizon, the value function $V^\\pi (\\tilde{x}_{t+\\Delta t_{H_s}})$ needs to be evaluated numerically for trajectory optimization.", "Since only the data from the safely executed trips is added into the buffer and there is no penalty mechanism for the constraint violation, the estimation of the critic network is valid only within the safe set and is subject to extrapolation error outside.", "Although the long receding horizon ensures the feasibility of the actual trajectory regardless of the use of safe set, the training is subject to instability and the learned performance can be significantly deteriorated without the constraint from the safe set.", "This effect is shown in Fig.", "REF .", "Here, the two subplots on top show the optimized trajectories with and without the use of safe set, respectively.", "The three curves in each plot are the trajectories from the optimizer at three consecutive seconds.", "During the first two seconds, the vehicle is more than $200m$ away from the traffic light, and thus the constraint from Eqn.", "() is not considered in the trajectory optimization.", "The subplots on the bottom show the safety status of the terminal state in the dimension of the vehicle velocity and the time at which it reaches the end of the receding horizon before the signalized intersection appears in the receding horizon.", "Here, green means the state is considered safe, i.e.", "$p_\\psi (x) \\ge \\delta $ , and red otherwise.", "Although the actual trajectories, with or without safe set, are able to slow down in time to avoid trespassing the red light thanks to the sufficiently long receding horizon, the terminal state without the safe set constraint has a speed of $20 m/s$ with $20 m$ left before a red light, which is clearly unsafe.", "Meanwhile, comparing the bottom two subplots, the terminal velocity constrained by the safe set progressively reduces as the vehicle approaches the intersection in red phase.", "In addition, given speed limit here is set to $v_\\mathrm {lim}=22 m/s$ , any state with velocity higher than $22 m/s$ is considered unsafe.", "It can be noticed that the red region on the top right corners are incorrectly considered as unsafe (false positive).", "This is because, by optimality, the agent rarely crosses an intersection in green phase with low speed, therefore, these false positive regions do not affect the performance.", "Figure: The Effect Of the Safe Set on Trajectory Optimization.As a summary for all the implementation details, the hyperparameters are listed in Tab.", "REF .", "Table: Hyperparameters of the Q Learning and Safe Set Estimation" ], [ "Results", "Both the PDDP optimizer and the neural network training requires GPU.", "To get the results to be shown, the training took 24 hours on a node with a NVIDIA Volta V100 GPU and $2.4 GHz$ Intel CPU from Ohio Supercomputer Center [45].", "As domain randomization is used during training, 5 trips out of the 1000 randomly generated trips are repeatedly selected for every 25 training episodes to evaluate the performance of the controller and to quantify the progress of training.", "During evaluation, domain randomization and epsilon greedy are both deactivated.", "Fig.", "REF shows the evolution of the total costs, fuel economy and average speed of the 5 evaluation trips.", "Comparing to the model-free on-policy method in [16], which takes 80,000 episodes to converge, the sample efficiency of the off-policy model-based method is significantly improved.", "In the meantime, with the constrained optimization formulation and the safe set, the training quickly learns to respect the constraints imposed by the terminal $SoC$ and signalized intersections.", "Furthermore, the fact that the agent does not need any extrinsic penalty from constraint violation and is still capable of learning to operate within the safe region significantly simplifies the design and tuning process as deployed in the previous reinforcement learning attempts on eco-driving [16], [15], [14].", "Figure: The Evolution of the Total Costs, Fuel Economy, and Average Speed of the 5 Evaluation Trips.Statistically, the performance of the agent trained with SMORL is compared against the three other controllers, a baseline EDM driver representing a human driver with comparable average speed [46], the Model-free DRL (MFDRL) agent proposed in [16] and the wait-and-see solution.", "The wait-and-see solution assumes the speed limits and the sequences of all traffic lights of the entire trip are known a priori, and it is solved by PDDP solver as well.", "Despite being non-causal and computational intractable for online implementation, the solution serves as the upper bound for the causal control strategies.", "The four strategies are evaluated on the 100 random trips as shown in Fig.", "REF , and the fuel economy, average speed and variance of the battery $SoC$ are listed in Tab.", "REF .", "Table: Fuel Economy, Average Speed and SoC Variance for Baseline, Model-free DRL, SMORL and Wait-and-see SolutionsHere, the proposed model-based method has a dominant performance in average speed and fuel economy compared to the previously trained MFDRL strategy, which is primarily due to two aspects.", "First, the online trajectory optimization solved by PDDP guarantees the global optimality within the receding horizon.", "Compared to the actions generated from the one-step stochastic policy from neural network, the solution from trajectory optimization is more accurate and reliable.", "Second, the fact that there is no extrinsic penalty to assist constraint satisfaction ensures that the agent focuses on learning only the objective of the OCP formulation, i.e.", "weighted sum of the trip time and fuel consumption, instead of a carefully designed yet delicate surrogate learning objective.", "Compared to the baseline EDM driver, SMORL agent consumes $21.8 \\%$ less total fuels while maintaining a comparable average speed.", "The benefit in fuel economy is achieved by avoiding unnecessary acceleration events and by taking advantage of a wider range of battery capacity as indicated by the higher $SoC$ variance.", "In Fig.", "REF , the mean and standard deviation of the vehicle speed and the fuel economy of each trip are plotted against the traffic light density.", "As the wait-and-see solution calculates the global optimal solution with the knowledge of the full trip, it is able to navigate among the traffic lights accordingly, as indicated by the non-increasing standard deviation of the vehicle speed.", "In fact, the overall fuel economy increases as the traffic density increases.", "This can be due to the fact that when there are more traffic lights, the vehicle is forced to operate with a lower speed and lower fuel consumption condition.", "On the other hand, as the baseline driver has limited line-of-sight [46] and only the SPaT of the upcoming traffic light is available to SMORL, the standard deviation of the vehicle speed increases and the fuel economy decreases as the traffic light density increases.", "Nevertheless, as indicated by the slope of the fitted curve, the fuel economy of SMORL is slightly less affected by the increase of the traffic light density compared to the baseline.", "Figure: The Variation of the Average Speed and the Fuel Economy against Traffic Light Density for Baseline, SMORL and Wait-and-see SolutionFig.", "REF shows the comparison among the baseline, SMORL and the wait-and-see solution on a specific testing trip.", "For this specific trip, while trip time are within $3s$ , SMORL and the wait-and-see solution consume $24.7\\%$ and $40.0\\%$ less fuels compared to the baseline strategy, respectively.", "While SMORL demonstrates some merits similar to the wait-and-see solution, its inferiority is primarily due to the fact that only the SPaTs from the upcoming intersection is available to the controller.", "In this trip specifically, the wait-and-see solution deployed a constant low-speed strategy from $\\sim 1,700m$ all the way till the end of the trip.", "Although it was behind the other two at the intersection at $4,300m$ by one red phase, it was able to eventually catch by not having to waste anytime at stop.", "While knowing the SPaTs from the entire trip is unrealistic and subject to uncertainties, the SMORL was able to save a significant amount of fuel compared to the baseline by decelerating ahead of time, decelerating with more energy recuperation using electric motor, avoiding unnecessary high speed within the connectivity range.", "Additional comparison among the three strategies are shown in Appendix .", "Figure: The Trajectory Comparison among Baseline, SMORL and Wait-and-See.Conclusion In this paper, a safe-critical model-based off-policy reinforcement learning algorithm SMORL is proposed.", "The algorithm is applied to the eco-driving problem for Connected and Automated Hybrid Electric Vehicles.", "Compared to the previous model-free attempts on eco-driving in the literature, the method does not require any extrinsic rewarding mechanism, and thus, greatly simplifies the design process and improves the final performance.", "With the online constrained optimization formulation and the approximate safe set, the learned strategy is capable of satisfying the constraints in the prediction horizon and restricting the state within the approximate safe set, which is an approximation to the robust control invariant set.", "The performance of the strategy trained with SMORL is compared to a baseline strategy representing human drivers' behavior over 100 randomly generated trips in Columbus, OH, US.", "With a comparable average speed, the strategy from SMORL consumes approximately $22\\%$ less fuel.", "While the demonstration of the algorithm is on the eco-driving problem, we believe it can be applied to many other real-world problems, in particular to those with well-studied system dynamics, such as robotics and autonomous driving.", "Future works include incorporating SPaT information broadcasted from multiple upcoming signalized intersections, incorporating surrounding vehicles into the model, and accordingly update the optimal control problem, releasing the deterministic assumption within the prediction horizon.", "Variational Autoencoder Let $X = \\left\\lbrace x_i \\right\\rbrace _{i=1}^N$ be some data set and $Z$ represent a set of low-dimensional latent variables, the objective is to maximize the marginal log-likelihood: $\\begin{aligned}\\log p(X) = \\sum _{i=1}^N \\log p(x_i) &= \\sum _{i=1}^N \\int _z \\log p(x_i|z)p(z) dz\\\\&= \\sum _{i=1}^N \\mathbb {E}_{z\\sim p(z)} \\log p(x_i|z),\\end{aligned} $ As Eqn.", "(REF ) is in general intractable, its variational lower bound is instead maximized: $\\begin{aligned}\\mathcal {L}(\\omega _1, \\omega _2, X) = & -D_\\mathrm {KL}\\left(q_{\\omega _1}(z|X)|| p(z)\\right) \\\\& + \\mathbb {E}_{q_{\\omega _1}(z|X)} \\left[\\log p_{\\omega _2}(X|z)\\right]\\end{aligned}.", "$ Here, $D_{\\mathrm {KL}}$ is the Kullback–Leibler (KL) divergence, and $p(z)$ is the prior distribution that is typically assumed to be a multivariate normal distribution.", "$q_{\\omega _1}(z|X)$ is the posterior distribution parametrized by $\\omega _1$ .", "To analytically evaluate the KL divergence, the posterior is typically constructed as $\\mathcal {N}(z|\\mu _{\\omega _1}(X), \\Sigma _{\\omega _1}(X))$ .", "From a coding theory perspective, $q_{\\omega _1}(z|X)$ and $p_{\\omega _2}(X|z)$ can be considered as a probabilistic encoder and a probabilistic decoder, respectively.", "To compute the $\\nabla _{\\omega _1} L(\\omega _1, \\omega _2, X)$ , policy gradient theorem [47] or Reparametrization trick [47], [37] can be used.", "The latter is often used in VAE as it typically leads to a lower variance.", "In practice, the encoder $q_{\\omega _1}(z|X)$ and the decoder $p_{\\omega _2}(X|z)$ can be any function approximator.", "An implementation of VAE as the generative model to sample actions can be found in https://github.com/sfujim/BCQ.", "In this work, the latent space dimension is selected to be 5, and the encoder and the decoder are both MLPs with 2 layers of 300 hidden units.", "Autoregressive Model with LSTM For any probability distribution, the joint distribution can be factorized as a product of conditional probabilities as follow: $\\begin{aligned}&p(x) = \\prod _{i=1}^K p(x^{(i)}|x^{(1)}, ..., x^{(i-1)}), \\\\\\rightarrow &\\log p(x) = \\sum _{i=1}^K \\log p(x^{(i)}|x^{(1)}, ..., x^{(i-1)}),\\end{aligned}$ where $x^{(i)}$ is the $i^{th}$ dimension of the discrete input vector and $K$ is the dimension of the input vector.", "As shown in Fig.", "REF , the input vector in one-hot vector form is fed to the LSTM network in sequence.", "The $i^{th}$ output of the LSTM network after the softmax operation becomes a proper conditional probability $p(x^{(i)}|x^{(1)}, ..., x^{(i-1)})$ .", "The model is trained by minimizing the KL divergence between the data distribution sampled from the experience replay buffer and the modeled distribution: $\\begin{aligned}& \\min _{\\psi } \\: D_{\\mathrm {KL}}\\left[p^*(x)||p_\\psi (x)\\right]\\\\=& \\min _{\\psi } \\: \\mathbb {E}_{x\\sim p^*(x)}\\left[-\\log p_\\psi (x) \\right] + \\text{constant} \\end{aligned}$ Figure: The Network Architecture of Recurrent Autoregressive Model.In this work, the LSTM network has a single layer and 50 hidden size.", "Additional Comparison among Strategies Here, we show the comparison on two additional trips.", "The trip shown in Fig.", "REF contains a large number of signalized intersections.", "As indicated by Fig.", "REF , the gap between SMORL and the baseline and the gap between the wait-and-see solution and SMORL are both amplified by the high traffic density.", "The trip shown in Fig.", "REF has a very low traffic density and the speed limits higher.", "In such case, the difference between SMORL and the wait-and-see solution becomes less noticeable.", "Meanwhile, SMORL was still able to consume less fuel by using the capacity of the battery more efficiently.", "Figure: Comparison for High-density Low-speed Scenario.Figure: Comparison for Low-density High-speed Scenario.", "Ablation In this part, we compare the full SMORL algorithm with the four intermediate algorithms.", "All the algorithms presented below use trajectory optimization solved via PDDP solver.", "Tab.", "REF shows the difference in configuration and compare the trained final performance over the 100 trips used for testing.", "Here, we see that safe set and BCQ both have positive impact on the trained performance.", "In fact, the native combination of TD3 and trajectory optimization (Config.", "2) does not provide any show any significant improvement over trajectory optimization only (Config.", "1).", "In addition, without the use of safe set, the controller will deplete the battery $SoC$ to $SoC^\\mathrm {min}$ at the end of the trip as the terminal state constraint cannot be considered unless with the help of extrinsic penalty.", "Table: Ablation Study for SMORL Acknowledgment The authors acknowledge the support from the United States Department of Energy, Advanced Research Projects Agency – Energy (ARPA-E) NEXTCAR project (Award Number DE-AR0000794) and Ohio Supercomputer Center." ], [ "Conclusion", "In this paper, a safe-critical model-based off-policy reinforcement learning algorithm SMORL is proposed.", "The algorithm is applied to the eco-driving problem for Connected and Automated Hybrid Electric Vehicles.", "Compared to the previous model-free attempts on eco-driving in the literature, the method does not require any extrinsic rewarding mechanism, and thus, greatly simplifies the design process and improves the final performance.", "With the online constrained optimization formulation and the approximate safe set, the learned strategy is capable of satisfying the constraints in the prediction horizon and restricting the state within the approximate safe set, which is an approximation to the robust control invariant set.", "The performance of the strategy trained with SMORL is compared to a baseline strategy representing human drivers' behavior over 100 randomly generated trips in Columbus, OH, US.", "With a comparable average speed, the strategy from SMORL consumes approximately $22\\%$ less fuel.", "While the demonstration of the algorithm is on the eco-driving problem, we believe it can be applied to many other real-world problems, in particular to those with well-studied system dynamics, such as robotics and autonomous driving.", "Future works include incorporating SPaT information broadcasted from multiple upcoming signalized intersections, incorporating surrounding vehicles into the model, and accordingly update the optimal control problem, releasing the deterministic assumption within the prediction horizon.", "Variational Autoencoder Let $X = \\left\\lbrace x_i \\right\\rbrace _{i=1}^N$ be some data set and $Z$ represent a set of low-dimensional latent variables, the objective is to maximize the marginal log-likelihood: $\\begin{aligned}\\log p(X) = \\sum _{i=1}^N \\log p(x_i) &= \\sum _{i=1}^N \\int _z \\log p(x_i|z)p(z) dz\\\\&= \\sum _{i=1}^N \\mathbb {E}_{z\\sim p(z)} \\log p(x_i|z),\\end{aligned} $ As Eqn.", "(REF ) is in general intractable, its variational lower bound is instead maximized: $\\begin{aligned}\\mathcal {L}(\\omega _1, \\omega _2, X) = & -D_\\mathrm {KL}\\left(q_{\\omega _1}(z|X)|| p(z)\\right) \\\\& + \\mathbb {E}_{q_{\\omega _1}(z|X)} \\left[\\log p_{\\omega _2}(X|z)\\right]\\end{aligned}.", "$ Here, $D_{\\mathrm {KL}}$ is the Kullback–Leibler (KL) divergence, and $p(z)$ is the prior distribution that is typically assumed to be a multivariate normal distribution.", "$q_{\\omega _1}(z|X)$ is the posterior distribution parametrized by $\\omega _1$ .", "To analytically evaluate the KL divergence, the posterior is typically constructed as $\\mathcal {N}(z|\\mu _{\\omega _1}(X), \\Sigma _{\\omega _1}(X))$ .", "From a coding theory perspective, $q_{\\omega _1}(z|X)$ and $p_{\\omega _2}(X|z)$ can be considered as a probabilistic encoder and a probabilistic decoder, respectively.", "To compute the $\\nabla _{\\omega _1} L(\\omega _1, \\omega _2, X)$ , policy gradient theorem [47] or Reparametrization trick [47], [37] can be used.", "The latter is often used in VAE as it typically leads to a lower variance.", "In practice, the encoder $q_{\\omega _1}(z|X)$ and the decoder $p_{\\omega _2}(X|z)$ can be any function approximator.", "An implementation of VAE as the generative model to sample actions can be found in https://github.com/sfujim/BCQ.", "In this work, the latent space dimension is selected to be 5, and the encoder and the decoder are both MLPs with 2 layers of 300 hidden units.", "Autoregressive Model with LSTM For any probability distribution, the joint distribution can be factorized as a product of conditional probabilities as follow: $\\begin{aligned}&p(x) = \\prod _{i=1}^K p(x^{(i)}|x^{(1)}, ..., x^{(i-1)}), \\\\\\rightarrow &\\log p(x) = \\sum _{i=1}^K \\log p(x^{(i)}|x^{(1)}, ..., x^{(i-1)}),\\end{aligned}$ where $x^{(i)}$ is the $i^{th}$ dimension of the discrete input vector and $K$ is the dimension of the input vector.", "As shown in Fig.", "REF , the input vector in one-hot vector form is fed to the LSTM network in sequence.", "The $i^{th}$ output of the LSTM network after the softmax operation becomes a proper conditional probability $p(x^{(i)}|x^{(1)}, ..., x^{(i-1)})$ .", "The model is trained by minimizing the KL divergence between the data distribution sampled from the experience replay buffer and the modeled distribution: $\\begin{aligned}& \\min _{\\psi } \\: D_{\\mathrm {KL}}\\left[p^*(x)||p_\\psi (x)\\right]\\\\=& \\min _{\\psi } \\: \\mathbb {E}_{x\\sim p^*(x)}\\left[-\\log p_\\psi (x) \\right] + \\text{constant} \\end{aligned}$ Figure: The Network Architecture of Recurrent Autoregressive Model.In this work, the LSTM network has a single layer and 50 hidden size.", "Additional Comparison among Strategies Here, we show the comparison on two additional trips.", "The trip shown in Fig.", "REF contains a large number of signalized intersections.", "As indicated by Fig.", "REF , the gap between SMORL and the baseline and the gap between the wait-and-see solution and SMORL are both amplified by the high traffic density.", "The trip shown in Fig.", "REF has a very low traffic density and the speed limits higher.", "In such case, the difference between SMORL and the wait-and-see solution becomes less noticeable.", "Meanwhile, SMORL was still able to consume less fuel by using the capacity of the battery more efficiently.", "Figure: Comparison for High-density Low-speed Scenario.Figure: Comparison for Low-density High-speed Scenario.", "Ablation In this part, we compare the full SMORL algorithm with the four intermediate algorithms.", "All the algorithms presented below use trajectory optimization solved via PDDP solver.", "Tab.", "REF shows the difference in configuration and compare the trained final performance over the 100 trips used for testing.", "Here, we see that safe set and BCQ both have positive impact on the trained performance.", "In fact, the native combination of TD3 and trajectory optimization (Config.", "2) does not provide any show any significant improvement over trajectory optimization only (Config.", "1).", "In addition, without the use of safe set, the controller will deplete the battery $SoC$ to $SoC^\\mathrm {min}$ at the end of the trip as the terminal state constraint cannot be considered unless with the help of extrinsic penalty.", "Table: Ablation Study for SMORL Acknowledgment The authors acknowledge the support from the United States Department of Energy, Advanced Research Projects Agency – Energy (ARPA-E) NEXTCAR project (Award Number DE-AR0000794) and Ohio Supercomputer Center." ], [ "Variational Autoencoder", "Let $X = \\left\\lbrace x_i \\right\\rbrace _{i=1}^N$ be some data set and $Z$ represent a set of low-dimensional latent variables, the objective is to maximize the marginal log-likelihood: $\\begin{aligned}\\log p(X) = \\sum _{i=1}^N \\log p(x_i) &= \\sum _{i=1}^N \\int _z \\log p(x_i|z)p(z) dz\\\\&= \\sum _{i=1}^N \\mathbb {E}_{z\\sim p(z)} \\log p(x_i|z),\\end{aligned} $ As Eqn.", "(REF ) is in general intractable, its variational lower bound is instead maximized: $\\begin{aligned}\\mathcal {L}(\\omega _1, \\omega _2, X) = & -D_\\mathrm {KL}\\left(q_{\\omega _1}(z|X)|| p(z)\\right) \\\\& + \\mathbb {E}_{q_{\\omega _1}(z|X)} \\left[\\log p_{\\omega _2}(X|z)\\right]\\end{aligned}.", "$ Here, $D_{\\mathrm {KL}}$ is the Kullback–Leibler (KL) divergence, and $p(z)$ is the prior distribution that is typically assumed to be a multivariate normal distribution.", "$q_{\\omega _1}(z|X)$ is the posterior distribution parametrized by $\\omega _1$ .", "To analytically evaluate the KL divergence, the posterior is typically constructed as $\\mathcal {N}(z|\\mu _{\\omega _1}(X), \\Sigma _{\\omega _1}(X))$ .", "From a coding theory perspective, $q_{\\omega _1}(z|X)$ and $p_{\\omega _2}(X|z)$ can be considered as a probabilistic encoder and a probabilistic decoder, respectively.", "To compute the $\\nabla _{\\omega _1} L(\\omega _1, \\omega _2, X)$ , policy gradient theorem [47] or Reparametrization trick [47], [37] can be used.", "The latter is often used in VAE as it typically leads to a lower variance.", "In practice, the encoder $q_{\\omega _1}(z|X)$ and the decoder $p_{\\omega _2}(X|z)$ can be any function approximator.", "An implementation of VAE as the generative model to sample actions can be found in https://github.com/sfujim/BCQ.", "In this work, the latent space dimension is selected to be 5, and the encoder and the decoder are both MLPs with 2 layers of 300 hidden units." ], [ "Autoregressive Model with LSTM", "For any probability distribution, the joint distribution can be factorized as a product of conditional probabilities as follow: $\\begin{aligned}&p(x) = \\prod _{i=1}^K p(x^{(i)}|x^{(1)}, ..., x^{(i-1)}), \\\\\\rightarrow &\\log p(x) = \\sum _{i=1}^K \\log p(x^{(i)}|x^{(1)}, ..., x^{(i-1)}),\\end{aligned}$ where $x^{(i)}$ is the $i^{th}$ dimension of the discrete input vector and $K$ is the dimension of the input vector.", "As shown in Fig.", "REF , the input vector in one-hot vector form is fed to the LSTM network in sequence.", "The $i^{th}$ output of the LSTM network after the softmax operation becomes a proper conditional probability $p(x^{(i)}|x^{(1)}, ..., x^{(i-1)})$ .", "The model is trained by minimizing the KL divergence between the data distribution sampled from the experience replay buffer and the modeled distribution: $\\begin{aligned}& \\min _{\\psi } \\: D_{\\mathrm {KL}}\\left[p^*(x)||p_\\psi (x)\\right]\\\\=& \\min _{\\psi } \\: \\mathbb {E}_{x\\sim p^*(x)}\\left[-\\log p_\\psi (x) \\right] + \\text{constant} \\end{aligned}$ Figure: The Network Architecture of Recurrent Autoregressive Model.In this work, the LSTM network has a single layer and 50 hidden size." ], [ "Additional Comparison among Strategies", "Here, we show the comparison on two additional trips.", "The trip shown in Fig.", "REF contains a large number of signalized intersections.", "As indicated by Fig.", "REF , the gap between SMORL and the baseline and the gap between the wait-and-see solution and SMORL are both amplified by the high traffic density.", "The trip shown in Fig.", "REF has a very low traffic density and the speed limits higher.", "In such case, the difference between SMORL and the wait-and-see solution becomes less noticeable.", "Meanwhile, SMORL was still able to consume less fuel by using the capacity of the battery more efficiently.", "Figure: Comparison for High-density Low-speed Scenario.Figure: Comparison for Low-density High-speed Scenario." ], [ "Ablation", "In this part, we compare the full SMORL algorithm with the four intermediate algorithms.", "All the algorithms presented below use trajectory optimization solved via PDDP solver.", "Tab.", "REF shows the difference in configuration and compare the trained final performance over the 100 trips used for testing.", "Here, we see that safe set and BCQ both have positive impact on the trained performance.", "In fact, the native combination of TD3 and trajectory optimization (Config.", "2) does not provide any show any significant improvement over trajectory optimization only (Config.", "1).", "In addition, without the use of safe set, the controller will deplete the battery $SoC$ to $SoC^\\mathrm {min}$ at the end of the trip as the terminal state constraint cannot be considered unless with the help of extrinsic penalty.", "Table: Ablation Study for SMORL" ], [ "Acknowledgment", "The authors acknowledge the support from the United States Department of Energy, Advanced Research Projects Agency – Energy (ARPA-E) NEXTCAR project (Award Number DE-AR0000794) and Ohio Supercomputer Center." ] ]
2105.11640
[ [ "Multifunctional high-frequency circuit capabilities of ambipolar carbon\n nanotube FETs" ], [ "Abstract An experimentally-calibrated carbon nanotube compact transistor model has been used here to design two high-frequency (HF) circuits with two different functionalities each: a phase configurable amplifier (PCA) and a frequency configurable amplifier (FCA).", "The former design involves an in-phase amplifier and an inverting amplifier while the latter design embraces a frequency doubler as well as a distinct inverting amplifier.", "The specific functionality selection of each of the two HF circuit designs is enabled mainly by the inherent ambipolar feature at a device level.", "Furthermore, at a circuit level the matching networks are the same regardless the operation mode.", "In-phase and inverting amplification are enabled in the PCA by switching the gate-to-source voltage (VGS) from -0.3 V to 0.9 V while the drain-to-source voltage (VDS) remains at 3 V. By designing carefully the matching and stability networks, power gains of 4.5 dB and 6.7 dB at 2.4 GHz for the in-phase and inverting operation mode have been achieved, respectively.", "The FCA, in its frequency doubler operation mode, exhibits 20 dBc of fundamental-harmonic suppression at 2.4 GHz when an input signal at 1.2 GHz is considered.", "This frequency doubler functionality is enabled at VGS=0.3 V, whereas at VGS=0.9 V amplification of 4.5 dB is obtained while VDS remains at 3 V in both cases.", "In both configurable circuits the stabilization and matching networks are the same regardless the bias-chosen operation mode.", "The circuits performance degradation due to metallic tubes in the device channel is studied as well as the impact of non-ideal inductors in each design.", "PCA and FCA operation modes are further exploited in high-frequency modulators." ], [ "Introduction", "Carbon nanotubes (CNTs) have emerged as one of the options to replace the silicon (Si)-based channel of novel field-effect transistors (FETs) in high-performance applications beyond the limits of conventional technologies [1], [2] while improving further the device footprint and fabrication costs of related circuits [3].", "The quasi-ballistic transport, an outstanding gate control over the channel, as well as an inherent device linearity associated to CNTFETs, makes them suitable for low-power high-frequency (HF) applications [2], [4], [5] as recently demonstrated by devices operating at frequencies of around 100 [6], [7], [8].", "One feature of carbon-based FETs is their ambipolar behavior [9], [10], [11], [12], i.e., the device performance in the active region can be of n-type-like or p-type-like according to the bias.", "Notice that CNTFETs in this work have only three terminals (source, drain and gate), in contrast to previous reports where an additional control gate has been used to control the ambipolarity [13], [14].", "In-phase and inverting amplification can be enabled in ambipolar transistors by applying an input AC signal and biasing the device in the linear n-type region and linear p-type region, respectively.", "Furthermore, if the DC device operation is around the minimum transconductance point, the frequency ($f$ ) of an input AC signal can be multiplied at the output [15].", "Therefore, on-chip designs can benefit from ambipolar devices since stages of radio-frequency (RF) integrated circuits, including frequency multipliers and amplifiers, can be fabricated with the same transistor technology biased at different regions, hence reducing the fabrication process and production costs.", "CNTFETs ambipolarity [10], [17], [16], [18], [19] has been already used in proof-of-concept analog circuits [15], [21], [22], [20] focused on the frequency multiplying characteristic rather than tackling efficient circuit design to exploit the device multifunctionality for analog circuits.", "In this work, two CNTFET-based HF circuit designs, identified here as a phase configurable amplifier (PCA) and a frequency configurable amplifier (FCA), are presented.", "The device biasing conditions enable two different circuit operation modes in each design: an in-phase amplifier (PA) and an inverting amplifier (IA) are possible in the PCA while a frequency doubler (FD) and a distinct IA can be selected in the FCA.", "The different functionalities for the PCA and the FCA have been demonstrated with identical matching networks and further exploited for the proposal of CNTFET-based modulators.", "Hence, this is an efficient on-chip solution for CNTFET-based configurable HF analog circuits by using a suitable compact large-signal model [23].", "This work intends to show the capabilities of ambipolar CNTFETs in high-frequency circuits by considering optimal and quasi-optimal conditions at both device and circuit level.", "The outcome of this study should be considered as a support and motivation for the CNTFET community related to fabrication and circuit design since these results have been obtained considering feasible scenarios." ], [ "Compact model and device description", "Electron and hole transport is inherently allowed simultaneously in CNTs.", "The dominant carrier type in CNTFETs is determined by the contacts physics [10], [16], [17].", "In the transfer curves shown in Fig.", "REF (a), the drain current ($I_{\\rm D}$ ) is comprised mainly by tunneling electron current at the saturation regime of the n-branch, due to a strong energy bands bending (thin source potential barrier).", "The ohmic region of the p-branch is due to a thin drain barrier induced by high negative gate-to-source voltage $V_{\\rm GS}$ enabling a major contribution of tunneling hole current.", "Figure: (a) Simulated transfer characteristics from CCAM for the reference MT-CNTFET without metallic tubes.", "Markers indicate the different bias points considered in this work.", "(b) Representation of the v in v_{\\rm in} (black) and ideal v out v_{\\rm out} (colored) for the device biased at the different operation regions.In this work, a semi-physical large-signal compact model for CNTFETs, named CCAM [23], [24] has been used.", "It describes the ambipolar behavior of a fabricated CNTFET technology considering, by different modules, both the contribution of semiconductor (s-) and metallic (m-) tubes in the channel.", "These modules correspond to the intrinsic part of the device as identified in the compact model equivalent circuit shown in Fig.", "REF (a).", "Discussions on each model parameter and the physics behind each of them have been provided in [23], [24].", "High-frequency noise modules, described elsewhere [28] have been activated in this work.", "The compact model has been calibrated (including both intrinsic part and parasitics) to hysteresis-free DC and dynamic experimental data [23], [25] from a fabricated top-gate multi-tube (MT) CNTFET technology [26], hence, the model used here is directly related to a manufacturable optimized device technologyThe compact model used here is one of the few able to correctly describe the performance of fabricated RF CNTFET technology.", "CNTFET-based RF applications have been already developed using this technology [26], [29], [30], [31].", "HF performance projections for amplifiers using the considered CNTFET technology have been developed using CCAM [27], [28], however, the ambipolarity features of CNTFETs have not been exploited in such works.", "The reference device considered here has a channel length of 700 and a top-gate length of 250, a total of eight gate fingers, 50 of width each, yielding a device total width of 400.", "The device cross-section is shown in Fig.", "REF (b).", "Diameters of the tubes have been estimated of $\\sim $ 1.8 [23], leading to a channel bandgap of $\\sim $ 0.47.", "The channel array has s- and m-tubes grown on SiO$_{2}$ substrate via chemical vapor deposition.", "Fabrication process details can be found in [26].", "Figure: (a) Equivalent circuit of the large-signal model CCAM.", "The intrinsic part (inside the dashed box) includes a module for parameters associated to semiconducting tubes (in red in the electronic version) and another one related to metallic tubes (in blue in the electronic version).", "See , for detailed discussions on each parameter as well as for the experimental validation of the model.", "(b) Schematic cross-section of the CNTFET used for the CCAM calibration used in this work (not drawn to scale).A high number of s-CNTs in the channel is required towards improving the HF device performance, as demonstrated by an s-tube sensitivy study for RF circuits presented elsewhere [27] and fabricated RF state-of-the-art devices [6], [7], [32], [8].", "Hence, the number of m-CNTs in the compact model has been set to zero while other parameters remain unchanged for enhancing the HF performance, i.e., an optimized device with only 2000 s-CNTs in the channel, leading to a tube density $n_{\\rm t}$ of 5 CNTs$/{}{}$ , has been considered in this study.", "Transfer characteristics of the simulated ambipolar MT-CNTFET, considering only s-tubes in the channel, are shown in Fig.", "REF (a) where three bias points, related to a different operation region each, have been indicated.", "Region I corresponds to the p-type behavior, region II embraces the polarity transition (low-transconductance regime) and region III includes the n-type dominant branch.", "At high $V_{\\rm DS}$ , tunneling processes are more pronounced as observed by the higher value of the minimum current at the corresponding transfer characteristic in contrast to curves at lower $V_{\\rm DS}$ .", "The expected outcome of an input AC signal for the device biased differently is qualitatively shown in Fig.REF (b)." ], [ "CNTFET-based multifunctional circuits", "The ambipolar behavior of CNTFETs enhances the development of multifunctional analog circuits by biasing the device in one of the three regions identified in Fig.", "REF .", "In regions I and III, the dependence of $I_{\\rm D}$ on $V_{\\rm GS}$ is ideally described by a linear function with a negative (region I) or positive (region III) slope, i.e., ${\\textstyle I_{\\rm D}=-g_{\\rm m}V_{\\rm GS}+B}$ or ${\\textstyle I_{\\rm D}=g_{\\rm m}V_{\\rm GS}+E}$ , respectively.", "In region II, ${\\textstyle I_{\\rm D}}$ has a parabolic-like $V_{\\rm GS}$ -dependence, i.e., ${\\textstyle I_{\\rm D}=C+D(V_{\\rm GS}-V_{\\rm GS_0})^2}$ , with the constants $B$ , $C$ , $D$ and $E$ and ${\\textstyle V_{\\rm GS_0}=V_{\\rm GS}\\vert _{\\rm {min}(\\mathit {I}_{\\rm D}\\rm )}}$ .", "Thus, considering ${\\textstyle V_{\\rm in}=V_{\\rm GS}+v_{\\rm in}=V_{\\rm GS}+A\\sin \\omega t }$ , in regions I and III, the device operates as a $g_{\\rm m}$ -controlled amplifier: being an in-phase amplifier for ${\\textstyle g_{\\rm m}<0}$ (in region I), while in region III (${\\textstyle g_{\\rm m}>0}$ ) the device works as an inverting amplifier.", "An improved device linearity (constant $g_{\\rm m}$ [4]) enhance the amplifying performance.", "Furthermore, if the device is biased in region II, it works during half period of the input signal in the $p$ -like region and during the other half period in the $n$ -like region, yielding a doubled frequency in the output signal enhanced by the parabolic-shape transfer characteristics within this bias.", "Due to the common source topology, by using a drain resistor $R_{\\rm D}$ to apply $V_{\\rm DS}$ from a $V_{\\rm DD}$ voltage source (see Fig.", "REF ), the output voltage in region II is ${\\textstyle V_{\\rm out}=V_{\\rm DS}+v_{\\rm out}=}$ ${\\textstyle V_{\\rm DD}-BR_{\\rm D}-1/2\\left(R_{\\rm D}CA^2\\right)+1/2\\left(R_{\\rm D}CA^2\\cos 2\\omega t \\right)}$ , i.e., the doubled frequency is explained by the fourth term.", "A similar analysis of the output signals for different operation modes of ambipolar CNTFETs has been provided elsewhere [15]." ], [ "Phase configurable amplifier design", "The selected bias points of the PCA are ${\\textstyle V_{\\rm GS}=-{0.3}{}}$ (region I) for the PA operation mode and ${\\textstyle V_{\\rm GS}={0.9}{}}$ (region III) for the IA operation mode, in both cases ${\\textstyle V_{\\rm DS}={3}{}}$ .", "Cutoff and maximum oscillation frequencies obtained at the bias point in region I(III) are 4(8.5) and 14.6(23.5), respectively.", "Both amplifiers have been designed for maximum gain at 2.4.", "The device unilateral power gain at 2.4 are 19 and 10 for the PCA-PA and PCA-IA, respectively.", "Ambipolar CNTFETs have been used previously to design in-phase and inverting amplifiers at low operation frequencies [15], biasing the device at high $V_{\\rm DD}$ (12), lacking gain for the in-phase configuration and requiring specific matching networks for each operation mode.", "In contrast, in this work the considered device is used to design a PCA suitable for RF applications at lower $V_{\\rm DD}$ ($=V_{\\rm DS}={3}{})$ and with identical matching and stabilization networks regardless the operation mode, i.e., a more efficient design for multifunctional CNTFET-based circuits is proposed here.", "Fig.", "REF shows the proposed PCA design operating at a $f$ of 2.4.", "The parameters values are listed in Table REF (a).", "The design has been developed using the Verilog-A implementation [24] of CCAM [23] into the Keysight Advance Design System software.", "Based on the internal transport mechanisms at each contact, source and drain contacts remain physically the same regardless the operation mode.", "However, at a circuit level and considering the difference in the external potentials, a common-source (common-drain) circuit topology is related for the CNTFET working in region I (III).", "A feedback topology using a resistor $R_1$ , a capacitor $C_1$ and an inductor $L_1$ has been designed in order to find stability conditions (${\\textstyle K>1}$ and ${\\textstyle \\vert \\Delta \\vert <1}$ , see [33]).", "The PCA is unconditionally stable at 2.4, and up to 3, while $V_{\\rm GS}$ changes indistinctly the operation mode (see Fig.", "REF (a)).", "Furthermore, matching networks have been designed towards a transparent implementation with established 50-RF systems.", "Notice that the stability and matching networks remain the same regardless the operation mode (bias).", "For higher frequencies however, matching and stability network designs, different to the ones considered here, are required to avoid non-desired oscillations.", "Figure: Schematic of the PCA and FCA.", "Possible operation modes for the PCA (R D =0{\\textstyle R_{\\rm D}={0}{}}) are as an in-phase or as an inverting amplifier.", "The FCA (R D ≠0{\\textstyle R_{\\rm D}\\ne {0}{}}) can operate as a frequency doubler or as an inverting amplifier.", "All operation modes are controlled by V GS V_{\\rm GS} in each configuration.", "Matching (dotted and dash-dotted line boxes) and stabilization (continuous line box) networks are the same regardless the operation mode of each design.", "Parameter values are reported in Table .Table: Parameter values of the designed circuitsFigure: (a) Stability metrics and (b) |S 11 |\\vert S_{11}\\vert and |S 22 |\\vert S_{22}\\vert for PCA operation modes: in-phase amplifier (V GS =-0.3{\\textstyle V_{\\rm GS}={-0.3}{}}) and inverting amplifier (V GS =0.9{\\textstyle V_{\\rm GS}={0.9}{}}).The PCA is able to change its operation mode only by switching the device bias between regions I and III.", "In both cases, $\\vert S_{11}\\vert $ and ${\\textstyle \\vert S_{22}\\vert \\lesssim -{10}{}}$ at 2.4 as shows Fig.", "REF (b).", "The power gain for the in-phase amplifier operation mode is $\\sim $ 4.5 whereas for the inverting amplifier mode is $\\sim $ 6.7 (see Fig.", "REF (a)).", "In contrast to voltage-gainless fabricated CNTFET-based ambipolar amplifiers designed for specific functionalities [15], voltage gain of $\\sim $ 1.7/ and $\\sim $ 2.16/ has been obtained here, for the in-phase and inverting amplifier, respectively.", "By switching operation modes, the amplified output signal phase is shifted from 0 (${\\textstyle V_{\\rm GS}={-0.3}{}}$ ) to 180 (${\\textstyle V_{\\rm GS}={0.9}{}}$ ) (see Fig.", "REF (b)).", "The input power ($P_{\\rm in}$ ) has been of $-{15}{m}$ at 2.4.", "Figure: (a) |S 21 |\\vert S_{21}\\vert and (b) v in v_{\\rm in} and v out v_{\\rm out} signals for the two PCA operation modes: in-phase amplifier (V GS =-0.3V_{\\rm GS}={-0.3}{}) and inverting amplifier (V GS =0.9V_{\\rm GS}={0.9}{}).", "Circles in (a) indicate the gain at 2.4.", "(a slight phase shift caused by the matching networks, the same in both operation modes, has been corrected for a better visualization of the circuit performance).In order to characterize entirely the proposed amplifiers, the noise performances is reported next.", "However notice that the matching and stability networks have been designed under the consideration of achieving maximum gain.", "The minimum noise figure $NF_{\\rm min}$ , noise resistance $R_{\\rm n}$ and optimum reflection coefficient $\\Gamma _{\\rm opt}$ are of 9.2, 103, 0.201 ($-{101.1}{}^{\\circ }$ ), respectively, for the PCA-PA and of 6.7, 85, 0.213 (${4.3}{}^{\\circ }$ ), respectively, for the PCA-IA.", "Low-noise amplifiers have been designed elsewhere [28] with the same model and technology used here." ], [ "Frequency configurable amplifier design", "The schematic of the FCA is similar to the PCA design, (Fig.", "REF ), however, parameter values are different (see Table I(b)).", "Notice specially the inclusion of $R_{\\rm D}$ in this design.", "The selected bias points are $V_{\\rm GS}={0.3}{}$ (region II) for the frequency doubler operation mode and $V_{\\rm GS}={0.9}{}$ (region III) for the inverting amplifier operation mode, in both cases ${\\textstyle V_{\\rm DS}={3}{}}$ .", "The FCA name has been proposed since the output signal is at doubled or similar $f$ as the input signal, depending on the operation mode.", "The feedback network has been designed in order to have both stability conditions and an improved output signal in the FD operation mode.", "The matching network has been designed towards the enhancement of both operation modes.", "$P_{\\rm in}$ is $-{10}{m}$ at 1.2.", "The frequency doubler performance ($V_{\\rm GS}={0.3}{}$ ), can be observed by comparing the input and output signals frequency shown in Fig.", "REF (a).", "The slight difference in the amplitudes between two successive periods is related to a non-symmetric ambipolar device performance (see Fig.", "REF (a)).", "The reference device has not been optimized towards an ideal ambipolar symmetry since this technology has been proposed for general RF applications [26], e.g., [29]-[31].", "The output power ($P_{\\rm out}$ ) for the frequency doubler operation mode is $-{14.6}{m}$ at 2.4 leading to a circuit conversion frequency loss of ${4.6}{}$ , while almost all the power is concentrated at 2.4 as shown in Fig.", "REF (b).", "The FCA design is a more efficient frequency doubler than the reported circuits in [21] with considerable lower conversion losses (-35 and -55, regardless the different $n_{\\rm t}$ ) at similar operation frequency.", "Additionally, the performance in $P_{\\rm out}$ and operation frequency obtained here are higher than the ones of CNTFET-based frequency doublers presented elsewhere [22], [34].", "Furthermore, the voltage gain of the frequency doubler operation mode is of $\\sim $ 0.6/ which, despite its low value, is almost 4 times higher than the one achieved at 2 with a different CNTFET technology [15].", "Multi-stage HF amplifiers solutions, with the same CNTFET technology [27], [28], can be proposed towards improving the FCA-FD performance.", "Figure: (a) v in v_{\\rm in} and v out v_{\\rm out} signals for the two operation modes of the FCA: frequency doubler (V GS =0.3V_{\\rm GS}={0.3}{}) and inverting amplifier (V GS =0.9V_{\\rm GS}={0.9}{}) (a slight phase shift, caused by the matching networks, has been corrected in the plot for a better visualization of the circuit performance).", "(b) P out P_{\\rm out} spectrum of the frequency doubler operation mode for a P in P_{\\rm in} of -10m-{10}{m} at 1.2.Fundamental-harmonic suppression for the frequency doubler operation mode is 19.65c while the 3rd and 4th harmonic suppression are 20.14c and 25.64c, respectively.", "This indicates a high spectral purity for the doubled frequency $P_{\\rm out}$ .", "Notice that this harmonic suppression is higher than in previous CNTFET-related works [21], [22] where devices with high $n_{\\rm t}$ have been considered.", "Device parasitics have been considered in the compact model, hence the achieved harmonic suppression is due to a proper matching network design, which produces $\\vert S_{11}\\vert $ and $\\vert S_{22}\\vert $ $<-{10}{}$ at 1.2 and 2.4, respectively (see Fig.", "REF (a)), in addition to the device properties.", "Notice that the results obtained for this circuit with this specific operation mode can be further improved if a more symmetric ambipolar response of the device can be achieved by, e.g., a modulation of the internal charge, however this is out of the scope of this study.", "The FCA can be switched into amplification mode by biasing the device in region III ($V_{\\rm GS}={0.9}{}$ ).", "The device is stable at 1.2 and $\\vert S_{11}\\vert $ and $\\vert S_{22}\\vert $ $\\lesssim $ $-{10}{}$ at 1.2 as shows Fig.REF (a).", "The unilateral power gain of the device in this bias point and at 1.2 is of 7.4.", "Power gain of $\\sim $ 4.5 at 1.2 (see Fig.REF (b)) is achieved with the FCA inverting amplifier operation mode.", "Noise figures of merit $NF_{\\rm min}$ , $R_{\\rm n}$ and $\\Gamma _{\\rm opt}$ ) of the FCA-IA are 10.3, 191.4 and 0.186 ($-{21.88}{}^{\\circ }$ ), respectively.", "As discussed in Section .A, CNTFET-based amplifiers can be designed towards optimum noise performance as shown elsewhere [28].", "Figure: (a) |S 11 |\\vert S_{11}\\vert and |S 22 |\\vert S_{22}\\vert for the two operation modes of the FCA.", "(b) |S 21 |\\vert S_{21}\\vert for the FCA-IA (the circle indicates the gain at 1.2).The performance of the FCA circuit can be controlled by the selected bias point without changes in the stability and matching networks resulting in a $V_{\\rm GS}$ -controlled circuit capable to operate as an amplifier at 1.2 or as a frequency doubler with an output $f$ of 2.4.", "A third operation mode for the multifunctional circuits presented here, involves a more intricate trade-off between impedance matching and stability for each functionality and it is out of the scope of this work." ], [ "Impact of m-tubes and inductor quality factor", "In order to consider a non-optimized techology, a study of the impact of m-tubes in the device channel on multifunctional circuits has been carried out here by considering a semiconducting-to-metallic tube ratio (s:m) of 3:1, i.e., the original calibration of the compact model (cf.", "Section ) [23] has been used.", "Fig.", "REF (a) shows that device operation regions remain similar in shape with higher $I_{\\rm D}$ values at the same bias in comparison to the optimal case (cf.", "Fig.", "REF (a)).", "At a circuit level however, higher $V_{\\rm DD}$ are required for the FCA design (8.6 for the FD and 7.6 for the IA).", "The overall performance of both multifunctional circuits is degraded with the presence of m-tubes (Figs.", "REF (b)-(d)) in contrast to the CNTFET with s-tubes only, considered in previous sections.", "E.g., for the non-optimzed technology, gains of 5.7 and 3.3 have been obtained for the PCA-IA and PCA-PA, respectively, and a conversion loss of 6.2 is achieved for the FCA-FD.", "The latter is obtained at the cost of more power consumption.", "Figure: Results with a s:m-ratio of 3:1 in the device channel: (a) CNTFET transfer characteristics, (b) v in v_{\\rm in} and v out v_{\\rm out} signals and (d) |S 21 |\\vert S_{21}\\vert for the PCA operation modes, and (c) output spectrum of the FCA frequency doubler mode.The impact of inductor quality factor ($Q$ ) on the performance of multifunctional circuits has been studied here by considering high-$Q$ on-chip CNT-based inductors [35] in the proposed designs ($L_1$ in Fig.", "REF ).", "Results are shown in Table REF for $Q$ of 8;35 (see Fig.", "12 in [35]) as well as the ideal case of CNT inductors.", "All inductors can be assumed to be over SiO$_2$ substrates towards an integrated fabrication.", "The gain of the PCA's both modes is degraded $\\sim {3}{}$ in the worst scenario (lowest $Q$ ).", "Similarly, the FCA-IA gain is diminished $\\sim {1.5}{}$ for $Q=8$ while the FCA-FD conversion loss is $\\sim {3}{}$ more than the ideal case.", "In the case of $Q=35$ the performance is slightly deviated from the ideal case response.", "Table: Impact of QQ-factor on PCA and FCA operation modes at 2.4{2.4}{}" ], [ "CNTFET-based modulators enabled by the multifunctional circuits", "In order to show potential applications of the PCA and FCA with their different operation modes in high-data rate communication systems, two different modulation schemes have been demonstrated: phase-shift keying (PSK) and frequency-shift keying (FSK).", "The device compact model with semiconducting CNTs in the channel (Sections II and III) has been considered in this part of the work.", "Figure: Modulation schemes achieved with CNTFET-based multifucntional circuits.", "(a) PSK modlated signal (v out v_{\\rm out}) achieved with the PCA by varying the operation mode between IA (V GS =0.9V_{\\rm GS}={0.9}{} equivalent to a digital \"1\") and PA (V GS =-0.3V_{\\rm GS}=-{0.3}{} equivalent to a digital \"0\").", "(b) FSK modulated signal (v out v_{\\rm out}) achieved with the FCA by varying the operation mode between IA (V GS =0.9V_{\\rm GS}={0.9}{} equivalent to a digital \"1\") and FD (V GS =0.3V_{\\rm GS}={0.3}{} equivalent to a digital \"0\").The input has been fed with AC carrier signals (not shown here) with frequency of 2.4 for the PSK and of 1.2 for the FSK as well as the baseband data given by the variation of $V_{\\rm GS}$ .", "In all cases, $V_{\\rm DS}={3}{}$ .", "By switching $V_{\\rm GS}$ alternatively between 0.9 (IA operation mode) to -0.3 (PA operation mode) the binary values \"1\" and \"0\" can be stablished with the PCA leading to a PSK modulated signal at the output as shown in Fig.", "REF (a).", "Similarly, FSK modulation has been achieved by considering the two functionalities of the FCA as shown in Fig.", "REF (b).", "Hence, the CNTFET-based circuit designs proposed in this work enable to perform modulation schemes, PSK with PCA or FSK with FCA, revealing them as suitable solutions for on-chip communication systems since only one single device and circuit have been used for each modulation.", "According to the authors' knowledge, this is the first proposal of CNTFET-based modulators." ], [ "Conclusions", "CNTFETs are promising candidates to develop multifunctional HF applications due to their intrinsic ambipolar transport.", "The circuits presented here include a PCA and a FCA.", "The PCA works at a $f$ of 2.4 for its two operation modes: as an in-phase amplifier and as an inverting amplifier.", "The FCA works at 1.2 for its two operation modes: as a frequency doubler and as an inverting amplifier.", "Similar matching networks have been used for both functionalities of each design.", "According to the authors' knowledge, this is the first proposal of CNTFET-based HF multifunctional circuits which configurability is only bias-controlled and without the need of changing matching and stabilization networks.", "These networks however, should be designed considering an enhanced performance for the two functionalities of each circuit.", "The results obtained here show the feasibility of multifunctional ambipolar CNTFET-based circuits for RF applications in S-band using a systematic study based on a device compact model calibrated with reproducible experimental data from a manufacturable CNTFET technology.", "This is of special interest towards multifunctional high-performance power-efficient on-chip solutions, e.g., in FSK or PSK modulators using a single device as shown in this work.", "Hence, this work's proposal can alleviate fabrication and production costs in HF integrated circuits based in CNTFETs or other ambipolar devices, e.g., graphene FETs.", "In the case of the PCA, voltage and power gains ($\\gtrsim $ 1.7/ and $\\gtrsim $ 4.5, respectively) have been achieved through a carefully bias point selection and stabilization and matching networks design.", "Moreover, fundamental-harmonic suppression close to 20c has been observed for the frequency doubler operation mode of the FCA, which indicates that doubler frequency circuits based on CNTFETs can be used without additional filtering stages.", "Moreover, both the impact of m-tubes in the device channel and CNTs-based inductors with different $Q$ on the circuits response have been studied.", "Results indicate that the multifuctionality feature of the designed circuits can be achieved with discrete success, in comparison to the optimized case, with at least ${1}{}/{3}{}$ of m-tubes in the channel array and with non-ideal CNT-based inductors." ] ]
2105.11710
[ [ "ConSERT: A Contrastive Framework for Self-Supervised Sentence\n Representation Transfer" ], [ "Abstract Learning high-quality sentence representations benefits a wide range of natural language processing tasks.", "Though BERT-based pre-trained language models achieve high performance on many downstream tasks, the native derived sentence representations are proved to be collapsed and thus produce a poor performance on the semantic textual similarity (STS) tasks.", "In this paper, we present ConSERT, a Contrastive Framework for Self-Supervised Sentence Representation Transfer, that adopts contrastive learning to fine-tune BERT in an unsupervised and effective way.", "By making use of unlabeled texts, ConSERT solves the collapse issue of BERT-derived sentence representations and make them more applicable for downstream tasks.", "Experiments on STS datasets demonstrate that ConSERT achieves an 8\\% relative improvement over the previous state-of-the-art, even comparable to the supervised SBERT-NLI.", "And when further incorporating NLI supervision, we achieve new state-of-the-art performance on STS tasks.", "Moreover, ConSERT obtains comparable results with only 1000 samples available, showing its robustness in data scarcity scenarios." ], [ "Introduction", "Sentence representation learning plays a vital role in natural language processing tasks , , , .", "Good sentence representations benefit a wide range of downstream tasks, especially for computationally expensive ones, including large-scale semantic similarity comparison and information retrieval.", "Figure: The correlation diagram between the gold similarity score (x-axis) and the model predicted cosine similarity score (y-axis) on the STS benchmark dataset.Recently, BERT-based pre-trained language models have achieved high performance on many downstream tasks with additional supervision.", "However, the native sentence representations derived from BERTTypically, we take the output of the [CLS] token or average token embeddings at the last few layers as the sentence representations.", "are proved to be of low-quality , .", "As shown in Figure REF a, when directly adopt BERT-based sentence representations to semantic textual similarity (STS) tasks, almost all pairs of sentences achieved a similarity score between 0.6 to 1.0 , even if some pairs are regarded as completely unrelated by the human annotators.", "In other words, the BERT-derived native sentence representations are somehow collapsed , which means almost all sentences are mapped into a small area and therefore produce high similarity.", "Such phenomenon is also observed in several previous works , , .", "They find the word representation space of BERT is anisotropic, the high-frequency words are clustered and close to the origin, while low-frequency words disperse sparsely.", "When averaging token embeddings, those high-frequency words dominate the sentence representations, inducing biases against their real semantics We also empirically prove this hypothesis, please refer to Section REF for more details.. As a result, it is inappropriate to directly apply BERT's native sentence representations for semantic matching or text retrieval.", "Traditional methods usually fine-tune BERT with additional supervision.", "However, human annotation is costly and often unavailable in real-world scenarios.", "To alleviate the collapse issue of BERT as well as reduce the requirement for labeled data, we propose a novel sentence-level training objective based on contrastive learning , , .", "By encouraging two augmented views from the same sentence to be closer while keeping views from other sentences away, we reshape the BERT-derived sentence representation space and successfully solve the collapse issue (shown in Figure REF b).", "Moreover, we propose multiple data augmentation strategies for contrastive learning, including adversarial attack , , token shuffling, cutoff and dropout , that effectively transfer the sentence representations to downstream tasks.", "We name our approach ConSERT, a Contrastive Framework for SEntence Representation Transfer.", "ConSERT has several advantages over previous approaches.", "Firstly, it introduces no extra structure or specialized implementation during inference.", "The parameter size of ConSERT keeps the same as BERT, making it easy to use.", "Secondly, compared with pre-training approaches, ConSERT is more efficient.", "With only 1,000 unlabeled texts drawn from the target distribution (which is easy to collect in real-world applications), we achieve 35% relative performance gain over BERT, and the training stage takes only a few minutes (1-2k steps) on a single V100 GPU.", "Finally, it includes several effective and convenient data augmentation methods with minimal semantic impact.", "Their effects are validated and analyzed in the ablation studies.", "Our contributions can be summarized as follows: 1) We propose a simple but effective sentence-level training objective based on contrastive learning.", "It mitigates the collapse of BERT-derived representations and transfers them to downstream tasks.", "2) We explore various effective text augmentation strategies to generate views for contrastive learning and analyze their effects on unsupervised sentence representation transfer.", "3) With only fine-tuning on unsupervised target datasets, our approach achieves significant improvement on STS tasks.", "When further incorporating with NLI supervision, our approach achieves new state-of-the-art performance.", "We also show the robustness of our approach in data scarcity scenarios and intuitive analysis of the transferred representations.Our code is available at https://github.com/yym6472/ConSERT." ], [ "Sentence Representation Learning", "Supervised Approaches Several works use supervised datasets for sentence representation learning.", "finds the supervised Natural Language Inference (NLI) task is useful to train good sentence representations.", "They use a BiLSTM-based encoder and train it on two NLI datasets, Stanford NLI (SNLI) and Multi-Genre NLI (MNLI) .", "Universal Sentence Encoder adopts a Transformer-based architecture and uses the SNLI dataset to augment the unsupervised training.", "SBERT proposes a siamese architecture with a shared BERT encoder and is also trained on SNLI and MNLI datasets.", "Self-supervised Objectives for Pre-training BERT proposes a bi-directional Transformer encoder for language model pre-training.", "It includes a sentence-level training objective, namely next sentence prediction (NSP), which predicts whether two sentences are adjacent or not.", "However, NSP is proved to be weak and has little contribution to the final performance .", "After that, various self-supervised objectives are proposed for pre-training BERT-like sentence encoders.", "Cross-Thought and CMLM are two similar objectives that recover masked tokens in one sentence conditioned on the representations of its contextual sentences.", "SLM proposes an objective that reconstructs the correct sentence ordering given the shuffled sentences as the input.", "However, all these objectives need document-level corpus and are thus not applicable to downstream tasks with only short texts.", "Unsupervised Approaches BERT-flow proposes a flow-based approach that maps BERT embeddings to a standard Gaussian latent space, where embeddings are more suitable for comparison.", "However, this approach introduces extra model structures and need specialized implementation, which may limit its application." ], [ "Contrastive Learning", "Contrastive Learning for Visual Representation Learning Recently, contrastive learning has become a very popular technique in unsupervised visual representation learning with solid performance , , .", "They believe that good representation should be able to identify the same object while distinguishing itself from other objects.", "Based on this intuition, they apply image transformations (e.g.", "cropping, rotation, cutout, etc.)", "to randomly generate two augmented versions for each image and make them close in the representation space.", "Such approaches can be regarded as the invariance modeling to the input samples.", "proposes SimCLR, a simple framework for contrastive learning.", "They use the normalized temperature-scaled cross-entropy loss (NT-Xent) as the training loss, which is also called InfoNCE in the previous literature .", "Contrastive Learning for Textual Representation Learning Recently, contrastive learning has been widely applied in NLP tasks.", "Many works use it for language model pre-training.", "IS-BERT proposes to add 1-D convolutional neural network (CNN) layers on top of BERT and train the CNNs by maximizing the mutual information (MI) between the global sentence embedding and its corresponding local contexts embeddings.", "CERT adopts a similar structure as MoCo and uses back-translation for data augmentation.", "However, the momentum encoder needs extra memory and back-translation may produce false positives.", "BERT-CT uses two individual encoders for contrastive learning, which also needs extra memory.", "Besides, they only sample 7 negatives, resulting in low training efficiency.", "DeCLUTR adopts the architecture of SimCLR and jointly trains the model with contrastive objective and masked language model objective.", "However, they only use spans for contrastive learning, which is fragmented in semantics.", "CLEAR uses the same architecture and objectives as DeCLUTR.", "Both of them are used to pre-train the language model, which needs a large corpus and takes a lot of resources." ], [ "Approach", "In this section, we present ConSERT for sentence representation transfer.", "Given a BERT-like pre-trained language model $\\mathbf {M}$ and an unsupervised dataset $\\mathcal {D}$ drawn from the target distribution, we aim at fine-tuning $\\mathbf {M}$ on $\\mathcal {D}$ to make the sentence representation more task-relevant and applicable to downstream tasks.", "We first present the general framework of our approach, then we introduce several data augmentation strategies for contrastive learning.", "Finally, we talk about three ways to further incorporate supervision signals." ], [ "General Framework", "Our approach is mainly inspired by SimCLR .", "As shown in Figure REF , there are three major components in our framework: A data augmentation module that generates different views for input samples at the token embedding layer.", "A shared BERT encoder that computes sentence representations for each input text.", "During training, we use the average pooling of the token embeddings at the last layer to obtain sentence representations.", "A contrastive loss layer on top of the BERT encoder.", "It maximizes the agreement between one representation and its corresponding version that is augmented from the same sentence while keeping it distant from other sentence representations in the same batch.", "For each input text $x$ , we first pass it to the data augmentation module, in which two transformations $T_1$ and $T_2$ are applied to generate two versions of token embeddings: $e_i = T_1(x), e_j = T_2(x)$ , where $e_i, e_j \\in \\mathbb {R}^{L \\times d}$ , $L$ is the sequence length and $d$ is the hidden dimension.", "After that, both $e_i$ and $e_j$ will be encoded by multi-layer transformer blocks in BERT and produce the sentence representations $r_i$ and $r_j$ through average pooling.", "Following , we adopt the normalized temperature-scaled cross-entropy loss (NT-Xent) as the contrastive objective.", "During each training step, we randomly sample $N$ texts from $\\mathcal {D}$ to construct a mini-batch, resulting in $2N$ representations after augmentation.", "Each data point is trained to find out its counterpart among $2(N-1)$ in-batch negative samples: $\\mathcal {L}_{i, j} = - \\log \\frac{\\exp (\\text{sim}(r_i, r_j)/\\tau )}{\\sum _{k=1}^{2N} \\mathbb {1}_{[k \\ne i]}\\exp (\\text{sim}(r_i, r_k)/\\tau )}$ , where $\\text{sim}(\\cdot )$ indicates the cosine similarity function, $\\tau $ controls the temperature and $\\mathbb {1}$ is the indicator.", "Finally, we average all $2N$ in-batch classification losses to obtain the final contrastive loss $\\mathcal {L}_\\text{con}$ ." ], [ "Data Augmentation Strategies", "We explore four different data augmentation strategies to generate views for contrastive learning, including adversarial attack , , token shuffling, cutoff and dropout , as illustrated in Figure REF .", "Adversarial Attack Adversarial training is generally used to improve the model's robustness.", "They generate adversarial samples by adding a worst-case perturbation to the input sample.", "We implement this strategy with Fast Gradient Value (FGV) , which directly uses the gradient to compute the perturbation and thus is faster than two-step alternative methods.", "Note that this strategy is only applicable when jointly training with supervision since it relies on supervised loss to compute adversarial perturbations.", "Token Shuffling In this strategy, we aim to randomly shuffle the order of the tokens in the input sequences.", "Since the bag-of-words nature in the transformer architecture, the position encoding is the only factor about the sequential information.", "Thus, similar to , we implement this strategy by passing the shuffled position ids to the embedding layer while keeping the order of the token ids unchanged.", "Cutoff proposes a simple and efficient data augmentation strategy called cutoff.", "They randomly erase some tokens (for token cutoff), feature dimensions (for feature cutoff), or token spans (for span cutoff) in the $L \\times d$ feature matrix.", "In our experiments, we only use token cutoff and feature cutoff and apply them to the token embeddings for view generation.", "Dropout Dropout is a widely used regularization method that avoids overfitting.", "However, in our experiments, we also show its effectiveness as an augmentation strategy for contrastive learning.", "For this setting, we randomly drop elements in the token embedding layer by a specific probability and set their values to zero.", "Note that this strategy is different from Cutoff since each element is considered individually.", "Table: The statistics of STS datasets." ], [ "Incorporating Supervision Signals", "Besides unsupervised transfer, our approach can also be incorporated with supervised learning.", "We take the NLI supervision as an example.", "It is a sentence pair classification task, where the model are trained to distinguish the relation between two sentences among contradiction, entailment and neutral.", "The classification objective can be expressed as following: $\\begin{split}f = \\text{Concat}(r_1, r_2, |r_1 - r_2|) \\\\\\mathcal {L}_\\text{ce} = \\text{CrossEntropy}(Wf + b, y)\\end{split}$ , where $r_1$ and $r_2$ denote two sentence representations.", "We propose three ways for incorporating additional supervised signals: Joint training (joint) We jointly train the model with the supervised and unsupervised objectives $\\mathcal {L}_\\text{joint} = \\mathcal {L}_\\text{ce} + \\alpha \\mathcal {L}_\\text{con}$ on NLI dataset.", "$\\alpha $ is a hyper-parameter to balance two objectives.", "Supervised training then unsupervised transfer (sup-unsup) We first train the model with $\\mathcal {L}_\\text{ce}$ on NLI dataset, then use $\\mathcal {L}_\\text{con}$ to fine-tune it on the target dataset.", "Joint training then unsupervised transfer (joint-unsup) We first train the model with the $\\mathcal {L}_\\text{joint}$ on NLI dataset, then use $\\mathcal {L}_\\text{con}$ to fine-tune it on the target dataset." ], [ "Experiments", "To verify the effectiveness of our proposed approach, we conduct experiments on Semantic Textual Similarity (STS) tasks under the unsupervised and supervised settings." ], [ "Setups", "Dataset Following previous works, , , we evaluate our approach on multiple STS datasets, including STS tasks 2012 - 2016 (STS12 - STS16) , , , , , STS benchmark (STSb) and SICK-Relatedness (SICK-R) .", "Each sample in these datasets contains a pair of sentences as well as a gold score between 0 and 5 to indicate their semantic similarity.", "For our unsupervised experiments, we mix the unlabeled texts from these datasets to fine-tune our model.", "We obtain all 7 datasets through the SentEval toolkit .", "The statistics is shown in Table REF .", "For supervised experiments, we use the combination of SNLI (570k samples) and MNLI (430k samples) to train our model.", "In the joint training setting, the NLI texts are also used for contrastive objectives.", "Baselines To show our effectiveness on unsupervised sentence representation transfer, we mainly select BERT-flow for comparison, since it shares the same setting as our approach.", "For unsupervised comparison, we use the average of GloVe embeddings, the BERT-derived native embeddings, CLEAR (trained on BookCorpus and English Wikipedia corpus), IS-BERT (trained on unlabeled texts from NLI datasets), BERT-CT (trained on English Wikipedia corpus).", "For comparison with supervised methods, we select InferSent , Universal Sentence Encoder , SBERT and BERT-CT as baselines.", "They are all trained with NLI supervision.", "Table: The performance comparison of ConSERT with other methods in an unsupervised setting.", "We report the spearman correlation ρ×100\\rho \\times 100 on 7 STS datasets.", "Methods with † ^\\dagger indicate that we directly report the scores from the corresponding paper, while methods with ‡ ^\\ddagger indicate our implementation.Evaluation When evaluating the trained model, we first obtain the representation of sentences by averaging the token embeddings at the last two layersAs shown in , averaging the last two layers of BERT achieves slightly better results than averaging the last one layer., then we report the spearman correlation between the cosine similarity scores of sentence representations and the human-annotated gold scores.", "When calculating spearman correlation, we merge all sentences together (even if some STS datasets have multiple splits) and calculate spearman correlation for only onceNote that such evaluation procedure is different from SentEval toolkit, which calculates spearman correlation for each split and reports the mean or weighted mean scores..", "Implementation Details Our implementation is based on the Sentence-BERThttps://github.com/UKPLab/sentence-transformers .", "We use both the BERT-base and BERT-large for our experiments.", "The max sequence length is set to 64 and we remove the default dropout layer in BERT architecture considering the cutoff and dropout data augmentation strategies used in our framework.", "The ratio of token cutoff and feature cutoff is set to 0.15 and 0.2 respectively, as suggested in .", "The ratio of dropout is set to 0.2.", "The temperature $\\tau $ of NT-Xent loss is set to 0.1, and the $\\alpha $ is set to 0.15 for the joint training setting.", "We adopt Adam optimizer and set the learning rate to 5e-7.", "We use a linear learning rate warm-up over 10% of the training steps.", "The batch size is set to 96 in most of our experiments.", "We use the dev set of STSb to tune the hyperparameters (including the augmentation strategies) and evaluate the model every 200 steps during training.", "The best checkpoint on the dev set of STSb is saved for test.", "We further discuss the influence of the batch size and the temperature in the subsequent section." ], [ "Unsupervised Results", "For unsupervised evaluation, we load the pre-trained BERT to initialize the BERT encoder in our framework.", "Then we randomly mix the unlabeled texts from 7 STS datasets and use them to fine-tune our model.", "The results are shown in Table REF .", "We can observe that both BERT-flow and ConSERT can improve the representation space and outperform the GloVe and BERT baselines with unlabeled texts from target datasets.", "However, ConSERT$_\\text{large}$ achieves the best performance among 6 STS datasets, significantly outperforming BERT$_\\text{large}$ -flow with an 8% relative performance gain on average (from 70.76 to 76.45).", "Moreover, it is worth noting that ConSERT$_\\text{large}$ even outperforms several supervised baselines (see Figure REF ) like InferSent (65.01) and Universal Sentence Encoder (71.72), and keeps comparable to the strong supervised method SBERT$_\\text{large}$ -NLI (76.55).", "For the BERT$_\\text{base}$ architecture, our approach ConSERT$_\\text{base}$ also outperforms BERT$_\\text{base}$ -flow with an improvement of 3.17 (from 69.57 to 72.74).", "Table: The performance comparison of ConSERT with other methods in a supervised setting.", "We report the spearman correlation ρ×100\\rho \\times 100 on 7 STS datasets.", "Methods with † ^\\dagger indicate that we directly report the scores from the corresponding paper, while methods with ‡ ^\\ddagger indicate our implementation." ], [ "Supervised Results", "For supervised evaluation, we consider the three settings described in Section REF .", "Note that in the joint setting, only NLI texts are used for contrastive learning, making it comparable to SBERT-NLI.", "We use the model trained under the joint setting as the initial checkpoint in the joint-unsup setting.", "We also re-implement the SBERT-NLI baselines and use them as the initial checkpoint in the sup-unsup setting.", "The results are illustrated in Table REF .", "For the models trained with NLI supervision, we find that ConSERT joint consistently performs better than SBERT, revealing the effectiveness of our proposed contrastive objective as well as the data augmentation strategies.", "On average, ConSERT$_\\text{base}$ joint achieves a performance gain of 2.88 over the re-implemented SBERT$_\\text{base}$ -NLI, and ConSERT$_\\text{large}$ joint achieves a performance gain of 2.70.", "When further performing representation transfer with STS unlabeled texts, our approach achieves even better performance.", "On average, ConSERT$_\\text{large}$ joint-unsup outperforms the initial checkpoint ConSERT$_\\text{large}$ joint with 1.84 performance gain, and outperforms the previous state-of-the-art BERT$_\\text{large}$ -flow with 2.92 performance gain.", "The results demonstrate that even for the models trained under supervision, there is still a huge potential of unsupervised representation transfer for improvement." ], [ "Analysis of BERT Embedding Space", "To prove the hypothesis that the collapse issue is mainly due to the anisotropic space that is sensitive to the token frequency, we conduct experiments that mask the embeddings of several most frequent tokens when applying average pooling to calculate the sentence representations.", "The relation between the number of removed top-k frequent tokens and the average spearman correlation is shown in Figure REF .", "We can observe that when removing a few top frequent tokens, the performance of BERT improves sharply on STS tasks.", "When removing 34 most frequent tokens, the best performance is achieved (61.66), and there is an improvement of 7.8 from the original performance (53.86).", "For ConSERT, we find that removing a few most frequent tokens only results in a small improvement of less than 0.3.", "The results show that our approach reshapes the BERT's original embedding space, reducing the influence of common tokens on sentence representations.", "Figure: The average spearman correlation on STS tasks w.r.t.", "the number of removed top-k frequent tokens.", "Note that we also considered the [CLS] and [SEP] tokens and they are the 2 most frequent tokens.", "The frequency of each token is calculated through the test split of the STS Benchmark dataset." ], [ "Effect of Data Augmentation Strategy", "In this section, we study the effect of data augmentation strategies for contrastive learning.", "We consider 5 options for each transformation, including None (i.e.", "doing nothing), Shuffle, Token Cutoff, Feature Cutoff, and Dropout, resulting in 5$\\times $ 5 combinations.", "Note that the Adversarial Attack strategy is not considered here, since it needs additional supervision to generate adversarial samples.", "All these experiments follow the unsupervised setting and use the BERT$_\\text{base}$ architecture.", "The results can be found in Figure REF .", "We can make the following observations.", "First, Shuffle and Token Cutoff are the two most effective strategies (where Shuffle is slightly better than Token Cutoff), significantly outperforming Feature Cutoff and Dropout.", "This is probably because Shuffle and Token Cutoff are more related to the downstream STS tasks since they are directly operated on the token level and change the structure of the sentence to produce hard examples.", "Secondly, Feature Cutoff and Dropout also improve performance by roughly 4 points when compared with the None-None baseline.", "Moreover, we find they work well as a complementary strategy.", "Combining with another strategy like Shuffle may further improve the performance.", "When combined Shuffle with Feature Cutoff, we achieve the best result.", "We argue that Feature Cutoff and Dropout are useful in modeling the invariance of the internal noise for the sentence encoder, and thus improve the model's robustness.", "Finally, we also observe that even without any data augmentation (the None-None combination), our contrastive framework can improve BERT's performance on STS tasks (from 53.86 to 63.84).", "This None-None combination has no effect on maximizing agreement between views since the representations of augmented views are exactly the same.", "On the contrary, it tunes the representation space by pushing each representation away from others.", "We believe that the improvement is mainly due to the collapse phenomenon of BERT's native representation space.", "To some extent, it also explains why our method works." ], [ "Performance under Few-shot Settings", "To validate the reliability and the robustness of ConSERT under the data scarcity scenarios, we conduct the few-shot experiments.", "We limit the number of unlabeled texts to 1, 10, 100, 1000, and 10000 respectively, and compare their performance with the full dataset.", "Figure REF presents the results.", "For both the unsupervised and the supervised settings, our approach can make a huge improvement over the baseline with only 100 samples available.", "When the training samples increase to 1000, our approach can basically achieve comparable results with the models trained on the full dataset.", "The results reveal the robustness and effectiveness of our approach under the data scarcity scenarios, which is common in reality.", "With only a small amount of unlabeled texts drawn from the target data distribution, our approach can also tune the representation space and benefit the downstream tasks." ], [ "Influence of Temperature", "The temperature $\\tau $ in NT-Xent loss (Equation REF ) is used to control the smoothness of the distribution normalized by softmax operation and thus influences the gradients when backpropagation.", "A large temperature smooths the distribution while a small temperature sharpens the distribution.", "In our experiments, we explore the influence of temperature and present the result in Figure REF .", "As shown in the figure, we find the performance is extremely sensitive to the temperature.", "Either too small or too large temperature will make our model perform badly.", "And the optimal temperature is obtained within a small range (from about 0.08 to 0.12).", "This phenomenon again demonstrates the collapse issue of BERT embeddings, as most sentences are close to each other, a large temperature may make this task too hard to learn.", "We select 0.1 as the temperature in most of our experiments.", "Figure: The influence of different temperatures in NT-Xent.", "The best performance is achieved when the temperature is set to 0.1." ], [ "Influence of Batch Size", "In some previous works of contrastive learning, it is reported that a large batch size benefits the final performance and accelerates the convergence of the model since it provides more in-batch negative samples for contrastive learning .", "Those in-batch negative samples improve the training efficiency.", "We also analyze the influence of the batch size for unsupervised sentence representation transfer.", "The results are illustrated in Table REF .", "We show both the spearman correlation and the corresponding training steps.", "We find that a larger batch size does achieve better performance.", "However, the improvement is not so significant.", "Meanwhile, a larger batch size does speed up the training process, but it also needs more GPU memories at the same time.", "Table: The average spearman correlation as well as the training steps of our unsupervised approach with different batch sizes." ], [ "Conclusion", "In this paper, we propose ConSERT, a self-supervised contrastive learning framework for transferring sentence representations to downstream tasks.", "The framework does not need extra structure and is easy to implement for any encoder.", "We demonstrate the effectiveness of our framework on various STS datasets, both our unsupervised and supervised methods achieve new state-of-the-art performance.", "Furthermore, few-shot experiments suggest that our framework is robust in the data scarcity scenarios.", "We also compare multiple combinations of data augmentation strategies and provide fine-grained analysis for interpreting how our approach works.", "We hope our work will provide a new perspective for future researches on sentence representation transfer." ], [ "Acknowledgements", "We thank Keqing He, Hongzhi Zhang and all anonymous reviewers for their helpful comments and suggestions.", "This work was partially supported by National Key R&D Program of China No.", "2019YFF0303300 and Subject II No.", "2019YFF0303302, DOCOMO Beijing Communications Laboratories Co., Ltd, MoE-CMCC “Artifical Intelligence\" Project No.", "MCM20190701.", "Sentence representation learning is a basic task in natural language processing and benefits many downstream tasks.", "This work proposes a contrastive learning based framework to solve the collapse issue of BERT and transfer BERT sentence representations to target data distribution.", "Our approach not only provides a new perspective about BERT's representation space, but is also useful in practical applications, especially for data scarcity scenarios.", "When applying our approach, the user should collect a few unlabeled texts from target data distribution and use our framework to fine-tune BERT encoder in a self-supervised manner.", "Since our approach is self-supervised, no bias will be introduced from human annotations.", "Moreover, our data augmentation strategies also have little probability to introduce extra biases since they are all based on random sampling.", "However, it is still possible to introduce data biases from the unlabeled texts.", "Therefore, users should pay special attention to ensure that the training data is ethical, unbiased, and closely related to downstream tasks." ] ]
2105.11741
[ [ "Transfer Learning and Curriculum Learning in Sokoban" ], [ "Abstract Transfer learning can speed up training in machine learning and is regularly used in classification tasks.", "It reuses prior knowledge from other tasks to pre-train networks for new tasks.", "In reinforcement learning, learning actions for a behavior policy that can be applied to new environments is still a challenge, especially for tasks that involve much planning.", "Sokoban is a challenging puzzle game.", "It has been used widely as a benchmark in planning-based reinforcement learning.", "In this paper, we show how prior knowledge improves learning in Sokoban tasks.", "We find that reusing feature representations learned previously can accelerate learning new, more complex, instances.", "In effect, we show how curriculum learning, from simple to complex tasks, works in Sokoban.", "Furthermore, feature representations learned in simpler instances are more general, and thus lead to positive transfers towards more complex tasks, but not vice versa.", "We have also studied which part of the knowledge is most important for transfer to succeed, and identify which layers should be used for pre-training." ], [ "Introduction", "Humans are good at reusing prior knowledge when facing new problems.", "As a consequence, we learn new tasks quickly, a skill of great interest in machine learning.", "In the human brain, information received by our sensors is first transformed into different forms, and different types of transformed information are stored in different areas of our brain.", "When another problem arrives later on, we retrieve useful information and adjust it to better suit solving this new problem.", "The knowledge stored in artifical neural networks is also re-usable and transferable [30].", "In supervised learning, pre-trained networks are commonly applied in computer vision [16], [24] and natural language processing [3], [8].", "Feature representations learned from images or words overlap to some extent, which makes such feature representations reusable and transferable.", "In reinforcement learning (RL), transfer learning is relatively new, although with the spread of deep neural networks, reusing pre-trained models becomes possible in RL as well [1], [7].", "Transfer learning works well in RL for recognition tasks, but tasks that rely heavily on planning are harder.", "Figure: An example instance of Sokoban.In this paper, we study transfer learning of behavior in Sokoban, a popular RL game in which planning is important [9], [11].", "An example instance from  [21] is shown in Fig.", "REF .", "The goal of Sokoban is to control a warehouse worker that pushes all boxes onto targets.", "Sokoban is a challenging game where one wrong move can lead to a dead end (after a box has been pushed, it can not be pulled, and we cannot undo an inadvertent push).", "This non-reversibility is known to make games harder for AI agents [5].", "Learning to solve Sokoban tasks is a challenge, especially in the multi-box scenario.", "For humans, if we have learned the basics of Sokoban (what is a box, what can an agent do), and if we are faced with a new, more complex instance, then we immediately focus on the new challenges in the instance, rather than re-learning the basics again.", "This building on prior knowledge saves time in the problem-solving process.", "We investigate if we can achieve this kind of pretraining/fine-tuning learning in an RL agent.", "Our main hypothesis is that feature representations learned in Sokoban instances can be reused to improve solving other instances, and that features learned in simpler instances are more general and better transferable.", "We test this hypothesis by means of different experiments, in which parts of the neural network that has previously been trained on one type of instances (e.g.", "one box one target) are taken over (unchanged) to a new type of instances (e.g.", "two boxes two targets), whereas the remaining part of the network is trained on these new instances from scratch.", "The overall idea is that we see successful transfer if the preserved knowledge (in terms of network layers) leads to a faster learning process on the new problem type.", "The main contributions of this paper are as follows: First, we show that feature representations learned in simple Sokoban instances can accelerate learning in more complex instances, indicating that curriculum learning can be used in Sokoban.", "Second, feature representations of simpler instances are more general and reusable than features learned in more complex instances.", "Third, our results confirm that in RL lower layers learn more general features.", "Interestingly, in some cases the best performance is achieved when more specific features are transfered, when source task and target task are similar enough to support these more specific features.", "Fourth, we found negative transfer from a simple supervised learning task, which tells us that choice and design of the source tasks are crucial.", "Fifth, we show that transferring top-fully-connected layers will not only be unhelpful but also harmful to the learning.", "We also used popular visualization techniques to explore potential reasons for successful transfers, which we explain in detail.", "Our code and test environments will be made available after blind review.", "The paper is structured as follows: we first briefly review related work on transfer learning and Sokoban in the next section; then the environment and methods we are using are described in Section ; Section  shows the experimental settings and results; in the last section, we conclude our work and discuss some potential future directions." ], [ "Related Work", "De la Cruz et al.", "[6] studied the reuse of feature representations between two similar games: Breakout and Pong, using DQN.", "They used a 3-layer convolutional network.", "Weights learned in one game were transferred to improve learning the other game; results showed positive transfer of features between the different games.", "Pong and Breakout do not require planning; in our experiments, in Sokoban, we study how a curriculum of simpler instances can benefit the learning of complex instances.", "Spector et al.", "[25] used self-transfer in a DQN grid-world task to identify which parts should be transferred and which parts should be fixed, showing significant benefit of knowledge transfer.", "Sokoban is a planning task that has been used as a benchmark for model-based reinforcement learning [21], [15].", "It has also been used in model-free RL [13], [14], achieving performance competitive with model-based methods.", "The efficiency of AlphaZero-style curriculum learning has been shown by solving hard single Sokoban instances [10], [11].", "Previous works were aimed at solving single Sokoban instances; our paper focuses on the transferability of learned knowledge among different instances.", "This transferability of learned feature representations was first studied in image classification problems [30].", "It was shown that bottom layers in CNNs extract more general features while ones extracted from back layers are more specific.", "In this paper, we verify this idea under RL settings.", "Reinforcement learning [26], [20] aims to reinforce behaviors of the learning agent by rewarding signals obtained from interactions with the environment.", "It has reached super-human performance in games such as Go [23], StarCraft [28], [19], as well as Atari games [2] and robotic tasks.", "In this paper we follow the conventional MDP notation for RL [26].", "Transfer learning reuses prior knowledge to improve the learning efficiency or performance in new tasks [29], [27].", "In reinforcement learning, higher-level knowledge such as macro actions, skills and lower-level knowledge such as reward functions, policies could be transferred.", "Transferring learned knowledge could take different approaches, such as reward shaping [4], learning from demonstration [18] and policy reuse [12]." ], [ "Experimental Setup", "The environment used in the paper is the Gym environment for Sokoban [22]; for the agent algorithms we follow Weber et al. [21].", "Examples are shown in Fig.", "REF .", "The game is solved by controlling the agent (green sprite) to push all boxes (yellow squares) onto corresponding targets(red squares).", "There's no hint about which boxes should on which targets, and boxes can only be pushed; some actions are irreversible, and can leave the game in an unsolvable state.", "The difficulty of the game can be increased easily by putting more boxes as well as targets into generated rooms.", "The agent can go up, down, left, and right.", "The agent gets a final reward of 10 by pushing all boxes on targets.", "Pushing a box on a target will result a reward of 1 and a penalty of -1 for pushing a box off a target.", "We also give a small penalty of 0.1 for each step the agent takes.", "We perform three types of experiments: (1) related tasks (source and target tasks are both RL tasks, while source tasks are to solve $n$ -boxes Sokoban instances and target tasks are to solve $m$ -boxes Sokoban instances, where $n\\ne m$ ), (2) different tasks (source tasks are SL tasks and target tasks are RL tasks), and (3) different texture appearance(source and target tasks are both RL tasks, while source tasks are to solve original Sokoban instances and target tasks are to solve Sokoban instances with different texture appearance).", "The agent was first pre-trained on source tasks and then fine-tuned on target tasks.", "RL tasks are to solve 100 randomly generated $n$ -boxes Sokoban instances.", "SL tasks are to recognize the location of the agent in Sokoban instances.", "The overall statistics of the maps are shown in Fig.", "REF .", "As the number of objectives increases, the number of steps for the optimal solution also increases, and so does the difficulty of solving the game.", "Figure: Distribution of optimal solutions in different Sokoban instances." ], [ "Neural Network Architecture", "The neural network we employ is taken from the DeepMind baseline [21] directly without hyper-parameter tuning.", "The model consists of 3 convolutional (Conv) layers with kernel size 8x8, 4x4, 3x3, strides of 4, 2, 1, and number of output channels 32, 64, 64.", "This is followed by a fully connected (FC) hidden layer with 512 units.", "The outputs of this FC layer will be fed into two heads: one for outputting the policy logits and one for outputting the state value.", "This is one of the most commonly-used architectures in RL, we selected it also in order to show what can be achieved with popular architecture.", "Details of architecture and hyper parameters we employ are found in Table REF .", "Table: Hyper-parameters of the neural network and training." ], [ "Transfer Approach", "The main idea of our transfer approach is to reuse feature representations from source tasks learned by the Conv layers in new unseen target tasks.", "As detailed in the last sub-section, our model consists of 3 Conv layers and 2 FC layers.", "The feature representations were transferred to new tasks by copying the weights of the first $k$ Conv layers trained in source tasks (where there are $n_s$ boxes/targets) to initialize the new learning model in target tasks (where there are $n_t$ boxes/targets).", "Then we froze these weights (they were no longer trainable) and retrained the remaining part of the model.", "In our experiments, $k \\in \\lbrace 1,2,3\\rbrace $ , $n_s \\in \\lbrace 1,2,3\\rbrace $ , $n_t \\in \\lbrace 1,2,3\\rbrace $ .", "Please refer to Fig.", "REF for an explanation of this approach.", "Different squares represent different layers of our neural network.", "The first 3 layers are Conv layers and the last two are FC layers.", "Reds are weights taken from pre-trained model and fixed, greens are weights reinitialized and trainable.", "Solved ratios were used for measuring agents' performances, and they were calculated every 1,000 environment steps.", "20 randomly selected test instances were performed by the current learning agent.", "We say the transfer is positive when the performance with the transfer is better than without (training from scratch), and negative when the performance with the transfer is worse than without.", "Figure: Three different transfer approaches, red layers are fixed while green layers are trainable.", "They correspond k=1,2,3k=1,2,3 from left to right respectively." ], [ "Experiments", "We designed experiments with different source, target tasks and $k$ , in order to verify the hypotheses we proposed.", "We experimented with Sokoban instances with 1, 2, and 3 boxes.", "All experiments were run for 1 million environment steps.", "Experimental details are shown in Table REF .", "We use abbreviations for each experiment.", "For instance, s1t1k1 means source tasks are 1-box instances, target tasks are 1-box instances and we transfer and fix the 1(first) layer.", "The neural networks were trained using A2C, a single threaded variant of A3C [17].", "All experiments were performed 5 times with different random seeds, and figures were drawn using averaged results with 0.95 confidence interval.", "Heavy fluctuations were caused by irreversible actions, one irreversible action during the game could make the whole game unsolvable.", "Table: Experimental design details.", "kk is the number of fixed layers." ], [ "Transfer Among Related Tasks", "Related tasks are tasks where the only difference between source and task is the difficulties of instances, i.e.", "the number of boxes and targets.", "(Recall that both source and task are trained on 100 different map-layouts, in all experiments.)", "Figure: Performance of transferring feature representations learned in 1-box, 2-boxes, 3-boxes instances to learning in 1-box with k=3k=3.", "n s =1,2,3n_s=1,2,3, n t =1n_t=1, k=3k=3.", "Pre-training on 1-box instances is much better than pre-training on 2 or 3 box instances when training new 1-box instances.Figure: Performance of transferring feature representations learned in 1-box, 2-boxes, 3-boxes instances to learning in 2-boxes (left) and 3-boxes (right) with k=3k=3.", "n s =1,2,3n_s=1,2,3, n t =2,3n_t=2,3, k=3k=3.Fig.", "REF and Fig.", "REF show results for training on 1-box, 2-boxes, 3-boxes instances with reusing features learned in different tasks, and we fix $k=3$ .", "All results showed that transferring feature representations learned in single-box instances is positive.", "Performance of agents (s1t1k3, s1t2k3, s1t3k3) who are using features learned from single-box instances always outperform other agents, including agents training from scratch and using features learned from other instances.", "The transfer, however, is not 'bi-directional', feature representations learned in multiple-box instances could not be successfully transferred to the learning in single-box instances.", "Their performance (s2t1k3, s3t1k3) converged to a relatively low solved ratio, which indicates that transferred features are not suitable for single-box instances.", "Just as humans learn more general knowledge in simpler cases, our agents also showed that the knowledge learned from single-box instances is more general and transferable than ones learned in multiple-box instances.", "To further enhance performances of transferring features learned in single-box instances, we tried different $k$ .", "We expected that the performance will be the best when $k=1$ since the first layer learn the most general features.", "However, the results in Fig.", "REF instead show that not $k=1$ but $k=2$ (s1t2k2, s1t3k2) perform the best.", "Similar to [6], features learned in the first 2 layers are still general enough for transfer; in addition, the difference between source tasks and target tasks is not as large as expected, and features learned between different instances are more overlapping than expected.", "It is also interesting to see the influence of how many layers are fixed on the success of the transfer.", "In particular, we want to know whether a smaller $k$ could change the negative transfer from multiple-box instances to single-box instances into positive.", "(We believe features from multiple and single-box instances are overlapping to some extent.)", "Results are shown in Fig.", "REF .", "We see that indeed the first layer (s2t1k1, s3t1k1) did learn enough general features from multiple-boxes instances to solve the single-box instances.", "Although agents with features only learned by the first layer could converge to decent performance in the end, the transfer is still negative.", "An interesting point is that $k=3$ (s2t1k3) performs better than $k=2$ (s2t1k2) when source tasks are 2-boxes instances.", "Note that $k=2$ (s3t1k2) performs better than $k=3$ (s3t1k3) when source tasks are 2-boxes instances.", "There are more overlapping features between the 2-boxes instances and single instances.", "Figure: Performance of transferring feature representations learned in 1-box instances to learning in 2-boxes (left) and 3-boxes (right) with different kk.", "n s =1n_s=1, n t =2,3n_t=2,3, k=1,2,3k=1,2,3.Figure: Performance of transferring feature representations learned in 2-boxes (left) and 3-boxes (right) instances to learning in 1-box instances with different kk.", "n s =2,3n_s=2,3, n t =1n_t=1, k=1,2,3k=1,2,3." ], [ "Transfer Among Different Tasks (SL/RL)", "Feature representations learned from previous tasks can either be helpful or harmful.", "In the previous subsection we saw some positive transfer to related Sokoban tasks, in this subsection we study if transfer between supervised and reinforcement learning tasks works.", "We follow prior work, Anderson et al.", "[1] showed that features can be transfered from hand-crafted supervised learning tasks to reinforcement learning.", "Their model was first trained to predict state dynamics of the environment, and then pre-trained hidden layers were helpful to accelerate solving RL tasks.", "For transfer to different (randomly chosen) instances in Sokoban, we also formed a supervised task, which was to train a prediction model to recognize the location of the agent, shown in Fig.", "REF .", "When humans are solving Sokoban, we first need to know where the agent is before we draw up a plan.", "If we already know the location of objectives, the solving process could be faster.", "After the prediction model could correctly recognize where the agent is, we took feature representations of the trained model and plug them into a new agent.", "The first layer of learned features is fixed, and we only train the remaining part.", "Fig.", "REF shows the performance of transferring and training from scratch.", "We find negative transfer for (sPt1k1): the performance is much worse compare with training from scratch.", "The bad performance is due to the choice of the source task, which is too different for the target tasks, and the expressive power that is learned is inadequate to overcome this difference.", "Figure: (a): How SL tasks work.", "Input states and neural network will learn to predict locations of the agent.", "(b): Performance of training from scratch and training with transferred feature representations from SL tasks." ], [ "Transfer To Different Appearance", "Experiments we described in previous subsections were all trying to transfer Conv layers which learned feature representations.", "In the next experiment, we try to make the agent utilize another part of the learned model, which are back FC layers of the whole model.", "The source and target tasks were both single-box instances, but the target tasks were instances with different appearances.", "Fig.", "REF is an example.", "The maps used for two groups of tasks were the same, the only difference was how they look like, the appearance was changed, with different textures, and we call it Game2.", "Fig.", "REF shows the transfer approach.", "We took FC layers trained in source tasks and fixed them, and retrained the remaining Conv layers.", "Since maps were the same, solutions of the instances were the same.", "When Conv layers learn new feature representations successfully, instances are solved then.", "Fig.", "REF shows the performance.", "One would expect that transferred FC layers(s1t1fc_game2) are faster because the agent only needs to learn new feature representations.", "However, the experiments did not show this result.", "Apparently, when the whole model is trained jointly, it has more flexibility to be trained into the final shape; when the last part of the model is fixed, the learning of the first part will be trying to cater for the last part in order to solve the problem, which made the learning slower.", "Figure: (a): Transfer approach for transfer to Game2.", "FC layers are taken from previously training and fixed, only conv layers will be retrained.", "(b): An example instance in Game2.", "We changed appearances in Game2 with different textures of objectives." ], [ "Visualizing Agent Detection", "In order to better understand what the network learned, we provide a visualization.", "We follow Yosinski et al.", "who showed that convolutional neural networks can detect latent objectives without explicit labels [30].", "Fig.", "REF shows a latent 'agent detector' for Sokoban.", "The neural network automatically learned to detect the agent without giving any labels or information.", "Left rows are pixel inputs, right rows are outputs of one specific feature map.", "Yellow-green units are detected agents.", "We note that although the network was trained in single-box instances, it still performed quite well in multiple-box instances, which is a potential reason for the successful transfer.", "The agent's abilities that were learned in source tasks are useful in target tasks.", "Figure: (a): Training on Game2 using transferred FC layers.", "Its performance is worse than training from scratch.", "(b): The agent detector.", "Outputs of the twenty third feature map of the first convolutional layer, which is an agent detector learned from 1-box instances, and it's still usable in multiple-boxes scenarios." ], [ "Conclusion and Future Work", "Our experiments showed that in a reinforcement learning setting the agent in Sokoban can learn four characteristics that are similar to humans.", "(1) Feature representations learned previously can accelerate the new learning in other Sokoban instances.", "Knowledge learned in previous related tasks could be reused to accelerate new learning, transfer learning is occurring, creating an implicit learning curriculum.", "(2) Feature representations learned in single-box instances are more general, and are more effective for learning in multiple-boxes instances, but not vice versa.", "Knowledge learned in simpler tasks is more general and more effective, even in more complex tasks.", "Further experiments showed negative learning, that confirms these results.", "(3) Feature representations learned in unrelated supervised learning tasks can hurt fine-tuning performance.", "If the learned knowledge is required to be helpful in new coming tasks, it's better to learn from similar tasks, otherwise the choice of tasks needs to be careful.", "(4) Fixing the top-fully-connected layers and retraining the bottom convolutional layers slows down learning and hurts performance.", "We conclude that learning should have explicit order, less flexibility will not only be unhelpful but also hurt the learning process and the performance.", "Our experiments showed that with a simple 5-layer convolutions/fully connected network (based on DeepMind's baseline [21]), transfer learning and curriculum learning of behavior to occur in Sokoban.", "This is surprising, since Sokoban is a planning-heavy problem, for which one would expect more elaborate network architectures to be necessary.", "Reusing pre-trained feature representations in RL fields is not well studied, and to the best of our knowledge, these are the first results show transfer learning and curriculum learning with such a simple network in such a planning-heavy behavioral task.", "In the future, we would like to see more utilization of pre-trained feature representations and of the enire pre-trained model in RL.", "We believe that reusing pre-trained model can significantly improve data-efficient reinforcement learning." ], [ "Acknowledgement", "The financial support to Zhao Yang is from the China Scholarship Council(CSC).", "Computation support is from ALICE and DSLab.", "The authors thank Hui Wang, Matthias Müller-Brockhausen, Michiel van der Meer, Thomas Moerland and all members from the Leiden Reinforcement Learning Group for helpful discussions." ] ]
2105.11702
[ [ "Quantum sensing of weak electric and magnetic fields by coherent\n amplification of energy level shift effects" ], [ "Abstract A method for measuring small energy level shifts in a qubit by coherent amplification of their effect is proposed.", "It is based on the repeated application of the same interaction pulse in two manners: with the same phase of each subsequent pulse, and with an alternating phase shift of $\\pi$ (i.e.", "a minus sign) from pulse to pulse.", "Two specific types of pulses are considered: a resonant $\\pi$ pulse and an adiabatic chirped pulse, both of which produce complete population inversion with high fidelity.", "In the presence of a weak ambient external electric or magnetic field, the ensuing Stark or Zeeman shift leads to an energy level shift and hence a static detuning.", "In both the resonant and adiabatic approaches, a small level shift does not alter the transition probability very much; however, it can significantly change the dynamical phases in the propagator.", "The repeated application of the same pulse greatly amplifies the changes in the dynamical phases and maps them onto the populations.", "Hence the effect of the level shift can be measured with good accuracy.", "It is found that sequences of pulses with alternating phases deliver much greater error amplification and much steeper excitation profiles around resonance, thereby providing much higher sensitivity to small energy level shifts.", "Explicit analytic estimates of the sensitivity are derived using the well-known non-crossing Rosen-Zener and Rabi models and the level-crossing Demkov-Kunike model.", "This recipe provides a simple tool for rapid and accurate sensing of weak electric and magnetic fields by using the same pulse generating an inversion quantum gate, without sophisticated tomography or entangling operations." ], [ "Introduction", "In scalable quantum computation [1], the quantum gates in a quantum circuit have to be implemented with very high fidelity, with the admissible error in the range of $10^{-3}$ to $10^{-4}$ , depending on the quantum error correction protocol [2].", "The current state of the art in trapped-ions experiments features errors of $10^{-5}$ [3], [4] and even $10^{-6}$ [5] for single-qubit gates, and errors of the order of $10^{-3}$ [6], [4] for two-qubit gates.", "The detrimental cross-talk to neighboring ion qubits has been suppressed to values of $10^{-5}$ [7] and $10^{-6}$ [8], and the errors in ion transport have been reduced below $10^{-5}$ too [9].", "In superconducting qubits, fidelities of 99.9% for single-qubit and 99.4% for two-qubit gates have been reported [10].", "Various steps in qubit readout and quantum gate tomography, e.g.", "population shelving [11], [12], [13], [14], have to be implemented with very high fidelity too.", "Such tiny errors require very good control of the driving field as well as the environment, e.g.", "compensation of ambient electric and magnetic fields to a very high degree.", "This is crucial in a quantum circuit where the entire propagator of the particular gate must be very well controlled, i.e.", "both the probability and the propagator phases have to be very stable.", "The phases, in particular, are very sensitive to a detuning shift, which can emerge due to both fluctuating frequency of the laser, microwave or radiofrequency generator, and energy level shifts caused by uncompensated magnetic or electric fields.", "In order to characterize such unwanted detuning shifts in the course of the measurement it is very useful to have a simple, fast and reliable method to detect and measure such shifts.", "Measuring the populations after the application of a single pulse, which generates the quantum gate, is not reliable because the populations are fairly insensitive to detuning shifts, whereas the gate phases, which are prone to these shifts, are invisible to such a measurement.", "Performing full quantum tomography would reveal the changes in the gate phases but such a tomography could be very time-consuming.", "To this end, I propose in this paper a conceptually very simple technique for sensing and measuring small detuning shifts.", "It is based on the repeated application of the same gate-generating pulse, which greatly amplifies the effect of the detuning shift due to quantum interference and maps it onto the populations.", "In this manner, one does not need any change in the experimental apparatus (i.e.", "the pulse amplitude, duration, shape, frequency and possibly chirp), and avoids the introduction of additional experimental parameters beyond the gate-generating pulse.", "In this manner, the proposed technique is simpler than other, more sophisticated quantum sensing methods [15] and quantum gate tomography [16].", "I consider the implementation of this sensing method by applying the pulses used in two major quantum control techniques for population inversion: a resonant $\\pi $ pulse and an adiabatic chirped pulse.", "The paper is organized as follows.", "The concept of the sensing method is introduced in Sec.", "and the general features of the transition probability generated the two types of sensing sequences are presented in Sec. .", "Three exactly soluble analytic models are presented in Secs.", ", , and .", "The last section wraps up the results and presents some discussion on the limitations of the method and an outlook of its possible extensions." ], [ "Concept ", "I consider two types of pulse sequences applied to a two-state quantum system, see Fig.", "REF .", "In one of them, the same pulse is applied $N$ times, see Fig.", "REF (top), and then the populations are measured.", "In the other, the same pulse is again applied $N$ times but the phase of every other pulse is shifted by $\\pi $ , i.e.", "a minus sign is applied to the Rabi frequency of every even-numbered pulse in the sequence, see Fig.", "REF (bottom).", "The sequences may contain an odd or even number of pulses.", "On exact resonance it is assumed that each pulse produces complete (for resonant pulse) or almost complete (for adiabatic chirped pulse) population inversion.", "This is achieved by a $\\pi $ pulse in the former case and a chirped pulse with a pulse area of a few $\\pi $ in the latter.", "Because the excitation profile has its maximum value (and hence a vanishing first-order derivative versus the detuning) at resonance, a small detuning shift has a very little effect on the populations.", "However, such a detuning shift changes the dynamic phases of the propagator much more significantly.", "If one is restricted to measuring populations, as it is assumed here, these phase shifts are invisible in a single-pulse interaction; however, they are mapped onto the populations by the interference generated by a train of pulses.", "These pulses are chosen to be identical to the single $\\pi $ pulse, or the single chirped adiabatic pulse, mentioned above, with the only difference being the sign flip of the Rabi frequency in the sign-alternating sequence.", "In this manner, no new parameters are introduced in addition to the dynamic parameters of the single-pulse propagator (the transition probability and the two dynamic phases).", "This fact is important because, in the absence of other uncertainties, the population changes are directly linked to the detuning shift.", "The rationale for the two pulse sequences considered here is the following.", "It might appear at first sight that repeating the same pulse, without phase shifts, is the most natural approach to coherent error amplification.", "Indeed, this case is carefully analyzed here.", "However, it turns out that a sequence of pulses with alternating Rabi frequency signs is a far better approach as far as detuning shift sensing is concerned.", "When considered more carefully, this is readily understood: a pair of two resonant pulses with a $\\pi $ phase shift cancel each other's effect exactly because the propagator for the second pulse is the Hermitean conjugate of the propagator of the first, thereby resulting in the identity operation.", "In other words, a pair of resonant pulses with Rabi frequencies $\\Omega $ and $-\\Omega $ will produce no overall change and exactly zero transition probability.", "A small detuning will break this symmetry and the cancellation will not occur, giving rise to a nonzero transition probability.", "When such a pair is repeated $N$ times, the nonzero probability is coherently amplified very quickly.", "Furthermore, sequences of both even and odd number of pulses $N$ are used.", "With the transition probability for zero static detuning being equal to one, or almost one, one should keep in mind that the multi-pass transition probability for an odd $N$ is equal to one, or almost one, while it is zero, or almost zero, for an even $N$ .", "Hence the sensing feature around the zero static detuning appears as a spike for odd $N$ and a dip for even $N$ .", "It is the width of this feature and its slope, which are important for the frequency shift sensing rather than whether it is a spike or a dip.", "In order to not only sense an energy level shift but accurately measure it using the pulse sequence scenario, an analytic relation between the single-pulse and $N$ -pulse probabilities is needed.", "Such a relation for a SU(2) propagator is available [17] and it has been used recently for quantum gate tomography [18], [19].", "Here it is used extensively, in combination with explicit analytic formulas for the propagator for three popular exactly soluble two-state models, the non-crossing Rosen-Zener [20], [21] and Rabi [22] models and the level-crossing Demkov-Kunike model [23], [24], [25], [26].", "They allow one to explicitly link the $N$ -pulse transition probability to the detuning shift." ], [ "Multi-pass transition probability", "The Hamiltonian of a coherently driven two-state quantum system, in the rotating-wave approximation [22], reads $ \\mathbf {H}(t) = \\tfrac{1}{2} \\left[\\begin{array}{cc} -\\Delta (t) & \\Omega (t) \\\\ \\Omega (t) & \\Delta (t) \\end{array}\\right], $ where $\\Delta (t)$ is the system-field frequency detuning and $\\Omega (t)$ is the Rabi frequency of the coupling between the two states.", "The Rabi frequency is supposed to be pulse-shaped and the detuning may contain a static part $\\Delta _0$ and a chirp, $ \\Delta (t) = \\Delta _0 + \\beta f(t).", "$ In this paper, the objective is to measure a static error in the detuning around zero, i.e.", "the static detuning $\\Delta _0$ will be the (unknown) quantity to be determined.", "For arbitrary $\\Omega (t)$ and $\\Delta (t)$ , the propagator corresponding to the traceless Hamiltonian (REF ) is a SU(2) matrix, which is expressed in terms of the complex-valued Cayley-Klein parameters $a$ and $b$ ($|a|^2 +|b|^2=1$ ) as $ \\mathbf {U}= \\left[\\begin{array}{cc} a & -b^* \\\\ b & a^* \\end{array}\\right].", "$ If the system is initially in state $\\vert 1\\rangle $ , the probabilities for remaining in state $\\vert 1\\rangle $ and for transition to state $\\vert 2\\rangle $ are $ q= |a|^2,\\quad p= |b|^2.", "$ with $p+q = 1$ .", "In the following sections, the connections between $\\Omega (t)$ and $\\Delta (t)$ and the Cayley-Klein parameters will be explicitly presented for three specific analytically soluble models.", "Instead of the two complex Cayley-Klein parameters $a$ and $b$ , the propagator (REF ) can be expressed in terms of three real parameters: the transition probability $p$ and the dynamical phases $\\xi $ and $\\eta $ (sometimes called Stückelberg phases), $ \\mathbf {U}= \\left[\\begin{array}{cc} e^{i \\xi } \\sqrt{1-p} & - e^{-i\\eta } \\sqrt{p} \\\\ e^{i\\eta } \\sqrt{p} & e^{-i \\xi } \\sqrt{1-p} \\end{array}\\right] , $ where $a = e^{i \\xi } \\sqrt{1-p}$ and $b = e^{i\\eta } \\sqrt{p}$ ." ], [ "Sequence of pulses with the same phases", "In order to determine the populations after $N$ passes we need to find the $N$ -pass propagator $\\mathbfcal {U}_N = \\mathbf {U}^{N}$ .", "It has been proved [17] that the $N$ -th power of any SU(2) propagator, parameterized as in Eq.", "(REF ), reads $ \\mathbfcal {U}_N = \\left[\\begin{array}{cc}\\cos N\\theta + ia_i \\dfrac{\\sin N\\theta }{\\sin \\theta } & -b^* \\dfrac{\\sin N\\theta }{\\sin \\theta } \\\\b \\dfrac{\\sin N\\theta }{\\sin \\theta } & \\cos N\\theta - ia_i \\dfrac{\\sin N\\theta }{\\sin \\theta }\\end{array}\\right], $ where $a = a_r + i a_i$ and $ \\theta = \\arccos (a_r) \\quad (0 \\leqq \\theta \\leqq \\pi ).", "$ Therefore, the transition probability after $N$ passes is $ \\mathcal {P}_N = p\\, \\frac{\\sin ^2 (N\\theta )}{\\sin ^2 (\\theta )}.", "$ For $N=2$ , after simple algebra one finds $ \\mathbf {U}_{\\Omega ,\\Delta } \\mathbf {U}_{\\Omega ,\\Delta } = \\left[\\begin{array}{cc} a^2 - |b|^2 & -2 b^* a_r \\\\ 2 b a_r & (a^*)^2 - |b|^2 \\end{array}\\right].", "$ The double-pass transition probability is equal to $\\mathcal {P}_2 = 4 p (1-p) \\cos ^2 \\xi $ ." ], [ "Sequence of pulses with alternating phases", "For the pulse sequence with alternating phases we need a modification of the above result as follows.", "Consider a second interaction with the same magnitudes of $\\Omega (t)$ and $\\Delta (t)$ , but with the opposite sign of $\\Omega (t)$ , see Fig.", "REF .", "The respective propagator can be obtained from Eq.", "(REF ) by simple algebraic operations [27], [18], [19] and, very importantly, can be expressed with the same Cayley-Klein parameters $a$ and $b$ , $ \\mathbf {U}_{-\\Omega ,\\Delta } = \\left[\\begin{array}{cc} a & b^* \\\\ -b & a^* \\end{array}\\right].", "$ The respective double-pass propagator reads $ \\mathbf {U}_{-\\Omega ,\\Delta } \\mathbf {U}_{\\Omega ,\\Delta } = \\left[\\begin{array}{cc} a^2 + |b|^2 & -2 i b^* a_i \\\\ -2 i b a_i & (a^*)^2 + |b|^2 \\end{array}\\right], $ where $\\mathbf {U}_{\\Omega ,\\Delta }$ is the same as $\\mathbf {U}$ of Eq.", "(REF ).", "The double-pass transition probability is equal to $\\mathcal {P}_2^\\pm = 4 p (1-p) \\sin ^2 \\xi $ , which is different from the double-pass transition probability of $\\mathcal {P}_2 = 4 p (1-p) \\cos ^2 \\xi $ in the case of the same phases due to the Stückelberg phase $\\xi $ .", "This difference will be greatly amplified in the $N$ -pass probabilities.", "For the sign-alternating sequence, we can use Eq.", "(REF ) as the basic building block and derive the $2n$ -pass propagator using the connection between Eqs.", "(REF ) and (REF ) above.", "Obviously, the parameter $a$ is now replaced by $a^2 + |b|^2 = 1 -2 a_i^2 + 2i a_r a_i $ , the parameter $b$ is replaced by $-2 i b a_i$ , and the parameter $\\theta $ is redefined as $ \\Theta = \\arccos (1-2a_i^2).", "$ Then the propagator for the sign-alternating sequence of $2N$ pulses reads $ \\mathbfcal {U}_{2n}^\\pm = \\left[\\begin{array}{cc}\\cos n\\Theta + 2i a_r a_i \\dfrac{\\sin n\\Theta }{\\sin \\Theta } & -2 i b^* a_i \\dfrac{\\sin n\\Theta }{\\sin \\Theta } \\\\-2 i b a_i \\dfrac{\\sin n\\Theta }{\\sin \\Theta } & \\cos n\\Theta - 2i a_r a_i \\dfrac{\\sin n\\Theta }{\\sin \\Theta }\\end{array}\\right].", "$ The propagator for $2n+1$ pulses can be obtained by multiplying $\\mathbfcal {U}_{2n}^\\pm $ by $\\mathbf {U}$ of Eq.", "(REF ), $ \\mathbfcal {U}_{2n+1}^\\pm = \\mathbf {U}\\mathbfcal {U}_{2n}^\\pm = \\left[\\begin{array}{cc}a C + 2i a_i S & - b^* C \\\\b C & a^* C - 2i a_i S\\end{array}\\right], $ with $ C= \\dfrac{\\cos (n+\\frac{1}{2})\\Theta }{\\cos \\frac{1}{2}\\Theta },\\quad S = \\dfrac{\\sin n\\Theta }{\\sin \\Theta }.", "$ Therefore, the transition probability after $2n$ and $2n+1$ pulses reads $\\mathcal {P}_{2n} &= p\\, \\frac{\\sin ^2 n\\Theta }{\\cos ^2 \\frac{1}{2}\\Theta }, \\\\\\mathcal {P}_{2n+1} &= p\\, \\frac{\\cos ^2 (n+\\frac{1}{2})\\Theta }{\\cos ^2 \\frac{1}{2}\\Theta } .", "$ In the derivation of Eq.", "(REF ), it is used that $a_i^2 = \\sin ^2 \\frac{1}{2}\\Theta $ , which follows from Eq.", "(REF ).", "In the limit of nearly complete population transfer by a single pulse, which is concerned here, we have $|b| \\approx 1$ and $|a| \\ll 1$ .", "Then $|a_r| \\ll 1$ and $|a_i| \\ll 1$ and hence $\\theta \\approx \\pi /2$ and $\\Theta \\ll 1$ .", "More accurately, we have $\\theta &\\approx \\tfrac{1}{2}\\pi - a_r - \\tfrac{1}{6} a_r^3 + \\cdots , \\\\\\Theta &\\approx 2|a_i| + \\tfrac{1}{3} |a_i|^3 + \\cdots $ Therefore, for sequences of pulses of the same phase, $\\mathcal {P}_{2n} &\\approx (2n)^2 a_r^2, \\\\\\mathcal {P}_{2n+1} &\\approx 1 - a_i^2 - (2n+1)^2 a_r^2, $ while for sequences of pulses of alternating phases, $\\mathcal {P}_{2n}^\\pm &\\approx (2n)^2 a_i^2, \\\\\\mathcal {P}_{2n+1}^\\pm &\\approx 1 - (2n+1)^2 a_i^2, $ where the relation $p = 1 - a_r^2 -a_i^2$ has been accounted for.", "While retaining the dominant asymptotic terms, especially in Eq.", "(), it has been also used that $a_i$ is of order $O(\\delta )$ and $a_r$ is of order $O(\\delta ^2)$ .", "Obviously, the difference between the two types of sequences is that for a sequence of pulses with the same phases, the $N$ -pulse transition probability (REF ) is controlled by the real part $a_r$ of the diagonal Cayley-Klein parameter $a$ , while for a sequence of pulses with alternating phases, the $N$ -pulse transition probability is controlled by the imaginary part $a_i$ of $a$ .", "As we will see, this is an important difference and these observations show the significance of the properties of the parameter $a$ ." ], [ "Case studies ", "In order to examine the performance of the general formulas (REF ) and (REF ) in regard to sensing, we consider three analytically soluble models, which provide explicit analytic expressions for the parameters $a_r$ and $a_i$ , from which one can find out the dependence of the populations on the static detuning shift $\\Delta _0$ .", "The first model is the Rosen-Zener model, which assumes a bell-shaped hyperbolic-secant pulse shape and a static detuning.", "It is suitable for studying the sensitivity of a population inverting resonant $\\pi $ pulse to a small detuning shift.", "The second model is the Rabi model, which assumes a rectangular pulse shape and a constant detuning.", "It is of the same type as, and simpler than the Rosen-Zener model.", "The comparison of the two models reveals the effects of the sharp pulse edges in the Rabi model.", "The third model is the Demkov-Kunike model, which assumes the same hyperbolic-secant pulse shape as the Rosen-Zener model but the detuning is a sum of a hyperbolic-tangent-shaped detuning and a static detuning.", "It is suitable for modelling adiabatic passage via a level crossing in the presence of a small detuning shift.", "Figure: Transition probability vs the static detuning shift Δ 0 \\Delta _0 for sequences of N=2N=2, 4, 8, and 16 identical pulses indicated by the boxed numbers.The pulse shape is hyperbolic secant, with a pulse area of π\\pi and width TT, and the detuning is constant (Rosen-Zener model).In each frame, the transition probability is plotted for a sequence of pulses with the same phase (thin blue solid curve), and a sequence of pulses with alternating phases (thick red solid curve).The dashed curve shows the no-transition probability for a single pulse and serves as a reference." ], [ "Resonant $\\pi $ pulses: Rosen-Zener model", "The non-crossing Rosen-Zener model is defined as [20] $ \\Omega (t) = \\Omega _0\\, \\text{sech}\\, (t/T),\\quad \\Delta (t) = \\Delta _0.", "$ The Cayley-Klein parameters of the Rosen-Zener propagator are [20], [21], [28] $a &= \\frac{\\Gamma (\\nu ) \\Gamma (\\nu -\\lambda -\\mu )}{ \\Gamma (\\nu -\\lambda ) \\Gamma (\\nu -\\mu )} , \\\\b &= -i \\frac{\\sin (\\pi \\alpha / 2)}{\\cosh (\\pi \\delta / 2)},$ where $\\Gamma (z)$ is Euler's gamma function [29] and $ \\lambda = \\alpha /2, \\quad \\mu = -\\alpha /2, \\quad \\nu = (1+i\\delta )/2, $ with $\\alpha = \\Omega _0 T$ and $\\delta = \\Delta _0 T$ .", "For $\\Delta _0 = 0$ , the Rosen-Zener model describes a resonant pulse of pulse area $\\pi \\Omega _0 T = \\pi \\alpha $ .", "Therefore, a resonant $\\pi $ pulse is realized with $\\alpha = 1$ .", "The presence of the static detuning $\\Delta _0$ allows to simulate the effect of a detuning shift on resonant excitation.", "Of particular interest in the present context is the behavior of the parameter $a$ for small detuning.", "For a $\\pi $ pulse ($\\alpha =1$ ) we have $ a \\approx i \\frac{\\pi \\delta }{2} + \\pi \\delta ^2\\ln 2 - i \\pi \\delta ^3 \\left[ \\frac{\\pi ^2}{24} + (\\ln 2)^2 \\right] + O(\\delta ^4).", "$ From here, it follows that $\\theta &\\approx \\frac{\\pi }{2} - \\pi \\delta ^2\\ln 2 + O(\\delta ^4), \\\\\\Theta &\\approx \\pi \\delta + O(\\delta ^3).$ Figure: The same as Fig.", "but for an odd number of pulses, N=3N=3, 5, 9, and 15.Here the dashed curve shows the transition probability for a single pulse and serves as a reference.Figure REF shows the transition probability vs the static detuning shift $\\Delta _0$ for sequences of an even number of hyperbolic-secant pulses, each with a pulse area of $\\pi $ , and Fig.", "REF shows the transition probability for sequences of an odd number of pulses.", "The probabilities are calculated from Eqs.", "(REF ) and (REF ).", "Around resonance, a narrow feature forms which gets more narrow as the number of pulses $N$ increases.", "For an even number of pulses, the feature shows up as a dip (Fig.", "REF ), and for an odd number of pulses, it appears as a spike (Fig.", "REF ).", "This feature is much more narrow for sequences of alternating phases (thick curves) than for sequences of the same phases (thin curves).", "Using Eqs.", "(REF ), (REF ) and (REF ), it is easy to derive the behavior of the transition probability for $|\\delta |\\ll 1$ .", "For a sequence of pulses with the same phases, the approximation is $P_{2n} &\\approx (2n \\pi \\ln 2)^2 \\delta ^4 , \\\\P_{2n+1} &\\approx 1 - \\tfrac{1}{4} \\pi ^2 \\delta ^2 + \\left[\\tfrac{1}{24}\\pi ^2 - 4n(n+1) (\\ln 2)^2 \\right] \\pi ^2\\delta ^4 .", "$ For a sequence of pulses of alternating phases, it reads $P_{2n} & \\approx n^2 \\pi ^2 \\delta ^2 , \\\\P_{2n+1} & \\approx 1 - (n+\\tfrac{1}{2})^2 \\pi ^2 \\delta ^2.", "$ Figure: Transition probability vs the static detuning shift Δ 0 \\Delta _0 for sequences of N=8N=8 and 9 identical pulses for sech pulses with a pulse area of π\\pi and width TT(Rosen-Zener model).The two frames are zoomed-in and modified versions of the corresponding frames in Figs.", "and .In each frame, the transition probability is plotted for a sequence of pulses with the same phase (thin blue solid curve), and a sequence of pulses with alternating phases (thick red solid curve).The dashed curves show the approximations for |δ|≪1|\\delta |\\ll 1: Eqs.", "() and () for N=8N=8 pulses, and Eqs.", "() and () for N=9N=9 pulses.The four approximate formulas (REF )-() are plotted in Fig.", "REF and compared to the exact values.", "A very good agreement between exact and approximate values is observed for $|\\delta | \\ll 1$ , as it should be the case.", "For a sequence of pulses with alternating phases, the dependence on $N$ and $\\delta $ near resonance is the same for an even and odd number of pulses, albeit inverted, see Eqs.", "(REF ) and (): the transition probability departs as $\\propto N^2 \\delta ^2$ from its resonant value.", "This means that the excitation profile can be squeezed as much as desired by merely increasing the number of pulses $N$ .", "The behavior of the transition probability for a sequence of pulses with the same phases is rather different: the transition probability departs as $\\propto N^2 \\delta ^4$ from zero for an even number of pulses [Eq.", "(REF )], while for an odd number of pulses, its departure from unity is described by a sum of a $N$ -independent term $\\propto \\delta ^2$ and a term $\\propto N^2 \\delta ^4$ [Eq. ()].", "Consequently, the transition probability is much flatter in the range near resonance than for sequences of pulses with alternating phases.", "Moreover, the presence of the $N$ -independent term $\\propto \\delta ^2$ for odd $N$ impedes the squeezing of the excitation profile by increasing $N$ .", "In either cases, due to the different departure law from resonance, the profile is much broader than for alternating-phase sequences, as indeed seen in Figs.", "REF -REF .", "The conclusion is that, as far as squeezing of the excitation profile near resonance is concerned, the sequences of pulses with alternating phases outperform the sequences of pulses with equal phases and therefore, are much more efficient for sensing of small detuning shifts.", "Moreover, the simple and accurate approximate formulas, Eqs.", "(REF )-(), allow ones to not only sense a detuning shift but also measure it by measuring populations.", "Any ambiguity, which may arise for a larger detuning shift due to the multiple oscillations for large $N$ , can be resolved be making measurements for different $N$ .", "In order to estimate the sensitivity of this technique, consider the sequences with alternating phases, for which the deviation from the resonant value is, in the lowest order, $(N\\pi \\delta /2)^2$ , see Eqs. ().", "By setting this deviation to $\\frac{1}{2}$ and recalling that $\\delta = \\Delta _0 T$ , we find that the half-width-at-half-maximum of the spike or the dip is $ \\Delta _{\\frac{1}{2}} = \\frac{\\sqrt{2}}{N\\pi T} \\approx \\frac{0.45}{N T} .", "$ Therefore, the sensitivity can be increased by increasing $T$ (which is the well-known Fourier bandwidth argument) or by increasing $N$ .", "For example, a sequence of ten $\\pi $ pulses with alternating phases, each of duration 10 $\\mu $ s, allows one to sense and measure a detuning shift of 4.5 kHz.", "If, instead of a deviation of $\\frac{1}{2}$ , we can measure a population deviation of $\\frac{1}{10}$ , then the sensitivity of the same arrangement improves to 2 kHz.", "As it is clear from the approximations (REF ) and (), the excitation profile can be squeezed by sequences of pulses with the same phases too.", "However, the scaling is much less efficient, $\\Delta _{\\frac{1}{2}} \\propto 1/N^{\\frac{1}{2}}$ , rather than $1/N$ .", "By setting the deviation to $\\frac{1}{2}$ , we find from Eq.", "(REF ) that $ \\Delta _{\\frac{1}{2}} = \\frac{1}{\\sqrt{N\\pi \\sqrt{2}\\, \\ln 2}\\, T} \\approx \\frac{0.57}{\\sqrt{N}\\, T}.", "$ Hence a sequence of ten $\\pi $ pulses with the same phases, each of duration 10 $\\mu $ s, allows one to sense and measure a detuning shift of 18 kHz, a factor of 4 larger than for sequences of alternating phases.", "Therefore, by merely flipping the phase of every other pulse in the sequence, one can achieve much stronger (quadratically enhanced) squeezing and hence much better sensitivity." ], [ "Rectangular pulses: Rabi model", "One can apply the same approach using pulses of rectangular shape, to the so-called Rabi model, $ \\Omega (t) = \\left\\lbrace \\begin{array}{cc} \\Omega _0, & |t|\\leqq T/2 \\\\ 0, & |t|> T/2 \\end{array} \\right.", ",\\quad \\Delta (t) = \\Delta _0.", "$ The Cayley-Klein parameters of the Rabi propagator are far simpler than for the Rosen-Zener model [22], $a &= \\cos \\left(\\tfrac{1}{2} \\sqrt{\\alpha ^2+\\delta ^2}\\right)+\\frac{i \\delta \\sin \\left(\\frac{1}{2} \\sqrt{\\alpha ^2+\\delta ^2}\\right)}{\\sqrt{\\alpha ^2+\\delta ^2}} , \\\\b &= -\\frac{i \\alpha \\sin \\left(\\frac{1}{2} \\sqrt{\\alpha ^2+\\delta ^2}\\right)}{\\sqrt{\\alpha ^2+\\delta ^2}},$ with $\\alpha = \\Omega _0 T$ and $\\delta = \\Delta _0 T$ .", "Note that the pulse area is equal to $\\alpha $ ; therefore, a resonant $\\pi $ pulse is realized with $\\alpha = \\pi $ .", "For such a $\\pi $ pulse, the Taylor expansion of the Cayley-Klein parameter $a$ reads $ a \\approx i\\frac{\\delta }{\\pi } - \\frac{\\delta ^2}{4\\pi } - i \\frac{\\delta ^3}{2\\pi ^3} + \\frac{\\delta ^4}{16\\pi ^3} + O(\\delta ^5).", "$ Hence to the lowest order in $\\delta $ we have $a_r = -\\delta ^2/(4\\pi )$ and $a_i = \\delta /\\pi $ .", "The transition probability for the pulse sequences with alternating phases can be calculated from Eqs.", "(REF ).", "For small $\\delta $ we find from Eqs.", "(REF ) $\\mathcal {P}_{2n}^\\pm &\\approx (2n)^2 \\frac{\\delta ^2}{\\pi ^2}, \\\\\\mathcal {P}_{2n+1}^\\pm &\\approx 1 - (2n+1)^2 \\frac{\\delta ^2}{\\pi ^2}, $ By setting the transition probability to $\\frac{1}{2}$ and recalling that $\\delta = \\Delta _0 T$ , we find for both odd and even $N$ $ \\Delta _{\\frac{1}{2}} \\approx \\frac{\\pi }{\\sqrt{2}\\, N T} \\approx \\frac{2.22}{N T} .", "$ This value is a factor of about 5 larger than that for the Rosen-Zener model, Eq.", "(REF ).", "Figure: Comparison of the Rosen-Zener and Rabi models for sequences of 10 pulses with alternating phases.The sech and rectangular pulse have the same peak amplitude and the same area π\\pi .Figure REF compares the excitation profiles for the Rabi and Rosen-Zener models generated by sequences of 10 $\\pi $ pulses of alternating phases.", "The pulse area of each pulse is $\\pi $ and the peak Rabi frequency is the same for each model.", "Obviously, the feature near resonance is much more narrow for the Rosen-Zener model (a factor of about 5, as noted above).", "The physical reason is that the rectangular pulses in the Rabi model exhibit typical power broadening, due to its sharp edges, while the smooth pulse in the Rosen-Zener model has no power broadening at all [30], [31], [32].", "Note that some pulse shapes, with wings vanishing as $t^{-n}$ exhibit even power narrowing [32] and they might provide even better sensitivity than sech pulses." ], [ "Adiabatic chirped pulses: Demkov-Kunike model", "The level-crossing Demkov-Kunike model is defined as [23] $ \\Omega (t) = \\Omega _0\\, \\text{sech}\\, (t/T),\\quad \\Delta (t) = \\Delta _0 + B \\tanh (t/T).", "$ For $\\Delta _0=0$ , the Demkov-Kunike model reduces to the Allen-Eberly-Hioe model [33], [34], [35], also known as the complex-sech pulse in NMR.", "It is the most beautiful example of chirped adiabatic passage involving a level crossing.", "The addition of the static detuning $\\Delta _0$ allows to simulate a detuning shift in this model.", "For $B=0$ , the Demkov-Kunike model reduces to the Rosen-Zener model.", "For $B=\\Delta _0$ , the Demkov-Kunike model turns into the Bambini-Berman model [36].", "The Cayley-Klein parameters for this model are [23], [28] $a &= \\frac{\\Gamma (\\nu ) \\Gamma (\\nu -\\lambda -\\mu )}{ \\Gamma (\\nu -\\lambda ) \\Gamma (\\nu -\\mu )} , \\\\b &= -\\frac{i\\alpha \\Gamma (1-\\nu ) \\Gamma (\\nu -\\lambda -\\mu )}{2 \\Gamma (1-\\lambda ) \\Gamma (1-\\mu )},$ where $\\lambda &= \\left(\\sqrt{\\alpha ^2-\\beta ^2} - i\\beta \\right) / 2, \\\\\\mu &= -\\left(\\sqrt{\\alpha ^2+\\beta ^2} + i\\beta \\right) / 2, \\\\\\nu &= (1+i\\delta -i\\beta ) / 2,$ with $\\alpha = \\Omega _0 T$ , $\\beta = B T$ , and $\\delta = \\Delta _0 T$ .", "In the Demkov-Kunike model, we have for $\\alpha =\\beta =2$ (corresponding to pulse area of $2\\pi $ and chirp rate of $B=2/T$ ) $ a \\approx 0.086 + 0.165i\\delta - 0.052\\delta ^2 + 0.036i\\delta ^3 + O(\\delta ^4), $ from where we find $\\theta &\\approx 1.484 + 0.052\\delta ^2 + O(\\delta ^4), \\\\\\Theta &\\approx 0.33 \\delta + O(\\delta ^3).$ Figure: Transition probability in the Demkov-Kunike model vs the static detuning shift Δ 0 \\Delta _0 for sequences of NN identical pulses indicated by the boxed numbers.The pulse shape is hyperbolic secant, with a pulse area of 2π2\\pi and width TT, and the detuning given by a hyperbolic-tangent chirp (with β=2/T\\beta = 2/T) and a constant term Δ 0 \\Delta _0.In each frame, the transition probability is plotted for a single pulse (dashed curve), a sequence of pulses with the same phase (thin blue solid curve), and a sequence of pulses with alternating phases (thick red solid curve).The dotted curves illustrate the approximations () for even NN and () for odd NN.The transition probability for the Demkov-Kunike model can be calculated from Eqs.", "(REF ) and (REF ) and it is plotted in Fig.", "REF .", "We are interested again in the feature near zero static detuning which emerges as a spike (for odd $N$ ) or a dip (for even $N$ ).", "Once again, the features produced by the sequences with alternating phases are much more narrow than the features produced by both the single pulse and the sequences of pulses with the same phases.", "The single-pulse profile is much broader near zero detuning compared to the single-pulse profile of the Rosen-Zener model in Figs.", "REF and REF , which manifests the robustness characteristic of adiabatic passage methods.", "Consequently, it is more difficult to squeeze the excitation profile with a small number of pulses (the frames with 2 and 3 pulses), but for longer pulse sequences the desired squeezing still occurs, especially for pulse sequences with alternating phases.", "As for resonant pulses, it is possible to sense a detuning shift by using the same chirped pulse used for population inversion, without changing anything except for the sign of the Rabi frequency.", "One can derive the behavior of the transition probability for small $\\delta $ using Eqs.", "(REF ), (REF ) and (REF ).", "For a sequence of pulses with the same phases, the picture is rather messy (see Fig.", "REF ) and the approximation is not very meaningful.", "For a sequence of pulses of alternating phases, Eqs.", "(REF ) and (REF ) give the asymptotics $(|\\delta |\\ll 1)$ $P_{2n} & \\approx (0.33n)^2 \\delta ^2 (1+0.438\\delta ^2) ,\\\\P_{2n+1} & \\approx 1 - [0.33 (n+\\tfrac{1}{2})]^2 \\delta ^2 (1+0.438\\delta ^2),$ where higher terms in $\\delta $ are retained for better accuracy.", "These approximations are plotted in Fig.", "REF by dashed curves.", "They allow one, as in the Rosen-Zener model, to estimate the sensitivity of this technique.", "By setting the transition probability to $\\frac{1}{2}$ and recalling that $\\delta = \\Delta _0 T$ , we find for both odd and even $N$ $ \\Delta _{\\frac{1}{2}} \\approx \\frac{4.3}{N T}.", "$ For example, a sequence of ten $\\pi $ pulses with alternating phases, each of duration 10 $\\mu $ s, allows one to sense and measure a detuning shift of 43 kHz.", "This is almost a factor of 10 larger than for resonant $\\pi $ pulses in the Rosen-Zener model, cf. Eq.", "(REF ) versus Eq.", "(REF ).", "This is not surprising because the adiabatic passage techniques, as described here by the Demkov-Kunike model, are resilient to parameter variations, including the detuning.", "For larger values of the pulse area and the chirp, the transition probability becomes even more robust to parameter errors and the sensitivity decreases even more." ], [ "Conclusions", "This paper presented a method for detection and measurement of small detuning shifts generated, e.g., by weak external electric or magnetic fields.", "The method uses coherent amplification of transition probability errors by a train of identical pulses in two setups: with the same phase of each subsequent pulse, and with an alternating phase shift of $\\pi $ from pulse to pulse.", "Two kinds of pulses were considered: a resonant $\\pi $ pulse and an adiabatic chirped pulse, both of which are standard quantum control tools for complete population inversion.", "In either cases, small detuning shifts do not change the transition probability very much; however, they modify the dynamical phases in the propagator much more significantly, which are amplified and mapped onto the populations by the repeated application of the same pulse.", "Explicit analytic estimates were derived using the well-known non-crossing Rosen-Zener and Rabi models and the level-crossing Demkov-Kunike model.", "Based on the analytical results and numerical simulations, it was concluded that sequences of pulses with alternating phases outperform those with the same phases, as far as sensing is concerned: they generate much steeper, and hence much narrower, excitation profiles around resonance, thereby providing much higher sensitivity to detuning shifts.", "Smooth resonant $\\pi $ pulses, exemplified by the Rosen-Zener model, are by far the best performer, with the greatest sensitivity.", "Alternatively, Gaussian pulses, for which analytic results (albeit not so simple) are also available [37], deliver similar performance.", "It is worth considering also pulses of Lorentzian-type shapes (with wings vanishing as $\\propto t^{-n}$ ), which exhibit power narrowing [32] and may deliver even better sensitivity than sech pulses.", "Rectangular pulses, represented by the Rabi model, exhibit broader profiles (by a factor of 5) than the Rosen-Zener model due to power broadening generated by the sharp edges of the pulse.", "Chirped adiabatic pulses, exemplified by the Demkov-Kunike model considered here, are far less suitable for sensing than resonant pulses because of the inherent robustness of adiabatic techniques to parameter variations.", "Similar results can be obtained for linearly chirped Gaussian pulses for which analytical results are available [38].", "Furthermore, using rectangular pulses with a linear chirp, as in the popular Landau-Zener-Stückelberg-Majorana model [39], [40], [41], [42] model in its finite version [43], is an inappropriate option for sensing either because it features both adiabaticity and sharp pulse edges.", "Therefore, sequences of smooth resonant $\\pi $ pulses with alternating phases are identified as the most suitable for sensing of small detuning shifts.", "It is worth emphasizing that the proposed sensing method uses identical pulses (except for the possible $\\pi $ phase shift from pulse to pulse).", "In this manner, no additional uncertainties and errors are introduced which might mask the effects of small level shifts.", "The proposed technique is very convenient for practical use because the same pulse used as a NOT gate in a quantum circuit can be used to detect detuning shifts.", "Hence this simple recipe provides an efficient tool for rapid sensing of weak electric and magnetic fields, without sophisticated tomography setups or entangling operations.", "It is applicable to all kinds of experimental platforms wherein the environment causes energy level shifts.", "Particularly promising are Rydberg atoms and ions [45], [46], [44], [47] due to their increased sensitivity to electric field variations.", "The present idea is basically the opposite of the idea that the NMR community is pursuing since many years, that is, alternating pulse phases such that the detuning effects are suppressed, termed dynamical rephasing or dynamical decoupling (DD).", "The simplest such sequence is the Carr-Purcell-Meiboom-Gill (CPMG) two-pulse sequence [48], [49].", "An important development is the XY-4 sequence [50], [51], which uses 4 phase-shifted pulses.", "The XY-4 sequence is used as a building block for periodic DD, in which it is applied sequentially [52], and concatenated DD (CDD) sequences, which concatenate lower-order CDDs recursively, starting from XY-4 [53], [54], [55].", "Another important development is the concept of robust DD sequences [56], [57], which are resilient to various pulse errors.", "Yet another development is the extension to quantum systems with more than two states [58].", "Further details can be found in a comprehensive review [59].", "It is likely that, taking inspiration from the vast DD literature, the straightforward flips in the pulse phase in the sensing method proposed here could be optimized further, by letting the relative phases from pulse to pulse be free control parameters.", "However, this task is well beyond the scope of the present work.", "It should be clear that the sensing method presented here is based upon quantum interference.", "Therefore, the measurement should be fast enough in order to avoid noise and hence dephasing.", "Finally, in this work only detuning errors have been considered.", "In a real experiment, Rabi frequency errors might occur too, which would turn the parameter estimation problem into a multi-parameter one.", "Therefore, the method is strictly applicable in the absence of such errors.", "However, the underlying assumption is that the $\\pi $ pulses used in the sensing sequence is of very high fidelity because it is used in some quantum circuit, which would be the main experiment, i.e.", "the loss of probability does not come from it but from the external ambient fields.", "The present method allows one to quickly measure, once in a while, whether the external field has changed.", "More importantly, a closer inspection shows that small Rabi frequency errors do not affect the method as far as detuning sensing is concerned.", "It is only important that the Rabi frequency does not change during the sensing sequence, even if it is slightly different from its nominal $\\pi $ pulse value.", "This work is supported by the Bulgarian Science Fund Grant DO02/3 (Quant-ERA Project ERyQSenS)." ] ]
2105.11661
[ [ "On the Ulam Type Stability of Nonlinear Volterra Integral Equations" ], [ "Abstract In this paper, we examine the Hyers-Ulam and Hyers-Ulam-Rassias stability of solutions of a general class of nonlinear Volterra integral equations.", "By using a fixed point alternative and improving a technique commonly used in similar problems, we extend and improve some well-known results on this problem.", "We also provide some examples visualizing the improvement of the results mentioned." ], [ "Introduction", "Hyers–Ulam stability, initiated with a speech of S. M. Ulam [1] at Wisconsin University, is a concept that provides an approximate solution for the exact solution in a simple form for differential equations.", "Ulam posed the following problem: “Under what conditions does there exists an homomorphism near an approximately homomorphism of a complete metric group?” More precisely: Given a metric group $\\left(G, \\cdot , d \\right)$ , a number $\\varepsilon >0$ and a mapping $f:G\\rightarrow G$ satisfying the inequality $d\\left( f(xy), f(x)f(y) \\right)<\\varepsilon $ for all $x,y\\in G$ , does there exist a homomorphism $g$ of $G$ and a constant $K$ , depending only on $G$ , such that $d\\left( f(x), g(x) \\right)\\le K\\varepsilon $ for all $x\\in G$ ?", "In the presence of affirmative answer, the equation $g(xy)=g(x)g(y)$ of the homomorphism is called stable, see [1] for details.", "One year later, Hyers [2] gave an answer to this problem for linear functional equations on Banach spaces and showed that the additive functional equation $f(x+y)=f(x)+f(y)$ is stable in the sense of Ulam with the following celebrated result.", "Theorem 1.1 ([2]) Let $E_1, E_2$ be real Banach spaces and $\\varepsilon >0$ .", "Then, for each mapping $f:E_1\\rightarrow E_2$ satisfying $\\left\\Vert f(x+y)-f(x)-f(y) \\right\\Vert \\le \\varepsilon $ for all $x,y\\in E_1$ , the limit $L(x):=\\lim \\limits _{n\\rightarrow \\infty }2^{-n}f\\left(2^nx\\right)$ exists for all $x\\in E_1$ and $L:E_1\\rightarrow E_2$ is the unique additive mapping satisfying $\\left\\Vert f(x)-L(x)\\right\\Vert \\le \\varepsilon $ for all $x\\in E_1$ .", "After this result of Hyers, a new concept of stability for functional equations established, called today Hyers-Ulam stability, and many papers devoted to this subject (see for example [3], [4], [5], [6], [7], [8], [9]).", "In 1978, T. Rassias [10] provided a remarkable generalization with the following well-known result, which known as Hyers-Ulam-Rassias stability today.", "Theorem 1.2 ([10]) Let $E_1, E_2$ be Banach spaces, $\\theta \\in (0,\\infty )$ and $p\\in [0,1)$ .", "Then, for each mapping $f:E_1\\rightarrow E_2$ satisfying $\\left\\Vert f(x+y)-f(x)-f(y) \\right\\Vert \\le \\varepsilon $ for all $x,y\\in E_1$ , there exists a unique additive mapping $L:E_1\\rightarrow E_2$ satisfying $\\left\\Vert f(x)-L(x)\\right\\Vert \\le \\frac{2\\theta }{2-2^p}\\left\\Vert x\\right\\Vert ^p$ for all $x\\in E_1$ .", "Moreover, if $f(tx)$ is continuous for all $t\\in \\mathbb {R}$ and each fixed $x\\in E_1$ , then $L$ is $\\mathbb {R}$ -linear.", "Apart from functional equations, this concept of stability has applied to various kind of equations such as differential equations, integral equations and integrodifferential equations.", "Stability problem of differential equations in the sense of Hyers-Ulam was initiated by the papers of Obloza [11], [12].", "Later Alsina and Ger [13], Takahasi et.", "al.", "[14], [15], [16] provided remarkable results in this topic.", "After these pioneering works, a large number of papers devoted to this subject have been published (see for example [17], [18], [19], [20], [21], [22], [23], [24] and references therein).", "Volterra integral equations have been studied extensively since the four fundamental papers of Vito Volterra in 1896, and specially since 1913 when Volterra's book “Leçons sur les Équations Intégrales et les Équations Intégro-différentielles” appeared.", "For a continuous function $f$ and a fixed constant $a\\in \\mathbb {R}$ , the integral equation $y(t)=\\int _{a}^{t}f(s,y(s))\\operatorname{d}\\!", "{s}$ is called Volterra integral equation of second kind.", "Jung [25] studied Hyers-Ulam and Hyers-Ulam-Rassias stability of the equation (REF ) by using a fixed point alternative.", "Further, Castro [26] investigated Ulam type stability criteria for the equations $y(t)=\\int _{a}^{t}f(t,s,y(s))\\operatorname{d}\\!", "{s}.$ Later Gachpazan and Baghani [27] and Akkouchi [28] considered the stability of Volterra integral equations $y(t)=f(t)+\\lambda \\int _{t_0}^{t}k(t,s)y(s)\\operatorname{d}\\!", "{s}\\quad \\textrm {and}\\quad y(t)=f(t)+\\lambda \\int _{t_0}^{t}f(t,y,f(y))\\operatorname{d}\\!", "{s}$ successively in the sense of Ulam.", "There are also some works in the literature on the stability problem of Volterra integral equation with delay, for example Castro [29] studied this problem for the equation $y(t)=f(t)+\\Psi \\left(\\int _{t_0}^{t}f(t,s, y(s),y(\\alpha (s))\\operatorname{d}\\!", "{s}\\right).$ Now, we give explicit definitions of the Hyers-Ulam and Hyers-Ulam-Rassias stability concepts for Volterra integral equation (REF ), these definitions can be adapted to the equations (REF ) and other equations mentioned above in similar ways.", "Definition 1.3 If for each function $y(t)$ satisfying $\\left| y(t)-\\int _{a}^{t}f(s,y(s))\\operatorname{d}\\!", "{s} \\right|\\le \\varepsilon ,$ for all $t$ and some $\\varepsilon \\ge 0$ , there exists a solution $y_0(t)$ of (REF ) and a constant $K>0$ independent of $y$ and $y_0$ satisfying $\\left| y(t)-y_0(t) \\right|\\le K\\varepsilon $ for all $t$ , then the Volterra integral equation (REF ) is said to be stable in the sense of Hyers-Ulam [25].", "Definition 1.4 If the statement of Definition REF is true after replacing the constant $\\varepsilon $ with the function $\\varphi (t)\\ge 0$ , where this function does not depend on $y$ and $y_0$ , then the Volterra integral equation (REF ) is said to be stable in the sense of Hyers-Ulam-Rassias [25].", "In [25], Jung proved following remarkable results on the Ulam type stability of the solutions of the Volterra integral equation (REF ).", "Theorem 1.5 ([25]) Let the function $f:[a,b]\\times \\mathbb {C}\\rightarrow \\mathbb {C}$ be a continuous function satisfying Lipschitz condition with the Lipschitz constant $L$ on $[a-r,a+r]$ with $Lr<1,$ then the Volterra integral equation (REF ) has Hyers-Ulam stability on $[a-r,a+r]$ .", "Theorem 1.6 ([25]) Let the function $f:[a,b]\\times \\mathbb {C}\\rightarrow \\mathbb {C}$ be a continuous function satisfying Lipschitz condition with the Lipschitz constant $L$ on $[a,b]$ .", "If the function $\\varphi :[a,b]\\rightarrow (0,\\infty )$ as in Definition REF satisfies $\\left| \\int _{a}^{t}\\varphi (s)\\operatorname{d}\\!", "{s} \\right|\\le K\\varphi (t)$ and $KL<1$ for a positive constant $K$ , then the Volterra integral equation (REF ) has Hyers-Ulam-Rassias stability on $[a,b]$ .", "In [26], Castro and Ramos obtained the following similar results on the stability problem of the Volterra integral equation (REF ).", "Theorem 1.7 ([26]) Let the function $f:[a,b]\\times [a,b]\\times \\mathbb {C}\\rightarrow \\mathbb {C}$ be a continuous function satisfying Lipschitz condition with the Lipschitz constant $L$ on $[a-r,a+r]$ satisfying (REF ), then the Volterra integral equation (REF ) has Hyers-Ulam stability on $[a-r,a+r]$ .", "Theorem 1.8 ([26]) Let the function $f:[a,b]\\times [a,b]\\times \\mathbb {C}\\rightarrow \\mathbb {C}$ be a continuous function satisfying Lipschitz condition with the Lipschitz constant $L$ on $[a,b]$ .", "If the function $\\varphi :[a,b]\\rightarrow (0,\\infty )$ as in Definition REF satisfies (REF ) and (REF ) for a positive constant $K$ , then the Volterra integral equation (REF ) has Hyers-Ulam-Rassias stability on $[a,b]$ .", "We remark that all the results mentioned above heavily depend on the conditions (REF ), (REF ) and (REF ).", "The main purpose of this paper is to examine the Hyers-Ulam and Hyers-Ulam-Rassias stability of Volterra integral equations given by (REF ) and (REF ).", "We will improve a technique seen in the work of Cădariu and Radu [30] and also used in the papers [25] and [26], this improvement will enable us to obtain more accurate results for the problem considered and to improve the results given in [25] and [26].", "More precisely, we will show that the conditions (REF ), (REF ) and (REF ) are not necessary." ], [ "Main Results", "We first introduce the concept of generalized metric which will be used in our proofs.", "Definition 2.1 For a nonempty set $X$ , a function $d:X\\times X\\rightarrow [0,\\infty ]$ is called a generalized metric on $X$ if and only if satisfies $d(x,y)=0$ if and only if $x=y$ , $d(x,y)=d(y,x)$ for all $x,y\\in X$ , $d(x,z)\\le d(x,y)+d(y,z)$ for all $x,y,z\\in X$ .", "It should be remarked that the only difference of the generalized metric from the usual metric is that the range of the former is permitted to be an unbounded interval.", "We will use the following fixed point result as main tool in our proofs, we refer to [31] for the proof of this result.", "Theorem 2.2 Let $(X,d)$ be a generalized complete metric space.", "Assume that $T:X\\rightarrow X$ is a strictly contractive operator with the Lipschitz constant $\\Lambda <1$ .", "If there is a nonnegative integer $k$ such that $d\\left(T^{k+1}x,T^kx\\right)<\\infty $ for some $x\\in X$ , then the following are true: The sequence $\\left\\lbrace T^nx\\right\\rbrace $ converges to a fixed point $x^*$ of $T$ , $x^*$ is the unique fixed point of $T$ in $X^*=\\left\\lbrace y\\in X \\; : \\; d\\left(T^kx,y\\right)<\\infty \\right\\rbrace ,$ If $y\\in X^*$ , then $d\\left(y,x^*\\right)\\le \\frac{1}{1-\\Lambda }d\\left(Ty,y\\right).$ In our proofs, we will use the metric defined in the following Lemma.", "In order to be able to use Theorem REF , we will need completeness of the space $X:=C\\left([a,b],\\mathbb {R}\\right)$ which is given in the following result (see [18]).", "Lemma 2.3 Let $X:=C\\left([a,b],\\mathbb {R}\\right)$ and define the function $d:X\\times X\\rightarrow [0,\\infty ]$ with $d\\left(f,g\\right):=\\inf \\left\\lbrace C\\in \\left[0,\\infty \\right] \\;:\\; \\left| f(t)-g(t) \\right| \\le C\\Phi (t),\\; t\\in [a,b]\\right\\rbrace $ where $\\Phi :[a,b]\\rightarrow \\left(0,\\infty \\right)$ is a given continuous function.", "Then $\\left(X,d\\right)$ is a generalized complete metric space.", "For the whole of this section, we define the interval $I:=[t_0, t_0+r]$ for real numbers $t_0$ and $r$ with $r>0$ , further we define the set $X$ of all continuous functions defined on $I$ , i.e.", "$X:=C(I,\\mathbb {R})$ .", "The following theorem is the first main result of this paper, which shows that the conditions (REF ) and (REF ) is not necessary for solutions of Volterra integral equation (REF ) to have Hyers-Ulam-Rassias stability on bounded intervals.", "Theorem 2.4 Suppose that the function $f:I\\times \\mathbb {R}\\rightarrow \\mathbb {R}$ is a continuous function satisfying the Lipschitz condition $\\left|f(t,y_1)-f(t,y_2)\\right|\\le L\\left| y_1-y_2 \\right|$ for all $t\\in I$ , all $y_1,y_2\\in \\mathbb {R}$ and some $L>0$ .", "If a continuous function $y:I\\rightarrow \\mathbb {R}$ satisfies $\\left|y(t)-\\int _{t_0}^{t}f(s,y(s))\\operatorname{d}\\!", "{s}\\right|\\le \\varphi (t)$ for all $t\\in I$ , where $\\varphi :I\\rightarrow (0,\\infty )$ is a nondecreasing continuous function, then there exists a unique solution $y_0:I\\rightarrow \\mathbb {R}$ of Volterra integral equation (REF ) satisfying $\\left|y(t)-y_0(t)\\right|\\le \\frac{\\varphi (t){\\rm e}^{\\eta r}}{1-L/\\eta }$ for all $t\\in I$ , where $\\eta $ is an arbitrary fixed real number with $\\eta >L$ .", "For any fixed $\\eta \\in \\mathbb {R}$ with $\\eta >L$ , let us introduce the generalized metric on $X$ by $d(f,g):=\\inf \\lbrace C\\in [0,\\infty ]\\;:\\; \\left| f(t)-g(t) \\right|{\\rm e}^{-\\eta (t-t_0)}\\le C\\varphi (t),\\;t\\in I \\rbrace .$ According to Lemma REF , $(X,d)$ is a generalized complete metric space.", "Now define the operator $\\Theta :X\\rightarrow X$ by $\\left(\\Theta y\\right)(t):=\\int _{t_0}^{t}f(s,y(s))\\operatorname{d}\\!", "{s}$ for all $t\\in I$ .", "Since $\\Theta y$ is continuously differentiable on $I$ , we remark that $\\Theta y\\in X$ .", "For a given $g_0\\in X$ , since $f$ and $g_0$ are bounded on $I$ and $\\min _{t\\in I}\\varphi (t)>0$ , there exists a constant $C_0<\\infty $ such that $\\left|(\\Theta g_0)(t)-g_0(t)\\right|{\\rm e}^{-\\eta (t-t_0)}=\\left|\\int _{t_0}^{t}f(s,g_0(s))\\operatorname{d}\\!", "{s}-g_0(t)\\right|{\\rm e}^{-\\eta (t-t_0)}\\le C_0\\varphi (t),$ for all $t\\in I$ , which implies $d(\\Theta g_0,g_0)<\\infty $ .", "Furthermore, for any give $g\\in X$ , since $g$ and $g_0$ are bounded on $I$ and $\\min _{t\\in I}\\varphi (t)>0$ , there exists a constant $C_1<\\infty $ such that $\\left| g_0(t)-g(t) \\right|{\\rm e}^{-\\eta (t-t_0)}\\le C_1\\varphi (t)$ for all $t\\in I$ , i.e.", "$d(g_0,g)<\\infty $ and hence $\\lbrace g\\in X\\,:\\,d(g_0,g)<\\infty \\rbrace =X$ .", "Now we will show that $\\Theta :X\\rightarrow X$ is strictly contractive on $X$ .", "First observe that, with integration by parts and using monotonicity of $\\varphi $ , $\\int _{t_0}^{t}\\varphi (s){\\rm e}^{\\eta (s-t_0)}\\operatorname{d}\\!", "{s}\\le \\frac{\\varphi (t)}{\\eta }{\\rm e}^{\\eta (t-t_0)}-\\frac{\\varphi (t)}{\\eta }\\int _{t_0}^{t}\\varphi ^{\\prime }(s){\\rm e}^{\\eta (s-t_0)}\\operatorname{d}\\!", "{s}\\le \\frac{\\varphi (t)}{\\eta }{\\rm e}^{\\eta (t-t_0)}$ for all $t\\in I$ .", "For any $g_1,g_2\\in X$ , let $C_{g_1,g_2}\\in [0,\\infty ]$ be an arbitrary constant such that $d(g_1,g_2)\\le C_{g_1,g_2}$ , i.e.", "$\\left|g_1(t)-g_2(t)\\right|{\\rm e}^{-\\eta (t-t_0)}\\le C_{g_1,g_2}\\varphi (t)$ for all $t\\in I$ .", "Then we have, by using (REF ) and (REF ), $\\left| (\\Theta g_1)(t)-(\\Theta g_2)(t) \\right| &=& \\left| \\int _{t_0}^{t}f(s,g_1(s))\\operatorname{d}\\!", "{s}-\\int _{t_0}^{t}f(s,g_2(s))\\operatorname{d}\\!", "{s} \\right|\\\\&\\le &\\int _{t_0}^{t}\\left|f(s,g_1(s))-f(s,g_2(s))\\right|\\operatorname{d}\\!", "{s}\\\\&\\le &L\\int _{t_0}^{t}\\left|g_1(s)-g_2(s)\\right|\\operatorname{d}\\!", "{s}\\\\&=&L\\int _{t_0}^{t}\\left|g_1(s)-g_2(s)\\right|{\\rm e}^{-\\eta (s-t_0)}{\\rm e}^{\\eta (s-t_0)}\\operatorname{d}\\!", "{s}\\\\&\\le &LC_{g_1,g_2}\\int _{t_0}^{t}\\varphi (s){\\rm e}^{\\eta (s-t_0)}\\operatorname{d}\\!", "{s}\\\\&\\le &\\frac{L}{\\eta }C_{g_1,g_2}\\varphi (t){\\rm e}^{\\eta (t-t_0)}$ for all $g_1,g_2\\in X$ and for all $t\\in I$ .", "Hence, we have $\\left| (\\Theta g_1)(t)-(\\Theta g_2)(t) \\right|{\\rm e}^{-\\eta (t-t_0)}\\le \\frac{L}{\\eta }C_{g_1,g_2}\\varphi (t),$ for all $g_1,g_2\\in X$ and all $t\\in I$ , that is $d(\\Theta g_1,\\Theta g_2)\\le \\frac{L}{\\eta }d(g_1,g_2)$ for all $g_1,g_2\\in X$ .", "Therefore, $\\Theta :X\\rightarrow X$ is strictly contractive and the assumptions of Theorem REF are satisfied with $k=1$ and $X^*=X$ .", "According the Theorem REF , there exists a unique solution $y_0:I\\rightarrow \\mathbb {R}$ of Volterra integral equation (REF ) satisfying $ d(y,y_0)\\le \\frac{1}{1-\\Lambda }d(\\Theta y, y) \\le \\frac{\\varphi (t)}{1-L/\\eta },$ which implies $\\left|y(t)-y_0(t)\\right|{\\rm e}^{-\\eta (t-t_0)}\\le \\frac{\\varphi (t)}{1-L/\\eta }$ for all $t\\in I$ .", "Thus, we have $\\left|y(t)-y_0(t)\\right|\\le \\frac{\\varphi (t)}{1-L/\\eta }{\\rm e}^{\\eta (t-t_0)}\\le \\frac{\\varphi (t){\\rm e}^{\\eta r}}{1-L/\\eta }$ for all $t\\in I$ , which completes the proof.", "As a corollary of Theorem REF , we obtain Hyers-Ulam stability of solutions of Volterra integral equation (REF ) on finite closed intervals without any restrictions on Lipschitz constant.", "Corollary 2.5 Suppose that the function $f:I\\times \\mathbb {R}\\rightarrow \\mathbb {R}$ is a continuous function satisfying the Lipschitz condition (REF ) for all $t\\in I$ , all $y_1,y_2\\in \\mathbb {R}$ and some $L>0$ .", "If a continuous function $y:I\\rightarrow \\mathbb {R}$ satisfies $\\left|y(t)-\\int _{t_0}^{t}f(s,y(s))\\operatorname{d}\\!", "{s}\\right|\\le \\varepsilon $ for all $t\\in I$ , then there exists a unique solution $y_0:I\\rightarrow \\mathbb {R}$ of Volterra integral equation (REF ) satisfying $\\left|y(t)-y_0(t)\\right|\\le \\frac{\\varepsilon {\\rm e}^{\\eta r}}{1-L/\\eta }$ for all $t\\in I$ , where $\\eta $ is an arbitrary fixed real number with $\\eta >L$ .", "Remark 2.6 In Theorem REF we can obtain Hyers-Ulam-Rassias stability of solutions of Volterra integral equation (REF ) without the assumptions $\\left| \\int _{a}^{t}\\varphi (s)\\operatorname{d}\\!", "{s} \\right|\\le K\\varphi (t)\\qquad \\textrm {and}\\qquad KL<1,$ while they are required in the result given in [25].", "We also note that result given in Theorem 3.1 on Hyers-Ulam stability of the paper [25] works only under the assumption $0<Lr<1$ , while our result Corollary REF is valid for all $L>0$ .", "Now we will give our second main result, which concerns the Ulam type stability of (REF ).", "In a similar way, in the following result, we show that the conditions (REF ) and (REF ) is not necessary for solutions of Volterra integral equation (REF ) to have Hyers-Ulam-Rassias stability on bounded intervals.", "Theorem 2.7 Suppose that the function $f:I\\times I\\times \\mathbb {R}\\rightarrow \\mathbb {R}$ is a continuous function satisfying the Lipschitz condition $\\left|f(t,y, z_1)-f(t,y, z_2)\\right|\\le L\\left| z_1-z_2 \\right|$ for all $t\\in I$ , all $z_1,z_2\\in \\mathbb {R}$ and some $L>0$ .", "If a continuous function $y:I\\rightarrow \\mathbb {R}$ satisfies $\\left|y(t)-\\int _{t_0}^{t}f(t, s,y(s))\\operatorname{d}\\!", "{s}\\right|\\le \\varphi (t)$ for all $t\\in I$ , where $\\varphi :I\\rightarrow (0,\\infty )$ is a nondecreasing continuous function, then there exists a unique solution $y_0:I\\rightarrow \\mathbb {R}$ of Volterra integral equation (REF ) satisfying (REF ) for all $t\\in I$ , where $\\eta $ is an arbitrary fixed real number with $\\eta >L$ .", "Proof of this result is very similar to proof of Theorem REF , hence we omit the details and give only the sketch of the proof.", "For any fixed $\\eta \\in \\mathbb {R}$ with $\\eta >L$ , define the generalized metric on $X$ by (REF ).", "According to Lemma REF , $(X,d)$ is a generalized complete metric space.", "Now define the operator $\\Theta :X\\rightarrow X$ by $\\left(\\Theta y\\right)(t):=\\int _{t_0}^{t}f(t,s,y(s))\\operatorname{d}\\!", "{s}$ for all $t\\in I$ .", "As in proof of Theorem REF , one can show that $d\\left(\\Theta g_0,g_0\\right)<\\infty $ for all $g_0\\in X$ and $\\lbrace g\\in X\\,:\\,d(g_0,g)<\\infty \\rbrace =X$ .", "Now we will show that the operator $\\Theta :X\\rightarrow X$ is strictly contractive on $X$ .", "For any $g_1,g_2\\in X$ , let $C_{g_1,g_2}\\in [0,\\infty ]$ be a constant such that $d(g_1,g_2)\\le C_{g_1,g_2}$ , i.e.", "$\\left|g_1(t)-g_2(t)\\right|$ $\\times {\\rm e}^{-\\eta (t-t_0)} \\le C_{g_1,g_2}\\varphi (t)$ for all $t\\in I$ .", "Then we have, by using (REF ) and (REF ), $\\left| (\\Theta g_1)(t)-(\\Theta g_2)(t) \\right| &=& \\left| \\int _{t_0}^{t}f(t,s,g_1(s))\\operatorname{d}\\!", "{s}-\\int _{t_0}^{t}f(t,s,g_2(s))\\operatorname{d}\\!", "{s} \\right|\\\\&\\le &\\int _{t_0}^{t}\\left|f(s,g_1(s))-f(s,g_2(s))\\right|\\operatorname{d}\\!", "{s}\\\\&\\le &L\\int _{t_0}^{t}\\left|g_1(s)-g_2(s)\\right|\\operatorname{d}\\!", "{s}\\\\&=&L\\int _{t_0}^{t}\\left|g_1(s)-g_2(s)\\right|{\\rm e}^{-\\eta (s-t_0)}{\\rm e}^{\\eta (s-t_0)}\\operatorname{d}\\!", "{s}\\\\&\\le &LC_{g_1,g_2}\\int _{t_0}^{t}\\varphi (s){\\rm e}^{\\eta (s-t_0)}\\operatorname{d}\\!", "{s}\\\\&\\le &\\frac{L}{\\eta }C_{g_1,g_2}\\varphi (t){\\rm e}^{\\eta (t-t_0)}$ for all $g_1,g_2\\in X$ and for all $t\\in I$ .", "Hence, we have $\\left| (\\Theta g_1)(t)-(\\Theta g_2)(t) \\right|{\\rm e}^{-\\eta (t-t_0)}\\le \\frac{L}{\\eta }C_{g_1,g_2}\\varphi (t),$ for all $g_1,g_2\\in X$ and all $t\\in I$ , that is $d(\\Theta g_1,\\Theta g_2)\\le \\frac{L}{\\eta }d(g_1,g_2)$ for all $g_1,g_2\\in X$ .", "Therefore, $\\Theta :X\\rightarrow X$ is strictly contractive and the assumptions of Theorem REF are satisfied with $k=1$ and $X^*=X$ .", "As in the proof of Theorem REF , by using Theorem REF , we conclude that there exists a unique solution $y_0:I\\rightarrow \\mathbb {R}$ of Volterra integral equation (REF ) satisfying $\\left|y(t)-y_0(t)\\right|\\le \\frac{\\varphi (t){\\rm e}^{\\eta r}}{1-L/\\eta }$ for all $t\\in I$ .", "As a corollary of Theorem REF , by taking $\\varphi (t):=\\varepsilon $ , we obtain Hyers-Ulam stability of solutions of Volterra integral equation (REF ) on finite closed intervals without the restriction (REF ).", "Corollary 2.8 Suppose that the function $f:I\\times I\\times \\mathbb {R}\\rightarrow \\mathbb {R}$ is a continuous function satisfying the Lipschitz condition (REF ) for all $t\\in I$ , all $z_1,z_2\\in \\mathbb {R}$ and some $L>0$ .", "If a continuous function $y:I\\rightarrow \\mathbb {R}$ satisfies $\\left|y(t)-\\int _{t_0}^{t}f(t,s,y(s))\\operatorname{d}\\!", "{s}\\right|\\le \\varepsilon $ for all $t\\in I$ , then there exists a unique solution $y_0:I\\rightarrow \\mathbb {R}$ of Volterra integral equation (REF ) satisfying (REF ) for all $t\\in I$ , where $\\eta $ is an arbitrary fixed real number with $\\eta >L$ ." ], [ "Examples", "Example 3.1 Consider the Volterra integral equation $y(t)=\\int _{0}^{t}sy(s)\\operatorname{d}\\!", "{s}$ on the interval $I:=[0,2]$ .", "The function $f(t,y(t))=ty(t)$ satisfies the Lipschitz condition on $I$ with the Lipschitz constant $L=2$ since $\\left|f(t,y_1)-f(t,y_2)\\right|=t\\left|y_1-y_2\\right|\\le 2\\left|y_1-y_2\\right|$ for all $t\\in I$ .", "According to Corollary REF , the Volterra integral equation (REF ) is stable in the sense of Hyers-Ulam on the interval $I$ .", "We remark that Theorem 3.1 of [25] and Theorem 5.1 of [26] do not work in this problem since $\\lambda r=2\\cdot 1=2>1$ .", "Example 3.2 Consider the Volterra integral equation (REF ) of Example REF on the interval $I:=[0,2]$ , it has been shown that the function $f(t,y(t))$ satisfies the Lipschitz condition with $L=2$ .", "If we choose $\\varphi (t):={\\rm e}^t$ , we have $\\left|\\int _{0}^{t}\\varphi (s)\\operatorname{d}\\!", "{s}\\right|=\\int _{0}^{t}{\\rm e}^s\\operatorname{d}\\!", "{s}={\\rm e}^t-1\\le {\\rm e}^t=\\varphi (t)$ for all $t\\in I$ , that is, the inequality (REF ) holds with $K=1$ .", "According to Theorem REF , the Volterra integral equation (REF ) is stable in the sense of Hyers-Ulam-Rassias.", "We remark that the condition (REF ) is not required in our result but this example shows that even if the inequality (REF ) holds, Theorem 2.1 of the paper [25] and Theorem 3.1 of the paper [26] do not work on this problem since $KL=2>1$ in this case." ] ]
2105.11778
[ [ "Planning Mm-Wave Access Networks With Reconfigurable Intelligent\n Surfaces" ], [ "Abstract With the capability to support gigabit data rates, millimetre-wave (mm-Wave) communication is unanimously considered a key technology of future cellular networks.", "However, the harsh propagation at such high frequencies makes these networks quite susceptible to failures due to obstacle blockages.", "Recently introduced Reconfigurable Intelligent Surfaces (RISs) can enhance the coverage of mm-Wave communications by improving the received signal power and offering an alternative radio path when the direct link is interrupted.", "While several works have addressed this possibility from a communication standpoint, none of these has yet investigated the impact of RISs on large-scale mm-Wave networks.", "Aiming to fill this literature gap, we propose a new mathematical formulation of the coverage planning problem that includes RISs.", "Using well-established planning methods, we have developed a new optimization model where RISs can be installed alongside base stations to assist the communications, creating what we have defined as Smart Radio Connections.", "Our simulation campaigns show that RISs effectively increase both throughput and coverage of access networks, while further numerical results highlight additional benefits that the simplified scenarios analyzed by previous works could not reveal." ], [ "Introduction", "Current and future mobile radio network generations are challenged to cope with ever-expanding mobile data demands, spurred by our increasingly connected society [1].", "At the same time, cellular communication systems based on sub-6GHz frequencies are currently experiencing a bandwidth shortage [2] as they struggle to deliver the required level of performance.", "Millimetre-wave (mm-wave) based cellular communications have been recognized as the key technology to address both these crucial issues, as they can fulfil the promise of supporting Gbps demands while also solving the spectrum scarcity issue [3].", "Although its standardization in cellular networks for mobile access began only recently with 3GPP Release 15, this technology has already been largely employed in satellite links and cellular backhauling [4] and its limitations are well known.", "In particular, mm-waves are affected by harsh propagation typical of such high frequency that leads to high free space attenuation.", "Simultaneously, high penetration losses and poor diffraction mean that any obstacle crossing the line of sight might easily cause mm-Wave communications to fail.", "While emergent technologies - such as massive MIMO and beamforming - can effectively compensate for the increased pathloss [5], the problem of blockage resiliency in mobile access has not encountered the same luck.", "Among the candidate technologies that can potentially address the issue above, the recent emerging concept of Reconfigurable Intelligent Surface (RIS) has gained extreme popularity among the academic community [6].", "RISs are described as quasi-passive planar structures whose electromagnetic properties can be electronically controlled to manipulate impinging radio waves in a variety of ways.", "While an RIS can produce several types of these electromagnetic manipulations, the ability to reflect and focus impinging waves in any direction has the potential of transforming these surfaces in passive relays [7].", "This ability is exciting for mm-Wave communications, as an RIS can increase the blockage resilience by creating an alternative electromagnetic path.", "As opposed to active relays, RISs also show significantly higher energy efficiency [8] and prototypal works [9] have shown how they can be effectively built with cheap materials.", "Indeed, part of the attention that RISs are generating might be well justified by the opportunity of reducing the cost of deploying and maintaining a resilient wireless access network as opposed to more traditional and expensive approaches [10].", "Theoretical works [11][12][13] have extensively analyzed this particular RIS configuration from a communication perspective, providing practical mathematical tools to model the propagation characteristics of such scenarios.", "However, these analyses are carried out at the link level with simplified network scenarios.", "In this work, instead, we focus on the planning of large-scale mm-Wave radio access networks employing intelligent surfaces and, to the best of our knowledge, it is the first to tackle this challenge.", "We have employed well-established coverage planning methods to develop a new mathematical formulation of the coverage planning problem where both base stations and RISs can be installed in given locations of an arbitrary geographic area.", "We have introduced the concept of Smart Radio Connection (SRC), a logical abstraction of the well-known concept of the RIS-enabled Smart Radio Environment [14].", "An SRC consists of a radio link assisted by an intelligent surface and, in our planning model, SRCs can be established alongside traditional connections between UEs and base stations to increase the coverage and system performance.", "Our extensive numerical analysis campaign testifies how the well known point-to-point benefits of employing RISs do scale well at the system level for mobile access.", "Results show that including RISs when planning a radio access network can simultaneously increase coverage, throughput and blockage resiliency.", "Additionally, our results give new interesting insights on the benefits of employing RISs for coverage planning of mm-wave networks that could not be noticed in the highly simplified scenarios of related works.", "In particular, our model can identify the RIS configurations and the deployment budget conditions that provide tangible performance advantages when RISs are considered.", "The rest of this paper is structured as follows: Sec.", "presents some relevant related works, Sec.", "details a baseline mm-wave coverage planning model that does not include the presence of RISs, Sec.", "describes the modeling choices that lead us to develop a RIS-aware planning model and presents the novel mathematical formulation.", "Finally, Sec.", "shows the simulation setup and the numerical results." ], [ "Related Works", "Reconfigurable Intelligent Surfaces represent the latest technological proposition in the domain of propagation waves control [15].", "Their use as passive relays has been proposed in [7], where preliminary link-level simulations have shown the potential benefits with respect to more traditional active relaying approaches.", "From a communication standpoint, the problem of jointly optimizing the base station pre-coding and the RIS elements phase shifts has been studied in [11], where an iterative algorithm addresses the non-convexity challenges.", "In [12], a closed-form solution of the same problem is derived exploiting the characteristic of mm-Wave channels.", "Finally, authors of [9] have shown how a prototype RIS can effectively enhance the coverage of indoor mm-Wave networks.", "Historically, the problem of coverage planning has been applied to different radio access technologies.", "However, mm-Wave coverage planning works have only lately appeared in the literature, given the relatively recent interest.", "Understandably, these works have studied the coverage problem with a focus on the network resilience against blockages.", "In particular, authors of [16] study the problem of optimizing the layout of an mm-Wave access network in a dense urban environment such that the LOS availability is maximized.", "A similar analysis is carried out in [17] for mm-Wave vehicular communication scenarios.", "In [18], the coverage planning problem is studied through a network cost minimization that employs a link availability stochastic model.", "Finally, authors of [10] have studied the impact of different network planning approaches on the blockage resiliency of mm-Wave deployments.", "None of the planning works mentioned above has included reconfigurable intelligent surfaces in their investigations.", "To the best of the authors' knowledge, this is the first published work to present such an analysis." ], [ "Basic mm-Wave Model", "In this section, we give a basic description of a mathematical programming model for mm-Wave access network coverage planning.", "Similarly to other coverage planning works [19], [10], we identify a set $ of candidate positions (i.e.", "Candidate Sites, CSs) over a given geographic area where Base Stations (BS) can be installed.", "A discrete set of Test Points (TP) $ represents the traffic/user distribution.", "Binary coverage parameter $\\Lambda _{t,c}$ captures the propagation characteristics between TP $t \\in and CS $ c .", "Particularly, $\\Lambda _{t,c}=1$ if a radio link between the two positions can be established and zero otherwise.", "These parameters are set according to physical considerations, such as distance, transmission power, receiver sensitivity, antenna gain, attenuation losses, and more.", "Additionally, blockages due to fixed and opaque obstructions between any pair of CS-TP can be modelled by setting the corresponding coverage parameter to 0.", "Given the fixed known position of any potential CS-TP pair, the maximum achievable downlink bit-rate can be pre-computed according to the transmitter and receiver characteristics and any propagation model of choice.", "Indeed, given the extreme directivity of mm-Wave downlink transmissions that can strongly limit any interference effect, we can reasonably assume this bit-rate to be independent of other simultaneous access transmissions [10].", "However, a well-known issue of millimetre-based communication is its high penetration loss and limited diffraction [20], resulting in frequent blockages due to obstacles transiting across the connection line of sight.", "Blocked radio links experience a dramatic reduction in throughput, and this can be taken into consideration by weighting the maximum achievable bit-rate of each link with the probability of the link being in a state where such bit-rate is actually available (i.e., not blocked)Specific blockage models, such as [21], express this probability as a decreasing function of the link length, allowing this quantity to be computed given the CS-TP distances.. Parameter $R_{t,c}^\\text{BS}$ denote this expected (blockage-weighted) maximum throughput between TP $t \\in and BS installed in $ c .", "Similarly, $R^\\text{MIN}$ identifies the minimum expected throughput that needs to be guaranteed to each TP for it to be considered as covered.", "Knowing the channel states $\\mathcal {S}$ , their probabilities $p_{s},s \\in \\mathcal {S}$ and the corresponding achievable rates $r_{s}, s \\in \\mathcal {S}$ , these parameters can be computed according to the following formula: $R = \\sum _{s \\in \\mathcal {S}}p_{s}r_{s}.$ Finally, the coverage planning is constrained to a budget value $B$ and parameter $P_c$ describes the cost of installing a BS in a particular CS $c \\in .\\newline The proposed planning model is based on the following decision variables:\\begin{itemize}\\item y_c^\\text{BS} \\in \\lbrace 0,1\\rbrace : installation variable equal to 1 if a BS is installed in site c \\in C and 0 otherwise,\\item x_{t,c} \\in \\lbrace 0,1\\rbrace : association variable equal to 1 if BS in c \\in is assigned for coverage of test point t \\in ,\\item \\tau _{t,c}^\\text{BS} \\in [0,1], time-sharing variable indicating the fraction of time during which BS in c \\in transmits to test point t \\in .", "This variable allows us to model the BS resource sharing as a time-sharing process, in accordance to 3GPP Rel.", "15 specifications.", "Note that the very same notation can be applied if the joint time and sub-carrier sharing has to be considered.\\end{itemize}Given the notation, the parameters and the variables described above, we now propose a basic MILP (Mixed Integer Linear Programming) formulation of the coverage planning problem:\\begin{maxi!", "}[2]{}{\\sum _{t \\in c \\in R_{t,c}^\\text{BS}\\cdot \\tau ^\\text{BS}_{t,c}}{}{}{\\sum _{c \\in x_{t,c}}{\\le 1}{\\forall t \\in {\\tau _{t,c}^\\text{BS}}{\\le \\Lambda _{t,c}\\cdot x_{t,c}\\quad }{\\forall t \\in c \\in {\\sum _{t \\in \\tau _{t,c}^\\text{BS}}{\\le y_c^\\text{BS}}{\\forall c \\in {\\sum _{c \\in R_{t,c}^\\text{BS}\\cdot \\tau _{t,c}^\\text{BS}}{\\ge R^\\text{MIN}\\quad }{\\forall t \\in T}{\\sum _{c \\in {P_c\\cdot y_c^\\text{BS}}}{\\le B}}The objective function in~(\\ref {opt1:obj}) expresses the goal of the planning model: the maximization of the sum-throughput.", "A per-user average throughput appears in the sum, which depends on both the nominal link capacity between BS and TP and the fraction of resources the BS dedicates to the specific TP.", "Also, note that we consider this objective function as one of the very many possible ones.", "Other approaches, such as the sum of throughput logarithms, the max-min throughput, etc., can be easily plugged in with minimal changes to the formulation.\\newline Constraints~(\\ref {opt1:1donor}) enforces each TP to be covered at most by 1 BS.", "Constraint~(\\ref {opt1:tau_act}) is such that a BS in c \\in can transmit to a TP t \\in for a strictly positive fraction of time only if such TP is associated with this particular BS (i.e.", "x_{t,c} = 1) and if a radio link can be established between the two (i.e.", "\\Lambda _{t,c} = 1).\\newline Constraint~(\\ref {opt1:tdm}) has a double function.", "First, it does not allow any transmission of strictly positive duration to originate from any BS which has not been installed.", "Additionally, it limits to 1 the overall sum of the fractions of time dedicated for transmissions towards specific TPs for each installed BS, effectively enforcing a time-based division of BS resources.", "Note that this constraint may imply single-beam BS transmissions.", "However, the goal of this formulation is not to provide a perfect user throughput figure, which is usually computed by system-level simulators, but rather to design a good network layout.", "The latter can be achieved even with approximated user throughput models that do not substantially change the optimal deployment.", "On top of that, multi-beam antenna patterns remarkably decrease link directivity, strongly limiting BS coverage.", "As such, we believe it is reasonable to assume that most of the downlink transmissions involve one user at a time.\\newline Constraint~(\\ref {opt1:min_rate}) simply bounds each TP^{\\prime }s throughput to be at least the minimum throughput R^\\text{MIN}.\\newline Finally, constraint~(\\ref {opt1:budget}) limits the deployment cost to the available planning budget B, with P_c^\\text{BS} indicating the cost of installing a BS in CS c \\in .\\section {Modelling Reconfigurable Intelligent Surfaces}\\begin{figure}[t]\\centering \\includegraphics [width=0.5]{content/figures/src.png}\\caption {Example of SRC with RIS orientation and lines of sight angles.", "}\\end{figure}In our modelling efforts, RISs behave as \\textit {passive beamformers}, focusing the impinging radio waves in specific directions and creating what is often identified as a \\textit {Smart Radio Environment}.", "In this way, a proper configuration of the RIS can actively assist the communication between a transmitter-receiver pair by increasing the Signal to Noise Ratio (SNR) at the receiver~\\cite {DiRenzo2020}.", "Following the same rationale, we introduce the novel concept of \\textit {Smart Radio Connection} (SRC): a triplet that comprises one transmitter (i.e.", "the BS), one receiver (i.e.", "the UE located in a specific TP) and a smart surface configured to assist this specific communication\\footnote {While it is possible for multiple RIS to be configured to assist a single TX-RX pair~\\cite {Cao2020}, in this work we focus on up to one surface per SRC.}.", "Any SRC is then modeled as a tuple <t,d,r>, where t \\in denotes the TP, d \\in denotes the BS installation site and r \\in denotes the RIS installation site, as the example pictured in Figure~\\ref {fig:src} shows.\\newline The problem of jointly optimizing the transmitter pre-coding and the RIS elements^{\\prime } phase shifts in a SRC is generally not convex~\\cite {Cao2020}.", "However, the inherent characteristics of a mm-Wave channel allow for significant simplifications and an optimal closed form expression of the average received power can be derived.", "In this work, we consider the average SRC channel gain expression developed in~\\cite {Wang2019} for mm-Wave communication, which we propose here in a compact form:\\begin{equation}\\gamma = f(\\mathbf {h}_{B,R},\\mathbf {h}_{R,P}) + f^{\\prime }(\\mathbf {h}_{B,R},\\mathbf {h}_{R,P}, \\mathbf {h}_{B,P}) + f^{\\prime \\prime }( \\mathbf {h}_{B,P}),\\end{equation}where \\mathbf {h}_{B,R} is the channel between the BS and the RIS, \\mathbf {h}_{R,P} is the channel between the RIS and the TP, \\mathbf {h}_{B,P} is the channel between the BS and the TP and f,f^{\\prime },f^{\\prime \\prime } are proper functions.\\newline The contribution of the RIS to the SRC channel gain is linearly separable from the contribution of the traditional direct link, meaning that the increment in SRC link capacity with respect to unassisted communication is directly dependent only on the terms f(\\mathbf {h}_{B,R},\\mathbf {h}_{R,P})+f^{\\prime }(\\mathbf {h}_{B,R},\\mathbf {h}_{R,P}, \\mathbf {h}_{B,P}).", "It follows that, by knowing the relative positions of the three components of a SRC, as well as the state probability of each channel, the performance of any SRC can be completely characterized.", "Indeed, we define R_{t,d,r}^\\text{SRC} as the expected (\\textit {blockage-weighted}) throughput when BS in d \\in transmits to TP t \\in , while being assisted by RIS in r \\in .\\newline In general, a RIS can be part of many SRCs, and we assume an instantaneous reconfiguration of the reflecting elements when the surface switches between different SRCs.", "However, we allow each surface to assist up to 1 TX-RX pair at a time, meaning that the RIS sharing takes the form of a time-sharing process.\\newline We are fully aware that the previous assumptions may represent some though technological challenges for RIS hardware manufacturers.", "However, we believe them to be consistent with a realistic technological maturity level that needs to be considered from the beginning if we want to investigate the potential benefits of RIS development.", "For instance, a similar evolution occurred in literature to beamforming reconfiguration assumptions.\\newline Similarly to what happens for uniform linear antenna arrays, RISs are expected to present a limited array field of view~\\cite {tan2018}.", "We consider this by defining a RIS orientation, coinciding with the vector normal to the surface.", "For a given orientation, the lines of sight of the base stations/test points of all SRCs which the RIS is assigned to have to fall inside the surface field of view.", "In this work, we define a horizontal field of view angle D and we discard the vertical field of view\\footnote {It usually has a limited impact on the network layout, however, if needed, a vertical field of view can be easily included in the model}.\\newline Finally, our proposed model maintains generality by not forcing any BS-TP pair to be RIS-assisted.", "However, including both SRCs and traditional direct-link radio connections in a planning model was found to require a cumbersome number of additional variables and constraints.", "We worked around this issue by including an additional candidate site \\tilde{c} where a \\textit {fake} RIS is always installed.", "This particular RIS has no cost, no time-sharing limitation and 360° field of view, but grants no additional throughput performance to any assisted BS-TP pair.", "After an optimal solution is found, a post-processing operation changes any SRC including the \\textit {fake} RIS into a traditional unassisted BS-TP communication.", "This way, we could maintain a leaner formulation by modelling SRCs only, while avoiding any loss of generality.\\newline According to the previously described modeling choices, the following variables were needed to extend the mm-Wave coverage planning model presented in sec.~\\ref {sec:base_model}:\\begin{itemize}\\item y_c^\\text{RIS}\\in \\lbrace 0,1\\rbrace : RIS installation variable, equal to 1 if a RIS is installed in site c\\in and 0 otherwise,\\item s_{t,d,r}\\in \\lbrace 0,1\\rbrace : SRC activation variable, equal to 1 if RIS in r \\in is assigned to assist the communication between BS in d \\in and TP t \\in ,\\item \\tau _{t,d,r}^\\text{SRC} \\in [0,1]: SRC time sharing variable, indicating the fraction of time during which BS in d \\in transmits to TP t \\in aided by a RIS installed in r \\in ,\\item \\phi _r \\in [0, 2\\pi ]: azimuth of RIS installed in CS r \\in computed with respect to a reference direction.\\end{itemize}We are now ready to introduce the coverage planning model extended to include Reconfigurable Intelligent Surfaces:\\begin{maxi!", "}[3]{}{\\sum _{t \\in d \\in r \\in R_{t,d,r}^\\text{SRC}\\cdot \\tau ^\\text{SRC}_{t,d,r}}{}{}{y_c^\\text{BS}+y_c^\\text{RIS}}{\\le 1}{\\forall c \\in {y_{\\tilde{c}}^\\text{RIS}}{\\ge 1} {\\sum _{d \\in r \\in s_{t,d,r}}{\\le 1}{\\forall t \\in {\\tau _{t,d,r}^\\text{SRC}}{\\le \\Lambda _{t,d,r}\\cdot s_{t,d,r}}{\\forall t \\in \\, d,r \\in {\\sum _{t \\in r \\in \\tau _{t,d,r}^\\text{SRC}}{\\le y_d^\\text{BS}}{\\forall d \\in {\\sum _{t \\in d \\in \\tau _{t,d,r}^\\text{SRC}}{\\le y_r^\\text{RIS}}{\\forall r \\in \\lbrace \\tilde{r}\\rbrace } {\\sum _{d \\in r \\in R_{t,d,r}^\\text{SRC}\\cdot \\tau ^\\text{SRC}_{t,d,r}}{\\ge R^\\text{MIN}}{\\forall t \\in T}{\\phi _r}{\\ge \\Phi ^{\\text{A}}_{r,t} - D/2 - 2\\pi (\\lnot s_{t,d,r})}{\\forall t \\in d,r \\in r \\ne \\tilde{c}}{\\phi _r}{\\le \\Phi ^{\\text{A}}_{r,t} + D/2 + 2\\pi (\\lnot s_{t,d,r})}{\\forall t \\in d,r \\in r \\ne \\tilde{c}}{\\phi _r}{\\ge \\Phi ^{\\text{B}}_{r,d} - D/2 - 2\\pi (\\lnot s_{t,d,r})}{\\forall t \\in d,r \\in r \\ne \\tilde{c}}{\\phi _r}{\\le \\Phi ^{\\text{B}}_{r,d} + D/2 + 2\\pi (\\lnot s_{t,d,r})}{\\forall t \\in d,r \\in r \\ne \\tilde{c}}{\\sum _{c \\in \\lbrace \\tilde{c}\\rbrace }{\\left(P_c^\\text{BS}\\cdot y_c^\\text{BS} + P_c^\\text{RIS}\\cdot y_c^\\text{RIS}\\right)}}{\\le B\\hspace{-14.22636pt}}}Objective function~(\\ref {opt2:obj}) is of the sum-throughput type.", "Constraint~(\\ref {opt2:install}) makes sure that a BS and a RIS cannot be installed in the same candidate site, while~(\\ref {opt2:fakeris}) forces the installation of the \\textit {fake} surface.", "Constraint~(\\ref {opt2:1src}) allows for up to 1 SRC to be active for each TP, meaning that each t \\in is covered by up to 1 BS and up to 1 RIS.", "In~(\\ref {opt2:tau_act}-\\ref {opt2:ris_tdm}) the BS and RIS time sharing is enforced.", "In particular, a strictly positive transmission duration is allowed only if the SRC is active, if both BS and RIS are installed and if a radio connection between the three network components can be established\\footnote {Note that, while the coverage parameter \\Lambda _{t,d} has been extended to also include a third index representing the RIS CS, its rationale remains unchanged.}.", "Constraints~(\\ref {opti2:or1}-\\ref {opti2:or4}) force the RIS azimuth to be such that the lines of sight of any associated BS and TP all fall inside its field of view.", "Parameters \\Phi _{r,t}^\\text{A} and \\Phi _{r,d}^\\text{B} indicate the angle between a reference vector originating from RIS r \\in and the connected TP t\\in and BS d \\in lines of sight, respectively.", "The reader can find an illustration in Figure~\\ref {fig:src}.", "Note that \\lnot s_{t,d,r}= (1-s_{t,d,r}).", "Finally, we have introduced a RIS cost parameter P_c^\\text{RIS} in the budget constraint~(\\ref {opt2:budget}).", "}\\section {Results}In this section, we numerically analyze the previously described models when applied to different instances.", "Such instances are characterized by parameters that vary according to the specific result or property intended to be highlighted.", "However, some assumptions will be valid throughout the entire section unless otherwise stated.\\newline We consider scenarios where the BS employs several uniform linear antenna arrays, such that the BS field of view is 360°.", "We assume 64 antennas per array and a transmit power of 30dBm.", "The receiver^{\\prime }s antenna is assumed to be omnidirectional, and RX sensitivity is set to -78dBm.\\newline Given that the size of the reflecting surface is directly related to the system performance~\\cite {Bjornson2020}, we show results for both 10^4 and 10^5 reflecting elements in each RIS.", "These are compatible with surface sizes of about 50\\text{x}50cm (i.e.", "small RIS) and 150\\text{x}150cm (i.e.", "large RIS), respectively, since the reflecting elements need around \\lambda /2 spacing~\\cite {Renzo2020}.", "Additionally, RIS field of view is set to 120°.\\newline Carrier frequency is set to 28GHz and both propagation and blockage models are taken from~\\cite {Akdeniz2014}.", "According to this model, the expected throughput decreases with the link-length, as longer links incur in higher blockage probabilities.\\newline The received power of SRCs has been computed with the formula derived in~\\cite {Wang2019} and summarized by Eq.~\\ref {eq:channel}.", "Traditional direct communication received powers have been computed using the same formula, but discarding the RIS contributions, without loss of generality.\\newline Maximum achievable bit-rates are computed according to realistic modulation and coding schemes, like those specified by IEEE 802.11ad standard~\\cite {sur2015}.\\newline In each instance, 52 CSs and 32 TPs are randomly but uniformly scattered on a 400\\text{x}300m area.\\newline The default planning budget is set to 10.6.", "BS cost is set to 1, while large and small RIS costs are set to 0.1 and 0.05, respectively.\\newline For any given set of parameters, numeric results have been computed by averaging on 30 random instances of TP and CS positions.\\newline We have used MATLAB to generate each instance and CPLEX to find an optimal solution.", "}\\begin{figure}[t]\\centering \\includegraphics [width=0.75]{content/figures/max800_tprate.eps}\\caption {Mean TP throughput varying R^\\text{MIN}}\\end{figure}The first result we intend to analyze is the performance in terms of expected throughput experienced at the test points for different values of R^\\text{MIN}.\\newline Figure~\\ref {fig:tp_vs_rmin} shows this value averaged over all TPs, for R^\\text{MIN} spanning from 0Mbps to 800Mbps, with 100Mbps increments.\\newline We note how, independently on the RIS size, the basic planning model is outperformed by the model that includes intelligent surfaces for any value of R^\\text{MIN}.\\newline Additionally, larger surfaces perform better than their smaller versions, and the performance difference between the 3 cases grows with the minimum guaranteed throughput.", "This suggests that the well studied link-level benefits of employing RIS in mm-Wave communication scale well also at system-level.\\newline Finally, the model without RIS becomes unfeasible when R^\\text{MIN} > 600 Mbps, while optimal solutions can still be found for both RIS sizes.", "This shows how re-configurable surfaces allow mm-Wave radio access networks to go beyond the coverage capabilities of traditional networks when a larger minimum guaranteed throughput is required.", "}\\begin{figure*}[!t]\\centering [Mean TP throughput]{\\includegraphics [width=2.6in]{content/figures/tp_vs_b.eps}}\\hfil [Active sites ]{\\includegraphics [width=4.0in]{content/figures/active_sites.eps}}\\caption {Budget variations, from 5 to 35 units with 4 units increments.", "}\\end{figure*}\\begin{figure}[t]\\centering \\includegraphics [width=0.8]{content/figures/distances_vs_b.eps}\\caption {Average TP-CS distances when varying budget.", "}\\end{figure}We have shown how intelligent surfaces effectively augment the coverage while also increasing the TP experienced throughput.", "In the following results, we further expand the analysis of the latter in order to establish the efficacy of RISs in boosting the raw network performance.\\newline We set R^\\text{MIN}=100Mbps and let B span from around 6 units to around 36 units with increments of 4 units.", "Note that B=36 is equivalent to an infinite budget since it allows the installation of the maximum number of BSs and RISs given the other parameters.\\newline Figure~\\ref {fig:rate} shows the impact of the available budget on the experienced TP throughput, while in Figure~\\ref {fig:active_sites} we have plotted the variations in the number of active sites where either BSs or RISs are installed.\\newline Interestingly, the number of active sites where RISs are installed - dotted and dashed blue curves in Figure~\\ref {fig:active_sites} - decreases as the budget increases from 4 until around 16 units, independently on the RIS size.", "For the same values of B, the number of installed base stations increases.\\newline Optimal solutions for lower budgets seem to favour a relatively larger number of RIS installations, which is reduced when BSs substitute RISs as more budget becomes available.", "However, while still being able to provide adequate coverage levels, the larger count of RISs has little impact on performance boosting for such low values of B, as Figure~\\ref {fig:rate} testifies.\\newline Indeed, this figure shows that a budget of 20 units or more is needed in order to experience a more substantial raw performance boost, which also coincides with an increased installed RISs count.", "The suggestion is that the sites where to install intelligent surfaces are chosen to increase coverage for lower values of B, while, as the budget increases, additional RISs are installed to increase the throughput.\\newline We confirm this by showing the average TP-RIS distances - dashed and dotted blue curves - against the budget variations in Figure~\\ref {fig:dist_vs_b}.", "Here is indeed evident how these distances decrease at first, as the budget increases, testifying that RISs are installed closer and closer to TPs in order to decrease the probability of blockage and thus guarantee a better coverage.\\newline However, when B\\ge 32 units, the average distances abruptly increase together with the average RIS installation count.", "Note indeed that only up to 32 BSs can be installed (i.e.", "one per TP), leaving the remaining budget to be spent entirely on RIS installations.\\newline This behaviour indirectly shows how RISs are most effective in boosting the radio access network performance when a portion of the planning budget can be dedicated to their installation or, in other words, when the BSs have been already installed.", "This is arguably an exciting result, as it suggests that intelligent surfaces might be quite effective in boosting the performance of mm-Wave access networks that have been already deployed.", "}We conclude this section by providing additional comments on Figure~\\ref {fig:dist_vs_b}.\\newline Consider the solid black line and both the dashed and dotted red lines.", "These represent the average optimal TP-BS distance for the model without RISs (solid black), the average optimal TP-BS distance in SRCs with small RIS size (dashed red) and the same quantity for larger RIS size (dotted red).\\newline In general, we can expect SRCs to be more robust against blockages because multiple lines of sight need to be interrupted at the same time for the connection to fail.\\newline This concept becomes evident when comparing the 3 curves above, as they show how base stations belonging to SRCs can be placed further away from the test points without reducing the \\textit {blockage-weighted} throughput as opposed to BS-TP distances found by solving the base model.", "Additionally, SRCs allow for a more efficient BS resource sharing, since on average more TPs are in the coverage range of each BS.\\newline As mentioned in Section~\\ref {sec:ris_model}, the RIS-aware model still allows for any TP to be covered by a traditional connection if such a choice is optimal.", "In this regard, the dashed and dotted black curves in Figure~\\ref {fig:dist_vs_b} show how those TPs which are covered through a traditional connection are, on average, remarkably closer to the assigned BS with respect to the test points involved in SRCs.", "This confirms that optimal TP-RIS assignments are chosen such that TPs which are further away from base stations are prioritized, while also suggesting that a heuristic approach based on such policy might yield satisfying results.", "\\section {Conclusion}To study the effect of RISs on large-scale mm-Wave access networks, we have developed a new mathematical formulation for the coverage planning problem that includes reconfigurable surfaces.", "In our models, RIS can be installed in given candidate sites of a geographic area to assist the communication between base stations and test points, effectively creating what we call a \\textit {Smart Radio Connection}.", "We have also formulated a baseline model where the coverage planning does not consider the presence of RISs.", "Our simulation campaigns show how RISs can effectively increase both performance and throughput of access networks.", "Numerical results also highlight the impact of the planning budget on the KPIs above.", "In particular, we have shown how RISs can offer better coverage even for relatively low budget values, while increasingly noticeable throughput gains are obtained for larger values.", "Finally, our analysis on the optimal distances between base stations, RISs and test points have shown which RIS positioning policies are the most effective.", "The study of different planning objectives, more complex deployment scenarios and more refined channel models might be subject of future works.", "\\section *{Acknowledgment}The research in this paper has been carried out in the framework ofHuawei-Politecnico di Milano Joint Research Lab.", "The Authorsacknowledge Huawei Milan research center for thecollaboration.}}}}\\end{maxi!}}}}}}}}\\end{maxi!", "}$" ] ]
2105.11755
[ [ "Optical trapping of nanoparticles in superfluid helium" ], [ "Abstract Optical tweezers, the three-dimensional confinement of a nanoparticle by a strongly focused beam of light, have been widely employed in investigating biomaterial nanomechanics, nanoscopic fluid properties, and ultrasensitive detections in various environments such as inside living cells, at gigapascal pressure, and under high vacuum.", "However, the cryogenic operation of solid-state-particle optical tweezers is poorly understood.", "In this study, we demonstrate the optical trapping of metallic and dielectric nanoparticles in superfluid helium below 2 K, which is two orders of magnitude lower than in the previous experiments.", "We prepare the nanoparticles via in-situ laser ablation.", "The nanoparticles are stably trapped with a single laser beam tightly focused in the superfluid helium.", "Our method provides a new approach for studying nanoscopic quantum hydrodynamic effects and interactions between quantum fluids and classical objects." ], [ "Introduction", "Superfluid helium is a peculiar quantum fluid appearing at low temperature (below 2.17K), and it comprises a non-viscous superfluid component with a viscous normal fluid component.", "The quantum nature of superfluid helium manifests itself in macroscopic phenomena such as unusually efficient heat transport, film flow, and vortex quantization.", "Owing to the higher transition temperature, a significantly larger number of superfluid helium atoms ($N\\sim 10^{25}$ ) can be prepared by just pumping helium vapor, which is advantageous compared with other superfluids, such as Bose-Einstein condensates of cold atoms($N\\lesssim 10^{7}$ ) that requires the complicated experimental techniques such as laser cooling.", "The large heat capacity ensures that any nanomaterial injected into the superfluid helium is quickly cooled and thermalized.", "This enables exploring the unprecedented interactions between quantum fluids and classical nanomaterials and nanoscopic quantum hydrodynamic effects[1], [2], [3].", "A major obstacle in such approaches is the lack of experimental techniques to suspend/levitate the target nanoobject in quantum fluid, and controlling and probing the object motion precisely.", "In this study, we demonstrate the stable suspension of nanoparticles in superfluid helium using optical trapping.", "Generally, optical trapping requires high numerical objective (NA) lenses[4], [5] comprising several optical components to realize suitable optical potentials.", "Irrespective of the research design, this is a desirable condition in biomaterial manipulation[5], [6], [7], micro hydrodynamic studies[8], [9], and trapping in extreme conditions[10], [11], [12], [13], [14], [15], [16], [17], [18], [19].", "However, cryogenic conditions prohibit the use of multi-element optics, due to the large thermal deformation and fragility of the bonding and housing materials.", "Optical-levitation-trap schemes have been proposed previously[20], where a long-working-distance and low-NA lens is placed outside cryostats to achieve optical trapping in low-temperature environments ($\\sim {180}{}$ ).", "In such techniques, an upward-directed laser beam pushes microparticles, thereby balancing the gravitational force.", "The low-NA results in the relatively weak transverse optical force, which makes it difficult to trap smaller nanoparticles.", "In this study, we use a high-NA moulded aspheric lens to implement simple all-optical single beam trapping[4].", "The lens was placed in the liquid helium cryostat and immersed in superfluid helium as shown in Fig.", "REF A.", "The strong gradient optical force combined with the very small thermal fluctuation energy ($\\sim $ e-4) realize stable optical trapping of solid nanoparticles in superfluid helium.", "Our study demonstrates the optical trapping at 1.4 K, which is two orders of magnitude lower than previously reported [20], [21].", "The all-optical method enables us to probe the suspended particle properties, such as the size of the particles.", "Moreover, the method would also provide an unprecedented scheme to study the nanoscopic quantum hydrodynamic phenomena such as Brownian motion in the quantum fluid[1].", "Figure REF A shows the predicted axial optical force, which is along the x-axis, and the light propagation direction; the forces shown in the figure were calculated using the parameters matched to the experimental setup, enabling us to investigate whether the stable optical trapping in superfluid helium is possible.", "The x-component of the optical force exerted on a gold nanoparticle is shown as a function of the nanoparticle position x on the optical axis (see Fig.", "REF A for the coordinate system).", "Gold nanoparticles are widely used in optical trapping, owing to the large polarizability [22] that enables stable optical trapping.", "The optical force is calculated based on the generalized Lorenz–Mie theory[23] for different particle sizes.", "The calculated optical force is 2 to 3 orders of magnitude higher than the gravitational force; therefore the gravitational effect is negligible here.", "Smaller particles remain at equilibrium, where the optical force is zero and the slope of the curve is negative.", "This ensures that the particles experience a force toward the equilibrium point.", "This can be seen more clearly in Fig REF B, where we show the effective potential energy by integrating the force curve.", "Note that the optical force normally includes a non-conservative force[24] and the calculated effective potential energy is just used for estimating the stability of the optical trapping.", "The existence of the potential minimum indicates the possibility of stable optical trapping for the smaller particles.", "Figure REF A reveals that there is no equilibrium point for the larger particles.", "The overall size dependency is consistent with the Rayleigh scattering cross section of the nanoparticles, and thus, the ’pushing’ scattering force scales with the sixth power of the particle size[25].", "In contrast, the optical dipole force scales with the third power of the particle size[26], which pulls the particle toward the focus of the light.", "Therefore, the scattering force is dominant for the larger particles, whereas for the smaller particles, the optical dipole force becomes dominant.", "The particle size dependence of the effective potential energy depth is depicted in Fig.", "REF E. It has been established that stable optical trapping requires the potential depth to be larger than $10\\times k_B T$ .", "The horizontal green line indicates the corresponding threshold energy for $T = {300}{}$ , showing the difficulty of the aspheric-lens-based optical trapping near room temperature.", "However, much lower threshold energy for superfluid helium (blue line in Fig.", "REF E) enables us the stable optical trapping.", "The lower threshold size of the particle is ${10}{nm}$ and the upper threshold size of the particles is ${77}{nm}$ .", "We also explore the possibility of optical trapping in superfluid helium for dielectric nanoparticles.", "Zinc oxide is a high-refractive index material with larger polarizability.", "Figure REF C and D respectively show the corresponding optical force and effective potential energy curves for zinc oxide.", "The resulting effective potential energy depth shown in Fig.", "REF E indicates that stable optical trapping is possible also for zinc oxide nanoparticles of size ${10}\\sim {120}{nm}$ .", "A major issue in nanoparticle optical trapping is preparing slow nanoparticles near the focal point.", "In particular, cryogenic conditions prohibit the use of the standard procedures such as the \"go and pick\" scheme[5] or using aerosols [27] to prepare nanoparticles.", "In this study, we utilize the laser ablation technique to load the nanoparticles.", "Gold nanoparticles (Methods) are coated on a glass coverslip, mounted in the liquid helium cryostat, and immersed in superfluid helium.", "The coverslip is irradiated by a pulsed laser light, which releases gold nanoparticles.", "Figure REF B and C are typical scanning and transmission electron microscope (SEM and TEM) images before and after laser ablation-induced ejection.", "Noticeable shape changes from octahedrons to spheres indicate that the ejection process accompanies melting and/or vaporization.", "Figure REF D shows the particle size distribution before and after laser ablation.", "The size distributions before (after) the laser ablation are well-fitted using a Gaussian function with a mean of ${60}{nm}$ (${77}{nm}$ ) and standard deviation of ${8}{nm}$ (${24}{nm}$ ).", "Although the size distribution of the ejected gold nanoparticles is broader than that of the original gold nanoparticles, half of the ejected particles lie within the range of the stability-size regime ${10}\\sim {77}{nm}$ illustrated in Fig.", "REF E. Dielectric nanoparticles can be synthesized in situ using laser ablation in the superfluid helium.", "Figure REF E shows the typical SEM image of the synthesized zinc oxide particles.", "The zinc oxide particles are highly spherical [28], which is suitable for optical trapping.", "Many synthesized zinc oxide particles have dimensionalities in the stability-size regime, although its difficult to control the synthesized particle size distribution.", "This is in contrast with the case of gold nanoparticles, where the loaded-particle size distribution reflects the original size distribution.", "Figure REF A illustrates the experimental arrangement.", "A gold-nanoparticle-coated coverslip is placed in a cuvette in the liquid helium cryostat.", "A lens for optical trapping (L1) is mounted in the middle of the cuvette sidewall.", "The cuvette has small gaps between the walls, ensuring the flow of liquid helium to and from the cuvette.", "After the liquid helium is transferred to the cryostat, the fluid is cooled using a vacuum pump, and the temperature falls below the superfluid transition temperature.", "We maintain this temperature ${1.4}{}$ during the experiment.", "The irradiation of a nanosecond laser pulse on the coverslip initiates the gold nanoparticle ejection process, and the ejected nanoparticles are dispersed in the cuvette.", "The gold nanoparticles exhibit random motion until landing on the cuvette surface or exiting the cuvette.", "Once a nanoparticle reaches the focal point of the tightly focused beam of light, it gets optically trapped.", "The strong light scattering enables visualizing the trapped nanoparticles, as shown in Fig.", "REF B.", "In addition, the zinc oxide nanoparticles can be optically trapped by replacing the gold-nanoparticle-coated coverslip with the zinc oxide bulk target.", "In situ laser-ablation synthesis allows dispersing zinc oxide nanoparticles in superfluid helium.", "The nanoparticles are captured by the optical force and stably trapped thereafter.", "The trapped nanoparticles are very stable and can remain in the trap typically for over 30 min.", "If we tentatively block the laser beam, the trapped particle escapes from the trapping site, ensuring that the trapping is truly due to the optical force.", "An animated visualization is provided as Supplementary Movie.", "Our method provides an optical-tweezers-based-approach[9], [8] to study nanoscopic quantum hydrodynamics and interaction between quantum fluids and classical nanoobjects[1], [2], [3].", "Determining the trapped particle size is very necessary because the size governs the interaction between the particle and the surrounding quantum fluid.", "In our study, zinc oxide particles of sizes ranging from nanometres to micrometres[28] were fabricated through the laser ablation process.", "We demonstrate the size determination of the actual trapped particle using the Mie scattering theory[29].", "The blue curve in Fig.", "REF D corresponds to the theoretically calculated relation between the particle size and the optical power scattered from the trapped particle into the observation solid angle.", "The red star marks in the figure indicate the detected actual scattering powers obtained from separate trapping events (see Supplementary Information).", "Note that the values are corrected in terms of the total detection efficiency, including the transmissivity of the detection optics.", "The scattering powers are well-below the Mie resonance region starting from $\\sim {1e5}{pW}$ , where the relation between the particle size and the scattering power is not straightforward.", "In other words, in our experimental condition, Rayleigh scattering is considered a good approximation, where the scattering power is proportional to the sixth power of the particle size.", "Accordingly, the curve is almost straight in the log–log plot.", "The trapped particle size was determined to be $30 \\sim 50\\, {nm}$ by comparing the experimental data and the theoretical curve, which exactly matches the calculated stability size range.", "The techniques developed in this study enable trapping and manipulating nanoparticles in superfluid helium and also characterizing the trapped particle all-optically.", "Our method will open up a new pathway to study quantum hydrodynamics, as our optical trapping scheme can be extended to monitor trapped nanoparticle positions with high spatial and temporal resolution[9], enabling the observation of the Brownian motion in the quantum fluid[1].", "Another possibility is combining our method with existing ingenious techniques for superfluid helium research.", "One exciting example is the quantum vortex visualization in superfluid helium[30].", "When a bunch of tracer nanoparticles is dispersed in superfluid helium, the particles are stabilized along the quantum vortex line.", "This is because the quantum vortex core is a local minimum of the pressure field.", "Therefore, quantum vortex dynamics can be visualized by imaging the scattered light from the nanoparticles, leading to recent experiments and studies on the hidden nature of the vortex-vortex interaction[31].", "These remarkable experiments, however, rely on the observation of the free motion of the quantum vortex, and the accidental collision of two vortices.", "Our results exhibit a striking possibility: optical trapping and control of the composite quantum vortex and tracer particle system.", "This technique could act as a new method to control the quantum vortex motion, and dynamically perturb and excite the quantum vortex states.", "Octahedral gold nanoparticles were synthesized in solution via a previously reported method with a modification[32].", "A 20 wt% poly (diallyldimethylammonium chloride) aqueous solution (Mw = 100000-200000) (80mm3) and a 0.1mol.dm-3 HCl aqueous solution (80mm3) were injected into diethylene glycol (4.0cm3) in a test tube with magnetic stirring at 30.", "A 19mm3 portion of 0.11mol.dm-3 HAuCl4 aqueous solution was added into the mixture, followed by heat treatment at 230 for 60 min with vigorous stirring in a N2 atmosphere.", "After cooling to room temperature, the synthesized octahedral gold nanoparticles were separated from the solution by adding acetone (4.0cm3), followed by centrifugation at 15000 rpm for 10 min.", "The precipitates of gold nanoparticles were dispersed in water, followed by centrifugation (15000 rpm, 10 min).", "This washing procedure was repeated three times.", "Finally, the purified octahedral gold nanoparticles were dispersed again in 1.0cm3 water.", "We placed a 3cm x 3cm x 3cm cuvette in superfluid helium.", "The cuvette sidewall has a mounting hole for the aspheric lens for optical trapping.", "The experimental process occurred entirely in this cuvette.", "The cuvette has small gaps between the walls, ensuring the flow of liquid helium to and from the cuvette.", "We mounted a coverslip coated with the gold nanoparticles and a bulk sintered semiconductor zinc oxide target in the cuvette.", "The liquid helium temperature is maintained at approximately 1.4 K during the experiment.", "Nanosecond light pulses from a frequency-doubled Q-switched Nd:YAG laser (wavelength 532 nm, pulse duration 10 ns, repetition rate 10 Hz, and pulse energy 1 mJ) were focused onto the target surface with a spot size of $\\sim $ 40, using a lens of 200 mm focal length.", "The ejected particles were optically trapped using continuous-wave laser (wavelength 785 nm).", "The typical power of the laser for the trapping is $\\sim {100}{mW}$ , ranging from ${50}{mW}$ to ${500}{mW}$ .", "Funding: This work was supported by the MEXT/JSPS KAKENHI Grant Number JP17K17841, JP16H06505, JP16H06507, and JP18KK0387 and by JST, PRESTO Grant Number JPMJPR18T5 and JPMJPR1909, Japan.", "Author contributions: Y.M.", "conceived and designed the project and wrote the paper.", "Y.M., X.G and K.K.", "performed the experiments and analyzed the data.", "K.S., T.K., and T.T.", "synthesized gold nanoparticles and characterized them.", "M.A.", "provided technical support, and assisted in writing and editing the manuscript.", "Competing interests: The authors declare that they have no competing interests.\"", "If this is not accurate, please list the competing interests.", "Data and materials availability: All data are available in the manuscript or supplementary materials.", "Supplementary Text Movie.", "S1" ] ]
2105.11631
[ [ "The Topology of Randomized Symmetry-Breaking Distributed Computing" ], [ "Abstract Studying distributed computing through the lens of algebraic topology has been the source of many significant breakthroughs during the last two decades, especially in the design of lower bounds or impossibility results for deterministic algorithms.", "This paper aims at studying randomized synchronous distributed computing through the lens of algebraic topology.", "We do so by studying the wide class of (input-free) symmetry-breaking tasks, e.g., leader election, in synchronous fault-free anonymous systems.", "We show that it is possible to redefine solvability of a task \"locally\", i.e., for each simplex of the protocol complex individually, without requiring any global consistency.", "However, this approach has a drawback: it eliminates the topological aspect of the computation, since a single facet has a trivial topological structure.", "To overcome this issue, we introduce a \"projection\" $\\pi$ of both protocol and output complexes, where every simplex $\\sigma$ is mapped to a complex $\\pi(\\sigma)$; the later has a rich structure that replaces the structure we lost by considering one single facet at a time.", "To show the significance and applicability of our topological approach, we derive necessary and sufficient conditions for solving leader election in synchronous fault-free anonymous shared-memory and message-passing models.", "In both models, we consider scenarios in which there might be correlations between the random values provided to the nodes.", "In particular, different parties might have access to the same randomness source so their randomness is not independent but equal.", "Interestingly, we find that solvability of leader election relates to the number of parties that possess correlated randomness, either directly or via their greatest common divisor, depending on the specific communication model." ], [ "Introduction", "There are two main categories of distributed algorithms: deterministic and randomized.", "The difference between them is stark as some tasks are solvable when randomness is present but cannot be solved deterministically.", "Further, randomness often yields faster and more efficient algorithms for tasks solvable deterministically.", "A main example is the task of electing a leader in anonymous systems [21], [14], where $n>1$ identical computing nodes need to designate a single node as their leader.", "For any deterministic algorithm running in symmetric systems where all nodes are identical, all nodes return exactly the same output.", "This is known as the impossibility of deterministic algorithms to break symmetry [3].", "On the other hand, if each node has access to an independent source of randomness, then symmetry can be broken, allowing electing a leader [4].", "It is widely agreed that the major breakthrough of computer science is the invention of the Turing machine [27], which abstracts all possible realistic computing machines into a single universal model.", "With this abstraction, researchers can ignore the system's specific properties and focus on the computational aspect of the problem rather than on its manifestation in a specific environment.", "In the distributed world, while each node is still a Turing machine, the environment plays a large role in setting the relevant model: nodes and communication links may be reliable or subject to various forms of failures.", "The system may be synchronous or experience arbitrary large communication delays.", "Also, all nodes might be connected as a single hop or connected in an arbitrary manner.", "This large variety of models required focusing on each model and analyzing each task and algorithm separately.", "In the early 1990s, a unifying framework was developed that captures all the above distributed models: distributed computing via algebraic topology [6], [16], [25].", "Similar to the Turing model, this framework enables analyzing the complexity of distributed tasks under a unique umbrella in which the same tools and methodologies apply, independently from the details of the actual distributed model of distributed computing.", "However, these pioneering works and the numerous subsequent work that followed them (see [15], and the references therein) mostly treated deterministic algorithms, and randomized algorithms remain so far excluded.", "The usual topology-analysis of deterministic distributed algorithms [15] involves a simplicial complex $\\mathcal {P}(t)$ , called protocol complex, that captures all the possible global states of the distributed system at time $t\\ge 0$ , where each global state is modeled as a simplex of that complex.See Appendix  for a summary of the topological notions used in this paper.", "Initially $\\mathcal {P}(0)=\\mathcal {I}$ , the so-called input complex that captures all initial global states of the system.", "The evolution of the system with time translates to the evolution of the complex $\\mathcal {P}(t)$ , $t\\ge 0$ .", "An algorithm solving a task in time $t$ is then modeled as a simplicial map $\\delta :\\mathcal {P}(t)\\rightarrow \\mathcal {O}$ from the protocol complex to the output complex $\\mathcal {O}$ , where the latter captures all legal final global states of the system.", "This mapping may need to satisfy some additional requirements in the case of input-output tasks (e.g., consensus) where the legality of the output depends on the input.", "It turns out that, for many distributed models, certain properties of a topological space, such as connectivity or homotopy type, are preserved during the evolution of the protocol complex with time.", "Comparing the topological properties of the protocol complex with those of the output complex is a fruitful approach for establishing impossibility results, that is, proving that the mapping $\\delta $ cannot exist.", "A typical example is the task of consensus in the crash-prone asynchronous shared-memory model in which the protocol complex remains connected throughout time while the output complex is disconnected, preventing the simplicial map $\\delta $ from existing.", "Extending this approach to the analysis of randomized algorithms requires overcoming several difficulties.", "In particular, for a large class of tasks that are not solvable deterministically in anonymous systems, including symmetry breaking tasks such as leader election, the solvability of these tasks using a randomized algorithm is only eventual.", "That is, there is no fixed $t\\ge 0$ for which there exists a simplicial map $\\delta :\\mathcal {P}(t)\\rightarrow \\mathcal {O}$ , even if we allow $t$ to be as large as we wish.", "Nevertheless, the task is eventually solvable, meaning that when $t$  goes to infinity, the probability for solving the task approaches 1.", "One may argue that, for a fixed $t\\ge 0$ , one can still compute the probability of solving the task at time $t$ by considering a mapping $\\delta _t:V(\\mathcal {P}(t))\\rightarrow V(\\mathcal {O})$ (not necessarily simplicial), then considering only the facets of $\\sigma \\in \\mathcal {P}(t)$ such that $\\delta _t(\\sigma )\\in \\mathcal {O}$ , and finally computing the sum of the probability of each of these facets to occur as a function of the outcomes of the random bits used in the protocol.", "This argument is valid.", "However, it may not be of practical use.", "Indeed, one is losing the whole point of using topology, as it might be difficult to analyze what topological properties (e.g., homotopy type) are preserved by such an arbitrary map $\\delta _t$ .", "Additionally, it might be difficult to relate the topological properties of the “best” map $\\delta _t$ (i.e., the one that maximizes the probability of solving the task at time $t$ ) with the properties of the best map $\\delta _{t+1}$ at time $t+1$ .", "In this paper, we give the first stab at developing a topological framework that captures synchronous randomized algorithms of certain types, which we now describe." ], [ "A Topological Approach for Randomized Algorithms", "We consider input-free symmetry-breaking tasks: tasks solely defined by their output complex $\\mathcal {O}$ , which we require to be symmetric (i.e., stable to permutation of the processing nodes names).", "For instance, this is the case of leader election in which there are no constraints on which node may be the leader.", "In this context, a crucial observation is that the solvability of a task at a time $t$ can be analyzed for each facet $\\sigma \\in \\mathcal {P}(t)$ separately.", "For each facet $\\sigma \\in \\mathcal {P}(t)$ , we can say that $\\sigma $ solves the task $\\mathcal {O}$ whenever there exists a simplicial map $\\delta :\\sigma \\rightarrow \\mathcal {O}$ which maps the state of each node to its output, such that the output value is independent of the actual name of the node.", "The interest of that observation is that it decouples the map $\\delta :\\sigma \\rightarrow \\mathcal {O}$ from the map $\\delta ^{\\prime }:\\sigma ^{\\prime }\\rightarrow \\mathcal {O}$ that may exist for another facet $\\sigma ^{\\prime }$ of $\\mathcal {P}(t)$ .", "The significant drawback of this approach is that, as the aforementioned general approach consisting of considering arbitrary maps $\\delta _t:V(\\mathcal {P}(t))\\rightarrow V(\\mathcal {O})$ , one is losing connection with topology.", "Indeed, the facet $\\sigma $ has a trivial topological structure, as it is just a simplex of $\\mathcal {P}(t)$ .", "Our approach, on the other hand, provides a structure to each simplex of $\\mathcal {P}(t)$ ; this allows us to keep analyzing the system using topological tools.", "To provide the simplices of $\\mathcal {P}(t)$ with a structure, we “project” each simplex $\\sigma \\in \\mathcal {P}(t)$ to a sub-complex $\\pi (\\sigma )$ of $\\sigma $ , the latter being viewed as a complex.To be precise, this complex is the induced subcomplex of $\\mathcal {P}(t)$ on $V(\\sigma )$ .", "Roughly, a set of nodes forms a simplex in $\\pi (\\sigma )$ if they have an identical individual state in $\\sigma $ , where the individual state of a node results from the outcomes of its source of randomness and from its knowledge about the outcomes of the other nodes (acquired during the communications up to time $t$ ).", "In a sense, a simplex in $\\pi (\\sigma )$ bears the meaning that the algorithm up to time $t$ did not break symmetry for the nodes associated with this simplex vertices.", "The sub-complex $\\pi (\\sigma )$ captures the internal state of each one of the parties in a specific execution and portrays similarity in the knowledge of the parties.", "The projection $\\pi $ is called consistency projection.", "Thanks to the consistency projection, we have regained structure, which we can now utilize to determine the ability of a facet $\\sigma \\in \\mathcal {P}(t)$ to solve a task $\\mathcal {O}$ .", "A facet $\\sigma $ solves the task $\\mathcal {O}$ whenever there exists a (name-preserving) simplicial map $\\delta :\\pi (\\sigma )\\rightarrow \\pi (\\tau )$ for some facet $\\tau \\in \\mathcal {O}$ , where $\\pi (\\tau )$ is the projection of $\\tau $ , that is, the sub-complex of $\\tau $ in which a set of vertices of $\\tau $ forms a simplex in $\\pi (\\tau )$ if they have identical individual output value in $\\tau $ .", "Granted with the notion of solvability for each facet of the protocol complex at time $t$ , we show how to define the eventual solvability of the task.", "Specifically, let us define the probability $p(t)$ that $\\mathcal {P}(t)$ solves $\\mathcal {O}$ as the sum, taken over all facets of $\\mathcal {P}(t)$ that solve $\\mathcal {O}$ , of the probability of each of these facets.", "Eventual solvability is then based on Borel-Cantelli's Lemma and on Kolmogorov's zero–one law stating that any tail event occurs with probability zero or one.", "As a consequence, we can show that $\\lim _{t\\rightarrow \\infty }p(t)\\in \\lbrace 0,1\\rbrace $ , and to forbid mixed answers where tasks are “solvable” with some probability $p\\in (0,1)$ .", "Therefore, this provides us with a complete (deterministic) characterization of eventual solvability, namely, an input-free symmetry-breaking task $\\mathcal {O}$ is eventually solvable if $\\lim _{t\\rightarrow \\infty }p(t)=1$ .", "To sum up this part, given an input-free symmetry-breaking task, i.e., a task represented by a symmetric output complex $\\mathcal {O}$ , we say that a global state $\\sigma \\in \\mathcal {P}(t)$ solves $\\mathcal {O}$ if there exists a name-preserving simplicial map $\\delta : \\pi (\\sigma )\\rightarrow \\pi (\\tau )$ for some $\\tau \\in \\mathcal {O}$ , where $\\pi $ is the consistency projection.", "For any $t\\ge 1$ , let $\\mathcal {S}(t)$ be the set of all the global states $\\sigma \\in \\mathcal {P}(t)$ that solves the task $\\mathcal {O}$ , and let us define $\\Pr [\\mathcal {S}(t)] = \\sum _{\\sigma \\in \\mathcal {S}(t)}\\Pr [\\sigma ]$ .", "We say that $\\mathcal {O}$ is eventually solvable if and only if $\\lim _{t\\rightarrow \\infty } \\Pr [\\mathcal {S}(t)]=1$ ." ], [ "Application to Leader Election in Anonymous Models", "To demonstrate the interest and usability of the topological approach of randomized algorithms sketched in the previous section, we study the arguably most prominent symmetry-breaking task, namely leader election, in two important models for anonymous computing, that is, the blackboard model and the message-passing model with port-numbers.", "While in this paper we focus on leader election, we note that the vast majority of other symmetry-breaking tasks in anonymous message-passing networks are trivially solvable once a leader can be elected; see [10], for instance.Another reason to focus on leader election is that many other important tasks, such as reaching consensus or $k$ -set agreement, are deterministically solvable in the fault-free setting.", "See also Appendix .", "Moreover, we consider an exhaustive range of randomness source assignments to the nodes, from private randomness (each node has its own independent source of randomness) to shared randomness (all nodes get their random bits from the same source).", "Indeed, in many real-life situations, we observe dependencies and correlations between the different randomness sources, and this is in particular noticeable in distributed systems as the parties are all deterministic machines that obtain their randomness via pseudo-random generators, which might cause the randomness of some (or all) parties to be correlated or even identical.", "This is not just a theoretical whim: recent work showed that the same random SSH keys were generated by more than 250,000 “independent” devices [22] and that 1 out of 172 RSA-based certificates found online (i.e., about 450,000 certificates) have a random RSA key that shares a factor with the key of another “independent” certificate, allowing the complete factorization of these RSA keys and breaking the security of these certificates [19].", "Specifically, we model our setting as follows.", "The system is composed of $n$ parties, where all parties are connected to each other either via an anonymous broadcast channel (blackboard model) or via private point-to-point channels (message-passing model).", "In the latter model, each private channel of a party is locally identified by an index in $\\lbrace 1,\\dots ,n-1\\rbrace $ , called port-number [18].", "Other than that, the $n$ parties are identical and anonymous (they have no IDs).", "There are $k$ , $1\\le k\\le n$ , independent sources of randomness $\\mathbf {R}_1,\\ldots , \\mathbf {R}_k$ , where each randomness source generates an infinite sequence of i.i.d random bits, each bit being picked uniformly at random.", "Each party is connected to a single randomness source.", "However, different parties may be connected to the same source, hence, their randomness is completely dependent (i.e., identical).", "This modeling is simple, but it allows us to capture both the case where all parties have independent randomness as well as the case where all parties share the same randomness source, or any situation in between—where some parties are correlated and others are not.", "One can easily verify that the correlations between parties bring up interesting questions.", "In particular, under which assumption on the correlations between the parties is leader election possible?", "There are trivial answers under specific scenarios.", "For instance, if each party is connected to a private source of randomness, then leader election is eventually solvable, even in the weak blackboard model (e.g., whenever one party gets a bit 1 while all the other parties get bits 0).", "Instead, if all parties are connected to the same randomness source, then symmetry cannot be broken in the blackboard model, and leader election is impossible.", "But what about intermediate cases?", "And what about the message-passing model in which the port-numbers may assist in breaking symmetry?", "We show that, in the blackboard model, leader election is eventually solvable if and only if there exists a randomness source $\\mathbf {R}_i$ , $i\\in \\lbrace 1,\\dots ,k\\rbrace $ , that is connected to a single party (Theorem REF ).", "In the message-passing model, we show that whenever $n_i\\ge 1$ parties are connected to the randomness source $\\mathbf {R}_i$ , $i\\in \\lbrace 1,\\dots ,k\\rbrace $ , leader election is eventually solvable if and only if $\\gcd (n_1,n_2,\\ldots ,n_k)=1$ (Theorem REF ).", "The latter result must be read as follows.", "If $\\gcd (n_1,n_2,\\ldots ,n_k)=1$ , then leader election is eventually solvable for every assignment of the port-numbers to the channels, and if $\\gcd (n_1,n_2,\\ldots ,n_k)>1$ , then there is an assignment of the port-numbers to the channels such that leader election is not eventually solvable.", "The intuition behind these two theorems is somewhat immediate.", "In the blackboard model, if two processes (or more) possess the same randomness source, then their states will be identical throughout the computation, hence none of them can be elected a leader.", "On the other hand, if there is a single process with a unique source of randomness, then with probability 1 this process eventually arrives at a state distinct from all other processes.", "In the next round, every process will see that this has happened and elect the distinct process as a leader.", "In the message-passing model, the GCD condition guarantees that processes with the same randomness source $\\mathbf {R}_i$ can “match” themselves with processes that have a different randomness source $\\mathbf {R}_j$ .", "At the end of this step, any process in the smaller set will have a match from the other set.", "The key observation is that if we deactivate all the matched-processes that belong to the larger set, then the GCD condition still holds.", "Thus, we can repeat this matching process recursively, similar to Euclid's GCD algorithm [11], until we reach a set with a size of 1.", "That process will be the leader.", "One can verify that this idea cannot work when the GCD is not 1, since eventually one of the sets will have a size of 0, while the size of the other set will be a multiple of the GCD.", "Indeed, if the GCD is $g$ , we show that every process belongs to a set (of size $c\\cdot g$ for some $c\\in \\mathbb {N}$ ) such that all processes within the same set share the exact same state.", "While the above intuition could lead to a direct proof for leader election solvability, it is almost straightforward to derive some of the above insights by applying our topological framework.", "The above intuition might seem trivial in hindsight, but such insights might not be simple to come up with a priori.", "As an example, consider the conditions for the solvability of 2-leader election, i.e., obtaining exactly two leaders.", "The topological framework immediately leads to the correct characterization of this problem and many others.", "We encourage the reader to find a direct characterization in both the blackboard and message-passing models, and then compare it with the characterization obtained via the topological framework.", "In order to establish the above characterization, we apply the topological approach described in the previous sub-section.", "Leader election is represented by the output complex $\\mathcal {O}_{\\mathsf {LE}}$ defined as follows.", "A set $\\lbrace (i,x_i):i\\in \\lbrace 1,\\dots ,n\\rbrace \\rbrace $ is a facet of $\\mathcal {O}_{\\mathsf {LE}}$ if $\\lbrace x_1,\\dots ,x_n\\rbrace =\\lbrace 0,1\\rbrace $ , and there is a unique $i\\in \\lbrace 1,\\dots ,n\\rbrace $ such that $x_i=1$ .", "See Figure REF for an illustration of $\\mathcal {O}_{\\mathsf {LE}}$ and its consistency projection.", "First, we characterize the global states $\\sigma \\in \\mathcal {P}(t)$ that solve $\\mathcal {O}_{\\mathsf {LE}}$ , that is, the states for which there exists a name-preserving simplicial map $\\delta : \\pi (\\sigma )\\rightarrow \\pi (\\tau )$ for some $\\tau \\in \\mathcal {O}_{\\mathsf {LE}}$ , where $\\pi $ is the consistency projection.", "Then, given the set $\\mathcal {S}(t)$ of the global states $\\sigma \\in \\mathcal {P}(t)$ that solves leader election, we compute $\\Pr [\\mathcal {S}(t)\\mid \\alpha ] = \\sum _{\\sigma \\in \\mathcal {S}(t)}\\Pr [\\sigma \\mid \\alpha ]$ for every configuration $\\alpha $ of the connections between the parties and the randomness sources.", "Finally, we compute $\\lim _{t\\rightarrow \\infty } \\Pr [\\mathcal {S}(t)\\mid \\alpha ]$ for figuring out under which condition this limit is 1, which establishes eventual solvability.", "Computing the probability of a global state $\\sigma \\in \\mathcal {P}(t)$ to occur (i.e., computing $\\Pr [\\sigma \\mid \\alpha ]$ ) is, however, non-trivial.", "So, we introduce another complex, called the realization complex at time $t$ , denoted by $\\mathcal {R}(t)$ .", "The vertices of $\\mathcal {R}(t)$ are pairs $(i,x_i)$ where $i\\in \\lbrace 1,\\dots ,n\\rbrace $ , and $x_i\\in \\lbrace 0,1\\rbrace ^t$ .", "Each set of vertices $\\lbrace (i,x_i):i\\in I\\rbrace $ is a simplex of $\\mathcal {R}(t)$ , for every non-empty set $I\\subseteq \\lbrace 1,\\dots ,n\\rbrace $ .", "The interest of $\\mathcal {R}(t)$ is that given an assignment $\\alpha $ of the randomness sources to the nodes and given a facet $\\rho $ of $\\mathcal {R}(t)$ , it is easy to compute $\\Pr [\\rho \\mid \\alpha ]$ .", "Interestingly, it turns out that the facets of $\\mathcal {R}(t)$ are isomorphic to the facets of $\\mathcal {P}(t)$ .", "That is, we show that a specific realization at time $t$ of the randomness sources fully and uniquely defines the states of the parties at time $t$ in the protocol complex $\\mathcal {P}(t)$ .", "This makes the computation of $\\Pr [\\sigma \\mid \\alpha ]$ easier for each facet $\\sigma $ of $\\mathcal {P}(t)$ , by computing the probability $\\Pr [\\rho \\mid \\alpha ]$ of the corresponding facet $\\rho $ of $\\mathcal {R}(t)$ .", "Thanks to the isomorphism between the facets of $\\mathcal {P}(t)$ and $\\mathcal {R}(t)$ , we can focus on determining whether a given facet $\\rho $ of $\\mathcal {R}(t)$ solves leader election.", "For this purpose, we introduce a variant of the consistency projection, denoted by $\\tilde{\\pi }$ , that applies to facets of $\\mathcal {R}(t)$ .", "Then we study the existence of a name-preserving simplicial map $\\delta :\\tilde{\\pi }(\\rho )\\rightarrow \\pi (\\tau )$ for some facet $\\tau \\in \\mathcal {O}_{\\mathsf {LE}}$ for determining whether the facet $\\rho $ of $\\mathcal {R}(t)$ solves leader election or not.", "Since for any facet $\\tau \\in \\mathcal {O}_{\\mathsf {LE}}$ , $\\pi (\\tau )$ contains an isolated vertex, for any realization $\\rho $ that potentially solves leader election, $\\tilde{\\pi }(\\rho )$ must contain an isolated vertex as well.", "In the blackboard model, the dimension of the smallest facet in $\\tilde{\\pi }(\\rho )$ is $\\min \\lbrace n_1,n_2,\\ldots ,n_k\\rbrace -1$ , and thus the presence of an isolated node in $\\tilde{\\pi }(\\rho )$ requires that $\\min \\lbrace n_1,n_2,\\ldots ,n_k\\rbrace =1$ .", "In the message-passing model, we prove that there exists some way to assign port-numbers to the channels such that, for any facet $\\gamma \\in \\tilde{\\pi }(\\rho )$ with dimension $d$ , it holds that $\\gcd (n_1,n_2,\\ldots ,n_k) \\mid d + 1$ .", "Therefore, if $\\gcd (n_1,n_2,\\ldots ,n_k)>1$ , then there are no isolated nodes in $\\tilde{\\pi }(\\rho )$ , and the task cannot be solved.", "In order to prove the other direction, i.e., that if $\\gcd (n_1,n_2,\\ldots ,n_k)=1$ then leader election is eventually solvable, we describe an algorithm that imitates Euclid's algorithm [11] for computing the greatest common divisor of the dimensions of the facets in $\\tilde{\\pi }(\\rho )$ , until reaching an isolated vertex." ], [ "Related Work", "As mentioned above, the pioneering work [6], [16], [25] formulated distributed computations in the language of algebraic topology in order to show impossibility results in the presence of failures.", "A tremendous amount of subsequent work is described in the book of Herlihy, Kozlov, and Rajsbaum [15] (see also dozens of citations therein).", "We mention that the language of algebraic topology was found useful to analyze systems both in the message-passing model as well in the shared-memory model, and it can even be extended to capture non-benign faults, like Byzantine failures [23].", "Leader election has been extensively studied as an interesting special case of symmetry-breaking between nodes, usually in anonymous systems.", "Angluin [3] showed that no deterministic algorithm could elect a leader in anonymous networks (usually, in a ring) while it is possible to elect a leader non-deterministically or probabilistically.", "A long line of leader election algorithms, as well as lower bounds, were developed for certain special cases [1], [2], [4], [9], [17], [20], [24], [26].", "Mostly related to this paper is the work of Yamashita and Kameda [28], which fully characterize the solvability of deterministic leader election over general graphs in the message-passing model; their characterization follows from considering various types of symmetries in the graph.", "Boldi et al.", "[5] also give a full characterization of solvability for leader election in networks with and without port-numbers, using a method of graph-homomorphisms known as graph covering or fibrations.", "This was later extended by Chalopin et al.", "[7] to families of graphs.", "(Codenotti et al.", "[8] mention that leader election in $K_{m,n}$ is possible if and only if $\\gcd (m,n)=1$ ).", "In contrast to [5], [7], [28], which require a complicated analysis of the structure of the network, our characterization is much more straightforward and intuitive, and is based only on the greatest common divisor of the sizes of the subsets of parties connected to the same randomness source — however, this clean characterization applies only for the clique.", "There is plenty of other work on leader election in various models and settings.", "We surveyed above only the most relevant work to our model.", "No prior work considered the interesting case of symmetry-breaking in correlated randomness settings to the best of our knowledge.", "We stress again that the analysis of leader election is merely a single example of our framework and machinery for topological analysis of randomized distributed algorithms." ], [ "Communication model.", "We consider $n\\ge 1$ identical fault-free processing nodes with no identifiers (i.e., they are anonymous) running the same algorithm.", "The nodes perform computation and communication in lockstep, that is, we assume synchronous rounds.", "For $r\\ge 1$ , the $r$ -th round occurs between time $r-1$ and time $r$ .", "During each round every node can send a message to each other node, and can receive messages from the other nodes.", "The size of the messages is finite but unbounded.", "Without loss of generality, we can assume that each node sends its entire history to the other nodes at every round, which is a complete description of all the information accumulated by the node during the previous rounds, including its inputs, the content of the messages that were sent and received, when these messages were sent and received, etc.", "We consider two sub-models regarding the way each node communicates with the other nodes in the system.", "The blackboard model: There is a shared memory called blackboard, and each node can send information to the other nodes by appending a message to the blackboard.", "Every message written on the blackboard by a node at the beginning of a round can be seen by all the other nodes at the end of the round.", "However, there are no indications about which node is the origin of a message written on the board.", "Furthermore, the order in which the messages appear on the blackboard during a single round is arbitrary; without loss of generality we will assume all the messages written to the board in a single round appear on it in a lexicographic order.", "The message-passing model: The nodes are connected as a clique $K_n$ , and they communicate by passing messages through the edges of $K_n$ .", "A message sent by a node $u$ through its incident edge $e$ at the beginning of a round is received by the other extremity of $e$ at the end of the round.", "The $n-1$ edges incident to every node $u$ are labeled by $n-1$ distinct integers in $\\lbrace 1,\\ldots , n-1\\rbrace $ .", "The label given to an edge $e$ incident to node $u$ is called the port number of $e$ at $u$ .", "The port numbers are arbitrary, and there are no correlations between the two port numbers of an edge at its two extremities.", "Our results hold for the worst case assignment of port numbers, that is, they can be assumed to be assigned by an adversary.", "In both cases, there are no restrictions on the amount of information that can be transmitted during a round, that is, there are no restrictions on the size of the messages to be written on the blackboard or to be sent through the links of the network.", "In other words, we assume full information protocols." ], [ "Randomness.", "Given a positive natural number $m$ we denote by $[m]$ the set $\\lbrace 1,\\ldots ,m\\rbrace $ .", "We assume that the system has access to $k$ independent sources of randomness, denoted by $\\mathbf {R}_1,\\ldots ,\\mathbf {R}_k$ , for $k\\in [n]$ .", "During every round, each source $\\mathbf {R}_i$ , $i\\in [k]$ , generates a single bit whose value is chosen uniformly at random in $\\lbrace 0,1\\rbrace $ .", "The random variable equal to the infinite binary string generated by $\\mathbf {R}_i$ is denoted by $R_i$ .", "Each node is connected to one of the $k$ sources $\\mathbf {R}_i$ , $i\\in [k]$ , and it may be the case that several nodes are connected to the same source of randomness (this necessarily happens whenever $k<n$ ).", "The random variable equal to the infinite binary string received by node $i\\in [n]$ , is denoted by $X_i$ .", "At time $t$ , node $i$ has received a prefix of length $t$ of $X_i$ , i.e., a $t$ -bit string $x_i(t)\\in \\lbrace 0,1\\rbrace ^t$ .", "Throughout the paper, random variables are denoted by uppercase letters (e.g., $R_i, X_i$ ), and their realizations are denoted by lowercase letters (e.g., $r_i, x_i$ ).", "A random variable $Z$ at round $t$ is denoted by $Z(t)$ , and for a string $S$ , we let $S(t,\\dots ,t^{\\prime })$ denote the sub-string $S(t)\\, S(t+1) \\dots S(t^{\\prime })$ ; the same holds for realizations of random variables, etc." ], [ "Knowledge", "For $t\\ge 0$ , let $K_i(t)$ be the knowledge of node $i\\in [n]$ at time $t$ , defined recursively as follows.", "For every $i\\in [n]$ , $K_i(0) = v_i$ , where $v_i$ is the input value given to node $i$ .", "If there are no inputs, then $K_i(0) = \\bot $ .", "In the blackboard model, for $t\\ge 1$ , we set $K_i(t) = \\Big (K_i(t-1), X_i(t), \\big \\lbrace K_j(t-1) : j\\in [n]\\setminus \\lbrace i\\rbrace \\big \\rbrace \\Big ).$ where the knowledge $\\lbrace K_j(t-1) : j\\in [n]\\setminus \\lbrace i\\rbrace \\rbrace $ received from the other nodes is a multi-set; this multiset corresponds to the entire content of the blackboard, up to the order which is lexicographic by assumption.", "In the message-passing model, we set $K_i(t) = \\Big (K_i(t-1), X_i(t), \\big ( K_{\\pi _i(1)}(t-1),\\dots , K_{\\pi _i(n-1)}(t-1) \\big ) \\Big ),$ where $\\pi _i(j)\\in [n]$ denotes the node connected to node $i$ by the edge with port-number $j$ at $i$ .", "Note that node $i$ does not know $i$ , nor does it know $\\pi _i(j)$ for $j=1,\\dots ,n-1$ .", "Note also that, at time $t$ , a node knows the $t$ random bits it received from its source of randomness during the first $t$ rounds, but only the $t-1$ bits received by every other node from its source of randomness during the first $t-1$ rounds." ], [ "Topological Description of Randomized Symmetry-Breaking Distributed Algorithms", "In this section, we describe the topological framework that enables the analysis of distributed algorithms, and extends it to capture the analysis of randomized algorithms.", "In Section  we will later show how to actually use this framework for analyzing the solvability of leader election as a function of the randomness given to each node, for both blackboard and message-passing models.", "The reader is referred to Appendix  for some basic topological definitions.", "Further information can be found, e.g., in [15]." ], [ "Topological Setting", "We recall the notion of tasks, and of solvability of tasks in fixed time, within the topological framework." ], [ "Tasks.", "A task is described by a triple $\\Pi =(\\mathcal {I},\\mathcal {O},\\Delta )$ , where $\\mathcal {I}$ is the input complex, $\\mathcal {O}$ is the output complex, and $\\Delta :\\mathcal {I}\\rightarrow 2^{\\mathcal {O}}$ is the input-output specification of the task (see, e.g., [15]).", "All the complexes are “colored”, in the sense that their vertices have the form $(i,x)$ with color $i\\in [n]$ , for some value $x$ , and their simplices include vertices with different colors.", "We rather refer to the color $i$ of a vertex $(i,x)$ as its name, i.e., the node named $i$ holds value $x$ ; we denote $\\mathsf {name}((i,x))=i$ , which can be extended to a set of nodes in a straightforward manner.", "In this paper, we focus on input-free symmetry breaking tasks, so $\\mathcal {I}$ is the trivial complex with a single facet $\\lbrace (i,\\bot ):i\\in [n]\\rbrace $ .", "For input-free tasks, the input-output specification is trivial, that is, given any input simplex $\\sigma \\in \\mathcal {I}$ , this simplex is mapped to all simplices of $\\mathcal {O}$ with same set of names as $\\sigma $ .", "A symmetry-breaking task is thus simply defined by its output complex $\\mathcal {O}$ .", "We only require that the output complex must be stable by permutation of the colors of the processes.", "That is, if $\\lbrace (i,v_i):i\\in I\\rbrace $ is a simplex of $\\mathcal {O}$ , with $\\varnothing \\ne I\\subseteq [n]$ , then, for every permutation $\\pi :I\\rightarrow I$ , it must be the case that $\\lbrace (i,v_{\\pi (i)}):i\\in I\\rbrace \\in \\mathcal {O}$ .", "For instance, the output complex $\\mathcal {O}_{\\mathsf {LE}}$ for leader election (LE) has $n$ facets $\\tau _i=\\lbrace (1,0),\\dots ,(i-1,0),(i,1),(i+1,0),\\dots ,(n,0)\\rbrace ,$ for $i=1,\\dots ,n$ .", "That is, $\\tau _i$ is the legal output state in which node $i$ is elected, and the $n-1$ other nodes are defeated.", "Note that $\\mathcal {O}_{\\mathsf {LE}}$ is symmetric." ], [ "Communication and randomness configuration.", "The exchanges of information between the nodes occurring throughout the execution is captured by the protocol complexes.", "The vertices of the protocol complex at time $t$ , denoted by $\\mathcal {P}(t)$ , are pairs $(i,K_i(t))$ , $i\\in [n]$ , where $K_i(t)$ denotes the knowledge acquired by node $i$ at time $t$ .", "See Figure REF for a demonstration of $\\mathcal {P}(t)$ for $t=0,1,2$ for a computation with two parties.", "Figure: The evolution of a 2-party algorithm for time steps t=0,1,2t=0,1,2.The knowledge of each party at a given time, written next to the respective node, consists of the party's previous knowledge, the random bit it has achieved in that round, and the knowledge of the other party sent to it in the previous round.", "In this figure, k 0 =(⊥,0,(⊥))k_0 = (\\bot ,0,(\\bot )), k 1 =(⊥,1,(⊥))k_1 = (\\bot ,1,(\\bot )).", "Each edge is a possible state of the system, whose probability is determined by the specific randomness configuration α∈𝒜\\alpha \\in \\mathcal {A} in a given execution.", "An edge at time tt (i.e., a facet of σ∈𝒫(t)\\sigma \\in \\mathcal {P}(t)) evolves into 4 possible facets (edges) of 𝒫(t+1)\\mathcal {P}(t+1).", "These correspond to the 4 possible values of the random bits obtained by the two parties at time t+1t+1.The knowledge acquired by the nodes however depends the randomness they obtain and on the way the $k$  randomness sources are assigned to the $n$  nodes.", "We also use a complex to formalize all possible assignments.", "This complex, denoted by $\\mathcal {A}$ (for “assignment”) is the pure $(n-1)$ -dimensional complex whose facets are of the form $\\alpha =\\lbrace (1,j_1), \\ldots , (n,j_n)\\rbrace ,$ where $\\bigcup _{i=1}^n \\lbrace j_i\\rbrace =[k]$ for some $k\\in [n]$ ; that is, without loss of generality we rename the $k$ different sources to be contiguous in $\\lbrace 1,\\ldots , k\\rbrace $ .", "For every $i\\in [n]$ , a pair $(i,j)$ means that node $i$ is connected to the randomness source $\\mathbf {R}_{j}$ .", "Every facet $\\alpha $ of $\\mathcal {A}$ is called a randomness-configuration, that is, a configuration determines which node is connected to which randomness source, for all the nodes.", "For a given configuration $\\alpha $ we denote by $k=k(\\alpha )$ the number of different sources actually connected to the systems.", "Note that our restriction on $\\mathcal {A}$ 's facets means that the $k$ sources actually connected to the system are exactly $\\mathbf {R}_1,\\ldots ,\\mathbf {R}_k$ .", "A set $\\lbrace (i,K_i(t)):i\\in [n]\\rbrace $ forms a facet of $\\mathcal {P}(t)$ whenever there exists a configuration $\\alpha \\in \\mathcal {A}$ such that, with non-zero probability, each node $i$ acquires knowledge $K_i(t)$ , $i=1,\\dots ,n$ , after $t$ rounds of communication." ], [ "Solvability in fixed time.", "Recall that a map $\\delta $ between the vertex sets of two complexes is simplicial if it preserves simplices.", "A simplicial map between two chromatic complexes is name-preserving if it preserves the names of the vertices (i.e., for every vertex $(i,x)$ , $\\delta (i,x)=(i,y)$ for some $y$ that may depend on $i$ and $x$ ), and it is name-independent if it is oblivious to the names (i.e., if $\\delta (i,x)=(i,y)$ , then $\\delta (j,x)=(j,y)$ for every $j$ , that is, $y$ depends solely on $x$ ).", "In this work all our complexes are chromatic and all the maps are name-preserving.", "In the standard topological setting, a task $(\\mathcal {I},\\mathcal {O},\\Delta )$ is solvable in $t$ rounds if there exists a name-preserving and name-independent simplicial map $\\delta :\\mathcal {P}(t) \\rightarrow \\mathcal {O}.$ This notion of solvability is not appropriate to our randomized setting, for two reasons.", "First, we want to discuss solvability as a function of the randomness-configuration $\\alpha \\in \\mathcal {A}$ of the randomness sources.", "Second, and more importantly, there might be no $t$ , even arbitrarily large, enabling such a simplicial map $\\delta :\\mathcal {P}(t) \\rightarrow \\mathcal {O}$ to exist.", "This holds even when the task is eventually solvable under the configuration $\\alpha $ .", "To better illustrate this point, assume two processes, each with its private and independent source of randomness.", "There is no $t$ for which one can guarantee that the two processes have received two different bits at some round $r\\in \\lbrace 1,\\dots ,t\\rbrace $ .", "Yet, leader election is almost surely solvable in this context as, eventually, the two processes will receive two different bits at some round." ], [ "Eventual Solvability", "A global state of the system at time $t$ is a facet $\\sigma $ of $\\mathcal {P}(t)$ .", "Definition 3.1 A global state $\\sigma \\in \\mathcal {P}(t)$ solves a symmetry-breaking task $\\mathcal {O}$ if there exists $\\tau \\in \\mathcal {O}$ and a name-preserving and name-independent simplicial map $\\delta : \\sigma \\rightarrow \\tau .$ This definition of solvability for a facet of $\\mathcal {P}(t)$ is motivated by the following observation.", "Let us assume that, for a facet $\\sigma =\\lbrace (i,K_i(t)):i\\in [n]\\rbrace $ of $\\mathcal {P}(t)$ , there is a name-preserving and name-independent simplicial map $\\delta : \\sigma \\rightarrow \\tau $ .", "This map can be written as $\\delta (i,K_i(t))=(i,f(K_i(t))$ , $i=1,\\dots ,n$ , for some function $f$ of the knowledge.", "Since the output complex $\\mathcal {O}$ is symmetric, the map $\\delta $ yields the existence of a simplicial map $\\delta ^{\\prime }: \\sigma ^{\\prime }\\rightarrow \\tau ^{\\prime }$ for every simplex $\\sigma ^{\\prime }=\\lbrace (i,K_{\\pi (i)}(t)):i\\in [n]\\rbrace $ of $\\mathcal {P}(t)$ where $\\pi :[n]\\rightarrow [n]$ is a permutation, letting $\\tau ^{\\prime }=\\lbrace (i,v_{\\pi (i)}) : (i,v_i)\\in \\tau \\rbrace \\in \\mathcal {O}$ and defining $\\delta ^{\\prime }(i,K_{\\pi (i)}(t))=(i,f(K_{\\pi (i)}(t)))$ .", "From this observation, we derive an algorithm solving $\\mathcal {O}$ whenever the global state of the system is of the form $\\sigma ^{\\prime }=\\lbrace (i,K_{\\pi (i)}(t)):i\\in [n]\\rbrace $ for some permutation $\\pi :[n]\\rightarrow [n]$ .", "Indeed, at time $t$ , the knowledge accumulated by nodes during the first $t$ rounds results in some global state $\\sigma ^{\\prime }\\in \\mathcal {P}(t)$ of the system.", "Each node may not be aware of $\\sigma ^{\\prime }$ as its individual knowledge may also be compatible with other global states.", "Nevertheless, after one more round, the nodes receives the knowledge of the other nodes in $\\sigma ^{\\prime }$ .", "This enables each node to reconstruct $\\sigma ^{\\prime }$ up to a permutation of the names of the other nodes.", "By applying $f$ on its knowledge, every node can then compute its output such that the collection of all outputs truly solves the task at time $t+1$ whenever the processes were in global state $\\sigma ^{\\prime }$ at time $t$ .", "Given an assignment $\\alpha \\in \\mathcal {A}$ of the randomness sources to the nodes, every global state $\\sigma \\in \\mathcal {P}(t)$ has some probability to occur at a given time.", "One can thus compute the probability of solving the task at time $t$ given $\\alpha $ as $\\Pr [\\mbox{$\\mathcal {P}(t)$ solves $\\mathcal {O}$} \\mid \\alpha ] = \\sum _{\\mbox{\\small $\\sigma $ solves $\\mathcal {O}$}} \\Pr [\\sigma \\mid \\alpha ].$ Observe that whenever $\\sigma \\in \\mathcal {P}(t)$ solves $\\mathcal {O}$ via some $\\delta _{\\sigma }:\\sigma \\rightarrow \\tau $ , every global state $\\sigma ^{\\prime }\\in \\mathcal {P}(t^{\\prime })$ for $t^{\\prime }\\ge t$ that results from $\\sigma $ after $t^{\\prime }-t$ additional rounds also solves $\\mathcal {O}$ .", "This is simply because the knowledge is cumulative, and one can discard all the additional information obtained by the nodes during the $t^{\\prime }-t$ additional rounds for defining $\\delta _{\\sigma ^{\\prime }}:\\sigma ^{\\prime }\\rightarrow \\tau $ using $\\delta _{\\sigma }$ .", "More importantly, the following holds.", "Lemma 3.2 For every input-free symmetry-breaking task $\\mathcal {O}$ , and every randomness-configuration $\\alpha \\in \\mathcal {A}$ , $\\lim _{t\\rightarrow \\infty } \\Pr [\\mbox{$\\mathcal {P}(t)$ solves $\\mathcal {O}$} \\mid \\alpha ]\\in \\lbrace 0,1\\rbrace .$ Let us assume that $\\Pr [\\mathcal {P}(t) \\; \\mbox{solves} \\; \\mathcal {O}\\mid \\alpha ]>0$ .", "Therefore, there exists a global state $\\sigma \\in \\mathcal {P}(t)$ that solves $\\mathcal {O}$ , where $\\Pr [\\sigma \\mid \\alpha ]>0$ .", "This means that there exists a set of nodes' knowledge $K = \\lbrace K_i(t)\\rbrace _{i\\in [n]}$ that yields a solution to the task.", "Note that in an input-free task, knowledge (at time $t$ ) stems only from the randomness and messages sent by time $t$ .", "That is, there exists $k$ realizations of randomness, each of length $t$ , that induce the set $ K$ , and these realizations have non-zero probability to occur, given $\\alpha $ .", "Denote these realizations by $R$ .", "For every $s\\ge 0$ , let $E_s$ be the event “$R$ occurred during rounds $s+1$ to $s+t$ ”.", "Since knowledge is cumulative, the occurrence of $E_s$ implies that, at time $s+t$ , the nodes hold a knowledge $\\lbrace K_i(s+t)\\rbrace _{i\\in [n]}$ where, for every $i\\in [n]$ , $K_i(t)$ is included in $K_i(s+t)$ .", "Thus, for any $s\\ge 0$ , if $E_s$ holds, the system reaches a global state that solves the task.", "Furthermore, note that two events $E_s$ and $E_{s^{\\prime }}$ are independent whenever $|s-s^{\\prime }|>t$ , since our sources are i.i.d across time.", "Recall that, given an infinite sequence $(X_i)_{i\\ge 1}$ of independent random variables, a tail event for $(X_i)_{i\\ge 1}$ is an event based on the realization of the $X_i$ 's, $i\\ge 1$ , which is probabilistically independent of any finite subset of $\\lbrace X_i:i\\ge 1\\rbrace $ .", "Kolmogorov's zero–one law [13] states that, for any tail event $E$ over $(X_i)_{i\\ge 1}$ , either [E]=0,   or   [E]=1.", "Let $E=\\bigcup _{s=0}^\\infty E_{s t}.$ We have $\\Pr [E\\mid \\alpha ] \\le \\lim _{t\\rightarrow \\infty } \\Pr [\\mbox{$\\mathcal {P}(t)$ solves $\\mathcal {O}$}\\mid \\alpha ].$ Moreover, $E$ is a tail event, and therefore its probability is either 0 or 1.", "Since $\\Pr [E_0\\mid \\alpha ]>0$ , it follows that $\\Pr [E\\mid \\alpha ]=1$ .", "We conclude that if $\\Pr [\\mathcal {P}(t)$ solves $\\mathcal {O}\\mid \\alpha ]>0$ , then $\\lim _{t\\rightarrow \\infty } \\Pr [\\mbox{$\\mathcal {P}(t)$ solves $\\mathcal {O}\\mid \\alpha $}]=1,$ as claimed.", "This result motivates the following definition.", "Definition 3.3 A task $\\mathcal {O}$ is eventually solvable given the randomness-configuration $\\alpha \\in \\mathcal {A}$ if and only if $\\lim _{t\\rightarrow \\infty } \\Pr [\\mbox{$\\mathcal {P}(t)$ solves $\\mathcal {O}$} \\mid \\alpha ]=1.$" ], [ "Realization and Consistency Complexes", "We now introduce new complexes, which are essentially reformulations of the protocol and output complexes $\\mathcal {P}(t)$ and $\\mathcal {O}$ , more suitable for the analysis of input-free symmetry-breaking tasks." ], [ "Realization complex.", "In the setting of this paper, namely the anonymous blackboard and message-passing models, the protocol complex is entirely determined by the values of the random bits produced by the $k$ sources of randomness.", "The realization complex at time $t$ is denoted by $\\mathcal {R}(t)$ , for any $t\\ge 1$ .", "The vertices of $\\mathcal {R}(t)$ are pairs $(i,x_i)$ where $x_i\\in \\lbrace 0,1\\rbrace ^t$ is a binary string of length $t$ (the random bits received by node $i$ during the first $t$ rounds).", "Specifically, $\\mathcal {R}(t)$ has vertex-set $V(\\mathcal {R}(t)) = \\lbrace (i,x_i) : i\\in [n] , x_i \\in \\lbrace 0,1\\rbrace ^t\\rbrace .$ For $I\\subseteq [n]$ , a set $\\rho = \\lbrace (i,x_i) : i\\in I\\rbrace \\subseteq V(\\mathcal {R}(t))$ is a simplex of $\\mathcal {R}(t)$ if there exists a randomness-configuration $\\alpha \\in \\mathcal {A}$ such that, with non-zero probability, each node $i\\in I$ may receive the random bit-string $x_i$ .", "See Figure REF for an illustration with three parties.", "Figure: A demonstration of ℛ(0)\\mathcal {R}(0) and ℛ(1)\\mathcal {R}(1) in a system with 3 processes.", "Each facet represents a possible state of the system described via the randomness received by the parties up to that time.", "The notation (w,b,r)(w,b,r) with w,b,r∈{0,1}w,b,r\\in \\lbrace 0,1\\rbrace describes the randomness of the white, black, and red nodes accordingly, i.e., the simplex {(1,w),(2,b),(3,r)}\\lbrace (1,w), (2,b), (3,r)\\rbrace ; ⊥\\bot is the empty string.We observe relations between $\\mathcal {P}(t)$ and $\\mathcal {R}(t)$ that allow us to analyze the algorithm via the more intuitive $\\mathcal {R}(t)$ .", "For any time $t\\ge 0$ there exists a simplicial map $h: \\mathcal {P}(t) \\rightarrow \\mathcal {R}(t)$ that takes each vertex $(i,K_i)\\in \\mathcal {P}(t)$ to $(i,x_i)\\in \\mathcal {R}(t)$ , where $x_i\\in \\lbrace 0,1\\rbrace ^t$ is the randomness received by party $i$ according to $K_i(t)$ ; recall that $K_i(t)$ indeed contains $x_i(t)$ by its definition (Eqs.", "(REF ) and (REF ) for the blackboard and message-passing models, respectively).", "Note that $K_i(t)$ contains also randomness received by all the other parties up to round $t-1$ , hence, $h$ maps multiple vertices to $(i,x_i)$ .", "Note that $h$ is name-preserving by construction.", "We observe that the simplicial map $h$ induces an isomorphism between facets of $\\mathcal {P}(t)$ and facets of $\\mathcal {R}(t)$ .", "Indeed, a facet $\\lbrace (i,K_i(t)):i\\in [n]\\rbrace \\in \\mathcal {P}(t)$ uniquely determines the randomness $(x_1,\\ldots , x_n)$ received by all parties up to round $t$ , and is mapped to the facet $\\lbrace (i,x_i) : i\\in [n]\\rbrace $ of $\\mathcal {R}(t)$ .", "Similarly, if one determines the randomness by time $t$ , $(x_1,\\ldots , x_n)$ , this uniquely defines the knowledge every party holds up to time $t$ , since each $K_i(t)$ consists of $x_i$ and $K_j(t-1)$ for $j\\in [n]$ , and these, by induction, are deterministic function of $(x_1,\\ldots , x_n)$ .", "With a slight abuse of notation we will commonly refer to $h$ as an isomorphism, implicitly restricting it to act on facets." ], [ "Consistency complexes.", "We now consider two general “consistency-projections” $\\pi $ and $\\tilde{\\pi }$ that apply on chromatic complexes.", "Let $\\mathcal {K}$ be a pure chromatic complex of dimension $n-1$ , that is, a complex whose vertices are pairs of the form $(i,v)$ , with $i\\in [n]$ , and $v$ a value.", "Let $\\sigma =\\lbrace (i,v_i): i\\in [n]\\rbrace $ be a facet of $\\mathcal {K}$ .", "We define the complex $\\pi (\\sigma )$ as the complex on vertex-set $\\lbrace (i,v_i) : i\\in [n]\\rbrace $ such that, for every non-empty $I\\subseteq [n]$ , $\\lbrace (i,v_i) : i\\in I\\rbrace \\in \\pi (\\sigma ) \\iff \\forall (i,j)\\in I\\times I, v_i=v_j.$ The projection $\\pi $ applied simultaneously to all the facets of $\\mathcal {K}$ results in the complex $\\pi (\\mathcal {K})=\\bigcup _{\\sigma \\in \\mathcal {K}}\\pi (\\sigma ),$ where the union is taken on the facets of $\\mathcal {K}$ .", "We note that $\\pi (\\mathcal {K})$ is a subcomplex of $\\mathcal {K}$ .", "As a simple illustrative example, in the case of leader election (LE), the complex $\\pi (\\mathcal {O}_{\\mathsf {LE}})$ has facets {(i,1)}    and    {(j,0):j[n]{i}} for every $i\\in [n]$ .", "See also Figure REF .", "Figure: 𝒪 𝖫𝖤 \\mathcal {O}_{\\mathsf {LE}} and π(𝒪 𝖫𝖤 )\\pi (\\mathcal {O}_{\\mathsf {LE}}).", "The facet τ 1 ∈𝒪 𝖫𝖤 \\tau _1\\in \\mathcal {O}_{\\mathsf {LE}} is mapped to the subcomplex π(τ 1 )⊆π(𝒪 𝖫𝖤 )\\pi (\\tau _1)\\subseteq \\pi (\\mathcal {O}_{\\mathsf {LE}}) that contains the edge {(2,0),(3,0)}\\lbrace (2,0),(3,0)\\rbrace and the isolated node {(1,1)}.\\lbrace (1,1) \\rbrace .The second consistency-projection, $\\tilde{\\pi }$ , is more specific and applies only to the realization complexes $\\mathcal {R}(t)$ , for $t\\ge 1$ .", "It is not using equality between values, but an equivalence relation between the vertices of $\\mathcal {R}(t)$ , defined as follows.", "Given specific realizations $x_i \\in \\lbrace 0,1\\rbrace ^t$ for the randomness of party $i\\in [n]$ at time $t$ , we say that nodes $i$ and $j$ are consistent at time $t$ , denoted by $i \\overset{t}{\\sim } j$ if $K_{i}(t) = K_{j}(t)$ .", "Note that once the randomness obtained by the parties up to round $t$ , $x_1,\\ldots , x_n\\in \\lbrace 0,1\\rbrace ^t$ , are fixed, the event $K_{i}(t) = K_{j}(t)$ is deterministic (it has probability either 0 or 1).", "In the blackboard model, this equality depends solely on the random bits received by nodes $i$ and $j$ during the first $t$ rounds.", "However, in the message-passing model, this equality also depends on the actual assignment of parties' port numbers.", "Also, it is worth observing that once $K_{i}(t) \\ne K_{j}(t)$ in a specific instance of randomness, the two nodes $i$ and $j$ become inconsistent for the rest of the execution.", "However, they become aware of this fact only at the next round, where knowledge is exchanged (in both the blackboard and the message-passing model)[12].", "Let $\\rho = \\lbrace (i,x_i) : i\\in [n]\\rbrace $ be a facet of $\\mathcal {R}(t)$ .", "We define the complex $\\tilde{\\pi }(\\rho )$ as the complex on vertex-set $V(\\rho )=\\lbrace (i,x_i) : i\\in [n]\\rbrace $ such that, for every non-empty $I\\in [n]$ , $\\lbrace (i,x_i) : i\\in I\\rbrace \\in \\tilde{\\pi }(\\rho ) \\iff \\forall (i,j)\\in I\\times I, \\ i\\overset{t}{\\sim } j.$ The consistency complex captures all the possible relations of consistency for all possible generations of random strings.", "The consistency-projection $\\tilde{\\pi }$ applied simultaneously to all the facets of $\\mathcal {R}(t)$ results in the complex $\\tilde{\\pi }(\\mathcal {R}(t))=\\bigcup _{\\rho \\in \\mathcal {R}(t)}\\tilde{\\pi }(\\rho ),$ where the union is taken on the facets of $\\mathcal {R}(t)$ .", "Note that $\\tilde{\\pi }(\\mathcal {R}(t))$ is a subcomplex of $\\mathcal {R}(t)$ ; its topological structure will be vital for our analysis." ], [ "Randomized Solvability of Tasks Revisited", "A facet of $\\mathcal {R}(t)$ is called a realization of the system at time $t$ .", "By definition, there are $2^{nt}$ different realizations at time $t$ .", "Also, since each source of randomness generates a single bit uniformly at random at each round, we have $\\Pr [R_i(1,\\dots ,t) = x]=2^{-t}$ for every $x \\in \\lbrace 0,1\\rbrace ^t$ .", "It directly follows that the probability that a node $i\\in [n]$ receives random string $x\\in \\lbrace 0,1\\rbrace ^t$ during the first $t$  rounds is $2^{-t}$ , regardless to which randomness source node $i$ is connected.", "However, there might be correlations between different nodes, whenever they are connected to the same randomness source (an information that is not given to the nodes a priori).", "Let $\\rho =\\lbrace (1,x_1),\\ldots (n,x_n) \\rbrace \\in \\mathcal {R}(t)$ be a realization of the system at time $t$ , and let $\\alpha \\in \\mathcal {A}$ be a randomness-configuration of the system.", "We have $\\Pr [\\rho \\mid \\alpha ]=\\Pr \\Big [\\bigwedge _{(i,j)\\in \\alpha } R_j(1,\\dots ,t)= x_i\\Big ].$ We introduce a novel definition for solvability.", "This definition will be shown to be equivalent to the definition using facets of $\\mathcal {P}(t)$ .", "Definition 3.4 A realization $\\rho \\in \\mathcal {R}(t)$ solves a symmetry-breaking task $\\mathcal {O}$ if there exists $\\tau \\in \\mathcal {O}$ and a name-preserving simplicial map $\\delta : \\tilde{\\pi }(\\rho ) \\rightarrow \\pi (\\tau ).$ Note that, in this definition, the map $\\delta $ is not asked to be name-independent, since this property will be provided by the structure the projections impose.", "Since every realization $\\rho \\in \\mathcal {R}(t)$ has some probability to occur at a given time, one can compute the probability of solving the task at time $t$ by summing up the probabilities of the realizations that solve the task at time $t$ .", "For any $t\\ge 1$ , let $\\mathcal {S}(t)$ be the set of all the realizations of the system after $t$ rounds that solves the task $\\mathcal {O}$ , and let us define the probability of $\\mathcal {S}(t)$ given some randomness-configuration $\\alpha \\in \\mathcal {A}$ as the sum of the probability of its facets, that is, $\\Pr \\left[\\mathcal {S}(t) \\mid \\alpha \\right] = \\sum _{\\sigma \\in \\mathcal {S}(t)}\\Pr \\left[\\sigma \\mid \\alpha \\right].$ Again, by Kolmogorov's zero–one law, the limit when $t$ goes to infinity of the probability of $\\mathcal {S}(t)$ is equal to 0 or 1.", "Moreover, this new notion of solvability is equivalent to the (algorithmic) notion of solvability of Definition REF .", "Lemma 3.5 An input-free symmetry-breaking task $\\mathcal {O}$ is eventually solvable given a randomness-configuration $\\alpha \\in \\mathcal {A}$ if and only if $\\lim _{t\\rightarrow \\infty } \\Pr \\left[\\mathcal {S}(t) \\mid \\alpha \\right]=1,$ where $\\mathcal {S}(t)$ is the set of all the realization $\\sigma \\in \\mathcal {R}(t)$ that solve $\\mathcal {O}$ at time $t$ .", "First, assume $\\sigma \\in \\mathcal {P}(t)$ solves $\\mathcal {O}$ .", "We show that $h(\\sigma )\\in \\mathcal {R}(t)$ solves $\\mathcal {O}$ for $h: \\mathcal {P}(t) \\rightarrow \\mathcal {R}(t)$ the name-preserving simplicial map defined in Section REF .", "Fix $t$ and $\\sigma \\in \\mathcal {P}(t)$ that solves the task, and set $\\rho = h(\\sigma )$ .", "According to Definition REF , there is a name-preserving name-independent simplicial map $\\delta :\\sigma \\rightarrow \\tau $ .", "where $\\tau =\\delta (\\sigma )$ is a facet of $\\mathcal {O}$ .", "Given $\\sigma $ and $\\rho $ , we can define a name-preserving simplicial map (in fact, an isomorphism) $\\tilde{h} :\\rho \\rightarrow \\sigma $ (being viewed as complexes) that for any $i\\in [n]$ takes $(i,x_i)\\in \\rho $ to $(i,K_i)\\in \\sigma $ ; note that $\\tilde{h}$ is the uniqe name-preserving simplicial map between $\\rho $ and $\\sigma $ .", "We claim that $\\lambda \\triangleq \\delta \\circ \\tilde{h}=\\delta (\\tilde{h}(\\cdot ))$ is a name-preserving simplicial map $\\lambda : \\tilde{\\pi }(\\rho )\\rightarrow \\pi (\\tau )$ .", "Indeed, (1) $\\lambda $  is name preserving: this follows immediately since $\\tilde{h},\\delta $ are both name-preserving.", "(2) $\\lambda $  preserves simplices: fix a realization $\\rho =\\lbrace (i,x_i) : i\\in [n]\\rbrace \\in \\mathcal {R}(t)$ and let $K_i(t)$ be the knowledge of party $i$ at time $t$ given the realization $\\rho $ .", "Then $\\sigma =\\tilde{h}(\\rho )=\\lbrace (i,K_i(t)): i \\in [n]\\rbrace $ ; in particular $\\tilde{h}((i,x_i))=(i,K_i(t))$ , since $\\tilde{h}$  preserves names.", "Let $\\rho ^{\\prime }$ be a facet in $\\tilde{\\pi }(\\rho )$ .", "By the definition of $\\tilde{\\pi }$ , for any two vertices ${(i,x_i),(j,x_j)}\\in \\rho ^{\\prime }$ it holds that $i\\overset{t}{\\sim } j$ .", "Thus, by the definition of the $\\sim $ relation, $K_i(t)=K_j(t)$ and thus $\\lbrace (i,K_i(t)),(j,K_j(t))\\rbrace \\in \\pi (\\sigma )$ .", "Since the consistency relation $\\sim $ is transitive, the same argument holds for any subset of vertices in $\\rho ^{\\prime }$ .", "Hence, $\\lambda (\\rho ^{\\prime })$ is a simplex in $\\pi (\\sigma )$ .", "We conclude that, if $\\Pr [\\sigma \\mid \\alpha ]>0$ , and $\\sigma $ solves $\\mathcal {O}$ by Definition REF via the map $\\delta $ , then for $\\rho =h(\\sigma )$ and $\\tau =\\delta (\\sigma )$ we have $\\Pr [\\rho \\mid \\alpha ]>0$ , and $\\rho $ solves $\\mathcal {O}$ by Definition REF via the map $\\lambda : \\tilde{\\pi }(\\rho )\\rightarrow \\pi (\\tau )$ .", "Conversely, assume that $\\rho \\in \\mathcal {R}(t)$ solves $\\mathcal {O}$ .", "We show that the facet $h^{-1}(\\rho ) \\in \\mathcal {P}(t)$ solves $\\mathcal {O}$ for $h: \\mathcal {P}(t) \\rightarrow \\mathcal {R}(t)$ the name-preserving simplicial map from Section REF (recall that $h$ induces an isomorphism on facets, thus it has a unique inverse for $\\rho $ ).", "Let $\\tau \\in \\mathcal {O}$ and $\\delta :\\tilde{\\pi }(\\rho )\\rightarrow \\pi (\\tau )$ be given for $\\rho $ by Definition REF .", "Let $\\sigma =h^{-1}(\\rho )$ be given by the isomorphism.", "We claim that $\\sigma $ solves $\\mathcal {O}$ via the map $\\lambda = \\delta \\circ h$ .", "First, note that $\\lambda :\\sigma \\rightarrow \\tau $ is a name-preserving simplicial map, which follows since $h$ and $\\delta $ are both name-preserving simplicial maps.", "This also implies that $\\lambda (\\sigma )=\\delta (h((\\sigma ))=\\tau $ is a facet of $\\mathcal {O}$ .", "Next, we argue that $\\lambda $ is name-independent, namely, that any two parties with the same knowledge give the same output.", "Let $(i,K_i(t)),(j_,K_j(t))$ be two vertices in $\\sigma $ , such that $K_i(t)=K_j(t)$ .", "By definition, $i\\overset{t}{\\sim } j$ .", "If so, then $\\lbrace (i,x_i),(j,x_j)\\rbrace $ is a simplex in $\\tilde{\\pi }(\\rho )$ , where $(i,x_i)=h((i,K_i(t)))$ and $(j,x_j)=h( (j,K_j(t)) )$ .", "This simplex must be mapped by the simplicial map $\\delta $ to some simplex $\\lbrace (i,v),(j,v)\\rbrace \\in \\pi (\\tau )$ (recall that simplices in $\\pi (\\tau )$ consist of nodes with similar outputs, Eq.", "(REF )).", "Since the argument holds for arbitrary two nodes, it easily extends to any subset of nodes in $\\sigma $ that have identical knowledge, which proves that $\\lambda $ is a name-independent.", "In Figure REF we illustrate the relations between the different complexes of this work.", "Figure: A summary of our topological complexes and the relations between them." ], [ "Solvability of Leader Election via Topology", "In the remainder of this paper we will consider the task of leader election, $\\mathcal {O}_\\mathsf {LE}$ .", "In this section we will discuss the conditions on randomness-configurations $\\alpha \\in \\mathcal {A}$ that make $\\mathcal {O}_\\mathsf {LE}$ eventually solvable.", "We begin in Section REF with the blackboard model.", "In Section REF we will consider the message-passing model.", "Recall that the leader election task is defined by $\\mathsf {LE}=(\\mathcal {I},\\mathcal {O}_\\mathsf {LE},\\Delta )$ with $\\mathcal {I}= \\lbrace (i,\\bot ):i\\in [n]\\rbrace $ and $\\Delta :\\mathcal {I}\\rightarrow 2^{\\mathcal {O}_\\mathsf {LE}}$ that maps the single facet of $\\mathcal {I}$ to the entire complex $2^{\\mathcal {O}_\\mathsf {LE}}$ .", "Further recall that $\\mathcal {O}_\\mathsf {LE}$ has facets $\\tau _i=\\lbrace (1,0),\\dots ,(i-1,0),(i,1),(i+1,0),\\dots ,(n,0)\\rbrace ,$ for any $i\\in [n]$ , and thus $\\pi (\\mathcal {O}_\\mathsf {LE}) = \\bigcup _{\\tau \\in \\mathcal {O}}\\pi (\\tau )$ has facets $\\lbrace (i,1)\\rbrace \\quad \\text{and} \\quad \\lbrace (j,0):j\\in [n]\\setminus \\lbrace i\\rbrace \\rbrace $ for any $i\\in [n]$ .", "To ease readability, we will denote $\\mathcal {O}_\\mathsf {LE}$ simply by $\\mathcal {O}$ from this point and on, as the task is clear from context.", "Intuitively, leader election is eventually solvable, in either model, if the algorithm can break symmetry, which in our topological view amounts to reaching some $\\rho \\in \\mathcal {R}(t)$ with positive probability, such that $\\tilde{\\pi }(\\rho )$ has an isolated vertex—that process will be the leader.", "Impossibility is obtained when for any time $t$ , for any $\\rho \\in \\mathcal {R}(t)$ with positive probability $\\tilde{\\pi }(\\rho )$ does not contain an isolated vertex and thus cannot be mapped to any $\\pi (\\tau )$ for $\\tau \\in \\mathcal {O}$ .", "In the blackboard model (Section REF ) we show that processes with the same randomness will always be connected in $\\tilde{\\pi }(\\rho )$ .", "Then, the only way to solve leader election is if there exists a process with its unique randomness source—eventually, the randomness will distinguish this node from any other node (with probability 1) so this node will be isolated in $\\tilde{\\pi }(\\rho )$ for any $\\rho \\in \\mathcal {R}(t)$ for which the realization of randomness that node obtained by time $t$ differs form all the realizations of all other nodes.", "In the message-passing model (Section REF ), we show that one plus the dimension of any facet in $\\tilde{\\pi }(\\rho )$ is a multiple of $g$ , the GCD of $n_1,n_2,\\ldots $ , where $n_i$ is the number of processors connected to source $\\mathbf {R}_i$ .", "Then, an isolated vertex $\\tilde{\\pi }(\\rho )$ cannot exist unless the GCD is 1.", "On the other hand, if the GCD is 1, we show an algorithm that leads to reducing the dimensions of certain facets (i.e., when looking on the evolution of $\\rho \\in \\mathcal {R}(t)$ as time goes by, that is, considering states $\\rho ^{\\prime }\\in \\mathcal {R}(t^{\\prime })$ for $t^{\\prime }>t$ with $\\Pr [\\rho ^{\\prime } \\mid \\rho ]>0$ ) until reaching a facet with dimension 0, that is, an isolated node." ], [ "The Blackboard Model", "With the above formulation we can determine the solvability of leader election in the blackboard model as a function of the specific configuration $\\alpha \\in \\mathcal {A}$ of the system: Theorem 4.1 Assume $k\\le n$ distinct randomness sources are available to $n$ parties, where for any $i\\in [k]$ , there are exactly $n_i$ parties connected to $\\mathbf {R}_i$ .", "Then, leader election in the blackboard model is eventually solvable if and only if there exists a source $i\\in [k]$ such that $n_i=1$ .", "We prove the theorem separately for $k=1$ and for $k>1$ .", "For each case we show that Eq.", "(REF ) holds if and only if $n_i=1$ .", "Base case ($k=1$ ): Let the randomness-configuration be such that $k=1$ , that is, all the parties are connected to $\\mathbf {R}_1$ and see exactly the same stream of randomness.", "In particular, any realization that gives two different parties different randomness strings, has zero probability.", "`if' direction: Since there is only a single source in the system ($\\mathbf {R}_1)$ , if $n_1=1$ then the entire network contains a single party, $n=1$ .", "In this case leader election is trivial: For any time $t>0$ , the complex $\\mathcal {R}(t)$ has only 0-dimension facets: $\\lbrace (1,x_1)\\rbrace $ for any $x_1\\in \\lbrace 0,1\\rbrace ^t$ .", "For any $\\rho \\in \\mathcal {R}(t)$ it holds that $\\tilde{\\pi }(\\rho )= \\rho $ .", "Further, $\\mathcal {O}$ reduces to a single isolated node $\\tau = \\lbrace (1,1)\\rbrace $ , and thus $\\pi (\\mathcal {O})=\\pi (\\tau )=\\lbrace (1,1)\\rbrace $ .", "It then follows that, for any $\\rho =\\lbrace (1,x_i)\\rbrace \\in \\mathcal {R}(t)$ there exists a name-preserving simplicial map $\\delta :\\tilde{\\pi }(\\rho ) \\rightarrow \\pi (\\tau )$ , i.e., the map that takes $(1,x_i)$ to $(1,1)$ .", "Thus, for every $t$ , it holds that $\\mathcal {S}(t)$ contains all the facets of $\\mathcal {R}(t)$ , because any such realization solves the task $\\mathsf {LE}$ .", "For any $t$ , as well as in the limit $t\\rightarrow \\infty $ , $\\Pr \\left[ \\mathcal {S}(t) \\mid \\alpha \\right]=1.$ `only if' direction: For the other direction, let $\\alpha ^{\\prime }\\in \\mathcal {A}$ be a randomness-configuration in which there is no source $i$ with $n_i=1$ .", "Since there is only a single source ($k(\\alpha ^{\\prime })=1$ ) we have $n_1=n>1$ and we need to show that leader election is not eventually solvable.", "Towards contradiction, assume that $\\rho \\in \\mathcal {R}(t)$ is a realization that solves LE, say, via a map $\\delta : \\tilde{\\pi }(\\rho ) \\rightarrow \\pi (\\tau _j)$ for some $j\\in [n], \\tau _j\\in \\mathcal {O}$ .", "By definition, any simplicial map $\\delta $ must preserve simplices.", "Let $(i,x_i)\\in \\tilde{\\pi }(\\rho )$ be the vertex that is mapped to $(j,1)\\in \\pi ({\\tau _j})$ .", "Since $\\lbrace (j,1)\\rbrace $ is a facet in $\\pi ({\\tau _j})$ , the vertex $(i,x_i)$ must be isolated in $\\tilde{\\pi }(\\rho )$ .", "However, this implies that $\\Pr [\\rho \\mid \\alpha ^{\\prime }]=0$ , since $\\alpha ^{\\prime }$ dictates that all parties share the same randomness source and their views (randomness and blackboard content) are identical.", "Therefore, for any realization $\\rho ^{\\prime }$ that has a non-zero probability (given $\\alpha ^{\\prime }$ ), the projection $\\tilde{\\pi }(\\rho ^{\\prime })$ must be a single facet of dimension exactly $n-1$ , that is, $\\lbrace (i,x) : {i\\in [n]}\\rbrace $ , for some value $x\\in \\lbrace 0,1\\rbrace ^t$ .", "On the other hand, $\\tilde{\\pi }(\\rho )$ has a 0-dimension facet, and since $n>1$ its probability is zero given $\\alpha ^{\\prime }$ .", "It follows that for any $t$ , as well as in the limit, $\\lim _{t\\rightarrow \\infty }\\Pr \\left[ \\mathcal {S}(t) \\mid \\alpha ^{\\prime } \\right]=0.$ General case ($k>1$ ): Assume we are given a randomness-configuration $\\alpha \\in \\mathcal {A}$ with $k=k(\\alpha )>1$ distinct randomness sources.", "Without loss of generality assume $0< n_1 \\le n_2 \\le \\cdots \\le n_k$ .", "`if' direction: Assume $\\alpha \\in \\mathcal {A}$ satisfies $n_1=1$ .", "At any time $t$ there are exactly $2^{kt}$ unique realizations $\\rho \\in \\mathcal {R}(t)$ with positive probability conditioned on $\\alpha $ , and recall that all such realizations are equiprobable (Lemma REF ).", "Let us define S1(t) def= { = {(1,x1),...,(n,xn)} R(t) i>1, xix1} to be the set of all the realizations $\\rho \\in \\mathcal {R}(t)$ in which the randomness obtained by the first party is unique, $x_1 \\ne x_i$ for all $i>1$ .", "There are $2^t\\cdot (2^t-1)^{k-1}$ such realizations with positive probability given $\\alpha $ .", "Note that each such realization solves the task $\\mathsf {LE}$ via the (unique) name-preserving map $\\delta :\\tilde{\\pi }(\\rho ) \\rightarrow \\pi (\\tau _1)$ , thus $\\mathcal {S}_1(t) \\subseteq \\mathcal {S}(t)$ .", "Since all the positive-probable realizations are equiprobable we have, [S(t) ] [S1(t) ] = 2t (2t-1)k-1 2-kt = (2t-1)k-12t(k-1) 1- (k-1)2t(k-2)2t(k-1) = 1-k-12t.", "From the above, $\\lim _{t\\rightarrow \\infty }\\Pr \\left[\\mathcal {S}(t)\\mid \\alpha \\right] = 1$ as required.", "`only if' direction: Assume a randomness-configuration $\\alpha ^{\\prime }$ (with $k(\\alpha ^{\\prime })>1$ ) in which $n_1>1$ (hence, there exists no $i$ with $n_i=1$ ).", "Let us fix a time $t$ , and let $\\rho \\in \\mathcal {R}(t)$ be a realization that solves the task $\\mathsf {LE}$ via $\\delta _j: \\tilde{\\pi }(\\rho ) \\rightarrow \\pi (\\tau _j)$ .", "In order for $\\delta _j$ to be a name-preserving simplicial-map, it is required that $(j,x_j)\\in V(\\tilde{\\pi }(\\rho ))$ is isolated in $\\tilde{\\pi }(\\rho )$ , since the vertex with name $j$ is a 0-dimensional facet of $\\pi (\\tau _j)$ .", "On the other hand, let $\\rho ^{\\prime }\\in \\mathcal {R}(t)$ be a realization with positive probability given $\\alpha ^{\\prime }$ , namely, $\\Pr [\\rho ^{\\prime }\\mid \\alpha ^{\\prime }]>0$ .", "Since in $\\alpha ^{\\prime }$ we have that $\\forall i, n_i>1$ , for any party $j$ there must exist another party $j^{\\prime }\\ne j$ that is connected to the same randomness source as $j$ .", "Hence, for any time $t$ , the randomness $j$ and $j^{\\prime }$ see is identical, $x_j = x_{j^{\\prime }}$ .", "Furthermore, in the blackboard model, equality of randomness is equivalent to equality of knowledge, since the knowledge of a party is just its randomness along with the content of the blackboard.", "Therefore, regardless of the realizations $\\lbrace x_i\\rbrace _{i\\notin \\lbrace j,j^{\\prime }\\rbrace }$ of the other parties, it holds that $j\\overset{t}{\\sim } j^{\\prime }$ and $\\lbrace (j,x_j),(j^{\\prime },x_{j^{\\prime }})\\rbrace $ is a simplex in $\\tilde{\\pi }(\\rho ^{\\prime })$ .", "So, for any $\\rho ^{\\prime }$ with $\\Pr [\\rho ^{\\prime }\\mid \\alpha ^{\\prime }]>0$ and any party $j$ , we get that $(j,x_j)$ is not isolated in $\\tilde{\\pi }(\\rho )$ .", "These two arguments imply that for any facet $\\rho \\in \\mathcal {R}(t)$ that solves the task $\\mathsf {LE}$ it holds that $\\Pr [\\rho \\mid \\alpha ^{\\prime }]=0.$ If we let $\\mathcal {S}(t)$ denote all the realizations the solve leader election at time $t$ , the above proves that for any $t$ , as well in the limit, $\\Pr \\left[\\mathcal {S}(t) \\mid \\alpha ^{\\prime } \\right]= 0.$ In particular, leader election is not eventually solvable in this case." ], [ "The Message-Passing Model", "We now turn to the message-passing model where nodes are indistinguishable and connected by point-to-point channels as a clique.", "Each node (privately) labels its $n-1$ neighbours with unique labels from $\\lbrace 1,\\ldots , n-1\\rbrace $ in an arbitrary way, which we refer to as the node's port-numbers.", "Then, when a node sends a message to some port $p$ it always reaches the same party; however, if a different node sends a message to (its own) port $p$ , it might reach a different party according to that node's private labeling.", "The main difference from the blackboard model is that a party's knowledge might be affected by its port-numbering in addition to its randomness.", "Unlike the blackboard model, where similar randomness means similar knowledge, here parties may have different knowledge while observing the same randomness (however, if their randomness is different, their knowledge will be different as well).", "We recall that the numbering of the ports is arbitrary.", "In the following we ask which configurations lead to solving leader-election regardless of the specific port numbers.", "Alternatively, we ask which configurations prevent any protocol from solving leader-election for at least one port-numbers labeling.", "That is, given a randomness-configuration, we look at the worst case for setting the port numbers and ask whether or not leader-election is eventually solvable.", "We term this question worst-case leader election.", "The following Theorem REF , which is this section's main theorem, shows that worst-case eventual solvability of leader election depends on the greatest common divisor (GCD) of the number of parties connected to each randomness source.", "Theorem 4.2 Assume $k\\le n$ distinct randomness sources are available to $n$ parties, where for any $i\\in [k]$ , exactly $n_i \\ge 1$ parties are connected to $\\mathbf {R}_i$ .", "Worst-case leader election is eventually solvable if and only if $\\gcd (n_1,n_2,\\ldots ,n_k)=1.$ We start by showing the following technical lemma that explains how the consistency-projected complex $\\tilde{\\pi }(\\mathcal {R}(t))$ changes with time.", "Lemma 4.3 For any $\\alpha \\in \\mathcal {A}$ such that $\\gcd (n_1,\\ldots ,n_k)=g$ there exists a way to number ports so that for any realization $\\rho \\in \\mathcal {R}(t)$ that has positive probability $\\Pr [\\rho \\mid \\alpha ]>0$ , the dimension $\\dim (\\gamma )$ of any facet $\\gamma \\in \\tilde{\\pi }(\\rho )$ satisfies $g \\mid \\dim (\\gamma )+1.$ Note that the above suggests that worst-case leader election is not eventually solvable when the GCD is larger then one, proving the “only if” direction of Theorem REF .", "Corollary 4.4 Assume $\\alpha ^{\\prime }\\in \\mathcal {A}$ such that $\\gcd (n_1,\\ldots ,n_k)=g$ and $g>1$ , then there exists a way to assign port-numbers to channels so that leader election is impossible.", "From Lemma REF we know that there exists a way to number ports such that any realization $\\rho \\in \\mathcal {R}(t)$ for which $\\Pr [\\rho \\mid \\alpha ^{\\prime }]>0$ , has facets of dimension at least $g-1 > 0$ .", "In particular, there is no isolated vertex in $\\tilde{\\pi }(\\rho )$ , as this will imply a facet $\\gamma $ with $\\dim (\\gamma )=0$ , but $g \\nmid \\dim (\\gamma )+1$ , which is a contradiction.", "Since there is no isolated vertex in $\\tilde{\\pi }(\\rho )$ , there exists no simplicial map from $\\tilde{\\pi }(\\rho )$ to $\\pi (\\tau _j)$ for any facet $\\tau _j\\in \\mathcal {O}$ and thus $\\rho $ does not solve leader election.", "We get that for any $t$ , no realization that solves leader election has positive probability, $\\Pr \\left[\\mathcal {S}(t) \\mid \\alpha ^{\\prime } \\right]=0$ .", "[Proof of Lemma REF .]", "Let $\\alpha \\in \\mathcal {A}$ be a randomness-configuration in which $\\gcd (n_1,\\ldots ,n_k)=g$ .", "Split the $n$  parties into $g$ disjoint subsets of nodes where subset $i\\in \\lbrace 1,\\ldots ,g\\rbrace $ holds exactly $n_j/g$ parties which are connected to $\\mathbf {R}_j$ (for all $j\\in [k]$ ).", "For this part only rename the parties as $0,1,\\ldots , n-1$ where the first $n_1$ parties are connected to the first source and the next $n_2$ parties are connected to the second one, etc.", "We assign the $j$ -th port ($j\\in [n-1]$ ) of party $i\\in \\lbrace 0,1,\\ldots , n-1\\rbrace $ to be connected to party number $\\big ( (i+j)\\text{mod }g + \\lceil i/g \\rceil \\cdot g + \\lceil j/g \\rceil \\cdot g\\big ) \\mod {n}.$ There exists an isomorphism $f:\\lbrace 0,\\cdots ,n-1\\rbrace \\rightarrow \\lbrace 0,\\cdots ,n-1\\rbrace $ that takes any party $i\\in \\lbrace 0,1,\\ldots , {n-1}\\rbrace $ written as $i=r+m\\cdot g$ for $r<g$ , $m\\in \\mathbb {N}$ , to the party $f(i)=(r+1 \\mod {g}) + mg$ .", "This isomorphism preserves the assignment of randomness source and it preserves port numbers.", "Thus, fixing any realization of randomness $(x_1,\\ldots , x_n)$ at round $t$ , for any party $i$ we get that $i \\overset{t}{\\sim } f(i)$ : they are connected to the same source and thus have the same randomness ($x_i=x_{f(i)}$ ).", "Moreover, every message that party $i$ receives from port $j$ , is also received by $f(i)$ from its own port $j$ , due to the way we number ports.", "This can be seen by induction; it trivially holds for round 0; now assume it holds for round $t-1$ .", "Note that if $i$ is connected to $p$ at its port $j$ , then $f(i)$ is connected to $f(p)$ in its port $j$ ; furthermore, if $p$ sees $i$ in its $j^{\\prime }$ port, then $f(p)$ sees $f(i)$ it its $j^{\\prime }$ port.", "Since $p,f(p)$ hold the same information at round $t-1$ by the induction hypothesis, $i$ and $f(i)$ will receive the same information from $p,f(p)$ respectively, and the claim will hold for time $t$ as well.", "Let us denote the equivalence class of party $i$ by i = { f(c)(i) c[g] }.", "where $f^{(c)}=f(f(\\ldots ))$ denotes the $c$ -th composition of $f$ with itself, and $f^{(0)}$ is the identity function (also note that $f^{(g)}=f^{(0)}$ ).", "Let $\\rho \\in \\mathcal {R}(t)$ be a realization such that $\\Pr [\\rho \\mid \\alpha ]>0$ , and let $\\gamma \\in \\tilde{\\pi }(\\rho )$ be a facet.", "Claim 4.5 If $(i,y_i)\\in \\gamma $ , then for any $j\\in \\llbracket i \\rrbracket $ we have that $(j,y_i) \\in \\gamma $ .", "Proof of claim.", "Suppose $(j,y_i)\\notin \\rho $ .", "Then there exists $y_j\\ne y_i$ such that $(j,y_j)\\in \\rho $ which implies that $\\Pr [\\rho \\mid \\alpha ]=0$ since $i$ and $j$ are connected to the same randomness source in $\\alpha $ according to the way we numbered parties and the definition of $f()$ .", "Therefore, $(j,y_i)$ is a node of $\\tilde{\\pi }(\\rho )$ , and we are left to show that $(j,y_i)\\in \\gamma $ .", "As argued above, we know that, for any time $t$ , $i \\overset{t}{\\sim } f(i)$ and thus (for the realization $\\rho $ ) $i \\overset{t}{\\sim } f(i) \\overset{t}{\\sim } \\cdots \\overset{t}{\\sim } f^{(g-1)}(i).$ Since the $\\overset{t}{\\sim }$ relation is transitive, all these vertices are consistent with one another.", "In particular, note that $i \\overset{t}{\\sim } j$ since $j\\in \\llbracket i\\rrbracket $ .", "Further, for any vertex $(x,y)\\in \\gamma $ , we have that $x\\overset{t}{\\sim } i$ by the definition of the projection $\\tilde{\\pi }$ (Eq.", "(REF )) and the fact that $\\gamma \\in \\tilde{\\pi }(\\rho )$ .", "Now, by the transitivity of the consistency operator and the fact that $i \\overset{t}{\\sim } j$ , we get that also $x \\overset{t}{\\sim } j$ .", "Finally, since $\\gamma $ is a facet of $\\tilde{\\pi }(\\rho )$ , the above implies that $(j,y_i)\\in \\gamma $ , which completes the proof of Claim REF .", "$\\diamond $ Finally, we conclude the proof of the lemma.", "Let $\\gamma \\in \\tilde{\\pi }(\\rho )$ be a facet.", "For any $(i,y_i)\\in \\gamma $ , Claim REF proves that also $(j,y_i)\\in \\gamma $ for any $j\\in \\llbracket i\\rrbracket $ .", "Since the sets $\\llbracket i\\rrbracket $ form a partition of $\\lbrace 0,...,n-1\\rbrace $ and since $|\\llbracket i\\rrbracket |=g$ for any $i$ , we conclude that $g\\mid \\dim (\\gamma )+1$ , which completes the proof of Lemma REF .", "We now move to proving the other direction of Theorem REF , that is, showing that if the GCD of the size of subsets that are assigned the same randomness source equals one, then leader election is solvable, regardless of the specific port numbers.", "Towards this goal, we show the following technical lemma that once again considers the structure of the projected complex $\\tilde{\\pi }(\\mathcal {R}(t))$ over time.", "The lemma suggests that facets in $\\tilde{\\pi }(\\mathcal {R}(t))$ (that stem from realizations with positive probability) eventually “split” into smaller facets.", "The change in their dimension is always a multiple of the GCD.", "Definition 4.6 We say that a realization $\\rho ^{\\prime }\\in \\mathcal {R}(t^{\\prime })$ succeeds a realization $\\rho \\in \\mathcal {R}(t)$ and denote $\\rho \\prec \\rho ^{\\prime }$ if (i) $t^{\\prime }>t$ and (ii) for any $i\\in [n]$ , if $(i,x_i)\\in \\rho ^{\\prime }$ then $(i,x_i(1,\\ldots ,t))\\in \\rho $ .", "Lemma 4.7 Fix a randomness-configuration $\\alpha \\in \\mathcal {A}$ .", "Assume a realization $\\rho \\in \\mathcal {R}(t)$ with $\\Pr [\\rho \\mid \\alpha ]>0$ .", "Let $\\gamma _1,\\gamma _2\\in \\tilde{\\pi }(\\rho )$ be two facets, where without loss of generality we assume $\\dim (\\gamma _1) \\le \\dim (\\gamma _2)$ .", "For any $t^{\\prime }>t$ , let $\\Gamma (t^{\\prime })$ be the set of realizations $\\rho ^{\\prime }\\in \\mathcal {R}(t^{\\prime })$ that satisfy the following three conditions: (1) $\\rho \\prec \\rho ^{\\prime }$ ; (2) $\\Pr [\\rho ^{\\prime } \\mid \\rho ,\\alpha ]>0$ and (3) if $\\gamma $ is the largest facet in $\\rho ^{\\prime }$ that contains a node with name from $\\textsf {names}(V(\\gamma _2))$ , then $\\dim (\\gamma )\\le \\max \\lbrace \\dim (\\gamma _1), \\dim (\\gamma _2)-\\dim (\\gamma _1)-1\\rbrace $ .", "It holds that $\\lim _{t^{\\prime }\\rightarrow \\infty } \\Pr \\left[\\Gamma (t^{\\prime })\\ \\vert \\ \\alpha , \\rho \\right] = 1.$ Let $V_1 = \\mathsf {names}(V(\\gamma _1))$ and $V_2 = \\mathsf {names}(V(\\gamma _2))$ be the names of all nodes in the respective set.", "Note that if at each round, parties send all their information to each other, this information can simulate any randomized protocol the network can run, in the sense that the parties can output the same output of any other protocol.", "In particular, it simulates the $\\textsc {CreateMatching()}$ procedure depicted in Algorithm REF , that essentially creates a matching between $V_1$ and $V_2$ .", "The matching is performed as follows: each party in $V_1$ randomly picks a node in $V_2$ , and sends a message to that node, asking to be matched with it.", "If a node in $V_2$ received only a single matching-request message, it accepts it.", "If it received more than a single request, it accepts just one request and rejects the others.", "This process continues with the remaining unmatched nodes until all the nodes in $V_1$ are matched (assuming $|V_1| \\le |V_2|$ ).", "[ht] The CreateMatching Procedure [1] Input: $n$ identical parties, connected as a clique, where $n_1$ parties connected to $\\mathbf {R}_i$ and $n_2$ parties connected to $\\mathbf {R}_j$ with $j\\ne i$ , and $\\gcd (n_1,n_2)=1$ .", "Initially: $V_1$ is the set of all nodes connected to $\\mathbf {R}_i$ and $V_2$ is the set of all parties connected to $\\mathbf {R}_j$ .", "The procedure assumes that this separation is already known to all the participating parties.", "Ignore any other parties in the network, if they exist.", "CreateMatching$V_1,V_2$ Assuming (without loss of generality) $|V_1| \\le |V_2|$ All nodes in $V_1,V_2$ set themselves as active.", "Each active node in $V_1$ randomly picks an active neighbour from $V_2$ and sends a messages to that selected neighbour.", "Each node in $V_2$ that have received at least one message, selects the minimal port from which a message has arrived, and sends an ACK message to that origin node (in $V_1$ ).", "Nodes from $V_1$ that received an ACK message in the previous step set themselves to done and broadcasts this event to all their neighbours.", "Nodes in $V_2$ that sent an ACK message in the previous step, set themselves to done and broadcast this event to all their neighbours.", "all $V_1$ nodes are done Lemma 4.8 Let $n$ identical nodes be connected as a clique where $n_1$ ($n_2$ ) nodes are connected to the randomness source $\\mathbf {R}_i$ ($\\mathbf {R}_j$ , $j\\ne i$ ).", "At the end of CreateMatching(), there exists a matching $M$ between all the $n_1$ parties connected to $\\mathbf {R}_i$ and $n_1$ parties connected to $\\mathbf {R}_j$ , and every party outputs whether it is matched or not.", "The proof is rather straightforward.", "Call $V_1$ ($V_2$ ) the set of parties connected to $\\mathbf {R}_i$ ($\\mathbf {R}_j$ ).", "Consider the state after the first time the procedure reaches Line REF .", "At this point, every node in $V_1$ has selected a single node in $V_2$ , which we consider as throwing $|V_1|$  balls into $|V_2|$  bins.", "In Line REF each non-empty bin selects exactly a single ball and “ignores” the other balls in it.", "Since each ball is uniquely identified with a node in $V_1$ , this creates a matching between all the non-empty bins (nodes) in $V_2$ to some nodes in $V_1$ .", "Nodes that belong to the matching now become done, and the matching continues recursively with the remaining nodes.", "Since in every iteration there must be at least one non-empty bin, every iteration increases the size of the matching by at least one.", "Eventually, all the parties in $V_1$ will be matched (since $|V_1|\\le |V_2|$ by our assumption).", "At this point, all the parties that are done participate in the matching and all the remaining active parties are unmatched.", "Note that all the parties learn when CreateMatching() has terminated (e.g., by counting the number of done parties in $V_1$ when reaching Line REF ).", "After the execution of CreateMatching(), the nodes whose names belong to $V_2$ has split into two subsets, $V_2 = V_m \\cup V_{um}$ of “matched” and “un-matched” nodes, of sizes $|V_m|=n_1$ and $|V_{um}|=n_2-n_1$ , respectively.", "Let $t^{\\prime }$ be a time after this split happens, and let $\\rho ^{\\prime }$ be the realization that corresponds this execution, so $\\rho \\prec \\rho ^{\\prime }$ .", "Note that the knowledge of nodes in these two subsets must be differentI.e., their state $\\in \\lbrace \\mathsf {matched},\\mathsf {unmatched}\\rbrace $ is function of their knowledge, hence their knowledge must be different in order to obtain different state.. By the definition of the projection $\\tilde{\\pi }$ , no facet in $\\tilde{\\pi }(\\rho ^{\\prime })$ can contain nodes from both $V_m$ and $V_{um}$ , as they are inconsistent.", "Hence, the maximal dimension of any facets of $\\tilde{\\pi }(\\rho ^{\\prime })$ that contain some nodes with name from $V_2$ is at most $\\max \\lbrace n_1-1, n_2-n_1-1\\rbrace $ .", "Note that CreateMatching() requires that the participating parties know their partition into $V_1,V_2$ .", "This holds in our case since $\\gamma _1,\\gamma _2$ are two different facets and parties that belong to different facets have different knowledge (and in a single rounds they can be aware of this).", "With the above lemma in hand, we can prove the `if' direction of Theorem REF .", "The intuition is that any realization $\\rho $ with positive probability whose projection $\\tilde{\\pi }(\\rho )$ has a facet of dimension $d\\ge 1$ is eventually succeeded by $\\rho ^{\\prime }$ whose projection $\\tilde{\\pi }(\\rho ^{\\prime })$ has facets with maximal dimension strictly less than $d$ .", "At the beginning of the computation, each set of parties that are connected to the same randomness source, are consistent in their knowledgeTo be more precise, the knowledge already might be different due to differences in parties' port numbers.", "This can only help us in reaching a leader.", "In the discussion we ignore this option and assume that knowledge differences only come from the randomness and its affect on being matched or unmatched., and form a facet in the projection of the relevant realization.", "The fact that the GCD of the sizes of these subsets is 1 implies that we can reduce the dimension of the (largest) facet again and again until we reach facets of dimension 0.", "To illustrate the above in an algorithmic manner, if we start with two sets of sizes $n_1<n_2$ , we can perform a matching between the sets, and turn off all the nodes that belong to $V_m$ in the matching.", "This leaves us with two new sets of parties, $V_1$ and $V_{um}$ of sizes $a=n_1, b=n_2-n_1$ , respectively, where parties that belong to the same set connect to the same randomness source.", "Repeating this process again and again yields a subset of size $\\gcd (n_1,n_2)$ .", "The fact that $\\gcd (n_1,n_2,\\ldots ,n_k)=1$ and that $\\gcd ()$ is an associative function, $\\gcd (a,b,c)=\\gcd (\\gcd (a,b),c)$ , means we can repeat the above process on different subsets of parties (in a similar way to the Euclidean algorithm) until the dimension of the maximal facet becomes 0.", "[Proof of Theorem REF , `if' direction] We just need to show that for any $\\alpha \\in \\mathcal {A}$ with $\\gcd (n_1,n_2,\\ldots ,n_k)=1$ there exists a realization $\\rho \\in \\mathcal {S}$ with positive probability given $\\alpha $ .", "Then, by Kolmogorov's zero–one Eq.", "(REF ) holds.", "We will assume, without loss of generality, that at the onset of the analysis shown here, $\\tilde{\\pi }(\\mathcal {R}(t))$ contains facets of dimensions $n_1-1,n_2-1,\\ldots ,n_k-1$ .", "This can easily be achieved by letting the parties exchange their randomness until $k$ differences appear (if $k$ is known), or until the parties' randomness distinguish $k^{\\prime }$ subsets whose sizes' GCD is 1 (if $k$ is unknown, restarting the protocol whenever future randomness exhibits new differences among parties that previously belonged to the same set).", "This step eventually succeeds with probability 1.", "We set time 0 to be after this step has completed, and below consider only $t>0$ .", "Let $\\alpha \\in \\mathcal {A}$ be a randomness-configuration with $k=k(\\alpha )$ in which $\\gcd (n_1,n_2,\\ldots ,n_k)=1$ .", "Let $\\mathcal {S}^*(t) = \\lbrace \\rho \\in \\mathcal {R}(t) \\mid \\dim (\\tilde{\\pi }(\\rho ))=0\\rbrace $ be the set of realizations whose consistency-projection has dimension 0, that is, it is composed of $n$ isolated vertices.", "Clearly any realization in $\\mathcal {S}^*$ solves leader election, $\\mathcal {S}^*\\subseteq \\mathcal {S}(t)$ , so finding a realization in $\\mathcal {S}^*$ with positive probability given $\\alpha $ suffices to complete the proof.", "Assume a realization $\\rho \\notin \\mathcal {S}^*(t)$ with positive probability, $\\Pr [\\rho \\mid \\alpha ]>0$ .", "Let $d_1,d_2,\\ldots $ be the dimensions of $\\tilde{\\pi }(\\rho )$ 's facets.", "Since $\\rho \\notin \\mathcal {S}^*(t)$ we know that $d=\\max (d_1,d_2,\\ldots ) > 0$ .", "We argue that $\\gcd (d_1+1,d_2+1,\\ldots )=1$ .", "This will follow from the following lemma Lemma 4.9 For any $\\alpha \\in \\mathcal {A}$ , $t^{\\prime }>t$ and realizations $\\sigma \\in \\mathcal {R}(t), \\sigma ^{\\prime }\\in \\mathcal {R}(t^{\\prime })$ , such that $\\sigma \\prec \\sigma ^{\\prime }$ , the (unique) name-preserving map $\\delta :\\tilde{\\pi }(\\sigma ^{\\prime })\\rightarrow \\tilde{\\pi }(\\sigma )$ is a simplicial map.", "Let $\\tau ^{\\prime }\\in \\tilde{\\pi }(\\sigma ^{\\prime })$ be a facet.", "By the properties of $\\tilde{\\pi }$ we have that for any two nodes $(i,x_i),(j,x_j) \\in V(\\tau ^{\\prime })$ we have $i\\overset{t^{\\prime }}{\\sim } j$ , with respect to the realization $\\sigma ^{\\prime }$ .", "Note that $(i,x_i(1,\\ldots ,t)),(j,x_j(1,\\ldots ,t))\\in V(\\sigma )$ since $\\sigma \\prec \\sigma ^{\\prime }$ .", "It thus also holds that $i \\overset{t}{\\sim } j$ with respect to the realization $\\sigma $ since knowledge is cumulative (i.e., if the knowledge in time $t$ makes them inconsistent, this holds also for any time $t^{\\prime }>t$ ).", "Thus, $\\delta (\\tau ^{\\prime })$ is a simplex in $\\tilde{\\pi }(\\sigma )$ .", "Assume, towards contradiction, that $\\gcd (d_1+1,d_2+1,\\ldots )=g >0$ , and let $\\sigma _1,\\sigma _2,\\ldots ,\\sigma _t=\\rho $ be the sequence of realizations that correspond to the computation, $\\sigma _i\\prec \\sigma _{i+1}$ for any $i<t$ .", "Note that due to the above lemma, facets in $\\tilde{\\pi }(\\sigma _{t-1})$ have dimension $\\in \\lbrace (-1)+\\sum D \\mid D\\in 2^{\\lbrace d_1+1,d_2+1,\\ldots \\rbrace } \\rbrace $ (where a sum of a set is the sum of the elements in the set), since facets in $\\tilde{\\pi }(\\sigma _{t-1})$ must be composed of one or more facets of $\\tilde{\\pi }(\\sigma _t)$ .", "Let $d^{\\prime }_1,d^{\\prime }_2,\\ldots $ be the dimensions of the facets of $\\tilde{\\pi }(\\sigma _1)$ .", "The above implies that $\\gcd (d^{\\prime }_1+1,d^{\\prime }_2+1,\\ldots )=g >0$ , which is a contradiction.", "Therefore, unless all the facets of $\\tilde{\\pi }(\\rho )$ are of dimension 0, there must exist a facet $\\gamma _1$ with dimension strictly less than $d$ (otherwise, if all facets have dimension $d$ , then $\\gcd (d_1+1,d_2+1,\\ldots )=d+1>1$ ).", "Now, we apply Lemma REF on $\\rho $ , where $\\gamma _1$ is as defined above (i.e., the fact with $\\dim (\\gamma _1)<d$ ), and $\\gamma _2$ is any facet with maximal dimension $\\dim (\\gamma _2)=d$ .", "The lemma suggests that there exists a time $t^{\\prime }$ and a realization $\\rho ^{\\prime }\\in \\mathcal {R}(t^{\\prime })$ that succeeds $\\rho $ , $\\Pr [\\rho ^{\\prime } \\mid \\rho ,\\alpha ]>0$ , where the maximal facet in $\\tilde{\\pi }(\\rho ^{\\prime })$ has dimension strictly less than $d$ , that is, $\\dim (\\tilde{\\pi }(\\rho ^{\\prime }))<d$ .Note that multiple applications of Lemma REF might be needed, e.g., when multiple facets in $\\tilde{\\pi }(\\rho )$ are of dimension $d$ .", "Note that, [ ' ] [',][] >0.", "Hence, $\\rho ^{\\prime }$ is a realization with positive probability, where facets in $\\tilde{\\pi }(\\rho ^{\\prime })$ have dimension strictly less than the ones in $\\tilde{\\pi }(\\rho )$ we began with.", "We can now apply the same reasoning on $\\rho ^{\\prime }$ and keep reducing the dimension of the projected complex, until we reach a realization $\\rho ^*$ with $\\Pr [\\rho ^* \\mid \\alpha ]>0$ , for which all facets of $\\tilde{\\pi }(\\rho ^*)$ are of dimension 0." ], [ "Conclusion", "This paper focuses on input-free symmetry-breaking tasks, and presents a topological framework for studying the solvability of such tasks by randomized algorithms.", "The applicability of this framework was demonstrated by studying the solvability of leader election in environments where correlations may exist between the randomness sources assigned to the processing nodes.", "Thus, the paper is expected to help move the study of randomized algorithms under the umbrella of algebraic topology.", "Our analysis resulted in a complete characterization of the solvability of leader election by randomized algorithms.", "In Appendix  we show that the same conditions required for solving leader election also suffice for solving any name-independent task (but they are not necessary).", "Extending this work to any task $(\\mathcal {I},\\mathcal {O},\\Delta )$ is an appealing research direction.", "A first step may consist of extending this paper's framework to input-free tasks for which the output complex $\\mathcal {O}$ is not symmetric.", "An intriguing example is electing a leader and a deputy leader (where the latter is to be used as an immediate backup in case the leader fails), under the constraint that some nodes may only be leaders, some nodes may only be deputy leaders, some nodes may be either of the two, and some nodes may be neither.", "Another compelling direction is extending the communication model to networks with arbitrary structure." ], [ "Acknowledgements", "The authors would like to thank Ami Paz for fruitful discussions and insightful explanations on topology in distributed computations.", "P. Fraigniaud is supported in part by ANR projects DESCARTES and FREDDA.", "R. Gelles is supported in part by ISF grant 1078/17." ], [ "Algebraic Topology: Basic Definitions", "We give here a brief survey of the topological terms used in this paper, and refer the reader to [15] for a more complete treatment of the subject.", "An abstract simplicial complex $\\mathcal {K}$ is a nonempty set of sets (simplices) $\\mathcal {K}=\\lbrace \\sigma _i\\rbrace _i$ and it holds that if $\\sigma $ is a simplex then any non empty subset $\\rho \\subseteq \\sigma $ is also a simplex, $\\rho \\in \\mathcal {K}$ .", "The elements of a simplex $\\sigma $ are called nodes or vertices, and are denoted by $V(\\sigma )$ .", "The set of all nodes, $V(\\mathcal {K})=\\bigcup _{\\sigma \\in \\mathcal {K}} V(\\sigma )$ is called the node-set of the complex.", "The dimension of a simplex is $\\dim (\\sigma )=|V(\\sigma )|-1$ .", "In particular, a single node $\\sigma =\\lbrace v\\rbrace $ has dimension 0.", "A facet of $\\mathcal {K}$ is a simplex that is not contained in any other simplex of $\\mathcal {K}$ .", "Note that the set of facets fully defines the complex.", "A facet of dimension 0 is called an isolated node.", "The dimension of a complex is the maximal dimension of its facets.", "A complex whose all facets have the same dimension is called pure.", "For two complexes, $\\mathcal {K}$ and $\\mathcal {L}$ , we say that $\\mathcal {K}$ is a subcomplex of $\\mathcal {L}$ if $\\mathcal {K}\\subseteq \\mathcal {L}$ .", "For a set $X\\subseteq V(\\mathcal {K})$ , the induced complex of $\\mathcal {K}$ on $X$ is the complex $\\lbrace \\sigma \\in \\mathcal {K}\\mid V(\\sigma )\\subseteq X\\rbrace $ .", "A vertex map from $\\mathcal {K}$ to $\\mathcal {L}$ is any function $f: V(\\mathcal {K})\\rightarrow V(\\mathcal {L})$ .", "A simplicial map $\\delta :\\mathcal {K}\\rightarrow \\mathcal {L}$ is a vertex map such that for any $\\sigma \\in \\mathcal {K}$ it holds that $\\delta (\\sigma )=\\lbrace \\delta (v)\\mid v\\in \\sigma \\rbrace \\in \\mathcal {L}$ , that is, it maps simplices in $\\mathcal {K}$ to simplices in $\\mathcal {L}$ (i.e., it preserves simplices).", "Complexes $\\mathcal {K}$ and $\\mathcal {L}$ are said to be isomorphic if there exist simplicial maps $f:\\mathcal {K}\\rightarrow \\mathcal {L}$ and $f^{-1}:\\mathcal {L}\\rightarrow \\mathcal {K}$ , such that for any $\\sigma \\in \\mathcal {K}$ , $\\sigma =f^{-1}(f(\\sigma ))$ , and for any $\\rho \\in \\mathcal {L}$ , $\\rho =f(f^{-1}(\\rho ))$ .", "A chromatic complex $\\mathcal {K}$ is a complex augmented with a naming function $\\mathsf {name}: V(\\mathcal {K}) \\rightarrow C$ where $C$ is called the set of names (colors).", "A vertex map $f: V(\\mathcal {K})\\rightarrow V(\\mathcal {L})$ preserves names if $\\mathcal {K}$ and $\\mathcal {L}$ are chromatic, and for any $v\\in V(\\mathcal {K})$ we have $\\mathsf {name}(v)=\\mathsf {name}(f(v))$ .", "In this paper all complexes are chromatic and all maps are name-preserving." ], [ "Technical Lemmas", "The following lemma shows that, for a given time and randomness-configuration, all the global states with positive probability are equiprobable.", "Lemma B.1 Given $t>0$ and $\\alpha \\in \\mathcal {A}$ where exactly $k=k(\\mathcal {A})$ different randomness sources are connected to the parties in $\\alpha $ , define the set of $\\alpha $ -inconsistent randomness, $\\textsf {B}_\\alpha = \\Big \\lbrace (x_1,\\ldots ,x_n)\\in (\\lbrace 0,1\\rbrace ^{t})^{n}\\ \\Big \\vert \\ \\exists i,j\\in [n], c\\in [k] \\text{ s.t. }", "x_i\\ne x_j \\text{ but } (i,c),(j,c)\\in \\alpha \\Big \\rbrace .$ For any facet $\\sigma \\in \\mathcal {R}(t)$ with nodes $V(\\sigma )=\\lbrace (i,x_i) \\mid i \\in [n]\\rbrace $ , $\\Pr [\\sigma \\mid \\alpha ] ={\\left\\lbrace \\begin{array}{ll}0 & ( x_1,\\cdots , x_n) \\in \\textsf {B}_\\alpha \\\\2^{-tk} & ( x_1,\\cdots , x_n) \\notin \\textsf {B}_\\alpha \\end{array}\\right.", "}.$ The proof follows directly from the definition since $\\Pr [\\sigma \\mid \\alpha ] = \\Pr [(R_1,\\ldots ,R_n)(1,\\ldots ,t)=(x_1,\\ldots ,x_n) \\mid \\alpha ]$ .", "Recalling that the mapping $h$ defined in Section REF induces a name-preserving isomorphism on the facets of $\\mathcal {P}(t)$ and $\\mathcal {R}(t)$ , we conclude: Corollary B.2 Given $t>0$ and $\\alpha \\in \\mathcal {A}$ with exactly $k=k(\\mathcal {A})$ randomness sources connected to the parties, for any $\\sigma \\in \\mathcal {P}(t)$ we have $\\Pr [ \\sigma \\mid \\alpha ] = \\Pr [ h(\\sigma ) \\mid \\alpha ] \\in \\lbrace 0, 2^{-tk}\\rbrace .$" ], [ "Name-independent\nInput-Output Tasks Reduce to Leader Election", "As a consequence of the above, any name-independent task can be solved in our model as long as leader election is possible.", "A task $(\\mathcal {I},\\mathcal {O}, \\Delta )$ is name-independent if $\\Delta $ maps inputs to outputs in a name-oblivious way.", "Namely, for any possible input for the system, $\\sigma \\in \\mathcal {I}$ , parties with the same input-value compute the same output-value, i.e., $(i,x),(j,x)\\in \\sigma \\Rightarrow (i,o),(j,o) \\in \\Delta (\\sigma )$ .", "by reducing the task to choosing a leader, who in turn computes the output in a centralized way.", "Theorem C.1 If Leader election is solvable by an anonymous network in the blackboard or message-passing model, then any distributed name-independent input-output task can be solved over the same model.", "Given a task $(\\mathcal {I},\\mathcal {O},\\Delta )$ , the parties perform leader election.", "Every party then sends the leader its inputs, either directly or via the blackboard.", "The leader collects all the inputs (and records the respective port-number for each in the message passing model).", "Then, the leader solves the task $(\\mathcal {I},\\mathcal {O},\\Delta )$ by himself and distributes the outputs to its neighbours, either by publishing the respective output of each input on the blackboard or by sending the appropriate output to the corresponding port.", "It is obvious that the other direction is invalid, as there are tasks that can be deterministically solved in both these models, regardless of the solvability of leader election." ] ]
2105.11713
[ [ "A Publicly Available Dataset of Out-of-Field Dose Profiles of a 6 MV\n Linear Accelerator" ], [ "Abstract An increase in radiotherapy-induced secondary malignancies has led to recent developments in analytical modelling of out-of-field dose.", "These models must be validated against measurements, but currently available datasets are outdated or limited in scope.", "This study aimed to address these shortcomings by producing a large dataset of out-of-field dose profiles measured with modern equipment.", "A novel method was developed with the intention of allowing physicists in all clinics to perform these measurements themselves using commonly available dosimetry equipment.", "A standard 3D scanning water tank was used to collect 36 extended profiles.", "Each profile was measured in two sections, with the inner section measured with the beam directly incident on the tank, and the outer section with the beam incident on a water-equivalent phantom abutted next to the tank.", "The two sections were then stitched using a novel feature-matching approach.", "The profiles were compared against linac commissioning data and manually inspected for discontinuities in the overlap region.", "The dataset is presented as a publicly accessible comma separated variable file containing off-axis ratios at a range of off-axis distances.", "This dataset may be applied to the development and validation of analytical models of out-of-field dose.", "Additionally, it may be used to inform dose estimates to radiosensitive implants and anatomy.", "Physicists are encouraged to perform these out-of-field measurements in their own clinics and share their results with the community." ], [ "I.Introduction Improvements in diagnostic and treatment technologies have resulted in overall cancer survival rates increasing in recent times[1].", "However, this increase in radiation therapy survivorship has coincided with a commensurate increase in the rate of radiation induced secondary malignancies later in life[2].", "In response, there has been increasing effort to better understand out-of-field doses delivered during radiation therapy treatments[3].", "Beyond secondary cancer induction, accurate out-of-field dose estimates are also necessary for assessing doses to radiosensitive implants such as cardiac devices[4], [5] and radiosensitive anatomy as in the case of pregnant patients[6], [7].", "There is also a need for increasingly accurate out-of-field dose estimates to inform epidemiological studies[8].", "Methods of estimating out-of-field doses include consulting simple reference data in the literature[6], [9], [10], [11], Monte Carlo simulations[12], [13], [14], and increasingly refined analytical models[15], [16], [17], [18].", "These models may need to consider the physical geometry of clinical linacs, including jaws, primary collimator, MLC leaves and carriage, and the arrangement of additional head shielding.", "This is further complicated by the relative positions of the linac treatment head and patient, such as during non-coplanar cranial radiotherapy[19].", "Modelling and verifying the effects of these factors would require the measurement of a large number of out-of-field profiles with many collimator orientations.", "As out-of-field dose modelling matures, it is also reasonable to expect that computational models will augment traditional dose calculation algorithms in commercial treatment planning systems.", "Commissioning these models may require the measurement of profiles much farther outside the field than physicists are currently accustomed.", "Available datasets in the literature generally contain older model linacs and MLCs that are rarely seen today [6], [9] and present coarse resolution dose profiles in a limited set of collimator orientations.", "Physicists wishing to use these profiles are forced to interpolate values from the printed figures by hand.", "This is a sub-optimal approach, and a contemporary solution using modern dosimetry equipment should achieve much greater spatial resolution while also presenting the data digitally in a manner that is computationally digestible.", "In this dataset article we establish a method of measuring high-quality out-of-field dose profiles using typically available clinical physics equipment.", "This method is then followed to produce a comprehensive and publicly available dataset of out-of-field dose profiles of our clinical linac.", "II.Acquisition and Validation Methods II.A.Equipment The linac under investigation in this study was a Varian Clinac iX (Varian Medical Systems, Palo Alto, USA) with a Millennium 120 MLC.", "All measurements were performed using the 6 MV photon energy with 600 MU/min dose rate.", "A PTW Semiflex 0.3 cm$^3$ 31013 (PTW-Freiburg, Freiburg, Germany) ionisation chamber was used for all profile scans.", "The dimensions of this chamber afforded lower volume averaging compared to larger thimble chambers, while still maintaining the sensitivity necessary for far out-of-field measurements.", "The chamber was affixed to a PTW BEAMSCAN water tank using the TRUFIX chamber positioning system and steered with version 4.3 of the control software.", "The BEAMSCAN water tank had a usable scanning range of 500 mm (horiz.)", "$\\times $ 500 mm (horiz.)", "$\\times $ 415 mm (vert.", "), and 15 mm thick PMMA walls.", "A PTW Semiflex 3D 0.07 cm$^3$ 31021 ionisation chamber was used as a reference to correct for fluctuations in the beam output during each scan.", "The chamber voltages were set to 400 V as per manufacturer recommendations, and the integrated water tank electrometer was set to low range.", "The control software was programmed to measure profiles in continuous scanning mode with a chamber speed of 5 mm/s and data points reported every 2 mm.", "A 30 $\\times $ 30 $\\times $ 40 cm$^3$ stack of Virtual Water (Standard Imaging, Middleton, USA) was used as additional scattering material.", "II.B.Measurement Setup Each profile was measured as a piece-wise combination of two scans to construct a much longer profile measurement than a typical water tank would allow.", "The first section acquired the in-field and near out-of-field region of each profile, while the second section captured the far out-of-field region.", "An overlap area of approximately 15 cm was included in each pair of profile sections.", "The first measurement geometry can be seen in Figure REF a.", "To begin, the tank was positioned such that the long axis of the chamber was orthogonal to the scanning direction, with a source to surface distance of 90 cm.", "The tank was then translated such that the central axis of the beam was 15 cm from the maximum chamber travel position on one side, and 35 cm from the maximum chamber travel position on the opposite side.", "This allowed the inner section of each profile to be collected out to 35 cm from the central axis while still allowing adequate scatter around the field.", "The measurement geometry was then altered to capture the far out-of-field section of each profile, as seen in Figure REF b.", "The tank was translated 31.5 cm in the direction of the profile, and a stack of Virtual Water was positioned in the field at 90 cm source to surface distance to re-establish full scattering conditions.", "Care was taken to ensure that the Virtual Water was abutted firmly against the tank wall with minimal air gaps.", "Figure: Equipment setup for the measurement of profiles in this study.", "(a) Inner profile section: The field is directed into the tank near the wall.", "(b) Outer profile section: The tank is shifted outside of the field and abutted to a stack of Virtual Water to ensure full scattering conditions.II.C.Post-processing The two sections of each profile were stitched together to create a whole.", "This was non-trivial, as the overlap regions did not necessarily coincide due to small air gaps between the Virtual Water and the tank wall, the non-water-equivalence of the tank wall, and positioning errors in the tank.", "To overcome this issue, each overlap region was searched for a prominent feature common to both sections of the profile.", "The far out-of-field section was then progressively shifted in 1 mm increments and re-scaled until the identified feature matched the inner profile section as well as possible.", "The two sections were then combined into one full profile, with an average being taken in the overlap region.", "A visual example of this process can be seen in Figure REF .", "No additional smoothing or filtering was applied.", "Figure: The process of stitching two profile sections to make a whole.", "(a) The two raw profile sections with a strong feature for matching.", "(b) The outer section shifted such that the features overlap.", "(c) The two sections conjoined, with the average value taken in the overlap region.II.D.Data Validation In order to establish that the irradiation conditions and measurement setup were representative of normal practice, the inner component of the 10 $\\times $ 10 cm$^2$ X and Y profiles were compared against the same profiles gathered during commissioning of the linac.", "A gamma comparison of the in-field sections, bounded by the 50% isodose lines, showed 100% agreement with criteria of 1% dose difference (local normalisation) and 1 mm distance-to-agreement.", "This gave good confidence that the measurement equipment was set up correctly, that the linac was behaving nominally, and that ultimately the data gathered in the session was representative of clinical practice.", "As mentioned in the previous section, the outer section of each profile was shifted to coincide with the inner section of the profile.", "Given that the equipment was not moved during the measurement of all outer profile sections within a given orientation (radial or transverse), it follows that the ideal shift should be the same across all profiles.", "This was indeed found to be the case with the largest difference in ideal shifts being 1 mm.", "After combining the profile halves and taking the average in the overlap region, all full profiles were then manually inspected to ensure that there were no obvious discontinuities in the overlap region.", "None were found.", "II.E.Summary of Collected Profiles In total, 36 profiles were collected (see summary in Table REF ).", "Profiles were gathered at three nominal field sizes: 5 $\\times $ 5 cm$^2$ , 10 $\\times $ 10 cm$^2$ , and 15 $\\times $ 15 cm$^2$ .", "Profiles in both the X and Y directions were measured with the jaws defining the field and the MLC in a 'parked' state (leaf tips fully retracted to approximately 21 cm off axis).", "These were repeated with the MLC defining the field and the jaws at the Varian recommended positions (X jaws 8 mm retracted from nominal field edge, Y jaws 2 mm retracted from nominal field edge).", "All profiles were measured with a source to surface distance of 90 cm and depth of 10 cm, except for the 10 $\\times $ 10 cm$^2$ MLC defined profiles which were additionally measured at depths of 15, 20, and 25 cm.", "Every profile was measured twice: once aligning with the radial (gun-target) direction and once aligning with the transverse (left-right) direction.", "The collimator angle was set appropriately to achieve this, for example, a transverse X profile required collimator 0$^{\\circ }$ , while a radial X profile required collimator 90$^{\\circ }$ (or 270$^{\\circ }$ ).", "The radial direction corresponds to the cranio-caudal axis of a patient receiving an coplanar treatment with no couch rotation and the transverse direction corresponds to the cranio-caudal axis of a patient receiving a treatment with a 90$^{\\circ }$ couch rotation.", "The radial and transverse profiles were collected in two separate measurement sessions as they required a complete reorientation of the tank.", "In total, the measurement time was approximately eight hours, including equipment setup/cleanup and data collection.", "Table: Field configurations measured in this study.", "For each row in the table, X and Y profiles were gathered in both the radial and transverse orientations,III.Data Format and Usage Notes The dataset had been made publicly accessible through the Zenodo platform and released under a Creative Commons Attribution 4.0 licence[20].", "The dataset consists of a single comma seperated variable (CSV) file containing 37 columns, with the first column storing the off-axis distance of each measurement point, and the remaining 36 columns storing the off-axis ratios at these points for each profile.", "The first row contains a label for each profile with the format '[$field\\_size$ ] [$mlc\\_state$ ] [$depth$ ] [$direction$ ] [$orientation$ ]' where $field\\_size$ is the nominal field size at the isocentre, $mlc\\_state$ designates the position of the MLC, taking the value `MLC defined' when the MLC defines the field, and `MLC parked' when the MLC is fully retracted, $depth$ is the depth of measurement, $direction$ indicates the direction of the profile, either X or Y, and $orientation$ indicates whether the measurement orientation was radial (GT) or transverse (LR).", "For example, a profile under the Y jaw of a 10 $\\times $ 10 cm$^2$ MLC defined field measured at a depth of 10 cm in the radial direction would have the label '10 x 10 MLC defined Y d10cm (GT)'.", "Each profile was normalised to the value at the central axis at the time of measurement so no further processing is needed to recover off-axis ratios.", "Users should be aware that the profiles are of unequal length, depending on whether they were acquired in the radial or transverse measurement orientation.", "IV.Discussion IV.A.An Exploration of Dose Outside the Treatment Field This investigation produced a large dataset with many curious features.", "In this section we compare and contrast a number of profiles that illuminate the underlying geometry and radiation interactions.", "However, these are just a representative selection of the profiles, and we encourage interested readers to further examine the dataset for themselves.", "Figure REF presents two 10 $\\times $ 10 cm$^2$ X profiles, one of which is defined by the MLC, and the other is defined by the jaw (MLC retracted).", "Both profiles were measured in the transverse direction.", "Feature A corresponds to a sharp decrease in relative dose in the jaw defined profile due to the combined effect of the primary collimator and the MLC leaves being retracted to this position.", "This feature is not seen in the MLC defined profile.", "Feature B denotes a large difference in the profiles very far out-of-field.", "This may be explained by this region being shielded by the tails of the MLC leaves when they are fully retracted, but not shielded when the leaves are moved in to define the field.", "Figure: Two 10 ×\\times 10 cm 2 ^2 X profiles, one MLC defined, and one jaw defined (MLC fully retracted).", "Marked features are explained in the article text.Two 10 $\\times $ 10 cm$^2$ Y profiles, one of which is defined by the jaw (MLC retracted), and the other is defined by the MLC, are shown in Figure REF .", "Both of these profiles were collected in the radial direction.", "Notably, the Y jaw defined profile is the only field arrangement giving a pure jaw profile uncoupled from additional MLC shielding effects.", "Three features are marked in the figure.", "Feature A denotes a sharp increase in relative dose at approximately 20 cm off axis, due to the finite lateral extent of the MLC.", "This feature has been observed in an earlier study[10].", "This is immediately followed by a decrease in relative dose due to the primary collimator, which can also be seen on the jaw defined profile.", "Feature B denotes an area in which the two profiles have the same shielding conditions (jaw, primary collimator, no MLC) and yet have different relative doses.", "This may be explained by a difference in lateral phantom scatter from leakage more centrally in the field.", "The profile with both the jaw and MLC shielding the beam has less leakage relative to the jaw only profile, and therefore also has less lateral phantom scatter from this leakage.", "This is supported by the profiles coming into agreement in region C very far out-of-field.", "By this point most of the phantom scatter has been attenuated, leaving only leakage through the primary collimator and head shielding.", "Figure: Two 10 ×\\times 10 cm 2 ^2 Y profiles, one MLC defined, and one jaw defined.", "Marked features are explained in the article text.The 10 $\\times $ 10 cm$^2$ MLC defined profiles were measured at depths of 10, 15, 20, and 25 cm to investigate the variation of off-axis ratio with depth.", "Figure REF displays the X profiles normalised to the central axis dose of the profile at 10 cm depth.", "Remarkably, the out-of-field dose differs only marginally between depths.", "This indicates that out-of-field doses are largely independent of depth, at least at the distances and depths measured in this study.", "This finding is in line with earlier studies[6], but may not hold closer to the surface due to electrons scattered from the treatment head[12].", "Figure: 10 ×\\times 10 cm 2 ^2 MLC defined X profiles measured at four different depths.", "All four profiles have been normalised to the central axis value of the 10 cm depth profile.As shown in Figures REF and REF , many features in the out-of-field dose are related to the MLC.", "Figure REF shows X and Y profiles for a 5 $\\times $ 5 cm$^2$ MLC defined field.", "The two features A and B have been seen earlier (the former related to the limited extent of the MLC bank in the Y direction exemplified in figure REF , and the latter related to the finite length of the MLC leaves exemplified in figure REF ), however, the magnitude of the features relative to the central axis are larger for the smaller field compared to the 10 $\\times $ 10 or 15 $\\times $ 15 cm$^2$ fields.", "Due to the smaller field size, there is less phantom and collimator scatter to wash out these features in the relative dose.", "Figure REF presents the 10 $\\times $ 10 cm$^2$ X profile where the MLC is fully retracted with the 10 $\\times $ 10 cm$^2$ Y profile in which there is no MLC present.", "From the central axis to approximately 21 cm off-axis, denoted feature A, there is a small difference in the relative dose likely due to the difference in vertical position of the X and Y jaws.", "Beyond 21 cm from the central axis (feature B), the presence of the retracted MLC reduces the X profile out-of-field dose to about 50% to 75% of the Y profile with no MLC present.", "This demonstrates the substantial effect that the MLC can have on shielding radiation far out-of-field, as noted by other authors[9].", "Figure: X and Y profile of the 5 ×\\times 5 cm 2 ^2 MLC defined field.", "Marked features are explained in the article text.Figure: Comparison of a 10 ×\\times 10 cm 2 ^2 jaw defined Y profile with an MLC defined X profile.", "Marked features are explained in the article text.All profiles were measured in both the radial and transverse directions.", "Differences were only noted very far out-of-field.", "In all cases, the general shape of the relative dose out-of-field was similar between radial and transverse measurements, however, the transverse relative doses were of greater magnitude.", "This was best exemplified in the 5 $\\times $ 5 cm$^2$ MLC defined configuration, shown in Figure REF (feature A).", "This effect may be explained by an asymmetry in the linac head shielding.", "Figure: Comparison of a 5 ×\\times 5 cm 2 ^2 MLC defined X profile measured in both the radial (GT) and transverse (LR) orientations.", "Marked features are explained in the article text.IV.B.Limitations Many aspects of the above discussion involve observing features in out-of-field dose profiles and relating those features to the internal geometry of the linac head.", "There are many assumed relationships that cannot be stringently verified without Monte Carlo simulations that include a high-fidelity reproduction of the linac head, including the dimensions and locations of shielding blocks.", "Linac vendors would need to be willing to share detailed 3D models to facilitate this as the phase spaces and simplified geometries commonly shared would not be suitable for modelling complex interactions in the head shielding.", "Such an arrangement has been possible in the past[12], [13], [14], and we encourage all vendors to be open to sharing this information into the future.", "The measurements reported in this study were performed in a radiotherapy treatment clinic with a single model of linac, and so the extent to which this dataset can be extrapolated to other linac models is not obvious.", "It is reasonable to expect that the near out-of-field results may align with other models with similar tertiary collimation systems, such as the Varian Truebeam with Millennium 120 MLC.", "However, far out-of-field the results are more likely to depend on the exact head shielding arrangements, and so the similarity between systems in unclear.", "Linacs with markedly different collimation systems, such as those produced by Elekta (Stockholm, Sweden), may have substantially different out-of-field features.", "Ultimately, we recommend that clinics gather their own dataset using the techniques presented in this study.", "We note however that our method would need substantial modification for use with cylindrical water tanks such as the 3D SCANNER (Sun Nuclear Corporation, Melbourne, USA), but should be directly transferable to other square tanks.", "IV.C.Recommendations We recommend that this dataset is used to further investigate and understand features of out-of-field dose distributions, particularly by assisting in the creation and validation of computational models.", "Furthermore, we encourage readers to perform their own measurements and compare and contrast them with this dataset.", "When planning the tank shift, we recommend that the overlap region of the inner and outer profile sections be centred around 20 cm from the central axis.", "This region contains several strong features, such as the beginning of the primary collimator, which allows the profiles to be stitched with confidence.", "For clinics wishing to save time by collecting a more limited dataset for radiation protection calculations, we recommend collecting a single profile for a variety of field sizes.", "We believe the ideal profile is under the Y jaw, with the MLC fully retracted, in the radial orientation.", "This specific profile is compared against all others in Figure REF .", "From the figure it can seen that this profile represents an upper bound of the out-of-field dose for much of the range covered in this study, and can therefore be used as a conservative estimate of the dose across this range.", "Being in the radial direction, it is also representative of the majority of radiotherapy treatments.", "We also make the suggestion that this same profile may be a good candidate for aiding in the development of out-of-field dose calculation models.", "As it is the only profile uncoupled from extra MLC shielding, it is a good basic test case.", "MLC shielding may then be added as a second order effect.", "Figure: The jaw defined Y profiles for the three field sizes in this study (solid lines).", "The shaded areas correspond to the complete range of profiles measured for each field size.V.Conclusion This dataset article has presented a comprehensive dataset of dose profiles outside of the treatment field.", "These profiles were collected in a non-academic radiotherapy treatment centre, with standard equipment, using a technique that should be repeatable elsewhere.", "Ultimately, we encourage other clinics to collect their own datasets and make them freely available to the community.", "Access to high-quality collections of out-of-field profiles for a range of contemporary linacs would be extremely useful for risk assessments, radiation protection studies, and the development and commissioning of out-of-field dose calculation models.", "VI.Acknowledgements The authors would like to thank Steven R. Sylvander for his helpful insights during the planning of this study.", "VII.References K. D. Miller, L. Nogueira, A. B.", "Mariotto, J. H. Rowland, K. R. Yabroff, C. M. Alfano, A. Jemal, J. L. Kramer, and R. L. Siegel, Cancer treatment and survivorship statistics, 2019, CA: A Cancer Journal for Clinicians 69, 363–385 (2019).", "C. B. Dracham, A. Shankar, and R. Madan, Radiation induced secondary malignancies: a review article, Radiation oncology journal 36, 85 (2018).", "S. F. Kry, B. Bednarz, R. M. Howell, L. Dauer, D. Followill, E. Klein, H. Paganetti, B. Wang, C.-S. Wuu, and X. G. Xu, AAPM TG 158: measurement and calculation of doses outside the treated volume from external-beam radiation therapy, Medical physics 44, e391–e429 (2017).", "M. Miften et al., Management of radiotherapy patients with implanted cardiac pacemakers and defibrillators: A Report of the AAPM TG-203, Medical physics 46, e757–e788 (2019).", "S. C. Peet, R. Wilks, T. Kairn, and S. B. Crowe, Measuring dose from radiotherapy treatments in the vicinity of a cardiac pacemaker, Physica Medica 32, 1529–1536 (2016).", "M. Stovall, C. R. Blackwell, J. Cundiff, D. H. Novack, J. R. Palta, L. K. Wagner, E. W. Webster, and R. J. Shalek, Fetal dose from radiotherapy with photon beams: report of AAPM Radiation Therapy Committee Task Group No.", "36, Medical physics 22, 63–82 (1995).", "S. C. Peet, T. Kairn, C. M. Lancaster, J. V. Trapp, S. R. Sylvander, and S. B. Crowe, Measuring foetal dose from tomotherapy treatments, Medical Dosimetry in press (2021).", "R. Harrison, Out-of-field doses in radiotherapy: Input to epidemiological studies and dose-risk models, Physica Medica 42, 239–246 (2017).", "S. Mutic and E. E. Klein, A reduction in the AAPM TG-36 reported peripheral dose distributions with tertiary multileaf collimation, International Journal of Radiation Oncology* Biology* Physics 44, 947–953 (1999).", "J. D. Ruben, C. M. Lancaster, P. Jones, and R. L. Smith, A comparison of out-of-field dose and its constituent components for intensity-modulated radiation therapy versus conformal radiation therapy: implications for carcinogenesis, International Journal of Radiation Oncology* Biology* Physics 81, 1458–1464 (2011).", "S. Miljanić, J.-M. Bordy, F. d'Errico, R. Harrison, and P. Olko, Out-of-field dose measurements in radiotherapy–An overview of activity of EURADOS Working Group 9: Radiation protection in medicine, Radiation measurements 71, 270–275 (2014).", "S. F. Kry, U. Titt, F. Pönisch, D. Followill, O. N. Vassiliev, R. Allen White, R. Mohan, and M. Salehpour, A Monte Carlo model for calculating out-of-field dose from a Varian beam, Medical physics 33, 4405–4413 (2006).", "S. F. Kry, U. Titt, D. Followill, F. Pönisch, O. N. Vassiliev, R. A.", "White, M. Stovall, and M. Salehpour, A Monte Carlo model for out-of-field dose calculation from high-energy photon therapy, Medical physics 34, 3489–3499 (2007).", "B. Bednarz and X. G. Xu, Monte Carlo modeling of a 6 and 18 MV Varian Clinac medical accelerator for in-field and out-of-field dose calculations: development and validation, Physics in Medicine & Biology 54, N43 (2009).", "M. A. Benadjaoud, J. Bezin, A. Veres, D. Lefkopoulos, J. Chavaudra, A. Bridier, F. de Vathaire, and I. Diallo, A multi-plane source model for out-of-field head scatter dose calculations in external beam photon therapy, Physics in Medicine & Biology 57, 7725 (2012).", "P. J. Taddei, W. Jalbout, R. M. Howell, N. Khater, F. Geara, K. Homann, and W. D. Newhauser, Analytical model for out-of-field dose in photon craniospinal irradiation, Physics in Medicine and Biology 58, 7463–7479 (2013).", "C. W. Schneider, W. D. Newhauser, L. J. Wilson, and R.-P. Kapsch, A physics-based analytical model of absorbed dose from primary, leakage, and scattered photons from megavoltage radiotherapy with MLCs, Physics in Medicine & Biology 64, 185017 (2019).", "L. J. Wilson, W. D. Newhauser, C. W. Schneider, F. Kamp, M. Reiner, J. C. Martins, G. Landry, A. Giussani, R.-P. Kapsch, and K. Parodi, Method to quickly and accurately calculate absorbed dose from therapeutic and stray photon exposures throughout the entire body in individual patients, Medical physics 47, 2254–2266 (2020).", "T. Kairn, S. B. Crowe, and S. C. Peet, Linac Leakage Dose Received by Patients Treated Using Non-coplanar Radiotherapy Beams, in World Congress on Medical Physics and Biomedical Engineering 2018, edited by L. Lhotska, L. Sukupova, I. Lacković, and G. S. Ibbott, pages 549–551, Singapore, 2019, Springer Singapore.", "S. C. Peet, [dataset] Varian iX 6 MV Extended Profiles, 2021, https://doi.org/10.5281/zenodo.4722077 II.Acquisition and Validation Methods II.A.Equipment The linac under investigation in this study was a Varian Clinac iX (Varian Medical Systems, Palo Alto, USA) with a Millennium 120 MLC.", "All measurements were performed using the 6 MV photon energy with 600 MU/min dose rate.", "A PTW Semiflex 0.3 cm$^3$ 31013 (PTW-Freiburg, Freiburg, Germany) ionisation chamber was used for all profile scans.", "The dimensions of this chamber afforded lower volume averaging compared to larger thimble chambers, while still maintaining the sensitivity necessary for far out-of-field measurements.", "The chamber was affixed to a PTW BEAMSCAN water tank using the TRUFIX chamber positioning system and steered with version 4.3 of the control software.", "The BEAMSCAN water tank had a usable scanning range of 500 mm (horiz.)", "$\\times $ 500 mm (horiz.)", "$\\times $ 415 mm (vert.", "), and 15 mm thick PMMA walls.", "A PTW Semiflex 3D 0.07 cm$^3$ 31021 ionisation chamber was used as a reference to correct for fluctuations in the beam output during each scan.", "The chamber voltages were set to 400 V as per manufacturer recommendations, and the integrated water tank electrometer was set to low range.", "The control software was programmed to measure profiles in continuous scanning mode with a chamber speed of 5 mm/s and data points reported every 2 mm.", "A 30 $\\times $ 30 $\\times $ 40 cm$^3$ stack of Virtual Water (Standard Imaging, Middleton, USA) was used as additional scattering material.", "II.B.Measurement Setup Each profile was measured as a piece-wise combination of two scans to construct a much longer profile measurement than a typical water tank would allow.", "The first section acquired the in-field and near out-of-field region of each profile, while the second section captured the far out-of-field region.", "An overlap area of approximately 15 cm was included in each pair of profile sections.", "The first measurement geometry can be seen in Figure REF a.", "To begin, the tank was positioned such that the long axis of the chamber was orthogonal to the scanning direction, with a source to surface distance of 90 cm.", "The tank was then translated such that the central axis of the beam was 15 cm from the maximum chamber travel position on one side, and 35 cm from the maximum chamber travel position on the opposite side.", "This allowed the inner section of each profile to be collected out to 35 cm from the central axis while still allowing adequate scatter around the field.", "The measurement geometry was then altered to capture the far out-of-field section of each profile, as seen in Figure REF b.", "The tank was translated 31.5 cm in the direction of the profile, and a stack of Virtual Water was positioned in the field at 90 cm source to surface distance to re-establish full scattering conditions.", "Care was taken to ensure that the Virtual Water was abutted firmly against the tank wall with minimal air gaps.", "Figure: Equipment setup for the measurement of profiles in this study.", "(a) Inner profile section: The field is directed into the tank near the wall.", "(b) Outer profile section: The tank is shifted outside of the field and abutted to a stack of Virtual Water to ensure full scattering conditions.II.C.Post-processing The two sections of each profile were stitched together to create a whole.", "This was non-trivial, as the overlap regions did not necessarily coincide due to small air gaps between the Virtual Water and the tank wall, the non-water-equivalence of the tank wall, and positioning errors in the tank.", "To overcome this issue, each overlap region was searched for a prominent feature common to both sections of the profile.", "The far out-of-field section was then progressively shifted in 1 mm increments and re-scaled until the identified feature matched the inner profile section as well as possible.", "The two sections were then combined into one full profile, with an average being taken in the overlap region.", "A visual example of this process can be seen in Figure REF .", "No additional smoothing or filtering was applied.", "Figure: The process of stitching two profile sections to make a whole.", "(a) The two raw profile sections with a strong feature for matching.", "(b) The outer section shifted such that the features overlap.", "(c) The two sections conjoined, with the average value taken in the overlap region.II.D.Data Validation In order to establish that the irradiation conditions and measurement setup were representative of normal practice, the inner component of the 10 $\\times $ 10 cm$^2$ X and Y profiles were compared against the same profiles gathered during commissioning of the linac.", "A gamma comparison of the in-field sections, bounded by the 50% isodose lines, showed 100% agreement with criteria of 1% dose difference (local normalisation) and 1 mm distance-to-agreement.", "This gave good confidence that the measurement equipment was set up correctly, that the linac was behaving nominally, and that ultimately the data gathered in the session was representative of clinical practice.", "As mentioned in the previous section, the outer section of each profile was shifted to coincide with the inner section of the profile.", "Given that the equipment was not moved during the measurement of all outer profile sections within a given orientation (radial or transverse), it follows that the ideal shift should be the same across all profiles.", "This was indeed found to be the case with the largest difference in ideal shifts being 1 mm.", "After combining the profile halves and taking the average in the overlap region, all full profiles were then manually inspected to ensure that there were no obvious discontinuities in the overlap region.", "None were found.", "II.E.Summary of Collected Profiles In total, 36 profiles were collected (see summary in Table REF ).", "Profiles were gathered at three nominal field sizes: 5 $\\times $ 5 cm$^2$ , 10 $\\times $ 10 cm$^2$ , and 15 $\\times $ 15 cm$^2$ .", "Profiles in both the X and Y directions were measured with the jaws defining the field and the MLC in a 'parked' state (leaf tips fully retracted to approximately 21 cm off axis).", "These were repeated with the MLC defining the field and the jaws at the Varian recommended positions (X jaws 8 mm retracted from nominal field edge, Y jaws 2 mm retracted from nominal field edge).", "All profiles were measured with a source to surface distance of 90 cm and depth of 10 cm, except for the 10 $\\times $ 10 cm$^2$ MLC defined profiles which were additionally measured at depths of 15, 20, and 25 cm.", "Every profile was measured twice: once aligning with the radial (gun-target) direction and once aligning with the transverse (left-right) direction.", "The collimator angle was set appropriately to achieve this, for example, a transverse X profile required collimator 0$^{\\circ }$ , while a radial X profile required collimator 90$^{\\circ }$ (or 270$^{\\circ }$ ).", "The radial direction corresponds to the cranio-caudal axis of a patient receiving an coplanar treatment with no couch rotation and the transverse direction corresponds to the cranio-caudal axis of a patient receiving a treatment with a 90$^{\\circ }$ couch rotation.", "The radial and transverse profiles were collected in two separate measurement sessions as they required a complete reorientation of the tank.", "In total, the measurement time was approximately eight hours, including equipment setup/cleanup and data collection.", "Table: Field configurations measured in this study.", "For each row in the table, X and Y profiles were gathered in both the radial and transverse orientations,III.Data Format and Usage Notes The dataset had been made publicly accessible through the Zenodo platform and released under a Creative Commons Attribution 4.0 licence[20].", "The dataset consists of a single comma seperated variable (CSV) file containing 37 columns, with the first column storing the off-axis distance of each measurement point, and the remaining 36 columns storing the off-axis ratios at these points for each profile.", "The first row contains a label for each profile with the format '[$field\\_size$ ] [$mlc\\_state$ ] [$depth$ ] [$direction$ ] [$orientation$ ]' where $field\\_size$ is the nominal field size at the isocentre, $mlc\\_state$ designates the position of the MLC, taking the value `MLC defined' when the MLC defines the field, and `MLC parked' when the MLC is fully retracted, $depth$ is the depth of measurement, $direction$ indicates the direction of the profile, either X or Y, and $orientation$ indicates whether the measurement orientation was radial (GT) or transverse (LR).", "For example, a profile under the Y jaw of a 10 $\\times $ 10 cm$^2$ MLC defined field measured at a depth of 10 cm in the radial direction would have the label '10 x 10 MLC defined Y d10cm (GT)'.", "Each profile was normalised to the value at the central axis at the time of measurement so no further processing is needed to recover off-axis ratios.", "Users should be aware that the profiles are of unequal length, depending on whether they were acquired in the radial or transverse measurement orientation.", "IV.Discussion IV.A.An Exploration of Dose Outside the Treatment Field This investigation produced a large dataset with many curious features.", "In this section we compare and contrast a number of profiles that illuminate the underlying geometry and radiation interactions.", "However, these are just a representative selection of the profiles, and we encourage interested readers to further examine the dataset for themselves.", "Figure REF presents two 10 $\\times $ 10 cm$^2$ X profiles, one of which is defined by the MLC, and the other is defined by the jaw (MLC retracted).", "Both profiles were measured in the transverse direction.", "Feature A corresponds to a sharp decrease in relative dose in the jaw defined profile due to the combined effect of the primary collimator and the MLC leaves being retracted to this position.", "This feature is not seen in the MLC defined profile.", "Feature B denotes a large difference in the profiles very far out-of-field.", "This may be explained by this region being shielded by the tails of the MLC leaves when they are fully retracted, but not shielded when the leaves are moved in to define the field.", "Figure: Two 10 ×\\times 10 cm 2 ^2 X profiles, one MLC defined, and one jaw defined (MLC fully retracted).", "Marked features are explained in the article text.Two 10 $\\times $ 10 cm$^2$ Y profiles, one of which is defined by the jaw (MLC retracted), and the other is defined by the MLC, are shown in Figure REF .", "Both of these profiles were collected in the radial direction.", "Notably, the Y jaw defined profile is the only field arrangement giving a pure jaw profile uncoupled from additional MLC shielding effects.", "Three features are marked in the figure.", "Feature A denotes a sharp increase in relative dose at approximately 20 cm off axis, due to the finite lateral extent of the MLC.", "This feature has been observed in an earlier study[10].", "This is immediately followed by a decrease in relative dose due to the primary collimator, which can also be seen on the jaw defined profile.", "Feature B denotes an area in which the two profiles have the same shielding conditions (jaw, primary collimator, no MLC) and yet have different relative doses.", "This may be explained by a difference in lateral phantom scatter from leakage more centrally in the field.", "The profile with both the jaw and MLC shielding the beam has less leakage relative to the jaw only profile, and therefore also has less lateral phantom scatter from this leakage.", "This is supported by the profiles coming into agreement in region C very far out-of-field.", "By this point most of the phantom scatter has been attenuated, leaving only leakage through the primary collimator and head shielding.", "Figure: Two 10 ×\\times 10 cm 2 ^2 Y profiles, one MLC defined, and one jaw defined.", "Marked features are explained in the article text.The 10 $\\times $ 10 cm$^2$ MLC defined profiles were measured at depths of 10, 15, 20, and 25 cm to investigate the variation of off-axis ratio with depth.", "Figure REF displays the X profiles normalised to the central axis dose of the profile at 10 cm depth.", "Remarkably, the out-of-field dose differs only marginally between depths.", "This indicates that out-of-field doses are largely independent of depth, at least at the distances and depths measured in this study.", "This finding is in line with earlier studies[6], but may not hold closer to the surface due to electrons scattered from the treatment head[12].", "Figure: 10 ×\\times 10 cm 2 ^2 MLC defined X profiles measured at four different depths.", "All four profiles have been normalised to the central axis value of the 10 cm depth profile.As shown in Figures REF and REF , many features in the out-of-field dose are related to the MLC.", "Figure REF shows X and Y profiles for a 5 $\\times $ 5 cm$^2$ MLC defined field.", "The two features A and B have been seen earlier (the former related to the limited extent of the MLC bank in the Y direction exemplified in figure REF , and the latter related to the finite length of the MLC leaves exemplified in figure REF ), however, the magnitude of the features relative to the central axis are larger for the smaller field compared to the 10 $\\times $ 10 or 15 $\\times $ 15 cm$^2$ fields.", "Due to the smaller field size, there is less phantom and collimator scatter to wash out these features in the relative dose.", "Figure REF presents the 10 $\\times $ 10 cm$^2$ X profile where the MLC is fully retracted with the 10 $\\times $ 10 cm$^2$ Y profile in which there is no MLC present.", "From the central axis to approximately 21 cm off-axis, denoted feature A, there is a small difference in the relative dose likely due to the difference in vertical position of the X and Y jaws.", "Beyond 21 cm from the central axis (feature B), the presence of the retracted MLC reduces the X profile out-of-field dose to about 50% to 75% of the Y profile with no MLC present.", "This demonstrates the substantial effect that the MLC can have on shielding radiation far out-of-field, as noted by other authors[9].", "Figure: X and Y profile of the 5 ×\\times 5 cm 2 ^2 MLC defined field.", "Marked features are explained in the article text.Figure: Comparison of a 10 ×\\times 10 cm 2 ^2 jaw defined Y profile with an MLC defined X profile.", "Marked features are explained in the article text.All profiles were measured in both the radial and transverse directions.", "Differences were only noted very far out-of-field.", "In all cases, the general shape of the relative dose out-of-field was similar between radial and transverse measurements, however, the transverse relative doses were of greater magnitude.", "This was best exemplified in the 5 $\\times $ 5 cm$^2$ MLC defined configuration, shown in Figure REF (feature A).", "This effect may be explained by an asymmetry in the linac head shielding.", "Figure: Comparison of a 5 ×\\times 5 cm 2 ^2 MLC defined X profile measured in both the radial (GT) and transverse (LR) orientations.", "Marked features are explained in the article text.IV.B.Limitations Many aspects of the above discussion involve observing features in out-of-field dose profiles and relating those features to the internal geometry of the linac head.", "There are many assumed relationships that cannot be stringently verified without Monte Carlo simulations that include a high-fidelity reproduction of the linac head, including the dimensions and locations of shielding blocks.", "Linac vendors would need to be willing to share detailed 3D models to facilitate this as the phase spaces and simplified geometries commonly shared would not be suitable for modelling complex interactions in the head shielding.", "Such an arrangement has been possible in the past[12], [13], [14], and we encourage all vendors to be open to sharing this information into the future.", "The measurements reported in this study were performed in a radiotherapy treatment clinic with a single model of linac, and so the extent to which this dataset can be extrapolated to other linac models is not obvious.", "It is reasonable to expect that the near out-of-field results may align with other models with similar tertiary collimation systems, such as the Varian Truebeam with Millennium 120 MLC.", "However, far out-of-field the results are more likely to depend on the exact head shielding arrangements, and so the similarity between systems in unclear.", "Linacs with markedly different collimation systems, such as those produced by Elekta (Stockholm, Sweden), may have substantially different out-of-field features.", "Ultimately, we recommend that clinics gather their own dataset using the techniques presented in this study.", "We note however that our method would need substantial modification for use with cylindrical water tanks such as the 3D SCANNER (Sun Nuclear Corporation, Melbourne, USA), but should be directly transferable to other square tanks.", "IV.C.Recommendations We recommend that this dataset is used to further investigate and understand features of out-of-field dose distributions, particularly by assisting in the creation and validation of computational models.", "Furthermore, we encourage readers to perform their own measurements and compare and contrast them with this dataset.", "When planning the tank shift, we recommend that the overlap region of the inner and outer profile sections be centred around 20 cm from the central axis.", "This region contains several strong features, such as the beginning of the primary collimator, which allows the profiles to be stitched with confidence.", "For clinics wishing to save time by collecting a more limited dataset for radiation protection calculations, we recommend collecting a single profile for a variety of field sizes.", "We believe the ideal profile is under the Y jaw, with the MLC fully retracted, in the radial orientation.", "This specific profile is compared against all others in Figure REF .", "From the figure it can seen that this profile represents an upper bound of the out-of-field dose for much of the range covered in this study, and can therefore be used as a conservative estimate of the dose across this range.", "Being in the radial direction, it is also representative of the majority of radiotherapy treatments.", "We also make the suggestion that this same profile may be a good candidate for aiding in the development of out-of-field dose calculation models.", "As it is the only profile uncoupled from extra MLC shielding, it is a good basic test case.", "MLC shielding may then be added as a second order effect.", "Figure: The jaw defined Y profiles for the three field sizes in this study (solid lines).", "The shaded areas correspond to the complete range of profiles measured for each field size.V.Conclusion This dataset article has presented a comprehensive dataset of dose profiles outside of the treatment field.", "These profiles were collected in a non-academic radiotherapy treatment centre, with standard equipment, using a technique that should be repeatable elsewhere.", "Ultimately, we encourage other clinics to collect their own datasets and make them freely available to the community.", "Access to high-quality collections of out-of-field profiles for a range of contemporary linacs would be extremely useful for risk assessments, radiation protection studies, and the development and commissioning of out-of-field dose calculation models.", "VI.Acknowledgements The authors would like to thank Steven R. Sylvander for his helpful insights during the planning of this study.", "VII.References K. D. Miller, L. Nogueira, A. B.", "Mariotto, J. H. Rowland, K. R. Yabroff, C. M. Alfano, A. Jemal, J. L. Kramer, and R. L. Siegel, Cancer treatment and survivorship statistics, 2019, CA: A Cancer Journal for Clinicians 69, 363–385 (2019).", "C. B. Dracham, A. Shankar, and R. Madan, Radiation induced secondary malignancies: a review article, Radiation oncology journal 36, 85 (2018).", "S. F. Kry, B. Bednarz, R. M. Howell, L. Dauer, D. Followill, E. Klein, H. Paganetti, B. Wang, C.-S. Wuu, and X. G. Xu, AAPM TG 158: measurement and calculation of doses outside the treated volume from external-beam radiation therapy, Medical physics 44, e391–e429 (2017).", "M. Miften et al., Management of radiotherapy patients with implanted cardiac pacemakers and defibrillators: A Report of the AAPM TG-203, Medical physics 46, e757–e788 (2019).", "S. C. Peet, R. Wilks, T. Kairn, and S. B. Crowe, Measuring dose from radiotherapy treatments in the vicinity of a cardiac pacemaker, Physica Medica 32, 1529–1536 (2016).", "M. Stovall, C. R. Blackwell, J. Cundiff, D. H. Novack, J. R. Palta, L. K. Wagner, E. W. Webster, and R. J. Shalek, Fetal dose from radiotherapy with photon beams: report of AAPM Radiation Therapy Committee Task Group No.", "36, Medical physics 22, 63–82 (1995).", "S. C. Peet, T. Kairn, C. M. Lancaster, J. V. Trapp, S. R. Sylvander, and S. B. Crowe, Measuring foetal dose from tomotherapy treatments, Medical Dosimetry in press (2021).", "R. Harrison, Out-of-field doses in radiotherapy: Input to epidemiological studies and dose-risk models, Physica Medica 42, 239–246 (2017).", "S. Mutic and E. E. Klein, A reduction in the AAPM TG-36 reported peripheral dose distributions with tertiary multileaf collimation, International Journal of Radiation Oncology* Biology* Physics 44, 947–953 (1999).", "J. D. Ruben, C. M. Lancaster, P. Jones, and R. L. Smith, A comparison of out-of-field dose and its constituent components for intensity-modulated radiation therapy versus conformal radiation therapy: implications for carcinogenesis, International Journal of Radiation Oncology* Biology* Physics 81, 1458–1464 (2011).", "S. Miljanić, J.-M. Bordy, F. d'Errico, R. Harrison, and P. Olko, Out-of-field dose measurements in radiotherapy–An overview of activity of EURADOS Working Group 9: Radiation protection in medicine, Radiation measurements 71, 270–275 (2014).", "S. F. Kry, U. Titt, F. Pönisch, D. Followill, O. N. Vassiliev, R. Allen White, R. Mohan, and M. Salehpour, A Monte Carlo model for calculating out-of-field dose from a Varian beam, Medical physics 33, 4405–4413 (2006).", "S. F. Kry, U. Titt, D. Followill, F. Pönisch, O. N. Vassiliev, R. A.", "White, M. Stovall, and M. Salehpour, A Monte Carlo model for out-of-field dose calculation from high-energy photon therapy, Medical physics 34, 3489–3499 (2007).", "B. Bednarz and X. G. Xu, Monte Carlo modeling of a 6 and 18 MV Varian Clinac medical accelerator for in-field and out-of-field dose calculations: development and validation, Physics in Medicine & Biology 54, N43 (2009).", "M. A. Benadjaoud, J. Bezin, A. Veres, D. Lefkopoulos, J. Chavaudra, A. Bridier, F. de Vathaire, and I. Diallo, A multi-plane source model for out-of-field head scatter dose calculations in external beam photon therapy, Physics in Medicine & Biology 57, 7725 (2012).", "P. J. Taddei, W. Jalbout, R. M. Howell, N. Khater, F. Geara, K. Homann, and W. D. Newhauser, Analytical model for out-of-field dose in photon craniospinal irradiation, Physics in Medicine and Biology 58, 7463–7479 (2013).", "C. W. Schneider, W. D. Newhauser, L. J. Wilson, and R.-P. Kapsch, A physics-based analytical model of absorbed dose from primary, leakage, and scattered photons from megavoltage radiotherapy with MLCs, Physics in Medicine & Biology 64, 185017 (2019).", "L. J. Wilson, W. D. Newhauser, C. W. Schneider, F. Kamp, M. Reiner, J. C. Martins, G. Landry, A. Giussani, R.-P. Kapsch, and K. Parodi, Method to quickly and accurately calculate absorbed dose from therapeutic and stray photon exposures throughout the entire body in individual patients, Medical physics 47, 2254–2266 (2020).", "T. Kairn, S. B. Crowe, and S. C. Peet, Linac Leakage Dose Received by Patients Treated Using Non-coplanar Radiotherapy Beams, in World Congress on Medical Physics and Biomedical Engineering 2018, edited by L. Lhotska, L. Sukupova, I. Lacković, and G. S. Ibbott, pages 549–551, Singapore, 2019, Springer Singapore.", "S. C. Peet, [dataset] Varian iX 6 MV Extended Profiles, 2021, https://doi.org/10.5281/zenodo.4722077 III.Data Format and Usage Notes The dataset had been made publicly accessible through the Zenodo platform and released under a Creative Commons Attribution 4.0 licence[20].", "The dataset consists of a single comma seperated variable (CSV) file containing 37 columns, with the first column storing the off-axis distance of each measurement point, and the remaining 36 columns storing the off-axis ratios at these points for each profile.", "The first row contains a label for each profile with the format '[$field\\_size$ ] [$mlc\\_state$ ] [$depth$ ] [$direction$ ] [$orientation$ ]' where $field\\_size$ is the nominal field size at the isocentre, $mlc\\_state$ designates the position of the MLC, taking the value `MLC defined' when the MLC defines the field, and `MLC parked' when the MLC is fully retracted, $depth$ is the depth of measurement, $direction$ indicates the direction of the profile, either X or Y, and $orientation$ indicates whether the measurement orientation was radial (GT) or transverse (LR).", "For example, a profile under the Y jaw of a 10 $\\times $ 10 cm$^2$ MLC defined field measured at a depth of 10 cm in the radial direction would have the label '10 x 10 MLC defined Y d10cm (GT)'.", "Each profile was normalised to the value at the central axis at the time of measurement so no further processing is needed to recover off-axis ratios.", "Users should be aware that the profiles are of unequal length, depending on whether they were acquired in the radial or transverse measurement orientation.", "IV.Discussion IV.A.An Exploration of Dose Outside the Treatment Field This investigation produced a large dataset with many curious features.", "In this section we compare and contrast a number of profiles that illuminate the underlying geometry and radiation interactions.", "However, these are just a representative selection of the profiles, and we encourage interested readers to further examine the dataset for themselves.", "Figure REF presents two 10 $\\times $ 10 cm$^2$ X profiles, one of which is defined by the MLC, and the other is defined by the jaw (MLC retracted).", "Both profiles were measured in the transverse direction.", "Feature A corresponds to a sharp decrease in relative dose in the jaw defined profile due to the combined effect of the primary collimator and the MLC leaves being retracted to this position.", "This feature is not seen in the MLC defined profile.", "Feature B denotes a large difference in the profiles very far out-of-field.", "This may be explained by this region being shielded by the tails of the MLC leaves when they are fully retracted, but not shielded when the leaves are moved in to define the field.", "Figure: Two 10 ×\\times 10 cm 2 ^2 X profiles, one MLC defined, and one jaw defined (MLC fully retracted).", "Marked features are explained in the article text.Two 10 $\\times $ 10 cm$^2$ Y profiles, one of which is defined by the jaw (MLC retracted), and the other is defined by the MLC, are shown in Figure REF .", "Both of these profiles were collected in the radial direction.", "Notably, the Y jaw defined profile is the only field arrangement giving a pure jaw profile uncoupled from additional MLC shielding effects.", "Three features are marked in the figure.", "Feature A denotes a sharp increase in relative dose at approximately 20 cm off axis, due to the finite lateral extent of the MLC.", "This feature has been observed in an earlier study[10].", "This is immediately followed by a decrease in relative dose due to the primary collimator, which can also be seen on the jaw defined profile.", "Feature B denotes an area in which the two profiles have the same shielding conditions (jaw, primary collimator, no MLC) and yet have different relative doses.", "This may be explained by a difference in lateral phantom scatter from leakage more centrally in the field.", "The profile with both the jaw and MLC shielding the beam has less leakage relative to the jaw only profile, and therefore also has less lateral phantom scatter from this leakage.", "This is supported by the profiles coming into agreement in region C very far out-of-field.", "By this point most of the phantom scatter has been attenuated, leaving only leakage through the primary collimator and head shielding.", "Figure: Two 10 ×\\times 10 cm 2 ^2 Y profiles, one MLC defined, and one jaw defined.", "Marked features are explained in the article text.The 10 $\\times $ 10 cm$^2$ MLC defined profiles were measured at depths of 10, 15, 20, and 25 cm to investigate the variation of off-axis ratio with depth.", "Figure REF displays the X profiles normalised to the central axis dose of the profile at 10 cm depth.", "Remarkably, the out-of-field dose differs only marginally between depths.", "This indicates that out-of-field doses are largely independent of depth, at least at the distances and depths measured in this study.", "This finding is in line with earlier studies[6], but may not hold closer to the surface due to electrons scattered from the treatment head[12].", "Figure: 10 ×\\times 10 cm 2 ^2 MLC defined X profiles measured at four different depths.", "All four profiles have been normalised to the central axis value of the 10 cm depth profile.As shown in Figures REF and REF , many features in the out-of-field dose are related to the MLC.", "Figure REF shows X and Y profiles for a 5 $\\times $ 5 cm$^2$ MLC defined field.", "The two features A and B have been seen earlier (the former related to the limited extent of the MLC bank in the Y direction exemplified in figure REF , and the latter related to the finite length of the MLC leaves exemplified in figure REF ), however, the magnitude of the features relative to the central axis are larger for the smaller field compared to the 10 $\\times $ 10 or 15 $\\times $ 15 cm$^2$ fields.", "Due to the smaller field size, there is less phantom and collimator scatter to wash out these features in the relative dose.", "Figure REF presents the 10 $\\times $ 10 cm$^2$ X profile where the MLC is fully retracted with the 10 $\\times $ 10 cm$^2$ Y profile in which there is no MLC present.", "From the central axis to approximately 21 cm off-axis, denoted feature A, there is a small difference in the relative dose likely due to the difference in vertical position of the X and Y jaws.", "Beyond 21 cm from the central axis (feature B), the presence of the retracted MLC reduces the X profile out-of-field dose to about 50% to 75% of the Y profile with no MLC present.", "This demonstrates the substantial effect that the MLC can have on shielding radiation far out-of-field, as noted by other authors[9].", "Figure: X and Y profile of the 5 ×\\times 5 cm 2 ^2 MLC defined field.", "Marked features are explained in the article text.Figure: Comparison of a 10 ×\\times 10 cm 2 ^2 jaw defined Y profile with an MLC defined X profile.", "Marked features are explained in the article text.All profiles were measured in both the radial and transverse directions.", "Differences were only noted very far out-of-field.", "In all cases, the general shape of the relative dose out-of-field was similar between radial and transverse measurements, however, the transverse relative doses were of greater magnitude.", "This was best exemplified in the 5 $\\times $ 5 cm$^2$ MLC defined configuration, shown in Figure REF (feature A).", "This effect may be explained by an asymmetry in the linac head shielding.", "Figure: Comparison of a 5 ×\\times 5 cm 2 ^2 MLC defined X profile measured in both the radial (GT) and transverse (LR) orientations.", "Marked features are explained in the article text.IV.B.Limitations Many aspects of the above discussion involve observing features in out-of-field dose profiles and relating those features to the internal geometry of the linac head.", "There are many assumed relationships that cannot be stringently verified without Monte Carlo simulations that include a high-fidelity reproduction of the linac head, including the dimensions and locations of shielding blocks.", "Linac vendors would need to be willing to share detailed 3D models to facilitate this as the phase spaces and simplified geometries commonly shared would not be suitable for modelling complex interactions in the head shielding.", "Such an arrangement has been possible in the past[12], [13], [14], and we encourage all vendors to be open to sharing this information into the future.", "The measurements reported in this study were performed in a radiotherapy treatment clinic with a single model of linac, and so the extent to which this dataset can be extrapolated to other linac models is not obvious.", "It is reasonable to expect that the near out-of-field results may align with other models with similar tertiary collimation systems, such as the Varian Truebeam with Millennium 120 MLC.", "However, far out-of-field the results are more likely to depend on the exact head shielding arrangements, and so the similarity between systems in unclear.", "Linacs with markedly different collimation systems, such as those produced by Elekta (Stockholm, Sweden), may have substantially different out-of-field features.", "Ultimately, we recommend that clinics gather their own dataset using the techniques presented in this study.", "We note however that our method would need substantial modification for use with cylindrical water tanks such as the 3D SCANNER (Sun Nuclear Corporation, Melbourne, USA), but should be directly transferable to other square tanks.", "IV.C.Recommendations We recommend that this dataset is used to further investigate and understand features of out-of-field dose distributions, particularly by assisting in the creation and validation of computational models.", "Furthermore, we encourage readers to perform their own measurements and compare and contrast them with this dataset.", "When planning the tank shift, we recommend that the overlap region of the inner and outer profile sections be centred around 20 cm from the central axis.", "This region contains several strong features, such as the beginning of the primary collimator, which allows the profiles to be stitched with confidence.", "For clinics wishing to save time by collecting a more limited dataset for radiation protection calculations, we recommend collecting a single profile for a variety of field sizes.", "We believe the ideal profile is under the Y jaw, with the MLC fully retracted, in the radial orientation.", "This specific profile is compared against all others in Figure REF .", "From the figure it can seen that this profile represents an upper bound of the out-of-field dose for much of the range covered in this study, and can therefore be used as a conservative estimate of the dose across this range.", "Being in the radial direction, it is also representative of the majority of radiotherapy treatments.", "We also make the suggestion that this same profile may be a good candidate for aiding in the development of out-of-field dose calculation models.", "As it is the only profile uncoupled from extra MLC shielding, it is a good basic test case.", "MLC shielding may then be added as a second order effect.", "Figure: The jaw defined Y profiles for the three field sizes in this study (solid lines).", "The shaded areas correspond to the complete range of profiles measured for each field size.V.Conclusion This dataset article has presented a comprehensive dataset of dose profiles outside of the treatment field.", "These profiles were collected in a non-academic radiotherapy treatment centre, with standard equipment, using a technique that should be repeatable elsewhere.", "Ultimately, we encourage other clinics to collect their own datasets and make them freely available to the community.", "Access to high-quality collections of out-of-field profiles for a range of contemporary linacs would be extremely useful for risk assessments, radiation protection studies, and the development and commissioning of out-of-field dose calculation models.", "VI.Acknowledgements The authors would like to thank Steven R. Sylvander for his helpful insights during the planning of this study.", "VII.References K. D. Miller, L. Nogueira, A. B.", "Mariotto, J. H. Rowland, K. R. Yabroff, C. M. Alfano, A. Jemal, J. L. Kramer, and R. L. Siegel, Cancer treatment and survivorship statistics, 2019, CA: A Cancer Journal for Clinicians 69, 363–385 (2019).", "C. B. Dracham, A. Shankar, and R. Madan, Radiation induced secondary malignancies: a review article, Radiation oncology journal 36, 85 (2018).", "S. F. Kry, B. Bednarz, R. M. Howell, L. Dauer, D. Followill, E. Klein, H. Paganetti, B. Wang, C.-S. Wuu, and X. G. Xu, AAPM TG 158: measurement and calculation of doses outside the treated volume from external-beam radiation therapy, Medical physics 44, e391–e429 (2017).", "M. Miften et al., Management of radiotherapy patients with implanted cardiac pacemakers and defibrillators: A Report of the AAPM TG-203, Medical physics 46, e757–e788 (2019).", "S. C. Peet, R. Wilks, T. Kairn, and S. B. Crowe, Measuring dose from radiotherapy treatments in the vicinity of a cardiac pacemaker, Physica Medica 32, 1529–1536 (2016).", "M. Stovall, C. R. Blackwell, J. Cundiff, D. H. Novack, J. R. Palta, L. K. Wagner, E. W. Webster, and R. J. Shalek, Fetal dose from radiotherapy with photon beams: report of AAPM Radiation Therapy Committee Task Group No.", "36, Medical physics 22, 63–82 (1995).", "S. C. Peet, T. Kairn, C. M. Lancaster, J. V. Trapp, S. R. Sylvander, and S. B. Crowe, Measuring foetal dose from tomotherapy treatments, Medical Dosimetry in press (2021).", "R. Harrison, Out-of-field doses in radiotherapy: Input to epidemiological studies and dose-risk models, Physica Medica 42, 239–246 (2017).", "S. Mutic and E. E. Klein, A reduction in the AAPM TG-36 reported peripheral dose distributions with tertiary multileaf collimation, International Journal of Radiation Oncology* Biology* Physics 44, 947–953 (1999).", "J. D. Ruben, C. M. Lancaster, P. Jones, and R. L. Smith, A comparison of out-of-field dose and its constituent components for intensity-modulated radiation therapy versus conformal radiation therapy: implications for carcinogenesis, International Journal of Radiation Oncology* Biology* Physics 81, 1458–1464 (2011).", "S. Miljanić, J.-M. Bordy, F. d'Errico, R. Harrison, and P. Olko, Out-of-field dose measurements in radiotherapy–An overview of activity of EURADOS Working Group 9: Radiation protection in medicine, Radiation measurements 71, 270–275 (2014).", "S. F. Kry, U. Titt, F. Pönisch, D. Followill, O. N. Vassiliev, R. Allen White, R. Mohan, and M. Salehpour, A Monte Carlo model for calculating out-of-field dose from a Varian beam, Medical physics 33, 4405–4413 (2006).", "S. F. Kry, U. Titt, D. Followill, F. Pönisch, O. N. Vassiliev, R. A.", "White, M. Stovall, and M. Salehpour, A Monte Carlo model for out-of-field dose calculation from high-energy photon therapy, Medical physics 34, 3489–3499 (2007).", "B. Bednarz and X. G. Xu, Monte Carlo modeling of a 6 and 18 MV Varian Clinac medical accelerator for in-field and out-of-field dose calculations: development and validation, Physics in Medicine & Biology 54, N43 (2009).", "M. A. Benadjaoud, J. Bezin, A. Veres, D. Lefkopoulos, J. Chavaudra, A. Bridier, F. de Vathaire, and I. Diallo, A multi-plane source model for out-of-field head scatter dose calculations in external beam photon therapy, Physics in Medicine & Biology 57, 7725 (2012).", "P. J. Taddei, W. Jalbout, R. M. Howell, N. Khater, F. Geara, K. Homann, and W. D. Newhauser, Analytical model for out-of-field dose in photon craniospinal irradiation, Physics in Medicine and Biology 58, 7463–7479 (2013).", "C. W. Schneider, W. D. Newhauser, L. J. Wilson, and R.-P. Kapsch, A physics-based analytical model of absorbed dose from primary, leakage, and scattered photons from megavoltage radiotherapy with MLCs, Physics in Medicine & Biology 64, 185017 (2019).", "L. J. Wilson, W. D. Newhauser, C. W. Schneider, F. Kamp, M. Reiner, J. C. Martins, G. Landry, A. Giussani, R.-P. Kapsch, and K. Parodi, Method to quickly and accurately calculate absorbed dose from therapeutic and stray photon exposures throughout the entire body in individual patients, Medical physics 47, 2254–2266 (2020).", "T. Kairn, S. B. Crowe, and S. C. Peet, Linac Leakage Dose Received by Patients Treated Using Non-coplanar Radiotherapy Beams, in World Congress on Medical Physics and Biomedical Engineering 2018, edited by L. Lhotska, L. Sukupova, I. Lacković, and G. S. Ibbott, pages 549–551, Singapore, 2019, Springer Singapore.", "S. C. Peet, [dataset] Varian iX 6 MV Extended Profiles, 2021, https://doi.org/10.5281/zenodo.4722077 IV.Discussion IV.A.An Exploration of Dose Outside the Treatment Field This investigation produced a large dataset with many curious features.", "In this section we compare and contrast a number of profiles that illuminate the underlying geometry and radiation interactions.", "However, these are just a representative selection of the profiles, and we encourage interested readers to further examine the dataset for themselves.", "Figure REF presents two 10 $\\times $ 10 cm$^2$ X profiles, one of which is defined by the MLC, and the other is defined by the jaw (MLC retracted).", "Both profiles were measured in the transverse direction.", "Feature A corresponds to a sharp decrease in relative dose in the jaw defined profile due to the combined effect of the primary collimator and the MLC leaves being retracted to this position.", "This feature is not seen in the MLC defined profile.", "Feature B denotes a large difference in the profiles very far out-of-field.", "This may be explained by this region being shielded by the tails of the MLC leaves when they are fully retracted, but not shielded when the leaves are moved in to define the field.", "Figure: Two 10 ×\\times 10 cm 2 ^2 X profiles, one MLC defined, and one jaw defined (MLC fully retracted).", "Marked features are explained in the article text.Two 10 $\\times $ 10 cm$^2$ Y profiles, one of which is defined by the jaw (MLC retracted), and the other is defined by the MLC, are shown in Figure REF .", "Both of these profiles were collected in the radial direction.", "Notably, the Y jaw defined profile is the only field arrangement giving a pure jaw profile uncoupled from additional MLC shielding effects.", "Three features are marked in the figure.", "Feature A denotes a sharp increase in relative dose at approximately 20 cm off axis, due to the finite lateral extent of the MLC.", "This feature has been observed in an earlier study[10].", "This is immediately followed by a decrease in relative dose due to the primary collimator, which can also be seen on the jaw defined profile.", "Feature B denotes an area in which the two profiles have the same shielding conditions (jaw, primary collimator, no MLC) and yet have different relative doses.", "This may be explained by a difference in lateral phantom scatter from leakage more centrally in the field.", "The profile with both the jaw and MLC shielding the beam has less leakage relative to the jaw only profile, and therefore also has less lateral phantom scatter from this leakage.", "This is supported by the profiles coming into agreement in region C very far out-of-field.", "By this point most of the phantom scatter has been attenuated, leaving only leakage through the primary collimator and head shielding.", "Figure: Two 10 ×\\times 10 cm 2 ^2 Y profiles, one MLC defined, and one jaw defined.", "Marked features are explained in the article text.The 10 $\\times $ 10 cm$^2$ MLC defined profiles were measured at depths of 10, 15, 20, and 25 cm to investigate the variation of off-axis ratio with depth.", "Figure REF displays the X profiles normalised to the central axis dose of the profile at 10 cm depth.", "Remarkably, the out-of-field dose differs only marginally between depths.", "This indicates that out-of-field doses are largely independent of depth, at least at the distances and depths measured in this study.", "This finding is in line with earlier studies[6], but may not hold closer to the surface due to electrons scattered from the treatment head[12].", "Figure: 10 ×\\times 10 cm 2 ^2 MLC defined X profiles measured at four different depths.", "All four profiles have been normalised to the central axis value of the 10 cm depth profile.As shown in Figures REF and REF , many features in the out-of-field dose are related to the MLC.", "Figure REF shows X and Y profiles for a 5 $\\times $ 5 cm$^2$ MLC defined field.", "The two features A and B have been seen earlier (the former related to the limited extent of the MLC bank in the Y direction exemplified in figure REF , and the latter related to the finite length of the MLC leaves exemplified in figure REF ), however, the magnitude of the features relative to the central axis are larger for the smaller field compared to the 10 $\\times $ 10 or 15 $\\times $ 15 cm$^2$ fields.", "Due to the smaller field size, there is less phantom and collimator scatter to wash out these features in the relative dose.", "Figure REF presents the 10 $\\times $ 10 cm$^2$ X profile where the MLC is fully retracted with the 10 $\\times $ 10 cm$^2$ Y profile in which there is no MLC present.", "From the central axis to approximately 21 cm off-axis, denoted feature A, there is a small difference in the relative dose likely due to the difference in vertical position of the X and Y jaws.", "Beyond 21 cm from the central axis (feature B), the presence of the retracted MLC reduces the X profile out-of-field dose to about 50% to 75% of the Y profile with no MLC present.", "This demonstrates the substantial effect that the MLC can have on shielding radiation far out-of-field, as noted by other authors[9].", "Figure: X and Y profile of the 5 ×\\times 5 cm 2 ^2 MLC defined field.", "Marked features are explained in the article text.Figure: Comparison of a 10 ×\\times 10 cm 2 ^2 jaw defined Y profile with an MLC defined X profile.", "Marked features are explained in the article text.All profiles were measured in both the radial and transverse directions.", "Differences were only noted very far out-of-field.", "In all cases, the general shape of the relative dose out-of-field was similar between radial and transverse measurements, however, the transverse relative doses were of greater magnitude.", "This was best exemplified in the 5 $\\times $ 5 cm$^2$ MLC defined configuration, shown in Figure REF (feature A).", "This effect may be explained by an asymmetry in the linac head shielding.", "Figure: Comparison of a 5 ×\\times 5 cm 2 ^2 MLC defined X profile measured in both the radial (GT) and transverse (LR) orientations.", "Marked features are explained in the article text.IV.B.Limitations Many aspects of the above discussion involve observing features in out-of-field dose profiles and relating those features to the internal geometry of the linac head.", "There are many assumed relationships that cannot be stringently verified without Monte Carlo simulations that include a high-fidelity reproduction of the linac head, including the dimensions and locations of shielding blocks.", "Linac vendors would need to be willing to share detailed 3D models to facilitate this as the phase spaces and simplified geometries commonly shared would not be suitable for modelling complex interactions in the head shielding.", "Such an arrangement has been possible in the past[12], [13], [14], and we encourage all vendors to be open to sharing this information into the future.", "The measurements reported in this study were performed in a radiotherapy treatment clinic with a single model of linac, and so the extent to which this dataset can be extrapolated to other linac models is not obvious.", "It is reasonable to expect that the near out-of-field results may align with other models with similar tertiary collimation systems, such as the Varian Truebeam with Millennium 120 MLC.", "However, far out-of-field the results are more likely to depend on the exact head shielding arrangements, and so the similarity between systems in unclear.", "Linacs with markedly different collimation systems, such as those produced by Elekta (Stockholm, Sweden), may have substantially different out-of-field features.", "Ultimately, we recommend that clinics gather their own dataset using the techniques presented in this study.", "We note however that our method would need substantial modification for use with cylindrical water tanks such as the 3D SCANNER (Sun Nuclear Corporation, Melbourne, USA), but should be directly transferable to other square tanks.", "IV.C.Recommendations We recommend that this dataset is used to further investigate and understand features of out-of-field dose distributions, particularly by assisting in the creation and validation of computational models.", "Furthermore, we encourage readers to perform their own measurements and compare and contrast them with this dataset.", "When planning the tank shift, we recommend that the overlap region of the inner and outer profile sections be centred around 20 cm from the central axis.", "This region contains several strong features, such as the beginning of the primary collimator, which allows the profiles to be stitched with confidence.", "For clinics wishing to save time by collecting a more limited dataset for radiation protection calculations, we recommend collecting a single profile for a variety of field sizes.", "We believe the ideal profile is under the Y jaw, with the MLC fully retracted, in the radial orientation.", "This specific profile is compared against all others in Figure REF .", "From the figure it can seen that this profile represents an upper bound of the out-of-field dose for much of the range covered in this study, and can therefore be used as a conservative estimate of the dose across this range.", "Being in the radial direction, it is also representative of the majority of radiotherapy treatments.", "We also make the suggestion that this same profile may be a good candidate for aiding in the development of out-of-field dose calculation models.", "As it is the only profile uncoupled from extra MLC shielding, it is a good basic test case.", "MLC shielding may then be added as a second order effect.", "Figure: The jaw defined Y profiles for the three field sizes in this study (solid lines).", "The shaded areas correspond to the complete range of profiles measured for each field size.V.Conclusion This dataset article has presented a comprehensive dataset of dose profiles outside of the treatment field.", "These profiles were collected in a non-academic radiotherapy treatment centre, with standard equipment, using a technique that should be repeatable elsewhere.", "Ultimately, we encourage other clinics to collect their own datasets and make them freely available to the community.", "Access to high-quality collections of out-of-field profiles for a range of contemporary linacs would be extremely useful for risk assessments, radiation protection studies, and the development and commissioning of out-of-field dose calculation models.", "VI.Acknowledgements The authors would like to thank Steven R. Sylvander for his helpful insights during the planning of this study.", "VII.References K. D. Miller, L. Nogueira, A. B.", "Mariotto, J. H. Rowland, K. R. Yabroff, C. M. Alfano, A. Jemal, J. L. Kramer, and R. L. Siegel, Cancer treatment and survivorship statistics, 2019, CA: A Cancer Journal for Clinicians 69, 363–385 (2019).", "C. B. Dracham, A. Shankar, and R. Madan, Radiation induced secondary malignancies: a review article, Radiation oncology journal 36, 85 (2018).", "S. F. Kry, B. Bednarz, R. M. Howell, L. Dauer, D. Followill, E. Klein, H. Paganetti, B. Wang, C.-S. Wuu, and X. G. Xu, AAPM TG 158: measurement and calculation of doses outside the treated volume from external-beam radiation therapy, Medical physics 44, e391–e429 (2017).", "M. Miften et al., Management of radiotherapy patients with implanted cardiac pacemakers and defibrillators: A Report of the AAPM TG-203, Medical physics 46, e757–e788 (2019).", "S. C. Peet, R. Wilks, T. Kairn, and S. B. Crowe, Measuring dose from radiotherapy treatments in the vicinity of a cardiac pacemaker, Physica Medica 32, 1529–1536 (2016).", "M. Stovall, C. R. Blackwell, J. Cundiff, D. H. Novack, J. R. Palta, L. K. Wagner, E. W. Webster, and R. J. Shalek, Fetal dose from radiotherapy with photon beams: report of AAPM Radiation Therapy Committee Task Group No.", "36, Medical physics 22, 63–82 (1995).", "S. C. Peet, T. Kairn, C. M. Lancaster, J. V. Trapp, S. R. Sylvander, and S. B. Crowe, Measuring foetal dose from tomotherapy treatments, Medical Dosimetry in press (2021).", "R. Harrison, Out-of-field doses in radiotherapy: Input to epidemiological studies and dose-risk models, Physica Medica 42, 239–246 (2017).", "S. Mutic and E. E. Klein, A reduction in the AAPM TG-36 reported peripheral dose distributions with tertiary multileaf collimation, International Journal of Radiation Oncology* Biology* Physics 44, 947–953 (1999).", "J. D. Ruben, C. M. Lancaster, P. Jones, and R. L. Smith, A comparison of out-of-field dose and its constituent components for intensity-modulated radiation therapy versus conformal radiation therapy: implications for carcinogenesis, International Journal of Radiation Oncology* Biology* Physics 81, 1458–1464 (2011).", "S. Miljanić, J.-M. Bordy, F. d'Errico, R. Harrison, and P. Olko, Out-of-field dose measurements in radiotherapy–An overview of activity of EURADOS Working Group 9: Radiation protection in medicine, Radiation measurements 71, 270–275 (2014).", "S. F. Kry, U. Titt, F. Pönisch, D. Followill, O. N. Vassiliev, R. Allen White, R. Mohan, and M. Salehpour, A Monte Carlo model for calculating out-of-field dose from a Varian beam, Medical physics 33, 4405–4413 (2006).", "S. F. Kry, U. Titt, D. Followill, F. Pönisch, O. N. Vassiliev, R. A.", "White, M. Stovall, and M. Salehpour, A Monte Carlo model for out-of-field dose calculation from high-energy photon therapy, Medical physics 34, 3489–3499 (2007).", "B. Bednarz and X. G. Xu, Monte Carlo modeling of a 6 and 18 MV Varian Clinac medical accelerator for in-field and out-of-field dose calculations: development and validation, Physics in Medicine & Biology 54, N43 (2009).", "M. A. Benadjaoud, J. Bezin, A. Veres, D. Lefkopoulos, J. Chavaudra, A. Bridier, F. de Vathaire, and I. Diallo, A multi-plane source model for out-of-field head scatter dose calculations in external beam photon therapy, Physics in Medicine & Biology 57, 7725 (2012).", "P. J. Taddei, W. Jalbout, R. M. Howell, N. Khater, F. Geara, K. Homann, and W. D. Newhauser, Analytical model for out-of-field dose in photon craniospinal irradiation, Physics in Medicine and Biology 58, 7463–7479 (2013).", "C. W. Schneider, W. D. Newhauser, L. J. Wilson, and R.-P. Kapsch, A physics-based analytical model of absorbed dose from primary, leakage, and scattered photons from megavoltage radiotherapy with MLCs, Physics in Medicine & Biology 64, 185017 (2019).", "L. J. Wilson, W. D. Newhauser, C. W. Schneider, F. Kamp, M. Reiner, J. C. Martins, G. Landry, A. Giussani, R.-P. Kapsch, and K. Parodi, Method to quickly and accurately calculate absorbed dose from therapeutic and stray photon exposures throughout the entire body in individual patients, Medical physics 47, 2254–2266 (2020).", "T. Kairn, S. B. Crowe, and S. C. Peet, Linac Leakage Dose Received by Patients Treated Using Non-coplanar Radiotherapy Beams, in World Congress on Medical Physics and Biomedical Engineering 2018, edited by L. Lhotska, L. Sukupova, I. Lacković, and G. S. Ibbott, pages 549–551, Singapore, 2019, Springer Singapore.", "S. C. Peet, [dataset] Varian iX 6 MV Extended Profiles, 2021, https://doi.org/10.5281/zenodo.4722077 V.Conclusion This dataset article has presented a comprehensive dataset of dose profiles outside of the treatment field.", "These profiles were collected in a non-academic radiotherapy treatment centre, with standard equipment, using a technique that should be repeatable elsewhere.", "Ultimately, we encourage other clinics to collect their own datasets and make them freely available to the community.", "Access to high-quality collections of out-of-field profiles for a range of contemporary linacs would be extremely useful for risk assessments, radiation protection studies, and the development and commissioning of out-of-field dose calculation models.", "VI.Acknowledgements The authors would like to thank Steven R. Sylvander for his helpful insights during the planning of this study.", "VII.References K. D. Miller, L. Nogueira, A. B.", "Mariotto, J. H. Rowland, K. R. Yabroff, C. M. Alfano, A. Jemal, J. L. Kramer, and R. L. Siegel, Cancer treatment and survivorship statistics, 2019, CA: A Cancer Journal for Clinicians 69, 363–385 (2019).", "C. B. Dracham, A. Shankar, and R. Madan, Radiation induced secondary malignancies: a review article, Radiation oncology journal 36, 85 (2018).", "S. F. Kry, B. Bednarz, R. M. Howell, L. Dauer, D. Followill, E. Klein, H. Paganetti, B. Wang, C.-S. Wuu, and X. G. Xu, AAPM TG 158: measurement and calculation of doses outside the treated volume from external-beam radiation therapy, Medical physics 44, e391–e429 (2017).", "M. Miften et al., Management of radiotherapy patients with implanted cardiac pacemakers and defibrillators: A Report of the AAPM TG-203, Medical physics 46, e757–e788 (2019).", "S. C. Peet, R. Wilks, T. Kairn, and S. B. Crowe, Measuring dose from radiotherapy treatments in the vicinity of a cardiac pacemaker, Physica Medica 32, 1529–1536 (2016).", "M. Stovall, C. R. Blackwell, J. Cundiff, D. H. Novack, J. R. Palta, L. K. Wagner, E. W. Webster, and R. J. Shalek, Fetal dose from radiotherapy with photon beams: report of AAPM Radiation Therapy Committee Task Group No.", "36, Medical physics 22, 63–82 (1995).", "S. C. Peet, T. Kairn, C. M. Lancaster, J. V. Trapp, S. R. Sylvander, and S. B. Crowe, Measuring foetal dose from tomotherapy treatments, Medical Dosimetry in press (2021).", "R. Harrison, Out-of-field doses in radiotherapy: Input to epidemiological studies and dose-risk models, Physica Medica 42, 239–246 (2017).", "S. Mutic and E. E. Klein, A reduction in the AAPM TG-36 reported peripheral dose distributions with tertiary multileaf collimation, International Journal of Radiation Oncology* Biology* Physics 44, 947–953 (1999).", "J. D. Ruben, C. M. Lancaster, P. Jones, and R. L. Smith, A comparison of out-of-field dose and its constituent components for intensity-modulated radiation therapy versus conformal radiation therapy: implications for carcinogenesis, International Journal of Radiation Oncology* Biology* Physics 81, 1458–1464 (2011).", "S. Miljanić, J.-M. Bordy, F. d'Errico, R. Harrison, and P. Olko, Out-of-field dose measurements in radiotherapy–An overview of activity of EURADOS Working Group 9: Radiation protection in medicine, Radiation measurements 71, 270–275 (2014).", "S. F. Kry, U. Titt, F. Pönisch, D. Followill, O. N. Vassiliev, R. Allen White, R. Mohan, and M. Salehpour, A Monte Carlo model for calculating out-of-field dose from a Varian beam, Medical physics 33, 4405–4413 (2006).", "S. F. Kry, U. Titt, D. Followill, F. Pönisch, O. N. Vassiliev, R. A.", "White, M. Stovall, and M. Salehpour, A Monte Carlo model for out-of-field dose calculation from high-energy photon therapy, Medical physics 34, 3489–3499 (2007).", "B. Bednarz and X. G. Xu, Monte Carlo modeling of a 6 and 18 MV Varian Clinac medical accelerator for in-field and out-of-field dose calculations: development and validation, Physics in Medicine & Biology 54, N43 (2009).", "M. A. Benadjaoud, J. Bezin, A. Veres, D. Lefkopoulos, J. Chavaudra, A. Bridier, F. de Vathaire, and I. Diallo, A multi-plane source model for out-of-field head scatter dose calculations in external beam photon therapy, Physics in Medicine & Biology 57, 7725 (2012).", "P. J. Taddei, W. Jalbout, R. M. Howell, N. Khater, F. Geara, K. Homann, and W. D. Newhauser, Analytical model for out-of-field dose in photon craniospinal irradiation, Physics in Medicine and Biology 58, 7463–7479 (2013).", "C. W. Schneider, W. D. Newhauser, L. J. Wilson, and R.-P. Kapsch, A physics-based analytical model of absorbed dose from primary, leakage, and scattered photons from megavoltage radiotherapy with MLCs, Physics in Medicine & Biology 64, 185017 (2019).", "L. J. Wilson, W. D. Newhauser, C. W. Schneider, F. Kamp, M. Reiner, J. C. Martins, G. Landry, A. Giussani, R.-P. Kapsch, and K. Parodi, Method to quickly and accurately calculate absorbed dose from therapeutic and stray photon exposures throughout the entire body in individual patients, Medical physics 47, 2254–2266 (2020).", "T. Kairn, S. B. Crowe, and S. C. Peet, Linac Leakage Dose Received by Patients Treated Using Non-coplanar Radiotherapy Beams, in World Congress on Medical Physics and Biomedical Engineering 2018, edited by L. Lhotska, L. Sukupova, I. Lacković, and G. S. Ibbott, pages 549–551, Singapore, 2019, Springer Singapore.", "S. C. Peet, [dataset] Varian iX 6 MV Extended Profiles, 2021, https://doi.org/10.5281/zenodo.4722077 VI.Acknowledgements The authors would like to thank Steven R. Sylvander for his helpful insights during the planning of this study.", "VII.References K. D. Miller, L. Nogueira, A. B.", "Mariotto, J. H. Rowland, K. R. Yabroff, C. M. Alfano, A. Jemal, J. L. Kramer, and R. L. Siegel, Cancer treatment and survivorship statistics, 2019, CA: A Cancer Journal for Clinicians 69, 363–385 (2019).", "C. B. Dracham, A. Shankar, and R. Madan, Radiation induced secondary malignancies: a review article, Radiation oncology journal 36, 85 (2018).", "S. F. Kry, B. Bednarz, R. M. Howell, L. Dauer, D. Followill, E. Klein, H. Paganetti, B. Wang, C.-S. Wuu, and X. G. Xu, AAPM TG 158: measurement and calculation of doses outside the treated volume from external-beam radiation therapy, Medical physics 44, e391–e429 (2017).", "M. Miften et al., Management of radiotherapy patients with implanted cardiac pacemakers and defibrillators: A Report of the AAPM TG-203, Medical physics 46, e757–e788 (2019).", "S. C. Peet, R. Wilks, T. Kairn, and S. B. Crowe, Measuring dose from radiotherapy treatments in the vicinity of a cardiac pacemaker, Physica Medica 32, 1529–1536 (2016).", "M. Stovall, C. R. Blackwell, J. Cundiff, D. H. Novack, J. R. Palta, L. K. Wagner, E. W. Webster, and R. J. Shalek, Fetal dose from radiotherapy with photon beams: report of AAPM Radiation Therapy Committee Task Group No.", "36, Medical physics 22, 63–82 (1995).", "S. C. Peet, T. Kairn, C. M. Lancaster, J. V. Trapp, S. R. Sylvander, and S. B. Crowe, Measuring foetal dose from tomotherapy treatments, Medical Dosimetry in press (2021).", "R. Harrison, Out-of-field doses in radiotherapy: Input to epidemiological studies and dose-risk models, Physica Medica 42, 239–246 (2017).", "S. Mutic and E. E. Klein, A reduction in the AAPM TG-36 reported peripheral dose distributions with tertiary multileaf collimation, International Journal of Radiation Oncology* Biology* Physics 44, 947–953 (1999).", "J. D. Ruben, C. M. Lancaster, P. Jones, and R. L. Smith, A comparison of out-of-field dose and its constituent components for intensity-modulated radiation therapy versus conformal radiation therapy: implications for carcinogenesis, International Journal of Radiation Oncology* Biology* Physics 81, 1458–1464 (2011).", "S. Miljanić, J.-M. Bordy, F. d'Errico, R. Harrison, and P. Olko, Out-of-field dose measurements in radiotherapy–An overview of activity of EURADOS Working Group 9: Radiation protection in medicine, Radiation measurements 71, 270–275 (2014).", "S. F. Kry, U. Titt, F. Pönisch, D. Followill, O. N. Vassiliev, R. Allen White, R. Mohan, and M. Salehpour, A Monte Carlo model for calculating out-of-field dose from a Varian beam, Medical physics 33, 4405–4413 (2006).", "S. F. Kry, U. Titt, D. Followill, F. Pönisch, O. N. Vassiliev, R. A.", "White, M. Stovall, and M. Salehpour, A Monte Carlo model for out-of-field dose calculation from high-energy photon therapy, Medical physics 34, 3489–3499 (2007).", "B. Bednarz and X. G. Xu, Monte Carlo modeling of a 6 and 18 MV Varian Clinac medical accelerator for in-field and out-of-field dose calculations: development and validation, Physics in Medicine & Biology 54, N43 (2009).", "M. A. Benadjaoud, J. Bezin, A. Veres, D. Lefkopoulos, J. Chavaudra, A. Bridier, F. de Vathaire, and I. Diallo, A multi-plane source model for out-of-field head scatter dose calculations in external beam photon therapy, Physics in Medicine & Biology 57, 7725 (2012).", "P. J. Taddei, W. Jalbout, R. M. Howell, N. Khater, F. Geara, K. Homann, and W. D. Newhauser, Analytical model for out-of-field dose in photon craniospinal irradiation, Physics in Medicine and Biology 58, 7463–7479 (2013).", "C. W. Schneider, W. D. Newhauser, L. J. Wilson, and R.-P. Kapsch, A physics-based analytical model of absorbed dose from primary, leakage, and scattered photons from megavoltage radiotherapy with MLCs, Physics in Medicine & Biology 64, 185017 (2019).", "L. J. Wilson, W. D. Newhauser, C. W. Schneider, F. Kamp, M. Reiner, J. C. Martins, G. Landry, A. Giussani, R.-P. Kapsch, and K. Parodi, Method to quickly and accurately calculate absorbed dose from therapeutic and stray photon exposures throughout the entire body in individual patients, Medical physics 47, 2254–2266 (2020).", "T. Kairn, S. B. Crowe, and S. C. Peet, Linac Leakage Dose Received by Patients Treated Using Non-coplanar Radiotherapy Beams, in World Congress on Medical Physics and Biomedical Engineering 2018, edited by L. Lhotska, L. Sukupova, I. Lacković, and G. S. Ibbott, pages 549–551, Singapore, 2019, Springer Singapore.", "S. C. Peet, [dataset] Varian iX 6 MV Extended Profiles, 2021, https://doi.org/10.5281/zenodo.4722077 VII.References K. D. Miller, L. Nogueira, A. B.", "Mariotto, J. H. Rowland, K. R. Yabroff, C. M. Alfano, A. Jemal, J. L. Kramer, and R. L. Siegel, Cancer treatment and survivorship statistics, 2019, CA: A Cancer Journal for Clinicians 69, 363–385 (2019).", "C. B. Dracham, A. Shankar, and R. Madan, Radiation induced secondary malignancies: a review article, Radiation oncology journal 36, 85 (2018).", "S. F. Kry, B. Bednarz, R. M. Howell, L. Dauer, D. Followill, E. Klein, H. Paganetti, B. Wang, C.-S. Wuu, and X. G. Xu, AAPM TG 158: measurement and calculation of doses outside the treated volume from external-beam radiation therapy, Medical physics 44, e391–e429 (2017).", "M. Miften et al., Management of radiotherapy patients with implanted cardiac pacemakers and defibrillators: A Report of the AAPM TG-203, Medical physics 46, e757–e788 (2019).", "S. C. Peet, R. Wilks, T. Kairn, and S. B. Crowe, Measuring dose from radiotherapy treatments in the vicinity of a cardiac pacemaker, Physica Medica 32, 1529–1536 (2016).", "M. Stovall, C. R. Blackwell, J. Cundiff, D. H. Novack, J. R. Palta, L. K. Wagner, E. W. Webster, and R. J. Shalek, Fetal dose from radiotherapy with photon beams: report of AAPM Radiation Therapy Committee Task Group No.", "36, Medical physics 22, 63–82 (1995).", "S. C. Peet, T. Kairn, C. M. Lancaster, J. V. Trapp, S. R. Sylvander, and S. B. Crowe, Measuring foetal dose from tomotherapy treatments, Medical Dosimetry in press (2021).", "R. Harrison, Out-of-field doses in radiotherapy: Input to epidemiological studies and dose-risk models, Physica Medica 42, 239–246 (2017).", "S. Mutic and E. E. Klein, A reduction in the AAPM TG-36 reported peripheral dose distributions with tertiary multileaf collimation, International Journal of Radiation Oncology* Biology* Physics 44, 947–953 (1999).", "J. D. Ruben, C. M. Lancaster, P. Jones, and R. L. Smith, A comparison of out-of-field dose and its constituent components for intensity-modulated radiation therapy versus conformal radiation therapy: implications for carcinogenesis, International Journal of Radiation Oncology* Biology* Physics 81, 1458–1464 (2011).", "S. Miljanić, J.-M. Bordy, F. d'Errico, R. Harrison, and P. Olko, Out-of-field dose measurements in radiotherapy–An overview of activity of EURADOS Working Group 9: Radiation protection in medicine, Radiation measurements 71, 270–275 (2014).", "S. F. Kry, U. Titt, F. Pönisch, D. Followill, O. N. Vassiliev, R. Allen White, R. Mohan, and M. Salehpour, A Monte Carlo model for calculating out-of-field dose from a Varian beam, Medical physics 33, 4405–4413 (2006).", "S. F. Kry, U. Titt, D. Followill, F. Pönisch, O. N. Vassiliev, R. A.", "White, M. Stovall, and M. Salehpour, A Monte Carlo model for out-of-field dose calculation from high-energy photon therapy, Medical physics 34, 3489–3499 (2007).", "B. Bednarz and X. G. Xu, Monte Carlo modeling of a 6 and 18 MV Varian Clinac medical accelerator for in-field and out-of-field dose calculations: development and validation, Physics in Medicine & Biology 54, N43 (2009).", "M. A. Benadjaoud, J. Bezin, A. Veres, D. Lefkopoulos, J. Chavaudra, A. Bridier, F. de Vathaire, and I. Diallo, A multi-plane source model for out-of-field head scatter dose calculations in external beam photon therapy, Physics in Medicine & Biology 57, 7725 (2012).", "P. J. Taddei, W. Jalbout, R. M. Howell, N. Khater, F. Geara, K. Homann, and W. D. Newhauser, Analytical model for out-of-field dose in photon craniospinal irradiation, Physics in Medicine and Biology 58, 7463–7479 (2013).", "C. W. Schneider, W. D. Newhauser, L. J. Wilson, and R.-P. Kapsch, A physics-based analytical model of absorbed dose from primary, leakage, and scattered photons from megavoltage radiotherapy with MLCs, Physics in Medicine & Biology 64, 185017 (2019).", "L. J. Wilson, W. D. Newhauser, C. W. Schneider, F. Kamp, M. Reiner, J. C. Martins, G. Landry, A. Giussani, R.-P. Kapsch, and K. Parodi, Method to quickly and accurately calculate absorbed dose from therapeutic and stray photon exposures throughout the entire body in individual patients, Medical physics 47, 2254–2266 (2020).", "T. Kairn, S. B. Crowe, and S. C. Peet, Linac Leakage Dose Received by Patients Treated Using Non-coplanar Radiotherapy Beams, in World Congress on Medical Physics and Biomedical Engineering 2018, edited by L. Lhotska, L. Sukupova, I. Lacković, and G. S. Ibbott, pages 549–551, Singapore, 2019, Springer Singapore.", "S. C. Peet, [dataset] Varian iX 6 MV Extended Profiles, 2021, https://doi.org/10.5281/zenodo.4722077" ] ]
2105.11666
[ [ "Deep learning-based bias transfer for overcoming laboratory differences\n of microscopic images" ], [ "Abstract The automated analysis of medical images is currently limited by technical and biological noise and bias.", "The same source tissue can be represented by vastly different images if the image acquisition or processing protocols vary.", "For an image analysis pipeline, it is crucial to compensate such biases to avoid misinterpretations.", "Here, we evaluate, compare, and improve existing generative model architectures to overcome domain shifts for immunofluorescence (IF) and Hematoxylin and Eosin (H&E) stained microscopy images.", "To determine the performance of the generative models, the original and transformed images were segmented or classified by deep neural networks that were trained only on images of the target bias.", "In the scope of our analysis, U-Net cycleGANs trained with an additional identity and an MS-SSIM-based loss and Fixed-Point GANs trained with an additional structure loss led to the best results for the IF and H&E stained samples, respectively.", "Adapting the bias of the samples significantly improved the pixel-level segmentation for human kidney glomeruli and podocytes and improved the classification accuracy for human prostate biopsies by up to 14%." ], [ "Introduction", "Deep learning (DL) applications play an increasingly important role in medicine, yet, “everyone participating in medical image evaluation with machine learning is data starved\" [15].", "When the same imaging technique is used, e.g.", "confocal microscopy of a specific tissue type with a matching staining protocol, image analysis networks should be applicable to datasets not seen during training without a substantial drop in performance.", "However, due to bias introduced during the acquisition or processing of datasets (`domain shift'), the generalizability of deep neural networks is negatively affected.", "If bias cannot be accounted for, models have to be re-trained every time a new dataset becomes available.", "In consequence, new data can only be used if a large cohort of labeled samples exists or can be created.", "Traditional techniques try to compensate domain shift through normalization or simple image transformations, like an increase of the contrast through histogram equalization [12].", "However, as indicated by de Bel et al.", "[4], simple approaches are limited in how much bias they can capture and adjust, leading to insufficient transformations.", "Recent evidence suggests that deep generative models could be utilized to modify samples, creating novel images that are close to the reference domain while not altering the original content.", "This way, images of a new dataset can be adjusted to the bias of the reference domain as a pre-processing step (`bias transfer'), in order to avoid re-training large image analysis networks.", "The goal is that new data of the same modality can be handled correctly and without a large drop in performance.", "Necessarily, content preservation is of the utmost importance since hallucination artifacts introduced by the generative models could lead to misdiagnosis.", "Reliable bias transfer approaches would also enable the usage of DL in settings that do not allow for frequent creation and retraining of models, such as decision support systems in hospitals.", "In this paper, we aim to improve existing generative models to enable stable bias transfer for medical histopathology images.", "In addition, we propose guidelines for testing and evaluating bias transfer models in settings with similar transformation goals.", "To benchmark the quality of the bias transfer for three state-of-the-art generative models, cycle-consistent generative adversarial networks (cycleGANs) [24], U-Net cycleGANs [4], and Fixed-Point GANs [6], we measured the content preservation, target domain adaptation, and impact on image segmentation and classification performance.", "To increase the performance of the models, we tested three additional losses designed to improve the quality in terms of content (`MS-SSIM loss') [1], structure integrity (`structure loss') [16] and intensity of transformation (`additional identity loss') [4].", "As a baseline, histogram matching [12] for color correction in a decorrelated color space [18] (`color transfer') is included in our evaluation, which utilizes a single, random image to represent the target domain." ], [ "Related work", "In a medical context, most datasets that require bias transfer are unpaired, meaning that a ground truth for the transformed images does not exist.", "A well-established approach for learning unpaired image-to-image translations is using CycleGANs [24].", "CycleGANs have already been used for stain transforming renal tissue sections [4] or histological images of breast cancer [20].", "Unfortunately, every paper introduces a new variation of cycleGAN.", "To the best of our knowledge, an extensive comparison of bias transfer algorithms for microscopy images does not exist yet.", "Selecting a fitting approach is a non-trivial task.", "One common enhancement is using a U-Net structure for the generators [17], [4].", "In a U-Net, the encoder and decoder structures of the neural network are connected via skip connections between layers of the same size [19].", "Thus, the generators can pass information directly without transitioning the bottleneck, improving the level of detail of the images.", "In this paper, we refer to the modified cycleGAN approach as U-Net cycleGAN.", "Besides architecture modifications or simple hyperparameter adaptations like changing the learning rate [4], [17], or the number of images per batch [4], [20], a frequent change is adding additional losses to incorporate demands the network has to fulfill.", "Armanious et al.", "used the multi-scale structural similarity index (MS-SSIM) [23] in [1] as an additional cycle loss between the original and the cycle-reconstructed images.", "Their goal was to penalize structural discrepancies between the images.", "In their experiments, the additional loss led to sharper results with better textural details.", "Another loss has been proposed by de Bel et al.", "in [4].", "They included an additional identity loss that is decreased to zero over the first 20 epochs of training.", "The loss is an addition to the original identity loss of cycleGANs [24].", "In their experiments, the loss stabilized the training process and led to faster convergence since it forces the generator to look for transformations close to identity first, effectively shrinking the solution space.", "Moreover, Ma et al.", "proposed a loss based on the sub-part of the structural similarity index (SSIM) that only evaluates local structure changes [16].", "They proposed the loss to enhance the quality of endoscopy and confocal microscopy images that suffer from “intensity inhomogeneity, noticeable blur and poor contrast” [16].", "The structure loss directly compares the original and transformed images.", "Unfortunately, cycleGANs are designed for transforming between two domains only.", "Therefore, transforming multiple domains to the reference domain requires individual models and training runs.", "In [6], Choi et al.", "proposed StarGANs, which transform between any number of domains with a single generator and discriminator.", "The generator produces samples based on the input image and the label of the target domain.", "Even if only two domains exist, StarGANs can potentially outperform cycleGAN-based approaches due to the benefit that all data samples are used to train a single generator.", "Siddiquee et al.", "proposed Fixed-Point GANs (FPGs), which extend StarGANs with a conditional identity loss, thus reducing hallucination artifacts [21].", "If a new dataset could be adjusted by adding a new condition to FPG, bias transfer would be an exceptionally fast solution for the generalizability problem of image analysis networks." ], [ "Methodology", "In the following, the overall workflow, as well as the generative models (Subsection REF ) and our datasets (Subsection REF ) are introduced.", "Figure: Overview of the image analysis pipeline including bias transfer: First, the original input images are downsampled a and transformed by the generative models b trained with or without additional losses c. Afterwards, they are upsampled to the original size d. The transformed images are then used as inputs for the segmentation or classification networks e. NEW k \\textrm {NEW}_{k} and TAR k \\textrm {TAR}_{k} refer to the domains of the kidney samples, whereas NEW p \\textrm {NEW}_{p} and TAR p \\textrm {TAR}_{p} are the domains of the prostate samples.As part of this work, several generative approaches are introduced into an existing image analysis pipeline.", "An overview of the complete workflow with bias transfer is shown in Figure REF .", "As bias transfer is used as a pre-processing step here, the image size of the transformed images should be identical to the original size.", "However, generative models are usually not designed for images larger than $256\\times 256$ pixels.", "We used the Laplacian pyramid-based approach introduced in [11] to downsample the images to $256\\times 256$ pixels pre- (Figure REF a $\\rightarrow $ b), and back to the original size post-transformation (Figure REF c $\\rightarrow $ d), which is $1024\\times 1024$ pixels for the kidney (upper row) and $2048\\times 2048$ for the prostate images (lower row).", "The approach involves replacing the smallest layer of the pyramids with the respective transformed image for upsampling.", "Hence, the edges of the original image, which are typically lost at a low resolution, are added back into the image, boosting the quality of the transformed image." ], [ "Generative approaches", "For this paper, we implemented, tested, and modified three state-of-the-art generative models for image generation, as shown in Figure REF b.", "For this purpose, we re-created the original implementation by Zhu et al.", "for cycleGANs [24] in Tensorflow 2.0 with a PatchGAN discriminator that judges $16\\times 16$ patches per image.", "The U-Net cycleGAN architecture has been constructed by adding skip connections to the generators.", "For FPG, the generator input consists of an image and the one-hot encoded label of the target domain.", "Otherwise the generator architecture is identical to cycleGAN.", "Instead of using the discriminator structure described in the original FPG paper, we created a modified version of the cycleGAN discriminator by replacing the original output layer with two outputs, one for predicting the image's domain and one for the authenticity of the individual image patches (real vs. generated).", "The modified cycleGAN discriminator has fewer trainable parameters and judges more patches per image than the original FPG discriminator.", "In [14] it has been shown that judging more patches per image leads to images with more detail for cycleGANs.", "It seems evident that FPGs could also benefit from this modification.", "The hyperparameters for cycleGAN and U-Net cycleGAN are mainly based on the original cycleGAN implementation.", "The networks are trained for up to 200 epochs with an initial steady learning rate of $0.0005$ (original: $0.0002$ ) for the first 100 epochs and a linearly decaying learning rate that reaches 0 after 200 epochs.", "For FPG, an initial learning rate of $0.0001$ has been used, as proposed by Choi et al.", "[6].", "We used a batch size of one.", "To improve the generative performance and reduce hallucination artifacts of the three models, we included additional terms in the loss functions (see Figure REF c) specifically the additional identity loss $\\mathcal {L}_{\\textrm {+id}}$  [4], the MS-SSIM loss $\\mathcal {L}_{\\textrm {+ms-ssim}}$  [1] and the structure loss $\\mathcal {L}_{\\textrm {+struc}}$  [16].", "The original losses of (U-Net) cycleGANs are the adversarial loss ($adv$ ), the cycle-reconstruction loss ($cyc$ ) and the identity loss ($id$ ).", "For FPGs, the original losses are the cycle-reconstruction loss ($cyc$ ), the domain-classification loss ($domain$ ), the gradient penalty loss ($gp$ ) and the conditional identity loss ($id$ ).", "For our experiments, the losses have been weighted with $\\lambda _{\\textrm {adv}} =1$ , $\\lambda _{\\textrm {cyc}} =10$ , and $\\lambda _{\\textrm {id}}=10$ (original: $\\lambda _{\\textrm {id}}=5$ ) for cycleGAN and U-Net cycleGAN and with $\\lambda _{\\textrm {cyc}}=\\lambda _{\\textrm {gp}}= \\lambda _{\\textrm {id}} = 10$ and $\\lambda _{\\textrm {domain}}=1$ for FPG.", "All additional losses have been weighted with $\\lambda =5$ .", "We added the structure loss and the MS-SSIM loss individually and, for the MS-SSIM loss, in combination with the additional identity loss (`combined losses').", "The structure loss was not combined with the other additional losses since the direct comparison of the original and transformed images covers similar objectives as the `combined losses'.", "Our implementation is publicly available on GitHub.https://github.com/imsb-uke/bias-transfer-microscopy" ], [ "Data", "In this paper, we perform bias transfer for two modalities, IF images of kidney biopsies and H&E stained tissue microarray (TMA) spots of prostate biopsies.", "In Figure REF , the upper row shows the workflow for the kidney and the lower row for the prostate biopsy samples.", "Each modality includes two sub-datasets (domains) originating from different hospitals and has a specified transformation direction.", "This is due to the fact that bias transfer is applied here to overcome the domain shift between a new and a target domain which has been used for training a segmentation or classification network, as outlined in Figure REF e." ], [ "Kidney biopsies", "The kidney dataset consists of 2D kidney IF images [25].", "The images have been used to train a modified U-Net with dual output for the automatic segmentation of the glomeruli and their podocytes, an important cell type inside the glomeruli that is indicative of the health of the kidney [25].", "To highlight the glomeruli and podocytes, three different biomarkers were used: DAPI for cell nuclei (blue), WT1 for podocyte cytoplasm (red), and DACH1 for podocyte nuclei (green).", "The images originate from two distinct hospitals (two domains: the target domain `$\\textrm {TAR}_{k}$ ' and the new domain `$\\textrm {NEW}_{k}$ '), which were imaged by two different operators with differing confocal laser scanning microscopes (Nikon A1 Confocal and Zeiss Confocal LSM 800).", "In total, subset $\\textrm {TAR}_{k}$ contains 269 images and subset $\\textrm {NEW}_{k}$ contains 375 images.", "Training and evaluating the segmentation requires annotations.", "For this purpose, a subset of the images has been annotated by medical experts ($\\textrm {TAR}_{k}$ : 109, $\\textrm {NEW}_{k}$ : 90).", "Further details on the datasets and their biological meaning can be found in [25].", "For bias transfer, the split into training, validation, and test sets was performed randomly, with the constraints that images originating from the same patient (up to 16) remain in the same set, that a ratio of approximately 70% for training and 15% each for validation and test sets is reached and that the validation and test sets only consist of annotated images.", "While bias transfer does not require a ground truth, masks are necessary to evaluate the segmentation quality achieved on the transformed images.", "Accordingly, the $\\textrm {TAR}_{k}$ images were split into 180 for training, 44 for validation, and 45 for testing.", "The $\\textrm {NEW}_{k}$ dataset was split into 285 images for training, 46 for validation, and 44 for testing." ], [ "Prostate biopsies", "The prostate dataset consists of circular biopsies (spots) originating from radical prostatectomies (RPE) that have been assembled with the help of TMAs.", "The images originate from two different hospitals (`$\\textrm {TAR}_{p}$ ' and `$\\textrm {NEW}_{p}$ ').", "The $\\textrm {TAR}_{p}$ dataset consists of 2866 and the $\\textrm {NEW}_{p}$ dataset of 886 images.", "Both datasets were created for staging prostate cancer with the help of Gleason patterns, which categorize the shape of the glands.", "Tissue can be classified as benign or as Gleason pattern 3, 4, or 5, whereas a higher number represents worse tumor tissue.", "The Gleason score (GS) is then calculated as the sum of the most prevalent and the worst occurring patterns in a sample [5].", "Unfortunately, the inter-pathologist agreement on GSs is usually low [10], which is why automated and stable GS prediction is of high clinical value.", "We trained an InceptionNet-V3-based classification network, pre-trained on ImageNet [8], on images of the $\\textrm {TAR}_{p}$ dataset, reaching a test accuracy of $81.6\\%$ .", "To exclude image background only the innermost $2048\\times 2048$ pixels of the spots have been used.", "The classification was limited to single Gleason patterns only (`0' = benign, `3+3', `4+4', and `5+5') since TMA spots with two (or more) differing Gleason patterns could potentially contain patterns that occur solely outside of the innermost pixels.", "During training, all images are pre-processed with normalization, and the data is augmented with random rotations, shearing, shifting, and flipping.", "After pre-processing, the images are resized to $224\\times 224$ pixels, which is the input size of the InceptionNet.", "The $\\textrm {NEW}_{p}$ dataset is publicly available [2] and does not include Gleason score annotations.", "However, the images have been annotated via segmentation masks to identify areas containing a specific Gleason pattern.", "To evaluate the classification performance of our network, we used the segmentation masks to calculate the Gleason scores of the image centers.", "We then used the same split into training, validation, and test sets that was defined for the dataset in the original publication [3].", "The training and validation sets have been annotated by one pathologist (pathologist 1) and the test set has been annotated independently by two pathologists (pathologist 1 and pathologist 2), resulting in 506 images (429 with a single Gleason pattern) for training, 133 (111) for validation, and 245 (199 – pathologist 1, 156 – pathologist 2) images for testing.", "For the $\\textrm {TAR}_{p}$ dataset, the split was determined according to a random stratified sampling strategy, resulting in the same label distribution for the training, validation, and test sets.", "The resulting split consists of 2000 (1136 with a single Gleason pattern) images for training, 432 (246) images for validation, and 434 (245) images for testing.", "All images were annotated by the same pathologist (pathologist 3).", "Nonetheless, the label distribution is not uniform, $\\textrm {TAR}_{p}$ overrepresents 3+3 samples.", "In contrast, $\\textrm {NEW}_{p}$ mostly contains 4+4 samples." ], [ "Results and discussion", "We based the evaluation of the generative approaches on three factors: content preservation, target domain imitation, and impact on the segmentation or classification performance.", "Two metrics have been selected for a quantitative evaluation of the transformation quality: the structural similarity index (SSIM) [22] and the Fréchet Inception Distance (FID) [13].", "The SSIM is a well-established metric that has been used here to calculate the degradation of structural information (content) between the original (Figure REF a) and the transformed (Figure REF d) input images.", "The FID measures the feature-wise distance between two domains.", "While bias transfer can be trained on all images, for the prostate dataset, the impact on the classification can only be evaluated on the single Gleason pattern images.", "Nonetheless, we decided to evaluate all images regarding the SSIM and FID since the differing annotations given by pathologist 1 and 2 create two non-identical subsets of test images.", "For our evaluation, the FID between the target domain and the transformed images is calculated and compared to the FID between the original domains.", "A lower FID implies that images are visually closer to the target domain after bias transfer.", "Since the FID highly depends on the image content, validation and test scores should be considered separately.", "Finally, for the kidney data, the segmentation accuracy pre- and post-bias transfer is compared.", "The segmentation network predicts one segmentation mask for the glomerulus as a whole and one for the podocytes (see Figure REF e).", "The quality of the segmentation is measured with three Dice scores [9], the pixel-wise segmentation of the glomeruli and podocytes and the object-wise segmentation of the podocytes.", "For the prostate data, the classification accuracy and the macro F1 scores pre- and post-bias transfer are compared.", "The macro F1 score can give insights into the classification performance for underrepresented classes.", "Considering class imbalance is especially important here since the label distributions of $\\textrm {TAR}_{p}$ and $\\textrm {NEW}_{p}$ do not match.", "Every variation has been trained five times with different random seeds, including the baseline.", "The training epoch with the lowest generator validation loss for transformations from $\\textrm {NEW}$ to $\\textrm {TAR}$ has been used for the evaluation of the individual runs.", "For each approach, the run with the best results on the validation set has been evaluated on the test set." ], [ "Kidney biopsies", "The SSIM and FID scores for the validation set are visualized in Figure REF .", "They indicate that the transformations performed by the U-Net cycleGAN and FPG variations had the highest quality, with U-Net cycleGAN with structure loss leading to the overall best and most stable SSIM scores.", "Regarding the FID, U-Net cycleGAN with combined losses largely improved the distance to the target domain for the validation and test sets.", "The combination of additional identity and MS-SSIM loss (combined losses) had a stabilizing effect on the training process of cycleGAN and U-Net cycleGAN.", "The structure loss had a positive effect on U-Net cycleGAN and FPG, however, it led to mode collapse for cycleGAN, only producing a single output irrespective of the input.", "Figure: Boxplots visualizing the transformation metrics 1- SSIM 1-\\textrm {SSIM} (a) and FID (b) for the validation and test sets of the kidney biopsies.", "The dashed lines highlight the original FID scores and the red dots show the values achieved by the runs that performed best on the validation set for each variation.Table: Means (μ test \\mu _{test}) and standard deviations (σ test \\sigma _{test}) of the Dice scores on the original test sets and the relative performance for the transformed images.", "`Dice glom.", "pix.'", "evaluates the pixel-wise segmentation of the glomeruli, and `Dice podo.", "pix.'", "of the podocytes.", "`Dice podo.", "obj.'", "evaluates the object-wise segmentation of the podocytes.", "Significance (p<0.05p<0.05) is marked with **.The segmentation scores on the test set are shown in Table REF .", "Here, `Test $\\textrm {TAR}_{k}$ ' only includes images not used for training the segmentation U-Net (21 images).", "Color transfer worsened the segmentation scores, despite improving the FID.", "As for the SSIM and FID scores, the U-Net cycleGAN and Fixed-Point GAN variations led to the best results.", "For those approaches, three out of four variations significantly improved the pixel-level Dice score for the glomeruli and all four for the podocytes.", "However, unlike FPG, not all runs of U-Net cycleGAN worsened the object-level Dice score for the podocytes.", "Overall, U-Net cycleGANs with combined losses performed the best for the task at hand.", "The aforementioned approach produced hallucination artifact-free images that significantly improved the pixel-level segmentation scores of the glomeruli (0.909 to 0.923, $p=0.005$ ), and podocytes (0.730 to 0.797, $p<0.0001$ ), due to a strong adaption to the target domain.", "An example transformation performed by U-Net cycleGAN and its effect on the segmentation result can be found in [25]." ], [ "Prostate biopsies", "For the prostate biopsies, the boxplots in Figure REF show that FPG trained with the structure loss outperformed the other approaches regarding content preservation.", "While the overall lowest FID score on the test set was achieved by cycleGAN with structure loss, this result is not reproducible for different random seeds – the training process is not stable enough.", "Figure: Boxplots visualizing the transformation metrics 1- SSIM 1-\\textrm {SSIM} (a) and FID (b) for the validation and test sets of the prostate biopsies.", "The dashed lines highlight the original FID scores and the red dots mark the results achieved by the runs that performed best on the validation set for each variation.Table: Means (μ val \\mu _{val}) and standard deviations (σ val \\sigma _{val}) of the accuracy and macro-weighted F1 scores for the classification of the Gleason scores on the original validation and test sets and the relative performance for the transformed images.", "For the test set, the predicted Gleason scores are compared to the annotations of two medical professionals (μ test1 \\mu _{test 1} and μ test2 \\mu _{test 2}).", "The best results are bold.Figure: (a) Relative confusion matrices for the classification of the test sets for TAR p \\textrm {TAR}_{p} (top) and NEW p \\textrm {NEW}_{p} annotated by pathologist 1 (bottom), as well as transformed test images with and without hallucination artifacts and their corresponding transformation heatmaps in (b) and (c).When we look at the accuracies achieved on the validation set in Table REF , FPG with structure loss achieved the largest improvement (from $0.576$ to $0.698$ ).", "However, this was not reproduced on the test set or for the macro-weighted F1 scores.", "Regarding the test accuracies, cycleGAN with combined losses and FPG resulted in the best scores.", "While FPG achieved the best accuracy and F1 scores for the images annotated by pathologist 1, cycleGAN with combined losses outperformed all other approaches for the images annotated by pathologist 2.", "Yet, the images transformed by cycleGAN with combined losses contained hallucination artifacts, as is reflected in the SSIM scores (Figure REF ).", "Unfortunately, for the classification network at hand, a direct link between transformation and classification quality does not seem to exist.", "In Figure REF an image transformed by cycleGAN with combined losses can be seen.", "The transformation adds a purple streak across the upper third of the image, which is also reflected in the heatmap of the changes in pixel intensities through the transformation.", "In contrast, the change heatmap of the images transformed by FPG with structure loss indicates that the color intensity of the background was reduced and that small cell structures remained intact (see Figure REF ).", "Hence, to avoid basing the classification on hallucination artifacts, the best bias-transfer approach cannot be judged based on classification accuracies only.", "Another factor that could contribute to the mismatch of transformation and classification quality is the very low accuracy on the original $\\textrm {NEW}_{p}$ test images, which is $25.6\\%$ for the images annotated by pathologist 1 and $21.7\\%$ for pathologist 2, compared to $81.6\\%$ achieved on the $\\textrm {TAR}_{p}$ test set.", "The samples of the two domains were annotated by different pathologists, who might interpret structures disparately, resulting in different Gleason scores.", "As a result, a transformation to the target domain might not be sufficient to reach high accuracy on this dataset.", "This is amplified by the fact that the classification network has the lowest accuracy for 4+4 samples (see Figure REF ), which is the most frequent class in $\\textrm {NEW}_{p}$ .", "For the original images of $\\textrm {NEW}_{p}$ most of the 4+4 samples were misclassified as 5+5 or 0 (benign) (see Figure REF ).", "As a consequence, an accidental improvement of the classification accuracy due to hallucination artifacts cannot be ruled out.", "All those reasons indicate that FPG with structure loss may still be considered the best performing bias transfer for the prostate biopsy samples, despite not resulting in the highest classification scores.", "It stably led to improved accuracies and F1 scores across all runs while keeping the balance between not adding hallucination artifacts to the images and imitating the target domain.", "Currently, bias can be overcome with hand-engineered methods [4] or with deep generative models, which are universal function approximators.", "While our datasets do not require a strong transformation, the simple baseline approach was not a sufficient solution.", "Like for many simple approaches, the transformation is applied uniformly across the whole image.", "Therefore, non-linear bias cannot be captured.", "The resulting, tinted images hindered the segmentation of the glomeruli and their podocytes, as well as the classification of prostate cancer.", "In contrast, nine generative approaches significantly improved the pixel-wise segmentation of the podocytes and six the segmentation of the glomeruli.", "The classification accuracies and F1 scores were improved by every variation.", "However, as indicated by the varying SSIM scores, not all approaches resulted in artifact-free images.", "In our experiments, the widely used cycleGAN architecture did not perform well for bias transfer without any modifications.", "The training process was unstable and often led to hallucination artifacts.", "This matches reports in previous studies.", "Manakov et al.", "described that, in their experiments, vanilla cycleGANs generated blurry images with checkerboard-like artifacts [17].", "CycleGANs were created for general image-to-image transformation.", "While goals like identity-transforming images that belong to the target domain and reconstructing the original from a transformed image are already incorporated in the basic model, other objectives have to be added explicitly and balanced carefully to achieve maximum performance for the task at hand.", "Otherwise the instability of the cycleGAN training process can be amplified, leading to complete mode collapse, as we experienced with the kidney biopsies.", "FPGs were invented to convert diseased to healthy samples, which requires removing structures from the images.", "For bias transfer, however, the content of the image has to stay the same.", "Since this is not explicitly safeguarded by FPG, it is not surprising that the additional losses had a positive impact on the transformation.", "U-Net cycleGANs, on the other hand, did not benefit as much from the losses as the other approaches.", "The structure loss (and the combined losses) did however further stabilize the training process since the transformation goals were explicitly incorporated into the loss function.", "In our evaluation, the transformation quality was well reflected in the segmentation scores, but only partially for the classification.", "It is particularly important to note that if the best approach is selected solely based on the resulting classification or segmentation scores, hallucination artifacts might be missed.", "The artifacts can introduce a new type of bias to the images which could `accidentally' improve the segmentation or classification scores.", "Finally, another factor that contributed to the mismatch of transformation quality and classification scores for the prostate biopsies is the inter-annotator variance.", "Multiple factors play a role in deciding which of the three architectures should be used to perform bias transfer.", "When a large number of domains are available, the training time can become a limiting factor.", "Since U-Net cycleGANs only transform between two domains, every target domain requires a separate model and training.", "Another common problem is a lack of data.", "If few samples exist (e.g.", "$<100$ per domain), U-Net cycleGANs might have too many trainable parameters to learn an adequate transformation.", "Both of these issues would warrant selecting FPGs instead since all domains are used to train a single generator.", "The cycleGAN variations did not match the performance of U-Net cycleGAN and FPG.", "Therefore, we do not recommend using vanilla cycleGANs for bias transfer in medicine.", "Regarding the evaluation of generative approaches, no consensus on adequate evaluation metrics exists.", "Here, the complementary metrics SSIM and FID were good indicators of hallucination artifacts and therefore suitable for evaluating bias transfer." ], [ "Limitations", "The selection of the models and the additional losses in this paper focused on bias transfer for domains with small differences between them, i.e., a shift in staining.", "Therefore, the results might not be reproducible for less similar domains.", "Additionally, bias transfer has to be performed with caution if the domains have an inherent content bias.", "A disease can become part of the `style' if one domain contains more healthy and the other more malformed samples [7]." ], [ "Conclusion and outlook", "The goal of this paper was to determine which deep generative approach results in the best bias transfer for medical images originating from IF confocal microscopy of kidney biopsies and H&E stained microscopy images of prostate biopsies.", "The performance on the test set mostly corroborated our findings obtained on the validation data.", "U-Net cycleGAN with combined losses and FPG with structure loss had a stable training process and created hallucination artifact-free images that imitated the target domain, improving the segmentation or classification performance.", "Our results show that bias transfer for histopathological images benefits from adding structure-based losses since they help with content preservation.", "The combination of MS-SSIM loss and additional identity loss is especially helpful if bias transfer is performed for domains that only require small changes.", "For medical image datasets with larger differences, e.g.", "a different staining type, the additional identity loss should only be used if it is weighted much lower than the MS-SSIM loss.", "Furthermore, additional losses are not a universal solution.", "The individual network architectures and approaches have to be adapted to the task at hand, as our differing results for the two modalities have shown.", "In future projects, similar datasets from additional sites could be investigated to get a firmer grasp on which transformation goals can be covered by the selected additional losses and which further limitations might prevent using deep learning-based bias transfer for medical datasets." ], [ "Acknowledgements", "This work was supported by DFG (SFB 1192 projects B8, B9 and C3) and by BMBF (eMed Consortia `Fibromap')." ] ]
2105.11765
[ [ "Scaling Hierarchical Agglomerative Clustering to Billion-sized Datasets" ], [ "Abstract Hierarchical Agglomerative Clustering (HAC) is one of the oldest but still most widely used clustering methods.", "However, HAC is notoriously hard to scale to large data sets as the underlying complexity is at least quadratic in the number of data points and many algorithms to solve HAC are inherently sequential.", "In this paper, we propose {Reciprocal Agglomerative Clustering (RAC)}, a distributed algorithm for HAC, that uses a novel strategy to efficiently merge clusters in parallel.", "We prove theoretically that RAC recovers the exact solution of HAC.", "Furthermore, under clusterability and balancedness assumption we show provable speedups in total runtime due to the parallelism.", "We also show that these speedups are achievable for certain probabilistic data models.", "In extensive experiments, we show that this parallelism is achieved on real world data sets and that the proposed RAC algorithm can recover the HAC hierarchy on billions of data points connected by trillions of edges in less than an hour." ], [ "Introduction", "The recent, unprecedented rise in large data sets has been largely fueled by the growth in unstructured and unlabeled data.", "A fundamental method for attaching structure to these types of datasets is through clustering, and advancing the state of the art in clustering continues to attract both practical and theoretical interest.", "Despite many advances, scaling clustering methods to massive data sets remains a hard task.", "Hierarchical Agglomerative Clustering (HAC) is one of the oldest but still most widely used clustering methods.", "Starting from a state in which every datapoint is in its own cluster, it successively merges pairs of clusters together to produce a hierarchy.", "The popularity of HAC stems from its ease of implementation coupled with a variety of useful properties.", "Unlike popular parametric methods such as $k$ -Means, HAC can generate flat clusterings by halting the merging process at a desired point, rather than requiring the number of clusters to be specified in advance.", "HAC's non-parametric nature also frees the user from the need to make assumptions about data distributions.", "Parametric clustering approaches can fail in high dimensional spaces since the underlying geometric shapes, i.e., Voronoi partitions in the case of $k$ -Means, become meaningless.", "HAC is also easily tunable through the use of different “linkage functions” that control how dissimilarities are updated when clusters merge.", "Finally, HAC enjoys the reputation for producing quality clusterings, and this behavior recently gained theoretical support [12].", "Scaling HAC has proven to be challenging.", "The sequentiality and random memory access requirements of HAC make it hard to parallelize computation, a critical aspect where data sets do not fit on a single machine.", "Previous efforts have been limited to inputs containing at most hundreds of thousands of datapoints [9].", "Single linkage HAC is the exception because of its unique connection to the minimum spanning tree problem [15], but has the drawback that it produces hierarchies which are highly sensitive to outliers [19].", "Another scalable approach to hierarchical clustering is affinity clustering [3], which is capable of scaling to billions of datapoints, but produces hierarchies which differ from HAC.", "We start with a brief introduction to HAC in Section .", "In Section we describe a massively scalable HAC algorithm (Algorithm ), which we call Reciprocal Agglomerative Clustering (RAC).", "RAC computes the exact HAC hierarchy for any reducible linkage (defined in Section , examples include average and complete linkage), and is scalable to billions of datapoints connected by trillions of edges.", "In Section , we give a rigorous proof of its correctness in Theorem REF .", "We note that while idea of merging reciprocal nearest neighbors appears in [13], there is no proof of correctness, speedup theory, or any details on parallel implementation.", "We next give a negative example (Theorem REF ) where RAC's parallelism does not guarantee speedup.", "We complement this result however by giving theoretical guarantees of runtime speedups in two scenarios: (i) Assuming the datapoints satisfy a notion of stability as defined in [2], and (ii) An average-case analysis of speedup when the graph of the data is an instance of a natural class of random graph models.", "We provide details of our implementation in Section , and in Section we use RAC to cluster billion-sized real world data sets and give scaling results with respect to CPUs and machines." ], [ "Hierarchical Agglomerative Clustering", "Hierarchical agglomerative clustering (HAC) produces a hierarchy of clusters, providing relationships between groups of datapoints at increasingly granular levels.", "An important parameter of this algorithm is the linkage function, which we will denote as $W$ , which defines the dissimilarity between two clusters of datapoints as a function of the dissimilarities between their constituent elements.", "Many commonly used clustering algorithms, e.g.", "SLINK [17], CLINK [6], and UPGMA [18] correspond to HAC with a specific linkage function.", "The three previously mentioned algorithms use single, complete, and average linkages, which are defined in Table REF .", "Table: Linkage functionsStarting from a state in which every data point is in a singleton cluster, HAC creates a hierarchy by sequentially combining the two clusters which are most similar according to the linkage function.", "This process is repeated until all of the original data points have been coalesced into a single cluster, and the unordered list of mergers, representing the cluster hierarchy, is returned (Algorithm ).", "[t] Hierarchical Agglomerative Clustering (HAC) [1] Set of clusters, $\\mathcal {C}$ , and a set of merges, $\\mathcal {M}$ , with default value $\\emptyset $ $|\\mathcal {C}| = 1$ Return $\\mathcal {M}$ $A, B \\leftarrow $ Pair of distinct clusters in $C$ which minimize $W(A,B)$ $\\mathcal {C} \\leftarrow \\lbrace A \\cup B\\rbrace \\cup \\mathcal {C}\\setminus \\lbrace A, B\\rbrace $ $\\mathcal {M} \\leftarrow \\mathcal {M} \\cup \\lbrace \\lbrace A,B\\rbrace \\rbrace $ Return $\\text{HAC}(\\mathcal {C}, \\mathcal {M})$ The linkage functions defined in Table REF all belong to the more specific class of reducible, Lance-Williams [10] linkage functions.", "These linkage functions have the property that the dissimilarities between clusters may be computed recursively in terms of dissimilarities between their refinements, i.e.", "given clusters $A, B$ , and $C$ , along with all their pairwise dissimilarities, we can compute $W(A\\cup B, C)$ in $O(1)$ time rather than by iterating over all of individual pairs of datapoints in the two sets, which would require $O(|C|(|A|+|B|))$ time.", "Reducible linkage functions have the property that merging two clusters does not make the resulting union more similar to any other cluster, i.e.", "for any three disjoint clusters $A, B,$ and $C$ , $W(A\\cup B, C)\\ge \\min (W(A,C), W(B,C))$ .", "Centroid linkage, where the dissimilarity between clusters is the distance between their centroids, is an example of a linkage function which is not reducible.", "If the linkage function used in HAC is reducible, then the dissimilarities between the clusters which are being merged in Algorithm  will be non-decreasing.", "There has been work on parallelizing portions of the HAC algorithm in the past.", "[4] keep the merges sequential but parallelizes the work done in each merge, e.g.", "finding the least dissimilar clusters and updating the cluster dissimilarities.", "[9] parallelizes the nearest neighbor chain algorithm [14] by using multiple threads to follow different chains but collisions between these chains could potentially create thread contention.", "In both cases, the algorithms were demonstrated on single machines for datasets with under a million datapoints." ], [ "Reciprocal Agglomerative Clustering", "When the HAC procedure is run using a reducible linkage function, the order in which clusters are merged can be relaxed [5], [13], [16].", "In particular, rather than merging the pair of clusters whose dissimilarity globally minimizes $W(A,B)$ , with a reducible linkage function, it is possible to merge any pair of clusters, $A$ and $B$ , that are reciprocal nearest neighbors.", "Reciprocal nearest neighbors are clusters that are nearest neighbors of one another, i.e., there exists no other cluster $C\\in \\mathcal {C}$ such that $W(A,C) < W(A,B)$ or $W(B,C) < W(A,B)$ .", "The key idea behind the nearest neighbor chain algorithm is to follow the nearest neighbor graph to identify such reciprocal nearest neighbors and to sequentially merge them.", "Inspired by this idea of reciprocal nearest neighbors, we propose a distributed algorithm called Reciprocal Agglomerative Clustering (RAC) to scale HAC to billions of data points.", "We describe a high level overview of RAC in this Section.", "RAC proceed in rounds where in each round, we merge all reciprocal nearest clusters in parallel on a set of machines instead of sequentially following a nearest neighbor chain.", "Algorithm presents a high level overview of RAC.", "Each round of RAC proceeds in two phases of finding reciprocal nearest neighbors and merging them, each of which is parallelized.", "The merge phase may be further subdivided into updating cluster dissimilarities and updating nearest neighbors (see Section  for a full distributed implementation).", "1.", "Find Reciprocal Nearest Neighbors: Given a set of clusters $\\mathcal {C}$ , we find and return in parallel all merges: pairs of clusters $\\lbrace C_i, D_i\\rbrace $ such that $C_i$ 's nearest neighbor is $D_i$ and vice-versa.", "2a.", "Update Cluster Dissimilarities: Once we merge the pairs of reciprocal nearest neighbor clusters, we update in parallel the dissimilarities of all pairs of clusters that have been changed and delete the original clusters that were merged.", "2b.", "Update Nearest Neighbors: For each cluster $C$ that belongs to a reciprocal nearest neighbor pair, and for each of $C$ 's neighbors $N$ , we find and update in parallel the new nearest neighbor of $N$ (since after merging $C$ with its reciprocal nearest neighbor, $C$ may no longer be $N$ 's nearest neighbor).", "Reciprocal Agglomerative Clustering (RAC) [1] Require: Set of clusters, $\\mathcal {C}$ , and a set of merges, $\\mathcal {M}$ , with default value $\\emptyset $ $\\mathcal {M} \\leftarrow \\mathcal {M} \\cup \\text{Find Reciprocal Nearest Neighbors}(C)$ $\\text{Update Cluster Dissimilarities}(C)$ [overlay,remember picture] right) ; [overlay,remember picture] top) ; $\\text{Update Nearest Neighbors}(C)$ [overlay,remember picture] bottom) ; Return RAC($\\mathcal {C}, \\mathcal {M}$ ) [overlay, remember picture] [decoration=brace,amplitude=0.5em,decorate, thick] ($(right)!(top.north)!", "($ (right)-(0,1)$)$ ) – ($(right)!(bottom.south)!", "($ (right)-(0,1)$)$ ) node [text width=2.5cm, pos=0.5, anchor=west] Merge;" ], [ "Theoretical Results", "In this section we first prove the correctness of RAC.", "RAC, as we have seen, proceeds in rounds, where each round performs a set of parallel merges.", "RAC can hope to perform significantly better than sequential implementations of HAC only if the number of rounds is significantly smaller than $n$ (the number of input datapoints).", "We first present a negative example showing that in the worse case, RAC can take $O(n)$ rounds even when the cluster tree is of depth $O(\\log n)$ (intuitively, a case with high parallelism).", "We then complement this by giving theoretical guarantees in two scenarios where the number of rounds is $O(\\log n)$ : We provide a characterization where the number of rounds of RAC is the same as the tree height, providing an explanation of its excellent performance in practical scenarios.", "We also show that RAC can achieve a provable average-case speedup in a probabilistic setting when the edges of the data are drawn from a random graph model.", "Lastly we provide a connection between the number of merges per-round and the overall complexity of RAC, proving that RAC can perform in near linear time under reasonable assumptions (in the experimental section we provide empirical proof that these assumptions are reasonable for many real applications)." ], [ "Proof of Correctness", "We start with the correctness proof, showing that given a reducible linkage and a set of clusters $\\mathcal {C}$ , $\\textsc {HAC}(\\mathcal {C}) = \\textsc {RAC}(\\mathcal {C})$ .", "The main idea in this proof is that at any state of HAC, there exists a set of pairs of clusters that are reciprocal nearest neighbors.", "We will first show that given a pair of clusters that are reciprocal nearest neighbors, HAC will merge them.", "We then show that the order in which HAC merges these reciprocal nearest neighbors does not matter.", "The combination of these results implies that all reciprocal nearest neighbors may be merged in parallel, with the resultant clusters matching the ones produced by HAC.", "Theorem 1 For reducible linkages $W$ , Hierarchical Agglomerative Clustering and Reciprocal Agglomerative Clustering produce the same result.", "That is, for a set of clusters $\\mathcal {C}$ , $\\textsc {HAC}(\\mathcal {C}) = \\textsc {RAC}(\\mathcal {C}).$ We prove this by strong induction on $|\\mathcal {C}|$ .", "The claim holds for $|\\mathcal {C}|=1$ .", "We will use the following two lemmas: Lemma 2 Let clusters $A, B\\in \\mathcal {C}$ be a pair of reciprocal nearest neighbors and let $\\epsilon = W(A, B)$ .", "Then the merge between $A$ and $B$ , $[A, B]$ , will appear in $\\textsc {HAC}(\\mathcal {C})$ .", "Let $\\mathcal {C}^*$ be any non-empty subset of $\\mathcal {C}$ which does not contain $A$ or $B$ and let $D$ be the union of all the clusters in $\\mathcal {C}^*$ .", "Using the fact that $W$ is reducible and that $\\epsilon < W(A,X)$ for any $X\\in \\mathcal {C}$ that is not $A$ or $B$ , we have that $W(A,D) = W\\left(A,\\cup _{X\\in \\mathcal {C}^*}X\\right)\\ge \\min _{X\\in \\mathcal {C}^*}\\left(W(A, X)\\right)> \\epsilon .$ This strict inequality means that $A$ cannot merge with $D$ while $B$ is still unmerged.", "A similar argument applies to $B$ .", "Since all clusters in $\\mathcal {C}$ will eventually be merged, the only possibility remaining is that $A$ and $B$ will be merged.", "We now show that the order in which HAC merges reciprocal nearest neighbors in $\\mathcal {C}$ does not matter.", "Lemma 3 Let $A, B\\in \\mathcal {C}$ be two clusters which are merged at some point in $\\text{HAC}(\\mathcal {C})$ .", "Let $\\mathcal {C^{\\prime }} = \\lbrace A \\cup B\\rbrace \\cup \\mathcal {C} \\setminus \\lbrace A, B\\rbrace $ .", "Then $\\textsc {HAC}(\\mathcal {C}, \\mathcal {M}) = \\textsc {HAC}(\\mathcal {C^{\\prime }}, \\mathcal {M}\\cup [A, B]).$ Figure: Step jj in HAC(𝒞,ℳ)\\textsc {HAC}(\\mathcal {C}, \\mathcal {M}) and in HAC(𝒞 ' ,ℳ∪[A,B])\\textsc {HAC}(\\mathcal {C^{\\prime }}, \\mathcal {M}\\cup [A, B]) Let $\\mathcal {C}_k$ (resp.", "$\\mathcal {C}^{\\prime }_k$ ) be the set of clusters produced after $k$ iterations of $\\text{HAC}(\\mathcal {C})$ (resp.", "$\\textsc {HAC}(\\mathcal {C}^{\\prime }, \\mathcal {M}\\cup [A, B])$ ).", "Let $s$ be the iteration in which the merge between $A$ and $B$ occurs in $\\text{HAC}(\\mathcal {C})$ .", "We will now show that $\\mathcal {C}_s = \\mathcal {C}^{\\prime }_s$ .", "Suppose by contradiction that $\\mathcal {C}_s \\ne \\mathcal {C}^{\\prime }_s$ .", "Consider the first iteration $j < s$ in the respective procedures, s.t.", "$\\mathcal {C}_j \\ne \\mathcal {C}^{\\prime }_j$ .", "Let ($C_{1}$ , $C_{2}$ ) (resp.", "($C^{\\prime }_{1}$ , $C^{\\prime }_{2}$ )) be the clusters that were merged in the creation of $\\mathcal {C}_j$ (resp.", "$\\mathcal {C}^{\\prime }_j$ ), where $C_{1} \\cup C_{2} \\ne C^{\\prime }_{1} \\cup C^{\\prime }_{2}$ .", "Since $\\mathcal {C}_{j-1} = \\mathcal {C}^{\\prime }_{j - 1} \\setminus \\lbrace A \\cup B\\rbrace \\cup \\lbrace A, B\\rbrace $ , it must be the case that $C^{\\prime }_{1} = A \\cup B$ or $C^{\\prime }_{2} = A \\cup B$ (see Figure REF ).", "Suppose without loss of generality that $C^{\\prime }_{1} = A \\cup B$ .", "We know that $W(A \\cup B, C^{\\prime }_{2}) < W(C_{1}, C_{2})$ since otherwise $C_{1}$ and $C_{2}$ would have merged in the creation $\\mathcal {C}^{\\prime }_j$ (right side of Figure REF ).", "Further, we know that $W(C_{1}, C_{2}) < W(A, C^{\\prime }_{2})$ since otherwise $A$ and $C^{\\prime }_{2}$ would have merged in the creation of $\\mathcal {C}_j$ (left side of Figure REF ).", "By similar logic, we have that $W(A \\cup B, C^{\\prime }_{2}) < W(B, C^{\\prime }_{2})$ .", "Thus, $W(A \\cup B, C^{\\prime }_{2}) < \\min (W(A, C^{\\prime }_{2}), W(B, C^{\\prime }_{2}))$ , a contradiction to $W$ being reducible.", "To finish the proof of Theorem REF , assume $|\\mathcal {C}|\\ge 2$ .", "Let $\\mathcal {R} = \\lbrace R_1, R_2, R_3, R_4, \\dots \\rbrace $ , be a set of clusters in $\\mathcal {C}$ where $\\lbrace R_{2i - 1}, R_{2i}\\rbrace , i = 1, 2, 3, \\dots $ are reciprocal nearest neighbor pairs.", "Let $\\mathcal {R^{\\prime }} = \\lbrace R_1 \\cup R_2, R_3 \\cup R_4, \\dots \\rbrace $ be their unions.", "By repeatedly applying the results from the above two lemmas, the inductive hypothesis and finally, using the definition of RAC, we get that $\\text{HAC}(\\mathcal {C})& = \\text{HAC}(\\mathcal {R^{\\prime }}\\cup \\mathcal {C} \\setminus \\mathcal {R}, \\lbrace [R_1, R_2], [R_3, R_4] \\dots \\rbrace ) \\\\&= \\text{RAC}(\\mathcal {R^{\\prime }}\\cup \\mathcal {C} \\setminus \\mathcal {R}, \\lbrace [R_1, R_2], [R_3, R_4] \\dots \\rbrace ) \\\\&= \\text{RAC}(\\mathcal {C})$" ], [ "Speedup guarantees", "We start by analyzing the number of merge rounds required by RAC.", "First, a negative result.", "Clearly, the number of rounds RAC requires is bounded from below by the height of the resulting dendrogram.", "One might hope that this bound is tight, in which we can use the tree height to characterize the parallelism inherent in a clustering problem.", "However, we show below there are inputs which require $O(n)$ rounds of RAC, even though the resulting dendrogram has only $O(\\log (n))$ height.", "Theorem 4 For any $n\\in \\mathbb {N}$ , set $X =\\lbrace 1 + \\epsilon , 2 + 4\\epsilon , 3 + 9\\epsilon , \\dots , 2^n + 2^{2n} \\epsilon \\rbrace $ , where $\\epsilon = 2^{-4n}$ .", "Then $\\textsc {RAC}(X)$ returns a dendrogram of height $n$ , but requires $\\Omega (2^n)$ rounds of reciprocal nearest neighbor merges to complete.", "[Proof in supplementary materials.]" ], [ "Stable Trees", "Notwithstanding this negative result, we can show that if the average linkage cluster tree has certain \"clusterability\" properties, then the number of rounds RAC requires is the same as the tree height.", "In [2], the authors introduce clusterability notion for flat clusters called stability.", "We adapt this definition for cluster trees as follows: Definition 1 A given cluster tree $T$ is stable if for any non-overlapping nodes $X, Y \\in T$ (i.e.", "the set of leaves of $X$ and $Y$ have empty intersection), and any $A \\subset X$ , $B \\subseteq Y$ $d(A, X \\setminus A) < d(A, B)$ Theorem 5 On stable cluster trees, RAC completes in a number of rounds equal to the tree height.", "Let the set of datapoints in the cluster tree be $P$ , and $h$ be the height of the tree.", "Define $\\rho (X)$ to be the parent of $X$ in the cluster tree.", "Define a Leveled Clustering at level l, $C_l$ for $0 \\le l < h$ of the tree recursively as follows: for a level $l$ , let $M_l = \\lbrace X \\in C_l : \\exists Y \\in C_l, \\rho (X) = \\rho (Y)\\rbrace $ , i.e., nodes in $C_l$ that merge with another node in $C_l$ , and let $ C_l ={\\left\\lbrace \\begin{array}{ll}P,& \\text{if } l = 0\\\\C_{l-1} \\setminus M_{l-1} \\cup \\lbrace \\rho (X): X \\in M_{l-1}\\rbrace , & \\text{otherwise}\\end{array}\\right.", "}$ We claim that the state of the RAC algorithm after $l$ rounds of merging is $C_l$ .", "Note that this implies that the number of rounds required is $h$ , proving the theorem.", "Let us prove this recursively, the base case being obvious.", "For the inductive step, assume the cluster state is $C_l$ .", "Let $|M_l| = 2 \\cdot M$ for an integer $M$ (it is clear that $|M_l|$ is even) and let the members of $M_l$ be $m_0, m_1 \\ldots m_{2M-1}$ .", "Assume without loss of generality that $m_{2i}$ and $m_{2\\cdot (i+1)}$ are siblings (for all $i \\le M - 1$ ), i.e., $\\rho (m_{2i}) = \\rho (m_{2(i+1)})$ .", "Since we know that RAC produces the correct cluster tree, all we need to demonstrate is that all $m_{2i}$ and $m_{2\\cdot (i+1)}$ get merged simultaneously in the next step of RAC, or equivalently, that $m_{2i}$ and $m_{2\\cdot (i+1)}$ are reciprocal nearest neighbors.", "Take any other node $Y \\in C_l$ .", "Set $X = m_{2i} \\cup m_{2\\cdot (i+1)}$ , $A = m_{2i}$ and $B = Y$ , and by the definition of stability, we see that $d(m_{2i}, m_{2i+1}) < d(m_{2i}, Y)$ .", "The same argument can be shown to imply that $d(m_{2i}, m_{2i+1}) < d(m_{2i+1}, Y)$ and thus $m_{2i}$ and $m_{2\\cdot (i+1)}$ are reciprocal nearest neighbors." ], [ "Probabilistic setting", "We can extend the conclusion of Theorem REF to probabilistic settings which do not require the assumption of stability for the case of single linkage.", "In particular, we present a probabilistic graphical model in which the expected number of merges in each round is a constant fraction of the number of clusters in that round.", "Moreover, we show that this implies that RAC terminates in $O(\\log n)$ rounds both in expectation and with high probability.", "We start with a proof of the latter result.", "Theorem 6 Fix integer $n > 0$ and real $\\alpha > 0$ .", "Let $(X_k)_{k\\ge 0}$ be a sequence of random variables with $X_0 = n$ and $X_{k+1} = 1$ if $X_k = 1$ and $X_{k+1} = X_k - Z_k$ otherwise, with $Z_k$ satisfying the following conditions: $Z_k$ is integer valued and $1\\le Z_k\\le X_k-1$ and $E[Z_k | X_k] \\ge \\alpha X_k$ .", "Let $\\tau = \\arg \\min _{k}\\lbrace X_k=1\\rbrace $ .", "Then $\\tau = O(\\log n)$ in expectation and with high probability.", "The application of Theorem REF to RAC is as follows: Let $X_k$ be the number of clusters and $Z_k$ be the number of merges in round $k$ respectively.", "Then, if we can show that at least $\\alpha $ fraction of merges occur in each round in expectation, then RAC terminates in $O(\\log n)$ rounds of merges.", "Begin by letting $Y_k = X_k/(1-\\alpha )^k$ .", "We use the optional stopping theorem [7] applied to the $Y_k$ 's to prove the result.", "In this context, the theorem states that if (a) $\\tau $ is bounded and (b) $E[Y_{k+1}|Y_k] \\le Y_k$ , then $E[Y_{\\tau }] \\le E[Y_0] = n$ .", "Assuming for now that (a) and (b) hold, by Jensen's inequality, $E[Y_{\\tau }] = E[1/(1-\\alpha )^{\\tau }] \\ge 1/(1-\\alpha )^{E[\\tau ]}$ which gives $E[\\tau ] \\le \\log n/\\log (1/(1-\\alpha ))$ .", "The high probability guarantee follows from Markov's inequality: $P[\\tau > (1+\\beta )\\frac{\\log n}{\\log (1/(1-\\alpha ))}] &= P[1/(1-\\alpha )^{\\tau } > n^{1+\\beta }] \\\\&\\le E[1/(1-\\alpha )^{\\tau }]/n^{1 + \\beta }\\\\&\\le n^{1-\\beta }.$ We turn to proving (a) and (b).", "To see (a), note that by condition (i), $X_k$ decreases by positive integer values in each round, and hence $\\tau \\le n-1$ .", "(b) follows from the following calculation.", "$E[Y_{k+1}|Y_k] &= E[Y_{k+1}|X_k]\\\\&= (1/(1-\\alpha )^{k+1})E[X_{k+1}|X_k]\\\\&= (1/(1-\\alpha )^{k+1})E[X_k-Z_k|X_k]\\\\&\\le (1/(1-\\alpha )^{k+1})X_k(1-\\alpha )\\\\&= Y_k.$ We now apply this result to two probabilistic settings.", "Single Linkage, 1-dimensional grid: Consider points $\\lbrace x_1, x_2, \\ldots , x_n\\rbrace $ on the real axis with $x_1 < x_2 < \\ldots < x_n$ such that the dissimilarities between consecutive points are sorted uniformly at random.", "Such a model can be generated by choosing the $x_i$ 's iid on the unit interval $[0, 1]$ and then relabeling them in increasing order.", "Suppose after some number of rounds, $k$ clusters $C_1, C_2, \\ldots , C_k$ remain with $k > 2$ .", "It is clear that each $C_i$ consists of a subset of points with contiguous indices, thus $C_i = \\lbrace x_{j_i}, x_{j_i + 1}, \\ldots , x_{j_i+ n_i}\\rbrace $ .", "Denoting the dissimilarity between $C_i$ and $C_{i+1}$ by $d(C_i, C_{i+1})$ , the probability that $C_i$ merges with $C_{i+1}$ is then given by $P[d(C_i, C_{i+1}) < d(C_{i-1}, C_i)]$ if $i = k-1$ , $P[d(C_i, C_{i+1}) < d(C_{i+1}, C_{i+2})]$ if $i = 2$ , and $P[d(C_i, C_{i+1}) < \\min (d(C_{i-1}, C_i), d(C_{i+1}, C_{i+2}))]$ otherwise.", "Under the assumption of randomly sorted edge weights, this simplifies to $P[C_i\\text{ merges with }C_{i+1}] ={\\left\\lbrace \\begin{array}{ll}1/3 &\\quad \\text{if }i = 2, 3, \\ldots , k-2 \\\\1/2 &\\quad \\text{if }i = 1\\text{ or }i = k-1.\\end{array}\\right.", "}$ By linearity of expectation, the number of merges is at least $k/3$ (if $k = 2$ , then a single merge occurs which also satisfies this).", "Thus, we can apply Theorem REF with $\\alpha = 1/3$ .", "Single Linkage, Bounded Degree Probabilistic Graph: We can generalize the previous result as follows.", "Let $G = (V, E)$ be a weighted graph on $n$ vertices with the weights sorted at random and each vertex initially being a singleton cluster.", "Let $E(C_i, C_j)$ denote the edges between clusters $C_i$ and $C_j$ .", "In a given round, the pair of clusters $C_i$ and $C_j$ merge if $\\min _{e\\in E(C_i, C_j)} e < \\min (\\min _{l\\ne j}\\min _{e\\in E(C_i, C_l)}e, \\min _{l\\ne i}\\min _{e\\in E(C_l, C_j)}e),$ i.e.", "according to the Single Linkage rule.", "Suppose that after some number of rounds, $k$ clusters remain.", "We can compute the expected number of merges as follows.", "Let $d_{ij} = |E(C_i, C_j)|$ and let $d_i = \\sum _{j\\ne i}d_{ij}$ .", "Then the probability that $C_i$ and $C_j$ merge is given by $P_{ij} = \\frac{d_{ij}}{d_i + d_j + d_{ij}}.$ By linearity of expectation, the expected number of merges is given by $M(G) = \\sum _{i=1}^k\\sum _{j=i+1}^kP_{ij}$ .", "Now suppose that the cluster graph is of bounded degree, i.e., $d_i \\le d = O(1)$ (as mentioned before, this is a reasonable assumption and supported by experiments).", "Note that since eventually all clusters merge, $d_i \\ge 1$ for all $i$ .", "Thus, $|\\lbrace \\lbrace i, j\\rbrace |d_{ij} \\ge 1\\rbrace | \\ge k-1$ .", "For each such $(i, j)$ , $P_{ij} \\ge 1/(d_i + d_j) \\ge 1/(2d)$ from which we have $M(G) \\ge (k-1)/(2d) \\ge k/(4d)$ and Theorem REF applies with $\\alpha = 1/(4d)$ ." ], [ "Runtime analysis", "We now study the time complexity of RAC which depends on the number of rounds and the complexity of each round.", "In each round, the algorithm needs to: a) Find reciprocal nearest neighbors b) Perform the resultant merges c) Update nearest neighbors for every remaining cluster.", "Finding the reciprocal nearest neighbors for every cluster is $O(n)$ operation, totaling $O(n^2)$ over all rounds.", "Merging a pair costs $O(n)$ , and there are $O(n)$ merges across all rounds, gaining another $O(n^2)$ .", "Finally, recomputing nearest neighbors is a $O(\\log n)$ operation if we use a min-heap.", "However, in practice, we simply iterate over a unsorted list in $O(n)$ (due to its improved cache-locality whose benefits outweighs the theoretical downsides).", "Nevertheless, this gives us $O(n^2)$ cost per-round for an overall cost of $O(n^3)$ , which is rather large.", "However, this makes really worst-case assumptions about the number of merges and character of the data distribution that do not hold in practice.", "In particular, as we will see in practice in the experimental section, many merges happen simultaneously and the number of rounds are much smaller than $n$ .", "We characterize this in the following theorem.", "Theorem 7 Assume that a constant $\\alpha $ fraction of nodes are merged in each round.", "Then, RAC runs in expected time $O(n^2)$ (as opposed to $O(n^3))$ .", "Let the runtime be $f(n)$ .", "Then we have the following: $f(n) &\\propto \\overbrace{\\alpha n \\cdot n}^{\\text{merges}} + \\overbrace{\\left(\\left(1-\\frac{\\alpha }{2}\\right) n\\right)^2}^{\\text{update nearest neighbors}} + f\\left(\\left(1 - \\frac{\\alpha }{2}\\right)n\\right) \\\\&\\le n^2 \\left(\\alpha + \\left(1 - \\frac{\\alpha }{2}\\right)^2\\right) \\sum _{i=0}^\\infty \\left(1 - \\frac{\\alpha }{2}\\right) ^ i = O(n^2).$ In view of the probabilistic models in the Subsection REF , we have a probabilistic analog of Theorem REF : Theorem 8 For the probabilistic graph model of Subsection REF with an expected constant fraction of nodes merging in each round, RAC runs in expected time $O(n^2)$ .", "[Proof in supplementary materials.]", "Now we argue that under reasonable assumptions, the runtime of the distributed algorithm can be nearly linear.", "Consider a problem with $n$ datapoints where every cluster has a degree bounded by $k$ (for large datasets this is a reasonable assumption – See Section ).", "Consider a round in which $m$ merges occur.", "Then there are $O(\\min (n, m \\cdot k))$ non-merging nodes that have their neighborhoods changed.", "Assume a fraction, $\\beta $ , of these have their nearest neighbor updated (equivalently, their nearest neighbor was merged).", "Each of these need $O(k)$ time to find the new nearest neighbor.", "Then the run-time of RAC can be decomposed as in Table REF .", "Table: Breakdown of run-times into phases.It is clear that for efficient runtimes we want $m$ to be large and $\\beta $ to be small.", "Assume as before that $m = \\alpha \\cdot n$ .", "Then the complexity of the $i^{\\text{th}}$ round (following arguments similar to Theorem REF ) is $O(\\alpha \\cdot n \\cdot k + \\alpha \\beta n k^2)$ .", "If $\\beta = O(k^{-1})$ then the total run time will be $O(n \\cdot k)$ .", "This assumption on $\\beta $ means that a merge pair can change at most a constant number of their neighbors nearest neighbor.", "We show experimental support for this assumption in the next section.", "Assuming a parallelism factor of $P$ (in the form of machines/CPU etc), we can state the following: Theorem 9 If $\\alpha = \\Omega (1)$ (constant fraction of nodes merge in each round) and $\\beta = \\Theta (\\frac{1}{k}$ ) (a merge pair changes at most a constant number of non-merging nodes nearest neighbors), then the runtime of RAC with $P$ parallelism factor is $O(\\frac{n \\cdot k}{P})$ .", "For very large datasets $k$ is often bounded by a few hundred or thousands, and $P$ can scale up to many hundreds.", "Thus in practical situations, we can often hit nearly linear time performance with RAC." ], [ "Implementation", "In this section, we provide a detailed implementation of RAC.", "In RAC, clusters are distributed across machines and have the following methods and attributes: id - a unique numeric identifier for the cluster, will_merge - a boolean flag which indicates whether the cluster will be merging in the current round, nn (abbreviation for nearest_neighbor) - a cached reference to the closest cluster, neighbors - the set of clusters with dissimilarities to the current cluster, update_dissimilarity(other_cluster, new_dissimilarity) - updates the stored dissimilarity value between this cluster and the other cluster to new value, get_dissimilarity(other_cluster) - returns the cached value of the dissimilarity between this cluster and the other cluster, and update_nearest_neighbor() - iterates over this cluster's neighbors to find and cache the reference to the neighboring cluster that this cluster is closest to.", "In each round of RAC, we first find all reciprocal nearest neighbors and then perform the merges in parallel.", "These steps are repeated until no merges are possible.", "Between each step, we wait for all machines to finish.", "We perform these parallel merges efficiently by avoiding contention as follows: for parallel merging clusters, $A\\cup B$ and $C \\cup D$ , the value of $W(A \\cup B, C \\cup D)$ will be computed twice, once for $A\\cup B$ and once for $C \\cup D$ .", "However, neither process needs to wait on the other.", "In practice, we find that this results in much greater throughput.", "When two clusters are to be merged, the machine owning the cluster with the lower id will be responsible for the merge operation and will own the merge result.", "In practice, the cluster with the lower id will be overwritten by the merge result, and the cluster with the higher id will be deleted.", "Because the clusters are sharded over machines, information about a cluster's nearest neighbor, such as its nearest neighbor, can typically only be found via a remote call.", "Rather than blocking on this, we buffer remote calls and only issue them in batches.", "This allows us to pipeline the computation and communication processes, which in turn allows us to scale the number of machines.", "For simplicity, Algorithm does not show batching of remote calls.", "Algorithm , along with the following procedures used in the algorithm, provide a full implementation of RAC.", "[H] 0 *Find Reciprocal Nearest Neighbors [1] Require: Set of clusters, $\\mathcal {C}$ $\\mathcal {M} \\leftarrow \\emptyset $ Set of merges to return $|\\mathcal {C}|=1$ Return $\\mathcal {M}$ $C\\in \\mathcal {C}$ $C.$ will_merge$\\leftarrow C$ .nn.nn is $C$ $C\\in \\mathcal {C}: C$ .will_merge and $C.\\text{id} < C$ .nn.id $\\mathcal {M} \\leftarrow \\mathcal {M}\\cup \\lbrace \\lbrace C, C.\\text{nn}\\rbrace \\rbrace $ Return $\\mathcal {M}$ [H] *Update Cluster Dissimilarities [1] Require: Set of clusters, $\\mathcal {C}$ $C\\in \\mathcal {C}: C$ .will_merge and $C.\\text{id} < C$ .nn.id $\\mathcal {N}\\leftarrow \\emptyset $ $C^{\\prime }\\in C.\\text{neighbors} \\cup C.\\text{nn}.\\text{neighbors}$ $C^{\\prime }.\\text{will\\_merge}$ and $C^{\\prime }\\text{.id}< C^{\\prime }$ .nn.id $w\\leftarrow W(C\\cup C.\\text{nn}, C^{\\prime } \\cup C^{\\prime }.\\text{nn})$ $C.\\text{update\\_dissimilarity}(C^{\\prime }, w)$ $\\mathcal {N}\\leftarrow \\mathcal {N} \\cup \\lbrace C^{\\prime }\\rbrace $ not $C^{\\prime }.\\text{will\\_merge}$ $w\\leftarrow W(C\\cup C.\\text{nn}, C^{\\prime })$ $C.\\text{update\\_dissimilarity}(C^{\\prime }, w)$ $\\mathcal {N}\\leftarrow \\mathcal {N} \\cup \\lbrace C^{\\prime }\\rbrace $ $C.\\text{neighbors}\\leftarrow \\mathcal {N}$ $C\\in \\mathcal {C}: C$ .will_merge and $C.\\text{id} > C$ .nn.id Delete $C$ $C\\in \\mathcal {C}: C$ .will_merge $C^{\\prime }\\in C.\\text{neighbors}$ $C^{\\prime }.\\text{update\\_dissimilarity}(C, C.\\text{get\\_dissimilarity($C^{\\prime }$)})$ [H] *Update Nearest Neighbors [1] Require: Set of clusters, $\\mathcal {C}$ $C\\in \\mathcal {C}$ $C$ .will_merge or $C$ .nn.will_merge $C$ .update_nn()" ], [ "Experimental evaluation", "In this section we experimentally evaluate RAC's scalability.", "In particular, since RAC produces the same hierarchy as HAC, we do not compare RAC's output to any other clustering algorithms.", "We run our experiments on a modern cloud infrastructure with access to hundreds of multi-core machines connected by a fast network.", "In all experiments below, we share the run-times in a relative scale or as speedup compared to a baseline.", "We first demonstrate the validity of our assumptions by investigating the merge characteristics on real datasets.", "Next we show that RAC can scale to very large real datasets (eg.", "1B nodes and 1T edges).", "We demonstrate that RAC effectively uses the parallelism available to it by running experiments with different machine/CPU configurations.", "Finally, we analyze the runtime of the merging phase of RAC, giving insights into its performance characteristic.", "Table: DatasetsWe study datasets of objects represented as vectors and construct graphs on them using a suitable distance metric.", "Our datasets are listed in Table REF .", "News20 [11] and RCV1 [1] are well known medium sized datasets.", "While they are too small for RAC's benefits to be visible, we use them to study the possibilities of parallelism in RAC.", "We use the large SIFT datasets [8] and WEB88M to demonstrate the scaling capacities of RAC.", "Note that the last SIFT dataset (SIFT200K) was constructed for this work by sampling randomly from the SIFT1M dataset.", "We generated WEB88M by crawling popular web pages and generating bag of words sparse features.", "To demonstrate that RAC indeed enables agglomerative clustering at unprecedented scales, we cluster datasets whose sizes are toward the limits of available infrastructure.", "We experiment with both a) complete graphs on the datasets b) sparse graphs (i.e.", "graphs with (much) fewer than $n^2$ edges).", "These sparse graphs are constructed by keeping first $k$ nearest neighbors or only neighbors within an $\\epsilon $ ball.", "In particular, SIFT1M is complete while SIFT1B is sparse.", "The sparse graph setting is common in practical applications.", "Merge Characteristics ($\\alpha $ and $\\beta $ ): Recall that RAC proceeds in rounds, merging a number of reciprocal nearest neighbors simultaneously.", "Figure REF shows the number of merges per round for 4 datasets, demonstrating high parallelism (specially for initial rounds).", "What perhaps is non-intuitive is hump for the SIFT datasets suggesting that RAC goes through a bottleneck before finding opportunities for parallelism again.", "For News20 and RCV1 dataset, we also show (Figure REF (a)) that the number of nearest neighbor updates per merge is bounded.", "Recall that this was an important part of our complexity analysis in Theorem REF (formulated as parameter $\\beta $ ).", "Figure: Merge characteristics: a) Number of nearest neighbor updates per merge for News20 and RCV1b) Number of merges per round for News20 and RCV1.", "c) and d) Number of merges per round for SIFT1B and SIFT1M respectively.", "Experiments on Large Datasets: Table REF shows representative timings for the 4 large datasets we achieved for complete linkage.", "Table: Performance of RAC on large datasets.We have specified run times normalized relative to WEB88M's runtime, but note that all experiments finished in under a few hours.", "The times presented are for merging only and do not include time to load the edges from disk.", "Edge loading accounts for 15% to 50% of the total run time in these datasets.", "A few observations about the results in Table REF .", "First, the number of merge rounds for all datasets are small compared to the size of the dataset, providing further support of the efficacy of the RAC approach.", "We note that the complete graphs register slower timing (compare SIFT1M vs SIFT1B), which is not surprising since the neighborhoods shuttling across the network is much larger.", "In Figure REF we demonstrate how RAC scales with access to more machines and CPUs, showing that the RAC algorithm is able to effectively use parallelism in the algorithm and the infrastructure to solve HAC at scale.", "Figure: Improved runtimes with increased machines and CPU in Figures (a-c).", "Figures (a) and (b) show how runtime improves with number of machines for SIFT200K and SIFT1B datasets respectively.", "Figures (c) shows speedups on SIFT1B when running RAC using 200 machines but increasing the number CPUs per machine.", "Figure (d) shows a log-log plot of merge time as a function of the number of mergers occurring in a round (for SIFT1M and SIFT200K).", "Merge time scales roughly linearly in number of merges.", "Finally, let us dig a bit deeper into the merge phase of the algorithm.", "We already mentioned that neither compute nor network dominates the run-time.", "This indicates that the merge phase, which uses network and compute equally (see Table REF ), would dominate the runtime.", "This is indeed what we find experimentally, i.e., that run-time for merge rounds is nearly linear in the number of merges (Figure REF (d)) in that round." ], [ "Conclusion", "We have introduced Reciprocal Agglomerative Clustering (RAC), a distributed variation of Hierarchical Agglomerative Clustering (HAC) that can be efficiently parallelized.", "RAC achieves efficiency by identifying reciprocal nearest neighbors and merging them in parallel.", "We have proved the correctness of RAC and that it achieves speedup guarantees in stable cluster trees as well as in certain probabilistic settings.", "Lastly, we have experimentally shown that RAC can scale to billions of nodes and trillions of edges.", "Supplementary material" ], [ " We provide proofs of Theorems 4 and 8 in the following sections.", "We fix $n\\in \\mathbb {N}$ and use the notation $\\epsilon =2^{-4n}$ , $P_k = (k+1) + \\epsilon (k+1)^2$ , and $X = \\lbrace P_i|i=0, \\ldots , 2^n-1\\rbrace $ .", "We first prove that RAC with average linkage takes $\\Omega (2^n)$ rounds with input $X$ .", "We do this by showing that in each round, at most one merge involves the singleton clusters $\\lbrace P_i\\rbrace $ (in fact, we show that they are merged in order of increasing $i$ ).", "We first note that clusters always consist of a contiguous set of points.", "In a given round, let $r$ be the smallest index such that for $i\\ge r$ , each $P_i$ is in a singleton cluster.", "Let $C_r=\\lbrace \\lbrace P_i\\rbrace | i = r, \\ldots , 2^n-1\\rbrace $ denote these singleton clusters.", "For $i > r$ , cluster $\\lbrace P_i\\rbrace $ 's nearest neighbor is $\\lbrace P_{i-1}\\rbrace $ .", "As for $\\lbrace P_r\\rbrace $ , because $P_{r-1}$ is not in a singleton cluster, $\\lbrace P_r\\rbrace $ 's nearest neighbor must be $\\lbrace P_{r+1}\\rbrace $ .", "As a result, the only reciprocal nearest neighbor involving clusters in $C_r$ is the pair $(\\lbrace P_r\\rbrace , \\lbrace P_{r+1}\\rbrace )$ .", "This completes the proof of the lower bound.", "In the rest of the proof, we show that HAC (and hence RAC) with average linkage produces a dendrogram of height $n$ with $X$ as input.", "Consider the natural complete binary tree $\\mathcal {T}$ on $X$ .", "Specifically, define nodes at the bottom level $N^0_i = P_i$ .", "For higher levels, define $N^l_i$ recursively, i.e.", "$N^l_i$ will have $N^{l-1}_{2i}$ and $N^{l-1}_{2i + 1}$ as its children.", "We note following simple properties of $\\mathcal {T}$ below, where we use the notation $X > Y$ for sets $X$ and $Y$ to mean that every point of $X$ is greater than every point in $Y$ .", "$\\mathcal {T}$ has height $n$ .", "There are $2^{n-l}$ nodes at level $l$ .", "The subtree rooted at $N^l_i$ contains the following $2^l$ points as leaves: $\\lbrace P_k | i \\cdot 2^l \\le k \\le (i + 1) \\cdot 2^l - 1 \\rbrace $ .", "$N^l_{i + 1} > N^l_i$ .", "We will show that HAC with average linkage produces $\\mathcal {T}$ .", "Since $\\mathcal {T}$ has height $n$ , this will complete the proof.", "Let $d(P_k, P_i) =|P_k - P_i|$ be the Euclidean distance between two points.", "For sets $X$ and $Y$ , define their distance according to average linkage, i.e., $d(X, Y) = \\frac{1}{|X||Y|} \\sum _{P_k \\in X, P_i \\in Y} d(P_k, P_i)$ .", "The following is easy to see: Lemma 10 Let $X, Y, Z$ be such that $X > Y > Z$ .", "Then $d(X, Y) < d(X, Z)$ .", "We will need the following series of somewhat tedious but straightforward lemmas regarding the distances between pairs of nodes in $\\mathcal {T}$ .", "Lemma 11 $d(N^l_i, N^l_{i + 1})$ is monotonically increasing in $i$ .", "We evaluate the more general distance between nodes $N^l_i$ and $N^l_{i + u}$ in the same level $l$ for any $u > 0$ : $& d(N^l_i, N^l_{i+u}) = \\frac{1}{2^{2l}}\\sum ^{2^l-1}_{r = 0} \\sum ^{2^l-1}_{s = 0} d\\left(P_{2^l\\cdot i + r}, P_{2^l\\cdot (i + u) + s}\\right) \\nonumber \\\\&= \\frac{1}{2^{2l}}\\sum _{r, s=0}^{2^l-1} (2^l (i+u) \\cdot s + (2^l (i+u) \\cdot s)^2 \\epsilon - 2^l i+ \\cdot r - (2^l i+ \\cdot r)^2 \\epsilon ) \\nonumber \\\\&= \\frac{1}{2^{2l}} \\sum _{r,s=0}^{2^l-1} ( (2^l \\cdot u + s - r) + (2^l \\cdot u + s - r)\\cdot (2^{l + 1} i + 2^l \\cdot u + s - r) \\cdot \\epsilon )$ Setting $u = 1$ and noting that $s$ and $r$ are upper bounded by $2^l - 1$ , we have that $(2^l + s - r)$ is positive, and hence the last expression is monotonically increasing in $i$ .", "The following lemma relates the nodes in two different levels: Lemma 12 Consider the nodes $N^l_{2 i+2}$ and $N^{l+1}_j$ for $j \\le i$ .", "Then $d(N^l_{2 i+2}, N^l_{2 i+3}) < d(N^l_{2 i+2}, N^{l+1}_j)$ .", "Informally, the distance from a node to its neighbor at the same level on its right is less than the distance to its neighbor on a higher level on its left.", "Since $N^{l+1}_{j-1} < N^{l+1}_j$ , it suffices to prove the lemma for $j = i$ .", "Now, by construction, $N^{l+1}_i$ is the result of the merging $N^{l}_{2 i}$ and $N^{l}_{2 i + 1}$ and thus $d(N^{l+1}_i, N^l_{2 i+2}) = \\frac{1}{2} (d(N^{l}_{2 i}, N^l_{2 i+2}) + d( N^{l}_{2 i + 1}, N^l_{2 i+2})).$ Thus, we need to prove that $\\frac{1}{2} (d(N^{l}_{2 i}, N^l_{2 i+2}) + d( N^{l}_{2 i + 1}, N^l_{2 i+2})) > d(N^l_{2 i+2}, N^l_{2 i+3})$ , which (replacing $2i$ with $i$ ) is equivalent to proving $\\frac{1}{2} (d(N^{l}_{i}, N^l_{i+2}) + d( N^{l}_{i + 1}, N^l_{i+2})) > d(N^l_{i+2}, N^l_{i+3})$ for any $i$ and $l$ .", "Rewriting each distance term using Equation (REF ), it suffices to prove for any $r$ and $s$ that $& (2^l \\cdot 2 + s - r) + (2^l \\cdot 2 + s - r)\\cdot (2^{l + 1} i + 2^l \\cdot 2 + s - r) \\cdot \\epsilon \\nonumber \\\\& + (2^l + s - r) + (2^l + s - r)\\cdot (2^{l + 1} (i + 1) + 2^l + s - r) \\cdot \\epsilon > \\nonumber \\\\& 2 ((2^l + s - r) + (2^l u + s - r)\\cdot (2^{l + 1} (i + 2 )+ 2^l + s - r) \\cdot \\epsilon ).$ Focusing first on the terms not involving $\\epsilon $ , we have $(2^l \\cdot 2 + s - r) + (2^l + s - r) - 2 (2^l + s - r) &= (2^l \\cdot 2 + s - r) - (2^l + s - r) \\\\&= 2^l \\ge 1.$ Since terms including $\\epsilon $ are much smaller than 1, the lemma follows.", "Lastly, this lemma states that adjacent distances at a higher level are larger than adjacent distances at a lower level.", "Lemma 13 If $l > l^{\\prime }$ , then $d(N^l_i, N^l_{i+1}) > d(N^{l^{\\prime }}_j, N^{l^{\\prime }}_{j+1})$ Using Equation (REF ) with $u = 1$ , we have $d(N^l_i, N^l_{i+1}) &= \\frac{1}{2^{2l}} \\sum _{r,s=0}^{2^l-1} (2^l + s - r) + O(\\epsilon ) \\\\&=2^l + \\frac{1}{2^{2l}} \\sum _{r,s=0}^{2^l-1} (s - r) + O(\\epsilon ).$ Note that the second term vanishes by symmetry.", "Thus, $d(N^l_i, N^l_{i+1}) = 2^l + O(\\epsilon )$ and for $l^{\\prime }\\ge l+1$ , $d(N^{l^{\\prime }}_j, N^{l^{\\prime }}_{j+1}) - d(N^l_i, N^l_{i+1}) = 2^{l^{\\prime }} - 2^l + O(\\epsilon ) \\ge 2^l + O(\\epsilon ) > 0$ for sufficiently small epsilon.", "To see that our choice of $\\epsilon = 2^{-4n}$ suffices, we can bound the term of Equation (REF ) involving $\\epsilon $ as $(2^l+ s - r)\\cdot (2^{l + 1} i + 2^l + s - r) \\cdot \\epsilon < 2^{4n-1} \\epsilon ,$ where we have used $i\\le 2^{n-l}$ , $s-r \\le 2^l - 1$ , and $l\\le n-1$ .", "Since the average of these terms is also bounded by the same quantity, the proof is complete.", "We now show that HAC with average linkage produces $\\mathcal {T}$ .", "Consider the state $S$ of the clustering algorithm after a merge.", "It consists of a partition of the nodes into a number of sets.", "We define $S$ to be good if there exists $l$ and $k$ with $0 \\le k < 2^{n-l}$ such that $S = \\lbrace N^l_i : i < k\\rbrace \\cup \\lbrace N^{l-1}_j : j \\ge 2k\\rbrace .$ Informally, this is a clustering of nodes covered by one or two levels where the nodes in each level consists of a single contiguous span.", "Note that the case $k = 0$ corresponds to a single layer.", "We claim that after each step of merging in HAC, the state is good.", "Note that this shows that $\\mathcal {T}$ is indeed the tree produced by HAC with average linkage.", "We proceed by induction.", "The base case is clear, since the state consisting of each $P_i$ as a single node is good.", "Now, consider an arbitrary state for a given $l$ and $k$ .", "We claim that nodes $N^{l-1}_{2k}$ and $N^{l-1}_{2k + 1}$ will be merged next.", "The fact that no other pair at level $l-1$ will be merged is implied by Lemma REF .", "Lemma REF implies that no pair at level $l$ is merged.", "Finally, a pair in different levels (one in $l$ and another in $l-1$ ) merging is ruled out by Lemma REF .", "It is clear that the merge of $N^{l-1}_{2k}$ and $N^{l-1}_{2k + 1}$ results in a good state, completing the proof.", "We prove that for the bounded probabilistic graph model, the expected runtime of RAC is $O(n^2)$ .", "Since the total number of rounds is $O(\\log n)$ in expectation, and each round takes $O(n^2)$ time, we easily obtain the weaker expected runtime guarantee of $O(n^2\\log n)$ .", "The following analysis strengthens this result by shaving off the $\\log n$ factor.", "Let $n_j$ be the number of clusters at the end of round $j$ , with $n_0 = n$ , and let $X_j = n_j/n_{j-1}$ be the fraction by which the number of clusters reduces in round $j$ .", "Thus, $n_j = n\\prod _{i=1}^j X_i$ .", "Since the number of merges, $M_j$ , in each round satisfies $E[M_j] \\ge \\alpha n_{j-1}$ , and $n_j = n_{j-1} - M_j$ , it follows that $E[X_j] \\le 1 - \\alpha $ .", "A key component of the proof is bounding the variance of the $X_j$ 's: Lemma 14 $\\operatorname{Var}(X_j|X_1, \\ldots , X_{j-1}) \\le B/n_{j-1}$ , where $B$ is a constant.", "Assuming Lemma REF , the rest of the proof is as follows.", "From the equation $E[X^2] = E[X]^2 + \\operatorname{Var}(X)$ , we have $E[X_j^2|X_1, \\ldots , X_{j-1}] \\le (1-\\alpha )^2 + B/n_{j-1}$ .", "Define $k$ such that $n_{k+1} < B/\\alpha \\le n_k$ .", "We decompose RAC's runtime into two parts - the time taken for the first $k$ rounds, and the remaining time.", "Using the fact that the number of points after the first $k$ rounds is the constant $B/\\alpha $ , we can write the expected runtime of RAC as $E[\\sum _{j=0}^k O(n_j)^2 + O(1)] &= \\sum _{j=0}^k E[O(n\\prod _{i=1}^j X_i)^2] \\\\&= O(n^2)\\sum _{j=0}^k E[\\prod _{i=1}^j X_i^2],$ where we use the convention that the empty product is 1.", "It suffices to show that $\\sum _{j=0}^k E[\\prod _{i=1}^j X_i^2] = O(1)$ .", "Using Lemma REF , we have $E[X_j^2|X_1, \\ldots , X_{j-1}] &\\le (1-\\alpha )^2 + B/n_{j-1} \\\\&\\le (1-\\alpha )^2 + \\alpha &\\text{ (using } B/\\alpha \\le n_{j-1}\\text{)}\\\\&= 1-\\alpha +\\alpha ^2 < 1.$ Defining $\\rho = 1-\\alpha +\\alpha ^2$ , we have $E[\\prod _{i=1}^j X_i^2] &= E[\\prod _{i=1}^{j-1} X_i^2E[X_j^2|X_1, \\ldots , X_{j-1}]] &\\text{ (by law of total expectation)}\\\\&\\le \\rho E[\\prod _{i=1}^{j-1} X_i^2]\\\\&\\le \\rho ^j &\\text{ (by induction)}$ and $\\sum _{j=0}^k E[\\prod _{i=1}^j X_i^2] \\le \\sum _{j=0}^{\\infty }\\rho ^j = 1/(1-\\rho ) = O(1)$ .", "[Proof of Lemma REF ] The main idea behind the bound is that the number of merges, $M$ , is the sum of Bernoulli random variables with most pairs having non-positive covariance.", "Let $G = (V, E)$ be a graph with $m$ edges $e_1, \\ldots , e_m$ and vertex degrees $d_v \\le d$ for $v\\in V$ .", "Note that the bounded degree property implies the edge bound $m \\le d|V|/2$ .", "We assign weights to the edges so that they are sorted at random.", "For $i = 1, \\ldots , m$ let $Y_i$ be the indicator random variable for the event that edge $e_i$ is merged (i.e., the weight of $e_i$ is smaller than the weights of its adjacent edges).", "Then $M = \\sum _{i=1}^m Y_i$ and $\\operatorname{Var}(M) = \\sum _{i=1}^m \\operatorname{Var}(Y_i) + 2\\sum _{i<j}\\operatorname{Cov}(Y_i, Y_j)$ .", "To bound the pairwise covariances, $\\operatorname{Cov}(Y_i, Y_j)$ , we consider three cases: $e_i$ and $e_j$ are adjacent (i.e., have a vertex in common): In this case, $E[Y_i Y_j] = 0$ since both $e_i$ and $e_j$ cannot be merged.", "Thus, $\\operatorname{Cov}(Y_i, Y_j) = -E[Y_i]E[Y_j] < 0$ .", "$e_i$ and $e_j$ are not adjacent to a common edge: In this case, $Y_i$ and $Y_j$ are independent, since the random variables that define them are disjoint.", "Hence, $\\operatorname{Cov}(Y_i, Y_j) = 0$ .", "$e_i$ and $e_j$ are adjacent to a common edge: In this case, we can bound the number of such pairs by $d^2m$ since each common edge $e_i = (a, b)$ gives rise to at most $d_a d_b \\le d^2$ such pairs for $i = 1, \\ldots , m$ .", "Breaking up the covariance term $2\\sum _{i < j}\\operatorname{Cov}(Y_i, Y_j)$ according to the above cases, we have $\\operatorname{Var}(M) &= \\sum _{i=1}^m \\operatorname{Var}(Y_i) + 2\\sum _{i < j}\\operatorname{Cov}(Y_i, Y_j)\\\\&\\le m + d^2m = (d^2+1)m\\\\&\\le (d^2+1)dn_{j-1}/2,$ where we have used the fact that for the indicator random variables $Y_i$ , $\\operatorname{Var}(Y_i)\\le 1$ and $\\operatorname{Cov}(Y_i, Y_j)\\le 1$ , and that $m \\le dn_{j-1}/2$ .", "We apply this in the case $M=M_j$ .", "Since $X_j = 1 - M_j/n_{j-1}$ , we have $\\operatorname{Var}(X_j|X_1, \\ldots , X_{j-1}) &= \\operatorname{Var}(M_j/n_{j-1})\\\\&\\le (1/n_{j-1}^2)\\operatorname{Var}(M_j)\\\\&\\le (d^2+1)d/(2n_{j-1}).$" ] ]
2105.11653
[ [ "A Taxonomy Study on Securing Blockchain-based Industrial Applications:\n An Overview, Application Perspectives, Requirements, Attacks,\n Countermeasures, and Open Issues" ], [ "Abstract Blockchain technology has taken on a leading role in today's industrial applications by providing salient features and showing significant performance since its beginning.", "Blockchain began its journey from the concept of cryptocurrency and is now part of a range of core applications to achieve resilience and automation between various tasks.", "With the integration of Blockchain technology into different industrial applications, many application designs, security and privacy challenges present themselves, posing serious threats to users and their data.", "Although several approaches have been proposed to address the specific security and privacy needs of targeted applications with functional parameters, there is still a need for a research study on the application, security and privacy challenges, and requirements of Blockchain-based industrial applications, along with possible security threats and countermeasures.", "This study presents a state-of-the-art survey of Blockchain-based Industry 4.0 applications, focusing on crucial application and security and privacy requirements, as well as corresponding attacks on Blockchain systems with potential countermeasures.", "We also analyse and provide the classification of different security and privacy techniques used in these applications to enhance the advancement of security features.", "Furthermore, we highlight some open issues in industrial applications that help to design secure Blockchain-based applications as future directions." ], [ "Introduction", "The widespread adoption of the Internet of Things (IoT) and related network and communication technologies drives the modern industrial revolution known as Industry 4.0 [1].", "Sensors, actuators, and embedded systems used in the IoT for sensing, computing, and communicating data for industrial automation have a significant impact on Industry 4.0 [2].", "As stated, Industry 4.0 is a series of cutting-edge technologies based on advanced knowledge and communication standards and industry guidelines applied to manufacturing to help manufacturers accomplish their goals more effectively [3].", "With an emerging trend of new disruptive technologies being used in Industry 4.0, academics and researchers have focused their efforts on developing Industry 4.0-based applications for the benefit of society.", "This emerging trend provides an interconnected platform for exchanging large amounts of data used in different processes.", "However, as the number of users increases rapidly, the network often experiences bottlenecks, resulting in scalability and single point of failure issues.", "Furthermore, it is often vulnerable to different types of security and privacy threats [4].", "Additionally, due to the amount of data exchanged over such an unsecured network, ensuring the confidentiality, privacy, and integrity of data becomes a major concern in Industry 4.0.", "Blockchain seems to be an excellent solution for dealing with the aforementioned problems and issues [5].", "Blockchain technology aims to eliminate the central third party between communication parties and to provide an equal opportunity to all network nodes for controlling and managing the operations over the network.", "In general, Blockchain technology stipulates a trusted P2P platform with an apparent motive to design the decentralised applications for performing secure computations on transactions using cryptography algorithms.", "In addition to secure computations, Blockchain technology also offers a promising solution for storing the verified transactions on a shared, immutable ledger.", "This immutable feature is an embellished concept of Blockchain technology which provides the irreversible guarantee of transactions stored at distributed databases [6].", "After earning remarkable success in the field of digital cryptocurrencies, Blockchain technology has gained much momentum between different business communities, and even the interest of different industrial application domains such as IoT [7], banking [8] and financial services [9], Smart Grid (SG) [10], logistics [11] and medical [12].", "Recent years are witnesses to the flexible nature of Blockchain technology utilised by many applications to provide ease of automation of different manufacturing tasks with the utilisation of inherent features of Blockchain, such as decentralised topology, distributed ledger, transparency, traceability and auditability of data.", "Considering the aforementioned core features of Blockchain that can be handy for different businesses and fit in Industry 4.0, this technology was characterised by a new revolution for the applications mentioned earlier.", "In the banking and financial sector, for example, a high level of security is required to keep the exchange of customers’ money, data and information secure.", "Practically, this needs many mediators to move the money and assets over a network infrastructure, making the transactions more expensive and prone to errors, fraud and misinterpretations [13].", "Blockchain holds the potential to transform and innovate the way of transferring transactions and assets securely without a trusted party.", "Thus, it streamlines the transactions, as well as reducing their complexities and associated costs with full transparency and accountability.", "However, many application domains are still hesitant to adopt Blockchain because of these aspects of security and privacy.", "The focus of the previous surveys presented by Yang et al.", "[14] and Li et al.", "[15] is on providing a high-level overview of security and privacy aspects without discussing the implementation scenario for different application domains.", "With the advancements of Bitcoin and the related cryptocurrency applications, the research and development communities moved their focus to investigate the security and privacy aspects of these applications.", "For example, Khalilov and Levi [16] targeted two security properties (that is, anonymity and privacy) for Bitcoin-like Digital Cash Systems; Conti et al.", "[17] identified the security and privacy needs in Bitcoin and their related cryptocurrency applications and Zhang et al.", "[18] covered the security and privacy needs, and requirements of the Bitcoin-like cryptocurrency systems.", "However, these studies are specific surveys for exploring the security and privacy of financial transactions in different models.", "In the same context, [19] conducted a state-of-the-art survey for studying the importance of anonymity and transactional privacy in finance-related applications.", "This survey only covered some of the security attacks and provided only limited cryptography solutions.", "Considering the completion of a successful journey made by leading Blockchain versions, a new version of Blockchain, that is Blockchain 4.0, has been introduced to address the challenges and limitations of many real-world applications.", "The recent surveys covered security and privacy aspects with different application domains using Blockchain technology.", "Joshi et al.", "[20] studied security and privacy issues in some Blockchain-based applications such as finance, healthcare, mobile, defence and IoT.", "Salman et al.", "[21] investigated the importance of different security services such as confidentiality, authentication, access control in IoT, healthcare and some cloud computing applications.", "Dasgupta et al.", "[22] outlined the different security services covering the numerous Blockchain-based applications such as big data, medical and social networks.", "Hassan at al.", "[23] presented a comprehensive survey that highlights the privacy issues which arise with the integration of IoT and Blockchain technology for the services available publicly.", "However, these surveys do not provide a comprehensive study for security and privacy requirements, or for the challenges and their mapping to corresponding attacks, with potential solutions for Blockchain-based industrial applications.", "Accordingly, there is a need to research the landscape of security and privacy issues related to Blockchain-based industrial applications in order to support the design setup of these applications and meet the needs of secure environment.", "Such research reduces any reluctance to embrace and adopt Blockchain technology in Industry 4.0.", "To assist in this topic and provide directions for both developers and research communities to implement secure industrial Blockchain-based systems that can meet security and privacy requirements in the industrial scenario, we present a state-of-the-art research survey that focuses specifically on security and privacy issues in Blockchain-based Industry 4.0 applications and then discusses potential security techniques and solutions used to address them.", "Our concrete contributions to this paper are as follows: A detailed comparison of existing state-of-the-art research studies focusing on design, security and privacy issues in different Blockchain-based Industry 4.0 applications drives the research enhancement guidelines for our survey study.", "We examine the need of developing secure Blockchain-based Industry 4.0 applications, focusing on the design requirements, measuring criteria, and security and privacy requirements.", "We provide a comprehensive discussion on Blockchain-based industrial applications to meet security and privacy requirements and we further elaborate on this to achieve these by utilising security enhancement solutions.", "We explore, discuss and analyse the various types of security attacks detected on Blockchain-based Industry 4.0 applications, in conjunction with attack categories, attackers’ objectives, vulnerabilities exploited and target applications.", "We identify some open issues of integrating Blockchain technology into Blockchain-based Industry 4.0 applications on a larger scale, which provide researchers with fuel to develop potential future solutions." ], [ "Paper Organization", "The organisation of this paper is as follows.", "Section provides a related work that includes a detailed comparison of existing published surveys on security and privacy for Blockchain-based applications and highlights the limitations in them.", "In section , we provide an overview of Blockchain technology in its introduction, features, layers, types, evolution, storage structure and transaction models.", "Section classifies the design, security and privacy requirements of Blockchain-based Industry 4.0 applications.", "A detailed discussion on security and privacy requirements for Blockchain-based Industry 4.0 applications is provided in section .", "In section , we illustrates and categorises the security and privacy enhancement techniques used in different Blockchain applications to fulfil the security and privacy objectives.", "Section describes the different security and privacy attacks on Blockchain-based Industry 4.0 applications.", "Furthermore, section highlights the open issues required to address the development of secure Blockchain applications.", "Finally, we conclude our paper and provide some future research directions in section .", "This section compares the related surveys that explicitly focus on security and privacy issues in different Blockchain-enabled applications.", "We make a detailed comparison of existing state-of-the-art studies related to the security and privacy domain in Blockchain-based Industry 4.0 applications based on various properties, including published year, publisher, paper title, applications covered, problems addressed, existing threats and vulnerabilities, attacks detected, techniques and solutions proposed, and future directions.", "Table shows a detailed comparison of these surveys.", "Existing Surveys on Security and Privacy of Blockchain-based Industry 4.0 Applications Table: NO_CAPTION Table: NO_CAPTION Table: NO_CAPTION Table: NO_CAPTIONThe R3-Zcash organisation presents an initial security and privacy survey on the Blockchain in a technical report.", "In this report, Yang et al.", "[14] addressed the security challenges in Blockchain by focusing on confidentiality and privacy in Blockchain applications.", "The authors also described the basic attacks on Blockchain services like denial of service and 51% attacks, discussed some solutions and proposed approaches, such as Hawk and Enigma , to overcome these attacks.", "However, this survey only highlighted some of the security properties and objectives, and their solutions, without further discussion regarding the most recent vulnerabilities.", "Li et al.", "[15] highlighted the security and privacy threats in Blockchain systems.", "In this survey, the authors discussed the different vulnerabilities and attacks in these systems.", "Similar to [14], some of the security solutions, that is, smart pool, Oyente, Towncrier, and Hawk, are discussed to address the fundamental security weaknesses and privacy challenges in Blockchain systems.", "However, these solutions only deal with smart contract applications to enforce the security policies of the Blockchain systems.", "To address the anonymity and privacy challenges in Blockchain-based digital cash systems, Khalilov and Levi [16] provided a detailed survey to cover the given problems.", "Bitcoin and its further extension of digital cash systems aim to work with different community studies to resolve the various limitations of address mappings in digital cash systems.", "The authors also covered the multiple attacks on these systems with their prospective solutions.", "However, this research work is a specific study on the security and privacy of financial transactions with various models.", "Joshi et al.", "[20] presented a survey to highlight the security and privacy required in some of the Blockchain-based applications such as finance, healthcare, mobile, defence and IoT.", "However, the drawback of this study is that the authors only addressed two forms of attacks, including denial of service and 51% attacks, and proposed the specific cryptography primitives as a solution.", "Conti et al.", "[17] outlined the security and privacy needs in Bitcoin and its related applications.", "This survey identified and mapped the Bitcoin’s system’s significant loopholes in their associated threats and categorised each threat with their own proposed solutions and techniques.", "Even though the study considered the Bitcoin system’s significant vulnerabilities in the literature part, it only focused on addressing the various security requirements and challenges of the financial system.", "To cover the privacy issues in Blockchain technology, Feng et al.", "[19] presented a survey study highlighting the importance of anonymity and transactional privacy in finance-related applications.", "Moreover, the limited cryptography solutions regarding denial of service and Sybil attacks were discussed in this study.", "Salman et al.", "[21] summarised the importance of different security services such as confidentiality, authentication, access control in IoT, healthcare and some cloud computing applications.", "This study’s major limitation is that the discussion provided security services and challenges intended for a limited number of applications in Blockchain areas.", "Dasgupta at al.", "[22] outlined the different security services covering the numerous Blockchain-based applications such as big data, medical and social networks.", "However, this research only highlighted the security requirements and challenges for essential aspects of privacy, along with a limited number of cryptography solutions.", "Hassan at al.", "[23] presented the privacy issues that arise with the integration of IoT and Blockchain technologies for making services publicly available.", "This survey considered basic privacy parameters to secure communication in different Blockchain-based applications.", "This survey is a preliminary study for privacy, preserving strategies of IoT-based applications with a limited target scope.", "The recent research work by Zhang et al.", "[18] covered the requirements of security and privacy for Bitcoin-like cryptocurrency systems.", "Different security attacks on Blockchain services, such as the denial of service and mining attacks with multiple security solutions, were discussed.", "The limitation of this research work is that it only explored the security and privacy requirements, focusing on different models for financial transactions.", "In a further study, Wang et al.", "[30] discussed privacy issues related to user identity and transactional privacy in Blockchain systems.", "This research study covered the traditional security mechanisms used to protect privacy, such as zero-knowledge proof and ring signatures, channel protocol, encryption and coin mixing mechanisms (Mix coin, Blind coin, Coin join, and so on) based on Blockchain technology.", "However, this study only covered limited privacy protection solutions which were mainly based on Blockchain technology.", "Casino et al.", "[24] underlined the importance of Blockchain technology and its underlying features in various real-time applications, ranging from industrial to business perspectives.", "This study’s applications were health care, IoT, voting, supply chain and a few in the business sector, such as data management, banking and insurance.", "However, this study did not address security threats and privacy issues in these Blockchain applications, nor did it discuss further solutions to overcome them.", "Akram et al.", "[27] presented a systematic review of existing security solutions specifically designed for Industry 4.0 applications.", "However, this study’s scope only covered a few Blockchain-based security approaches, described their merits and demerits, and discussed the challenges of interoperability and governance.", "In another study, Maesa and Mori [28] explored the use of Blockchain from an Industrial 3.0 perspective and its links to underlying applications.", "They then further discussed the problem and solution requirements of Blockchain adoption in Industry 3.0.", "However, this study’s limitation was that it focused only on the importance of Blockchain technology in Industry 3.0 and did not cover the needs of security and the privacy issues.", "Mohanta et al.", "[25] discussed the importance of Blockchain in various Blockchain applications, including healthcare, finance, IoT, cloud computing, power grids, smart transport and so on, and then highlighted the different security and privacy issues and challenges in those applications.", "The limitation found in this survey study was that it focused only on security and privacy challenges and did not discuss security solutions to overcome those challenges.", "In a recent survey, Perera et al.", "[29] explored the possibility of adopting Blockchain technology in Industry 3.0 application sectors, particularly in the construction sector, by demonstrating its relevance with different use-case perspectives.", "However, this study’s focus was solely on exploring and mapping the various aspects and features of Blockchain in the industrial sectors and did not cover in detail the security threats and issues related to these applications or possible countermeasures.", "Fernandez-Carames and Fraga-Lamas [26] presented a survey to analyse the advantages and disadvantages of using Blockchain and smart contracts to build Industry 4.0 applications.", "However, this study primarily focused on describing a general roadmap for Industry 4.0 researchers to illustrate how to use Blockchain for more cybersecure industries.", "Bodkhe et al.", "[4] conducted a recent survey to investigate emerging Blockchain-based solutions and their applicability for various smart applications, especially in Industry 4.0.", "However, this study’s focus covered only the merit and demerits of available solutions with a few countermeasures.", "Furthermore, this study did not go into detail about security risks and privacy attacks in Blockchain applications.", "The Industry 4.0 revolution has brought new paradigms to the manufacturing industry, for example Cyber-Physical Production Systems (CPPSs), which can provide many advantages and future opportunities, such as self-awareness, self-prediction and self-reconfiguration.", "CPPSs attempts to connect the virtual and physical production realms but an integrated computational platform is necessary to execute these systems in the real world.", "To achieve this, Lee et al.", "[3] investigated the possible consequences of introducing Blockchain in real-world cyber-physical systems for creation and implementation perspectives.", "Moreover, a three-tier Blockchain architecture was also provided to direct industrial researchers to clearly define the role of Blockchain technology in next-generation manufacturing processes.", "To achieve the security and privacy of the devices and networks in industrial manufacturing processes under a smart factory setup, Lin et al.", "[31] presented a Blockchain-based secure mutual authentication system to enforce fine-grained access policies.", "Business process management (BPM) integrates with Industry 4.0 and Blockchain technology features such as decentralisation, immutability and accountability to digitise and automate business process workflows and to support open inter-operations of service providers, in order to achieve asset trustworthiness.", "To accomplish this goal, Viriyasitavat et al.", "[32] investigated a business process management method in the composition services in which Blockchain technology is used to identify the best possible combinations and determine partner businesses’ trustworthiness, using automated process management solutions.", "Moreover, a middleware approach is provided in [5] for leveraging Blockchain tools and capabilities to allow for more stable and transparent autonomous smart manufacturing applications, enabling different parties to build trust in the manufacturing process." ], [ "A Generalised Overview of Blockchain", "Blockchain is a decentralised and distributed ledger technology that follows the peer-to-peer (P2P) network fashion in which participating nodes can interact and communicate with others, without having trusted third parties.", "The distributed ledger is a shared, timestamped, immutable and append-only database that keeps a record of transactions in a block structure.", "Each block is connected to its predecessor block by a cryptography hash stored in the block header to form a full chain called a Blockchain.", "Each block structure contains multiple information, such as timestamp, nonce and transaction-related, to a specific event.", "A timestamp indicates the time of creating each block, whereas nonce is a unique random number generated to each block and used in different cryptography operations.", "In a Blockchain, each block can contain multiple verified transactions stored as hash values that cannot be changed or modified regardless of the need for a lot of computing power [6], [33].", "Blockchain allows the network’s participating nodes to interact and communicate with others without a significant third party to manage and provide verification services.", "Communication between network nodes is first validated and then stored as a transaction in a Blockchain database.", "Different cryptography primitives, such as digital signatures, are used in Blockchain to determine the level of trust for broadcasting transactions between nodes.", "Usually, there are two types of nodes involved in the Blockchain network which are responsible for creating and validating blocks.", "One is a simple node that can create the account wallets and transactions in the network.", "Simultaneously, the others are full nodes (also called miner nodes) responsible for verifying or validating transactions before grouping and adding them to the Blockchain.", "Although both types of nodes can access all the blocks in the distributed ledger, no one has full control of the blocks and cannot modify them [34].", "To ensure the reliability of data and transactions and to maintain trust between decentralised nodes, Blockchain systems follow the consensus concept, in which nodes do not accept any trusted third party’s services to manage their behaviour and interactions.", "Each interaction between the communicating nodes is cryptographically secured and recorded in the distributed ledger.", "By receiving broadcast transactions, full nodes or miner nodes on the Blockchain network can verify transactions using computational procedures.", "After verification, the miner nodes build a new block of validated transactions and add them to the Blockchain.", "To conclude, the complete process of validating and adding transactions to the Blockchain is called mining.", "followed by some decision-making or consensus mechanism.", "Each consensus mechanism is associated with miners’ rewards for their effort and computation [35].", "Depending on the Blockchain systems and their types, several consensus mechanisms have been proposed.", "Nevertheless, the commonly used consensus mechanisms in most Blockchain systems are PoW (Proof of Work) [36], PoS (Proof of Stake) [37], PBFT (Practical Byzantine Fault Tolerance) [38] and DPoS (Delegated Proof of Stake) [39].", "The PoW consensus mechanism is generally used by the Bitcoin cryptocurrency, while the Ethereum Blockchain systems use the PoS.", "Apart from these consensus mechanisms, several other consensus mechanisms have also been developed, such as PoA (Proof of Authentication) [40], PoET (Proof of Elapsed Time) [41], PoSpace (Proof of Space) [42] and PoI (Proof of Importance) [43].", "Blockchain technology can be classified into the following set of properties that may vary depending on the design perspectives of each application, ranging from single user level to business level.", "These properties include evolution, layered architecture, Blockchain types, storage structure and transaction models.", "A generalised overview of Blockchain, which illustrates its features, evolution, layers, types, storage structure and transaction models, is shown in Fig.", "REF ." ], [ "Features", "The overall Blockchain technology can be summarised with the following features: decentralisation, immutability, open source, anonymity, autonomy and transparency which is used to achieve a set of security features for different applications.", "Decentralisation feature allows a group of nodes to be organised in a P2P manner and is responsible for maintaining the network’s overall structure, rather than relying on a single governing authority to control and manage network-wide operations [44]." ], [ "Immutability", "Blockchain’s immutability feature relates to the distributed ledger, which means that the state of Blockchain remains unchanged.", "Since the data stored in the distributed ledger cannot be modified or changed once the majority of the nodes have been verified, immutability ensures the integrity and traceability of Blockchain data in a verifiable manner [45]." ], [ "Open Source", "An open-source feature of Blockchain technology allows developers to build trust between network nodes and their data, using some of the available code features constructed.", "It can also provide a way to create new decentralised applications to govern the code and adopt a flexible approach [46]." ], [ "Anonymity", "Anonymity applies to an entity’s status as being secret and unrevealed means that no one can access the users’ true identity from their behaviour or their transactions in the system [47]." ], [ "Autonomy", "Autonomy can be defined as self-governing in any system capable of performing functions independently to achieve specific objectives.", "The anonymity feature of Blockchain enables users to participate in a self-organising system and gives them the freedom to verify transactions without involving any centralised third party [48]." ], [ "Transparency", "Transparency is one of the most appealing features of Blockchain technology as it allows any user to join the network and verify transactions before adding to the distributed ledger.", "In Bitcoin, transparency allows users to track the history of all transactions, for example, who created them and who verified them [49]." ], [ "Evolution", "Blockchain technology continues to evolve its underlying architecture through a sequence of phases or evolution for developing a variety of applications, as illustrated in Fig.", "REF .", "In each phase, Blockchain technology identifies the various inherited challenges and has proposed splendid solutions to overcome them.", "To this end, the Blockchain evolution phases (1.0 to 4.0) are designed to provide a variety of lookouts, such as functionality, features, strengths, challenges, and security issues.", "Version 5.0 is currently under development, and research communities are working on it to improve its functionality for different business models.", "Table REF summarises the different Blockchain generations (from 1.0 to 5.0) with respect to their applications, consensus mechanisms and features for each generation." ], [ "Blockchain 1.0", "Following this, the first application of Blockchain technology was a very famous cryptocurrency named Bitcoin proposed by Satoshi Nakamoto in 2009 under the first evolution phase called Blockchain 1.0 [50].", "The Bitcoin concept is very famous with the most commonly used terms on the internet being “Cryptocurrency” [51], “Cash for the internet” [52], and “Internet of money” [53].", "Bitcoin used the concept of distributed ledger technology to transfer money without the need for a trusted third party.", "On the scene, this technology has become a fast and rapid growing digital payment system adopted by most of the financial organisations around the globe [54].", "At present, Bitcoin is not just a currency system; it also changed the economic models and working structure of different organisations, for example, government sectors [55], banking [56] and accounting.", "For security purposes, Bitcoin utilises the immutable feature of distributed ledger, to ensure the integrity of recorded transactions and to guarantee that no one can change or modify the transactions.", "In addition, advanced cryptography protocols, such as hashing algorithms and digital signatures, provide the authentication trust and privacy of users in the Blockchain environment [57].", "However, at present, in Blockchain 1.0, there are a few issues about computational cost, extended waiting times, lack of inter-operability and versatility which are recognised as major barriers to wider adoption." ], [ "Blockchain 2.0", "Blockchain technology is considered a fast-growing technology that has been revolutionised by continuous improvements and rapid progression in the distributed ledger to develop smart applications for society and businesses.", "Blockchain version 2.0 comes with the concept of smart contracts, small executable user programs which run in the Blockchain environment called Ethereum Blockchain to carry out different automatic tasks and make valid decisions [58].", "The key features of such programs are that they execute automatically, based on defined logics and conditions in them, for example, time, performance, the decision and verification policies [59].", "It is equally important to describe here that these small programs (or contracts) run with the autonomous identities of users to protect personal information in the Blockchain network [60].", "The advantage of the smart contracts is that they can possibly reduce execution and verification times without requiring additional system resources to perform computation.", "Further, it can also allow the users to write smart contracts in a transparent way which prevents different fraud and hazard problems [13].", "To summarise, the Ethereum Blockchain [61] is the most prominent feature of Blockchain version 2.0 in which the users are allowed to write and execute smart contracts in a secure way.", "Figure: Blockchain Evolution" ], [ "Blockchain 3.0", "The major limitations found in previous Blockchain versions (1.0 and 2.0) are that they mostly rely on the public Blockchain network and cannot store a massive amount of data in the distributed ledger of Blockchain technology.", "Bitcoin and Ethereum are open to everyone and the data are produced and recorded on the Blockchain daily.", "Therefore, the primary need is to store a large amount of data in different storage places, such as data servers and clouds [62].", "For this purpose, a new version of the Blockchain has proposed a Blockchain 3.0 in which the decentralisation concept is utilised to store a huge amount of data and to legally support a wide variety of communication mediums [63].", "Indeed, the code in decentralised applications supports multiple servers to run and compile it; whereas a single server with limited storage only runs limited applications [64].", "The advantage of Blockchain 3.0 is that it allows the developer to write the code of applications in any language since it requires system calls to communicate with the decentralised system for the execution of the program.", "Apart from the disadvantages, there are various security challenges faced by these decentralised networks such as authentication, authorisation and access control of users and their data.", "The privacy of users and their transactions in a decentralised network is also a challenging task, along with other security requirements [65].", "To illustrate the concept of Blockchain 3.0, the developers of smart contracts introduced Genaro [66], a first Turing machine-based public Blockchain, which permits the users to write and deploy native smart contracts in decentralised storage systems with the support of different network modules in the one place.", "Different Blockchain Generations along with their Applications, Consensus Mechanism and Unique Features Table: NO_CAPTION" ], [ "Blockchain 4.0", "With the completion of a successful journey made by leading Blockchain versions (from 1.0 to 3.0), the new version of Blockchain 4.0 is presented to address the industrial challenges and limitations of real-world applications.", "Blockchain 4.0 is a new generation or version of Blockchain technology that aims to introduce Blockchain into the industrial world and make it practical for developing and running real-world applications in a secure and decentralised way.", "The new version also enables us to propose new solutions and fills the gap between business and information technology industries [77].", "Furthermore, Blockchain 4.0 enables the industry and business sectors to transition their entire structure and processes (or parts of them) transparently, to stable, self-recording applications built on a decentralised, distributed and immutable ledger.", "As Industry 4.0 is known as a revolutionary technological wave for the interconnectivity between people and machines, it provides substantial industry growth and productivity change that positively affects both the human quality of life and the environment [75].", "The convergence of Industry 4.0 and the Blockchain 4.0 generation creates a joined paradigm based on trusted networks that eliminate the need for a third party.", "Individual manual processes are transformed into linked systems using automated, autonomous systems, which are also underpinned by Blockchain technology.", "This convergence is primarily centred on the use of Blockchain features such as public ledgers and distributed databases, as well as the implementation of smart contracts in industry processes to remove the need for paper-based contracts and to control the network through consensus [26].", "Moreover, introducing Blockchain version 4.0 into Industry 4.0 aims to achieve transparency in the industrial processes from planning to implementation, and to establish the relationship between industry policies and underlying Blockchain features [78].", "There are a few examples of Industry 4.0 which have recently adopted this new version into their business processes: financial services [8], IoT [79], Transport and Logistics [80], SG [81], [82] and eHealth [83]." ], [ "Blockchain 5.0", "Although Blockchain technology is relatively new, it has advanced dramatically.", "It is now used in a broad range of industrial sectors, including banking, healthcare, IoT and supply chain management.", "After achieving considerable success in earlier versions, Blockchain 5.0 is designed to serve the needs of the next generation business peoples’ by formalising and standardising digital lifelines.", "Therefore, it is becoming extremely important to have Blockchain 5.0 in the today's world.", "The aim of Blockchain 5.0 is to concentrate on the integration of AI and DLT in order to develop the next generation of decentralised Web 3.0 applications to achieve data privacy, security, and interoperability.", "By making this option, a project called \"Relictum Pro\" is well on its way to achieving success in the new age of Blockchain technology, which is characterised by Blockchain 5.0.", "The “Relictum Pro” Project has advanced technology to use Blockchain 5.0 to build virtual channels on this dedicated network.", "As a result, there is a significant increase in transfer rates and the introduction of a seamless system with smaller block sizes and faster transactions [84].", "Figure: Layered Architecture of Blockchain" ], [ "Layers", "The layered architecture of Blockchain can be divided into the following categories from top to bottom: application layer, smart contract layer, incentive layer, consensus layer, network layer and data layer [85], [86].", "Fig.", "REF illustrates the layered architecture of Blockchain." ], [ "Application Layer", "The application layer is devoted to creating a wide range of Blockchain applications for use in many businesses and industrial sectors.", "This layer is an essential component of any architecture because it allows humans to communicate with the existing system and facilitates communication between an individual or a system over a network.", "The application layer comprises smart contracts, chain code, scripts, application program interface (APIs), user interfaces and frameworks.", "Further, it is also responsible for delivering specific user interface components and encompasses all that makes an application work, such as protocols and code." ], [ "Smart Contract Layer", "The smart contract layer is the second layer of the Blockchain layer architecture, containing smart contract script and algorithmic logic for performing specific tasks inside the Blockchain application.", "In general, a smart contract script is a piece of code that is written and stored on the distributed ledger, and network nodes automatically execute it.", "Algorithmic logic is a set of rules and conditions that control how parties interact and communicate.", "When certain predefined conditions are met, the agreement is enforced and executed automatically." ], [ "Incentive Layer", "The incentive layer is the third layer in the layered architecture of Blockchain.", "It is responsible for distributing incentives to nodes that contributed to the Blockchain by inserting valid blocks.", "The incentive method is primarily composed of two mechanisms: the issuance of incentives and the allocation of incentives.", "Aside from that, this layer enables nodes to participate in Blockchain verification by providing incentives.", "For instance, in Bitcoin, miners are rewarded with bitcoins, allowing additional users to join the network and mine the blocks.", "Similarly, ethers are used as mining incentives in Ethereum." ], [ "Consensus Layer", "This layer is responsible for enforcing network rules that specify how nodes within the network should behave in order to achieve consensus on broadcasted transactions.", "It also ensures the integrity of records stored on the Blockchain as the fundamental layer of the Blockchain architecture.", "To accomplish this goal, the consensus layer incorporates several consensus protocols that enable Blockchain nodes to agree on the authenticity and legitimacy of newly created data blocks.", "The consensus layer contains specifications that define the rules for achieving consensus and how they can be applied depending on the consensus process.", "Various consensus mechanisms, such as PoW, PoS, DPoS, PBFT, DBFT, etc., have been proposed and used by various Blockchain-based applications.", "PoW algorithm was the first Blockchain algorithm to be implemented into the Blockchain network.", "PoS is a Blockchain consensus algorithm that allows miners to participate in the mining process by staking their coins.", "DPoS is a variant of PoS in which the stakeholders' problem is fully resolved, and any component on the network may act as a delegate.", "PBFT is primarily concerned with the state machine because it can replicate the system while avoiding the primary Byzantine general issue.", "DBFT is one of the most well-known consensus algorithms, and it is created to address the shortcomings of PBFT." ], [ "Network Layer", "The network layer is the fifth layer in the Blockchain layer architecture, and it is primarily responsible for information exchange between Blockchain nodes.", "Although various components constitute the network layer and allow nodes to communicate on a Blockchain network, three primary components are considered primary components: the P2P network, the broadcasting protocol and the validation mechanism.", "In a P2P network, all nodes communicate using simple rules, and each node has an equal opportunity to create a new block in the Blockchain network.", "Following the generation of a block, each node broadcast the data to the P2P network for validation.", "All nodes do not have to receive the block data during broadcasting, but the primary node must accept it and connect it to the Blockchain to form a chain structure.", "The nodes in the validation mechanism obtain a new block containing information from other peer nodes and then verify the information before adding it to the Blockchain.", "The node agreed to add the new block to the Blockchain based on the validity of the information." ], [ "Data Layer", "The data layer is responsible for handling and storing Blockchain data since it manages the data structure and physical storage space.", "As we know, Blockchain is based on distributed ledger technology; it enables the secure and efficient storage of data on a shared digital database.", "The ledger is constructed using a linked list of blocks, referred to as Merkle trees, that are encrypted using asymmetric encryption.", "The following components comprise the data layer: hash function, asymmetric cryptography, Merkle tree, transaction, block structure and chain structure.", "A hash function is used to convert the transactions into hash values since transactions are stored in the block in the form of hashes.", "Asymmetric encryption, such as public and private key pairs, is often used to secure the transfer of blocks through a network.", "The Merkle tree is used to arrange transactions as a tree and store them on the Blockchain.", "A transaction is any piece of data that is stored on the Blockchain.", "Blocks are primarily used as data structures, with the primary function of grouping all transactions and then distributing them to all nodes in the P2P network for verification.", "The transactions specified by the user are linked together in the chain structure by storing the hash of the previous block, in which each block stores the root hash." ], [ "Types", "Starting with the types, there are three Blockchain types: public, private and consortium.", "These types are divided according to their assessment criteria and permission rules, all of which require access to the Blockchain network." ], [ "Public Blockchain", "The public Blockchain is the most fundamental type of Blockchain network in which any user can participate, send and receive transactions, and validate (mine) the network’s transactions [87].", "The validation process is performed by specifically designated nodes called miners which run the consensus algorithm to verify the network’s transactions.", "The miners also add updated and validated blocks in the existing Blockchain [88].", "Indeed, the consensus algorithms such as PoW [36] and PoS [37] are mostly employed in a public Blockchain, in which a reward is given to the miners for their services (hashing or computations) in the network.", "The distribution of reward in the public network is directly proportional to the effort made by each miner; everyone has an equal opportunity to validate the blocks [89].", "Moreover, different cryptography protocols are utilised in the public Blockchain to authenticate and secure users’ transactions [90].", "For privacy purposes, the identity of each user remains anonymous in the public Blockchain.", "By confirming all the above features, Bitcoin [50], Ethereum [61], Litecoin [68] and Monero [91] are the most common and well-known examples of the public Blockchain network." ], [ "Private Blockchain", "In contrast to the public Blockchain, a private Blockchain is a permission-based Blockchain network that manages and specifies access for an organisation or group of people to read or write the blocks from the Blockchain [92].", "Unlike a public Blockchain, a single authority is responsible for managing and updating the complete Blockchain setup by defining the rules and policies.", "In particular, a private Blockchain is designed for those organisations which want to keep their data safe within the given, defined boundaries, such as finance and audit companies [15].", "In a private Blockchain, the miners are the special nodes or trusted agents in which other nodes of the Blockchain network can blindly trust.", "The encrypted immutable ledger is shared with all other organisation members to keep the data safe [93].", "Generally speaking, a private Blockchain can solve challenging problems by providing secure solutions for different corporate sectors [6] and governmental organisations [94]." ], [ "Consortium Blockchain", "The hybrid form of the Blockchain network is referred to as a consortium Blockchain.", "In simpler terms, the consortium Blockchain combines different features and properties used in both public and private Blockchain networks.", "In most cases, the read request to access the specific block could be from a public Blockchain, whereas the write request is allowed only to private Blockchain nodes [95].", "Fig.", "REF depicts an example of various access policies (read, write and approve) on a consortium Blockchain as executed by other Blockchain types, such as public and private [96].", "However, the consortium Blockchain’s consensus mechanism is run by those specific nodes that are initially defined to control and maintain the Blockchain setup [97].", "For instance, an organisation consists of ten departments involved in different processes and activities; however, seven departments are the most authentic departments responsible for creating the organisation’s laws." ], [ "Storage Structure", "Blockchain technology consists of different valuable and dominant features of cryptography to build real-time and networked applications based on decentralised and distributed databases.", "As the real-time applications gather and produce a large volume of data, it is essential to manage and secure the storage locations used either internally (that is, a local hard drive) or externally, such as a server or cloud.", "In general, there are two types of storage models utilised by Blockchain applications in order to store real-time data.", "The first model used in most of the applications is the on-chain storage model.", "The other is the off-chain model, which is employed by those applications which produced the data in a streaming fashion, such as IoT, SG, vehicular network and financial services applications." ], [ "On-Chain Storage", "Built on distributed systems to manage the data across the network, the primary storage of Blockchain is called on-chain storage.", "In the Blockchain, the data is stored as confirmed transactions in the form of blocks.", "A block is linked to the previous block to form a complete chain.", "The special nodes (or miners) are responsible for undertaking the validation tasks, in order to add the existing Blockchain’s confirmed blocks [98].", "For this purpose, the miners are rewarded with some financial benefits for their services provided to the Blockchain network.", "However, all these tasks, such as transaction execution and verification, distribution of reward and decentralised execution, can incur an extra storage overhead on the system.", "Moreover, in a public Blockchain, the transaction structure is open to everyone and, therefore, not completely anonymous.", "There are a few advantages to storing the data on the on-chain Blockchain; for instance, the users are not required to maintain both storage locations, thus reducing the computation and storage cost for off-chain storage.", "Also, users have complete control over their data in the on-chain storage [99]." ], [ "Off-Chain Storage", "In the Blockchain, off-line storage is also referred to as off-chain storage in which user transactions are stored on another system (or storage) other than the actual storage, in order to restrict users’ access.", "In fact, the off-chain system is utilised to restrict the read access to the Blockchain using different access control policies.", "Blockchain transactions are stored in a hash containing the actual information about the transactions stored inside them and no one can obtain the information from this hash [63].", "Only the verifying party can access the hash information with the information about stored transactions on the Blockchain.", "There are many advantages in using the off-chain storage concept in Blockchain; for example, it can provide greater privacy for user transactions when user access is controlled and managed by access control policies.", "In this way, the user keeps personal information separate from the other Blockchain users in the network.", "However, using the off-chain method in the Blockchain, there are some disadvantages, such as a lack of user confidence and the need to distribute information across multiple storage locations through connecting hash references.", "Moreover, the cost of storing data in the off-chain system, along with the actual storage, is very high because users have to manage the record with extra computation power." ], [ "Transaction Models", "One of the prominent features of Blockchain is distributed ledger in which the different users record the online generated transactions in an immutable way.", "The transactions in the Blockchain and their related applications are designed and maintained in a specific way.", "The basic idea behind the design of different transaction models is that they can resist a wide variety of attacks on Blockchain applications.", "The two most commonly used models in Blockchain applications are the Unspent Transaction Outputs (UTXO) model and the Account-based Transaction model." ], [ "UTXO Model", "This model is the fundamental transaction model commonly used in Bitcoin and related cryptocurrencies applications to represent the currency transactions.", "In Bitcoin, the currency is in Bitcoins being recorded as a transaction in the user’s wallet.", "For the representation of Bitcoin in the wallet, a list consisting of unspent transactions is maintained by the user.", "This includes all details such as value, owner and time [100].", "Anyone can see all the unspent transactions in the user’s wallet in the group, as the total balance of that user.", "The owners sign Bitcoin transactions with their private keys.", "Anyone in the group can prove the transaction and the user’s ownership and authentication, with the owner’s public key.", "The UTXO model has become very successful in cryptocurrencies, especially in Bitcoin, for reasons of privacy and scalability of transactions [101]." ], [ "Account-based Online Transaction Model", "This model represents the Blockchain transactions in a different format designed as an Account-based Online Transaction, a model in which the address (or account) of the sender is utilised to represent the transactions, as opposed to the unspent transactions output in the UTXO model [94].", "The account-based model is employed by the Ethereum application to generate and deploy the smart contracts on the Blockchain with the available accounts.", "The basic aim of the account-based model is to enhance the consensus algorithms’ efficiency and reliability by improving the verification time for blocks [102].", "As with the UTXO model, the balance in Ethereum applications are stored as the transaction and called ethers (or gas) in the Ethereum Blockchain with approved properties about the sender, such as signature, approval and balance of the sender.", "Unlike the UTXO model, the account-based online transaction model has unlimited space store the information of other users because it does not store any unnecessary details of ethers compared with the coins in the previous model.", "The account-based online transaction model is widely accepted in most Blockchain applications due to its many features, such as design simplicity, ease of understanding and security enhancement [103]." ], [ "Requirements for Blockchain-based Industry 4.0 Applications", "We are in the industrial revolution 4.0, and this profoundly changes the way we live, work and communicate with each other.", "The term “Industry 4.0” is interchangeable with \"fourth industrial revolution\" and refers to a new stage in the coordination and planning of the industrial production process.", "With advanced digital technologies and the transition of Industry 4.0, the manufacturing process of Industry 4.0 has now changed completely, through a series of digital transformations, to achieve productivity and automation of the entire process.", "This transition of industrial processes enables traditional enterprises and factories to move towards the emergence of smart factories, referred to as Industry 4.0.", "The smart factories can be developed by integrating artefacts, operators, and the provision of background information into an industry framework through the internet [104], [105].", "The internet is the most important technology in Industry 4.0 since it is the foundation for most other technology drivers.", "By utilizing this technology, IoT and other associated innovations, including distributed network, fully automation, and competitive production networks, are primarily driving Industry 4.0.", "However, the scope of Industry 4.0 can be expanded by bringing leading technological players, propelling this advancement to the next level to benefit both industry and partners.", "IoT, Blockchain, big data, edge and cloud computing, robotics, artificial intelligence, and open-source software are the key technological players [106].", "The advantage of integrating these players with Industry 4.0 is the creation of an integrated and automated system, such as a cyber-physical system (CPS), that can transform the underlying industrial infrastructure and production processes into an autonomous and dynamic system in order to achieve process resiliency.", "The components in these highly integrated CPSs must interact and act intelligently to collaborate autonomously and accomplish a shared objective [107].", "Furthermore, real-time information sharing between different components can be achieved by the use of the web or a computing and data infrastructure that allows warehouse-scale computers to communicate with each another, which benefits organisations to improve their productivity and collaboration [108].", "Ultimately, the goal of Industry 4.0 is to speed up manufacturing process, enhance operational effectiveness, and better serve customers while opening the door to new business models and possibilities that go beyond automation and to aid in real-time discovery.", "In Industry 4.0, the manufacturing process is integrated with smart devices, sensors, machines and humans, along with their behaviours, to periodically generate data used in different production processes, in order to achieve better quality and more sustainable environmental decision-making.", "In the manufacturing process, the modules collect data from the sensors embedded in the machine equipment, perform some computations and send them to the next module as an output transaction.", "Each module in the manufacturing application is linked to others, using some communication mediums.", "For most industrial setups, a series of processes carried out at various sites contribute to the system’s functionality as a whole [1], [109].", "Industry 4.0 is not only concerned with the digital transformation of production or manufacturing, but also with the digital transformation of other industrial sectors and value creation processes.", "With the emergence of new technologies such as mobile data networking and network protocols, as well as the IoT stack components and security features used in Industry 4.0, it is possible to take a step forwards and leverage Industry 4.0 features, which can then be connected with other industrial sectors to form Industry 4.0-based applications.", "This transformation in Industry 4.0 has encouraged business and research communities to look beyond manufacturing processes and expand their horizons to include Industry 4.0 applications in other industry sectors such as healthcare, energy, financial, logistics and supply chain [110].", "With the increased interest in the development of Industry 4.0-based applications, different critical challenges have been raised.", "These issues are related to the design and performance of the application, security and privacy of users [111].An example for design challenge that industries face is the use of centralised architecture, which can become a bottleneck and often results in problems with scalability and single point of failure in large industry setups.", "Additionally, since industrial applications process and store a vast amount of data, it is critical to consider storage-related issues such as data heterogeneity and data redundancy and security and privacy concerns regarding to data confidentiality and data integrity.", "As different industries continue to utilise unique features of Industry 4.0, they are seen as an appealing target for attackers.", "Therefore, security is considered as a critical factor in the successful implementation of Industry 4.0-based applications [5], [26].", "Nowadays, Blockchain technology is becoming more popular in multiple Industrial-based 4.0 applications due to its promising features, such as decentralisation, distributed immutable ledger, transparency, anonymity, autonomy, open source, verifiability and security.", "Such innovative Blockchain features can solve a number of problems which has risen dramatically in Industry 4.0 applications.", "Each application, which is explicitly based on the Blockchain structure, must comply with the requirements set out for the industrial system model, following the design phase, security and privacy of both users and data [2], [75].", "Numerous blockchain architectures have been suggested for use in a variety of Industry 4.0-based applications.", "The summary of Blockchain architectures being used across different industry sectors is as follows, but these architectures are not limited to: Codefi [112] is a blockchain-based financial architecture launched in September 2019.", "It consists of various product modules that work together to fuel the next generation of commerce and finance.", "Codefi's blockchain suite offers a significant solution for resolving the issues associated with traditional finance, scaling decentralised networks, and better access to web-based technology.", "MedRec [113] is a well-known implementation of the blockchain architecture for health care applications, which enables the secure and efficient storage of health-related data.", "In this architecture, each entity that participates in the case, such as the patient, the doctor, and the patient's insurance company, may update the patient's health record.", "Medicalchain [114] is another decentralised health care architecture focused on blockchain technology used in the United Kingdom to manage patient data.", "This architecture prioritises the users' needs while maintaining a single reliable version of the users' data on a distributed ledger.", "PowerLedger [115] architecture is based on Blockchain technology that facilitates the buying and selling of energy resources based on an allocation market and prioritises this surplus energy within micro-grids or around the distribution.", "Bankymoon [116] is another blockchain-based energy architecture that offers blockchain-enabled smart prepaid energy metres to schools and communities worldwide that lack access to affordable energy.", "Electronic Product Code Information Services (EPCIS) [117] is a blockchain-based architecture for food traceability that uses Electronic Product Code Information Services (EPCIS) services and demonstrates its advantages.", "In this architecture, the blockchain is used to keep track of data to a higher degree since less information is on-chain and more is off-chain using EPCIS.", "OriginTrail [118] is a blockchain-based architecture established in 2013 with the aim of bringing transparency to complex global supply chains.", "This platform is widely used in the food industry and informs users about their food products' location.", "The IBM Watson IoT platform [119] supports isolated blockchains for IoT data sharing, adding an extra layer of protection and integrity to IoT transaction flows.", "Another Blockchain architecture, ADEPT (Autonomous Decentralized Peer-to-Peer Telemetry) [120], uses Blockchain technology in IoT network and employs Ethereum, Telehash's functions and BitTorrent.", "Developing secure Blockchain-based applications using Industry 4.0 guidelines, is a critical challenge that usually requires an appropriate relationship between the architectural components of the underlying domain and the Blockchain features in order to achieve optimal usability at various levels [121].", "Furthermore, security and privacy requirements are critical challenges for developers and researchers in order to ensure the proper compliance between users and industry partners [122], [123].", "For example, the Blockchain-based IoT application needs to accomplish domain requirements, as well as the security and privacy requirements for the Blockchain [79].", "To meet the requirements of the Blockchain-based application in the Industry 4.0 domain, we show the example of the car manufacturing process in the smart factory and how it works under the Industry 4.0 definition and Blockchain in which each module involved in the manufacturing process is integrated with Blockchain technology, as shown in Fig REF .", "With distributed ledger technology, the transactions of each module during the car manufacturing process are stored on a distributed, immutable ledger.", "A complete record of chassis designing, body manufacturing, painting, quality servicing and delivery of successful units is maintained throughout the process, followed by the distributed ledger’s data.", "From this Industry 4.0 example, we draw two requirement perspectives for the design of secure Blockchain-based Industry 4.0 applications, including application and security.", "From an application perspective, we cover the design requirements of industrial applications related to their architecture, functional and non-functional, and performance that need to be met for decentralised and distributed Industry 4.0-based applications, together with security and privacy for both users and processes in Blockchain perspectives.", "The design requirements of industrial applications are discussed in detail in subsection REF .", "We discuss users’ security and privacy requirements of Blockchain-based Industry 4.0 applications from a security point of view.", "Security and privacy are critical issues for Industry 4.0-based applications since there is a high chance of unauthorised data breaches or information leakage, resulting in critical data loss for Industry 4.0 based applications.", "For example, malicious attacks on sensing devices and between supply chain processes may disrupt the communication and services of the overall manufacturing processes and disclose personal information related to their identity and transaction.", "Therefore, the security of integrated modules and their generated data in the industrial production process is essential, requiring users’ security and privacy of their transactions in the field of application development Industry 4.0.", "The security and privacy requirements are discussed in detail in subsection REF .", "Figure: A Taxonomy of Design Requirements for Blockchain-based Industry 4.0 Applications" ], [ "Application Design Requirements", "This section goes into depth about the requirements and sub-requirements for designing Blockchain-based Industry 4.0 applications.", "We categorise each design requirement into its potential sub-requirements and describe them in accordance with industry perspectives.", "The requirements for the design of secure Blockchain-based Industry 4.0 applications include decentralisation, scalability, correctness, efficiency, interoperability, consistency, usability, flexibility, protection, modularity, fairness, completeness and transparency.", "We also investigated and outlined the potential sub-requirements for each design requirement and classified them rationally.", "The high-level taxonomy of requirements and sub-requirements for designing Blockchain-based Industry 4.0 applications is illustrated in Fig.", "REF .", "Along with specifying the requirements and sub-requirements, we conducted an in-depth review and exploration of the various measuring criteria that state how to achieve these requirements rationally, as shown in table REF .", "Decentralisation is one of the key requirements for a Blockchain-based application to distribute the loads among the entities involved in the manufacturing process [124].", "Traditionally, the manufacturing process of the industrial setup was managed by a single centralised party responsible for distributing and maintaining the overall computational loads on different modules involved in the process.", "This centralised server faced bottleneck and overhead storage issues if multiple requests were received from other processes at the same time [125].", "Decentralised architecture removes a centralised entity’s requirement from the overall process, saving middlemen’s costs to ensure process verifiability and auditability.", "Decentralised architecture in Blockchain systems is designed according to the three types of Blockchain networks, public, private and consortium, managing access control for all network users [126].", "Aside from that, distributed storage is an essential prerequisite for any decentralised architecture that recognises the value of equality in a P2P network in which each node has equal rights to validate and verify transactions stored in the database [65], [127].", "Since there is no central authority in the Blockchain network, each node has a dual responsibility to perform data computing and send updated copies of data to all other nodes in order to ensure consistency across the network.", "In this respect, each node manages its storage locally without relying on centrally controlled storage systems [128].", "The decentralised requirement can be further subdivided into sub-requirements such as: Fully Decentralised: In a fully decentralised network, anyone can join the network as an individual node and send transactions to other nodes without the need for a trusted third party.", "A consensus process is used to create a trusting relationship between nodes.", "A public Blockchain is an example of a fully decentralised network.", "Partially Decentralised: In a partially decentralised network, some nodes are assigned to monitor and manage network operations, while other nodes may participate in the same way as in a completely decentralised network.", "A consensus mechanism, for example, is managed by a pre-selected set of nodes in this type of network.", "An example of a partially decentralised network is a consortium Blockchain.", "Distributed Storage: Distributed storage is a requirement for any decentralised architecture that emphasises equality between P2P nodes in order to validate and verify transactions stored in the database." ], [ "Scalability", "With the rapid pace of development of Blockchain applications and their popularity in Industry 4.0 over the last few years, scalability is emerging as a critical requirement [129], [130], [131].", "In real-time applications, data or transactions generated on a continuing process, each module will send multiple transactions to other modules.", "As a result, the entire network could be constrained under certain circumstances, such as traffic bottleneck and storage overflow [45].", "For example, Bitcoin handles an average of five transactions per second in one stream, extended to hundreds or thousands of transactions per second for financial applications [132].", "Hence, scalability is an essential requirement for Blockchain-based applications to support the maximum number of users and transactions in the system.", "The scalability requirement can be further subdivided into sub-requirements such as: System Scalability: The system scalability addresses the requirements for scalable server capabilities to meet the expected future number of available interactions, as well as the response time per user request.", "User Scalability: The user scalability requirement is defined as the maximum number of concurrent users the system can accommodate without hindering its performance.", "Transaction Scalability: Transaction scalability is often associated with some storage structures such as databases, in which scalability is described as the ability to react to an increasing number of user queries within a specific time frame in comparison to the increasing database's output performed using multi-processing systems." ], [ "Correctness", "Correctness is a crucial requirement to analyse computational results and to measure the behavioural aspects of any real-time system [133], [129].", "There are many ways to determine the system’s correctness, such as experiments, simulations, mathematical analysis, logical evidence and formal modelling [134].", "In Blockchain applications, correctness can be measured by the number of transactions executed, block generation time, consensus time, transparency level and the integrity of the proposed system [31].", "Similar to the performance parameters, the correctness of security and privacy-preserving schemes for Blockchain-based applications is achieved by performing detailed security analysis and robustness against different types of privacy attacks.", "The correctness requirement can be further subdivided into many sub-requirements, including the following: Functional Correctness: A functional correctness requirement is concerned with properties that include deciding the relationship between inputs and outputs from industrial processes instead of other systems efficiency settings such as computation time, communication, and memory overhead.", "Transaction Correctness: The correctness of a transaction is concerned with the correct execution of the operations specified in that transaction in terms of their abstract meanings and data structure.", "Decision-Making Correctness: The requirement for correct decision-making is often based on a rational decision-making model, which ensures that decisions are made based on facts, systematic data collection, and analysis.", "Furthermore, this requirement ensures that the user is aware of the possible and reasonable values for the input data." ], [ "Efficiency", "Efficiency is the system’s competence to perform a variety of tasks with minimal effort but higher output.", "The basic parameters used to measure system efficiency are computational power, bandwidth communication and storage capacity [9], [135].", "The most prominent mode of measuring the efficiency of Blockchain applications is the miner’s ability to solve the challenges in a given time [131].", "For example, in Bitcoin, PoW takes almost 10 minutes to solve a block, including the time needed to add it to the existing Blockchain.", "Although many consensus algorithms have been proven to be more efficient than others, their energy consumption is of concern [136].", "For example, each consensus algorithm consumes a certain amount of energy (or power) from adherent hardware resources in order to solve a computational challenge such as a puzzle.", "The efficiency requirement can be further divided into many sub-requirements, including the following: System Efficiency: The term \"system efficiency\" refers to the highest performance level of any computation system, in which optimal results are obtained with the least number of inputs.", "Network Efficiency: Network efficiency is characterised as the efficient sharing or transmission of information to local and global networks such as the internet while maintaining the acceptable bandwidth.", "Storage Efficiency: The storage efficiency requirement focuses on storing and processing data in such a way that it requires the least amount of space while having little effect on output.", "Energy Efficiency: The degree of energy efficiency measures the extent to which particular results are attained with less energy expenditure." ], [ "Interoperability", "The interoperability design requirement enforces process integration between the different components of decentralised applications in order to facilitate efficient interaction and communication [137], [130], [138].", "Since interoperability derives from the idea of secure communication between different network components, it enables decentralised applications to exchange information over the network using a secure Blockchain [139].", "Thus, the Blockchain platform needs to propose a seamless exchange of data between different Blockchains.", "Furthermore, the inter-operability feature is more challenging when considering that Blockchain applications must not cross the boundaries defined for the inter-operability of the system employing fair access mechanisms.", "The interoperability requirement can be further subdivided into sub-requirements such as: Data Interoperability: Data interoperability enables various common frameworks to construct, exchange, and handle data in order to share definitions, comprehend context, and accept collective responsibility.", "Platform Interoperability: Interoperability across platforms allows individuals and applications to explore, access, integrate, and analyse data on a single platform.", "Furthermore, it promotes system flexibility through the use of standardisation software bundles, metadata, and identifiers.", "Infrastructure Interoperability: Infrastructure interoperability is concerned with enhancing and expanding information technology infrastructure to a broader scale, such as clouds, to ensure that all applications and related technologies work seamlessly.", "Furthermore, infrastructure interoperability is based on securely connecting new and existing systems to guarantee data consistency and data security." ], [ "Consistency", "Blockchain is recognised as a leading technology for maintaining the consistency of transactions stored in the distributed databases [140], [130].", "As transactions in Blockchain applications are generated periodically, each node is responsible for sending updated copies of these transactions to other nodes over the network in order to maintain transaction consistency [141], [142].", "Therefore, maintaining and ensuring the Blockchain system’s degree of consistency is a challenging requirement for real-time industrial applications.", "The consistency requirement can be further subdivided into many sub-requirements, including the following: Data Consistency: In an industrial environment, data consistency ensures that the original logics can be correctly simulated.", "Furthermore, it implies that all operating processes have access to the same data in order to satisfy data integrity constraints.", "Agreement Consistency: Agreement consistency is characterised as the realisation of rules and regulations between different processes implemented by some regulatory authorities, such as smart contracts in Blockchain.", "Network Consistency: Network consistency is an essential requirement that focuses on evaluating system behaviour and functionality using index measures.", "These measurements are based on three factors: network hardware, software, and technology configurations." ], [ "Usability", "With the decentralisation aspect of Blockchain technology, many users choose to solve the scalability and efficiency issues in traditional IoT systems [11].", "However, most Blockchain systems achieve the essential functionality needed to process and validate transactions over the network without an in-depth look at usability issues that may prevent users from using such systems in their domains.", "Therefore, usability is one of the key requirements, specified to meet customers’ essential needs at first sight, so that users can feel more secure in communicating with different Blockchain systems [127].", "As a result, the current requirement is to provide users with an easy-to-use interface to enhance customer experience and satisfaction.", "The usability requirement is further subdivided into various sub-requirements such as: Application-Level Usability: Application-level usability is concerned with an analysis of user experiences with the application.", "The resulting data is used to enhance the system's capabilities and suggest further changes to make the interface more interactive and accessible.", "Service-Level Usability: In comparison to application-level usability, service level usability is characterised as any system's ability to react to user requests in order to measure the user's expectation on some scale.", "At this level of usability, each user request is weighed against the services offered by the system." ], [ "Flexibility", "Blockchain technology has proven to be a potential solution for the requirements of various business and Industry 4.0 that have arisen in their existing systems, such as the efficiency of secure and reliable transactions without a central entity.", "A flexible Blockchain system can provide a basic platform for other technologies to integrate effectively, deploy different modules and deliver effective results [133], [139], [31].", "Besides, there is also a need to optimise performance for integrated Blockchain systems; therefore, addressing the flexibility of different applications is a challenging requirement.", "The system should have inherent features provided by the different Blockchain technologies [143].", "The flexibility requirement can be further subdivided into the following sub-requirements: Process Flexibility: In the industrial context, process flexibility is an essential requirement that uses the principle of process management to effectively respond to critical system operations concerning outside factors such as increases or decreases in supply or demand.", "Using process flexibility can increase system outputs in terms of goods while also lowering the cost of external factors such as time.", "Product Flexibility: Product flexibility requirement can be measured in terms of adaptability for any potential changes in the product, including new designs and variations.", "Flexible product design reduces the costs of redesign and allows for quick customer response through better efficiency.", "Resource Flexibility: Being resource-flexible is frequently evidenced by the capabilities of the resources to handle a wide variety of manufacturing activities in an efficient way.", "Network Flexibility: In the industrial context, network flexibility is defined as the underlying system's ability to effectively handle processes that must be executed and migrated between different modules." ], [ "Protection", "Blockchain technology is designed as protected since it uses the immutable feature to store data in an append-only fashion.", "It is not, therefore, practical for everyone to modify the data stored in the Blockchain databases [133], [132], [144], [140], [130].", "Besides, Blockchain technology has demonstrated its potential to achieve data integrity by using the Merkle hash tree concept, in which block hashes are interlinked so that all network nodes can easily detect any change in hash values.", "Given this, the key requirement here is to develop a security model that protects the user from unauthorised attempts to obtain personal information.", "The protection requirement can be further categorized into two sub-requirements such as: Data Protection: The data protection requirement focuses on preventing sensitive information from being altered, corrupted, or lost.", "Nowadays, data protection is becoming a significant issue in industry as the volume of data reaches an unprecedented scale.", "User's Information Protection: User information protection aims to safeguard the personal information of individuals involved in the overall processing of industrial setups." ], [ "Modularity", "To maximise flexibility and decouple the hierarchy of modules, modularity allows different interrelated organisations to join and use network resources in order to provide comprehensive services with the power of reusability [129], [140], [145].", "Over the years, Blockchain technology has become more popular among the public to solve various problems; developers can build and develop decentralised applications using different languages that run on heterogeneous platforms [130].", "Efforts are being made by industry and research communities to address the issue of modularity.", "One example is the the Komodo [146], an open source Blockchain modular framework designed to facilitate the process of integrating different end-to-end communication modules between users, in order to address the issues of scalability, protection and inter-operability in the Blockchain network.", "The modular requirement is further subdivided into two sub-requirements, which include the following: Process Modularity: Process modularity is a requirement that focuses on improving system efficiency by breaking down a single extensive process into multiple sub-processes that can operate in parallel on multiple machines.", "Component Modularity: In terms of component-level modularity, a modular design is often interpreted as partitioning functions into multiple discrete, compact, and scalable modules, in which extensive use of well-defined standardised interfaces is needed." ], [ "Fairness", "Fairness is one of the key requirements for Blockchain applications to achieve trust between different industries and the proposed security models [90], [147], [140].", "In other words, fairness is achieved by providing middleman agreements, that is smart contracts, which comply with the rules and conditions in order to facilitate the communication process between sender and receiver [148].", "These contracts define the logic and conditions used to allow the parties to interact with each other without trusted third parties.", "As a result, there is a critical need to design security schemes for Blockchain applications that provide fairness to all users, keeping them actively involved and remaining part of the Blockchain.", "The fairness requirement can be further subdivided into many sub-requirements, including the following: Resource Fairness: Resource fairness is an essential requirement in industrial setup since it enables a system to allocate resources equally among the various processes running in the system.", "A single user may completely overload the system with its transactions, leading to poor performance for other users.", "Transaction Fairness: In the Blockchain, transaction fairness is often associated with some payment systems, in which fair transactions are needed for promoting fair payment to those who participate in and join the network for mining.", "Service-level Fairness: Service-level fairness encourages the equality of resources and software running on the system to all network users." ], [ "Completeness", "Completeness, as a design requirement, aims to ensure users’ specific needs and requirements in order to complete any application.", "In Blockchain-based applications, the security and privacy models are deemed complete if they prove the satisfactory computational requirement and comprehensive security analysis, using multiple proofs and logic [90], [147].", "The completeness requirement can be further subdivided into many sub-requirements, including the following: Record or Information Completeness: Data completeness is expressed as an expected degree of completion of data, in which optional data is often discarded at some level.", "As a result, as long as the data follows the standard and specifications, it is considered complete.", "Requirements Completeness: An individual requirement is considered complete if it contains all required information to communicate the message to prevent uncertainty and requires no amplification to maintain adequate implementation and verification.", "Functions Completeness: A function completeness must ensure." ], [ "Transparency", "Transparency is one of the most demanding requirements of Blockchain applications, specifically for public Blockchain users [132].", "Although completely different from the implementation point of view [140], many users are confused with the concepts of privacy and transparency, For example, in Blockchain applications, cryptography primitives are considered the most powerful means of achieving transparency of transactions linked to the respective users.", "Therefore, regardless of the Blockchain’s open nature, the users’ transactions must be transparent and invisible to other users on the network.", "The transparency requirement can be further subdivided into many sub-requirements, including the following: Data Transparency: Data transparency is an essential requirement that many industries today integrate into their everyday routine processes in order to facilitate open communication, so everyone has access to the same information.", "Access Transparency: Access transparency enables objects to be accessed using the same access functions regardless of whether they are static or mobile.", "Furthermore, the interface required to access an object should be identical to the object's position in the system.", "Location Transparency: The ability to access entities and system resources without knowing their specific location is referred to as location transparency.", "Designing Blockchain-based Industry 4.0 Applications: Requirements, Sub-Requirements and Measuring Criteria Table: NO_CAPTION Table: NO_CAPTION Table: NO_CAPTIONThis subsection explains various security requirements for Blockchain-based applications in detail.", "These requirements are subdivided into different security categories, such as objectives, measures, properties, services and operations, as shown in Fig.", "REF .", "In privacy requirements, we discussed two key types of privacy: identity privacy and transaction privacy.", "Figure: A Taxonomy of Security and Privacy Requirements for Blockchain-based Industry 4.0 Applications" ], [ "This subsection explains various security requirements for Blockchain-based applications in detail.", "These requirements are subdivided into different security categories, such as objectives, measures, properties, services and operations, in order to ensure secure communication between other network nodes." ], [ "Security Objectives", "Confidentiality, integrity and availability are the primary security objectives of any system to provide a secure environment for communication and ensure the safe utilisation of the system and network resources efficiently.", "Here, we discuss each security objective set to achieve security measures in Blockchain-based applications.", "Confidentiality: This is the critical security objective of any system defined to protect data from unauthorised access.", "Confidentiality allows individuals or groups of people to communicate with each other securely and prevents any unauthorised person from accessing their data [198].", "Since the data contains users’ personal and classified information, it is necessary to protect personal information from others.", "In the Blockchain, a public network is open for everyone and it allows the users to send their transactions publicly to others in the network.", "Therefore, the key challenge of the public Blockchain network is to keep data secure.", "No unauthorised person is allowed to read and access data from the network [199].", "Integrity: Integrity is another essential security objective that ensures the validity and reliability of data throughout its life cycle [200].", "An adversary can change or modify the contents of data stored in some databases or communicate through a network channel.", "Blockchain technology combines many elastic features to provide security for Blockchain data; the immutable distributed ledger provides sophisticated services resistant to data modifications.", "Once transactions are recorded on an immutable ledger, it is impossible to change or delete them later [201].", "Availability: This refers to the state of any resource or system’ available to perform certain functions within a given timeframe [202].", "The availability of Blockchain can be defined as the interaction of validated Blockchain transactions.", "Interaction mainly depends on the availability of two Blockchain functions, such as reading and writing.", "The read-availability response is always higher than the written-availability response in the Blockchain system.", "Therefore, variation in the measured time can lead to a Blockchain system failure in terms of availability [203]." ], [ "Security Measures", "Security measures define the criteria for limiting unauthorised users’ access to personal information from some storage locations, thereby protecting system resources from malicious attempts.", "These measures enable users to perform legitimate actions.", "Defining security measures for any system, authentication, authorisation and access control are the primary methods to define essential access criteria that help protect the system from malicious activities.", "Authentication: Authentication is defined as an essential security measure for any security system that integrates specific human details, such as name, age and password.", "To access a complete system or some resources, it is important to describe here that authentication is an essential security measure for all Blockchain systems to grant permission to authorised users [204].", "Digital Identities (IDs) are used in Blockchain systems with several other features, such as public and private keys, to access system resources [205].", "Authorisation: This security measure is intimately connected to the authentication process to gain access to system resources.", "In most cases, authentication is the first critical step to determine whether a particular user is permitted to access system resources [206].", "For authorisation in Blockchain systems, each system defines its own rules and policies that apply to users who complete the authentication procedure [207].", "Access Control: Like authorisation, access control is another critical security measure that defines an authorised user’s conditions for accessing system resources.", "Access control policies usually provide users with information and control of the resources of critical systems [208].", "For example, Blockchain network types, including public, private and consortium Blockchains, are based on different access control policies used to restrict access by a person or a group of users [209]." ], [ "Security Properties", "Security properties are defined as a set of features that enforce security on the computer system.", "They are the challenging design part of most real-world applications.", "For example, from the Blockchain perspective, being tamper-proof is the desirable feature that encourages the users to save a record on distributed databases in an immutable way.", "Similarly, non-repudiation and unforgeability are also fundamental security properties for any Blockchain system.", "Details on security properties, along with their related characteristics, are given below.", "Non-Repudiation: In modern conversation systems, non-repudiation is an agreement to secure transforming assets between two parties [210].", "In non-repudiation, both the sender and receiver cannot deny when they are committed to send and receive the messages from each other [211].", "For example, non-repudiation assures others about the creator and origin of data when each user is responsible for creating their data.", "Besides this, non-repudiation also confirms the data integrity of the users in the network [212].", "Unforgeability: Unforgeability is a security property that validates the authenticity of both the user and the data, and confirms that it is sent by legitimate users in the network [46].", "In the Blockchain, the user generates and broadcasts the transactions to all other users in the network for verification purposes [213].", "The transactions are verified and added to the Blockchain using some mechanisms called consensus algorithms.", "Tamper-Proof: Blockchain utilises the distributed ledger concept for maintaining and storing the transactions data in a permanent and tamper-proof way.", "In general, the transactions stored in containers consist of a header unit and a data unit called blocks.", "Each subsequent block is connected to previous blocks using hash functions in order to form a complete chain of blocks.", "Once the data is added into the Blockchain, it is computationally impossible for the nodes to change the data unless the majority of the network nodes agree to do so.", "This requires much computational power and many energy sources.", "Therefore, the tamper-proof security requirement is desirable for the Blockchain.", "Most of the applications based on this technology prevent fake access from unauthorised users which would harm the stored data." ], [ "Security Services / Operations", "Security services are defined as basic operations and protocols, and they require any system or application to provide an adequate, secure environment for data communication over the network.", "These services can be derived from the available third parties or the system can inherently implement the required services for each operation or task.", "In this sub-section, we place robustness, reliability, accounting and key-revocation as the underlying security services for Blockchain-based applications.", "Robustness: The robustness feature of Blockchain technology provides guarantees to individuals or groups of people about their data stored on distributed ledger in which each node maintains and updates the distributed ledger [214].", "Blockchain uses the decentralised and distributed network feature to store identical information across the whole network in a P2P fashion [215].", "Reliability: Reliability is the primary feature of Blockchain technology and it is supposed to be a key element of each Blockchain application.", "In general, Blockchain’s reliability depends upon many constructive factors such as decentralisation, distributed ledger, immutability and security [216].", "Collectively, these factors play an important role in achieving the reliability of Blockchain and they store data in a better way [44].", "Anonymity: This refers to hiding the personal information or details of users from the outside world.", "In other words, anonymity means remaining part of the system without showing any real identity to others [217].", "In a public Blockchain, all transactions are stored at the immutable ledger and are publicly available to all other network users.", "Taking Bitcoin as an example, the pseudo-anonymous technique is used to link an actual person’s transactions with their given addresses.", "However, this is not a completely anonymous method for hiding the details of transactions [218], [219].", "Accounting: This service refers to keeping the record of all system resources used by authorised users on the network.", "In the Blockchain system, accounting refers to maintaining the following system activities, such as node behaviour, session maintenance, network resources and wallet information [220], [221].", "Key-Revocation: Key Revocation is a cryptography mechanism that requires users to update secret keys periodically to make it impossible for adversaries to know and obtain keys from valid users.", "In a Blockchain, the key revocation approach requires key certification authorities to update each certificate before sending it to valid users [222].", "Based on the above-mentioned security and privacy challenges in Blockchain-based applications, we provide a detailed description and analysis for each of them and conclude that confidentiality, integrity, availability, authentication, authorisation, access control, non-repudiation and privacy are the main security challenges mostly encountered by Blockchain-based applications.", "However, there are some additional security challenges such as unforgeability, robustness, reliability and key-revocation that require researchers to address them in Blockchain-based security and privacy solutions." ], [ "Privacy Requirements", "In this subsection, we study two broader categories of the privacy requirements of Blockchain-based applications: identity privacy and transactional privacy.", "Identity Privacy: In the network world, each object is identified by means of unique information referred to as the identity of a specific object.", "The object can be a user or a device in the network.", "In Blockchain systems, identity leakage threat is the main concern because an attacker can use different methods such as behaviour analysis and pattern matching to obtain the user’s personal information from the transactions [57].", "Personal information refers to the unique credentials related to the users such as ID, password, address, private key and balance, in some of the cases using the Bitcoin and Ethereum applications.", "Therefore, this issue can reveal the identity of Blockchain users and impose a serious threat to identity privacy [223].", "Transaction Privacy: In a Blockchain, the distributed immutable ledger consists of several verified transactions updated and maintained by the network users.", "Each user is responsible for sending the updated transaction to all other network users and updating the distributed ledger.", "However, during the transmission or interchange of transactions among Blockchain users, the transaction’s information can easily be obtained by an adversary in the network [223].", "The adversary follows different approaches to steal personal information from a transaction related to a specific user.", "The transaction graph pattern method is one of the common methods that link and retrieve users’ personal information from a transaction [224].", "Considering the given scenario, the privacy of transactions is the first and foremost requirement before sending an update to the Blockchain’s distributed ledger.", "The interest in Blockchain technology and its implementation in Industry 4.0 has evolved to capture the new opportunities.", "Many Blockchain-based applications have been developed and deployed across industries such as energy, finance and banking, healthcare and supply chain and logistics.", "Also, the adaptation of Blockchain technology in IoT, big data and cloud, crowdsensing and eCommerce technologies has been explored.", "With the use of Blockchain technology across industries, considerations about security and privacy requirements are crucial.", "This section presents the security and privacy requirements of various Blockchain-based Industry 4.0 applications.", "For each application, we provide an overview, present integration challenges with Blockchain technology and discuss further security and privacy requirements to meet the needs of a secure environment.", "We also discuss how these requirements could be met effectively using security enhancement techniques." ], [ "The industrial age influence is particularly recognised in financial services, including online payments, digital loans, currency trading and so on.", "The financial sector has greatly benefited from industrial automation 4.0 [225].", "In the financial sector under Industry 4.0, different banks, insurance companies, brokerage firms, investment firms and other financial institutions benefit from the broad positive growth of digital innovation [226].", "The changes in Industry 4.0 have affected the financial services sector in several respects.", "For example, it offers the quickest way to carry out global financial services with less human effort and lower costs [227].", "Additionally, the emergence of Blockchain-based financial services has an enormous effect on financial institutions.", "For example, with Blockchain technology and smart contracts, intermediaries, centralised administration, fraud and burglary can also be minimised.", "Blockchain technology also makes it easier for the transparency mechanism to reduce the liability of financial services [228], [229].", "One of the key principles of Blockchain technology is to build a trustworthy and transparent relationship across multiple platforms from which users can obtain and share their data anonymously, storing transactions on a tamper-proof distributed ledger.", "Furthermore, the distributed ledger also plays an important role in many cross-organisations in which people can trust each other and contribute to making the financing process easier and more efficient.", "Banking organisations are now using Blockchain as a key objective of transferring their digital assets to other banks located nationally or internationally through distributed ledgers.", "In the context of financial industry, the role of decentralised Blockchain network is to eliminate financial intermediaries by allowing each financial party to contribute to the network with the implementation of a P2P network in order to facilitate direct money transfers.", "Additionally, each financial group maintains a distributed ledger in which each block comprises a finite volume of financial transactions, such as in banking.", "Fig.", "REF shows the difference between traditional financial systems based on centralised architecture and current Blockchain-based financial systems.", "This figure clearly shows that the intermediary bank was solely responsible for managing all transactions between various organisations in conventional banking systems.", "However, with the introduction of Blockchain to the financial sector, the role of the middleman is removed, and the majority of nodes in the network must verify the transaction's validity.", "With the integration of public Blockchain and distributed ledgers, the stored transactions may reveal “sensitive and trading information” to others.", "Another critical challenge the distributed ledgers face is to audit or verify the stored transactions since there is no central party within the whole process that can verify the transactions later.", "To address the integration problems of distributed ledgers with Blockchain technology, Wang and Kogan [132] designed a Blockchain-based privacy preserving scheme to “protect the financial information” of banking users.", "The proposed system utilises zero-knowledge proof and homomorphic encryption to secure financial transactions in a private Blockchain network.", "Most of the Blockchain systems specially designed for financial transactions face the “issues of transaction cost, propagation delay and high latency” in the network.", "To address these critical issues in the Blockchain network, Zhong et al.", "[140] suggested a secure and lightweight payment scheme to make efficient use of Blockchain-based financial system services.", "The proposed scheme takes advantage of both digital signatures and a one-way hash function to provide a guarantee to achieve security properties and the robustness of the system.", "In the proposed system, off-chain storage is utilised to store and access effectively the data from different remote locations.", "A further improvement in the proposal for Blockchain technology is the concept of smart contracts that allow users to share information and perform automatic tasks over a decentralised network without any trusted third party.", "However, users in a decentralised network face a vital challenge as “distributed privacy” requires significant effort from the researchers to overcome this challenge.", "With this intention, Kosba et al.", "[90] presented the Blockchain-based cryptography model to preserve the privacy of the user’s financial transactions.", "The proposed model has become very popular and was given the name “Hawk”.", "In Hawk, the zero-knowledge proofs used to store the transactions on the Ethereum Blockchain in unclear format achieve the security and privacy of users’ transactions.", "The Hawk project’s distinguishing feature is its exclusion of cryptography operations from the actual program, as the cryptography compiler generates the operations for the specific program on run time to protect the data inside the transaction.", "Another work [127] is proposed by Kopp et al.", "to overcome the problem of “transactional privacy” in the payment system.", "The system combines the features of the distributed system, that is, cloud, with privacy enhancement mechanisms such as ring signatures and one-time addressing to protect users’ privacy and their financial transactions over a public Blockchain network.", "In Blockchain-based financial systems, two types of clients, (i) a full client and (ii) a light client, are involved in the payment process to efficiently perform the financial transactions.", "The full client’s responsibility is to authenticate those light clients who have a valid address with which to do so.", "Since the full clients send the payment to the light clients, the attacker can easily track and obtain the full clients’ details by performing attacks on the light clients.", "To illustrate this concept, Kanemura et al.", "[230] put forward the idea of “deniability” to preserve the privacy of the light client in the Blockchain payment system.", "In addition to the privacy metric, the bloom filters also prove deniability using identical patterns of addresses matching metric parameters.", "Another major challenge found in the Blockchain-based finance system (primarily based on the principle of Know-your-Customer) is the “validation” of the customers’ transactions with which respective bank authorities can disclose the personal and confidential information of the customers involved in the verification process.", "To meet this challenge, Bhaskaran et al.", "[231] proposed the public Blockchain-based data validation scheme which utilises smart contracts in a combination of double-blind sharing scheme to protect customer data.", "Another similar work was presented by Biryukov et al.", "[232] to protect the “identity of customers” in a decentralised public network.", "The proposed system utilised the Ethereum smart contracts to solve the challenges faced by a centralised system, such as identity leakage and disclosure of other personal information.", "With regards to privacy issues, most of the research aims to solve transactional privacy in finance and payment related systems with high accuracy.", "However, a few privacy-preserving schemes face an issue of trade-off; the more the use of expensive protocols, the less the speed of the transaction.", "An efficient Blockchain-based approach called FPPP (Fast and Privacy Preserving Blockchain) is modelled by Li et al.", "[233] to achieve the performance of a system with guaranteed privacy of transactions.", "In this approach, an Ethereum Blockchain is enforced with an off-chain storage system to record many transactions which can significantly enhance the computation time and speed of the transaction process.", "Moving further to another approach, Ziegeldorf et al.", "[129] proposed the CoinParty scheme which efficiently utilised the distributed mixing method to combine the different identities for the protection of “financial privacy”.", "The CoinParty method used both mix nets and threshold signatures to preserve users’ anonymity and scalability in a decentralised public environment.", "Another coin mixing scheme for Bitcoin and their related cryptocurrencies was proposed by Liu et al.", "[234], enabling the users to unlink their “identities” from the coins without the need of any central party.", "Furthermore, the scheme utilises the different cryptography primitives such as ring signatures and elliptic curve digital signature schemes to preserve the privacy of the transactions.", "Figure: Blockchain-based Healthcare Industry Model" ], [ "The existing industry healthcare models, particularly those using centralised architectures [235], have identified many vulnerabilities, including single-point failure and unauthorised modifications, as well as further security and privacy issues [236].", "Since the conventional models are no longer reliable and stable, patient data is no longer maintained in this situation.", "Privacy of patient data is also key to effective health care management [237].", "These problems and challenges can be addressed by applying Blockchain technology sponsored by Industry 4.0 to achieve data security, including integrity and privacy, and eliminating a single point of failure issue [238].", "Industry 4.0 is also transforming the healthcare sector, in a similar way to other industries, in order to embrace the adoption of innovative technologies.", "In the healthcare industry, patient data is considered the most valuable source of information to regularly monitor patient health and make critical decisions that assist doctors and researchers to improve the diagnostic learning process effectively, providing an efficient way to address health-related issues.", "The healthcare industry makes use of the IoT, along with cloud computing and big data services, to collect and store a massive amount of patient data that enables doctors, researchers and health workers around the world to build a digital global healthcare ecosystem called Healthcare 4.0 [239], [240].", "Blockchain technology has proved to be a promising technology due to its characteristics of transparency, immutability and security, as well as its ability to connect multiple organisations through the decentralised and distributed aspects of the network.", "The rise of Blockchain technology in the healthcare industry empowers electronic health records (EHRs) and telecare medicine by keeping patient data secure and anonymous while opening the door to medical researchers to perform reliable analysis.", "Moreover, Blockchain has made healthcare transactions more transparent and accessible, enabling patients to know more about their treatment options and providers [238], [241].", "The role of Blockchain in the healthcare industry is multifaceted.", "For example, Blockchain is reliable in terms of network structure since there is no centralised structure for a malicious user to target the patient data stored in a single location.", "Further, Blockchain technology enables patients to have full access to their medical records and history in a secure way.", "Accessing medical records in conventional systems can be difficult since they are usually spread across multiple healthcare facilities; however, Blockchain technology and primarily distributed ledger technology can be applied to safely access and exchange patient medical records [242].", "The distributed ledger technology enables the secure transfer of patient health records between doctors, researchers, and government agencies [243].", "This ledger also helps in the effective and safe management of medication supplies and assists healthcare researchers in unlocking the human genome.", "Additionally, Blockchain technology can enhance the security and quality of mobile apps and remote monitoring machines used in the healthcare industry.", "Fig.", "REF reflects the Blockchain-based EHR model to provide a better understanding of the Blockchain role in the healthcare industry, and the involvement of individuals in the system, such as patients, doctors, medical researchers, and government organisations who can interact and communicate securely with each other using Blockchain technology.", "By following the idea of implementing Blockchain technology for the healthcare industry, Zhang et al.", "[130] designed the FHIRChain, a Blockchain-based scheme to meet the specific needs and requirements of national health infrastructure organisations who “control and manage” further health-related organisations and sectors.", "The proposed scheme carefully determines the specifications of health organisations in order to bridge the gap between patients and service providers, such as hospitals.", "In EHR systems, “access control” is a crucial task that determines the access of personal data given to the right person for the right purposes.", "Therefore, to solve the access control challenge specifically for EHR, Hussein, et al.", "[144] proposed a private Blockchain-based data-sharing scheme that allows doctors to access the sensitive data of patients who have granted access rights.", "This scheme uses the discrete wavelet transformation technique to ensure the privacy and anonymisation of clinical data.", "The proposed scheme also employs the query service interface at which the genetic algorithm technique can access and optimise the Blockchain data.", "The experimental result determines that the scheme is scalable and robust against different types of security attacks.", "Similar to this scheme [144], Dagher et al.", "[139] implemented the Ancile, a Blockchain-based “access control” scheme to utilise the private and sensitive information of patients without disclosing this information to others.", "In addition to access control, the scheme also confirms patients’ data privacy, which would not be disclosed to others while accessing the data via multiple platforms.", "Another significant contribution is put forward by Yue et al.", "[243] to solve the problems between patients and doctors about the “controlling of sensitive information”.", "The proposed scheme empowers the patients by giving them the appropriate control over their data and allows them to send and securely receive data.", "Another “secure data sharing” scheme (MeDShare) was proposed by Xia et al.", "[244] to solve the disputing correspondence among different public parties using Blockchain technology.", "“Data reliability” is another major challenge in electronic health data that requires substantial contributions from the researchers to propose secure solutions for the reporting of patients’ data at different levels.", "Kuo and Ohno-Machado proposed a solution to a particular challenge [137] that implements the Blockchain-based privacy preserving public model called “ModelChain”, in order to preserve the privacy of patient data by using machine learning techniques.", "Sun et al.", "[245] proposed the Blockchain-based privacy preserving scheme that utilised the patient’s attributes in attribute-based signatures so as to protect personal information.", "In the proposed model, both on-chain and off-chain storage systems are employed with the private Blockchain system so that on-chain is used to store the original data, whereas off-chain is used to store indexes of data.", "Another privacy preserving scheme called BSPP (Blockchain-based secure and privacy preserving) was proposed by Zhang and Lin [138] to “protect the personal data” from different health-related organisations involved in the whole process.", "In the proposed scheme, both private and consortium Blockchains construct the data structure, which can securely store the patient’s data.", "Similar to the scheme presented in [138], Guo et al.", "[246] utilised the attribute-based signature scheme to provide the “validation of health record” stored on the public Blockchain.", "In the proposed scheme, many authorities can sign and send data without disclosing patients’ personal information unless it requires evidence.", "To solve the “authentication and accountability challenges” of the patient health record, Azaria et al.", "[247] proposed the MedRec, a novel Blockchain-based methodology to secure the medical data of patients in a decentralised public environment.", "In the proposed architecture, patients can easily control and access their own data from remote places.", "Ji et al.", "[248] made a significant contribution to the Blockchain-based EHR application, investigating a “location sharing problem” regarding patients in a telecare medical information system.", "To identify location issues of health data, a Blockchain-based location preserving scheme called BMPLS was proposed to achieve multi-level location sharing protection by employing a Merkle hash tree which can store the patient data in an hierarchical form.", "Apart from the different security challenges in the EHR, there have been many issues in distributed applications such as “retrieving, indexing, and aggregating” of data collected from multiple domains.", "To solve the given challenges, Zhou et al.", "[249] proposed the distributed “data vending” framework that utilises indexing and embedding methods to save similar data into different locations with the calculated indexes." ], [ "Industry 4.0 is driving substantial and pervasive change within the transport and logistics sector because of a rise in supply chain demand and usage of modern and emerging technologies, as well as the potential to build the industry process’s digital supply chain [250], [251], [252].", "Transport and logistics sectors are following a trend towards increased automation levels so that these sectors integrate their business models to access different market segments.", "The increased availability of information through the physical internet and open standards encourages manufacturers to reposition their supply chains to support social and data-driven market dynamics and innovation in traceability and acquisitiveness [253], [254].", "In supply chains, formal concepts being applied to real-world shipping processes, with flexible, sustainable and online shipping varying from a large container to a small box, are becoming industry standards worldwide.", "These containers are constantly tracked and monitored and guided through the IoT [255].", "The shipping industry is an important player in virtually every industry and operation that manages the shipment of vast containers from one place to another.", "These containers consist of various items sent to a specific destination, assuming that it does not include illegal or mislabelled items.", "However, the central agencies cannot check and perform audit procedures on each container’s items due to bulk quantities and time-restrictions [256].", "Moreover, the audit procedure includes tracking and selecting random items from the container with detailed information, such as item id, company name and addresses of both sending and receiving parties.", "Mostly, the auditing agencies stored the data about all items at a single location or server, which is easily accessible for every auditing party to audit.", "On the other hand, centralised systems are more at risk when authorised access or a single point of failure attack can breach the privacy of shipping information.", "However, with the advent of Blockchain technology, decentralisation and immutability features allow the freight system to transfer freight items from one place to another in a secure manner.", "The role of Blockchain technology in transportation and logistics is to ensure data integrity and security in the ecosystem, as the entire network contributes to data validation.", "Moreover, Blockchain-based logistics systems facilitate document sharing through a shared distributed ledger, obviating the need for manual paper-based processes.", "The use of smart contracts speeds up and automates the customs clearance and approval process, resulting in reduced processing times for items at customs checkpoints.", "Fig.", "REF illustrates the Blockchain-based transport and logistics framework, which shows the complete process of transferring the items from one destination to others, through the shipping agencies.", "To protect the “data privacy” of shipping items, Vos et al.", "[257] proposed DEFEND, a secure decentralised Blockchain-based platform that protects the privacy of containers and the store items.", "In this system, the sending agency carries out the following tasks: make a claim of sending items, encrypt the claim and send it to other destination agencies.", "At the destination side, the claim can only be accessed and decrypted by the destination custom agency.", "The proposed scheme’s main contribution is the partitioning of the data among the different parties involved in the Blockchain system.", "Moreover, experimental results and performance analysis claim that the proposed scheme is efficient for both customs agencies and economic operators so as to perform risk analysis on items without causing the delay." ], [ "Industry 4.0 has provided the path for continuous and widespread grid-based use of alternative energy sources, by implementing a flexible framework called the SG in order to manage increasing demands in overall energy consumption.", "More precisely, in the area of Industry 4.0, the SG is developed by the convergence of electricity networks and state-of-the-art information and communication technologies (ICTs), including smart meters, smart information processing units and advanced communication protocols to address numerous barriers and vulnerabilities such as increased energy consumption, faulty transmission, increased construction costs and less efficient delivery in conventional metering systems.", "The purpose of designing the SG is to control and track the energy resources of users and suppliers using smart digital metres in a more effective, accurate, functional, scalable, safe and cost-effective manner [258], [259].", "In the SG scenario, one unit called the grid collects the real-time data (also referred to as meter reading in a conventional meter) from smart meters deployed at different locations such as homes and industries.", "Most of the data collected are in low-level operations and this allows data analysts to discover significant data outcomes and help coordinate subsequent usage, followed by even more complex analytics and planning [260].", "This technology has become a significant part of any country’s success, encouraging its power consumers to use smart meters to manage and efficiently control power consumption.", "However, one crucial challenge found in the SG is the “leakage of personal information” through the streaming data.", "Personal information can include the details of household users, such as billing information, metering units and address, and these can cause severe security threats and loss of privacy for both consumers and providers [261].", "Blockchain technology is an emerging technology that can be utilised to provide a solution that enables secure, transparent and efficient energy transactions.", "Indeed, it provides a decentralised management and P2P energy trading.", "In SG architectures, the primary role of the Blockchain technology is to handle network transactions.", "Each entity in the SG, including producers, customers, distributors, and managing authorities, communicates with others operating as Blockchain nodes, and their interactions are recorded on the Blockchain as a transaction.", "There are usually two types of entities in the Blockchain-based SG architecture: light nodes and full nodes.", "Light nodes are typically customers that use electricity and pay their bills, while full nodes are those nodes that handle electricity and participate in the Blockchain mining process.", "Further, smart contracts are typically used to enforce transactions in Blockchain-based SG.", "These transactions include the payer's and payee's remaining balances, balance deductions, benefit or loss on the grid side, and so on [262].", "Fig.12 indicates the Blockchain-based SG model in which each grid is acting as a miner to manage and control the group of homes using the Blockchain architecture.", "As we mentioned earlier, streaming data can disclose the household users’ private information, which causes further security and privacy problems in the SG environment.", "An attacker can also obtain power consumption history by tracking and analysing the different behaviour patterns of users, such as on and off timing for a job, switching off the house lights and recharging smart meters.", "To tackle these issues, Guan et al.", "[135] proposed the Blockchain-based “privacy” preserving and data aggregation method for the SG in which the users are divided into separate subgroups and each subgroup head can record the data of their sub-users.", "To protect the users’ information, each head employs the pseudonym technique to hide the streaming data during data transmission to other neighbouring heads.", "Similar to the previous scheme [135], Aitzhan and Svetinovic [65] designed a private Blockchain-based decentralised system that provides secure “end-to-end communication” between the smart meter and the SG without the need for any trusted third party.", "A multi-signature scheme and anonymous encrypted messages are used to transform the trading transactions anonymously between the different end-users to preserve transaction privacy.", "Rottondi and Verticale [141] made another contribution towards the “privacy preservation” proposal in a SG network when they implemented the Blockchain-based smart metering architecture in which public users can transform their data to the SG in a secure way.", "In the proposed system, a secure multi-party protocol which is utilised for encryption and authentication purposes, guarantees users the correctness and authenticity of their data without them being disclosed by the SG." ], [ "Technology Industry", "The technology industry includes internet-of-things, big data and cloud computing, crowdsensing, and eCommerce as Blockchain-based Industry 4.0 applications." ], [ "Internet of Things", "Industry 4.0 has used the IoT and its associated technologies and protocols to conduct digital manufacturing in which almost all embedded devices, such as robots, machines and tools, have sensors to gather data from the environment and function accordingly.", "IoT has changed the lifestyle of humans by implicating ubiquitous applications at every step, which help to perform daily tasks automatically and more efficiently.", "In conjunction with Fog computing, IoT plays a significant role in offering time-sensitive services such as disaster related services, smart transportation and smart health services [263], [264].", "However, the improvement of the IoT’s current paradigms can only occur with continuous research and development in the underlying technology.", "Although this technology simplifies and facilitates human life, it also introduces numerous performance, security, and privacy concerns, as well as other risks to human life.", "Thanks to Blockchain technology that filled the missing gap between IoT services and security challenges due to its tremendous features such as decentralisation, immutable ledger, transparency and data auditability.", "The role of Blockchain technology in IoT has the ability to alleviate security and performance concerns.", "The use of Blockchain technology in IoT-based applications adds another security level, which is almost impossible for the attackers to get access to the network.", "The overall contribution of Blockchain to IoT systems, either full or partial, can be summarised as follows: the distributed ledger in a Blockchain system is immutable, eliminating the need for trust between the parties involved.", "No one organisation has complete control over the massive amounts of data produced by IoT devices.", "Blockchain technology offers a significantly higher degree of security, making it almost impossible to bypass existing data records.", "The transparency features of Blockchain allow anyone with authorisation to access the network to monitor previous transactions.", "By facilitating trust between stakeholders, Blockchain enables IoT companies to cut costs by removing the processing overhead associated with IoT gateways [265].", "Fig.", "REF illustrates one example of integration of a decentralised IoT network with Blockchain technology.", "In this illustration, the IoT devices act as the light nodes that are resource-constrained in nature and whose security such as authentication and authorisation is desirable and the scalability challenge.", "The miners' nodes are the full nodes and work as gateway nodes responsible for providing IoT devices' secure interaction to the Blockchain.", "Miners nodes are also responsible for mining the transactions for validation and adding purposes into the Blockchain.", "Administrator authority is responsible for deploying the entire system and managing the behaviour of nodes in the network.", "To solve the “user's privacy” challenge in IoT, Cha et al.", "[266] proposed the Blockchain-based scheme for IoT users that provides a secure gateway to achieve the privacy of users in the IoT network.", "The secure gateway restricts unauthorised access to users’ personal data from the Blockchain in the proposed scheme.", "To achieve the ”authentication of users” in IoT, a digital signature scheme is utilised to manage users’ access to resources.", "Another Blockchain-based IoT scheme is proposed by Wan et al.", "[267] to achieve security and privacy among different industrial processes.", "In the proposed scheme, existing industrial models are studied and then the weaknesses analysed in order to overcome any potential challenges of industrial applications.", "The devices in IoT are subject to different types of constraints, such as computation and limited memory.", "Therefore, there is a need for third parties or cloud service providers to process and store a massive amount of data on them.", "Although cloud computing is considered the most potent resource for computation and storage of data, this technology has its security and privacy concerns that need to be overcome.", "To address the “access control challenges” for IoT devices, Le and Mutka [268] proposed the CapChain, a novel access control scheme that allows the IoT devices to store and manage their data on a public cloud without disclosing any personal information.", "An anonymisation technique is utilised to protect sensitive information, such as identities, from adversaries in order to preserve user privacy.", "For the “protection of sensor data”, Chanson et al.", "[269] proposed the certification-based Blockchain scheme to achieve data integrity in the IoT network.", "The certification authority allows the users to perform authentication steps in three different stages to prevent malicious activities in the system.", "Nowadays, IoT-based smart home applications are becoming very popular and many home appliances are connected to the internet to control and manage the home environment remotely.", "The increasing demand for smart home devices raises different problems in terms of “security and privacy”, efficiency and scalability.", "To cover such needs for designing a smart home system, Singh et al.", "[270] proposed the Blockchain-based smart home network to achieve “secure communication” between IoT devices.", "In the proposed system, the multi-correlation technique analyses the network traffic that contains malicious data and information.", "The security analysis of the proposed system claims the high efficiency and throughput of the system.", "Another potential design of IoT-based smart home was proposed by Dorri et al.", "[271] to guarantee the “security and privacy” of home users.", "In their proposed Blockchain design, three major components are used in a complete smart home setup: cloud, overlay network and home appliances.", "On-chain storage is used to store the local processing data, whereas off-chain storage, such as cloud, is utilised to store future data which can only be retrieved through separate transactions." ], [ "Big Data and Cloud Computing", "Big data analytics refers to the use of advanced computing techniques implemented by many enterprises and industries to discover significant patterns in broad datasets, in order to help businesses detect trends and the impact of consumer preferences.", "Within Industry 4.0, data analytics plays a critical role in smart factories, where equipment captures relevant machine data to predict when maintenance and operations are required.", "Manufacturing companies use big data analytics in the same way that manufacturers use it, with the exception of emphasis.", "In manufacturing, numerous distributed sensors, which are deployed through cloud computing and IoT technologies, help the manufacturer uncover patterns which enable them to manage the supply chain more efficiently [272].", "Cloud computing is a stack of on-demand different resources and services provided to users to deliver different operations appropriately.", "The resources and services provided to Cloud users are controlled and managed by Cloud Services Providers (CSPs), which monitor and determine the applicability of on-demand access with the available resources.", "With the rapid pace of the development of cloud computing applications, numerous industrial organisations utilise cloud computing services for their data’s extensive computations and storage.", "The partnership between the Cloud and Industry 4.0 is a winning one, as both technologies required time to develop and gain acceptance within the broader industry communities.", "This incorporation enables businesses to fundamentally and profoundly reconsider their entire spectrum of digitisation processes and modify their current architectures to accommodate a broader industry market.", "Additionally, all of this occurred with increased versatility across the globe, from consumer response to cost control and proper management.", "With the convergence of big data and cloud computing services into Industry 4.0, millions of people today can use devices and applications daily that contain highly complicated data in an ever-changing technological environment.", "The growing trend of digital innovation, such as cloud computing, big data, and the IoT, creates new connectivity and knowledge-sharing opportunities.", "However, with such a large volume of sensitive data, it must be handled and secured effectively and continuously.", "The convergence of Blockchain technology and cloud computing brings the industrial community into a new era of data protection and service availability.", "The majority of cloud research problems can be resolved by leveraging Blockchain characteristics such as decentralisation, immutability, and transparency.", "The role of Blockchain technology into cloud computing and big data is many-fold.", "For instance, there is no single authority responsible for data management, eliminating the risk of a single point of failure.", "Depending on the level of data protection needed, Blockchain technology offers better deals than traditional providers in terms of security using an immutable ledger.", "In terms of storage, a cloud storage network could be managed by the nodes that assist with transaction facilitation, with the nodes granting the user access to storage on their devices.", "Additionally, advanced cryptographic technologies used in Blockchain can partition user data stored in the cloud into small bits, encrypt them for an extra layer of data security, and distribute them across the network.", "From the security point of view, CSPs are solely responsible for giving proper access to cloud users using the different authentication and authorisation services available.", "However, the prevalent issue revealed in cloud computing services is a single point of failure, which mostly leads to disclosing the personal information of cloud users.", "For the alleviation of the \"authentication issue\" occuring in cloud computing, Lu.", "et al.", "[273] proposed the Blockchain-based decentralised authentication system for storing a complete access control list of users on the Blockchain.", "The proposed system provides authorisation and accounting features which utilise virtual coin features commonly available for the digital currencies system.", "For security enhancement, a simple one-way hash function is employed to secure the links between users’ transactions; this confirms transparency at a higher level.", "Another Blockchain-based scheme for trusted “data sharin” with the cloud service provider was designed by Zheng et al.", "[45], in which unauthorised users were restricted from performing the malicious modifications on data.", "In the proposed scheme, Paillier cryptography was utilised to achieve the confidentiality of data stored at distributed databases.", "For the “protection of data” from unauthorised access, Fan et al.", "[274] proposed the Blockchain-based privacy preserving scheme to protect users’ information being communicated over a content-centric 5G mobile network.", "The proposed scheme successfully established a secure connection between different service providers and users for transferring data.", "In addition to data protection, the access control mechanism is also used to ensure access to cloud resources.", "Figure: Blockchain-based Crowdsensing Framework" ], [ "Crowdsensing", "Giving freedom of knowledge to people in order to obtain information through some valuable resources has become a popular concept, used to discover innovations in information and communication technologies.", "One of the most popular ways of providing leverage to the crowd about real knowledge discovery is crowdsensing [275].", "In Industry 4.0, crowdsensing has been proven to solve diverse problems effectively, while lowering costs and extending the reach of ideas.", "The use of individuals’ data to implement processes and projects differs greatly in every sector, from production to delivery, depending on intent.", "Computational frameworks with high throughput are important in streamlining organisational processes and also assist in connecting decisions to crowd-sourced information and expertise through decision making projects.", "Crowdsensing aids in monitoring ecosystems and mapping and exchanging information and knowledge get by the peoples.", "For instance, in energy sector, crowdsensing can help minimising the building's energy consumption by monitoring users behaviour and thermal comfort.", "In industry, crowdsensing can minimise maintenance cycles and help fix machinery issues by tracking environmental conditions and failure.", "Crowdsensing also assists in monitoring ecosystems and mapping, and exchanging information and knowledge gained by the people [276].", "Human beings are equipped with sensing devices in the crowdsensing approach to sense the data from surrounding environments and to take useful actions over data in the industrial process.", "The quality of data captured by sensing devices depends on the number of people and their level of competency, such as primary, average and high skilled users.", "However, the limitation found in the crowdsensing process is that the data of users who participated in the data capturing process can be leaked during the sensing, which prevents further users from joining the crowdsensing network.", "Furthermore, the crowdsensing process faces several challenges, including fact discovery, knowledge quality, and estimation quality using sensing data [277].", "To overcome these limitation, the Blockchain technology is introduced in crowdsensing that can support the joining of the maximum number of highly qualified users in the crowdsensing method, using a rewarding mechanism that attracts and motivates skilled users to participate in the data collection process, in order to receive high rewards as an incentive for their services for the crowdsensing process.", "The aim of integrating the Blockchain network with current crowdsensing systems is to use the characteristics of Blockchain, such as decentralisation, to provide a way for emerging decentralised systems to solve the issues of a single point of failure in traditional systems and to provide equal opportunity to contribute in the fair data collection.", "Furthermore, the distributed ledger, as a core component of Blockchain in crowdsensing, ensures the immutability and traceability of users' data and their feedback for use in different processes.", "Thus, the overall role of Blockchain technology in crowdsensing can be summarised in terms of achieving the following objectives: increasing worker efficiency, implementing a fair compensation system, protecting confidential data, and lowering deployment costs [278].", "Fig.REF describes the Blockchain-based crowdsensing framework, consisting of different nodes such as assigners, groups of users and miners to participate and control the overall crowdsensing process.", "For instance, Wang et al.", "[279] proposed the Blockchain-based “privacy-preserving” incentive scheme for crowdsensing applications that allows highly skilled users to participate in the sensing process publically and securely, in order to gain high incentives.", "In the proposed mechanism, the k-anonymity scheme is utilised to anonymise skill users’ profiles to protect privacy.", "Another similar approach presented by Cai et al.", "[280] to “protect the personal information” of the crowd uses the knowledge discovery method, without disclosing personal information.", "In this method, the public Blockchain platform collects knowledge from different sensing users in different distributed places.", "To provide the “guaranteed privacy” of mobile users and crowdsensing providers, Chatzopoulos et al.", "[128] proposed the Blockchain-based crowdsensing scheme that specifies the smart contracts to ensure the secure relationship ?", "between them.", "The proposed scheme uses the secure multi-party computation algorithm in conjunction with smart contracts to protect users’ privacy and incentive payments." ], [ "E-Commerce", "In modern times, electronic commerce (eCommerce) business has been widely accepted as a leading trading platform for the purchase and sale of online goods or services to promote their business via the Internet.", "With the widespread adoption of Blockchain technology in every field, traditional eCommerce platforms have been shifted to Blockchain technology to allow customers to carry out fair transactions without having a trusted party between them.", "However, one of the key issues identified in these systems is that it does not protect customer transactional privacy, such as content, addresses, account details and trading information.", "While several Blockchain-based privacy security mechanisms have been put in place to protect financial transactions, there is still a trade-off between privacy and the speed at which transactions are processed.", "To resolve this ongoing challenge, Li and Wang [281] proposed RZKPB – a Blockchain-based “Privacy Preservation” scheme that does not allow financial information to be stored in a plain-text format on the Blockchain.", "Multiple cryptography primitives, such as hashing and signatures, are used in the proposed methodology to verify transactions to establish a secure relationship between trading partners." ], [ "Security and Privacy Techniques for Blockchain-based Industry 4.0 Applications", "This section provides a detailed description of security and privacy preserving methods used in different Blockchain-based Industry 4.0 applications.", "We categorise them as cryptography, mixing, anonymisation, artificial intelligence and others, such as discrete wavelet, clustering and bloom filters.", "These categories are further divided into sub-categories, giving a better understanding of security and privacy methods and their usage in Blockchain-based Industry 4.0 applications.", "A taxonomy of security and privacy techniques is illustrated in Fig.", "REF ." ], [ "Cryptography", "Cryptography is a security technique that consists of a mathematical set of rules and logic used to secure the communication between parties.", "In cryptography, a plaintext is converted into some hidden text using a logic known only to both the sender and the receiver.", "The objective of the cryptography technique is to send the information securely only to those who are authorised to see and read the information.", "Cryptography techniques are used in many ways, such as protecting information from theft or modification, providing authentication and ensuring user and data access [282].", "In cryptography, encryption is the most frequently used technique to transform simple text or plain data into an encoded format that can only be read by those persons who have access to decode it.", "It is important to describe here that this encoding/decoding process would not be possible without a shared key, which two or more parties mutually set for secure communication.", "The key is an important encryption element that provides data security between end-to-end components and should not be disclosed to other network components.", "There have been several other techniques proposed that are purely based on the encryption used to solve the security and privacy challenges in different Blockchain application domains.", "Here, we divide the encryption technique into further sub-extensions of encryption.", "Simple Encryption: As stated above, encryption is a method to convert the simple text format into another format, which others cannot easily understand.", "There are two types of encryption methods, that is symmetric and asymmetric.", "In the symmetric method, one secret key is shared between two parties to encrypt and decrypt the data.", "However, in the asymmetric approach, each party has two types of keys, such as public and private, in order to encode and decode the data, respectively.", "In the Blockchain, each transaction is encrypted in other forms, which hide content details.", "Homomorphic Encryption: This is an extended version of encryption that allows the user to perform computations, such as addition and exponents of encrypted text.", "In contrast, simple encryption does not allow users to perform any useful computations on the ciphertext.", "Therefore, the main advantage of homomorphic encryption is two-fold: (i) to perform complex mathematical functions on encrypted data and (ii) to analyse encrypted data without losing the original data.", "Additionally, homomorphic encryption is the most common technique used in cloud computing, in which different organisations are involved in-service analysis of their encrypted data stored in the public cloud.", "In Blockchain-based applications [45], [132], homomorphic encryption is used to preserve users’ privacy without revealing personal and sensitive information to others.", "Multi-Party Encryption: Multi-party encryption or multiparty computation is another encryption form with which multiple users jointly perform the encryption of data.", "However, data remain private during the multi-party process and are not disclosed to others in the group.", "The advantage of using multi-party encryption in secure computations is that it can prevent the attacker from obtaining the secret information of any targeted users in the network.", "To illustrate this concept, [147] utilise multi-party computation to provide users’ guaranteed security and privacy in the decentralised Blockchain network.", "Proxy Re-Encryption: In cryptosystems, proxy re-encryption is a third-party encryption technique in which the third-party (medium party) changes the plaintext into ciphertext, without knowing the actual content inside it.", "Generally, proxy re-encryption is considered a public-key cryptography technique that uses public and private keys to encrypt and decrypt the data.", "The proxy re-encryption scheme is commonly used in those applications when the different users want to exchange encrypted data without sharing their secret key.", "In Blockchain-based IoT applications [11], [65], [139], [138], [246], [248], proxy re-encryption is used to share private contracts between users so as to control and manage the IoT devices.", "Paillier Encryption: The Paillier cryptography or Paillier encryption is a key- pair based algorithm that utilises two keys, that is public and private, to encrypt and decrypt the data, respectively.", "It is also known as probabilistic asymmetric cryptography since it performs n-th computations on multiple residue classes.", "In Paillier cryptography, additive homomorphic encryption applies to the given set of messages, and each message is encoded/decoded with the key pairs of respective users.", "Considering the implementation of Blockchain in IoT [266] and cloud computing [45] domains, Paillier cryptography is used to achieve privacy and anonymity in such decentralised applications." ], [ "Hashing", "A hashing technique is used to compact or digest any arbitrary size input into a fixed-size output.", "The input can be given in any sizes and formats such as integer, character, string and binary.", "Hash functions work according to a specific data format, which is often called a hash table.", "This is used to map the input data values on stored values to produce the output.", "A hash function’s strength is that it is designed to be a one-way function, which means that a user cannot change the input back from a given output value.", "In cryptography, the hash functions are utilised to achieve the integrity of data messages because any single change in data value can be detected easily by a change in output value.", "Apart from data integrity, the cryptography hash functions are also used in digital signatures and different checksum protocols to achieve users’ authentication.", "The best cryptography hash function must have the following important key features.", "Firstly, it should be collision-resistant which means that two of the same inputs must produce different output values.", "Secondly, it should be impossible for everyone to regenerate the same input from the output values.", "Finally, the hash function should easily detect any small modification in data values.", "Many families of hash functions in cryptography have been proposed but the most commonly used hash families are MD5 and SHA-0 to SHA-3 with different output sizes.", "In Blockchain, cryptography hash functions are extensively used to link and maintain the integrity of blocks.", "The blocks are linked with other blocks so that the hash of the previous block is stored in the header of the next block to form a complete hash chain.", "Moreover, hash functions are used in different Blockchain applications such as financial systems [231], [283], eHealth [249], IoT [266], [268], IoT [11], [284], [285] and crowdsensing [257] to achieve the security and privacy of transactions and users." ], [ "Signatures", "In the past, the signature method was utilised as the simplest method to authenticate any document by placing hand-written signatures at the bottom of documents [286].", "In the digital world, the signature method is used to protect software ownership and digital communication in a well-defined way.", "Also, this is an essential tool in information security to achieve the authenticity, integrity and non-repudiation of messages [287].", "We classify the signatures into the following sub-categories, such as digital signatures, multi-signatures, threshold signatures, ring signatures, blind signatures, attribute-based signatures, identity-based signatures and elliptic curve digital signatures algorithms.", "Digital Signature: This is a public-key cryptography technique that binds the identity of users to their digital data using a signature mechanism.", "In digital signatures, a private key (or secret key) is only known to the specific user and is utilised to sign the messages [288]..", "The digital signature method confirms the authenticity of the text from the sender, to the receiver and he/she cannot repudiate the origin of the message.", "In addition, digital signatures are also used to check the integrity of a message and ensure that an adversary in the communication does not modify it [289].", "Multi-Signature: Similar to digital signatures, multi-signature is also a digital signature technique used to prove the authenticity of digital documents.", "However, multi-signature allows people to sign one single document instead of a single user per document.", "For example, in government organisations, one document is passed on from many people having different ranks (bottom-up) to authenticate and prove it.", "Typically, a multi-signature scheme generates one single joint signature from a group of people, rather than individual signatures [290].", "In the Blockchain, multi-signatures are used to sign cryptocurrency transactions in order to add extra security protection to them [21].", "The total number of signatures required for one document is decided before the generation of addresses.", "In [65], a multi-signature scheme is utilised in energy trading systems to protect users’ privacy and their related energy consumption data.", "Threshold Signature: The working of threshold signatures is similar to a multi-signature where a group of people signs one document to ensure its authenticity.", "However, the only difference in threshold signatures comparing to the multi-signature is that it requires a fixed number of peoples to produce a valid signature for the document [291].", "To illustrate the proof of threshold signatures, [129], [292] proposed the Blockchain-based E-voting scheme to achieve the security and privacy of voters.", "Ring Signature: Another most important type of digital signature is a ring signature that works in a group pattern arranged in a ring shape.", "Ring signatures enhance the idea of a group signature to provide better security and privacy to group users.", "In a group, the signature can be generated by any group member who is assigned valid cryptography keys.", "In this way, it is challenging for others to determine the group’s actual user who generated the signature with their public key [293].", "In Blockchain-based applications, ring signatures are employed to protect the input transactions signatures with the public key of any node.", "Therefore, it is complicated for adversaries to find the correct identity of the group [127], [234].", "Blind Signature: In this signature algorithm, the user acts as a blind person in the overall signature process and generates a signature without knowing the actual content; thus, it is called a blind signature.", "However, the result of the blind signature is publicly available to everyone to verify the original contents.", "It is essential to describe here that blind signatures are mostly used in privacy-preserving protocols to achieve the anonymity of users (or signers) belonging to different parties [294].", "In the Blockchain scenario, blind signatures are widely used in e-voting applications to achieve the security and privacy of voters and candidates [231], [295].", "Attribute-Based Signature (ABS): This is a modern technique in digital signature methods that allows signature parties to sign documents with users’ information on fine-grained access control policies.", "This known information is formally called attributes given by the central authority in attribute-based signatures [296].", "Each user in the attribute-based signature comes with different types of attributes that are not identical to others.", "Therefore, the changing nature of attributes can also generate different signatures.", "The ABS is mostly employed in Blockchain-based E-health applications in which concern for users’ privacy is essential [245], [31].", "Identity-Based Signature (IBS): Certificates follow the idea of a public-key certificate to the user for signing documents.", "In this way, no one can go beyond their credential limits as defined by certification authorities.", "Compared with other digital signature schemes, the IBS scheme has some advantages in terms of implementation and computation [297].", "However, the drawback of the IBS scheme is that it can increase the length of the signatures by combining the two different signatures, that is one from the user and the other from the certification authority.", "Similarly, it needs two verifiers to verify and prove the generated signatures.", "In the Blockchain, the IBS scheme is frequently used in authentication systems in which users require authentication before using system resources [298].", "Elliptic Curve Digital Signature Algorithm (ECDSA): This algorithm combines the elliptic curve cryptography and digital signature algorithms to generate the signature of data contents.", "Therefore, it is considered the most powerful digital signature algorithm and mostly used in IoT based-applications [299].", "In Blockchain-based applications, the ECDSA is utilised to ensure the integrity and authenticity of transactions [57].", "However, the drawback of ECDSA is that the key size needs to be double that of the other available cryptography algorithms." ], [ "Secret Sharing", "In cryptography, secret sharing is a common technique used in distributed computing when one secret is shared equally among all of the group participants.", "In particular, this secret sharing scheme builds the trust of group participants by fulfilling the following criteria: a sufficient number of participants and the conditions and types of shares to reconstruct the share later.", "To build the secret, an (n, m) - threshold method is used, which is often called the (n, m) - threshold scheme [300].", "The advantage of using secret sharing with Blockchain-based applications is that it can reduce the communication and storage costs for sharing and storing data on distributed ledgers.", "In cryptography, secret sharing is a common technique used in distributed computing where one secret is shared equally among all of the group participants.", "Especially, the secret sharing scheme builds the trust of group participants by fulfilling the following criteria, such as a sufficient number of participants, conditions and types of shares to reconstruct the share later.", "To build the secret, an (n, m) - threshold method is used, which is often called (n, m)- threshold scheme .", "The advantage of using secret sharing with Blockchain-based applications is that it can reduce the communication and storage costs for sharing and storing data on distributed ledgers.", "Shamir Secret Sharing: This is a security scheme that uses the concept of encryption to provide evidence securely from the majority of community participants.", "The Shamir secret scheme works in the form of a hierarchy in which most participants trust each other to maintain the trusted relationship between them [301].", "For instance, one participant must be reliable in distributing the private keys to other group members.", "In the Blockchain, the secret Shamir scheme requires a number (or secret) to define the threshold value to reconstruct the secret used by the miners [302].", "Additive Secret Sharing: Additive secret sharing is another cryptography primitive used to achieve privacy by employing the multi-party computation method.", "Similar to the Shamir secret sharing scheme, no one can recover the total value of a secret using their shared secret [303].", "The advantage of using the additive secret sharing scheme in Blockchain applications is that it follows homomorphic encryption for bitwise transactions processing." ], [ "Zero-Knowledge Proof", "Zero-Knowledge Proof (ZKP) is one of the fundamental concepts of applied cryptography used to ensure security properties, for example anonymity, privacy and verification of transactions.", "In ZKP, a verifier party verifies the proof of the claimant and provides proof of knowledge without disclosing personal information to others [304].", "The advantage of ZKP in the Blockchain is that each verifier (or miner) can prove a shared secret challenge if the claimant does not provide any information (zero-knowledge).", "ZKP is most commonly used in business applications in which parties exchange confidential information without fear of personal information leakage [305]." ], [ "Mixing", "Mixing service plays a vital role in privacy to conceal the senders’ and receivers’ transactions in such a way that no one can know what is actually inside them.", "In the mixing technique, both incoming and outgoing transactions are mixed up with the same type of transactions of others.", "This approach aims to separate the transaction detail from the identities of the sender/receiver.", "Generally, there are two types of mixing techniques used in Blockchain applications, that is transaction mixing and address mixing." ], [ "Transaction Mixing", "In the Blockchain, transactions are stored in the distributed ledger and are open to everyone for graph analysis and other purposes.", "An adversary can easily track individuals by knowing the information stored in the Blockchain.", "To protect the information, especially financial information (bitcoins) from adversaries, transaction mixing services help users to mix their financial transactions so that it would be difficult for others to trace the original user who is involved in the transactions.", "According to defined criteria, these services accept the transactions as inputs from different users, mix or shuffle them according to defined criteria, and send them as outputs at different addresses.", "The mixing services are mostly charge-based services that charge the users to mix their transactions.", "However, the main issue identified in existing mixing services is that it depends on third parties that hold the transactions (coins) for some time.", "These services are often attacked or operated by malicious parties that use attacking techniques such as DDOS and ransomware to steal coins from both parties’ accounts.", "To resolve such issues, many Blockchain-based mixing services, such as CoinJoin and CoinShuffle, have been proposed to remove the third party’s requirement to mix financial transactions between the sender and the receiver." ], [ "Address Mixing", "Address mixing (or address shuffling) is also a mixing service in which an input address of the transaction is related to the transaction’s output address.", "The address mixing service can be implemented in two ways, that is explicit address shuffling and implicit address shuffling.", "In explicit address shuffling, the mixing party explicitly knows the senders’ and their addresses for which the shuffling service is performing.", "CloakCoin [306] is the cryptocurrency that used the explicit address shuffling scheme to relate the input address to the output address.", "Simultaneously, the senders’ and senders’ addresses are completely hidden in implicit (or hidden) shuffling services.", "In an implicit shuffling service, the mixing server cannot relate the address of the sender with the address of the receiver.", "An example of implicit address shuffling is Maxwell’s CoinJoin [307] cryptocurrency that used blind signatures to implement this strategy." ], [ "Anonymisation", "Anonymisation is a commonly used data hiding technique that ensures that users are anonymous throughout the process.", "The anonymisation technique is designed to make it impossible for others to identify the specific user and their data stored in the database.", "The main aim of the anonymisation process is to protect users’ privacy with cryptography or generalisation methods." ], [ "K-Anonymity", "K-anonymity is the most popular technique used to achieve the anonymisation of users’ data.", "The goal of the k-anonymity technique is to protect users’ privacy while performing complex operations on data, for instance, data required to fulfil the k-anonymity property if it is not distinguished from the at-least remaining k-1 users involved in the computation process.", "In this way, the k-anonymity property ensures that the probability of identifying the users in a given data set must not exceed 1/k [308].", "To achieve k-anonymity, the two most commonly used approaches are generalisation and data suppression.", "In the Blockchain, k-anonymity is the basic method used to achieve users’ privacy in the public environment [309]." ], [ "Artificial Intelligence", "Artificial Intelligence (AI) is a combination of different intelligence features and practices for using native hardware in meaningful ways.", "At present, this technology can increase the level of thinking using the concepts of the neural network, machine learning and deep learning [310].", "Blockchain technology is designed to store the data on the immutable ledger by using different cryptography algorithms.", "In addition, AI algorithms have also been used to examine user activities in the Blockchain network and practise different heuristic approaches in a deterministic way [311]." ], [ "Machine Learning", "This is a basic and more frequently used technique in artificial intelligence that enables the system to learn from experience and make an automatic decision to improve it.", "The inputs to machine learning techniques are usually in learning data, collected heuristic observations and experiences.", "Thus, the only objective of machine learning technology is to allow the system to take automatic action without human intervention.", "Blockchain technology, with a combination of machine learning methods, plays a significant role in developing Blockchain-based applications.", "In fact, it changes the way of thinking about building decentralised applications [312].", "In addition, machine learning approaches also improve the Blockchain system’s security, using analytical learning methods, and provide a way to design new privacy-preserving models for decentralised applications [310]." ], [ "Deep Learning", "Deep Learning is another widely used technique of artificial intelligence (sometimes referred to as machine learning sub-field) that has a significant impact on some different areas capable of performing multiple tasks to produce accurate results [313].", "Similar to the machine learning technique, deep learning also gives instructions to the computer in some available datasets, such as text, images, audio and videos [314].", "As the deep learning method integrates and works with large data sets to produce high-quality results, data security is an important requirement that requires significant solutions in terms of security and privacy.", "Building on Blockchain technology, deep learning features can ensure user data security and can also meet the privacy needs of different applications.", "In addition, decentralised deep learning approaches are used with Blockchain applications to ensure consistency and transparency of Blockchain data [315]." ], [ "Others", "Many other approaches are available to respond to the security and privacy problems in the Blockchain-based application.", "For example, a solution to privacy problems in the E-Health application [144] is a discrete wavelength transformation method in which a conversion strategy is utilised to convert the wavelets into discrete sampling, in order to achieve accurate frequency and timing information of stored data.", "In Blockchain security, the discrete wavelength transformation is used with cryptography functions to generate the key pairs for encryption/decryption.", "In [11], a clustering technique, in which each vehicle is linked and tracked by the respective controlling unit, called RSU (Road-Side Unit), is used to protect vehicle users’ privacy.", "The unit authenticates the vehicles using some asymmetric cryptography primitives.", "Another privacy preserving approach, called the three-weight subjective logical [131] , is employed in the vehicular network to protect vehicles’ data.", "This approach is purely based on a probabilistic logic model in which data is assigned a different weight, in order to calculate the subjective logic in the decision-making process.", "There are also a few other methods, such as Range Query [136], Game Theory [128], Bilinear Maps [138], Bloom Filter [127] and Statistical Measures [316] which are used to deal with the problem of security and privacy in different Blockchain-based applications." ], [ "Security and Privacy Attacks on Blockchain-based Industry 4.0 Applications", "This section describes the various security and privacy attacks on Blockchain-based Industry 4.0 Applications, in which the attacker uses different approaches to obtain data and information.", "Since our survey paper aims to integrate Blockchain technology with various industrial applications, it is important to mention here that the attack surface includes attacks on both platforms, such as the Blockchain network and the industrial applications.", "We categorise the attacks based on their layers as follows: data layer, network layer, consensus layer, incentive layer, smart contract layer and application layer.", "We also categorise the security breaches into three different primary branches: (i) breach of confidentiality, (ii) breach of integrity and (iii) breach of availability.", "In breach of confidentiality, the attacker tries to listen to the communication between two parties without the consent of the owner of the data rights.", "In breach of integrity, the attacker aims to change or modify the original data into another form, after listening to the communication channel.", "Undoubtedly, breach of availability is one of the most severe breaches because the attacker’s intention is to disrupt the network or data services using some malicious attacks, such as a denial of services, to make these services unavailable to legitimate users.", "Moreover, we explain each attack with the attacker’s goals and objectives to expose vulnerabilities and threats in the system.", "We also sort attacks and targeted applications whereby some attacks, such as 51% attacks, double spending attacks and selfish mining attacks, are specially designed for Bitcoin and Ethereum applications.", "However, most of the attacks can also be targeted generally to the other domains, including IoT, SG, medical and vehicles.", "We also present state-of-the-art solutions and techniques used to protect the applications and their underlying systems against a subset of such malicious attacks.", "Table presents a summary of the reviewed attacks on both the Blockchain network and industrial applications, along with their layer categories, attacker goals and objectives, security breaches and vulnerabilities, as exploited in the system or network.", "In addition, we also include the targeted applications suffering from these potential threats and vulnerabilities, and prevention methods and security approaches in the table.", "Security and Privacy Attacks on Blockchain-based Industry 4.0 Applications Table: NO_CAPTION Table: NO_CAPTION Table: NO_CAPTION" ], [ "Data Layer", "The data layer is the last layer of the Blockchain framework; it defines the physical structure and properties of a block and encapsulates the data and chain of connecting blocks to the Blockchain.", "The data layer is responsible for handling data that is stored on the Blockchain (on-chain) and in the database (off-chain).", "The attacks on the data layer of Blockchain architecture, which include malleability attack, time hijacking attack, quantum attack, replay attack, modification attack, fault injection attack and upgraded attack, are listed in detail.", "The malleability attack is a specific type of double-spending attack that often happens in networks due to the malleability of signatures.", "In this attack, the attacker broadcasts the two transactions to the Blockchain network, resulting in the appearance of double-spending.", "For instance, attackers can monitor transactions on the Bitcoin network and change their signatures while still authenticating the transactions.", "The signature uses the secure sockets layer protocol, which means that even though an attacker modifies any bytes, the signature remains valid.", "Upon modifying the signature of a transaction, a new transaction identifier is generated [317].", "Numerous solutions have been proposed to this attack in order to prevent the network from being malleable, including segregated witness [318], modification of the bitcoin specification [319] and time commitment methods [320]." ], [ "Time Hijacking Attack", "Time hijacking attacks occur as a result of a loophole discovered in the time stamp protocol of Bitcoin and their related cryptocurrencies.", "In this attack, the attacker's goal is to change the time counters of both the node and the network.", "One strategy for resolving this problem is implementing hardware-oriented systems, which benefit from replacing older network time technologies.", "Time hijacking attacks can also be mitigated by using tolerance range constraints and improved network time protocols [321]." ], [ "Quantum Attack", "Quantum computers are explicitly developed to solve cryptographic problems based on complex mathematical problems.", "Quantum attacks are generally aimed at the Blockchain's cryptographic component, with the primary objective of resolving the mathematical problem of cryptographic dependency.", "For instance, attackers may conduct quantum attacks against Blockchain in order to perform hash collisions against consensus protocols such as PoW.", "Khalifa et al.", "[322] suggested general security steps against quantum elastic Blockchain using the quantum post-signature scheme to minimise the problem of quantum attacks on Blockchain." ], [ "Replay Attack", "In Blockchain systems, the replay attack is the most common issue faced by Blockchain transactions and can cause a long delay in the communication between the two parties.", "In response to the attack, the opponent holds certain transactions in the network and does not send them to the miners for verification.", "As a result, each node has to wait a long time to acknowledge its transactions and results.", "Some significant works have shown that this problem is addressed by using multiple approaches, such as adding a nonce to each transaction, mixing techniques and digital signature methods [323]." ], [ "Modification Attack", "An adversary always tries to change the broadcasted transactions in the Blockchain network before sending them to the miners for verification.", "Furthermore, it can also involve the modification of acknowledgment messages received from miners.", "As a result, the attacker breaches the system’s integrity to launch harmful activities and can take complete control of the underlying system.", "To overcome this problem, some approaches [137], [147], [31] use cryptography operations as attribute-based signatures and consensus algorithms utilised to provide resistance to modification attacks in Blockchain systems." ], [ "Fault Injection Attack", "In a fault injection attack, the adversary can change the programme’s execution by inserting malicious or fake code inside the program.", "The purpose of the fault injection is to either stop the execution of specific instructions or disrupt the complete code.", "This vulnerability usually occurs in the system due to the improper use of code validations in the program.", "In the Blockchain, the adversary attempts to impersonate the Blockchain by adding fake data or blocks to the existing Blockchain, using fault injection methods.", "To give the idea of a fault injection attack, [136] proposed the Blockchain-based vehicular system in which different cryptography methods, such as hashing and digital signatures, were used to protect the system from injection attacks." ], [ "Upgraded Attack", "The upgraded attack is a type of data attack in which the adversary involves changing the trust values defined as threshold values of the system’s participant amount.", "The system’s trust level can reach higher than the current level with many users in the Blockchain network.", "In an upgraded attack, the Blockchain miner controls the overall network to launch the network’s upgraded attack.", "Multiple fake miners can change the current threshold value to perform malicious activity in the system.", "A possible solution to overcome this problem was proposed in [324], in which digital signatures were utilised to verify the defined threshold values in the vehicular network.", "In this way, the modification in the threshold values can easily be detected and mitigated." ], [ "Network Layer", "The network layer is the fifth layer in the Blockchain layered architecture and it is primarily responsible for information transmission between Blockchain nodes.", "As we all know, Blockchain operates on a network known as a P2P network, in which peers exchange knowledge about the state of the network.", "For example, any node in the public Blockchain may enter the network.", "That node can be any ordinary home computer or mobile device; therefore, network layer protection must be implemented to prevent further network attacks.", "Therefore, security and privacy are also important components of the network layer.", "The network layer is subject to the following attacks such as 51% attack, denial-of-service attack, distributed denial-of-service attack, eclipse attack, Sybil attack, BGP Hijacking attack, phishing attack, liveness attack, routing attack and man-in-the-middle attack.", "These attacks are described in detail below." ], [ "51% Attack", "One of the common vulnerabilities found in the Blockchain network, especially in Bitcoin and Ethereum applications, is the 51% attack.", "A group of miners wants to control the network with more than 50% mining (or computing) power [325].", "The mining group prevents the newly created transactions and would not allow them to go to other miners to pass the confirmation successfully.", "In this case, the 51% controlling group takes control of the overall Blockchain and creates wrong decisions to dispute the network’s reputation.", "Moreover, 51% of miners would be able to transfer all of the bitcoins from the user account to their targeted accounts.", "One approach to solve this problem involves selecting random miners and restricting them from recycling their bitcoins to participate in the consensus process [326]." ], [ "Denial of Services (DoS) Attack", "Denial of Services is a common attack on centralised systems; this prevents the host’s network communication or services from performing some legitimate actions.", "In this way, the particular node or system cannot provide legal services to others until it recovers so as to provide the same services again.", "In the Blockchain systems, the DoS attack could restrict one particular node from sending and receiving updates from other nodes in the P2P network.", "Generally, applications that are specifically based on centralised architectures may suffer most from a DoS attack.", "The standard solutions to this problem are decentralised networks and optimal consensus protocols that better protect the systems from DoS attack [327], [328]." ], [ "Distributed Denial of Services (DDoS) Attack", "Compared with the DoS attack, the attacker’s main aim in a DDoS attack is to interrupt the complete services of the integrated network, instead of an attack on a specific node in the network [329].", "In most cases, this attack happened in the network due to poorly managed cache records of Domain Name Services (DDoS), in which the nodes could not receive the updates from other nodes on time.", "This attack is most dangerous for network applications when all legal services go down at once; therefore, it requires significant solutions to overcome this problem.", "In Blockchain-based cryptocurrencies, the consensus mechanism plays a vital role in a decentralised and distributed environment to make an optimal decision at some common points [330]." ], [ "Eclipse Attack", "In an eclipse attack, the attacker aims to target a single specific node rather than capture all the P2P network nodes.", "In this way, the attacker could stop the targeted node from receiving the new updates from the other nodes.", "Indeed, the attacker wants to connect the target node with the other malicious captured nodes in the network.", "The major difference between the eclipse and Sybil attack is that the eclipse attack only targets the specific node.", "In contrast, the Sybil attack captures and takes control of all network nodes at once.", "If the attacker successfully launches the eclipse attack, he can govern his own rules in the network.", "Several possible solutions can resist the eclipse attack by applying the following methodologies, such as using the private network, a random selection of miners, static IP address and limiting the number of incoming/outgoing connections [331], [332]." ], [ "Sybil Attack", "Sybil attack refers to the most important issue of the P2P network.", "An adversary exploits the performance of the network by creating multiple fake identities of the same user.", "Similarly, a malicious node in the Blockchain can cover a large portion of the network by creating multiple fake profiles of the same node.", "In this scenario, the honest nodes in the same network cannot detect fraudulent behaviour and seem to receive the transactions from other honest nodes in the network.", "This malicious behaviour of the network nodes infers that the attacker aims to control the overall Blockchain network.", "There are many solutions available to reduce the risk of Sybil attack in Blockchain-based applications such as eHealth [137],smart vehicles [324], trusted computing [57] and online digital platforms [333].", "However, one simple technique that can restrict the access of a malicious user is to apply some identity-based mechanisms in the systems.", "Moreover, consensus algorithms, such as PoW, are also used in many cryptocurrencies to protect the Sybil attack.", "Each node requires solving the expensive puzzle problem to participate in the mining process." ], [ "BGP Hijacking Attack", "In a BGP (Border Gateway Protocol) hijacking attack, the adversary takes advantage of a vulnerability found in network operators to intercept and manipulate the network traffic routing through gateways [334].", "In Blockchain systems, the BGP attack controls miners’ mining power (or mining pool server) by splitting them into different groups to cause the propagation delay of blocks in the network.", "Thus, all traffic from the Blockchain nodes is directed towards the malicious server to gain others’ bitcoins.", "The common strategy used to tackle the BGP hijacking is the BGPsec protocol that prevents the malicious traffic from gaining access to the system [335].", "Moreover, monitoring and verifying network traffic after some intervals can also be a useful solution." ], [ "Phishing Attack", "A phishing attack is a type of social engineering activity that is often used to obtain financial benefits by stealing user personal information, login credentials and banking information such as credit card numbers.", "This attack is typically carried out when an attacker attempts to pose a trustworthy party and persuades a victim to act in various ways, such as opening an email or responding to a text message.", "The most successful way to prevent phishing attacks on the system is to install anti-spyware software and periodically upgrade firewall settings.", "Furthermore, firewall security can prevent unauthorised file access by blocking malicious attempts [336]." ], [ "Liveness Attack", "This attack happens in Bitcoin and Ethereum applications, when the attacker can hold the broadcasted transactions longer than their confirmation time, in order to cause a delay in the network.", "As a result, an attacker can build a single chain consisting of transactions that are not transferred to honest miners on the network.", "The size of the private chain is longer than the public chain maintained by honest nodes in the network [337].", "The round-trip time (RTT) is used to solve the liveness attacks in Blockchain-based applications in order to overcome this problem." ], [ "Routing Attack", "In a routing attack, an attacker compromises and intercepts the network channel to perform malicious modifications to the data packets.", "An attacker in a routing attack aims to temper with the data values inside the packet before transmitting to other nodes in the network.", "The adversary in the routing attack routes the data traffic towards the malicious server by changing the packet header’s destination addresses.", "This attack is a common attack on client-server-based applications, and especially on those applications based on a P2P network.", "As Blockchain technology follows the idea of a P2P network, the attacker can change the destination address of broadcasted transactions to get the maximum reward from the system [338], [339], [340].", "In the Blockchain, the standard solution used to detect the routing attack is simply discarding those updates that do not match the other received updates.", "Moreover, the network parameters, such as round-trip time (RTT) and irregular patterns, can also help users identify and detect the routing attacks in the network." ], [ "Man-in-the-Middle (MITM) Attack", "This is one of the common vulnerabilities found in network systems in which the attacker plays the role of a middleman to bypass the network traffic to obtain users’ personal information or secrets.", "Once the attacker successfully captures the communication channel’s data, he can use this data in further malevolent activities.", "The MITM attack also takes advantage of vulnerabilities found in key-agreement protocols and storage systems to retrieve the secret keys from them.", "In Blockchain-based cryptocurrency systems, the attacker uses the MITM attack to steal money from the victim’s wallet by changing the destination address with their fake wallet address.", "The most significant way to overcome the MITM attack is using an advanced authentication mechanism which does not allow the adversary to enter into the system." ], [ "Consensus Layer", "The consensus layer is regarded as the foundation layer of the layered architecture of Blockchain.", "Further, it contains numerous consensus algorithms essential to the operation of all Blockchain networks; for example, they allow Blockchain nodes to agree on the validity of newly generated data blocks.", "The security of the Blockchain is based on the participation of each node in the network.", "The security of Bitcoin, for example, is dependent on the high hash power of the nodes that participated in the PoW.", "There are several distinct consensus protocols, for example, PoW, PoS, PBFT and DPoS.", "The attacks on consensus layers are described below, including the double-spending attack and the stake bleeding attack." ], [ "Double Spending Attack", "In Blockchain-based cryptocurrencies, especially in Bitcoin, an attacker with some bitcoins tries to collude with the network by sending a transaction of already consumed bitcoins to others, with the intention of a newly generated transaction.", "In this case, the attacker can use and spend the double bitcoins to collude with someone in the network.", "This type of vulnerability is only found in digital cryptocurrencies due to the limitation of miners’ abilities during the verification process.", "Therefore, recently proposed cryptocurrencies and different Blockchain systems [65], [140], [142] are trying to overcome the double-spending attack by using the latest consensus protocols and cryptography mechanisms, such as digital signatures." ], [ "Stake Bleeding Attack", "As the name implies, a stake-bleeding attack is a type of attack against the PoS consensus mechanism.", "The attacker used transaction fees and processed transactions out of context in this attack, enabling attackers to track the newly added block to the Blockchain.", "Stake-bleeding attacks on Blockchain networks grew in probability as the number of legitimate transactions and adversarial shares increased.", "To address this problem in Blockchain networks, Gazi et al.", "[341] suggested a protocol focused on perspective transactions for validating low-growth chains in order to avoid stake-bleeding attacks." ], [ "Incentive Layer", "The incentive layer in the Blockchain architecture is intended to provide rewards to nodes for participating in the mining process in order to ensure security and verification of blocks added to the Blockchain.", "The security of the Blockchain is determined by a few factors such as the number of miner nodes, the consensus protocol and the mining method.", "This layer, however, is vulnerable to the following attacks, as described below: selfish mining attack, bribery attack, refund attack, block withholding attack and balance attack." ], [ "Selfish Mining Attack", "In a selfish mining attack, an adversary acting as a miner shows selfish behaviour by holding the confirmed blocks, without broadcasting to the other miners in the pool network.", "More than one miner is involved in selfish mining behaviour in order to govern their own rules and policies.", "In this way, the selfish miners collecting the validated blocks can demonstrate and claim more reward against their PoW (hashing power) than other honest miners in the mining pool.", "Recently, one solution was proposed to tackle the selfish mining attack when a fair mining mechanism is adapted to determine the scale and height of the block.", "It also allows the network to block the selfish miners in the event of a discrepancy in the blocks [342]." ], [ "Bribery Attack", "In a bribery attack, the attackers attempt to obtain a temporary majority of miners to increase mining ability on the network.", "This attack is made by renting it from the nominal owners and mining on the fork that does not require this transaction.", "Consequently, the attackers may execute a transaction first and the supplier will then await transaction confirmation.", "Attackers used various methods to bribe miners and increase mining power, including direct payment, fraudulent mining pools and inside payout via tokens.", "An efficient strategy to alleviate this problem is using the PoW consensus mechanism as attackers have to pay considerable costs to discredit the miner [343]." ], [ "Refund Attack", "In a refund attack, the attacker aims to refund the transactions (or payments) made to illegal users by honest users and then fairly deny them to participate in the refund transaction phase.", "In such an attack, the intruder impersonates unauthorised traders in order to exploit the entire network by rejecting all sent transactions.", "McCorry et al.", "[344] suggest an effective solution to this problem in which users are asked to include payment request message along with a few verifiable pieces of evidence such as a delivery address, which reduces the attacker's incentive for profitable attacks." ], [ "Block Withholding Attack", "A block withholding attack is a form of resource squandering attack in which miners violate the mining rules by disguising the hash of the puzzle rather than returning it to the mining pool for a greater self-reward.", "In a real-world example, as miners engage in mining activities through a mining pool, all miners share the reward for successfully solving the computing puzzle based on their computing power quota.", "However, block withholding attacks waste computing resources and reduces the overall mining pool's income, as only malicious miners profit from this attack and collect additional rewards.", "A wide range of solutions to this problem have been suggested, including silent time stamps [345], zero determinant methods [346], contribution of smaller pools [347] and a game model based on the consensus protocol and Nash equilibrium [348]." ], [ "Balance Attack", "As the name suggests, a balance attack is an attack on some consensus mechanism to increase the balance by using some unfair means.", "Balance Attack is a special type of attack on a PoW-based consensus protocol.", "An attacker with low balance or power attempts to delay communication between those subgroups of miners who have the same hashing power (or balance) [349].", "In this way, the attacker captures some of the information from their communications and allows for other types of attacks, such as double-spending.", "This type of attack is most common in Bitcoin and Ethereum-based applications that have coins and ethers to spend.", "Generally speaking, this problem is resolved by limiting the number of more balanced miners in the network.", "The contract layer is the fifth layer of the Blockchain network and comprises three major components: the smart contract itself, scripting code and an algorithm or logic.", "These three elements represent the key logic and conditions in the executed contract.", "These logics are generally written in Solidity, a programming language.", "The smart contract layer attacks are as follows: integer overflow attack, re-entrancy attack and short address attack, all of which are described further below." ], [ "Integer Overflow Attack", "Integer overflow is a common security vulnerability in many applications, especially ethereum smart contracts on the Blockchain, which occurred primarily due to a lack of code validations.", "Smart contracts are a series of programme codes in which unique numbers determine the upper and lower limits of an integer.", "When the value executed reaches their prescribed limits, an integer overflow problem occurs, causing the machine to halt for specific errors.", "A few solutions have been suggested to mitigate the risk of integer overflow attack in smart contracts; however, most of them focus on careful analysis, rewriting, verification of codes writing [350], [142]." ], [ "Re-Entrancy Attack", "Reentrancy attacks are normally triggered by those functions that are not meant to be re-entered by developers.", "In this attack, attackers can create malicious contracts that call these functions reentrantly with the intent of stealing Ether from an honest user's account, causing the user to lose his credentials and all Ethers.", "An example of this type of attack is the DAO attack on smart contracts, which occurred in 2016 and resulted in the loss of 60 million Ether.", "Many solutions have been suggested to overcome the re-entrancy attack on smart contracts.", "For example, one approach called Sereum [351] is proposed to solve the re-entrance attack, allowing for dynamic taint tracking of smart contract data flows.", "ReGuard [352] is an automatic detection system that conducts fuzzing tests in order to fix the issue of re-entry attacks." ], [ "Short Address Attack", "A short address attack is actually a bug in developer-side code that causes users to enter a short address instead of the full address.", "For example, if a user uses the transfer method to withdraw coins and is needed to enter a short address.", "If the size of the address entered by the user is not checked due to a lack of validation measures, a short address attack may occur.", "A solution to this problem is suggested by a technique called SmartScopy [353]—automatically synthesising adversarial contracts to achieve smart contract stability." ], [ "Application Layer", "The application layer is responsible for the execution of applications used by end-users to communicate with the Blockchain network.", "Application layer security refers to the protection of this layer and the users who communicate with it.", "Since this layer is a combination of different Blockchain components and third-party technologies to develop an application, it is vulnerable to a wide range of attacks, including location cheating attack, ballot stuffing attack, badmouthing attack, guess attack, chosen ciphertext attack, impersonation attack, linking attack and collusion attack.", "These attacks are detailed below." ], [ "Location Cheating Attack", "A location cheating attack is a common attack on most vehicular networks when both passengers and drivers are involved in the location cheating activities by sending false locations to a central authority called a road-side unit (RSU).", "In the case of passengers, an adversary launches the location cheating attack by sending a false and long-distance location to an RSU to pick up the passengers from some point.", "On the other hand, the driver can also be involved in the location cheating attack by spoofing the identity of an RSU to cheat passengers.", "To rectify the issue of location cheating in vehicular networks, [233] proposed a robust and secure Blockchain-based method to calculate drivers’ and passengers’ authenticity." ], [ "Ballot Stuffing Attack", "Ballot stuffing (or ballot-box stuffing) is an attack on the electronic voting system.", "An adversary tries to carry out this illegal activity by casting several votes (ballots) favouring a target.", "This fraudulent behaviour triggers the system’s breach of integrity to increase the number of votes for one candidate, thus reducing another candidate’s credibility.", "In this case, the attacker may be able to exploit the system’s vulnerability by taking control of the overall election process.", "One way to overcome this problem is to use digital signatures in the voting process to validate voters’ and candidates’ authenticity.", "In addition, there are a variety of Blockchain-based e-voting applications designed to detect ballot stuffing attacks in the systems [295], [354], [355]." ], [ "Badmouthing Attack", "A badmouthing attack is a widespread attack on rating or feedback systems in which the opponent is trying to respond negatively to the target party.", "In this attack, the opponent’s main objectives are twofold: (i) to degrade the reputation of a particular user in the system by making the scale system negative or (ii) to increase the rating of a favourable user.", "As a result, a badmouth attack can significantly reduce a network’s reputation and , or corrupt the entire system.", "To solve this issue, the self-organising maps technique [356] is used to detect and prevent users’ malicious behaviour." ], [ "Guess Attack", "In this attack, the attacker’s goal is to discover a specific user’s personal information from some stored data using brute force or other matching techniques.", "The guess attack can occur during the searching process, when an attacker tries to match the randomly typed keywords with the stored information.", "If the attacker is successful in his actions, he can easily steal the record, that is, the private keys and personal information of any user.", "To solve these issues, [138] proposed the Blockchain-based E-health scheme in which cryptography primitives, such as encryption and hashing, are employed to protect the personal information of both patients and doctors." ], [ "Chosen Ciphertext Attack", "The chosen ciphertext attack aims to obtain the target user’s private key by analysing the different chosen ciphertexts obtained from the communication channel [357].", "In this attack, an adversary uses some packet sniffing tools to access the network’s ciphertexts and attempts to retrieve the secret keys for decryption.", "This problem can overcome by using advanced cryptography methods, such as order-preserving encryption and ECDSA in different systems, to transform the secret key to others [248]." ], [ "Impersonation Attack", "This is an illegal attempt against an individual or a group to retrieve personal information from databases.", "In this attack, the attacker first creates a fake profile of the legitimate user in the network and then uses some social engineering methods, such as email and links, to reach a targeted system.", "In Blockchain systems, the miners involved in the consensus process gain some incentives to reward their computations.", "Thus, the attacker uses some common techniques to access miner’s systems to maximise profit.", "To reduce the chances of impersonation attacks on Blockchain systems, that is transport and crowdsensing applications, [298], [279], [31] proposed the security mechanism that prevents the miners from impersonation attack and only validates those miners who have the valid attributes to log in to the system." ], [ "Linking Attack", "In a linking attack, the adversary’s goal is to create the link between the external data and stored data by using some de-anonymisation techniques to expose the personal information of users.", "The linking attack is a severe type of attack that is applied over many Blockchain systems such as Bitcoin [234], online digital platforms [333], IoT and vehicular networks [11], in order to extract the secret data from stored transactions.", "This problem has been widely discussed and several approaches have been proposed to resolve a link attack but the most accurate approach is to create a new key pair each time so as to encrypt the data." ], [ "Collusion Attack", "In a collusion attack, the adversary aims to retrieve users’ secrets by combining several different copies of data obtained from the network.", "For this purpose, the attacker examines the network traffic using some packet sniffing tools to obtain data packets containing users’ personal information.", "The same issue could occur in the Blockchain network if multiple miners try to collude with the network to get the maximum reward for their mining activities.", "To address collusion activity that occurred in Blockchain-based eHealth and eCommerce applications, [295], [246] utilise pseudo-random methods in which the random seed is shared between two users in such a way that it can resist N-1 other fake users in the system." ], [ "Analysis and Discussion on Attacks", "We provide an in-depth analysis of the security and privacy attacks covered in this survey paper.", "For analysis, we compared the attacks with the various parameters listed in the table, such as layers, security breaches, applications and solutions.", "Figure REF shows various graphical representations of security and privacy attacks in percentages, with several parameters such as layers, security breaches, impacted applications and security solutions.", "Fig.", "REF shows the percentage of each Blockchain layer affected by security and privacy attacks included in the paper.", "The Blockchain layers are the network layer, data layer, consensus layer, incentive layer, smart contract layer and application layer.", "Overall, it is clear that the network layer of Blockchain architecture is the target of a large number of attacks, accounting for 26% of all attacks.", "It is also worth noting that the application layer and the incentive layer are the most affected layers by these attacks, with just a 2% gap between the two.", "Attacks on these layers account for 21% and 19% of all attacks, respectively.", "In comparison, the data layer is impacted by these attacks by 17%, while the smart contract layer is only impacted by 12%.", "Finally, only a small percentage of attacks are countered at the consensus layer, approximately 5%.", "Figure REF depicts the percentage of different security breaches that cause the security and privacy attacks discussed in our paper.", "In our work, we mainly considered three types of security breaches: breach of confidentiality, breach of integrity and breach of availability.", "Additionally, attacks may occur as a result of a combination of those security breaches.", "Overall, it is clear that a significant proportion of attacks on various Blockchain and industrial applications are the result of a breach of integrity, which is regarded as the primary security breach in this analysis part.", "Furthermore, it is worth noting that nearly half of all attacks for breach of integrity result from a combination of these security breaches.", "A combination of security breaches may come in the form of a breach of confidentiality and a breach of integrity, or a breach of confidentiality and a breach of availability.", "On the other side, there is just a 6% difference between the remaining two breaches (breach of confidentiality and breach of integrity).", "Fig.", "REF illustrates the percentage of various industrial applications susceptible to security and privacy attacks explored in our work.", "Financial services, health industry, energy industry, transport and logistics and technology industry are among the industrial applications covered by our research.", "At first glance, it appears that financial services are the most affected application by these security attacks.", "The primary goal of the attacker is to obtain financial benefits in the form of Bitcoins, Ether and other related cryptocurrencies.", "The percentage of financial applications affected is 29%, which is less than a third of the overall percentages.", "However, it is clear that the technology industry is also affected by these attacks, with just a 5% difference from the financial industry.", "Furthermore, the vehicle industry and transport and logistics have been exposed to these attacks, with just a 3% gap between the two.", "Finally, the same comparison applied to the other two industrial applications, such as the energy industry and the health industry, with just a 2% disparity between the two.", "Fig.", "REF represents the percentages of various types of security solutions used to mitigate security and privacy threats on Blockchain-based Industry 4.0 applications.", "These solutions are classified as cryptographic methods, address mixing, anonymization, artificial intelligence and a few others.", "Overall, it is clear that the most effective solution for securing industrial applications from attacks is the use of various cryptographic techniques such as encryption, hashes, digital signatures, which accounts for nearly two-fifths of all other security solutions.", "By leading this, nearly a third used a combination of security solutions such as bilinear maps, three weight models, time ranks and consensus mechanisms to minimise the risks of attacks on Blockchain-based applications.", "Anonymization techniques, which account for 16% of the total, also play an important role in securing those applications from attacks.", "Address mixing and artificial intelligence-based solutions are also used to solve security issues, but these are in the minority." ], [ "Open Issues", "Despite the fact that potential Blockchain features offer enormous benefits to industry users, some issues must be addressed in order to broaden the scope of industry and business applications.", "In this section, we highlight open issues that arise from integrating Blockchain technology into industrial applications, limiting its applicability to Blockchain technology adoption at a broader industry level.", "The open issues of Blockchain adoption in Industry 4.0-based applications are summarised in Fig.", "REF ." ], [ "Interoperability and Governance", "As the scope of Industry 4.0 is not limited to some applications, it includes a wide range of applications and businesses, in order to interact and share valuable data or assets with others.", "To achieve this interaction between different industrial applications, interoperability is defined as a network infrastructure or capacity that enables the industry partner to exchange information over the network.", "Similarly, Blockchain interoperability allows different Blockchain systems to interact by sending messages and sharing trusted values with others [358].", "However, the issues arising during the interoperability of Blockchain systems are the security of the exchanged data and the appropriate industry guidelines for the regulation and control of applications [157].", "Thus, there is a need to design a mechanism that can ensure interaction between different Blockchain platforms and define the appropriate rules and regulations, in line with industry principles and guidelines [168]." ], [ "Legal and Compliance Issues", "Some of the significant obstacles to the deployment of Blockchain technology in the industrial context is using a different standard, agreement laws and some uncertainty around government regulatory bodies, which prevents this technology from being deployed in larger industrial domains.", "For example, stakeholders in the manufacturing industry use Blockchain tools to ensure that their internal processes and goods adhere to legal and regulatory frameworks [359].", "According to commercial law, there must be specific contracts between industry users and government regulations relating to negotiation, execution, administration and Blockchain management.", "Furthermore, liability must be acknowledged if a contract were miscoded and is not carried out as intended [360].", "Another crucial problem between the parties’ legal and compliance arrangements is to adhere to substantive law, effective governance, jurisdiction and settlement, as well as ensuring the privacy of both consumers and the product.", "Additionally, the sharing of manufacturing data across various platforms can pose challenges for manufacturers and their products.", "Therefore, when developing industry platforms, it is imperative to protect both the users’ privacy and their data [361].", "In regard to the public interest, when making new laws, rules, creating guidelines and applying laws in the industries, governments should hold to account a specific government obligation.", "One way to achieve this aim is to use a private Blockchain to stop illicit activities such as money laundering or bypassing regulations.", "Additionally, there should be some mechanism to prevent new blocks from being produced by fake miners.", "Figure: Open Issues of Blockchain Adoption in Industry 4.0-based Applications" ], [ "Scalability", "Blockchain technology has been proven to be better due to its adherent features, such as decentralisation, in reforming current centralised systems.", "It provides a dynamic environment for P2P network nodes to mine blocks over a specified time and frequently updates transactions to other network nodes.", "For example, Bitcoin and Ethereum Blockchain systems have both their capabilities and their limitations in handling users and transactions.", "However, the issues of scalability in existing Blockchain technology discourage broad adoption and implementation.", "One such example is that VISA [362] can process about 2,000 transactions per second, while Bitcoin [2] can process just seven transactions per second.", "With the increased adoption and applicability of Blockchain technology in Industry 4.0, the length of a particular system increases, and thousands of nodes are needed to join the network for block creation and mining.", "Thus, the scalability of Blockchain is questioned; this is a significant challenge for network security systems and their related applications [363], [364]." ], [ "Storage Capacity", "Storage capacity has been questioned in many Blockchain-based applications as part of the storage constraint for storing data on a secure distributed ledger.", "As in Bitcoin, the chain is increasing by one megabyte every 10 minutes and each node in the network has a copy of the full chain.", "A full node can store the entire blocks but total storage requirements expand exponentially with transaction size, thereby growing the entire system’s capacity.", "Since the amount of manufacturing data in an industrial setup is enormous, incorporating Blockchain technology with many industrial processes is a difficult problem [365].", "Furthermore, the underlying Blockchain protocols cause significant traffic congestion in the system, increasing the system’s need for overall Blockchain storage space [366].", "A further problem with oversized chains is that they put unnecessary synchronisation overload on new users.", "For example, in the industrial IoT context, with the growing number of sensors and produced data, the problem is greatly amplified [367].", "Even though Ethereum-inspired frameworks have recently been developed, resource constrained IoT solutions are still in their infancy and are perhaps incapable of supporting more industrial applications at this time [368]." ], [ "Performance Overhead", "Most industrial applications, such as IoT, SG, eHealth, and so on, are simple in nature and have minimal computing, storage and energy capabilities.", "With the integration of Blockchain with those applications, the most challenging problems arise in node computation limitations [369].", "They require significant computation to mine blocks and perform extensive cryptography operations, including hashing, encryption/decryption and digital signatures.", "Many solutions have been proposed that distinguish mining nodes and simple nodes as full nodes and simple nodes in Blockchain-based IoT applications, respectively [370].", "Another performance issue arising in industrial applications is communication overhead, in which each mining node is responsible for performing multiple functions, such as mining and updating Blockchain, along with sending updated blocks to other peer nodes over the network [371].", "This issue creates an extra network overhead that affects the overall bandwidth and performance of the network.", "Thus, computation and communication overheads in Blockchain-based Industry 4.0 applications create barriers to their adoption at a wider industry level [372]." ], [ "Device Standards and Protocols", "The key part of any industrial setup is the use of various heterogeneous IoT devices that are involved in continuous monitoring and controlling of the environment in order to undertake some future measures.", "However, device standards and protocols have become another barrier for IoT and Industry 4.0, in which a vast range of smart devices are deployed.", "Integrating them is expensive and challenging since they all record data in different formats and use different protocols [373].", "Manufacturing companies such as Bosch and the Eclipse Foundation aim to create more common data formats and communication protocols for data sharing, such as MQTT [374].", "The main objective is to assist smart devices in providing popular data formats that enable them to interact seamlessly with one another; however, more data formats mean more complexity in developing a single data model [375]." ], [ "Cost-Effectiveness and Energy Sustainable Consensus Mechanisms", "Consensus algorithms such as POW and POS are considered very energy-intensive, as they tend to use more energy and computational resources [376], [377].", "As the Blockchain’s size grows over time, more successful miners, as well as some energy-efficient consensus protocols, such as DPoS [378] and Proof of Trust (PoT) [379], have been proposed recently to execute this process in cost-effective ways.", "The key advantage of using these protocols is that they can only store the most recent transaction data on the Blockchain.", "However, since energy and resource-scarce industrial IoT devices are surrounded by enormous amounts of data, more efficient consensus algorithms are required [380]." ], [ "Infrastructure / Implementation Cost", "Blockchain technology generally requires specialised infrastructure, such as additional storage and computationally intensive hardware resources to store the Blockchain.", "Blockchain storage is built on Distributed Ledger Technology (DLT) and serves as a shared database of knowledge regarding transactions between various parties.", "The DLT must be able to store an increasing number of blocks indefinitely and immutably.", "According to one estimation, between April 2019 and March 2021, Bitcoin is seen a growth rate of 340GB; however, this growth varies with the discovery rate of new blocks [381].", "Furthermore, as the number and size of Bitcoin miners have grown dramatically, mining the blocks requires more sophisticated hardware and computational power [382].", "As a result, using Bitcoin’s infrastructure cost as an example, one must consider various cost scenarios when implementing Blockchain technology in industrial settings.", "These scenarios can include: (i) the cost of implementing and deploying the Blockchain setup, (ii) the cost of replacing the current industry infrastructure, (iii) the cost of training employees to be familiar with Blockchain technology, and (iv) the cost of energy running resources as a backup [3]." ], [ "Privacy and Trust", "One of the most appealing features and benefits of Blockchain technology is to achieve anonymity of user identity and transactions, using pseudo-anonymity methods [18].", "However, the selection of public Blockchain type and the use of different pseudo-anonymous techniques in Blockchain applications may connect the identity of users to transactions such as public keys, thus increasing the chances of disclosing personal information to others [383].", "Therefore, there is a substantial need to implement pseudo-anonymity methods that must be fully anonymous and must achieve a higher level of privacy and trust among Blockchain users [384]." ], [ "Attack Surface", "With an increasing number of industrial applications that have adopted Blockchain technology, many attack surfaces have been targeted.", "These attacks take advantage of various methods and vulnerabilities found in applications to get into the environment and they have made some malicious changes to user data [385].", "For example, the Bitcoin Double Spending Attack comes in various ways that sometimes combine with other attacks, such as a Sybil attack, to obtain details of user wallets, balance and private keys [386].", "In addition, the vulnerabilities identified in the smart contract code and, in some cases, open source applications, have made Blockchain systems more vulnerable to other malicious users [387].", "Therefore, carefully designed Blockchain applications with proper guidelines and secure cryptography mechanisms across different layers can minimise the chances of attacks [86]." ], [ "Conclusion and Future Work", "The integration of Blockchain technology with various industrial applications such as IoT, finance, SG, eHealth, transport and logistics and the cloud is growing at a rapid pace, and it has changed human lives in a beneficial way.", "However, the integration of Blockchain into existing applications has increased many security and privacy challenges for the users and their data.", "Considering the design of secure Blockchain-based Industry 4.0 applications, we present a detailed study achieving a significant number of contributions regarding security and privacy for Blockchain-based Industry 4.0 applications.", "Firstly, we provide an overview of Blockchain features and properties that map and initiate the design requirements for secure Blockchain applications.", "Secondly, we categorise the design phase of such applications into different design, security and privacy requirements.", "Thirdly, we extend our survey with various types of security and privacy attacks on Blockchain applications, along with the categories of attacks, the attacker’s objectives, exploited vulnerabilities and targeted applications.", "Furthermore, we provide an in-depth review of Blockchain-based Industry 4.0 applications in terms of defined requirements.", "Then we explore different security and privacy enhancement solutions used to achieve security and privacy requirements in Blockchain-based Industry 4.0 applications.", "Finally, we discuss open design issues which provide the fuel to researchers and developers for the advancement of secure applications.", "In the latter part of the work, we present several future recommendations for design and security requirements, as well as open issues, providing the directions to both researchers and developers for designing secure, scalable, efficient and flexible Blockchain-based applications.", "Currently, most of the existing Blockchain-based schemes designed for applications are lacking in the achievement of the following requirements: scalability, interoperability, usability, flexibility, modularity and transparency.", "Scalability is the most critical requirement among all others for designing the Blockchain applications, especially in the industrial domain, in which systems perform poorly with the increasing number of users and their real-time transactions.", "However, many systems (such as Ethereum [61] and Hyperledger [71]) are subject to their restrictions regarding mining and validating transactions that influence the system’s scalability.", "Therefore, researchers and developers need to address the scalability challenge during the development of Blockchain applications.", "Another design requirement issue found in the Blockchain applications is the linking gap between the system internal modules, which is also referred to as an interoperability issue.", "In practice, industrial processes are dependent on each other and linked together to obtain recent updates about the system activities.", "Moreover, the interoperability of system components also helps the administrator to take instant security actions against malicious activities occurring in the system.", "Thus, the interoperability challenge needs to be addressed in the design course of secure Blockchain applications.", "Blockchain consists of many supporting and underlying features which provide ease of use for the completion of multiple tasks.", "However, not all ordinary users understand the technical detail and working of Blockchain technology.", "Thus, it stops them from adopting the Blockchain applications in different businesses and industries.", "As a result, a Blockchain model must endorse usability and flexibility requirements, while building the Industry 4.0 applications and business domains.", "Moving forward to the modularity design requirement, developers need to write the application code that achieves greater applicability among different Blockchain applications, and it should support the wide variety of services that offer network resources efficiently.", "As Blockchain applications are gaining popularity within the community, it becomes essential for researchers and developers to provide a secure environment in which users can share resources efficiently and communicate transparently with each other." ], [ "Declarations of Interest", "The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper." ] ]
2105.11665
[ [ "$\\tilde{A}$ and $\\tilde{D}$ type cluster algebras: Triangulated surfaces\n and friezes" ], [ "Abstract By viewing $\\tilde{A}$ and $\\tilde{D}$ type cluster algebras as triangulated surfaces, we find all cluster variables in terms of either (i) the frieze pattern (or bipartite belt) or (ii) the periodic quantities previously found for the cluster map associated with these frieze patterns.", "We show that these cluster variables form friezes which are precisely the ones found in [1] by applying the cluster character to the associated cluster category." ], [ "Introduction", "Cluster algebras were first defined in [12], they are generated by cluster variables.", "Starting with an initial seed; a quiver and a set of cluster variables, there is an operation called mutation, at any vertex, which gives a new quiver and a new cluster variable.", "The set of cluster variables is given by applying all possible sequences of mutations.", "By restricting so that we are only allowed to mutate at sinks or sources (where mutation becomes much simpler) we find cluster variables $X^i_n$ satisfying the frieze pattern: $X^i_nX^i_{n+1}=1+\\left(\\prod _{j\\rightarrow i}(X^j_n)^{|b_{ji}|}\\right)\\left(\\prod _{i\\rightarrow j}(X^j_{n+1})^{|b_{ji}|}\\right)$ for each vertex $i$ and $n\\in \\mathbb {Z}$ .", "Here $b_{ji}=-b_{ij}$ is the number of arrows from $j$ to $i$ in $Q$ .", "We consider (REF ) as an equation for determining $X^i_{n+1}$ .", "Remarkably, if the initial quiver is of Dynkin type, then this isn't a restriction at all, as proved in [13].", "Theorem 1.1 If the quiver $Q$ is of Dynkin type then there are finitely many cluster variables and each of them can be found on the frieze pattern.", "If $Q$ if not Dynkin then the frieze pattern does not contain all of the cluster variables.", "The aim of this work is to describe what appears outside of the frieze pattern for the affine quivers $\\tilde{A}$ and $\\tilde{D}$ .", "The frieze pattern was considered in [21] where, via the cluster category, the authors proved that linear relations exist between the frieze variables if and only if $Q$ is Dynkin or affine type.", "A more direct construction of these linear relations was given in [14], [24] by proving that there exists periodic quantities $J_n$ and $\\tilde{J}_n$ for $\\tilde{A}$ type and $J^{\\prime }_n$ for $\\tilde{D}$ type.", "These are functions of the $X^i_n$ that are fixed by sending $n \\mapsto n+a$ for an appropriate $a$ .", "For example, in $\\tilde{D}$ case, we have $J^{\\prime }_n:=\\frac{X^1_{n+1}+X^1_{n-1}}{X^2_n}=\\frac{X^1_{n+N-1}+X^1_{n+N-3}}{X^2_{n+N-2}}=J^{\\prime }_{n+N-2}$ where the vertices 1 and 2, that appear as superscripts, are given in Figure REF .", "The first result of this paper is that these periodic quantities are also cluster variables, but do not live on the frieze pattern.", "In order to find the rest of the cluster variables we rely on a method pioneered in [11] for viewing cluster variables as arcs of triangulations of particular surfaces.", "For $\\tilde{A}_{q,p}$ the surface is an annulus with $q$ and $p$ marked points on either boundary component.", "An example is shown in Figure REF which gives an annulus by gluing along the dotted lines.", "For $\\tilde{D}_N$ the surface is a disc with two internal marked points and $N-2$ marked points on the boundary, shown in Figure REF .", "By finding all of the possible arcs for $\\tilde{A}$ and $\\tilde{D}$ types, we prove that all cluster variables can be described in terms of the frieze sequence (REF ) and determinants: $D^m_a(F_n):=\\begin{vmatrix}F_n & 1 & 0 \\\\1 & F_{n+a} & 1 & 0 \\\\0 & 1 & F_{n+2a} & 1 & 0 \\\\& 0 & 1 & F_{n+3a} & \\ddots \\\\& & 0 & \\ddots & \\ddots & 1 \\\\& & & & 1 & F_{n+(m-1)a}\\end{vmatrix}$ where $F$ is a function of the periodic quantities as described in the following two theorems.", "Theorem 1.2 For $\\tilde{A}_{q,p}$ cluster algebras the arcs on the annulus are of three types: The arcs that connect the two boundary components.", "These are in bijection with the frieze variables $X^i_n$ .", "The arcs connecting the boundary component with $q$ marked points to itself, in bijection with the cluster variables $D^l_p(J_{jp})$ , for $j=0,\\ldots ,q-1$ and $l=1,\\ldots ,q-1$ .", "The arcs connecting the other boundary component (with $p$ marked points) to itself, in bijection with $D^l_q(\\tilde{J}_{jq})$ , for $j=0,\\ldots ,p-1$ and $l=1,\\ldots ,p-1$ .", "The corresponding result for $\\tilde{D}$ type is: Theorem 1.3 For $\\tilde{D}_{N}$ cluster algebras we divide the arcs of the twice punctured disk into four types.", "The arcs that connect the boundary vertices such that the two punctures lie on different sides of this arc.", "The arcs connecting the boundary to the punctures.", "These and the arcs of (i) are in bijection with the frieze variables $X^i_n$ .", "The arcs connecting the boundary component to itself, in bijection with the $D^l_1({J^{\\prime }}_{j})$ , for $j=0,\\ldots ,N-3$ and $l=1,\\ldots , N-3$ .", "The three exceptional arcs $\\Gamma _{\\mathrm {except}}$ : $\\begin{tikzpicture}[scale=0.9, every node/.style={fill=white}](0,0) circle [radius=3.0];[fill=black] (0,1) circle [radius=0.1cm];[fill=black] (0,-1) circle [radius=0.1cm];[green, thick] plot [smooth, tension=1] coordinates { (0,1) (0,-1) };[blue, thick] plot [smooth, tension=1] coordinates { (0,1) (-0.6,0) (0,-2) (1.2,0) (0,1) };[red, thick] plot [smooth, tension=1] coordinates {(0,-1) (-1.2,0) (0,2) (0.6,0) (0,-1)};\\end{tikzpicture}$ If we forget about arcs and surfaces these two theorems give a description of all of the cluster variables in $\\tilde{A}$ and $\\tilde{D}$ type: Theorem 1.4 For $\\tilde{A}_{q,p}$ cluster algebras the cluster variables are $\\left\\lbrace X^i_n\\:|\\:\\begin{aligned}&i=1,\\ldots ,N \\\\ &n\\in \\mathbb {Z}\\end{aligned}\\right\\rbrace \\cup \\left\\lbrace D^l_p(J_{jp})\\:|\\:\\begin{aligned}j=0,\\ldots ,q-1 \\\\ l=1,\\ldots ,q-1\\end{aligned}\\right\\rbrace \\cup \\left\\lbrace D^l_q(\\tilde{J}_{jq})\\:|\\:\\begin{aligned}j=0,\\ldots ,p-1 \\\\ l=1,\\ldots ,p-1\\end{aligned}\\right\\rbrace $ For $\\tilde{D}_N$ the cluster variables are $\\left\\lbrace X^i_n\\:|\\:\\begin{aligned}&i=1,\\ldots ,N+1 \\\\ &n\\in \\mathbb {Z}\\end{aligned}\\right\\rbrace \\cup \\left\\lbrace D^l_1(J^{\\prime }_{j})\\:|\\:\\begin{aligned}j=0,\\ldots ,N-3 \\\\ l=1,\\ldots ,N-3\\end{aligned}\\right\\rbrace \\cup \\Gamma _{\\mathrm {except}}$ Where $\\Gamma _{\\mathrm {except}}$ is the set of cluster variables associated with three exceptional arcs (REF ).", "Friezes were defined in [8] where they were shown to have connections with continued fractions and Farey series.", "Further links with diverse topics including triangulations of polygons [7], Auslander-Reiten theory [17] and moduli spaces of points in projective space [22] were later found.", "They were shown to be linked with cluster algebras [5] via the cluster category and a more direct link was studied in [2].", "Here we prove that in $\\tilde{A}$ and $\\tilde{D}$ cases the off-frieze pattern cluster variables form (slightly generalised) friezes.", "Proposition 1.5 In $\\tilde{A}_{q,p}$ and $\\tilde{D}_N$ cluster algebras the non-frieze pattern cluster variables defined by (REF ) form friezes: $\\begin{matrix}\\ldots & & 1 & & 1 & & 1 & & \\\\& D^1_a(F_n) & & D^1_a(F_{n+a}) & & D^1_a(F_{n+2a}) & & \\ldots & & \\\\\\ldots & & \\mathcal {D}^2_{a}(F_n) & & \\mathcal {D}^2_{a}(F_{n+a}) & & \\mathcal {D}^2_{a}(F_{n+2a}) & & \\\\& \\mathcal {D}^3_{a}(F_{n-a}) & & \\mathcal {D}^3_{a}(F_n) & & \\mathcal {D}^3_{a}(F_{n+a}) & & \\ldots \\\\\\ldots & & \\mathcal {D}^4_{a}(F_{n-a}) & & \\mathcal {D}^4_{a}(F_{n}) & & \\mathcal {D}^4_{a}(F_{n+a}) & & \\\\ \\\\& & \\vdots & & \\vdots && \\vdots \\\\ \\\\& \\mathcal {D}^L_{a}(F_{n-3a}) & & \\mathcal {D}^L_{a}(F_{n-2a}) & & \\mathcal {D}^L_{a}(F_{n-a}) & & \\ldots \\end{matrix}$ i.e.", "each diamond $\\begin{matrix}& \\beta & \\\\\\alpha & & \\delta \\\\& \\gamma &\\end{matrix}$ satisfies $\\alpha \\delta -\\beta \\gamma =1$ .", "In $\\tilde{A}_{q,p}$ type there are two of these friezes, given by $(i) \\qquad F_n=J_{jp}, \\quad a=p, \\quad L=q-1.$ $(ii) \\qquad F_n=\\tilde{J}_{jq}, \\quad a=q, \\quad L=p-1.$ In $\\tilde{D}_N$ type the single frieze is given by $F_n=J^{\\prime }_{j}, \\quad a=1, \\quad L=N-3.$ For these quivers this structure has previously been found [1] in terms of the cluster category.", "There is a map $X_{?", "}$ from the objects of the cluster category to the cluster algebra, such that the frieze pattern (REF ) is the image of the transjective component.", "It was also shown that the exceptional tubes form friezes under this map, and we link our work to this construction.", "Proposition 1.6 In $\\tilde{A}_{q,p}$ type the frieze pattern (REF ) and the friezes of Proposition REF give precisely the cluster frieze given in [1].", "In $\\tilde{D}_N$ type the frieze pattern (REF ) and Proposition REF give the cluster frieze except for the portion on the period 2 tubes, which we have not constructed.", "In this way we can see the $\\tilde{A}$ and $\\tilde{D}$ type cluster algebras as a union of friezes (3 friezes in the $\\tilde{A}$ case and 2 in the $\\tilde{D}$ case) linked by the relation between the expression in the frieze variables defining the periodic quantities $J$ , $\\tilde{J}$ and $J^{\\prime }$ .", "This paper is arranged as follows: In Section we explain the background material we need, including the construction of the frieze pattern (REF ) and some properties of the periodic quantities we will use to prove the determinant formula (REF ).", "We then describe how certain cluster algebras can be viewed as surfaces and how mutation works in this picture.", "Next we show that the $X^i_n$ defined by (REF ) form friezes on repetition quivers.", "We then define the cluster category and the cluster character and show that for $\\tilde{A}$ and $\\tilde{D}$ quivers the frieze (REF ) agrees with the frieze constructed on the transjective component of the cluster category via the cluster character, as given in [1].", "Finally we give the definition of a cluster frieze, which is a frieze given on the whole cluster category, not just the transjective component.", "In Section we construct the friezes of Proposition REF , while not yet proving that the frieze entries are cluster variables.", "In Section we describe the possible arcs (cluster variables) in $\\tilde{A}$ type, proving the $\\tilde{A}$ parts of Theorems REF and REF and Proposition REF .", "We then compare our friezes with those constructed in [1], proving Proposition REF in the $\\tilde{A}$ case.", "Section is analogous to Section , but for $\\tilde{D}$ type." ], [ "Review of cluster algebras, the cluster map, cluster algebras as triangulations of surfaces, friezes and the cluster category", "Here we describe the disparate elements that combine to give our results.", "Firstly we give the construction of cluster mutation and cluster algebras.", "We then discuss how the cluster map is defined for period 1 quivers and bipartite quivers before giving a general definition that gives the frieze pattern (REF ).", "Next we give previous results that have appeared for the frieze pattern in $\\tilde{A}$ and $\\tilde{D}$ types, including identifying periodic quantities for their respective cluster maps.", "We then show how certain cluster algebras can be described as triangulations of surfaces, since we aim to interpret these periodic quantities as arcs in these triangulations.", "We next define friezes and show how they relate to cluster algebras.", "Finally we give a very brief introduction to the cluster category and the cluster character, a map from the objects of this category to the set of cluster variables.", "We describe the cluster categories in $\\tilde{A}$ and $\\tilde{D}$ type and describe the frieze constructed in [1] on the objects." ], [ "Cluster algebras", "Here we give our definition of cluster algebras.", "In this paper we mean a cluster algebra without coefficients or frozen variables.", "A quiver $Q$ is a directed graph where multiple edges are allowed.", "We disallow loops or 2-cycles.", "Quiver mutation $\\mu _k$ at any vertex $k$ is defined in 3 steps: For each length two path $i\\rightarrow k \\rightarrow j$ add a new arrow $i\\rightarrow j$ .", "Reverse the direction of all arrows entering or exiting $k$ .", "Delete all 2-cycles that have appeared.", "This gives a new quiver $\\mu _k(Q)$ .", "The exchange matrix for a quiver $Q$ is the skew-symmetric matrix $B$ with entries $b_{ij}$ , the number of arrows from $i$ to $j$ , with $b_{ji}=-b_{ij}$ .", "The mutation $\\mu _k$ acts on the $b_{ij}$ as $\\mu _k(b_{ij})={\\left\\lbrace \\begin{array}{ll}-b_{ij} & \\mbox{ if } i=k \\mbox{ or } j=k, \\\\b_{ij}+\\frac{1}{2}(|b_{ik}|b_{kj}+b_{ik}|b_{kj}|) & \\mbox{ otherwise.", "}\\end{array}\\right.", "}$ In addition to this, we will also attach a cluster variable $x_i$ at each vertex $i$ .", "Cluster mutation at vertex $k$ , also denoted $\\mu _k$ , fixes all variables $x_i$ with $i\\ne k$ but transforms $x_k$ as follows: $\\mu _k(x_k):=\\frac{1}{x_k}\\left(\\prod _{i\\rightarrow k} x_i+\\prod _{i\\leftarrow k}x_i\\right).$ Here the two products run over the arrows into and out of $k$ respectively, e.g.", "in the first product we have an $x_i$ for every arrow from $i$ to $k$ .", "We consider $\\mu _k$ as a mutation both of the quiver and of the cluster variables.", "The cluster algebra $\\mathcal {A}_Q$ is the algebra over $\\mathbb {Z}$ generated by the cluster variables obtained by any sequence of mutations applied to $Q$ ." ], [ "Periodic quivers and the cluster map", "In this section we define periodic quivers and (a restricted version of) the cluster map that follows.", "We show how the cluster map has been extended to bipartite quivers before giving a general definition for all acyclic quivers.", "We then see how the frieze pattern (REF ) follows from this map.", "Period 1 quivers were defined and classified in [15].", "These are $N$ vertex quivers $Q$ , labelled $0,1,\\ldots ,N-1$ , satisfying $\\mu _0(Q)=\\rho (Q)$ , where $\\rho =(0,N,N-1,\\ldots ,2,1)$ is a permutation acting on the vertices of $Q$ .", "This means that mutation at vertex 0 is tantamount to a (specific) relabelling of the vertices of $Q$ .", "By taking initial cluster variables $x_0,x_1,\\ldots ,x_{N-1}$ mutation at 0 will give a new cluster variable, which we call $x_N$ , determined by $x_Nx_0=F(x_{N-1},x_{N-2},\\ldots ,x_2,x_1)$ for an appropriate function $F$ .", "Next mutation at 1 will give a new cluster variable called $x_{N+1}$ satisfying $x_{N+1}x_1=F(x_N,x_{N-1},\\ldots ,x_2)$ where, due to the quiver being period 1, this $F$ is the same as in (REF ).", "Continuing this process gives the recurrence $x_{n+N}x_n=F(x_{n+N-1},x_{n+N-2},\\ldots ,x_{n+2},x_{n+1})$ for all $n$ .", "An example of this is given by an orientation of an affine $A$ type diagram with $p$ and $q$ arrows pointing clockwise and anticlockwise, respectively, called $\\tilde{A}_{q,p}$ .", "This is given by taking the diagram shown in Figure REF , reducing the labels modulo $N$ , and orienting so that each arrow points from the lower label to the higher.", "Lemma 2.1 The $\\tilde{A}_{q,p}$ quiver, with $p$ and $q$ coprime, gives the recurrence $x_{n+p+q}x_n=x_{n+q}x_{n+p}+1.$ Figure: The A ˜\\tilde{A} type diagram used to obtain the A ˜\\tilde{A} type recurrence relation.Example 2.2 For $p=7$ and $q=8$ the recurrence of this lemma is given by the quiver shown in Figure REF .", "Figure: Our orientation of the A ˜ 8,7 \\tilde{A}_{8,7} quiver.The map defined by these mutations $\\varphi :(x_n,x_{n+1},\\ldots ,x_{n+N-1})\\mapsto (x_{n+1},x_{n+2},\\ldots ,x_{n+N})$ is know as the cluster map.", "In [24] it was generalised to include bipartite quivers $Q$ ; we take $\\mu _{\\mathrm {sink}}$ and $\\mu _{\\mathrm {source}}$ to be the compositions of mutations at every sink and source in $Q$ , respectively, then the product $\\mu :=\\mu _{\\mathrm {source}}\\circ \\mu _{\\mathrm {sink}}$ fixes the quiver but gives new cluster variables.", "We let $X^i_n$ be the cluster variable at vertex $i$ after $n$ applications of $\\mu $ (with $n=0$ giving the initial variables) then the map $\\varphi :(X^1_n,X^2_n,\\ldots ,X^{N+1}_n)\\mapsto (X^1_{n+1},X^2_{n+1},\\ldots ,X^{N+1}_{n+1})$ is also known as a cluster map.", "For a general acyclic quiver this map is obtained by first taking $X^i_0$ for each vertex $i$ .", "We then take $X^i_n$ , for $n\\ne 0$ , to satisfy $X^i_nX^i_{n+1}=1+\\left(\\prod _{j\\rightarrow i}(X^j_n)^{|b_{ji}|}\\right)\\left(\\prod _{i\\rightarrow j}(X^j_{n+1})^{|b_{ji}|}\\right)$ where the products is taken over all arrows in $Q$ .", "This is called a “generalised frieze pattern\" in [21].", "It can be shown that all of the $X^i_n$ can be obtained by cluster mutation, similarly to the construction of (REF ).", "The $\\tilde{A}$ type cluster variables obtained by (REF ) are also included here, with the identification $X^i_n\\leftrightarrow x_{i-nN}$ ." ], [ "Periodic quantities for the cluster map", "Here we examine the periodic quantities found for the cluster map where the quiver is of affine type.", "These immediately give linear relations between the frieze variables with periodic coefficients and, with some work, linear relations with constant coefficients can be obtained.", "Here we explain how this is done.", "In $\\tilde{A}$ type, both papers [15], [14] prove that the recurrence (REF ) has the periodic quantities $J_n:=\\frac{x_{n+2p}+x_n}{x_{n+p}}, \\qquad \\tilde{J}_n:=\\frac{x_{n+2q}+x_n}{x_{n+q}}$ with period $q$ and $p$ , respectively.", "By this we mean that $J_{n+q}=J_n$ and $\\tilde{J}_{n+p}=\\tilde{J}_n$ .", "Immediately we see that the $x_n$ satisfy the linear relations $x_{n+2p}-J_nx_{n+p}+x_n=0, \\qquad x_{n+2q}-\\tilde{J}_nx_{n+q}+x_n=0$ with periodic coefficients.", "The authors of [14] then use this to construct linear relations with constant coefficients.", "Theorem 2.3 The cluster variables $x_n$ satisfy the linear relation $x_{n+2qp}-\\mathcal {K}x_{n+qp}+x_n=0$ where $\\mathcal {K}$ is constant.", "We define the matrices $\\Psi _n:=\\begin{pmatrix}x_{n+p+q} & x_{n+q} \\\\x_{n+p} & x_{n}\\end{pmatrix}\\qquad L_n:=\\begin{pmatrix}J_n & 1 \\\\-1 & 0\\end{pmatrix}\\qquad \\tilde{L}_n:=\\begin{pmatrix}\\tilde{J}_n & -1 \\\\1 & 0\\end{pmatrix}$ such that, by (REF ), $\\Psi _{n+p}=\\Psi _nL_n$ and $\\Psi _{n+q}=\\tilde{L}_n\\Psi _n$ .", "By defining $M_n:=L_nL_{n+p}L_{n+2p}\\ldots L_{n+(q-1)p}, \\qquad \\tilde{M}_n:=\\tilde{L}_{n+(p-1)q}\\ldots \\tilde{L}_{n+2q}\\tilde{L}_{n+q}\\tilde{L}_n$ we have $\\Psi _nM_n=\\Psi _{n+qp}$ and $\\tilde{M}_n\\Psi _n=\\Psi _{n+qp}$ .", "We then apply the Cayley-Hamilton theorem to $M_n$ : $M_n^2-\\mathcal {K}M_n+I=0$ where ${\\mathcal {K}=\\mathrm {Tr}(M_n)=\\mathrm {Tr}(\\tilde{M}_n)}$ is constant.", "Multiplying this matrix equation by $\\Psi _n$ on the left gives the linear relation.", "The first question we ask is about the entries of $M_n$ and $\\tilde{M}_n$ .", "We define the matrices $M^m_n$ as the product of the first $m$ of these $L$ matrices: $M^m_n:=L_nL_{n+p}L_{n+2p}\\ldots L_{n+(m-1)p}.$ For example $M^3_n$ is given by $M^3_n=L_nL_{n+p}L_{n+2p}=\\begin{pmatrix}J_{n+2p}J_{n+p}J_n-J_{n+2p}-J_n & J_{n+p}J_n-1 \\\\-J_{n+2p}J_{n+p}+1 & -J_{n+p}\\end{pmatrix}$ Similarly we let $\\tilde{M}^m_n:=\\tilde{L}_{n+(m-1)q}\\ldots \\tilde{L}_{n+2q}\\tilde{L}_{n+q}\\tilde{L}_n.$ We prove the following in Section : Proposition 2.4 The matrices $M^m_n$ are given by $M^m_n=\\begin{pmatrix}A^m_n & A^{m-1}_n \\\\-A^{m-1}_{n+p} & -A^{m-2}_{n+p}\\end{pmatrix}$ where $A^m_n$ satisfies the recurrence in $m$ : $A^m_n=J_{n+(m-1)p}A^{m-1}_n-A^{m-2}_n, \\qquad A^1_n=J_n, \\qquad A^0_n=1$ for each $n$ .", "Conversely the matrices $\\tilde{M}^m_n$ are given by $\\tilde{M}^m_n=\\begin{pmatrix}\\tilde{A}^m_n & -\\tilde{A}^{m-1}_{n+q} \\\\\\tilde{A}^{m-1}_n & -\\tilde{A}^{m-2}_{n+q}\\end{pmatrix}$ satisfying $\\tilde{A}^m_n=\\tilde{J}_{n+(m-1)q}\\tilde{A}^{m-1}_n-\\tilde{A}^{m-2}_n, \\qquad \\tilde{A}^1_n=\\tilde{J}_n, \\qquad \\tilde{A}^0_n=1.$ In [24] periodic quantities were found for the cluster map (REF ) for $\\tilde{D}$ and $\\tilde{E}$ quivers, as shown in Figure REF .", "Figure: Periodic quantities found for the cluster map for D ˜\\tilde{D} and E ˜\\tilde{E} quivers.The linear relations that follow from these were also obtained in [21] using the cluster category.", "For $\\tilde{D}$ type, the period $N-2$ quantity can be written $J^{\\prime }_n:=\\frac{X^1_{n+1}+X^1_{n-1}}{X^2_n}$ and can also be used to give linear relations with constant coefficients, as in the $\\tilde{A}$ case.", "Proposition 2.5 In the $\\tilde{D}_N$ cases, for even $N$ the cluster variables at vertex 1 satisfy the constant coefficient linear relation: $X^1_{n+2N-4}-\\mathcal {K}^{\\prime }X^1_{n+N-2}+X^1_n=0$ and for odd $N$ : $X^1_{n+4N-8}-\\mathcal {K}^{\\prime }X^1_{n+2N-4}+X^1_n=0$ where vertex 1 is as shown in Figure REF .", "The proof of this is similar to the $\\tilde{A}$ case.", "For $N$ even the constant $\\mathcal {K}^{\\prime }$ is given by the trace of ${L^{\\prime }}_n{L^{\\prime }}_{n+1}\\ldots {L^{\\prime }}_{n+N-3}, \\qquad {L^{\\prime }}_n:=\\begin{pmatrix}0 & -1 \\\\1 & J^{\\prime }_n\\end{pmatrix}$ and for odd $N$ the trace of ${L^{\\prime }}_n{L^{\\prime }}_{n+1}\\ldots {L^{\\prime }}_{n+2N-5}.$ For the same ${L^{\\prime }}_n$ .", "Interestingly, the same product of matrices appears as in the $\\tilde{A}$ case.", "A similar result holds too.", "Proposition 2.6 The entries of the matrix products (REF ) and (REF ) are determined by their upper left entry, which we call ${A^{\\prime }}^m_n$ in either case, which satisfies the linear recurrence ${A^{\\prime }}^m_n=J^{\\prime }_{n+m-1}{A^{\\prime }}^{m-1}_n-{A^{\\prime }}^{m-2}_n, \\qquad {A^{\\prime }}^1_n=J^{\\prime }_n, \\qquad {A^{\\prime }}^0_n=1$ for each $n$ .", "The proof of this is similar to the one for $\\tilde{A}$ type given in Section .", "Due to this result we ask how the $A^m_n$ , $\\tilde{A}^m_n$ and ${A^{\\prime }}^m_n$ fit into their respective cluster algebras.", "Are they are cluster variables and, if so, what cluster variables remain outside of the $A^m_n$ , $\\tilde{A}^m_n$ and ${A^{\\prime }}^m_n$ and the variables obtained by the cluster map?", "To answer these we need an alternative way of viewing these cluster algebras." ], [ "Cluster algebras as triangulated surfaces", "In [11] the authors describe how to view both Dynkin and affine $A$ and $D$ type cluster algebras as triangulations of surfaces with marked points.", "In this section we briefly explain how this correspondence works: we describe how a quiver is obtained from a triangulation and what the analogues of quiver and cluster mutation are.", "For $\\tilde{A}_{q,p}$ cluster algebras we shall be dealing with triangulations of annuli with $q$ and $p$ marked points on opposite boundary components, an example of which can be seen in Figure REF , which gives an annulus by gluing along the dotted lines.", "In addition to the traditional triangles seen for this surface we also allow “self-folded\" triangles: $\\scalebox {0.75}{\\begin{tikzpicture}[every node/.style={fill=white}][fill=black] (0,0) circle [radius=0.15cm];[fill=gray] (0,2) circle [radius=0.15cm];[-] (0,0) to (0,2);plot [smooth, tension=1] coordinates {(0,0) (-0.9,2) (0,3.4) (0.9,2) (0,0)};\\node [scale=1.2] at (0,1) {i};\\node [scale=1.2] at (0,3.4) {k};\\end{tikzpicture}}$ that will appear in the $\\tilde{D}$ type triangulations, as in Figure REF .", "For $\\tilde{A}$ type some possible arcs (not a full triangulation) are displayed in Figure REF .", "Figure: The annulus associated with the A ˜ q,p \\tilde{A}_{q,p} cluster algebra, with some possible arcs.To return the quiver from the triangulation: Attach a vertex to every arc.", "For every pair of vertices $i$ and $j$ from step (i) that are part of the same non-self-folded triangle we draw an arrow $i\\mapsto j$ if, while travelling clockwise around the triangle, $j$ comes directly after $i$ .", "If no arcs lie on a boundary then the situation looks as follows: $\\begin{tikzpicture}[every node/.style={fill=white}][fill=black] (0,0) circle [radius=0.1cm];[fill=black] (4,0) circle [radius=0.1cm];[fill=black] (2,2) circle [radius=0.1cm];[fill=white] (1,1) circle [radius=0.15cm];[fill=white] (3,1) circle [radius=0.15cm];[fill=white] (2,0) circle [radius=0.15cm];[-] (0,0) to (2,2);[-] (0,0) to (4,0);[-] (2,2) to (4,0);[->] (1.2,1) to (2.8,1);[->] (1.8,0.2) to (1.2,0.8);[->] (2.8,0.8) to (2.2,0.2) ;\\end{tikzpicture}$ where the white circles are the vertices of the quiver.", "If an arc lies on a boundary component then we do not draw a vertex there.", "For every pair of arcs $k$ and $i$ forming a self-folded-triangle (REF ) and for every arrow $j\\mapsto k$ we add an arrow $j\\mapsto i$ .", "Also for every arrow $k\\mapsto j$ we add an arrow $i\\mapsto j$ .", "For arcs $k$ not part of self-folded triangles, quiver mutation is analogous to replacing the diagonal of a quadrilateral, $k$ , with the other diagonal, $k^{\\prime }$ : $\\begin{tikzpicture}[every node/.style={fill=white}][fill=black] (0,0) circle [radius=0.1cm];[fill=black] (2,0) circle [radius=0.1cm];[fill=black] (1,2) circle [radius=0.1cm];[fill=black] (3,2) circle [radius=0.1cm];[-] (0,0) to (2,0);[-] (1,2) to (3,2);[-] (0,0) to node[scale=0.7] {k} (3,2);[-] (2,0) to (3,2);[-] (0,0) to (1,2);\\node [scale=0.7] at (2,2){a};\\node [scale=0.7] at (2.5,1){b};\\node [scale=0.7] at (1,0){c};\\node [scale=0.7] at (0.5,1){d};\\end{tikzpicture}\\quad \\begin{tikzpicture}[every node/.style={fill=white}]\\node at (0,0) {\\;};\\node at (0,1) {\\xrightarrow{}};\\end{tikzpicture}\\quad \\begin{tikzpicture}[every node/.style={fill=white}][fill=black] (0,0) circle [radius=0.1cm];[fill=black] (2,0) circle [radius=0.1cm];[fill=black] (1,2) circle [radius=0.1cm];[fill=black] (3,2) circle [radius=0.1cm];[-] (0,0) to (2,0);[-] (1,2) to (3,2);[-] (2,0) to node[scale=0.7] {k^{\\prime }} (1,2);[-] (2,0) to (3,2);[-] (0,0) to (1,2);\\node [scale=0.7] at (2,2){a};\\node [scale=0.7] at (2.5,1){b};\\node [scale=0.7] at (1,0){c};\\node [scale=0.7] at (0.5,1){d};\\end{tikzpicture}$ known as a flip.", "Figures REF and REF show an example of this.", "In our work, for arcs $k$ that are the “outside\" of a self-folded triangle mutation looks like: $\\begin{tikzpicture}[every node/.style={fill=white}][fill=black] (0,-2) circle [radius=0.1cm];[fill=black] (0,-4) circle [radius=0.1cm];[fill=black] (3.46,-2) circle [radius=0.1cm];plot [smooth, tension=1] coordinates {(0,-4) (0,-2)};plot [smooth, tension=1] coordinates {(0,-4) (-0.7,-3) (0,-1.6) (0.7,-3) (0,-4)};plot [smooth, tension=0.9] coordinates {(0,-4) (-1,-3) (0,-1.4) (3.46,-2)};plot [smooth, tension=0.9] coordinates {(0,-4) (1.5,-3.5) (3.46,-2)};\\node [scale=0.7] at (0,-3){i};\\node [scale=0.7] at (0.7,-3){k};\\node at (4,-2.7){\\xrightarrow{}};\\node [scale=0.7] at (1.5,-3.5){1};\\node [scale=0.7] at (2.5,-1.75){j};\\end{tikzpicture}\\begin{tikzpicture}[every node/.style={fill=white}][fill=black] (0,-2) circle [radius=0.1cm];[fill=black] (0,-4) circle [radius=0.1cm];[fill=black] (3.46,-2) circle [radius=0.1cm];plot [smooth, tension=1] coordinates {(0,-4) (0,-2)};plot [smooth, tension=1] coordinates {(0,-2) (3.46,-2)};plot [smooth, tension=0.9] coordinates {(0,-4) (-1,-3) (0,-1.4) (3.46,-2)};plot [smooth, tension=0.9] coordinates {(0,-4) (1.5,-3.5) (3.46,-2)};\\node [scale=0.7] at (0,-3){i};\\node [scale=0.7] at (1.4,-2){k^{\\prime }};\\node [scale=0.7] at (1.5,-3.5){1};\\node [scale=0.7] at (2.5,-1.75){j};\\end{tikzpicture}$ where the arc labelled 1 lives on the boundary.", "This follows by gluing certain vertices (REF ).", "In either case we remove the arc $k$ and replace it with the unique other arc $k^{\\prime }$ that forms a triangulation.", "For arcs inside self-folded triangles we do not allow mutation.", "Under these rules mutation of triangulations mimics exactly quiver mutation.", "Remark 2.7 Due to the construction of quivers from triangulations, for any pair of arcs $i$ and $j$ forming a self-folded triangle (REF ) we have $b_{i,k}=b_{j,k}$ for all vertices $k$ .", "So $\\mu _i=\\sigma \\mu _j$ where $\\sigma $ is the relabelling $i\\leftrightarrow j$ .", "Due to this, not being able to mutate at the arc inside the self-folded triangle, $i$ , isn't really a restriction, we may as well just mutate at $j$ .", "In [11] this inability to mutate inside self-folded triangles is rectified with the introduction of “tagged arcs\".", "We also assign cluster variables to each arc of a triangulation.", "For this we let the boundary arcs have value 1.", "To each arc $i$ in an initial triangulation we give initial cluster variables $x_i$ .", "New cluster variables are defined to satisfy $x_{k}x_{k^{\\prime }}=x_ax_c+x_bx_d$ where the arcs are as labelled in (REF ).", "For quivers that are constructible as triangulated surfaces this matches the cluster mutation formula (REF ).", "For the situation described in (REF ) we have $x_{k}x_{k^{\\prime }}=1+x_j.$ Finally in the following situation $\\begin{tikzpicture}[every node/.style={fill=white}][fill=black] (0,-2) circle [radius=0.1cm];[fill=black] (0,-4) circle [radius=0.1cm];[fill=black] (3.46,-2) circle [radius=0.1cm];[fill=black] (-3.46,-2) circle [radius=0.1cm];plot [smooth, tension=1] coordinates {(0,-4) (0,-2)};plot [smooth, tension=1] coordinates {(0,-4) (-0.7,-3) (0,-1.6) (0.7,-3) (0,-4)};plot [smooth, tension=0.9] coordinates {(0,-4) (-1.5,-3.5) (-3.46,-2)};plot [smooth, tension=0.9] coordinates {(0,-4) (1.5,-3.5) (3.46,-2)};plot [smooth, tension=0.9] coordinates {(-3.46,-2) (0,-0.5) (3.46,-2)};plot [smooth, tension=0.9] coordinates {(0,-4) (-2,-2.5) (0,-1.3) (3.46,-2)};\\node [scale=0.7] at (0,-3){i};\\node [scale=0.7] at (0.7,-3){j};\\node at (4,-2.7){\\xrightarrow{}};\\node [scale=0.7] at (1.5,-3.5){1};\\node [scale=0.7] at (-1.5,-3.5){1};\\node [scale=0.7] at (0,-0.5){m};\\node [scale=0.7] at (2.5,-1.75){k};\\end{tikzpicture}\\begin{tikzpicture}[every node/.style={fill=white}][fill=black] (0,-2) circle [radius=0.1cm];[fill=black] (0,-4) circle [radius=0.1cm];[fill=black] (3.46,-2) circle [radius=0.1cm];[fill=black] (-3.46,-2) circle [radius=0.1cm];plot [smooth, tension=1] coordinates {(0,-4) (0,-2)};plot [smooth, tension=1] coordinates {(0,-4) (-0.7,-3) (0,-1.6) (0.7,-3) (0,-4)};plot [smooth, tension=0.9] coordinates {(0,-4) (-1.5,-3.5) (-3.46,-2)};plot [smooth, tension=0.9] coordinates {(0,-4) (1.5,-3.5) (3.46,-2)};plot [smooth, tension=0.9] coordinates {(-3.46,-2) (0,-0.5) (3.46,-2)};plot [smooth, tension=0.9] coordinates {(0,-4) (2,-2.5) (0,-1.3) (-3.46,-2)};\\node [scale=0.7] at (0,-3){i};\\node [scale=0.7] at (0.7,-3){j};\\node [scale=0.7] at (1.5,-3.5){1};\\node [scale=0.7] at (-1.5,-3.5){1};\\node [scale=0.7] at (0,-0.5){m};\\node [scale=0.7] at (0,-1.3){k^{\\prime }};\\end{tikzpicture}$ we have $x_kx_{k^{\\prime }}=1+x_ix_jx_m$ .", "Again, these are special cases of (REF ) by gluing (REF ) in the right way.", "We will use this construction of cluster algebras as triangulated surfaces to “see\" the cluster variables, in particular the frieze variables $X^i_n$ from (REF ).", "We will also use this picture to show that the $A^m_n$ of (REF ) are indeed cluster variables.", "The recurrence defining these also means that they form a frieze, which we now define." ], [ "Friezes", "In this section we give the definition of friezes, and then of repetition quivers.", "We show that in $\\tilde{A}$ type the repetition quiver gives rise to a frieze.", "We define friezes on repetition quivers and show that for any acyclic quiver the frieze variables $X^i_n$ of (REF ) form a frieze on its repetition quiver.", "Friezes were first defined in [8].", "They were originally arrays of integers in the plane organised like brickwork and sandwiched between two rows of zeroes and ones: $\\begin{matrix}& \\ldots & & 0 & & 0 & & 0 & & \\ldots \\\\\\ldots & & 1 & & 1 & & 1 & & 1 & & \\ldots \\\\& & \\vdots & & \\vdots & & \\vdots & & \\vdots & & \\\\& \\ldots & & a & & b & & c & & \\ldots \\\\\\ldots & & d & & e & & f & & g & & \\ldots \\\\& \\ldots & & h & & i & & j & & \\ldots \\\\& & \\vdots & & \\vdots & & \\vdots & & \\vdots & & \\\\\\ldots & & 1 & & 1 & & 1 & & 1 & & \\ldots \\\\& \\ldots & & 0 & & 0 & & 0 & & \\ldots \\\\\\end{matrix}$ such that each diamond $\\begin{matrix}& \\beta & \\\\\\alpha & & \\delta \\\\& \\gamma &\\end{matrix}$ satisfies $\\alpha \\delta -\\beta \\gamma =1$ .", "For our purposes we don't demand that the entries are integers and don't require the rows of zeroes and ones.", "We have, for example, the friezes formed by the periodic quantities in $\\tilde{A}$ and $\\tilde{D}$ type shown in Proposition REF , the proof of which we give in Section .", "In order to see the link with the frieze patterns of Section REF , we first need to make some definitions.", "Definition 2.8 For a quiver $Q$ , the repetition $\\mathbb {Z}Q$ is the quiver with vertices $(n,i)$ , with $n\\in \\mathbb {Z}$ and $i$ a vertex of $Q$ .", "For every arrow $i\\mapsto j$ in $Q$ we have arrows $(n,i)\\mapsto (n,j)$ and $(n-1,j)\\mapsto (n,i)$ for every $n$ .", "The repetition quiver is an example of a stable translation quiver: it has a map $\\tau :(n,i)\\mapsto (n-1,i)$ such that there is a $1:1$ correspondence between arrows $j\\mapsto i $ and $\\tau (i)\\mapsto j$ .", "Definition 2.9 A frieze on a repetition quiver is an assignment $f(n,i)$ for each vertex $(n,i)$ such that $f(n,i)f(n-1,i)=1+\\prod _{(m,j)\\mapsto (n,i)} f(m,j)$ where the product is taken over all arrows into $(n,i)$ .", "Example 2.10 We draw a frieze on the repetition quiver for $\\tilde{A}_{5,2}$ (the quiver $\\tilde{A}_{5,2}$ can be seen in red), where we identify the top and bottom rows: ${scale=0.9,center}{\\begin{tikzcd}&\\ldots &&x_{-7}&& \\color {red}x_0[dl] && x_7[dl]\\\\\\ldots &&\\ldots &&x_{-2}[dr][ul]&& \\color {red}x_5[dr][ul, red] &&x_{12}[ul]\\\\&\\ldots &&x_{-4}[dr][ur]&&\\color {red}x_3[dr] [ur, red] &&x_{10}[ur]\\\\\\ldots &&x_{-6}[ur]&& \\color {red}x_1[dl] [ur, red] &&x_{8}[dl][ur] && \\ldots \\\\&\\ldots &&x_{-1}[dr][ul]&& \\color {red}x_6[ll, \"\\tau \" description, blue][dr][ul, red] && x_{13}[ul]\\\\\\ldots &&x_{-3}[dr][ur]&&\\color {red}x_4 [dr][ur, red] &&x_{11}[ur] &&\\ldots \\\\&x_{-5}[dr][ur]&& \\color {red}x_2[dr] [ur, red] &&x_9[ur] &&\\ldots \\\\x_{-7}[ur]&&\\color {red}x_0 [ur, red] &&x_7[ur] && \\ldots & & \\ldots \\end{tikzcd}}$ here the map $\\tau $ acts by shifting each vertex left, as shown in blue for $\\tau (x_6)=x_{-1}$ .", "Furthermore this is a frieze due to the $\\tilde{A}$ type relation (REF ) $x_{n}x_{n+7}=1+x_{n+2}x_{n+5}.$ To define a frieze on the repetition quiver for $\\tilde{A}_{q,p}$ for general $q$ and $p$ we let $f(n,i)=x_{nN+ip}$ .", "The $x_n$ satisfy (REF ) so they give the frieze: $\\begin{matrix}\\ddots & & \\vdots & & \\vdots & & \\\\&x_{n-q} & & x_{n+p} & & x_{n+N+p} & & \\ldots \\\\\\ldots & & x_n & & x_{n+N} & & x_{n+2N} \\\\&x_{n-p} && x_{n+q} & & x_{n+N+q} & & \\ldots \\\\\\ldots &&x_{n+q-p} & & x_{n+2q} & & x_{n+N+2q} \\\\& &&\\vdots & & \\vdots & & \\ddots \\end{matrix}$ where, since $nN+ip=(n-p)N+(i+q+p)p$ we identify $(n,i)=(n-p,i+q+p)$ .", "Remark 2.11 The initial values for the associated dynamical system (REF ) are highlighted in red in Example REF .", "In [2] this layout of the initial values is called a frontier (after rotation so that it is horizontal) and a formula is given for the frieze entries in terms of this frontier.", "This type of construction can be found in [5] for $A$ type, with the initial cluster variables set to 1, thus forming a traditional frieze.", "For arbitrary quivers $Q$ (including $\\tilde{D}$ type) we can take $f(n,i)=X^i_n$ , as defined in (REF ) to give a frieze on the repetition quiver $\\mathbb {Z}Q$ ." ], [ "Cluster algebras and representation theory", "Here we give the necessary background to compare our results with [1], where friezes are constructed on the various components of the cluster category.", "An overview of most of the elements of this section, including the cluster category and the cluster character, can be found in the notes [20].", "Throughout this section the quiver $Q$ is taken to be acyclic and to have $N$ vertices and $\\mathbf {k}$ is an arbitrary algebraically closed field." ], [ "The cluster category", "In this section we define and briefly describe the Auslander-Reiten quivers for category of modules over the path algebra $\\mathbf {k}Q$ if $Q$ is Dynkin or affine.", "We then describe the objects of the cluster category in these cases.", "Definition 2.12 The Auslander-Reiten quiver $\\Gamma (\\mathbf {k}Q)$ of the module category $\\mathrm {mod}\\:\\mathbf {k}Q$ has isomorphism classes of finitely generated, indecomposable modules for vertices.", "The arrows are given by “irreducible maps\".", "There is an automorphism $\\tau :\\Gamma (\\mathbf {k}Q)\\rightarrow \\Gamma (\\mathbf {k}Q)$ called the Auslander-Reiten translate, such that $\\tau (P)=0$ for all projectives $P$ and $\\tau ^{-1}(I)=0$ for all injectives $I$ .", "All of the projective modules lie in the same component $\\mathbf {p}$ , which is called the preprojective component.", "Similarly the injectives all lie in $\\mathbf {q}$ , the preinjective component.", "The remaining components are called regular.", "See [19] for an overview or [3] for details.", "A quiver is called representation finite if, up to mutation, it has only finitely many indecomposable modules.", "Gabriel's theorem [16] says that these are precisely the Dynkin quivers.", "Moreover we have the following theorem, the first two points of which are from [18] and paraphrased in [19].", "The third point can be found in [9].", "Theorem 2.13 If $Q$ is Dynkin then $\\Gamma (\\mathbf {k}Q)=\\mathbf {p}=\\mathbf {q}$ is a full and finite subquiver of $\\mathbb {N}Q^{\\mathrm {op}}$ .", "If $Q$ is not Dynkin then $\\mathbf {p}=\\mathbb {N}Q^{\\mathrm {op}}$ and $\\mathbf {q}=-\\mathbb {N}Q^{\\mathrm {op}}$ with $\\mathbf {p}\\cap \\mathbf {q}=0$ .", "The modules of $\\mathbf {p}$ and $\\mathbf {q}$ are uniquely determined by their dimension vectors.", "Finally $\\mathbf {p}\\cup \\mathbf {q}\\ne \\Gamma (\\mathbf {k}Q)$ .", "If $Q$ is affine then $\\mathbf {p}=\\lbrace \\tau ^{-m}P_j\\mid m\\ge 0,\\: j=1,\\ldots , N\\rbrace , \\qquad \\mathbf {q}=\\lbrace \\tau ^{m}I_j\\mid m\\ge 0,\\: j=1,\\ldots , N\\rbrace $ and the regular modules $M$ are those that satisfy $\\tau ^m M\\ne 0$ for all $m\\in \\mathbb {Z}$ .", "The regular components are parametrised by $\\lambda \\in \\mathbb {P}^1(\\mathbf {k})$ .", "By defining the quiver $A_{\\infty }$ as $\\begin{tikzpicture}[every node/.style={fill=white}][->] (0,0) to (1.8,0);[->] (2,0) to (3.8,0);[->] (4,0) to (5.8,0);\\node at (6.2,0){\\ldots };\\node at (0,0.0){1};\\node at (2,0.0){2};\\node at (4,0.0){3};\\end{tikzpicture}$ we can describe the regular components as “tubes\".", "They are given by ${\\mathbb {Z}A_{\\infty }/<\\tau ^{p_{\\lambda }}>}$ for an appropriate $p_{\\lambda }\\in \\mathbb {Z}_{\\ge 0}$ , known as the width of the tube, and ${\\tau :(n,i)\\mapsto (n-1,i)}$ where $n\\in \\mathbb {Z}$ and $i$ is a vertex of $A_{\\infty }$ .", "The modules of the form $(n,1)$ for $n=1,2,\\ldots ,p_{\\lambda }$ are called quasi-simple.", "If $p_{\\lambda }=1$ then the tube is called homogeneous, otherwise it is called exceptional.", "We define $\\mathbb {P}^{\\mathcal {E}}=\\lbrace \\lambda \\in \\mathbb {P}^1(\\mathbf {k})\\mid p_{\\lambda }>1\\rbrace $ and let $T_{\\lambda }$ be the tube of width $p_{\\lambda }$ .", "A list of the widths of the exceptional tubes can be found in [9].", "In particular $|\\mathbb {P}^{\\mathcal {E}}|\\le 3$ .", "The cluster category $\\mathcal {C}_Q$ was first defined in [4].", "It has a suspension functor $[1]$ that coincides with the Auslander-Reiten translation $\\tau $ .", "The indecomposable objects of $\\mathcal {C}_Q$ can be identified with the disjoint union of the objects of $\\Gamma (\\mathbf {k}Q)$ and the shifts of the projectives $P_i[1]$ .", "The Auslander-Reiten quiver of $\\mathcal {C}_Q$ is a stable translation quiver with translate $\\tau $ .", "There is a connection between the cluster category and the cluster algebra given by the cluster character $X_?", ":\\mathrm {Ob}(\\mathcal {C}_Q)\\rightarrow \\mathcal {A}_Q$ which has a few components we need to define." ], [ "The cluster character", "For a $\\mathbf {k}Q$ module $M$ the quiver Grassmanian of dimension $\\underline{e}\\in \\mathbb {Z}_{\\ge 0}^N$ is $\\mathrm {Gr}_{\\underline{e}}(M)=\\lbrace N\\subset M \\mid \\underline{\\mathrm {dim}}(N)=\\underline{e}\\rbrace $ and $\\chi (\\mathrm {Gr}_{\\underline{e}}(M))$ is its Euler characteristic with respect to an appropriate cohomology.", "We let $b_{ij}$ be the number of arrows between vertices $i$ and $j$ in $Q$ , then the Euler form $<-\\:,\\:->$ acts on pairs $a,a^{\\prime }\\in \\mathbb {Z}^N$ by $<a\\:,\\:a^{\\prime }>=\\sum _{i=1}^N a_ia^{\\prime }_i+\\sum _{i,j=1}^N b_{ji}a_ia^{\\prime }_j.$ Definition 2.14 [5] For a cluster algebra with initial cluster variables $u_i$ , the cluster character is a map $X_?", ":\\mathrm {Ob}(\\mathcal {C}_Q)\\rightarrow \\mathcal {A}_Q$ defined by: If $M$ is an indecomposable $kQ$ module with $\\underline{m}=\\underline{\\mathrm {dim}}(M)$ then $X_M=\\sum _{\\underline{e}}\\chi (\\mathrm {Gr}_{\\underline{e}}(M))\\prod _i u_i^{-<\\underline{e}\\:,\\:\\underline{m}>-<\\alpha _i\\:,\\:\\underline{m}-\\underline{e}>}$ where $\\alpha _i=\\underline{\\mathrm {dim}}(S_i)$ , the dimension of the simple module at $i$ .", "The sum is taken over the $\\underline{e}\\in \\mathbb {Z}^N$ such that $\\chi (\\mathrm {Gr}_{\\underline{e}}(M))\\ne 0$ and the product is taken over all vertices $i$ .", "If $M=P_i[1]$ then $X_M=u_i$ .", "For any two objects $M$ and $N$ $X_{M\\bigoplus N}=X_MX_N.$ Theorem 2.15 [6] Theorem 4.", "The map $X_?$ gives a bijection between the set of isomorphism classes of rigid indecomposable modules in $\\mathcal {C}_Q$ and the set of cluster variables $\\mathcal {A}_Q$ .", "Proposition 2.16 [1] Proposition 2.2.", "For an acyclic quiver $Q$ the cluster character $X_?$ induces a frieze on the repetition quiver $\\Gamma (\\mathcal {C}_Q)$ ." ], [ "Friezes on the cluster category", "We are interested affine quivers, in which case the transjective component of $\\Gamma (\\mathcal {C}_Q)$ is isomorphic to the repetition quiver $\\mathbb {Z}Q$ and has the objects $P_i[1]$ at the vertices $(0,i)$ .", "The frieze on $\\mathbb {Z}Q$ given by (REF ) places the initial cluster variables $X^i_0$ at $(0,i)$ , so by [1], Corollary 3.2, the frieze $X^i_n$ coincides with the frieze given by $X_?$ on the transjective component.", "Throughout this section the vertex $e$ is taken to be a fixed extending vertex.", "In the $\\tilde{A}$ case the extending vertices are all vertices and in the $\\tilde{D}$ case these are the vertices labelled $1,2,N$ or $N+1$ in Figure REF .", "When we use these results in Sections REF and REF we choose $e$ explicitly.", "For any $\\lambda \\in \\mathbb {P}^1(k)$ there exists a unique quasi-simple module $M_{\\lambda }$ in $\\mathcal {T}_{\\lambda }$ such that $\\mathrm {dim}\\:M_{\\lambda }(e)=1$ .", "Set $N_{\\lambda }=M_{\\lambda }[1]$ if $e$ is a source or $N_{\\lambda }=M_{\\lambda }[-1]$ if $e$ is a sink.", "In [1] it is proved that there exists transjective $B_{\\lambda }$ and $B^{\\prime }_{\\lambda }$ that are uniquely determined by the existence of non-split triangles $\\begin{tikzcd}N_{\\lambda }[r] & B_{\\lambda }[r] & S_{e}[r] & N_{\\lambda }[1]\\end{tikzcd}\\qquad \\begin{tikzcd}S_e[r] & B^{\\prime }_{\\lambda }[r] & N_{\\lambda }[r] & S_e[1]\\end{tikzcd}$ for a source $e$ and $\\begin{tikzcd}N_{\\lambda }[r] & B^{\\prime }_{\\lambda }[r] & S_{e}[r] & N_{\\lambda }[1]\\end{tikzcd}\\qquad \\begin{tikzcd}S_e[r] & B_{\\lambda }[r] & N_{\\lambda }[r] & S_e[1]\\end{tikzcd}$ if $e$ is a sink.", "For $\\tilde{A}$ and $\\tilde{D}$ type these $B_{\\lambda }$ and $B^{\\prime }_{\\lambda }$ are then constructed explicitly.", "These triangles are used to allow consideration of friezes on the whole of $\\Gamma (\\mathcal {C}_Q)$ .", "Definition 2.17 For an affine quiver $Q$ , a cluster frieze on $\\Gamma (\\mathcal {C}_{Q})$ is a frieze $f$ on $\\Gamma (\\mathcal {C}_{Q})$ such that, for any $\\lambda \\in \\mathbb {P}^{\\mathcal {E}}$ we have $f(N_{\\lambda }[k])=\\frac{f(B_{\\lambda }[k])+f(B^{\\prime }_{\\lambda }[k])}{f(S_{e}[k])}$ for any $k=1,2,\\ldots p_{\\lambda }-1$ .", "We can consider this as “gluing\" the friezes on the exceptional tubes to the frieze on the transjective component.", "Due to (REF ) the values of a cluster frieze on the exceptional tubes are determined by the values on the transjective component.", "Proposition 2.18 [1], Proposition 2.2.", "If $f$ is a cluster frieze on $\\Gamma (\\mathcal {C}_Q)$ such that $f(P_i[1])=u_i$ for each $i$ then $f(M)=X_M$ for all $M\\in \\mathcal {C}_Q$ .", "For $\\tilde{A}$ and $\\tilde{D}$ quivers we have seen that the $X^i_j$ of (REF ) form a frieze on their transjective components.", "Our goal is to define friezes on the exceptional components, in terms of the periodic quantities $J_n$ and $\\tilde{J}_n$ ($\\tilde{A}$ type) and $J^{\\prime }_n$ ($\\tilde{D}$ type) satisfying the cluster frieze condition (REF ).", "This is done in Sections REF and REF .", "In this section we prove Propositions REF and REF which give linear relations between the $A^m_n$ and $\\tilde{A}^m_n$ in the $\\tilde{A}$ case and ${A^{\\prime }}^m_n$ in the $\\tilde{D}$ case, the matrix entries that appear in the construction of the constant coefficient linear relations.", "For both we show that we can construct friezes from these matrix entries.", "We describe the entries of $M_n^m$ as $M^m_n:=\\begin{pmatrix}A^m_n & B^m_n \\\\C^m_n & D^m_n\\end{pmatrix}$ These satisfy $M^{m+1}_n=M^m_nL_{n+mp}$ so we have $\\begin{pmatrix}A^{m+1}_{n} & B^{m+1}_{n} \\\\C^{m+1}_{n} & D^{m+1}_{n}\\end{pmatrix}=\\begin{pmatrix}A^m_n & B^m_n \\\\C^m_n & D^m_n\\end{pmatrix}\\begin{pmatrix}J_{n+mp} & 1\\\\-1 & 0\\end{pmatrix}=\\begin{pmatrix}A^m_nJ_{n+mp}-B^m_n & A^m_n \\\\C^m_nJ_{n+mp}-D_n & C^m_n\\end{pmatrix}$ This gives $B^m_{n}=A^{m-1}_{n}$ and $D^m_n=C^{m-1}_{n}$ , so $A^m_n$ and $C^m_n$ satisfy $A^{m+1}_{n}=A^m_nJ_{n+mp}-A^{m-1}_{n}, \\qquad C^{m+1}_{n}=C^m_nJ_{n+mp}-C^{m-1}_{n}$ with initial values $A^1_n=J_n, \\qquad A^0_n=1, \\qquad C^1_n=-1, \\qquad C^0_n=0$ since $M^0_n=I$ and $M^1_n=L_n$ .", "We have $C^2_n=-J_{n+p}=-A^1_{n+p}$ and $C^1_n=-1=-A^0_{n+p}$ so $C^m_n=-A^{m-1}_{n+p}$ for all $n,m$ .", "This proves Proposition REF for $M^m_n$ and the result for $\\tilde{M}^m_n$ is proved similarly.", "Remark 3.1 In [14], for $q=1$ , the traces $\\mathcal {K}^m_n=A^m_n+D^m_n$ are constructed in terms of a recursion operator $\\mathcal {R}^{(m)}_n=J_{n+m}J_{n+m+1}\\frac{\\partial ^2}{\\partial J_n\\partial J_{n+m-1}}-J_{n+m}\\frac{\\partial }{\\partial J_n}-J_{n+m+1}\\frac{\\partial }{\\partial J_{n+m-1}}+J_{n+m}J_{n+m+1}-1$ satisfying $\\mathcal {K}^{m+2}_n=\\mathcal {R}^m_n\\mathcal {K}^m_n$ .", "The recurrence (REF ) appears in [8], with the iterates forming a frieze with the boundaries of ones and zeroes.", "Coxeter gives a general solution, which still works in our case; we define the determinants ${D}^{m}_{a}(F_n):=\\begin{vmatrix}F_n & 1 & 0 \\\\1 & F_{n+a} & 1 & 0 \\\\0 & 1 & F_{n+2a} & 1 & 0 \\\\& 0 & 1 & F_{n+3a} & \\ddots \\\\& & 0 & \\ddots & \\ddots & 1 \\\\& & & & 1 & F_{n+(m-1)a}\\end{vmatrix}$ for an arbitrary function $F_n$ and integer $a$ .", "By expanding along the last row these satisfy ${D}^{m}_{a}(F_n)=F_{n+(m-1)a}{D}^{m-1}_{a}(F_n)-{D}^{m-2}_{a}(F_n)$ with ${D}^{0}_{a}(F_n)=1$ and ${D}^{1}_{a}(F_n)=F_n$ , hence we can express each of $A^m_n$ , $\\tilde{A}^m_n$ and ${A^{\\prime }}^m_n$ in terms of these determinants: ${D}^{m}_{p}(J_n)=A^m_n, \\qquad D^m_q(\\tilde{J}_n)=\\tilde{A}^m_n, \\qquad D^m_1(J^{\\prime }_n)={A^{\\prime }}^m_n.$ Furthermore applying the Desnanot-Jacobi identity to ${D}^{m}_{a}(F_n)$ gives $D^{m}_a(F_n)D^{m-2}_a(F_{n+a})=D^{m-1}_a(F_n)D^{m-1}_a(F_{n+a})-1$ so the determinants $D^m_a(F_n)$ form a frieze: $\\begin{matrix}&1 & & 1 & & 1 & & 1 & & \\ldots \\\\\\ldots & & F_n & & F_{n+a} & & F_{n+2a} & & F_{n+3a} \\\\&D^2_a(F_{n-a}) & & D^2_a(F_n) & & D^2_a(F_{n+a}) & & D^2_a(F_{n+2a}) & & \\ldots \\\\\\ldots && D^3_a(F_{n-a}) & & D^3_a(F_{n}) & & D^3_a(F_{n+a}) & & D^3_a(F_{n+2a}) \\\\&\\vdots & & \\vdots & & \\vdots && \\vdots \\end{matrix}$ This gives two friezes in $\\tilde{A}$ type: $(i) \\qquad \\:\\: F_n=J_n, \\qquad a=p.$ $(ii) \\qquad F_n=\\tilde{J}_n, \\qquad a=q.$ and one frieze in $\\tilde{D}$ type, $\\qquad \\quad \\:\\: F_n=J^{\\prime }_n, \\qquad a=1.$ Remark 3.2 The determinants (REF ) are examples of continuants.", "See [23], for example, where they are shown to be related to continued fractions.", "This determinant solution for Dynkin $A$ type appears in [10].", "There all cluster variables $x_n$ come from the cluster map and they form a frieze (REF ), with the $x_n$ replacing the $J_n$ in an appropriate way.", "There the $D^m_n$ are called “generalised Chebyshev polynomials\"." ], [ "Triangulated surfaces and the cluster frieze in $\\tilde{A}$ type", "In Section REF we construct $\\tilde{A}_{q,p}$ cluster algebras as triangulated surfaces.", "We first find the cluster variables $x_n$ , obtained by the cluster map, as arcs before proving that the remaining arcs are given by the determinants $D^l_p(J_{jp})$ and $D^l_q(\\tilde{J}_{jq})$ of Theorem REF , proving the $\\tilde{A}$ part of that theorem.", "Finally in Section REF we prove that the friezes constructed in this case, (REF ) for the $x_n$ and (REF ) and (REF ) for the $J_n$ and $\\tilde{J}_n$ , are precisely the cluster friezes given in [1]." ], [ "$\\tilde{A}$ type cluster algebras as triangulated surfaces", "The $\\tilde{A}_{q,p}$ cluster algebra is obtained by triangulations of an annulus with $q$ and $p$ vertices on the inner and outer boundaries, respectively.", "By cutting this annulus we represent the surface as a strip with $q$ vertices on the top and $p$ vertices on the bottom, as in Figure REF , where we have displayed some possible arcs (cluster variables) but not a full triangulation.", "In order to present the $\\tilde{A}_{q,p}$ quiver constructed in Section REF (for example Figure REF ) as a triangulation, we take a strip with vertices labelled as in Figure REF and draw the arcs $kp$ for ${k=0,1,\\ldots ,p+q-2}$ from vertex $k- \\lfloor {kp/N} \\rfloor $ (on the bottom) to vertex $q+\\left\\lfloor {kp/N}\\right\\rfloor $ (on the top).", "We also have an arc labelled $(p+q-1)p$ from 0 on the bottom to $p+q-1$ on the top.", "We then reduce the labels modulo $N$ .", "Example 4.1 In Figure REF we demonstrate this construction for the quiver from Figure REF .", "Before reducing modulo $N$ our vertex labels are $0,7,14,21,28,\\ldots , 98$ .", "The first three, $k=0,1,2$ , satisfy $\\lfloor {kp/N} \\rfloor =0$ so they give arcs $k\\mapsto q=8$ labelled $0,7$ and 14.", "The next, $k=3$ has $\\lfloor {kp/N} \\rfloor =1$ so it gives an arc $2\\mapsto q+1=9$ which we label 6, after reducing 21 modulo 15.", "Continuing this gives the rest of the triangulation.", "Figure: The triangulation for the A ˜ 8,7 \\tilde{A}_{8,7} quiver.We see that in this construction every arc connects the top boundary to the bottom.", "Of course, any arc will either do this or connect two vertices on the same boundary component.", "The following proposition tells us precisely how to obtain every arc of the first kind.", "Proposition 4.2 The arcs connecting the top of the strip to the bottom are in bijection with the cluster variables $x_n$ obtained by the recurrence (REF ).", "We temporarily forget the periodicity of the strip and consider the (now infinitely many) vertices on the top and bottom to be labelled by $v_i$ and $v^{\\prime }_i$ for $i\\in \\mathbb {Z}$ , as in Figure REF .", "Figure: Alternative labelling of the vertices on the strip.The mutation $\\mu _{k}$ will mutate each arc $k$ while it is a diagonal of a square (REF ) $\\begin{tikzpicture}[every node/.style={fill=white}][fill=black] (0,0) circle [radius=0.1cm];[fill=black] (2,0) circle [radius=0.1cm];[fill=black] (1,2) circle [radius=0.1cm];[fill=black] (3,2) circle [radius=0.1cm];[-] (-1,0) to (4,0);[-] (-1,2) to (4,2);[-] (0,0) to node[scale=0.7] {k} (3,2);[-] (2,0) to (3,2);[-] (0,0) to (1,2);\\node [scale=1] at (0,-0.5) {v_j};\\node [scale=1] at (2,-0.5) {v_{j+1}};\\node [scale=1] at (1,2.5) {v^{\\prime }_i};\\node [scale=1] at (3,2.5) {v^{\\prime }_{i+1}};\\end{tikzpicture}$ to give the arc on the other diagonal $v^{\\prime }_{i}\\:\\mbox{---}\\:v_{j+1}$ , for example mutating the triangulation of Figure REF at 0 gives Figure REF .", "The whole mutation sequence ${\\mu =\\mu _N\\mu _{N-1}\\ldots \\mu _1\\mu _0}$ will do this to every arc.", "With the labelling in Figure REF , the initial quiver has arcs $kp$ for ${k=0,1,\\ldots ,p+q-1}$ from vertex $k- \\lfloor {kp/N} \\rfloor $ (on the bottom) to vertex $\\left\\lfloor {kp/N}\\right\\rfloor $ (on the top).", "To show that a general arc $v_{i}\\:\\mbox{---}\\:v^{\\prime }_{j}$ appears due to the cluster map, we reinstate the periodicity on Figure REF and have that $v_{i}\\:\\mbox{---}\\:v^{\\prime }_{j}=v_{i+mp}\\:\\mbox{---}\\:v^{\\prime }_{j+mq}$ for all $m\\in \\mathbb {Z}$ , so by taking the correct $m$ we can assume that $0\\le i+j\\le p+q-1$ .", "In this case the arc $k=i+j$ that appears in the initial triangulation is from $k- \\lfloor {kp/N} \\rfloor $ on the bottom to $\\lfloor {kp/N}\\rfloor $ on the top.", "Applying $\\mu ^{\\lfloor {kp/N}\\rfloor -j}$ will give the arc $v_{i}\\:\\mbox{---}\\:v^{\\prime }_{j}$ .", "Figure: The triangulation for the A ˜ 8,7 \\tilde{A}_{8,7} quiver after mutation at 0.Aside from the arcs discussed in the previous proposition, we only have arcs connecting the top (or bottom) of the strip to itself, for example the arc $2\\:\\mbox{---}\\: p$ in Figure REF .", "We call these $J_{jp}$ and $\\tilde{J}_{-iq}$ , as shown in Figure REF , for $j=0,1,\\ldots ,q-1$ and $i=0,1,\\ldots ,p-1$ .", "Figure: The arcs J jp J_{jp} and J ˜ -iq \\tilde{J}_{-iq}.This naming is justified by the following lemma.", "Lemma 4.3 The arcs $J_{jp}$ and $\\tilde{J}_{-iq}$ are the arcs associated with the periodic quantities (REF ), which are cluster variables.", "We first prove that for any arc labelled $\\overline{kp}$ , for some $k$ , at vertex $j$ on the bottom boundary we have $\\overline{kp}\\equiv jp \\mod {q}$ .", "Here $\\overline{kp}$ denotes the label after reduction modulo $N$ .", "We prove this by induction and assume that the statement is true for the arcs labelled up to $\\overline{(k-1)p}$ .", "There are two possibilities: $\\begin{tikzpicture}[every node/.style={fill=white}][fill=black] (0,0) circle [radius=0.1cm];[fill=black] (4,0) circle [radius=0.1cm];[fill=black] (5,2) circle [radius=0.1cm];[-] (-1,0) to (6,0);[-] (3,2) to (6,2);[-] (4,0) to node[scale=0.7] {\\overline{kp}} (5,2);[-] (0,0) to node[scale=0.7] {\\overline{(k-1)p}} (5,2);\\node at (0,-0.4) {j-1};\\node at (4,-0.4) {j};\\end{tikzpicture}$ or $\\begin{tikzpicture}[every node/.style={fill=white}][fill=black] (4,0) circle [radius=0.1cm];[fill=black] (9,2) circle [radius=0.1cm];[fill=black] (5,2) circle [radius=0.1cm];[-] (3,0) to (7,0);[-] (3,2) to (9,2);[-] (4,0) to node[scale=0.7] {\\overline{(k-1)p}} (5,2);[-] (4,0) to node[scale=0.7] {\\overline{kp}} (9,2);\\node at (4,-0.4) {j-1};\\end{tikzpicture}$ In the first case $\\overline{kp}=\\overline{(k-1)p}+p$ so $\\overline{kp}\\equiv \\overline{(k-1)p}+p\\equiv (j-1)p+p\\equiv jp \\mod {q}$ where we have used the induction assumption.", "In the second case $\\overline{kp}=\\overline{(k-1)p}+p-N=\\overline{(k-1)p}-q$ so $\\overline{kp}\\equiv \\overline{(k-1)p} \\equiv (j-1)p \\mod {q}.$ To construct the $J_n$ we let $\\overline{kp}$ be the rightmost arc at $j$ .", "Near $j+1$ there are two situations: $\\begin{tikzpicture}[every node/.style={fill=white}][fill=black] (0,0) circle [radius=0.1cm];[fill=black] (4,0) circle [radius=0.1cm];[fill=black] (8,0) circle [radius=0.1cm];[fill=black] (5,2) circle [radius=0.1cm];[-] (-1,0) to (9,0);[-] (3,2) to (7,2);[-] (4,0) to node[scale=0.7] {\\overline{kp}+p} (5,2);[-] (0,0) to node[scale=0.7] {\\overline{kp}} (5,2);[-] (8,0) to node[scale=0.7] {\\overline{kp}+2p} (5,2);\\node at (0,-0.4) {j};\\node at (4,-0.4) {j+1};\\node at (8,-0.4) {j+2};\\end{tikzpicture}$ or $\\begin{tikzpicture}[every node/.style={fill=white}][fill=black] (0,0) circle [radius=0.1cm];[fill=black] (4,0) circle [radius=0.1cm];[fill=black] (8,0) circle [radius=0.1cm];[fill=black] (9,2) circle [radius=0.1cm];[fill=black] (5,2) circle [radius=0.1cm];[-] (-1,0) to (9,0);[-] (3,2) to (9,2);[-] (4,0) to node[scale=0.7] {\\overline{kp}+p} (5,2);[-] (0,0) to node[scale=0.7] {\\overline{kp}} (5,2);[-] (4,0) to node[scale=0.7] {\\overline{kp}+2p-N} (9,2);[-] (8,0) to node[scale=0.7] {\\overline{kp}+3p-N} (9,2);\\node at (0,-0.4) {j};\\node at (4,-0.4) {j+1};\\node at (8,-0.4) {j+2};\\end{tikzpicture}$ In the first case mutation at $\\overline{kp}+p$ gives $J_{\\overline{kp}}$ .", "Mutating the second case at $\\overline{kp}+2p-N$ gives the first case so we can obtain $J_{\\overline{kp}}$ between vertices $j$ and $j+2$ again.", "Since $\\overline{kp}\\equiv jp \\mod {q}$ and $J_n$ is period $q$ we see that the arc obtained is indeed $J_{jp}$ .", "For $\\tilde{J}$ we look at the leftmost arc at $q+i$ on the top boundary, labelled $\\overline{kp}$ .", "The situation is this: $\\begin{tikzpicture}[every node/.style={fill=white}][fill=black] (-2,0) circle [radius=0.1cm];[fill=black] (2,0) circle [radius=0.1cm];[fill=black] (4,0) circle [radius=0.1cm];[fill=black] (8,0) circle [radius=0.1cm];[fill=black] (9,2) circle [radius=0.1cm];[fill=black] (5,2) circle [radius=0.1cm];[fill=black] (-3,2) circle [radius=0.1cm];[-] (-3,0) to (9,0);[-] (-4,2) to (9,2);[-] (4,0) to node[scale=0.7] {\\overline{kp}+q-p} (5,2);[-] (2,0) to (5,2);[-] (-2,0) to (5,2);[-] (-2,0) to (-3,2);[-] (8,0) to node[scale=0.7] {\\overline{kp}+q} (5,2);[-] (8,0) to node[scale=0.7] {\\overline{kp}} (9,2);\\node at (5,2.4) {q+i-1};\\node at (2,0.7) {\\ldots };\\node at (0,0) {\\ldots };\\node at (9,2.4) {q+i};\\end{tikzpicture}$ We mutate at each of the arcs at $q+i-1$ from left to right, except for the arc labelled $\\overline{kp}+q$ which gives, near $q+i-1$ , $\\begin{tikzpicture}[every node/.style={fill=white}][fill=black] (8,0) circle [radius=0.1cm];[fill=black] (9,2) circle [radius=0.1cm];[fill=black] (5,2) circle [radius=0.1cm];[fill=black] (-3,2) circle [radius=0.1cm];[-] (-3,0) to (9,0);[-] (-4,2) to (9,2);[-] (8,0) to node[scale=0.7] {\\overline{kp}+2q} (-3,2);[-] (8,0) to node[scale=0.7] {\\overline{kp}+q} (5,2);[-] (8,0) to node[scale=0.7] {\\overline{kp}} (9,2);\\node at (5,2.4) {q+i-1};\\node at (-3,2.4) {q+i-2};\\node at (9,2.4) {q+i};\\node at (1,0.7) {\\ldots };\\node at (3,0) {\\ldots };\\end{tikzpicture}$ Mutation at $\\overline{kp}+q$ will then give $\\tilde{J}_{\\overline{kp}}$ .", "By a similar argument to the one at the start of this proof we have that $\\tilde{J}_{\\overline{kp}}=\\tilde{J}_{-iq}$ The $J$ and $\\tilde{J}$ connect vertices on the same boundary by “jumping\" over one vertex.", "The arcs we haven't discussed so far are those jumping over more than one vertex.", "We define these as $J^{l-1}_{j}$ , which starts at $j$ and jumps right over $l-1$ vertices to reach $j+l$ , and $\\tilde{J}_{i}^{m-1}$ which starts at $q+i$ and jumps left over $m-1$ vertices to reach $q+i-m$ , as shown in Figure REF .", "Again we define these for $j=0,1,\\ldots ,q-1$ and $i=0,1,\\ldots ,p-1$ .", "We remark that $J^{q-1}_{j}$ and $\\tilde{J}^{p-1}_{i}$ are the widest arcs, as $J^{q}_{j}$ and $\\tilde{J}^{p}_{i}$ self-intersect.", "Figure: The arcs J j l-1 J^{l-1}_{j} and J ˜ i m-1 \\tilde{J}^{m-1}_{i}.For small $l$ we have $J^0_{j}=1, \\qquad J^1_{j}=J_{jp}, \\qquad \\tilde{J}^0_{i}=1, \\qquad \\tilde{J}^1_{i}=\\tilde{J}_{-iq}$ and the following theorem allows us to calculate $J^l_{j}$ and $\\tilde{J}^l_i$ for $l>1$ .", "Theorem 4.4 The arcs $J^l_j$ satisfy the recurrence relation $J^{l-1}_{j}=J^{l-2}_{j}J_{(j+l-2)p}-J^{l-3}_{j}$ for $l=3,4,\\ldots ,q$ , with initial values $J^l_0=1$ and $J^1_{j}=J_{jp}$ .", "Similarly, the arcs $\\tilde{J}^m_i$ satisfy $\\tilde{J}_i^{m-1}=\\tilde{J}^{m-2}_i\\tilde{J}_{-(i-m+2)q}-\\tilde{J}^{m-3}_i$ for $m=3,4,\\ldots ,p$ , with initial values $\\tilde{J}^0_{i}=1$ and $\\tilde{J}^1_{i}=\\tilde{J}_{-iq}$ .", "Hence $J^l_{j}={D}^{l}_p(J_{jp})$ , as defined in (REF ), and they form a frieze (REF ).", "Similarly we have $\\tilde{J}^m_i=D^m_q(\\tilde{J}_{-iq})$ forming a frieze (REF ).", "We look at the quadrilateral with diagonals $J_{(j+l-2)p}$ and $J^{l-2}_{j}$ : $\\begin{tikzpicture}[fill=black] (0,0) circle [radius=0.1cm];[fill=black] (2,0) circle [radius=0.1cm];[fill=black] (4,0) circle [radius=0.1cm];[fill=black] (8,0) circle [radius=0.1cm];[fill=black] (10,0) circle [radius=0.1cm];[fill=black] (12,0) circle [radius=0.1cm];plot [smooth, tension=1] coordinates {(0,0) (6,3) (12,0)};plot [smooth, tension=1] coordinates {(0,0) (4,0.7) (8,0)};plot [smooth, tension=1] coordinates {(0,0) (5,1.7) (10,0)};plot [smooth, tension=1] coordinates {(8,0) (10,0.7) (12,0)};plot [smooth, tension=1] coordinates {(-1,0) (4,0)};plot [smooth, tension=1] coordinates {(7,0) (13,0)};\\node at (6,0) {\\ldots };\\node at (6,3.3) {J^{l-1}_{j}};\\node at (10,1.0) {J_{(j+l-2)p}};\\node at (4,1.0) {J^{l-3}_{j}};\\node at (5,2.0) {J^{l-2}_{j}};\\node at (0,-0.3) {j};\\node at (2,-0.3) {j+1};\\node at (4,-0.3) {j+2};\\node at (8,-0.3) {j+l-2};\\node at (10,-0.3) {j+l-1};\\node at (12,-0.3) {j+l};\\end{tikzpicture}$ In this case the Ptolemy relation (REF ) gives $J^{l-2}_{j}J_{(j+l-2)p}=J^{l-3}_{j}+J^{l-1}_{j}$ since the boundary arcs have the value 1.", "The widest arcs are $J^{q-1}_{j}$ , since the next arc, $J^q_{j}$ , would self-intersect.", "The result for the $\\tilde{J}^m_i$ arcs follows from a similar construction.", "This proves the $\\tilde{A}$ parts of Theorem REF and Proposition REF .", "What remains is to prove the $\\tilde{A}$ part of Proposition REF ." ], [ "$\\tilde{A}$ type cluster friezes", "Here we intend to show that the frieze pattern (REF ) and the friezes of Proposition REF give a cluster frieze on the Auslander-Reiten quiver $\\Gamma (\\mathcal {C}_Q)$ .", "They are already friezes, so we just need to ensure that they are connected by the relation (REF ).", "In this case there are two exceptional tubes given by $\\lambda =0,1$ .", "Firstly we need a description of the modules $B_{\\lambda }$ and $B^{\\prime }_{\\lambda }$ , as given in [1].", "The vertex $e$ is required to be a sink so we first mutate the $\\tilde{A}_{q,p}$ quiver at 0 and take $e=0$ .", "The portion of the quiver near 0 now looks like $\\begin{tikzcd}2q-N[r] & {q}[r] & {0} & p[l]\\end{tikzcd}$ with cluster variables $\\begin{tikzcd}x_{2q-N}[r] & x_{q}[r] & x_{N} & x_p[l]\\end{tikzcd}$ and [1] gives $B_0=P_{q}$ and $B^{\\prime }_0=P_{p}[1]$ .", "By definition $X_{P_{p}[1]}=x_p$ .", "$X_{P_0}=X_{S_0}$ has $x_N$ as its denominator, hence is given by mutating (REF ) at $N$ , so $X_{P_0}=x_0$ .", "Similarly $X_{P_q}$ has denominator $x_qx_N$ so is given by performing $\\mu _q\\mu _{N}$ on (REF ), so $X_{P_q}=x_{-p}$ .", "Collecting this we have that (REF ) is $X_{N_0}=\\frac{X_{P_{q}}+X_{P_{p}[1]}}{X_{P_{N}}}=\\frac{x_{-p}+x_{p}}{x_{0}}=J_{0}.$ The map $M\\mapsto M[1]$ gives an automorphism of the cluster algebra, so $X_{N_0[j]}=\\frac{X_{P_{q}[j]}+X_{P_{p}[j+1]}}{X_{P_{N}[j]}}=\\frac{x_{-p+jN}+x_{p+jN}}{x_{-1+jN}}=J_{jN}=J_{jp}.$ Conversely the modules $B_1=P_{p}$ and $B^{\\prime }_1=P_{q}[1]$ give $X_{N_1[j]}=\\frac{X_{P_{q}[j]}+X_{P_{p}[j+1]}}{X_{P_{N}[j]}}=\\frac{x_{-q+jN}+x_{q+jN}}{x_{jN}}=\\tilde{J}_{jN}=\\tilde{J}_{jq}.$ This proves Proposition REF in that $\\tilde{A}$ case." ], [ "Triangulated surfaces and the cluster frieze in $\\tilde{D}$ type", "In this section we look at the construction of $\\tilde{D}$ quivers as triangulations of discs with two punctures.", "We first show how to construct the arcs corresponding to the cluster map variables $X^i_n$ .", "We then construct a periodic frieze of cluster variables, with the first row given by the periodic quantities $J^{\\prime }_n$ , as in the $\\tilde{A}$ case.", "We show that there are only 3 exceptional arcs outside of these.", "Finally we identify the two friezes constructed here as the cluster friezes given in [1]." ], [ "$\\tilde{D}$ type cluster algebras as triangulated surfaces", "As discussed in Subsection REF we take the bipartite orientation of the $\\tilde{D}$ diagram as shown in Figure REF .", "To ensure that this is bipartite the orientation of the arrows at the right end of the diagram depend on the parity of $N$ , which we have signified with double ended arrows.", "Figure: The D ˜ N \\tilde{D}_N quiver.The cluster map $\\varphi :\\begin{pmatrix}X^1_n, & X^2_n, &\\ldots &X^{N+1}_n\\end{pmatrix}\\mapsto \\begin{pmatrix}X^1_{n+1}, & X^2_{n+1}, &\\ldots &X^{N+1}_{n+1}\\end{pmatrix}$ is obtained by applying $\\mu :=\\mu _{\\mathrm {source}}\\circ \\mu _{\\mathrm {sink}}$ to $Q$ , where $\\mu _{\\mathrm {source}}:=\\mu _1\\mu _2\\mu _4\\ldots , \\qquad \\mu _{\\mathrm {sink}}:=\\mu _3\\mu _5\\mu _7\\ldots $ which are the compositions of mutations at the sources and sinks in Figure REF .", "As shown in [11] we can see this quiver as a triangulation of a disk with 2 marked points inside and $N-2$ marked points on the boundary.", "An example of this is seen in Figure REF .", "In order to determine all of the arcs that can occur in a triangulation, we firstly note that every arc $\\gamma $ connecting boundary vertices splits the disc into two connected components.", "There are two possibilities: either the punctures $p_1$ and $p_2$ lie in the same connected component or in different components.", "We denote these sets of arcs by: $\\Gamma _{2,0}:=\\lbrace \\gamma \\mid \\gamma \\textrm { connects boundary vertices and } p_1 \\textrm { and } p_2 \\textrm { appear on the same side of } \\gamma \\rbrace $ $\\Gamma _{1,1}:=\\lbrace \\gamma \\mid \\gamma \\textrm { connects boundary vertices and } p_1 \\textrm { and } p_2 \\textrm { appear on different sides of } \\gamma \\rbrace $ There are also arcs connecting the boundary to the punctures, which we call $\\Gamma _{\\mathrm {punc}}:=\\lbrace \\gamma \\mid \\gamma \\textrm { connects a puncture to the boundary}\\rbrace $ Outside of these three sets there are three exceptional arcs involving only the punctures (REF ), which we call $\\Gamma _{\\mathrm {except}}$ .", "$\\begin{tikzpicture}[scale=0.9, every node/.style={fill=white}](0,0) circle [radius=3.0];[fill=black] (0,1) circle [radius=0.1cm];[fill=black] (0,-1) circle [radius=0.1cm];[green, thick] plot [smooth, tension=1] coordinates { (0,1) (0,-1) };[blue, thick] plot [smooth, tension=1] coordinates { (0,1) (-0.6,0) (0,-2) (1.2,0) (0,1) };[red, thick] plot [smooth, tension=1] coordinates {(0,-1) (-1.2,0) (0,2) (0.6,0) (0,-1)};\\end{tikzpicture}$ Lemma 5.1 The arcs of $\\Gamma _{1,1}$ look like those shown in Figure REF .", "They start from a vertex $v_i$ and encircle the line $L$ and the punctures $m\\in \\mathbb {Z}$ times, as shown in Figure REF in blue for $m>0$ .", "After this the curve crosses $L$ and takes the only possible path to $v_j$ (red).", "For $m<0$ the curve is shown by reflecting this picture in the vertical axis.", "We denote these by $\\gamma (v_i,v_j,m)$ .", "The arcs of $\\Gamma _{\\mathrm {punc}}$ are the arcs that form self-folded triangles with a $\\gamma (v_i,v_i,m)$ .", "We firstly look at an arc $\\gamma \\in \\Gamma _{1,1}$ connecting two boundary vertices $v_i$ and $v_j$ .", "Since this arc separates the two punctures it necessarily crosses the line joining them, which we call $L$ .", "The portion of $\\gamma $ before this crossing can only live in the annulus obtained by removing an area enclosing the two punctures and $L$ , as shown in Figure REF on the left, so it simply orbits the drawn ellipse $m\\in \\mathbb {Z}$ times, where we take $m>0$ to mean that the blue portion of the arc travels anticlockwise, starting from $v_i$ , while $m<0$ means clockwise.", "When $\\gamma $ finally crosses $L$ , as shown on the right of Figure REF (we draw this new portion of $\\gamma $ in red for clarity), there is only one choice, to follow the maze back to $v_j$ .", "The only remaining issue is that perhaps the curve crosses $L$ from the other direction, i.e.", "the curve of Figure REF is blue to the right of $L$ and red to the left.", "In this case we just need to permute $i\\leftrightarrow j$ to obtain a curve of the right form.", "Finally we note that every arc of $\\Gamma _{\\mathrm {punc}}$ appears as the internal arc of a self-folded triangle.", "The other arc of this self-folded triangle, as we have just shown, is of the form $\\gamma (v_i,v_i,m)$ .", "Figure: The portion of an arc (blue) before crossing the line LL and the behaviour just after crossing (red).Figure: The full curve after the red portion takes the only path to reach v j v_j.Next we try to identify the arcs obtained by the cluster map.", "Our initial triangulation is shown in Figure REF for $N=8$ .", "Its construction can be tentatively described in two steps: Draw two self folded triangles; one at $v_1$ and $p_1$ and one at $v_6$ and $p_2$ .", "Draw arcs $i$ from $v_{i-2}$ to $v_{i-1}$ for $i=3,4,\\ldots , N-1$ .", "We remark that in the second step there is only one choice for each of the arcs.", "We need to see how this triangulation changes under $\\mu $ , which we perform in several steps, as shown in Figure REF .", "Figure: A full application of μ\\mu to the initial quiver.In Figure REF , after a complete application of $\\mu $ , we can see that the boundary vertices of the self-folded triangles have moved clockwise.", "We have also rotated the boundary vertices to obtain a new labelling for the boundary of $\\mu (Q)$ , so now the new quiver can be described by the same two steps as before.", "In Figure REF we display the full triangulation for $\\mu ^2(Q)$ and some of the triangulation for $\\mu ^3(Q)$ (because the full picture would be too busy to be helpful) which are also constructed by the same two steps, if we relabel as before.", "Figure: Further applications: μ 2 (Q)\\mu ^2(Q) on the left and an incomplete drawing of μ 3 (Q)\\mu ^3(Q) on the right.Figure: The arcs of μ 6 (Q)\\mu ^6(Q) connecting the punctures to the boundary.The self folded triangles are wrapping around each other more with each $\\mu $ .", "In Figure REF we show the arcs connected to the punctures after 6 applications of $\\mu $ , which is enough to construct the rest of the triangulation, following step (ii).", "We see, however, that in both $Q$ and $\\mu ^6(Q)$ we have two self folded triangles, one at $v_1$ and $p_1$ and one at $v_6$ and $p_2$ , so the description of step (i) is not sufficient.", "To amend this we draw a line from $p_1$ that cuts the circle somewhere between $v_1$ and $v_2$ (as shown in Figure REF ) and consider how many times the arc from $v_1$ to $p_1$ crosses this line.", "This is the dashed line drawn Figure REF , where arc 1 crosses it once.", "In this example we shall have one crossing for every 6 applications of $\\mu $ .", "We update step (i) in our description of $\\mu ^l(Q)$ to capture this in the following lemma.", "Lemma 5.2 To describe the triangulation given by applying $\\mu ^l$ to (REF ), for any $l\\in \\mathbb {Z}$ , we first take the boundary vertex labelling and dashed line of Figure REF and rotate it by $\\frac{2\\pi \\overline{l}}{N-2}$ clockwise, where $0\\le \\overline{l}< N-2, \\qquad \\overline{l} \\equiv l \\mod {N}-2$ while fixing $p_1$ and $p_2$ .", "After this rotation the dashed line will still go from $p_1$ to the boundary circle between $v_1$ and $v_2$ .", "The triangulation is then given in three steps: Draw a self folded triangle at $v_1$ and $p_1$ , such that the arc inside the triangle crosses the dashed line $\\left\\lfloor {\\frac{l}{N-2}}\\right\\rfloor $ times and travels anticlockwise (clockwise) from $v_1$ to $p_1$ if $l$ is positive (negative).", "Draw a self-folded triangle at $v_{N-2}$ and $p_2$ .", "There is only one choice due to step (i).", "Draw arcs $i$ from $v_{i-2}$ to $v_{i-1}$ for $i=3,4,\\ldots , N-1$ .", "Again, there is only one choice for each of these.", "Figure: Labelling around the boundary.Analogously to Proposition REF we have the following result.", "Proposition 5.3 The set of cluster variables obtained by the cluster map (REF ), $\\lbrace X^i_n \\mid i=1,2,\\ldots ,N+1, \\quad n\\in \\mathbb {Z}\\rbrace $ is precisely the set of cluster variables associated with $\\Gamma _{1,1}\\cup \\Gamma _{\\mathrm {punc}}$ .", "For this proof we take a fixed labelling of the boundary vertices as shown in Figure REF .", "We first show that every $\\gamma (v_i,v_i,m)$ in $\\Gamma _{1,1}$ appears due to the cluster map.", "From Lemma REF each of these arcs will orbit the two punctures and the line between them $m$ times before crossing the line $L$ .", "For $i=1$ this arc appears on the outside of the self-folded triangle at $v_1$ in the quiver $\\mu ^{m(N-2)}(Q)$ , since this encircles $L$ once for every $N-2$ applications of $\\mu $ .", "A further $\\mu $ will then give $\\gamma (v_3,v_3,m)$ since $\\mu $ just moves the end points of $\\gamma (v_1,v_1,m)$ clockwise.", "More applications of $\\mu $ will give us $\\gamma (v_i,v_i,m)$ for $i=1,2,\\ldots , N-2$ (but not necessarily in this order).", "To show that $\\gamma (v_i,v_j,m)$ occurs for any $i,j$ we give another labelling, as shown in Figure REF .", "We have just shown that the arc $\\gamma (v_{\\left\\lfloor {\\frac{i+j}{2}}\\right\\rfloor },v_{\\left\\lfloor {\\frac{i+j}{2}}\\right\\rfloor },m)$ appears due to the cluster map.", "Our construction of Lemma REF then has us draw $\\gamma (v_{\\left\\lfloor {\\frac{i+j}{2}}\\right\\rfloor },v_{\\left\\lfloor {\\frac{i+j}{2}}\\right\\rfloor +1},m)$ since, away from the boundary, every arc drawn in Lemma REF is homotopic to the one drawn before.", "The next two we draw are $\\gamma (v_{\\left\\lfloor {\\frac{i+j}{2}}\\right\\rfloor -1},v_{\\left\\lfloor {\\frac{i+j}{2}}\\right\\rfloor +1},m),\\qquad \\gamma (v_{\\left\\lfloor {\\frac{i+j}{2}}\\right\\rfloor -1},v_{\\left\\lfloor {\\frac{i+j}{2}}\\right\\rfloor +2},m)$ continuing this will eventually give $\\gamma (v_i,v_j,m)$ .", "We also note that every arc of $\\Gamma _{\\mathrm {punc}}$ appears inside a $\\gamma (v_i,v_i,m)$ and so will also be given by the cluster map.", "Finally it is clear from Lemma REF that every $X^i_n$ is in $\\Gamma _{1,1}\\cup \\Gamma _{\\mathrm {punc}}$ .", "Figure: A different labelling around the boundary.Proposition 5.4 The arcs of $\\Gamma _{2,0}$ live on an annulus obtained by removing a region containing the two punctures and the line between them, so we can display them on a strip, as in Figure REF .", "$\\Gamma _{2,0}$ is precisely the arcs ${J^{\\prime }}^{l}_i$ for $i=0,1,\\ldots ,N-3$ and $l=1,2,\\ldots ,N-3$ .", "These satisfy the linear relation ${J^{\\prime }}^{l-1}_i={J^{\\prime }}^{l-2}_iJ^{\\prime }_{i+l-2}-{J^{\\prime }}^{l-3}_i$ where $J^{\\prime }_{i+l-2}$ is the period $N-2$ quantity (REF ).", "Hence ${J^{\\prime }}^l_j=D^l_1(J^{\\prime }_j)$ , where $D^l_1(J^{\\prime }_j)$ is given in (REF ), and these cluster variables form a frieze (REF ) but with final bottom row $\\begin{matrix}\\\\\\ldots & D^{N-3}_1(J^{\\prime }_{-2}) & & D^{N-3}_1(J^{\\prime }_{-1}) & & D^{N-3}_1(J^{\\prime }_0) & & D^{N-3}_1(J^{\\prime }_1) & & D^{N-3}_1(J^{\\prime }_2) & \\ldots \\\\\\end{matrix}$ These arcs divide the disc in two such that the punctures live in the same connected component after this division.", "Hence the arcs avoid the part of the disc containing the line $L$ between the two punctures, so they must live in the annulus obtained by removing a small region around $L$ .", "We draw this as in Figure REF , where we glue along the dotted lines, and define ${J^{\\prime }}^{l-1}_i$ as the arc starting at $i$ and “jumping over\" $l-1$ vertices to reach $i+l$ .", "We remark that this picture is the same as in the $\\tilde{A}$ case, Figure REF , but with 0 marked points on the internal boundary of the annulus.", "As such, the picture is the same as (REF ) (with $p=1$ ) so we have the relation ${J^{\\prime }}^{l-1}_j={J^{\\prime }}^{l-2}_j{J^{\\prime }}^1_{j+l-2}-{J^{\\prime }}^{l-3}_j.$ The periodic quantities for the cluster map, $J^{\\prime }_n$ , have many equivalent expressions given in [24].", "For this proof we use $J^{\\prime }_n=\\frac{X^3_{n+1}+X^5_n}{X^4_n}.$ We can obtain $J^{\\prime }_{-1}$ by mutating our initial quiver (triangulation), $Q$ , at 3 and then at 4, as shown in Figure REF .", "In general $J^{\\prime }_n$ will be given by applying $\\mu _4\\mu _3$ to $\\mu ^{n+1}(Q)$ .", "From this we see that $J^1_{j+l-2}=J^{\\prime }_{j+l-2}$ so (REF ) becomes (REF ).", "This linear relation is identical to the one in the $\\tilde{A}$ case, so the determinant and frieze construction are the same as in Section .", "Figure: Applying μ 3 \\mu _3 then μ 4 \\mu _4 to the initial triangulation gives the arcs J -1 ' J^{\\prime }_{-1}.Figure: The arcs J ' i l-1 {J^{\\prime }}^{l-1}_{i}.We have now found all the arcs in the $\\tilde{D}$ case, except for the three only involving the punctures, $\\Gamma _{\\mathrm {except}}$ .", "We collect these results in the following theorem.", "Theorem 5.5 The $\\tilde{D}$ type cluster variables are given by $\\left\\lbrace X^i_n\\:|\\:\\begin{aligned}&i=1,\\ldots ,N+1 \\\\ &n\\in \\mathbb {Z}\\end{aligned}\\right\\rbrace \\cup \\left\\lbrace D^l_1(J^{\\prime }_{j})\\:|\\:\\begin{aligned}j=0,\\ldots ,N-3 \\\\ l=1,\\ldots ,N-3\\end{aligned}\\right\\rbrace \\cup \\Gamma _{\\mathrm {except}}$ where the $X^i_n$ are obtained by the cluster map (REF ) and the ${J^{\\prime }}^{l}_i$ are defined in Proposition REF .", "The three arcs of $\\Gamma _{\\mathrm {except}}$ are shown in (REF ).", "We have now proven the $\\tilde{D}$ part of Theorem REF and Proposition REF .", "In the next section we complete our results for $\\tilde{D}$ type with a proof of Proposition REF ." ], [ "$\\tilde{D}$ type cluster friezes", "In $\\tilde{D}$ type there are 3 exceptional tubes, for $\\lambda =1,0,\\infty $ .", "whose periods are $N-2,2,2$ respectively.", "We have not constructed friezes of width two, only one of width $N-2$ , so our goal is to prove that our friezes agree with the friezes of [1] away from the width two tubes.", "To do this we prove that the relation (REF ), holds for $\\lambda =1$ Firstly we follow [1] to find the cluster variables associated with $B_1$ and $B^{\\prime }_1$ .", "The vertex $e$ is required to be a sink, so we first mutate Figure REF at 1 and 2, so the quiver near $e=2$ is $\\begin{tikzcd}X_{-1}^1 \\\\& X_0^3[ul][dl] & X_0^4[l] \\\\X_{-1}^2\\end{tikzcd}$ We have $B_1=P_3/P_1$ and $B^{\\prime }_1=P_1[1]$ with $X_{P_1[1]}=X^1_{-1}$ and $X_{S_2}=X^2_0$ .", "Performing $\\mu _3\\mu _2$ gives $\\begin{tikzcd}X_{-1}^1[dr] \\\\& (X_0^3)^{\\prime }[dl][r] & X_0^4[ull] \\\\X_{0}^2[uu]\\end{tikzcd}$ where $(X_0^3)^{\\prime }=\\frac{X^2_0X^4_0+X^1_{-1}}{X^3_0}=\\frac{1+X^3_0+X^2_{-1}X^1_{-1}}{X^2_{-1}X^3_0}$ is the cluster variable associated with $B_1$ .", "We can express this as $(X_0^3)^{\\prime }=\\frac{X^1_0X^2_0X^4_0+X^1_{-1}X^1_0}{X^1_0X^3_0}=\\frac{X^3_1X^3_0-1+X^3_0+1}{X^1_0X^3_0}=\\frac{X^3_1+1}{X^1_0}=X^1_1$ so we have $X_{N_{0}}=\\frac{X_{B_1}+X_{B^{\\prime }_1}}{X_{S_1}}=\\frac{X^1_1+X^1_{-1}}{X^2_0}=J^{\\prime }_0$ which gives $X_{N_{0}[j]}=\\frac{X_{B_1[j]}+X_{B^{\\prime }_1[j]}}{X_{S_1[j]}}=\\frac{X^1_{1-j}+X^1_{-1-j}}{X^2_{-j}}=J^{\\prime }_{-j}$ as desired." ] ]
2105.11682
[ [ "Self-Supervised Graph Representation Learning via Topology\n Transformations" ], [ "Abstract We present the Topology Transformation Equivariant Representation learning, a general paradigm of self-supervised learning for node representations of graph data to enable the wide applicability of Graph Convolutional Neural Networks (GCNNs).", "We formalize the proposed model from an information-theoretic perspective, by maximizing the mutual information between topology transformations and node representations before and after the transformations.", "We derive that maximizing such mutual information can be relaxed to minimizing the cross entropy between the applied topology transformation and its estimation from node representations.", "In particular, we seek to sample a subset of node pairs from the original graph and flip the edge connectivity between each pair to transform the graph topology.", "Then, we self-train a representation encoder to learn node representations by reconstructing the topology transformations from the feature representations of the original and transformed graphs.", "In experiments, we apply the proposed model to the downstream node classification, graph classification and link prediction tasks, and results show that the proposed method outperforms the state-of-the-art unsupervised approaches." ], [ "Introduction", "Graphs provide a natural and efficient representation for non-Euclidean data, such as brain networks, social networks, citation networks, and 3D point clouds.", "Graph Convolutional Neural Networks (GCNNs) [1] have been proposed to generalize the CNNs to learn representations from non-Euclidean data, which has made significant advances in various applications such as node classification [2], [3], [4] and graph classification [5].", "However, most existing GCNNs are trained in a supervised fashion, requiring a large amount of labeled data for network training.", "This limits the applications of the GCNNs since it is often costly to collect adequately labeled data, especially on large-scale graphs.", "Hence, self-supervised learning is required to learn graph feature representations by exploring the dependencies of unlabeled data in an unsupervised fashion, which enables the discovery of intrinsic graph structures and thus adapts to various downstream tasks.", "Various attempts have been made to explore self-supervisory signals for representation learning.", "The self-supervised learning framework requires only unlabeled data in order to design a pretext learning task, where the target objective is optimized without any supervision [6].", "Self-supervised learning models can be categorized into three classes [7]: generative, adversarial, and contrastive.", "Generative models are often based on Auto-regressive models [8], [9], [10], flow-based models [11], [12], and Auto-Encoding (AE) models [13], [14], [15] to generate or reconstruct data from latent representations.", "Adversarial models extract feature representations in an unsupervised fashion by generating data from input noises via a pair of generator and discriminator [16], [17].", "Contrastive models aim to train an encoder to be contrastive between the representations of positive samples and negative samples [18], [19], [20], [21].", "Recently, many approaches have sought to learn transformation equivariant representations (TERs) to further improve the quality of unsupervised representation learning.", "It assumes that the learned representations equivarying to transformations are able to encode the intrinsic structures of data such that the transformations can be reconstructed from the representations before and after transformations [22].", "Learning TERs traces back to Hinton's seminal work on learning transformation capsules [23], and embodies a variety of methods developed for Euclidean data [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34].", "Further, Gao et al.", "[35] extend transformation equivariant representation learning to non-Euclidean domain, which formalizes Graph Transformation Equivariant Representation (GraphTER) learning by auto-encoding node-wise transformations in an unsupervised fashion.", "Nevertheless, only transformations on node features are explored, while the underlying graph may vary implicitly.", "The graph topology has not been fully explored yet, which however is crucial in graph representation learning.", "To this end, we propose a self-supervised Topology Transformation Equivariant Representation learning to infer expressive graph feature representations by estimating topology transformations.", "Instead of transforming node features as in the GraphTER, the proposed method studies the transformation equivariant representation learning by transforming the graph topology, i.e., adding or removing edges to perturb the graph structure.", "Then the same input signals are attached to the resultant graph topologies, resulting in different graph representations.", "This provides an insight into how the same input signals associated with different graph topologies would lead to equivariant representations, enabling the fusion of node feature and graph topology in GCNNs.", "Formally, we formulate the proposed model from an information-theoretic perspective, aiming to maximize the mutual information between topology transformations and feature representations with respect to the original and transformed graphs.", "We derive that maximizing such mutual information can be relaxed to the cross entropy minimization between the applied topology transformations and the estimation from the learned representations of graph data under the topological transformations.", "Specifically, given an input graph and its associated node features, we first sample a subset of node pairs from the graph and flip the edge connectivity between each pair at a perturbation rate, leading to a transformed graph with attached node features.", "Then, we design a graph-convolutional auto-encoder architecture, where the encoder learns the node-wise representations over the original and transformed graphs respectively, and the decoder predicts the topology transformations of edge connectivity from both representations by minimizing the cross entropy between the applied and estimated transformations.", "Experimental results demonstrate that the proposed method outperforms the state-of-the-art unsupervised models, and even achieves comparable results to the (semi-)supervised approaches in node classification and graph classification tasks at times.", "The proposed method distinguishes from our previous work GraphTER [35] mainly in two aspects.", "1) We formulate our model from an information-theoretic perspective by maximizing the mutual information between representations and transformations, which provides a theoretical derivation for the training objective and generalizes transformations to more general forms.", "In contrast, GraphTER directly minimizes the MSE between the estimated and ground-truth transformations, which lacks theoretical explanation and is limited to parametric transformations; 2) We explicitly exploit transformations in the graph topology, which is crucial in graph representation learning and explores how the same input signals associated with different graph topologies would lead to equivariant representations, thus enabling deeper fusion of node features and the graph topology in GCNNs.", "In contrast, GraphTER focuses on learning equivariant representations of nodes under node-wise transformations.", "Our main contributions are summarized as follows.", "We propose a self-supervised paradigm of the Topology Transformation Equivariant Representation learning to infer expressive node feature representations, which characterizes the intrinsic structures of graphs and the associated features by exploring the graph transformations of connectivity topology.", "We formulate the Topology Transformation Equivariant Representation learning from an information-theoretic perspective, by maximizing the mutual information between feature representations and topology transformations, which is proved to relax to the cross entropy minimization between the applied transformations and the prediction in an end-to-end graph-convolutional auto-encoder architecture.", "Experiments demonstrate that the proposed method outperforms the state-of-the-art unsupervised methods in both node classification and graph classification.", "The remainder of this paper is organized as follows.", "We first review related works in Sec. .", "Then we formalize our model in Sec.", "and present the algorithm in Sec. .", "Finally, experimental results and conclusions are presented in Sec.", "and Sec.", ", respectively." ], [ "Related Work", "We review previous works on relevant unsupervised/self-supervised feature representation learning, including graph auto-encoders, graph generative models, graph contrastive learning, as well as transformation equivariant representation learning." ], [ "Graph Auto-Encoders", "Graph Auto-Encoders (GAEs) are the most representative unsupervised methods.", "GAEs encode graph data into feature space via an encoder and reconstruct the input graph data from the encoded feature representations via a decoder.", "Kipf et al.", "[15] first integrate the GCN [2] into an auto-encoder framework to learn graph representations in an unsupervised manner by reconstructing the adjacency matrix.", "Variational GAE (VGAE) [15] is a variational version of GAE to learn the distribution of data.", "Cao et al.", "[36] proposed to employ the stacked denoising auto-encoder [37] to reconstruct the positive pointwise mutual information (PPMI) matrix to capture the correlation of node pairs.", "Wang et al.", "[38] employ the stacked auto-encoders to preserve the first-order proximity and the second-order proximity of nodes jointly." ], [ "Graph Generative Networks", "Graph Generative Networks aim to learn the generative distribution of graphs by encoding graphs into hidden representations and generate graph structures given hidden representations.", "The graph generative networks can be classified into two categories [39]: sequential approaches and global approaches.", "Sequential approaches generate nodes and edges step by step.", "Deep Generative Model of Graphs (DeepGMG) [40] assumes that the probability of a graph is the sum over all possible node permutations, and generates graphs by making a sequence of decisions.", "You et al.", "[10] proposed GraphRNN model to generate nodes from a graph-level RNN and edges from an edge-level RNN.", "Global approaches generate an entire graph at once.", "Molecular GAN (MolGAN) [41] combines Relational Graph Convolutional Networks (R-GCNs) [42], GANs [43], and reinforcement learning objectives to generate graphs with desired properties.", "NetGAN [44] combines LSTMs [45] with the Wasserstein GANs [46] to generate graphs from a random-walk-based approach." ], [ "Graph Contrastive Learning", "An important paradigm called contrastive learning aims to train an encoder to be contrastive between the representations of positive samples and negative samples [18], [47], [48], [49], [50].", "Recent contrastive learning frameworks for graph data can be divided into two categories [7]: context-instance contrast and context-context contrast.", "Context-instance contrast focuses on modeling the relationships between the local feature of a sample and its global context representation.", "Deep InfoMax (DIM) [18] first maximizes the mutual information between a local patch and its global context through a contrastive learning task.", "Deep Graph InfoMax (DGI) [19] extends DIM to graph-structured data to learn node-level feature representations.", "Sun et al.", "[21] proposed an InfoGraph model to maximize the mutual information between the representations of entire graphs and the representations of substructures of different granularity.", "Peng et al.", "[20] proposed a Graphical Mutual Information (GMI) approach to maximize the mutual information of both features and edges between inputs and outputs.", "Compared with context-instance methods, context-context contrast studies the relationships between the global representations of different samples.", "Caron et al.", "[51] proposed a Deep Cluster approach to cluster encoded representations and produces pseudo labels for each sample, and then predicts whether two samples are from the same cluster.", "Sun et al.", "[52] adopts a self-supervised pre-training paradigm as in DeepCluster [51] for better semi-supervised prediction in GCNNs.", "Qiu et al.", "[53] designs the pre-training task as subgraph instance discrimination in and across networks to empower graph neural networks to learn intrinsic structural representations." ], [ "Transformation Equivariant Representations", "Many approaches have sought to learn transformation equivariant representations, which has been advocated in Hinton's seminal work on learning transformation capsules [23].", "Following this, a variety of approaches have been proposed to learn transformation equivariant representations [28], [30], [31], [54], [55].", "To generalize to generic transformations, Zhang et al.", "[32] proposed to learn unsupervised feature representations via Auto-Encoding Transformations (AET) by estimating transformations from the learned feature representations of both the original and transformed images, while Qi et al.", "[33] extend AET from an information-theoretic perspective by maximizing the lower bound of mutual information between transformations and representations.", "Wang et al.", "[34] extend the AET to Generative Adversarial Networks (GANs) for unsupervised image synthesis and representation learning.", "Gao et al.", "[35] introduce the GraphTER model that extends AET to graph-structured data, which is formalized by auto-encoding node-wise transformations in an unsupervised manner.", "De et al.", "[56] proposed Gauge Equivariant Mesh CNNs which generalize GCNNs to apply anisotropic gauge equivariant kernels.", "Fuchs et al.", "[57] introduce a self-attention mechanism specifically for 3D point cloud data, which adheres to equivariance constraints, improving robustness to nuisance transformations.", "Haan et al.", "[58] proposed a Natural Graph Network (NGN) that can be used to describe maximally flexible global and local equivariance.", "Satorras et al.", "[59] present an E(n) equivariant graph neural network that is translation, rotation and reflection equivariant.", "Gao et al.", "[60] proposed to learn multi-view representations by decoding the 3D transformations of 3D objects from multiple 2D views." ], [ "The Proposed Formulation", "In this section, we first introduce the preliminaries in Sec.", "REF , and define the topology transformation in Sec.", "REF .", "Then we formulate the proposed method in Sec.", "REF .", "Further, some analysis of the proposed model are presented in Sec.", "REF ." ], [ "Preliminary", "We consider an undirected graph $\\mathcal {G}=\\lbrace \\mathcal {V},\\mathcal {E},\\mathbf {A}\\rbrace $ composed of a node set $\\mathcal {V}$ of cardinality $|\\mathcal {V}|=N$ , an edge set $\\mathcal {E}$ connecting nodes of cardinality $|\\mathcal {E}|=M$ .", "$\\mathbf {A}$ is a real symmetric $N \\times N$ matrix that encodes the graph structure, where $a_{i,j}=1$ if there exists an edge $(i,j)$ between nodes $i$ and $j$ , and $a_{i,j}=0$ otherwise.", "Graph signal refers to data that reside on the nodes of a graph $\\mathcal {G}$ , denoted by $\\mathbf {X} \\in \\mathbb {R}^{N \\times C}$ with the $i$ -th row representing the $C$ -dimensional graph signal on the $i$ -th node of $\\mathcal {V}$ ." ], [ "Topology Transformation", "We define the topology transformation $\\mathbf {t}$ as adding or removing edges from the original edge set $\\mathcal {E}$ in graph $\\mathcal {G}$ .", "This can be done by sampling, i.i.d., a switch parameter $\\sigma _{i,j}$ as in [19], which determines whether to modify edge $(i,j)$ in the adjacency matrix.", "Assuming a Bernoulli distribution $\\mathcal {B}(p)$ , where $p$ denotes the probability of each edge being modified, we draw a random matrix $\\Sigma =\\left\\lbrace \\sigma _{i,j}\\right\\rbrace _{N \\times N}$ from $\\mathcal {B}(p)$ , i.e., $\\Sigma \\sim \\mathcal {B}(p)$ .", "We then acquire the perturbed adjacency matrix as $\\widetilde{\\mathbf {A}}= {\\mathbf {A}}\\oplus \\Sigma ,$ where $\\oplus $ is the exclusive OR (XOR) operation.", "This strategy produces a transformed graph through the topology transformation $\\mathbf {t}$ , i.e., $\\widetilde{\\mathbf {A}}= \\mathbf {t}({\\mathbf {A}})$ .", "Here, the edge perturbation probability of $p=0$ corresponds to a non-transformed adjacency matrix, which is a special case of an identity transformation to ${\\mathbf {A}}$ .", "The transformed adjacency matrix $\\widetilde{\\mathbf {A}}$ can also be written as the sum of the original adjacency matrix ${\\mathbf {A}}$ and a topology perturbation matrix $\\Delta \\mathbf {A}$ : $\\widetilde{\\mathbf {A}}= {\\mathbf {A}}+ \\Delta \\mathbf {A},$ where $\\Delta \\mathbf {A}= \\lbrace \\delta a_{i,j}\\rbrace _{N \\times N}$ encodes the perturbation of edges, with $\\delta a_{i,j} \\in \\lbrace -1, 0, 1\\rbrace $ .", "As shown in Fig.", "REF , when $\\delta a_{i,j}=0$ , the edge between node $i$ and node $j$ keeps unchanged (i.e., black solid lines); when $\\delta a_{i,j} = -1$ or 1, it means removing (i.e., orange dotted lines) or adding (i.e., blue solid lines) the edge between node $i$ and node $j$ , respectively." ], [ "The Formulation", "Definition 1 Given a pair of graph signal and adjacency matrix $({\\mathbf {X}},{\\mathbf {A}})$ , and a pair of graph signal and transformed adjacency matrix $({\\mathbf {X}},\\widetilde{\\mathbf {A}})$ by a topology transformation $\\mathbf {t}(\\cdot )$ , a function $E(\\cdot )$ is transformation equivariant if it satisfies $E({\\mathbf {X}},\\widetilde{\\mathbf {A}})=E\\left({\\mathbf {X}},\\mathbf {t}({\\mathbf {A}})\\right)=\\rho (\\mathbf {t})\\left[E({\\mathbf {X}},{\\mathbf {A}})\\right],$ where $\\rho (\\mathbf {t})[\\cdot ]$ is a homomorphism of transformation $\\mathbf {t}$ in the representation space.", "Let us denote $ {\\mathbf {H}}=E({\\mathbf {X}},{\\mathbf {A}}), \\; \\text{and} \\; \\widetilde{\\mathbf {H}}=E({\\mathbf {X}},\\widetilde{\\mathbf {A}})$ .", "We seek to learn an encoder $E: ({\\mathbf {X}},{\\mathbf {A}}) \\mapsto {\\mathbf {H}};({\\mathbf {X}},\\widetilde{\\mathbf {A}}) \\mapsto \\widetilde{\\mathbf {H}}$ that maps both the original and transformed sample to representations $\\lbrace {\\mathbf {H}},\\widetilde{\\mathbf {H}}\\rbrace $ equivariant to the sampled transformation $\\mathbf {t}$ , whose information can thus be inferred from the representations via a decoder $D: (\\widetilde{\\mathbf {H}},{\\mathbf {H}}) \\mapsto \\widehat{\\Delta \\mathbf {A}}$ as much as possible.", "From an information-theoretic perspective, this requires $({\\mathbf {H}},\\Delta \\mathbf {A})$ should jointly contain all necessary information about $\\widetilde{\\mathbf {H}}$ .", "Then a natural choice to formalize the topology transformation equivariance is the mutual information $I({\\mathbf {H}},\\Delta \\mathbf {A};\\widetilde{\\mathbf {H}})$ between $({\\mathbf {H}},\\Delta \\mathbf {A})$ and $\\widetilde{\\mathbf {H}}$ .", "The larger the mutual information is, the more knowledge about $\\Delta \\mathbf {A}$ can be inferred from the representations $\\lbrace {\\mathbf {H}},\\widetilde{\\mathbf {H}}\\rbrace $ .", "Hence, we propose to maximize the mutual information to learn the topology transformation equivariant representations as follows: $\\max _{\\theta } \\; I({\\mathbf {H}},\\Delta \\mathbf {A};\\widetilde{\\mathbf {H}}),$ where $\\theta $ denotes the parameters of the auto-encoder network.", "Nevertheless, it is difficult to compute the mutual information directly.", "Instead, we derive that maximizing the mutual information can be relaxed to minimizing the cross entropy, as described in the following theorem.", "Theorem 1 The maximization of the mutual information $I({\\mathbf {H}},\\Delta \\mathbf {A};\\widetilde{\\mathbf {H}})$ can be relaxed to the minimization of the cross entropy $H(p \\; \\Vert \\; q)$ between the probability distributions $p(\\Delta \\mathbf {A},\\widetilde{\\mathbf {H}},{\\mathbf {H}})$ and $q(\\widehat{\\Delta \\mathbf {A}}|\\widetilde{\\mathbf {H}},{\\mathbf {H}})$ : $\\begin{split}\\min _{\\theta } \\; & H\\left(p(\\Delta \\mathbf {A},\\widetilde{\\mathbf {H}},{\\mathbf {H}}) \\; \\Vert \\; q(\\widehat{\\Delta \\mathbf {A}}|\\widetilde{\\mathbf {H}},{\\mathbf {H}})\\right) \\\\\\end{split}$ Proof    By using the chain rule of mutual information, we have $\\nonumber I({\\mathbf {H}},\\Delta \\mathbf {A};\\widetilde{\\mathbf {H}}) = I(\\Delta \\mathbf {A};\\widetilde{\\mathbf {H}}|{\\mathbf {H}})+I({\\mathbf {H}};\\widetilde{\\mathbf {H}}) \\ge I(\\Delta \\mathbf {A};\\widetilde{\\mathbf {H}}|{\\mathbf {H}}).$ Thus the mutual information $I(\\Delta \\mathbf {A};\\widetilde{\\mathbf {H}}|{\\mathbf {H}})$ is the lower bound of the mutual information $I({\\mathbf {H}},\\Delta \\mathbf {A};\\widetilde{\\mathbf {H}})$ that attains its minimum value when $I({\\mathbf {H}};\\widetilde{\\mathbf {H}})=0$ .", "Therefore, we relax the objective to maximizing the lower bound mutual information $I(\\Delta \\mathbf {A};\\widetilde{\\mathbf {H}}|{\\mathbf {H}})$ between the transformed representation $\\widetilde{\\mathbf {H}}$ and the topology transformation $\\Delta \\mathbf {A}$ : $\\nonumber I(\\Delta \\mathbf {A};\\widetilde{\\mathbf {H}}|{\\mathbf {H}}) = H(\\Delta \\mathbf {A}|{\\mathbf {H}}) - H(\\Delta \\mathbf {A}|\\widetilde{\\mathbf {H}},{\\mathbf {H}}),$ where $H(\\cdot )$ denotes the conditional entropy.", "Since $\\Delta \\mathbf {A}$ and ${\\mathbf {H}}$ are independent, we have $H(\\Delta \\mathbf {A}|{\\mathbf {H}})=H(\\Delta \\mathbf {A})$ .", "Hence, maximizing $I(\\Delta \\mathbf {A};\\widetilde{\\mathbf {H}}|{\\mathbf {H}})$ becomes $\\min _{\\theta } \\; H(\\Delta \\mathbf {A}|\\widetilde{\\mathbf {H}},{\\mathbf {H}}).$ According to the chain rule of conditional entropy, we have $\\nonumber \\begin{split}H(\\Delta \\mathbf {A}|\\widetilde{\\mathbf {H}},{\\mathbf {H}})&=H(\\Delta \\mathbf {A},\\widetilde{\\mathbf {H}},{\\mathbf {H}})-H(\\widetilde{\\mathbf {H}},{\\mathbf {H}}) \\\\&\\le H(\\Delta \\mathbf {A},\\widetilde{\\mathbf {H}},{\\mathbf {H}}),\\end{split}$ where the conditional entropy $H(\\Delta \\mathbf {A}|\\widetilde{\\mathbf {H}},{\\mathbf {H}})$ is upper bounded by the joint entropy $H(\\Delta \\mathbf {A},\\widetilde{\\mathbf {H}},{\\mathbf {H}})$ .", "Thus, the minimization problem in Eq.", "(REF ) becomes $\\min _{\\theta } \\; H(\\Delta \\mathbf {A},\\widetilde{\\mathbf {H}},{\\mathbf {H}}).$ We next introduce a conditional probability distribution $q(\\widehat{\\Delta \\mathbf {A}}|\\widetilde{\\mathbf {H}},{\\mathbf {H}})$ to approximate the intractable posterior $\\tilde{q}(\\Delta \\mathbf {A}|\\widetilde{\\mathbf {H}},{\\mathbf {H}})$ with an estimated $\\widehat{\\Delta \\mathbf {A}}$ .", "According to the definition of the Kullback-Leibler divergence, we have $\\nonumber \\begin{split}H(\\Delta \\mathbf {A},\\widetilde{\\mathbf {H}},{\\mathbf {H}}) & = H(p) = H(p \\; \\Vert \\; q) -D_\\text{KL}(p \\; \\Vert \\; q) \\\\& \\le H(p \\; \\Vert \\; q),\\end{split}$ where $D_\\text{KL}(p \\; \\Vert \\; q)$ denotes the Kullback-Leibler divergence of $p$ and $q$ that is non-negative, and $H(p \\; \\Vert \\; q)$ is the cross entropy between $p$ and $q$ .", "Thus, Eq.", "(REF ) is converted to minimizing the cross entropy as the upper bound: $\\nonumber \\begin{split}\\min _{\\theta } \\; & H\\left(p(\\Delta \\mathbf {A},\\widetilde{\\mathbf {H}},{\\mathbf {H}}) \\; \\Vert \\; q(\\widehat{\\Delta \\mathbf {A}}|\\widetilde{\\mathbf {H}},{\\mathbf {H}})\\right) \\\\\\end{split}$ Hence, we relax the maximization problem in Eq.", "(REF ) to the optimization in Eq.", "(REF ).", "$\\square $ Based on Theorem 1, we train the decoder $D$ to learn the distribution $q(\\widehat{\\Delta \\mathbf {A}}|\\widetilde{\\mathbf {H}},{\\mathbf {H}})$ so as to estimate the topology transformation $\\widehat{\\Delta \\mathbf {A}}$ from the encoded $\\lbrace \\widetilde{\\mathbf {H}},{\\mathbf {H}}\\rbrace $ , where the input pairs of original and transformed graph representations $\\lbrace \\widetilde{\\mathbf {H}},{\\mathbf {H}}\\rbrace $ as well as the ground truth target $\\Delta \\mathbf {A}$ can be sampled tractably from the factorization of $p(\\Delta \\mathbf {A},\\widetilde{\\mathbf {H}},{\\mathbf {H}})\\triangleq p(\\Delta \\mathbf {A})p({\\mathbf {H}})p(\\widetilde{\\mathbf {H}}|\\Delta \\mathbf {A},{\\mathbf {H}})$ .", "This allows us to minimize the cross entropy between $p(\\Delta \\mathbf {A},\\widetilde{\\mathbf {H}},{\\mathbf {H}})$ and $q(\\widehat{\\Delta \\mathbf {A}}|\\widetilde{\\mathbf {H}},{\\mathbf {H}})$ as in (REF ) with the training triplets $(\\widetilde{\\mathbf {H}},{\\mathbf {H}};\\Delta \\mathbf {A})$ drawn from the tractable factorization of $p(\\Delta \\mathbf {A},\\widetilde{\\mathbf {H}},{\\mathbf {H}})$ .", "Hence, we formulate the Topology Transformation Equivariant Representation learning as the joint optimization of the representation encoder $E$ and the transformation decoder $D$ ." ], [ "Analysis", "Since we learn the topology transformation equivariant representations by maximizing the mutual information between representations and transformations as discussed in Sec.", "REF , the proposed model approximately learns equivariant representations for graph data as defined in Eq.", "(REF ) via the optimization in Eq.", "(REF ).", "This distinguishes from relevant attempts on equivariance for graph data [57], [56], [58], [59], which design exact equivariant kernels for graph data.", "Though equivariance is explicitly satisfied in [57], [56], [58], [59], the equivariant kernels depend on the modalities of the input data, and thus are not generalizable to various tasks.", "This is because they exploit representative information in different data modalities instead of their commonalities, e.g., 3D coordinates of point clouds [57] and the angles of two neighboring points in meshes [56].", "For instance, Fuchs et al.", "[57] proposed an SE(3)-equivariant convolutional network by constructing an attention-based SE(3)-Transformer specifically for 3D point clouds.", "Haan et al.", "[56] presented an anisotropic gauge equivariant kernel for mesh data, which is applied into graph convolutional networks and results in equivalent outputs regardless of the arbitrary choice of kernel orientation.", "Also, Haan et al.", "[58] proposed Natural Graph Networks for isomorphic graphs that are equivariant to node permutations.", "Further, Satorras et al.", "[59] came up with a new graph convolution kernel, which makes inputs equivariant to parameterized orthogonal transformations (e.g., rotations, translations and reflections) and permutations for data such as molecules.", "In all, these methods design equivariant network kernels tailored for specific graph data, which have been applied in supervised graph representation learning.", "In contrast, we propose an unsupervised model that generalizes to different downstream tasks of various graph data, without restrictions to transformations or types of graph data, while providing good approximations of equivariant representations via the proposed effective optimization.", "Figure: The architecture of the proposed model.Table: Node classification accuracies (with standard deviation) in percentage on three datasets.𝐗,𝐀,𝐘{\\mathbf {X}},{\\mathbf {A}},{\\mathbf {Y}} denote the input data, adjacency matrix and labels respectively.We design a graph-convolutional auto-encoder network for the proposed model, as illustrated in Fig.", "REF .", "Given a graph signal ${\\mathbf {X}}$ associated with a graph $\\mathcal {G}=\\lbrace \\mathcal {V},\\mathcal {E},\\mathbf {A}\\rbrace $ , the proposed unsupervised learning algorithm consists of three steps: 1) topology transformation, which samples and perturbs some edges from all node pairs to acquire a transformed adjacency matrix $\\widetilde{\\mathbf {A}}$ ; 2) representation encoding, which extracts the feature representations of graph signals before and after the topology transformation; 3) transformation decoding, which estimates the topology transformation parameters from the learned feature representations.", "We elaborate on the three steps as follows." ], [ "Topology Transformation", "We randomly sample a subset of node pairs from all node pairs for topology perturbation—adding or removing edges, which not only enables to characterize local graph structures at various scales, but also reduces the number of edge transformation parameters to estimate for computational efficiency.", "In practice, in each iteration of training, we sample all the node pairs with connected edges ${\\mathbf {S}}_1$ (i.e., ${\\mathbf {S}}_1=\\mathcal {E}$ ), and randomly sample a subset of disconnected node pairs ${\\mathbf {S}}_0$ , i.e., ${\\mathbf {S}}_0=\\left\\lbrace (i,j) \\big | a_{i,j}=0 \\right\\rbrace , {\\mathbf {S}}_1=\\left\\lbrace (i,j) \\big | a_{i,j}=1 \\right\\rbrace ,$ where $|{\\mathbf {S}}_0|=|{\\mathbf {S}}_1|=M$ .", "Next, we randomly split ${\\mathbf {S}}_0$ and ${\\mathbf {S}}_1$ into two disjoint sets, respectively, i.e., $\\begin{split}{\\mathbf {S}}_i=\\bigg \\lbrace {\\mathbf {S}}_i^{(1)},{\\mathbf {S}}_i^{(2)} \\; \\big | \\; & {\\mathbf {S}}_i^{(1)} \\cap {\\mathbf {S}}_i^{(2)} = \\varnothing , {\\mathbf {S}}_i^{(1)} \\cup {\\mathbf {S}}_i^{(2)} = {\\mathbf {S}}_i, \\\\& |{\\mathbf {S}}_i^{(1)}|=r \\cdot |{\\mathbf {S}}_i| \\bigg \\rbrace , i \\in \\lbrace 0,1\\rbrace ,\\end{split}$ where $r$ is the edge perturbation rate.", "Then, for each node pair $(i,j)$ in ${\\mathbf {S}}_0^{(1)}$ and ${\\mathbf {S}}_1^{(1)}$ , we flip the corresponding entry in the original graph adjacency matrix.", "That is, if $a_{i,j}=0$ , then we set $\\tilde{a}_{i,j}=1$ ; otherwise, we set $\\tilde{a}_{i,j}=0$ .", "For each node pair $(i,j)$ in ${\\mathbf {S}}_0^{(2)}$ and ${\\mathbf {S}}_1^{(2)}$ , we keep the original connectivities unchanged, i.e., $\\tilde{a}_{i,j}=a_{i,j}$ .", "This leads to the transformed adjacency matrix $\\widetilde{\\mathbf {A}}$ , as well as the sampled transformation parameters by accessing $\\Delta \\mathbf {A}$ at position $(i,j)$ from ${\\mathbf {S}}_0$ and ${\\mathbf {S}}_1$ .", "Also, we can category the sampled topology transformation parameters into four types: add an edge to a disconnected node pair, i.e., $\\lbrace \\mathbf {t}:a_{i,j}=0 \\mapsto \\tilde{a}_{i,j}=1, (i,j) \\in {\\mathbf {S}}_0^{(1)}\\rbrace $ ; delete the edge between a connected node pair, i.e., $\\lbrace \\mathbf {t}:a_{i,j}=1 \\mapsto \\tilde{a}_{i,j}=0, (i,j) \\in {\\mathbf {S}}_1^{(1)}\\rbrace $ ; keep the disconnection between node pairs in ${\\mathbf {S}}_0^{(2)}$ , i.e., $\\lbrace \\mathbf {t}:a_{i,j}=0 \\mapsto \\tilde{a}_{i,j}=0, (i,j) \\in {\\mathbf {S}}_0^{(2)}\\rbrace $ ; keep the connection between node pairs in ${\\mathbf {S}}_1^{(2)}$ , i.e., $\\lbrace \\mathbf {t}:a_{i,j}=1 \\mapsto \\tilde{a}_{i,j}=1, (i,j) \\in {\\mathbf {S}}_1^{(2)}\\rbrace $ .", "Thus, we cast the problem of estimating transformation parameters in $\\Delta \\mathbf {A}$ from $(\\widetilde{\\mathbf {H}},{\\mathbf {H}})$ as the classification problem of the transformation parameter types.", "The percentage of these four types is $r:r:(1-r):(1-r)$ ." ], [ "Representation Encoder", "We train an encoder $E: ({\\mathbf {X}},{\\mathbf {A}}) \\mapsto E({\\mathbf {X}},{\\mathbf {A}})$ to encode the feature representations of each node in the graph.", "As demonstrated in Fig.", "REF , we leverage GCNNs with shared weights to extract feature representations of each node in the graph signal.", "Taking the GCN [2] as an example, the graph convolution in the GCN is defined as ${\\mathbf {H}}=E({\\mathbf {X}},{\\mathbf {A}})={\\mathbf {D}}^{-\\frac{1}{2}}({\\mathbf {A}}+{\\mathbf {I}}){\\mathbf {D}}^{-\\frac{1}{2}}{\\mathbf {X}}{\\mathbf {W}},$ where ${\\mathbf {D}}$ is the degree matrix of ${\\mathbf {A}}+{\\mathbf {I}}$ , ${\\mathbf {W}}\\in \\mathbb {R}^{C \\times F}$ is a learnable parameter matrix, and ${\\mathbf {H}}= [{\\mathbf {h}}_1,...,{\\mathbf {h}}_N]^{\\top } \\in \\mathbb {R}^{N \\times F}$ denotes the node-wise feature matrix with $F$ output channels.", "From Eq.", "(REF ), we see that node-wise representations from GCN are updated in two steps: 1) feature propagation and aggregation, and 2) linear transformation.", "The first step aims to aggregate the features of each node $v_i$ and its local neighborhood, e.g., 1-hop neighborhood, $\\widehat{{\\mathbf {h}}}_i=\\frac{1}{d_i}{\\mathbf {x}}_i+\\sum _{i \\sim j}\\frac{a_{i,j}}{\\sqrt{d_i \\cdot d_j}}{\\mathbf {x}}_j,$ where $i \\sim j$ represents node $i$ and $j$ are connected, and $\\widehat{{\\mathbf {h}}}_i$ denotes the aggregated features of node $v_i$ .", "Eq.", "(REF ) can also be expressed over the entire graph by matrix multiplication, i.e., $\\widehat{{\\mathbf {H}}}={\\mathbf {D}}^{-\\frac{1}{2}}({\\mathbf {A}}+{\\mathbf {I}}){\\mathbf {D}}^{-\\frac{1}{2}}{\\mathbf {X}}$ .", "The second step performs a linear transformation with a learnable parameter matrix ${\\mathbf {W}}$ on the aggregated features $\\widehat{{\\mathbf {H}}}$ to generate node embeddings for the GCN layer, i.e., ${\\mathbf {H}}=\\widehat{{\\mathbf {H}}}{\\mathbf {W}}$ .", "Similarly, the node feature of the transformed counterpart is as follows with the shared weights ${\\mathbf {W}}$ .", "$\\begin{split}\\widetilde{\\mathbf {H}}& =E({\\mathbf {X}},\\widetilde{\\mathbf {A}}) =\\widetilde{\\mathbf {D}}^{-\\frac{1}{2}}(\\widetilde{\\mathbf {A}}+{\\mathbf {I}})\\widetilde{\\mathbf {D}}^{-\\frac{1}{2}}{\\mathbf {X}}{\\mathbf {W}}\\\\& =\\widetilde{\\mathbf {D}}^{-\\frac{1}{2}}({\\mathbf {A}}+{\\mathbf {I}})\\widetilde{\\mathbf {D}}^{-\\frac{1}{2}}{\\mathbf {X}}{\\mathbf {W}}+ \\widetilde{\\mathbf {D}}^{-\\frac{1}{2}}\\Delta \\mathbf {A}\\widetilde{\\mathbf {D}}^{-\\frac{1}{2}}{\\mathbf {X}}{\\mathbf {W}}.\\end{split}$ We thus acquire the feature representations ${\\mathbf {H}}$ and $\\widetilde{\\mathbf {H}}$ of graph signals before and after topology transformations." ], [ "Transformation Decoder", "Comparing Eq.", "(REF ) and Eq.", "(REF ), the prominent difference between $\\widetilde{\\mathbf {H}}$ and ${\\mathbf {H}}$ lies in the second term of Eq.", "(REF ) featuring $\\Delta \\mathbf {A}$ .", "This enables us to train a decoder $D: (\\widetilde{\\mathbf {H}},{\\mathbf {H}}) \\mapsto \\widehat{\\Delta \\mathbf {A}}$ to estimate the topology transformation from the joint representations before and after transformation.", "We first take the difference between the extracted feature representations before and after transformations along the feature channel, $\\Delta \\mathbf {H}= \\widetilde{\\mathbf {H}}- {\\mathbf {H}}= [\\delta \\mathbf {h}_1, ..., \\delta \\mathbf {h}_N]^{\\top } \\in \\mathbb {R}^{N \\times F}.$ Thus, we can predict the topology transformation between node $i$ and node $j$ through the node-wise feature difference $\\Delta \\mathbf {H}$ by constructing the edge representation as $\\begin{split}\\mathbf {e}_{i,j} = \\frac{\\exp \\lbrace -(\\delta \\mathbf {h}_i - \\delta \\mathbf {h}_j) \\odot (\\delta \\mathbf {h}_i - \\delta \\mathbf {h}_j)\\rbrace }{\\Vert \\exp \\lbrace -(\\delta \\mathbf {h}_i - \\delta \\mathbf {h}_j) \\odot (\\delta \\mathbf {h}_i - \\delta \\mathbf {h}_j)\\rbrace \\Vert _1} & , \\\\\\forall (i,j) \\in {\\mathbf {S}}_0 \\cup {\\mathbf {S}}_1 & ,\\end{split}$ where $\\odot $ denotes the Hadamard product of two vectors to capture the feature representation, and $\\Vert \\cdot \\Vert _1$ is the $\\ell _1$ -norm of a vector for normalization.", "The edge representation ${\\mathbf {e}}_{i,j}$ of node $i$ and $j$ is then fed into several linear layers for the prediction of the topology transformation, $\\widehat{\\mathbf {y}}_{i,j}=\\mathrm {softmax}\\left(\\mathrm {linear}({\\mathbf {e}}_{i,j})\\right), \\quad \\forall (i,j) \\in {\\mathbf {S}}_0 \\cup {\\mathbf {S}}_1,$ where $\\mathrm {softmax}(\\cdot )$ is an activation function.", "According to Eq.", "(REF ), the entire auto-encoder network is trained by minimizing the cross entropy $\\mathcal {L}=-\\underset{(i,j) \\in {\\mathbf {S}}_0 \\cup {\\mathbf {S}}_1}{\\mathbb {E}}\\sum _{f=1}^{4}\\mathbf {y}_{i,j}^{(f)} \\log \\widehat{\\mathbf {y}}_{i,j}^{(f)},$ where $f$ denotes the transformation type ($f \\in \\lbrace 1,2,3,4\\rbrace $ ), and $\\mathbf {y}$ is the ground-truth binary indicator (0 or 1) for each transformation parameter type." ], [ "Experiments", "In this section, we evaluate the proposed model on two representative downstream tasks: node classification and graph classification.", "Table: Graph classification accuracies (with standard deviation) in percentage on 6 datasets.“>1 Day\" represents that the computation exceeds 24 hours.", "“OOM\" is out of memory error." ], [ "Datasets", "We adopt three citation networks to evaluate our model: Cora, Citeseer, and Pubmed [66].", "The dataset statistics are reported in Tab.", "REF .", "The three datasets contain sparse bag-of-words feature vectors for each document and a list of citation links between documents.", "We treat documents as nodes, and the citation links as (undirected) edges, leading to a binary and symmetric adjacency matrix ${\\mathbf {A}}$ as in [2].", "We follow the standard train/test split in [2] to conduct the experiments, where the label rate denotes the number of labeled nodes that are used for training.", "Table: Dataset statistics of citation networks." ], [ "Implementation Details", "In this task, the auto-encoder network is trained via Adam optimizer, and the learning rate is set to $10^{-4}$ .", "We use the same early stopping strategy as DGI [19] on the observed training loss, with a patience of 20 epochs.", "We deploy one Simple Graph Convolution (SGC) layer [62] as our encoder, and the order of the adjacency matrix is set to 2.", "The LeakyReLU activation function with a negative slope of $0.1$ is employed after the SGC layer.", "Similar to DGI [19], we set the output channel $F=512$ for Cora and Citeseer dataset, and 256 for Pubmed dataset due to memory limitations.", "After the encoder, we use one linear layer to classify the transformation types.", "We set the edge perturbation rate in Eq.", "(REF ) as $r=\\lbrace 0.7, 0.4, 0.7\\rbrace $ for Cora, Citeseer, and Pubmed, respectively.", "During the training procedure of the classifier, the SGC layer in the encoder is used to extract graph feature representations with the weights frozen.", "After the SGC layer, we apply one linear layer to map the features to the classification scores." ], [ "Experimental Results", "We compare the proposed method with five unsupervised methods, including one node embedding method DeepWalk, two graph auto-encoders GAE and VGAE [15], and two contrastive learning methods DGI [19] and GMI [20].", "Additionally, we report the results of Raw Features and DeepWalk+Features [65] under the same settings.", "For fair comparison, the results of all other unsupervised methods are reproduced by using the same encoder architecture of the proposed method except DeepWalk and Raw Features.", "We report the mean classification accuracy (with standard deviation) on the test nodes for all methods after 50 runs of training.", "As reported in Tab.", "REF , the proposed method outperforms all other competing unsupervised methods on three datasets.", "Further, the proposed unsupervised method also achieves comparable performance with semi-supervised results.", "This significantly closes the gap between unsupervised approaches and the semi-supervised methods.", "Moreover, we compare the proposed method with two contrastive learning methods DGI and GMI in terms of the model complexity, as reported in Tab.", "REF .", "The number of parameters in our model is less than that of DGI and even less than half of that of GMI, which further shows the proposed model is lightweight.", "Table: Model size comparison among DGI, GMI, and Ours." ], [ "Experiments On Different Orders of The Adjacency Matrix", "As presented in Sec.", "REF , we perturb the 1-hop neighborhoods via the proposed topology transformations, leading to possibly significant changes in the graph topology.", "This increases the difficulties of predicting the topology transformations when using one-layer GCN [2] by aggregating the 1-hop neighborhood information.", "Therefore, we employ one Simple Graph Convolution (SGC) layer [62] with order $k$ as our encoder $E(\\cdot )$ , where the output feature representations aggregate multi-hop neighborhood information.", "Formally, the SGC layer is defined as ${\\mathbf {H}}=E({\\mathbf {X}},{\\mathbf {A}})=\\left({\\mathbf {D}}^{-\\frac{1}{2}}({\\mathbf {A}}+{\\mathbf {I}}){\\mathbf {D}}^{-\\frac{1}{2}}\\right)^{k}{\\mathbf {X}}{\\mathbf {W}},$ where ${\\mathbf {D}}$ is the degree matrix of ${\\mathbf {A}}+{\\mathbf {I}}$ , ${\\mathbf {W}}\\in \\mathbb {R}^{C \\times F}$ is a learnable parameter matrix, and $k$ is the order of the normalized adjacency matrix.", "To study the influence of different orders of the adjacency matrix, we adopt five orders from 1 to 5 to train five models on the node classification task.", "Fig.", "REF presents the node classification accuracy under different orders of the adjacency matrix for the proposed method and DGI respectively.", "As we can see, the proposed method achieves best classification performance when $k=\\lbrace 4,2,3\\rbrace $ on the three datasets respectively, and outperforms GAE in different orders.", "When $k=1$ , our model still achieves reasonable results although it is difficult to predict the topology transformations from 1-hop neighborhood information; when $k>1$ , our model outperforms DGI by a large margin on Cora and Pubmed dataset, and achieves comparable results to DGI on Citeseer dataset.", "This is because DGI adopts feature shuffling to generate negative samples, which is insufficient to learn contrastive feature representations when aggregating multi-hop neighborhood information, while the proposed method takes advantage of multi-hop neighborhood information to predict the topology transformations, leading to improved performance.", "Figure: Pubmed" ], [ "Robustness Test", "To evaluate the robustness of our model on the node classification task, we jitter the original node features with an additive noise model, namely, $\\widetilde{\\mathbf {X}}={\\mathbf {X}}+{\\mathbf {E}},$ where ${\\mathbf {X}}\\in \\mathbb {R}^{N\\times F}$ is the original node features, ${\\mathbf {E}}\\in \\mathbb {R}^{N\\times F}$ is a random matrix which is sampled from a random distribution (e.g., Gaussian or Laplace), and $\\widetilde{\\mathbf {X}}$ denotes the noise-corrupted node features.", "Specifically, we select the zero-mean Gaussian noise with a range of standard deviation $\\sigma $ from $0.01$ to $0.10$ at an interval of $0.01$ , as well as the zero-location Laplace noise with a range of scale parameter $s$ from $0.01$ to $0.10$ at an interval of $0.01$ , for extensive classification performance comparison.", "We employ one SGC layer as the encoder $E(\\cdot )$ of the proposed model and two representative self-supervised model DGI and GAE, where the order of the adjacency matrix is set to 1.", "We use the original node features ${\\mathbf {X}}$ to train the three models in the unsupervised training stage and the linear classifier in the supervised evaluation stage, and employ the corrupted node features $\\widetilde{\\mathbf {X}}$ to evaluate the classification accuracies.", "The classification performance under Gaussian and Laplace noises are presented in Fig.", "REF and Fig.", "REF , respectively.", "When the noise level is low, our model outperforms GAE by a large margin on the Cora and Citeseer datasets, and achieves comparable results to DGI on the three datasets.", "When the noise level is high, our model significantly outperforms GAE and DGI on the three datasets.", "This is because DGI takes the original graph topology to aggregate node features.", "When the original node features are seriously corrupted (the noise level is high), the aggregated features will change significantly.", "GAE reconstructs the adjacency matrix from the feature representations of individual nodes.", "The high dependency of the graph topology and node features leads to bad performance when the node features suffer from serious noise perturbations.", "In contrast, our model aims to predict the topology transformations from the feature representations of nodes before and after transformation, which not only employs the original graph topology information, but also explores how node features would change by applying a topology transformation, thus enhancing the robustness.", "Figure: PubmedFigure: Pubmed" ], [ "Experiments On Different Edge Perturbation Rates", "Further, we evaluate the influence of the edge perturbation rate in Eq.", "(REF ) on the node classification task.", "We choose 11 edge perturbation rates from $0.0$ to $1.0$ at an interval of $0.1$ to train the proposed model.", "We use one SGC layer as our encoder $E(\\cdot )$ , where the order of the adjacency matrix is set to 1.", "As presented in Fig.", "REF , the blue solid line with error bar shows the classification accuracy of our method under different edge perturbation rates.", "We also provide the classification accuracy on feature representations of graphs from a randomly initialized encoder $E(\\cdot )$ , denoted as Random Init., which serves as the lower bound of the performance.", "As we can see, the classification performance reaches the best when the graph is perturbed under a reasonable edge perturbation rate, e.g., $r=\\lbrace 0.6,0.5,0.6\\rbrace $ for the Cora, Citeseer, and Pubmed dataset, respectively.", "When the edge perturbation rate $r=0.0$ , the unsupervised training task of the proposed model becomes link prediction, which cannot take advantage of the proposed method by predicting the topology transformations; when the edge perturbation rate $r=1.0$ , our model still achieves reasonable classification results, which shows the stability of our model under high edge perturbation rates.", "At the same time, we observe that the proposed method outperforms Random Init.", "by a large margin, which validates the effectiveness of the proposed unsupervised training strategy.", "Figure: Pubmed" ], [ "Datasets", "We conduct graph classification on six well-known graph benchmark datasets [67], including two molecule datasets MUTAG and PTC, and four social network datasets REDDIT-BINARY, REDDIT-MULTI-5K, IMDB-BINARY, and IMDB-MULTI.", "In the two molecule datasets, graphs are molecules, where nodes represent atoms and edges represent chemical bonds, and the graph classification task is to classify the molecules.", "In the REDDIT dataset, a graph denotes a discussion thread, where nodes correspond to users, two of which are connected by an edge if one responded to a comment of the other.", "The graph classification task is to distinguish whether the subreddits is discussion-based or question/answer-based (REDDIT-BINARY), or predict the subreddit (REDDIT-MULTI-5K).", "The IMDB dataset consists of ego-networks derived from actor collaborations, and the graph classification task is to predict the genre, e.g., Action or Romance." ], [ "Implementation Details", "In this task, the entire network is trained via Adam optimizer with a batch size of 64, and the learning rate is set to $10^{-3}$ .", "For the encoder architecture, we follow the same encoder settings in the released code of InfoGraph [21], i.e., three Graph Isomorphism Network (GIN) layers [5] with batch normalization.", "We also use one linear layer to classify the transformation types.", "We set the sampling rate $r=0.5$ for all datasets.", "During the evaluation stage, the entire encoder will be frozen to extract node-level feature representations, which will go through a global add pooling layer to acquire global features.", "We then use LIBSVM to classify these global features to classification scores.", "We adopt the same procedure of previous works [21] to make a fair comparison and use 10-fold cross validation accuracy to report the classification performance, and the experiments are repeated five times." ], [ "Experimental Results", "We take six graph kernel approaches for comparison: Random Walk (RW) [68], Shortest Path Kernel (SP) [69], Graphlet Kernel (GK) [70], Weisfeiler-Lehman Sub-tree Kernel (WL) [71], Deep Graph Kernels (DGK) [67], and Multi-Scale Laplacian Kernel (MLG) [72].", "Aside from graph kernel methods, we also compare with three unsupervised graph-level representation learning methods: node2vec [73], sub2vec [74], and graph2vec [75], and one contrastive learning method: InfoGraph [21].", "The experimental results of unsupervised graph classification are preseted in Tab.", "REF .", "The proposed method outperforms all unsupervised baseline methods on the first five datasets, and achieves comparable results on the other dataset.", "Also, the proposed approach reaches the performance of supervised methods at times, thus validating the superiority of the proposed method." ], [ "Conclusion", "We propose a self-supervised paradigm of Topology Transformation Equivariant Representation for graph representation learning.", "By maximizing the mutual information between topology transformations and feature representations before and after transformations, the proposed method enforces the encoder to learn intrinsic graph feature representations that contain sufficient information about structures under applied topology transformations.", "We apply our model to node classification and graph classification tasks, and results demonstrate that the proposed method outperforms state-of-the-art unsupervised approaches and reaches the performance of supervised methods at times.", "We believe this model will have impact on applications such as social analysis, molecule property prediction, as well as applications in 3D computer vision." ] ]
2105.11689
[ [ "Dynamic analysis of influential stocks based on conserved networks" ], [ "Abstract Characterizing temporal evolution of stock markets is a fundamental and challenging problem.", "The literature on analyzing the dynamics of the markets has focused so far on macro measures with less predictive power.", "This paper addresses this issue from a micro point of view.", "Given an investigating period, a series of stock networks are constructed first by the moving-window method and the significance test of stock correlations.", "Then, several conserved networks are generated to extract different backbones of the market under different states.", "Finally, influential stocks and corresponding sectors are identified from each conserved network, based on which the longitudinal analysis is performed to describe the evolution of the market.", "The application of the above procedure to stocks belonging to Standard \\& Pool's 500 Index from January 2006 to April 2010 recovers the 2008 financial crisis from the evolutionary perspective." ], [ "Introduction", "Stock markets are well-defined complex systems consisting of multi heterogeneous stocks with complex relationships among them [1].", "The prices of the stocks evolve as a consequence of their internal and external interactions, and different assets present turbulent financial time series making the market behaviors even more difficult to be examined.", "Therefore, it is important to mine essential information from the market and build an efficient model to characterize its dynamic properties, which not only provides a fundamental understanding of financial systems but also provides practical insights for policymakers and practitioners [2].", "One seminal approach is the random matrix theory [3], [4], [5] which characterizes the eigenvalue distribution of the correlation coefficient matrix of time series of stocks and have unveiled many stylized facts of stock markets [6], [7], [8], [9].", "For instance, a stock market containing many business sectors (groups of stocks sharing common economic properties) with hierarchial organization [10].", "However, the random matrix could not draw interactions well among these sectors.", "To understand the market in a more exact way, the complex network theory [11] was adopted instead.", "Examples include the minimal spanning tree [12], the asset graph [13], the planar maximally filtered graph [14] and the threshold network [15].", "Based on these models, many topological characteristics have been observed for the markets such as New York Stock Exchange [16], [17], [18], [19], German Stock Exchange [20], Tokyo Stock Exchange [21], Hong Kong Stock Market [22] and Shanghai Stock Market [24].", "In general, a stock market evolves with economic states, resulting in sequential changes of stock prices from one state to another [23].", "To explain the development of the economic state from the perspective of the market, there is an increasing interest in characterizing temporal evolution of stock networks.", "Yet, most studies focused on the evolution of global topologies, such as the edge density, the average clustering coefficient and the average shortest path length, which only provides the macro topological information of the market corresponding to different states [19], [22].", "To get a deeper understanding of the dynamics of the market, it is essential to study the evolution of the market from a micro point of view [25], [26].", "Specially, how to identify influential stocks and evaluate their roles in the diffusion of microfinance is of great importance [27].", "There are two major approaches to identify influential stocks in the literature.", "The first approach is from the dynamical point of view.", "For instance, Wang et al.", "[28] and Benzaquen et al.", "[29] suggested the cross-response (impact) function to characterize the influence of a stock.", "That is, the stock with strong cross-response has large influence on other stocks, resulting in the rank of stocks.", "The second approach is from the structural point of view.", "Specially, the concept of centrality has been widely used to rank nodal influence.", "For example, Roy and Sarkar [30] employed the degree centrality to rank stocks.", "They compared top 10 influential stocks corresponding to pre- and post-crisis and observed that the change in ranks of top 3 influential stocks are relatively low compared to those ranked lower.", "Nevertheless, a subtle analysis remains demanding.", "The goal of this paper is to identify most influential stocks of a stock market from the evolutionary perspective such that it can recover a financial crisis efficiently.", "To this end, we first construct a series of threshold networks for stocks in an investigating period.", "Then, we consider different stages of the crisis and build a conserved network for each stage.", "Finally, we identify influential stocks and corresponding sectors to describe crisis propagation.", "To test its efficacy, we apply our framework to stocks belonging to Standard & Pool's (S&P) 500 Index from January 2006 to April 2010 and recovers the 2008 financial crisis in an evolutionary way." ], [ "Methodology", "Section REF introduces the Pearson correlation coefficient, the P-threshold method with multiple hypothesis testing and the moving-window method to construct the dynamic sequence of stock networks.", "Section REF presents conserved networks associated with different stages of a financial crisis.", "Section REF introduces four typical centralities to measure nodal influence.", "Section REF presents the order statistic to synthesize centralities of nodes to rank their influence." ], [ "Stock networks", "Let $p_i(\\tau )$ $(i=1,2,\\cdots ,N; \\tau =1,2,\\cdots ,M)$ be the daily closing price of stock $i$ at time $\\tau $ , one obtains the logarithmic return of $i$ over a time interval $\\Delta \\tau $ by $r_i(\\tau )=\\ln p_i(\\tau )-\\ln p_i(\\tau -\\Delta \\tau ).$ In this paper we set $\\Delta \\tau =1$ , so $r_i(\\tau )$ represents the daily return of stock $i$ at time $\\tau $ .", "Then, the correlation coefficient between stocks $i$ and $j$ is defined by $w_{ij}=\\frac{\\langle r_i r_j\\rangle - \\langle r_i\\rangle \\langle r_j\\rangle }{\\sigma _i \\sigma _j},$ where $\\langle r_i \\rangle =\\sum _{\\tau =1}^{M} r_i(\\tau )/M$ is the mean and $\\sigma _i=\\sqrt{\\sum _{\\tau =1}^{M}\\left[ r_i-\\langle r_i \\rangle \\right]^2/M}$ is the standard deviation.", "The ensemble of $w_{ij}$ forms the correlation matrix ${W}$ of a stock market in a window of width $M$ .", "To filter $w_{ij}$ , we use the P-threshold method and set the following hypothesis test [22], $H_0: & w_{ij}=0,\\\\H_1: & w_{ij}\\ne 0.$ The corresponding test statistic is $T_{ij}=w_{ij}\\sqrt{\\frac{n-2}{1-w_{ij}^2}} \\sim t_{n-2},$ where $n$ is the sample size and $n-2$ is the degree of freedom.", "Given a significance level $\\alpha $ , one should reject $H_0$ if the absolute value of the test statistic exceeds the cut-off value $t_{\\alpha /2}(n-2)$ , namely, if $|T_{ij}|>t_{\\alpha /2}(n-2).$ To maximize the number of discoveries while controlling the fraction of false discoveries, we perform the multiple hypothesis test based on the Bonferroni correction.", "Specially, for the significance level of $\\alpha =0.01$ , we include any interactions between stocks if $|T_{ij}|>t_{\\alpha /N(N-1)}(n-2)$ .", "Since the Bonferroni correction assumes complete independence between the tested p-values, one may consider further the False Discovery Rate (FDR) approach [31] to relax the assumption of independence.", "As a consequence, more edges among stocks will be maintained.", "For smoothing purpose, we adopt the moving-window method [32].", "Assuming the width of each window is $M$ and the sliding interval is $\\Delta M$ , one can obtain a series of windows overlap with each other for any oberving period with proper choices of $M$ and $\\Delta M$ .", "Inside each window, an edge is created between a pair of stocks $i$ and $j$ if $w_{ij}\\ne 0$ .", "This process is repeated throughout all the elements of the correlation matrix and finally a stock network is generated.", "Specially, we assume the network $G(V,E)$ is unweighted and undirected, which can be described by an adjacency matrix ${A}=(a_{uv})_{N \\times N}$ with elements $a_{uv}=\\biggl \\lbrace \\begin{array}{ll}1, & \\quad \\mbox{if $u$ and $v$ are connected,}\\\\0, & \\quad \\mbox{otherwise.", "}\\end{array}$" ], [ "Conserved networks", "A typical stock market usually experiences various financial situations, including bull and bear runs, business as usual and financial crises.", "Of great importance is to delve into reliable indicators of the crisis from the market.", "However, the 2008 financial crisis has highlighted the main limitations of standard models, as they cannot detect the crisis even by using posterior data [33].", "Here, we address this issue by means of conserved networks.", "According to different states associated with a crisis, we divide the whole investigating period into 5 stages: the normal stage before the crisis, the stage of the transition from the normal state to the crisis, the stage during the crisis, the stage of the transition from the crisis to the normal state and the stage after the crisis.", "Inside any stage, there are a number $K$ of consecutive windows overlap with each other, based on which $K$ stock networks are constructed.", "Furthermore, we assume that the interactions between significant stocks will persist while the interactions between insignificant stocks will vary with time, which leads to the idea of conserved networks: for any pair of stocks, an edge between them in the corresponding conserved network exists if and only if all these $K$ networks within the stage have this edge.", "As a consequence, we obtain 5 conserved networks, which characterize dynamic characteristics of the investigating period." ], [ "Centrality measures", "The most influential stocks may help us understand risk propagation in a stock market and design corresponding control measures.", "To represent nodal influence in each conserved network, we adopt the concept of centrality.", "In this paper, we consider four typical measures: degree centrality (DC), eigenvector centrality (EC), closeness centrality (CC) and betweenness centrality (BC).", "Given a stock network $G(V,E)$ , the DC of node $u \\in V$ is define by [34] $\\texttt {DC}(u)=k_{u},$ where $k_{u}=\\sum _{v=1}^{|V|}a_{uv}$ is the degree of node $u$ and $|V|$ is the number of nodes.", "Although the DC is the simplest centrality measure, it can be illuminating.", "In stock markets, it seems reasonable to suppose that stocks with connections to many others might have more access to information than those with fewer connections.", "A natural extension of the DC is EC, defined as [35] $\\texttt {EC}(u)={\\upsilon }_u^{\\texttt {max}},$ where ${\\upsilon }^{\\texttt {max}}$ is the eigenvector corresponding to the largest eigenvalue of the adjacency matrix ${A}$ and ${\\upsilon }_u^{\\texttt {max}}$ is the $u$ th element of ${\\upsilon }^{\\texttt {max}}$ corresponding to stock $u$ .", "As a consequence, the stock $u$ with larger $\\texttt {EC}(u)$ can be important because it has many neighbors (even though those stocks may not be important themselves) or because it has important neighbors of high degrees.", "Both the DC and EC only consider local information of a network.", "Regarding global information, some entirely different measures of centrality have been suggested incorporating shortest path lengths.", "One is CC, defined as [34] $\\texttt {CC}(u)=\\frac{|V|-1}{\\sum _{v \\in V}l(u,v)},$ where $l(u,v)$ is the shortest path length from node $u$ to node $v$ .", "According to Eq.", "(REF ), the smaller average distance from the stock $u$ to others, the larger value of $\\texttt {CC}(u)$ it has.", "BC is another different concept of centrality, initially proposed by Bavelas [36] and generalized by Freeman [37], $\\texttt {BC}(u)=\\frac{\\eta (s,u,t)}{\\sum _{s\\ne u\\ne t}\\eta (s,t)},$ where $\\eta (s,u,t)$ is the number of those shortest paths passing through $u$ and $\\eta (s,t)$ is the total number of the shortest paths from node $s$ to node $t$ .", "In contrast to the CC, the BC measures the extent to which a stock lies on paths between other stocks." ], [ "Order statistic", "Different centrality measures yield different ranks of nodal influence.", "To synthesize multiple centralities, we regard each rank as an order statistic [38] and obtain a Q-statistic from the joint cumulative distribution of the $n$ -dimensional order statistic: $Q(\\gamma _1,\\gamma _2,\\cdots ,\\gamma _n)=n!\\int _0^{\\gamma _1}\\int _{q_1}^{\\gamma _2}\\cdots \\int _{q_{n-1}}^{\\gamma _n}\\texttt {d}q_n\\texttt {d}q_{n-1}\\cdots \\texttt {d}q_1,$ where $\\gamma _i$ is the rank ratio for centrality $i$ and $q_{i}$ is the lower bound of the $(i+1)$ th order statistic.", "In present work, we use 4 centrality measures, hence $n=4$ .", "In fact, the above integration can be calculated in a fast way $Q(\\gamma _1,\\gamma _2,\\cdots ,\\gamma _n)=n!Q_n$ with $Q_k=\\sum _{i=1}^k(-1)^{i-1}Q_{k-i}\\gamma _{k-i+1}^i/i!$ .", "The larger value of $Q$ , the greater influence the node has." ], [ "Application to S&P 500 stocks", "To validate the above framework, we apply it to stocks belonging to S&P 500 Index.", "The data are daily records and the investigating period ranges from January 2006 to April 2010, yielding 1089 observations of 422 stocks.", "During the period, there was an economic crisis: the 2008 financial crisis.", "Figure: (Color online) Temporal evolution of the normalized DC, EC, BC and CC.", "The average correlation coefficient 〈w〉\\langle w\\rangle is presented for comparison." ], [ "Dynamic networks", "First of all, we divide the investigated period through moving windows.", "By setting the width of each window as $M=125$ (about half a year) and the moving step as $\\Delta M=5$ (about a week), we obtain 193 windows.", "For each window, we fix the significance level at $\\alpha =0.01$ , based on which a correlation network is constructed.", "As a result, we obtain 193 consecutive networks for the whole period.", "Then, we calculate the centrality of each node, the average of which is used to stand for the characteristics of the network.", "For each network, we consider the DC, EC, CC and BC, respectively.", "Figure 1 shows the evolution of four centralities in comparison to the evolution of the average correlation coefficient $\\langle w\\rangle $ .", "For comparison, we normalize each plot by the corresponding maximum of the investigating period.", "However, it is not essential.", "The dark gray interval corresponds to the middle stage of the crisis and the two light gray intervals respectively correspond to the early transition from the normal state to the crisis and the late transition from the crisis to the normal state.", "Remarkably, the DC, EC and CC display the same trend with $\\langle w\\rangle $ , while the BC evolves in the opposite way.", "According to Eq.", "(REF ), one has $\\langle \\texttt {DC}\\rangle =\\sum _{u\\in V}k_u/|V|=2|E|/|V|=(|V|-1)e$ , where $|E|$ is the number of edges and $e=2|E|/|V|/(|V|-1)$ is the density of edges.", "Moreover, the edge density is proportional to $\\langle w\\rangle $ .", "Therefore, both the DC and EC exhibit the same trend as $\\langle w\\rangle $ .", "As to the CC and BC (see Eqs.", "(10) and (11)), the higher density of edges, the smaller distance that a node reach all the others, hence the larger value of the CC.", "On the contrary, the number of connected pairs of nodes increases, resulting in the decrease of the BC.", "Overall, the four measures can serve as good indicators of the market evolution from the macro perspective.", "Table: Basic statistics of 5 conserved networks based on the Bonferroni correction." ], [ "Conserved networks", "Considering the 2008 financial crisis, we divide the period from January 2006 to April 2010 into 5 stages: the normal stage before the crisis, the stage of the transition from the normal state to the crisis, the stage during the crisis, the stage of the transition from the crisis to the normal state and the stage after the crisis.", "For each stage, we generate a conserved network.", "Table REF shows the basic characteristics of the 5 conserved networks, including the number of stocks $N$ , the number of edges $E$ , the average degree $\\langle k\\rangle $ , the average clustering coefficient $\\langle c\\rangle $ and the average shortest path length $\\langle l\\rangle $ .", "One notices apparent differences between these networks.", "For instance, the conserved network in the crisis is relatively dense because of the higher systemic risk.", "As a result, $\\langle k\\rangle $ and $\\langle c\\rangle $ are larger while $\\langle l\\rangle $ is smaller.", "But these topologies only provide macro information of the market development.", "Table: Top 10 influential stocks of the conserved network corresponding to the normal stage before the 2008 crisis." ], [ "Influential stocks", "The 2008 financial crisis is due to the U.S. financial problem of subprime mortgages.", "Mortgage is a loan taken from bank to buy a house, which is an agreement between homebuyers and banks.", "In general, people with low credit and low income can not get a loan from retail banks.", "But since year 2000's, a third party got involved, namely investment banks.", "These investment banks started buying mortgage agreements from the retail banks.", "In this way, the retail banks sold the loans to investment banks to have zero liability and the mortgages bought by the investment banks were used to form Mortgage Backed Security.", "Although the Mortgage Backed Security is a special category of subprime mortgages with higher risk, it yields higher return at the same time.", "As more people were becoming eligible for the mortgage, the demand for homes started increasing.", "More people had money (borrowed) to buy a new home.", "The price of residential properties only went up, hence creating the bubble.", "Table REF lists 10 most influential stocks identified from the conserved network corresponding to the normal stage before the crisis (from June 30th 2006 to September 25th 2007).", "As is expected, 6 stocks are from the real estate sector.", "Table: Top 10 influential stocks of the conserved network corresponding to the early transition from the normal state to the crisis.Investment banks made the loans were issued to people who had little capability to payback the loan.", "Inevitably, some of them eventually could not afford the monthly payments and their property went for foreclosure.", "At a point in 2007-2008, there were more houses on sale than there were buyers for it.", "This triggered a steady price fall.", "The housing bubble burst.", "When property prices started going down, people who had bought the property with the sole purpose of “buying low and selling high”stopped paying the mortgage.", "This led to more loan defaults and more foreclosures.", "As a result, the share price of the Mortgage Backed Security started to fall continuously and eventually started to affect on big investors that could not cover this urgency.", "So the crisis came into being.", "Table REF lists 10 most influential stocks identified from the conserved network corresponding to the early transition from the normal state to the crisis (from October 2nd 2007 to Marc 26th 2008), among which 6 stocks are financials, indicating the first shock of the crisis.", "Table: Top 10 influential stocks of the conserved network corresponding to the stage during the crisis.After the burst of the bubble in the housing market, many investment banks had more liabilities than assets and faced a big trouble of liquidity.", "For example, the New Century Financial Corporation filed for Chapter 11 bankruptcy protection in April 2008 because of repurchase agreements and the Lehman Brothers announced bankrupt in September 2008 due to asset deterioration.", "In addition to investment banks, many insurance companies and financial institutions have also been greatly impacted.", "A conspicuous example is the American International Group, which lost 250 billion dollars in the second quarter of 2008 and was taken over by the U.S. government finally.", "Soon after the earthquake in the financial market, the real economy was also shocked.", "On the one hand, the depression of the housing market caused correlated companies to fold.", "One the other hand, the rising unemployment and the shrink of personal wealth decreased consuming intention for industrial products.", "Table REF lists 10 most influential stocks identified from the conserved network corresponding to the stage during the crisis (from April 2nd 2008 to March 30th 2009).", "We find that not only financial institutions but also correlated industrial companies were involved with enormous losses.", "Table: Top 10 influential stocks of the conserved network corresponding to the late transition from the crisis to the normal state.To contain the crisis, the Troubled Assets Relief Program was carried out in October 2008, which authorized the United States Treasury to spend up to 700 billion dollars to purchase trouble assets both domestically and internationally.", "The act was widely credited with restoring stability and liquidity to the financial sector, unfreezing the markets for credit and capital and lowering borrowing costs for households and businesses.", "This, in turn, helped restore confidence in the financial system and restart economic growth.", "Another dose of fiscal stimulus is monetary easing.", "The lower interest rate spurred businesses to make new investments, spurred industrials to invest in renovations and spurred purchases of major durable goods like cars.", "Table REF lists 10 most influential stocks identified from the conserved network corresponding to the late transition from the crisis to the normal state (from April 6th 2009 to September 18th 2009), among which 8 stocks are industrials, implying the initial recovery.", "Table: Top 10 influential stocks of the conserved network corresponding to the normal state after the crisis.In fact, the speed of the recovery from the 2008 financial crisis has been unusually slow.", "Nevertheless, under the percolation of the stimulating policy of quantitative easing, the U.S. economy began to recover since the middle of 2009.", "With the reduction of the systematic risk and the rising opportunity for business, the stock market boomed again.", "Table REF lists 10 most influential stocks identified from the conserved network corresponding to the stage after the crisis (from September 25th 2009 to April 26th 2010), suggesting that the market is active across various sectors, including financials, industrials, materials, real estate and energy." ], [ "Discussion", "We have synthesized the DC, EC, CC and BC by means of the Q-statistic.", "Although it does rank stocks in each stage, the values of $Q$ are relative close.", "Therefore, is is nature to ask do top 10 influential stocks vary and how much is the variance if the procedure to construct the network is changed?", "To answer this question, we first consider the change in the P-value.", "We perform simulations for $\\alpha =0.02$ and shown the corresponding results in Appendix.", "Comparing Tables REF and REF , we find that all the basic characteristics of the generated networks are of the same order, indicating similar structure.", "We also compare top 10 influential stocks in each stage as $\\alpha $ increases from $0.01$ to $0.02$ .", "Specially, the stock AVB belonging to Real Estate (Table REF ) replaces the stock PH belonging to Industrials (Table REF ) in Stage 1; the stock GPC belonging to Consumer Discretionary (Table REF ) replaces the stock WY belonging to Real Estate (Table REF ) in Stage 2; the stock EFX belonging to Industrials (Table REF ) replaces the stock DD belonging to Materials (Table REF ) in Stage 3; the stock OKE belonging to Energy (Table REF ) replaces the stock TROW belonging to Financials (Table REF ) in Stage 4; and two stocks MRO and FTI belonging to Energy (Table REF ) replace stocks UNM and AFL belonging to Financials (Table REF ) in Stage 5.", "Overall, we notice tiny change in top 10 influential stocks in each stage, hence the robustness of our framework.", "Table: Basic statistics of 5 conserved networks based on the FDR correction.Then, we adopt the FDR for the multiple hypothesis test to relax the assumption of independence of the Bonferroni correction.", "As shown in Table REF , the number of edges of the 5 conserved networks are 3 to 10 times of those in Table REF .", "As a consequence, the networks are much dense.", "For example, conserved network 4, corresponding to the late transition from the crisis to the normal state, takes the value $1.2318$ of the average shortest path length.", "Intuitively, it is not the case of reality.", "In the stock market, the correlation matrix ${W}$ always contain much noise, and therefore a restrictive procedure may perform better.", "Table: Top 10 influential stocks identified by the PageRank algorithm from the conserved network corresponding to the normal stage before the 2008 crisis.Finally, we employ the Google's PageRank algorithm [39] to identify influential stocks.", "As a paradigmatic example of centrality-based ranking algorithm, PageRank has been found application in a vast range of real systems, although it was devised originally to rank web pages.", "In Table REF , we show the results via the PageRank algorithm for the conserved network corresponding to the normal stage before the 2008 crisis.", "As aforesaid, stocks belonging to Real Estate should be more influential in this stage.", "However, none of the top 10 influential stocks resulting from PageRank fall within that sector.", "We also observe contradictory results in other stages (not shown here)." ], [ "Conclusion", "The study of structure and dynamics of stock markets has attracted much attention from economists, mathematicians and physicists.", "An increasing interest is constructing reliable stock networks and analyzing their evolution [40], [41].", "Most approaches, however, have focused on the macro characteristics of the networks with less predictive power.", "In this paper, we have addressed this issue from the micro point of view by characterizing spreading influence of each stock and identifying most influential stocks in a dynamic way.", "For this purpose, we first divided the investigating period of a stock market into a large number of windows through the moving-window method.", "Then, we constructed a stock network for each window by the significance test of stock correlations.", "Furthermore, we generated several conserved networks to extract various backbones of the market under different stages.", "Finally, we used order statistics to rank nodal influence and identified most influential stocks for each conserved network.", "To illustrate its efficacy, we have applied this procedure to stocks belonging to S&P 500 Index from January 2006 to April 2010 and constructed 5 conserved networks, based on which we identified various influential stocks under different stages and recovered the 2008 financial crisis from the evolutionary perspective.", "We note, however, the stock market is too complex to be predicted.", "The present framework could be generalized by incorporating more physical and structural properties of the market.", "The comprehensive investigation of the market dynamics from both global and local aspects will be subjected to future research." ], [ "Acknowledgments", "We are grateful to referees for their valuable comments.", "This work was supported by the Natural Science Foundation of China under Grant Nos.", "12071281 and 11771277." ] ]
2105.11630
[ [ "On arithmetic functions orthogonal to deterministic sequences" ], [ "Abstract We prove Veech's conjecture on the equivalence of Sarnak's conjecture on M\\\"obius orthogonality with a Kolmogorov type property of Furstenberg systems of the M\\''obius function.", "This yields a combinatorial condition on the M\\\"obius function itself which is equivalent to Sarnak's conjecture.", "As a matter of fact, our arguments remain valid in a larger context: we characterize all bounded arithmetic functions orthogonal to all topological systems whose all ergodic measures yield systems from a fixed characteristic class (zero entropy class is an example of such a characteristic class) with the characterization persisting in the logarithmic setup.", "As a corollary, we obtain that the logarithmic Sarnak's conjecture holds if and only if the logarithmic M\\''obius orthogonality is satisfied for all dynamical systems whose ergodic measures yield nilsystems." ], [ "Introduction", "All transformations in this paper are assumed to be invertible.", "A topological dynamical system is a pair $(X,T)$ , where $T$ is a homeomorphism of a compact metric space $X$ .", "A measure-theoretic dynamical system is a system of the form $(Z,{\\cal B}(Z),\\kappa ,R)$ where $R$ is an automorphism (invertible measure-preserving transformation) of a standard Borel probability space $(Z,{\\cal B}(Z),\\kappa )$ ." ], [ "Sarnak's conjecture", "Given a topological dynamical system $(X,T)$ and a bounded arithmetic function $u\\colon {\\mathbb {N}}\\rightarrow , we consider the corresponding problem of orthogonality:\\begin{equation}\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n\\le N}f(T^nx)u(n)=0~\\mbox{ for all $f\\in C(X)$ and $x\\in X$.", "}\\end{equation}Once(\\ref {ort1}) holds for $ (X,T)$, we say that the system $ (X,T)$ {\\em satisfies the Sarnak property with respect to u} and write $ u(X,T)$.When a class $ C$ of topological dynamical systems is given, and $ u(X,T)$ for each $ (X,T)C$ then we write $ uC$.$ The main motivation to study this orthogonality problem is Sarnak's conjecture [46] on Möbius orthogonality in which $u$ is the Möbius function $\\mu $ (or, equivalently [15], the Liouville function $\\lambda $ ) and ${C}$ is the class ${C}_{\\rm ZE}$ of zero topological entropy dynamical systems.", "Sequences of the form $(f(T^nx))$ with $(X,T)$ running over ${C}_{\\rm ZE}$ , $f\\in C(X)$ and $x\\in X$ are often called deterministic sequences.", "Focusing on a special $u$ in (), say, being multiplicative, is important if we count on applications in number theory – the main motivation of Sarnak himself for the Möbius orthogonality conjecture was to “attack” the celebrated Chowla conjecture on auto-correlations of the Möbius function dynamically (indeed the Chowla conjecture implies Sarnak's conjecture [46], see also [49], [2]).On a potential equivalence of the Chowla and Sarnak's conjectures see [46], [49], [50], [51], [26] and a resumé of that in the survey articles [15], [37].", "However, if we aim at providing an internal characterization of $u$ being orthogonal to ${C}_{\\rm ZE}$ , then dropping the assumption of multiplicativity of $u$ seems to be reasonable.", "Let us give an argument for that.", "First, note that the class of deterministic sequences equipped with the coordinatewise multiplication and addition is a ring (this is an easy consequence of the fact that the zero entropy class is closed under taking joinings and factors).", "Moreover, the class of bounded sequences $u$ orthogonal to ${C}_{\\rm ZE}$ is a module over this ring.", "In other words, even if our “starting” $u$ displays some additional arithmetic property (like multiplicativity), the characterization which we aim at must still hold for $u\\cdot v$ , where $v$ is an arbitrary deterministic sequence.In what follows, we replace ${C}_{\\rm ZE}$ by ${C}_{{{F}}}$ , where ${{F}}$ is a general characteristic class.", "Note that the argument $u\\cdot v\\perp {C}_{{{F}}}$ whenever $u\\perp {C}_{{{F}}}$ and $v$ is an arbitrary ${{F}}$ -sequence (cf.", "Def.", "REF ) persists.", "Of course, properties like the multiplicativity of $u\\cdot v$ changes dramatically if $v$ is arbitrary, while the characterization we are looking for has to be stable under such a change of $u$ ." ], [ "Visible measures and Furstenberg systems", "Let us now briefly discuss the matter of the topological and measure-theoretic aspects of choosing a class ${C}$ .", "Recall that $M(X)$ stands for the set of probability (Borel) measures on $X$ and $M(X,T)$ is the (always non-empty) subset of $T$ -invariant measures.", "Both spaces are compact in the weak-$\\ast $ -topology.", "There is a third natural subspace $V(X,T)\\subset M(X,T)$ which is the set of ($T$ -invariant) visible measures, i.e.", "measures possessing a quasi-generic point.", "Formally, $\\nu \\in V(X,T)$ if, for some point $x\\in X$ and some increasing sequence $(N_{\\ell })$ , we have $\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{N_{\\ell }}\\sum _{n\\le N_{\\ell }}\\delta _{T^nx}=\\nu .\\footnote {Not all invariant measures, even in a transitive subshift, have to be visible: let X_y\\subset \\lbrace 0,1,2\\rbrace ^{{\\mathbb {N}}} be given by y=0^{k_1}1^{k_1}2^{k_1}1^{k_1} 0^{k_2}1^{k_2}2^{k_2}1^{k_2}\\dots with k_1<k_2<\\dots Let\\nu =\\frac{1}{2}\\delta _{\\underline{0}}+\\frac{1}{2}\\delta _{\\underline{2}} (two fixed points).", "If A:=\\lbrace x\\in X_y: x_0=0\\text{ or } x_0=2\\rbrace , then \\nu (A)=1 but no window in y has the property that the frequency of 0 jointly with 2 on it is close to the value~1.", "}$ We also write $\\nu \\in V_T(x)$ and say that $x$ is generic for $\\nu $ along $(N_{\\ell })$ .", "Note that the set $V(X,T)$ of visible measures contains the set $M^e(X,T)$ of all ergodic measures, but it may be strictly smaller than $M(X,T)$ .", "To illustrate this, consider the automorphism $(x,y)\\mapsto (x,x+y)$ of 2 in which each point is generic (i.e.", "generic along the whole sequence of natural numbers) and the corresponding measure-theoretic system is either a rational or an irrational rotation on the circle.", "Hence, each visible measure gives rise to a system with discrete spectrum, while there are many other invariant measures which yield partly continuous spectrum.", "Given a bounded $u\\colon {\\mathbb {N}}\\rightarrow , $ |u|L$, we first extend it to $ Z$ by setting $ u(-n)=u(n)$ for $ nN$, $ u(0)L:={z |z|L}$, and then take the closure$ Xu:={Sku:kZ}$ of the orbit of $ uLZ$ under the left shift $ SLZLZ$, $ S((vk)kZ)=(vk+1)kZ$ (recall that we only consider invertible systems).Hence, $ (Xu,S)$ is the {\\em subshift} determined by $ u$.", "By the very definition, $ uXu$.", "Each measure $ VS(u)$ yields a measure-theoretic system $ (Xu,B(Xu),,S)$ called a {\\em Furstenberg system} of $ u$ (by some abuse of vocabulary, we may call $$ itself a Furstenberg system of $ u$).", "Fix $ xX$ and consider the sequence $ (1NnN(Tnx,Snu))$.", "By compactness of $ M(XXu)$, we can select an increasing sequence $ (N)$ so that$$\\frac{1}{N_{\\ell }}\\sum _{n\\le N_{\\ell }}\\delta _{(T^nx,S^nu)}\\rightarrow \\rho ,$$where $ VTS((x,u)) V(XXu,TS)$.", "The projections of $$ on $ X$ and $ Xu$ are denoted by $$ and $$, respectively,so that $$ is a joining of $$ and $$.", "Since $$ is a visible measure, so are $$ and $$:\\begin{equation}\\nu \\in V_T(x),\\end{equation}and $$ yields a Furstenberg system of $ u$.Now, setting\\begin{equation}\\pi _0: {\\left\\lbrace \\begin{array}{ll}&X_{u}\\rightarrow &z=(z_n)_{n\\in {\\mathbb {Z}}}\\mapsto z_0,\\end{array}\\right.", "}\\end{equation}we have{\\begin{@align}{1}{-1}\\begin{split}\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{N_{\\ell }}\\sum _{n\\le N_{\\ell }}f(T^nx)u(n)&=\\lim _{\\ell \\rightarrow \\infty }\\int f\\otimes \\pi _0\\,d\\left(\\frac{1}{N_{\\ell }}\\sum _{n\\le N_{\\ell }} \\delta _{(T^nx,S^nu)}\\right)\\\\&=\\int f\\otimes \\pi _0\\,d\\rho =\\int {\\mathbb {E}}^\\rho (f|X_{u})\\cdot \\pi _0\\,d\\kappa .\\end{split}\\end{@align}}So, the fact that the limit is~0 in (\\ref {ort1}), i.e.\\ $ u(X,T)$, depends on the joinings of measures from $ V(X,T)$ with those from $ VS(u)$ -- the invariant measures from $ M(X,T)V(X,T)$ are irrelevant in this context.", "More precisely, what will matter is the ``geometric position^{\\prime \\prime } of the {\\bf single} continuous function $ 0$ in the $ L2$-space of all Furstenberg systems $ VS(u)$ for such joinings.", "Namely, inside the $ L2()$-space, we want $ 0$ to be orthogonal to $ L2()$ (in condition~(\\ref {three}) we use {\\bf all} continuous functions $ f$ on $ X$ and we take into account all $ V(X,T)$).Of course, whether this orthogonality can be established without referring to the (visible) joinings $$ (and remaining at the level of $ VS(u)$) is another question.", "A kind of surprise is that the answer to this question will turn out to be positive for $ CZE$ and some other characteristic classes, see Theorem~\\ref {tB}, below.$" ], [ "Characteristic classes and Veech's conjecture", "With the above in mind, a knowledge of some fundamental facts on joinings, especially on disjointness in the sense of Furstenberg in ergodic theory, strongly suggests to consider only the situation when the measures $\\nu $ appearing above yield measure-theoretic systems $(X,\\mathcal {B}(X),\\nu ,T)$ belonging to one of so called characteristic classes of measure-theoretic dynamical systems.", "Definition 1.1 A class ${{F}}$ of measure-theoretic dynamical systems is called characteristic if it is closed under taking isomorphisms, factors and (countable) joinings.", "Given a characteristic class ${{F}}$ , by an ${{F}}$ -factor of a measure-theoretic dynamical system $(Z,\\mathcal {D},\\kappa ,R)$ we call any factor sub-$\\sigma $ -algebra of $\\mathcal {D}$ on which the action of $R$ yields a system in the class ${{F}}$ .", "We denote by ${C}_{{{F}}}$ the class of topological systems $(X,T)$ for which we have $(X,\\mathcal {B}(X),\\nu ,T)\\in {{F}}$ for each $\\nu \\in V(X,T)$ .", "The sequences of the form $(f(T^nx))_{n\\in {\\mathbb {Z}}}$ with $f\\in C(X)$ , $x\\in X$ and $(X,T)\\in {C}_{{{F}}}$ are called ${{F}}$ -sequences.", "It follows from the above definition that: every measure-theoretic dynamical system $(Z,\\mathcal {D},\\kappa ,R)$ has a largest ${{F}}$ -factor (in the sense of inclusion of sub-$\\sigma $ -algebras), which we denote by $\\mathcal {D}_{{{F}}}$ , any joining of $(Z,\\mathcal {D},\\kappa ,R)$ with a system from ${{F}}$ is uniquely determined by its restriction to the joining of the largest ${{F}}$ -factor $\\mathcal {D}_{{{F}}}$ of $R$ with the given system from ${{F}}$ (see Section  for details).", "The class ZE of zero (Kolmogorov-Sinai) entropy is of course characteristic.", "The largest zero entropy factor $\\mathcal {D}_{\\rm ZE}$ of $(Z,\\mathcal {D},\\kappa ,R)$ is called the Pinsker factor and is denoted by $\\Pi (R)$ or $\\Pi (\\kappa )$ .", "Note that (via the variational principle) the family ${C}_{\\rm ZE}$ is precisely the family of all topological systems whose all invariant measures yield systems in ZE.", "Returning to Sarnak's conjecture, in [52], Veech proves the following result: Theorem 1.1 ([52]) If $\\pi _0\\perp L^2(\\Pi (\\kappa ))\\text{ for each Furstenberg system }\\kappa \\in V_S(\\mu )$ then Sarnak's conjecture holds, i.e.", "$\\mu \\perp {C}_{\\rm ZE}$ .An ergodic proof (which goes back to a suggestion of Sarnak in [46]) of this result is implicit in [2], where the implication “Chowla conjecture $\\Rightarrow $ Sarnak's conjecture” has been proved using joinings.", "Veech cites [2] (which was on arXiv two years before Veech's lecture notes [52] appeared) but instead he gives his own (slightly complicated) proof using the concept of quasi-factors of Glasner and Weiss.", "Then he formulates the following conjecture (Conjecture 24.3 page 88 in [52]): Conjecture 1 (Veech's conjecture) Condition (REF ) is equivalent to Sarnak's conjecture.", "One of motivations for the present work was to prove the above conjecture.", "Let us first formulate (REF ) in full generality.", "Definition 1.2 Given a characteristic class ${{F}}$ we say that a (bounded) arithmetic function $u\\colon {\\mathbb {N}}\\rightarrow satisfies the {\\em Veech condition} with respect to $ F$ if\\begin{equation}\\pi _0\\perp L^2((\\mathcal {B}(X_{u}),\\kappa )_{{{F}}})\\text{ for each Furstenberg system }\\kappa \\in V_S(u).\\end{equation}$ Given a characteristic class ${{F}}$ , we denote by ${{F}}_{\\rm ec}$ the class of those automorphisms $R$ of $(Z,{\\cal D},\\kappa )$ such that a.e.", "ergodic component of $\\kappa $ yields a system in ${{F}}$ .", "Then (see Section REF ), ${{F}}_{\\rm ec}$ is also a characteristic class, ${C}_{{{F}}}\\subset {C}_{{{F}}_{\\rm ec}}$ and ${C}_{{{F}}_{\\rm ec}}=\\lbrace (X,T):\\:(X,\\mathcal {B}(X),\\nu ,T)\\in {{F}}\\text{ for each {\\bf ergodic} } \\nu \\in M(X,T)\\rbrace .$ Our main result is the following: Theorem A Assume that ${{F}}$ is a characteristic class.", "Let $u\\colon {\\mathbb {N}}\\rightarrow be a bounded arithmetic function.", "Then $ uCFec$ if and only if $ u$ satisfies the Veech condition~(\\ref {veech3}) (with respect to $ Fec$).$ Remark 1.2 A subsequence version of this result also holds: If $(N_\\ell )$ is an increasing sequence of integers, then $u$ is $(N_\\ell )$ -orthogonal to ${C}_{{F}_{\\rm ec}}$ (i.e.", "$\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }f(T^nx)u(n)\\rightarrow 0$ for each $(X,T)\\in {C}_{{F}_{\\rm ec}}$ and all $f\\in C(X)$ , $x\\in X$ ) if and only if the Veech condition () holds for each measure $\\kappa $ for which $u$ is generic along some subsequence of $(N_\\ell )$ .", "The reason for the validity of this “local” version is that all tools used in the proof of Theorem REF work well on subsequences, see also Remarks REF and REF to cope with the strong $u$ -MOMO property along subsequences.", "See [8] for the validity of the alternative: either there are no Siegel zeros or there exists a (universal) subsequence along which Sarnak's conjecture holds and [27] for a density version of Sarnak's conjecture.", "Theorem REF for $u=\\mu $ and ${{F}}={\\rm ZE}$ (note that ${\\rm ZE}_{\\rm ec}={\\rm ZE}$ ) has the following consequence: Corollary 1.3 Veech's conjecture holds.", "Moreover, if Sarnak's conjecture holds then all Furstenberg systems $\\kappa \\in V_S(\\mu )$ have positive entropy.", "We also generalize Veech's theorem (Theorem REF ) to the setting of characteristic classes: Theorem B Assume that ${{F}}$ is a characteristic class.", "Let $u\\colon {\\mathbb {N}}\\rightarrow be a bounded arithmetic function.", "If $ u$ satisfies the Veech condition~(\\ref {veech3}) with respect to $ F$ then $ uCF$.$ Let us now briefly discuss the methods involved in the proofs of the above theorems.", "The proof of Theorem REF is provided in Section REF .", "It is a straightforward application of the above Property (b) of the largest ${{F}}$ -factor (see Proposition REF for more details).", "The proof of Theorem REF which we provide in Section REF relies on: so called Hansel models [29] of possibly highly non-ergodic measure-preserving systems, our version of a lifting lemma by Conze, Downarowicz, Serafin [9] on the existence of generic points for joinings, valid in a general context and using the concept of quasi-generic sequences (Proposition REF ), general joining techniques.", "Our first proof of Theorem REF was based on a different lifting lemma by Bergelson, Downarowicz and Vandehey (Theorem 5.16 in [6]) and a recent result by Downarowicz and Weiss showing the existence, for each zero-entropy measure-theoretic system, of a special Hansel model which is symbolic [13].", "The proof we finally chose to present here does not require the use of this symbolic model, as our lifting lemma works for all topological systems.", "Its additional advantage is that it is also valid in the context of logarithmic averages (see Section REF ).", "This makes all our results also true in the logarithmic set up: for example, the logarithmic Sarnak's conjecture (denoted as $\\mu \\perp _{\\rm log}{C}_{\\rm ZE}$ ) is equivalent to the Veech condition for $\\mu $ for all logarithmic Furstenberg systems $\\kappa \\in V_S^{\\rm log}(\\mu )$ , see Corollary REF for more.", "In Theorem REF , it is crucial that we deal with a characteristic class of the form ${{F}}_{\\rm ec}$ since, by Proposition REF , $u\\perp {C}_{{{F}}_{\\rm ec}}$ is equivalent to the so called strong $u$ -MOMO property for systems in ${C}_{{{F}}_{\\rm ec}}$ .", "This property, whose definition is recalled below, has been introduced in [3], [27].", "We leave as an open problem whether Theorem REF holds for an arbitrary characteristic class.", "Definition 1.3 A topological system $(X,T)$ satisfies the strong $u$ -MOMO property, if $\\lim _{K\\rightarrow \\infty }\\frac{1}{b_K}\\sum _{k<K}\\left\\Vert \\sum _{b_k\\le n<b_{k+1}}u(n)f\\circ T^n\\right\\Vert _{C(X)}=0$ for each $f\\in C(X)$ and each increasing sequence $(b_k)\\subset {\\mathbb {N}}$ such that $b_{k+1}-b_k\\rightarrow \\infty $ .", "Clearly, the strong $u$ -MOMO property implies () uniformly in $x\\in X$ .", "The concept of strong $u$ -MOMO is formally stronger than the usual orthogonality.", "To see the difference between the usual orthogonality and strong MOMO, consider the system $(x,y)\\mapsto (x,x+y)$ on 2 whose Möbius orhogonality follows easily from the DDKBSZ criterion,DDKBSZ stands for Daboussi-Delange-Kátai-Bourgain-Sarnak-Ziegler [7], [35].", "see e.g.", "[15]) (in fact, the orthogonality holds even uniformly due to the Davenport estimate [10]) , while the strong $\\mu $ -MOMO property (apply the definition to $f(x,y)=e^{2\\pi iy}$ ) yields $\\frac{1}{b_K}\\sum _{k<K}\\sup _{x\\in \\left|\\sum _{b_k\\le n<b_{k+1}}\\mu (n)e^{2\\pi inx}\\right|\\rightarrow 0when K\\rightarrow \\infty which is an open problem,\\footnote {In fact, it is open whether a non-periodic, zero entropy, continuous, algebraic automorphism of 2 satisfies the strong \\mu -MOMO property.}", "see Sections~\\ref {s:FSMOMO} and~\\ref {s:orthogonality} for more details.", "}$ Remark 1.4 Note that in the definition of the strong $u$ -MOMO property, convergence (REF ) can be replaced by $\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\left(\\sum _{k<K_{\\!N}}\\left\\Vert \\sum _{b_k\\le n<b_{k+1}}u(n)f\\circ T^n\\right\\Vert _{C(X)} \\right.", "\\\\\\left.", "+ \\left\\Vert \\sum _{b_{K_{\\!N}}\\le n < N} u(n)f\\circ T^n \\right\\Vert _{C(X)}\\right) =0,$ where $K_N:=\\max \\lbrace k:b_k<N\\rbrace $ .", "As a matter of fact, the definition is unchanged if we only restrict to sequences $(b_k)$ which further satisfy the condition $\\lim _{k\\rightarrow \\infty } \\frac{b_{k+1}-b_k}{b_k} = 0.$ (If we have to consider a sequence $(b_k)$ which does not satisfy the above condition, we can always add more integers to the set $\\lbrace b_k:k\\ge 1\\rbrace $ so that this convergence holds for the new sequence.", "And the validity of (REF ) or (REF ) for the new sequence is stronger than the same for the former sequence $(b_k)$ .)", "Then it is easy to define also the strong $u$ -MOMO property along an increasing sequence $(N_\\ell )$ , by restricting convergence (REF ) to the subsequence $(N_\\ell )$ ." ], [ "Veech condition and combinatorics", "Given a characteristic class ${{F}}$ , Theorems REF and REF determine a natural strategy to describe the arithmetic functions $u$ orthogonal to all ${{F}}$ -sequences.", "Namely, we need to describe the ${{F}}$ -factors and understand the orthogonality to their $L^2$ -space (i.e.", "the Veech condition), hoping that this description can be expressed (for $\\pi _0$ in $X_{u}$ and a Furstenberg system $\\kappa $ of $u$ ) by some asymptotics of the integrals of continuous functions on $X_{u}$ .", "The final step would be to use the definition of a Furstenberg system to obtain a combinatorial condition on $u$ itself.", "The first part of the strategy should be seen as an extension of the theory of characteristic factors $\\mathcal {Z}_s(T)$ (given an automorphism $T$ ) and the Gowers-Host-Kra (GHK in what follows) seminorms $\\Vert \\cdot \\Vert _{u^s}$ for $s\\ge 1$ [31].", "In this perspective the Veech condition on $\\pi _0$ is the counterpart of $\\Vert \\pi _0\\Vert _{u^s}=0$ for each Furstenberg system $\\kappa \\in V_S(u)$ (and has its combinatorial translation in terms of the GHK seminorm of $u$ ).", "We will give more details on this shortly.", "Let us discuss this strategy for the class ZE, see Section REF for details.", "Here, the characteristic factor of a measure-preserving system is the Pinsker factor.", "The reader has certainly noticed that even though we study dynamical properties of Furstenberg systems, as a matter of fact, at the end we deal with a process $(\\pi _0\\circ S^n)_{n\\in {\\mathbb {Z}}}$ , stationary with respect to $\\kappa \\in V_S(u)$ (each such measure is invariant in the subshift $(X_{u},S)$ ).", "Now, the Veech condition leads to the following concept.", "Definition 1.4 A centered stationary process $\\underline{X}=(X_n)$ taking finitely many values is called a Sarnak process if ${\\mathbb {E}}(X_0\\mid \\sigma (X_{N},X_{N+1},\\ldots ))\\rightarrow 0$ in $L^2$ (or a.e.", "); equivalently ${\\mathbb {E}}(X_0\\mid \\Pi (\\underline{X}))=0$ , where $\\Pi (\\underline{X})$ stands for the tail $\\sigma $ -algebra.", "Understanding the structure of Sarnak processes seems to be a problem of an independent interest and it will be studied elsewhere.", "Now, when $u$ takes finitely many values, our main result (Theorem REF applied to ZE) can be reformulated in the following manner: Corollary 1.5 Let $u\\colon {\\mathbb {N}}\\rightarrow be an arithmetic function taking finitely values.", "Then $ uCZE$ if and only if all stationary processes $ (0Sn)nZ$ determined by $ VS(u)$ are Sarnak.$ From the ergodic theory point of view we are close to the concept of the relative Kolmogorov property (K-property) which is however perturbed by the fact that we need this property for a single function.", "But even though only $\\pi _0$ is involved, the dynamical idea of the equivalence between the K-property and K-mixing (uniform mixing) works, and we can apply K-mixing of $\\pi _0$ against the family of functions depending on finitely many non-negative coordinates.", "This leads to studying the asymptotics of integrals of some continuous functions and finally gives the following combinatorial characterization of the orthogonality of $u$ to all deterministic sequences.", "In the following corollary, we use the fact that when $u$ takes its values in a finite set, a subset $A\\subset X_{u}$ depends on finitely many non-negative coordinates if and only if there exists $\\ell \\ge 1$ and a set $C$ of blocks of length $\\ell $ appearing in $u$ such that ${1}_{A}(y)={1}_{(y(0),y(1),\\ldots ,y(\\ell -1)) \\in C}$ .", "Corollary Let $u\\colon {\\mathbb {N}}\\rightarrow be an arithmetic function taking finitely many values.", "Then $ uCZE$ if and only if, for each subsequence $ (Nk)$ defining a Furstenberg system of $ u$ and each subset $ AXu$ depending on finitely many non-negative coordinates, we have the cancellation phenomenon of the values of $ u$ uniformly along sufficiently large shifts of the set of visits of $ u$ in $ A$: for each $ >0$, there exists $ M1$ such that for each $ 1$ and each set $ C$ of blocks of length $$, we have for each $ mM$\\begin{equation}\\lim _{k\\rightarrow \\infty }\\left|\\frac{1}{N_k}\\sum _{n\\le N_k}u(n)\\underbrace{{1}_{(u(m+n),u(m+n+1),\\ldots ,u(m+n+\\ell -1))\\in C}}_{{1}_{A}(S^{m+n}u)}\\right|\\le \\varepsilon .\\end{equation}$ Note that $M$ above depends on $(N_k)$ and $\\varepsilon $ .", "This combinatorial condition looks more attractive if we assume that $u$ is generic: Corollary C' Let $u\\colon {\\mathbb {N}}\\rightarrow be an arithmetic function taking finitely many values.", "If $ u$ is generic then $ uCZE$ if and only if$$\\lim _{m\\rightarrow \\infty }\\lim _{N\\rightarrow \\infty }\\left|\\frac{1}{N}\\sum _{n\\le N}u(n){1}_{(u(m+n),u(m+n+1),\\ldots ,u(m+n+\\ell -1))\\in C}\\right|=0$$uniformly in $ 1$ and in $ C$, a set of blocks of length $$.$ Remark 1.6 Note however that the above two corollaries do not say much if the (clopen) sets $C$ are already of small measures (e.g., in the most interesting case of blocks of large length).", "In fact, condition () in Corollary REF is equivalent to the following: for each subsequence $(N_k)$ defining a Furstenberg system of $u$ we have the conditional cancellation phenomenon of the values of $u$ uniformly along sufficiently large shifts of the set of visits of $u$ to “typical” blocks.", "More precisely, for each $\\varepsilon >0$ there exists $M\\ge 1$ such that $\\lim _{k\\rightarrow \\infty }\\frac{\\left|\\sum _{n\\le N_k}u(n){1}_{(u(m+n),u(m+n+1),\\ldots ,u(m+n+\\ell -1))=C}\\right|}{\\sum _{n\\le N_k}{1}_{(u(n),\\ldots ,u(n+\\ell -1))=C}}\\le \\varepsilon $ for all $m\\ge M$ , all $\\ell $ sufficiently large and blocks $C$ of length $\\ell $ forming a family of measure $>1-\\varepsilon $ .", "If, additionally, $u$ is generic, then the above condition reduces to $\\lim _{m\\rightarrow \\infty }\\lim _{N\\rightarrow \\infty }\\frac{\\left|\\sum _{n\\le N}u(n){1}_{(u(m+n),u(m+n+1),\\ldots ,u(m+n+\\ell -1))= C}\\right|}{\\sum _{n\\le N}{1}_{u(m+n),\\ldots ,u(m+n+\\ell -1)=C}}=0$ uniformly in $m$ , for “good” blocks of length $\\ell $ sufficiently large.", "Remark 1.7 Note the basic difference between the sums $\\sum _{n\\le N_k}u(n){1}_{(u(n),u(n+1),\\ldots ,u(n+\\ell -1))=C}=C[0]\\cdot \\sum _{n\\le N_k}{1}_{(u(n),u(n+1),\\ldots ,u(n+\\ell -1))=C}$ and $\\sum _{n\\le N_k}u(n){1}_{(u(m+n),u(m+n+1),\\ldots ,u(m+n+\\ell -1))= C},$ namely, the first one does not display any cancellation (that is, along the return times to a fixed block, we have no cancellation) and in the second one cancellations are possible and the fact that along further and further shifts of the set of return times we observe more and more cancellations, characterizes the Sarnak property." ], [ "Veech's and Sarnak's conjectures for other characteristic classes", "The family of all characteristic classes is enormous, see Section  for natural examples.", "Here, let us just notice that the discrete spectrum automorphisms form a characteristic class and it contains uncountably many characteristic subclasses whose pairwise intersections are equal to the class of all identities (indeed, discrete spectrum automorphisms whose group of eigenvalues is contained in ${\\mathbb {Z}}\\alpha $ , with $\\alpha $ irrational, is a characteristic class).", "Moreover, there are the largest proper and the smallest non-trivial characteristic classes.", "Indeed, although our study of the zero entropy class was originally motivated by Sarnak's conjecture, yet, ZE plays a special role since it is the largest (proper) characteristic class.", "In fact, we have $\\lbrace \\ast \\rbrace \\subset {\\rm ID}\\subset {{F}}\\subset {\\rm ZE}\\subset {\\rm ALL}$ for each characteristic class ${{F}}$ , where ${\\rm ID}$ stands for the (characteristic) class of identities (of all standard Borel probability spaces) and ALL stands for the (characteristic) class of all systems.", "Note that ${\\rm ID}=\\lbrace \\ast \\rbrace _{\\rm ec}\\text{ and } {\\rm ZE}={\\rm ZE}_{\\rm ec}.$ Clearly, the topological class ${C}_{\\rm ALL}$ consists of all topological systems and the only $u$ orthogonal to all of them is $u(n)=0$ on a subset of $n$ of full density which is compatible with the Veech condition (which in this setting means that $\\pi _0$ equals (a.e.)", "zero for each Furstenberg system).", "The topological class ${C}_{\\lbrace \\ast \\rbrace }$ consists of topological systems whose all visible measures are given by fixed points.", "The reader can check that the Veech condition here is just $\\int _{X_{u}}\\pi _0\\,d\\kappa =0$ for each $\\kappa \\in V_S(u)$ .", "The combinatorial condition () from Corollary REF (equivalent to the Veech condition) in this setting reduces to $\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n\\le N}u(n)=0$ .", "Clearly, in this setting, the latter is nothing but the Sarnak condition.", "It is not hard to see that the topological class ${C}_{\\rm ID}$ consists of topological systems whose only ergodic measures are Dirac measures at fixed points.", "The Veech condition here is $\\pi _0\\perp L^2(\\mathcal {I}_\\kappa )$ for each $\\kappa \\in V_S(u)$ , where $\\mathcal {I}_\\kappa $ stands for the $\\sigma $ -algebra of invariant sets.", "Finally, the counterparts of Corollaries REF and REF are the following.", "Corollary 1.8 Let $u\\colon {\\mathbb {N}}\\rightarrow be a bounded arithmetic function.Then $ uCID$ if and only if for each $ >0$ and each subsequence $ (Nk)$ defining a Furstenberg system of $ u$, there exists $ H01$ such that for each $ HH0$,\\begin{equation}\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}\\left|\\frac{1}{H}\\sum _{h\\le H}u(n+h)\\right|^2\\le \\varepsilon .\\end{equation}$ Note that if, additionally, $u$ is generic and it satisfies a non-quantitative version of the Matomäki-Radziwiłł [41] convergence on a typical short interval: $\\lim _{M, H\\rightarrow \\infty , H={\\rm o}(M)}\\frac{1}{M}\\sum _{n\\le M}\\left|\\frac{1}{H}\\sum _{h\\le H}u(n+h)\\right|^2=0$ then $u\\perp {C}_{\\rm ID}$ .", "Remark 1.9 It is also worth mentioning that the ID-sequences are precisely the mean slowly varying functions (see Proposition 5.1 in [28]), i.e.", "(bounded) arithmetic functions $v\\colon {\\mathbb {N}}\\rightarrow for which$$\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n<N}|v(n+1)-v(n)|=0.$$Therefore, sequences satisfying (\\ref {ortmsvf}) are precisely those orthogonal to all mean slowly varying functions.$ Notice that it follows from (REF ) that, for a non-trivial class ${{F}}$ , the “zero mean condition on a typical short interval” () is a necessary condition for $u\\perp {C}_{{{F}}}$ , whereas the condition given by Corollary REF is sufficient for $u\\perp {C}_{{{F}}}$ .", "In Section REF , we discuss the case ${{F}}={\\rm DISP}_{\\rm ec}$ , where DISP stands for the (characteristic) class of discrete spectrum automorphisms.", "In view of (REF ), ${C}_{{\\rm DISP}_{\\rm ec}}$ consists of homeomorphisms whose all ergodic measures yield systems with discrete spectrum.", "Let $u\\colon {\\mathbb {N}}\\rightarrow be bounded.", "For an increasing sequence of integers $ (Nk)$, we set\\begin{equation}\\Vert u\\Vert ^2_{u^1((N_k))}:=\\lim _{H\\rightarrow \\infty } \\frac{1}{H}\\sum _{h\\le H} \\Big (\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}u(n)\\overline{u(n+h)}\\Big )\\end{equation}and, for $ s>1$,\\begin{equation}\\Vert u\\Vert _{u^s((N_k))}^{2^{s+1}}:=\\lim _{H\\rightarrow \\infty } \\frac{1}{H}\\sum _{h\\le H}\\Vert u(\\cdot +h)\\cdot u(\\cdot )\\Vert _{u^s((N_k))}^{2^s},\\end{equation}whenever all the above limits exist.", "If $ Nk=k$, we set $ uus:=uus((Nk))$.$ Corollary 1.10 Let $u\\colon {\\mathbb {N}}\\rightarrow be bounded.Then$ uCDISPec$ if and only if $ uu2((Nk))=0$ for each subsequence $ (Nk)$ along which $ u$ is generic.", "In particular, if $ u$ is generic then $ uu2=0$.$ The main assertion of Corollary REF is equivalent to $\\Vert \\pi _0\\Vert _{u^2(\\kappa )}=0$ for each Furstenberg system $\\kappa \\in V_S(u)$ .", "The reason for the validity of this result is that given an automorphism $(Z,{\\cal D},\\rho ,R)$ , we have the equality ${\\cal D}_{{\\rm DISP}_{\\rm ec}}=\\mathcal {Z}_1(R,\\rho )$ which is a consequence of ${\\rm DISP}_{\\rm ec}={\\rm NIL}_1$ , where ${\\rm NIL}_s$ stands for the class of automorphisms whose a.a. ergodic components are inverse limits of $s$ -step nil-automorphisms (see Section  for more details).", "When we turn to the classes ${\\rm NIL}_s(=({\\rm NIL}_s)_{\\rm ec})$ and return to the original Sarnak's conjecture (for $u$ ) then clearly $\\pi _0\\perp L^2(\\Pi (\\kappa ))$ implies $\\pi _0\\perp L^2(\\mathcal {Z}_s(\\kappa ))$ for each $s\\ge 1$ .", "We hence obtain one more necessary condition for $u$ to be orthogonal to all deterministic sequences: Corollary 1.11 Let $u:{\\mathbb {N}}\\rightarrow be bounded.", "If $ uCZE$ then $ uus((Nk))=0$ for each $ sN$ (for each subsequence $ (Nk)$ along which $ u$ is generic).", "$ In Section REF , we prove that the Sarnak property of $u$ for the fundamental (in ergodic theory) class of distal automorphisms is equivalent to the Veech property of $u$ .", "We leave as an open problem whether the Veech property can be expressed combinatorially in the distal case." ], [ "The logarithmic Sarnak's conjecture", "As we have already noticed, our results also hold in the logarithmic case.", "In the corollary below we put together conditions which are equivalent to the logarithmic Sarnak's conjecture.Our thanks go to N. Frantizkinakis who pointed out to us one of crucial equivalences: (iv) $\\Leftrightarrow $ (v).", "Corollary 1.12 Let $u=\\mu $ or $\\lambda $ .", "The following conditions are equivalent: (i) $u\\perp _{\\rm log} {C}_{\\rm ZE}$ (i.e.", "zero entropy systems satisfy the logarithmic Sarnak property with respect to $u$ ), (ii) $u\\perp _{\\rm log} {C}_{\\rm NIL_s}$ for each $s\\ge 1$ , (iii) $\\pi _0\\perp L^2(\\Pi (\\kappa ))$ for each $\\kappa \\in V_S^{\\rm log}(u)$ (i.e.", "$u$ satisfies the Veech condition for each logarithmic Furstenberg system of $u$ ), (iv) $\\pi _0\\perp L^2(\\mathcal {Z}_s(\\kappa ))$ for each $\\kappa \\in V_S^{\\rm log}(u)$ and $s\\ge 1$ , (v) $u$ satisfies the logarithmic Chowla conjecture.", "The equivalence of (ii) and (iv) is due to Theorem REF (in its logarithmic form), the equivalence of (iv) and (v) (based on the facts proved by Tao [50] for the equivalence of the logarithmic Sarnak's and Chowla's conjectures for the Liouville function) is formally proved in [16] (implicit in Section 2.7 therein).", "Other (needed) implications are standard.", "Recall also that, by [27], (i) is equivalent to the logarithmic strong $u$ -MOMO property of all zero entropy systems.", "Note that the equivalence of (i) and (ii) in Corollary REF yields immediately the following.", "Corollary 1.13 The logarithmic Sarnak's conjecture holds if and only if for each $s\\ge 1$ , $\\mu $ is orthogonal to all systems whose all ergodic measures yield ${\\rm NIL}_s$ -systems.", "Remark 1.14 As Theorem REF is true in a larger context, also Corollary REF can be formulated for more general multiplicative functions bounded by 1, cf.", "Theorem 1.8 in [16]." ], [ "Averaged Chowla property", "The “iff” assertion of Theorem REF cannot be applied to the class ${C}_{\\rm DISP}$ .", "However, in Section REF , we will show that the Sarnak and Veech conditions are equivalent in this setting for each bounded $u\\colon {\\mathbb {N}}\\rightarrow such that\\begin{equation}\\mbox{all circle rotations satisfy the strong $u$-MOMO property}\\end{equation}(strong $ u$-MOMO property has been defined in Definition~\\ref {def:strongMOMO}).$ Corollary 1.15 Assume that $u\\colon {\\mathbb {N}}\\rightarrow is bounded by~1 and satisfies~(\\ref {avch}).", "Then $ u$ satisfies the Veech condition for the $ F=$DISP.", "In particular, the Sarnak and Veech conditions are equivalent for $ F=$DISP.", "Moreover, for every sequence $ (Nk)$ along which $ u$ is generic, $ u$ satisfies the averaged 2-Chowla property:\\begin{equation}\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\left|\\sum _{n\\le N_k}u(n)\\overline{u(n+h)}\\right|=0\\end{equation}and, for all sequences $ v1,...,vk$ (bounded by~1), we have\\begin{equation}\\lim _{H\\rightarrow \\infty }\\frac{1}{H^k}\\sum _{h_1,\\ldots ,h_k\\le H}\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\left|\\sum _{n\\le N_k}u(n)\\prod _{i=1}^kv_i(n+h_i)\\right|=0.\\end{equation}$ Property (), called the averaged Chowla property (cf.", "[42]), follows from () – this will be shown in Appendix .", "For an alternative approach to obtain the assertions of the above corollary, see the method in [19]: for () cf.", "Thm.", "4.1 and Prop.", "4.3 [19], and for () cf.", "Thm.", "2.1 and Prop.", "5.1 therein.", "Corollary 1.16 Let $u\\colon {\\mathbb {N}}\\rightarrow be a multiplicative function bounded by~1.", "If, for each Dirichlet character $$, $ u$ satisfies the short interval behaviour~(\\ref {ortmsvf})~\\footnote {This is equivalent to saying that along arithmetic sequences the averages on a typical short interval vanish.}", "then $ u$ satisfies the averaged Chowla property~(\\ref {avch2}) (along each sequence $ (Nk)$ for which $ u$ is generic).$ The above result with $(N_k)$ being all positive integers has been proved by Matomäki, Radziwiłł and Tao in [42].", "Note that to obtain Corollary REF it is enough to show that irrational rotations satisfy the strong $u$ -MOMO property.", "Via the DDKBSZ criterion, this follows from the ergodic property AOP introduced and proved to hold for totally ergodic rotations in [1].", "Thus, we obtain an ergodic proof of the averaged Chowla property for each sequence $(N_k)$ as above for all multiplicative $u$ enjoying the special short interval behavior.", "In particular, assuming that $u$ is generic, we get a non-quantitative version of [42].", "In Section , we prove (see Theorem REF ) that for each $u:{\\mathbb {N}}\\rightarrow taking finitely many values, if it satisfies the Sarnak property for the class $ CZE$, then no positive entropy system satisfies the strong $ u$-MOMO property (this was previously known for the Liouville function assuming the Chowla conjecture \\cite {Ab-Ku-Le-Ru2}).$" ], [ "Definition, examples, basic properties", "Recall that a class ${{F}}$ of measure-theoretic dynamical systems is characteristic if it is closed under taking isomorphisms, factors and (countable) joinings.", "Recall also the following classical result on such classes (see e.g.", "[45]).", "Proposition 2.1 Given a characteristic class ${{F}}$ , each automorphism $R$ on $(Z,\\mathcal {D},\\kappa )$ has a largest ${{F}}$ -factor, denoted by $\\mathcal {D}_{{{F}}}$ .", "The following result whose proof is based on a fundamental non-disjointness lemma from [39] will be crucial for us: Proposition 2.2 ([45]) Let $(X,{\\cal B},\\nu ,T)$ be a measure-theoretic dynamical system in the characteristic class ${{F}}$ , and let $(Z,{\\cal D},\\kappa ,R)$ be any measure-theoretic dynamical system.", "Then any joining of $R$ and $T$ is relatively independent over the largest ${{F}}$ -factor ${\\cal D}_{{{F}}}$ of $R$ .", "That is: if $g\\in L^2(Z,\\kappa )$ is such that ${\\mathbb {E}}^{\\kappa }[g|{\\cal D}_{{{F}}}]=0$ , and if $\\rho $ is a joining of $T$ and $R$ , then for any $f\\in L^2(X,\\nu )$ we have ${\\mathbb {E}}^\\rho (f\\otimes g)=0.$ Examples of characteristic classes (some acronyms are used for those which will be used in the sequel): ALL: all automorphisms of standard Borel probability spaces; $\\lbrace \\ast \\rbrace $ : the identity on the one-point space; ID: identity automorphisms (of all standard Borel probability spaces); DISP: discrete spectrum automorphisms; RDISP: rational discrete spectrum automorphisms; DISP($G$ ): discrete spectrum automorphisms whose group of eigenvalues is contained in fixed countable subgroup $G$ of the circle; ${\\rm NIL}_s$ : automorphisms whose a.a. ergodic components are inverse limits of $s$ -step nilautomorphisms.", "The fact that ergodic joinings of nilsystems remain nil, see Proposition 15, page 186 in the book [31], and the same holds for inverse limits (this is actually Lemma A.4 in [20]).", "Regarding factors of ergodic nilsystems, see Theorem 11 in page 230 [31].", "Here $s\\in {\\mathbb {N}}$ .", "DIST: distal automorphisms are those which are given as a transfinite (indexed by ordinals smaller than a fixed countable ordinal) sequence of consecutive extensions each of which either has relative discrete spectrum or (in case of a limit ordinal) is the corresponding inverse limit.", "The structural theorem Theorem 6.17 together with the concluding remark (for ${\\mathbb {Z}}$ -actions) on page 139 [22] tell us that each system has a largest distal factor, hence DIST is closed under countable joinings.", "In Lemma REF , we note that an automorphism is distal if and only if a.a. its ergodic components are distal.", "To see that this class is closed under taking factors, let us first recall that this fact holds for ergodic automorphisms (see Theorem 10.18 [25]).", "If now ${\\cal A}\\subset {\\cal D}$ is a factor of a distal automorphism $(Z,{\\cal D},\\kappa ,R)$ then ${\\cal A}$ (as an $R$ -invariant $\\sigma $ -algebra) is also a factor of a.e.", "of its ergodic components.", "So a double use of Lemma REF together with Theorem 10.18 from [25] gives that ${\\cal A}$ is also distal.", "ZE: zero entropy automorphisms; RIG$_{(q_n)}$ : automorphisms with a fixed sequence $(q_n)$ of rigidity; multipliers ${M}({D}^{\\perp })$ of a class ${D}^\\perp $ (${D}$ is any class of automorphisms and by ${D}^{\\perp }$ we mean the set of automorphisms disjoint from all systems from ${D}$ , and by ${M}({D}^{\\perp })$ we mean the set of systems whose all joinings with any element of ${D}^{\\perp }$ remain in ${D}^{\\perp }$ ); interesting classes of multipliers arise e.g.", "for ${D}$ =all weakly mixing (cf.", "Proposition 5.1 in [38]) or all mixing automorphisms; see [24], [38].", "the class of factors of all infinite self-joinings of a fixed automorphism $R$ (the smallest characteristic class containing $R$ ); especially in case of MSJ and simple automorphisms (cf.", "[25], Chapter 12).", "Characteristic classes of such type were used in [40].", "Note also that the intersection of any family of characteristic classes yields again a characteristic class.", "In Section REF , we will show that each characteristic class ${{F}}$ determines another characteristic class ${{F}}_{\\rm ec}$ consisting of those automorphisms whose ergodic components are in ${{F}}$ ." ], [ "The smallest nontrivial and the largest proper characteristic class", "An obvious observation has been made already in the introduction that the family ALL of all automorphisms is the largest characteristic class, while the one-element $\\lbrace \\ast \\rbrace $ family (which is the one-point space automorphism) is the smallest characteristic class.", "It is more interesting however that the smallest non-trivial and the largest proper characteristic classes exist.", "Proposition 2.3 ID is the smallest non-trivial characteristic class.", "Let us first notice that the system $([0,1],{\\rm Leb}, {\\rm Id})$ has all other identities as factors.", "Indeed, any standard Borel probability space is determined by a sequence $(t_i)_{i\\ge 0}$ of non-negative numbers such that $\\sum _{i\\ge 0}t_i=1$ and $t_0$ corresponds to the mass of the continuous part and $t_1,t_2,\\ldots $ correspond to the masses of atoms.", "Then, take the corresponding partition of $[0,1]$ into intervals $I_i$ of length $t_i$ and, for each $i\\ge 1$ , the factor map will glue points in $I_i$ .", "Now, notice that any non-trivial characteristic class ${{F}}$ contains a non-ergodic automorphism.", "Indeed, suppose that $T$ is ergodic, acting on a non-trivial space $(Y,\\nu )$ .", "Since $Y$ is non-trivial and so is $T$ , the graph joinings $\\Delta _{{\\rm Id}}$ and $\\Delta _T$ are ergodic and different, so any non-trivial convex combination of them yields a non-ergodic member of ${{F}}$ .", "It follows that by taking the factor $\\mathcal {I}_{\\nu }$ of (a.e.)", "$T$ -invariant sets (which belongs to ${{F}}$ ), we obtain the identity on a non-trivial standard Borel probability space $(\\overline{Y},\\overline{\\nu })$ .", "But then the infinite Cartesian product $(\\overline{Y}^{\\times \\infty },\\overline{\\nu }^{\\otimes \\infty })$ is also in ${{F}}$ and this infinite product is isomorphic to $([0,1],{\\rm Leb})$ , which completes the proof.", "In order to prove the existence of the largest characteristic (proper) class, we need to recall some results.", "Theorem 2.4 (non-ergodic Sinai's factor theorem [36], [48]) Assume that $R$ is an automorphism of $(Z,{\\cal D},\\rho )$ and let $\\rho =\\int _{X/\\mathcal {I}_{\\rho }}\\rho _{\\overline{x}}\\,dm(\\overline{x})$ stand for the ergodic decomposition of $\\rho $ .", "Assume that $m-{\\rm essinf}_{\\overline{x}\\in X/\\mathcal {I}_\\rho }h(\\rho _{\\overline{x}},R)\\ge \\alpha >0.$ Then a Bernoulli automorphism of entropy $\\alpha $ is a factor of $R$ .", "In [48] (see Theorem 4.3 therein) the above result is attributed to Kieffer and Rahe [36], see also [47] p.2 (The non-ergodic factor theorem).", "We also need the following well-known result (we include its proof for completeness).", "Proposition 2.5 Each automorphism $R$ is a factor of a self-joining of the (infinite entropy) Bernoulli automorphism $([0,1]^{{\\mathbb {Z}}},{\\rm Leb}^{\\otimes {\\mathbb {Z}}},S)$ .", "Remark 2.6 Before we prove the above result, let us notice that any automorphism $T$ of $(X,{\\cal B},\\mu )$ has an isomorphic copy in the space $([0,1]^{{\\mathbb {Z}}},\\kappa ,S)$ .The same arguments apply to $\\lbrace z\\in |z|\\le 1\\rbrace $ instead of $[0,1]$ .", "Consider first the aperiodic part of $T$ which is realized on a standard Borel space.", "This space is isomorphic to $[0,1]$ , via a Borel isomorphism $I$ .", "It follows that the distribution $\\mu ^{\\prime }$ of the process $(I\\circ S^k)_{k\\in {\\mathbb {Z}}}$ yields a realization of the aperiodic part.", "Now, $\\mu ^{\\prime }$ takes measure zero on the set of periodic points for the shift.", "Moreover, the set of periodic points of $S$ can be identified with a subset of $[0,1]$ , points of period 2 with a subset of $[0,1]^2$ etc., and we can easily settle an isomorphism of the fixed point subspace for $T$ with a subset of $[0,1]$ , period 2 points with a subset of $[0,1]^2$ , etc.", "Thus, it suffices to take $\\kappa $ equal to the sum of $\\mu ^{\\prime }$ and the relevant atomic measures corresponding to the periodic points.", "[Proof of Proposition REF ] Fix any automorphism $R$ of $(Z,{\\cal D},\\rho )$ and take its isomorphic copy in the space $([0,1]^{{\\mathbb {Z}}},\\kappa ,S)$ .", "Take the product space $([0,1]^{{\\mathbb {Z}}}\\times [0,1]^{{\\mathbb {Z}}},{\\rm Leb}^{\\otimes {\\mathbb {Z}}}\\otimes \\kappa )$ and consider the map $\\psi \\colon [0,1]^{{\\mathbb {Z}}}\\times [0,1]^{{\\mathbb {Z}}}\\rightarrow [0,1]^{{\\mathbb {Z}}}$ given by $(x_n,y_n)\\mapsto (x_n+y_n).$ Then $\\psi _\\ast ({\\rm Leb}^{\\otimes {\\mathbb {Z}}}\\otimes \\kappa )={\\rm Leb}^{\\otimes {\\mathbb {Z}}}$ and clearly the join of the $\\sigma $ -algebra of the first coordinate and of $\\psi ^{-1}(\\mathcal {B}([0,1]^{\\otimes {\\mathbb {Z}}}))$ is the product $\\sigma $ -algebra in $[0,1]^{\\otimes {\\mathbb {Z}}}\\times [0,1]^{\\otimes {\\mathbb {Z}}}$ .", "The result follows.", "We now prove the following.", "Lemma 2.7 Assume that ${{F}}$ is a characteristic class such that ${{F}}\\setminus {\\rm ZE}\\ne \\emptyset $ .", "Then ${{F}}={\\rm ALL}$ .", "Fix $T\\in {{F}}\\setminus {\\rm ZE}$ .", "Because of Proposition REF , we only need to prove that the infinite entropy Bernoulli automorphism is in ${{F}}$ .", "The first step is to consider the factor of $T$ that arises by gluing together the periodic part and the ergodic components from the aperiodic part whose entropy is smaller than $\\alpha =h(T)$ .", "Clearly, this factor remains in ${{F}}\\setminus {\\rm ZE}$ .", "Moreover, in its ergodic decomposition we have a single point and the remaining part (which may still be non-ergodic) consists of ergodic components of entropy at least $\\alpha $ .", "By Theorem REF , it follows that as a further factor $R\\in {{F}}\\setminus {\\rm ZE}$ we can obtain a non-ergodic automorphism with two ergodic components: one of them is a Bernoulli of entropy $\\alpha $ and the other one is a fixed point.", "Finally, we take the infinite Cartesian product $R^{\\times \\infty }$ .", "It is not hard to see that a.e.", "ergodic component of this automorphism is a Bernoulli with infinite entropy.", "Using once more Sinai's theorem (Theorem REF ), we obtain that a Bernoulli with infinite entropy belongs to ${{F}}$ which completes the proof.", "Now, using the lemma we obtain the following.", "Proposition 2.8 ZE is the largest proper characteristic class.$\\quad \\hbox{\\vrule \\vbox to 6pt {\\hrule width 4pt\\vfill \\hrule }\\vrule } $" ], [ "Characteristic classes given by ergodic components", "Assume that ${{F}}$ is a characteristic class.", "By ${{F}}_{\\rm ec}$ we denote the class of those automorphisms $R$ such that (a.e.)", "ergodic components of $R$ are in ${{F}}$ (or more precisely, in ${{F}}\\cap $  Erg, where Erg stands for the family of all ergodic automorphisms).", "Note that we have ${{F}}\\cap {\\rm Erg}={{F}}_{\\rm ec}\\cap {\\rm Erg}.$ Lemma 2.9 ${{F}}_{\\rm ec}$ is a characteristic class.", "The proof has two parts: we need to show that ${{F}}_{\\rm ec}$ is closed under taking factors and joinings." ], [ "Factors", "Let $R$ acting on $(Z,{\\cal D},\\kappa )$ belong to ${{F}}_{\\rm ec}$ and fix a factor ${\\cal A}\\subset {\\cal D}$ of $R$ .", "Let $\\kappa =\\int \\kappa _{\\overline{x}}\\,dP(\\overline{x})$ denote the ergodic decomposition of $\\kappa $ .", "Since the ergodic components $\\kappa _{\\overline{x}}$ are $R$ -invariant measures, ${\\cal A}$ (being an $R$ -invariant sub-$\\sigma $ -algebra) is also a factor of the automorphism $(Z,\\kappa _{\\overline{x}},R)$ and $\\kappa |_{{\\cal A}}=\\int \\kappa _{\\overline{x}}|_{{\\cal A}}\\,dP(\\overline{x})$ is the ergodic decomposition of $\\kappa |_{{\\cal A}}$ .", "It follows that the ergodic components of the factor are factors of ergodic components of $R$ , and since $R\\in {{F}}_{\\rm ec}$ , $(\\kappa _{\\overline{x}},R)\\in {{F}}$ , so also $(\\kappa _{\\overline{x}}|_{{\\cal A}},R|_{{\\cal A}})\\in {{F}}$ for $P$ -a.e.", "$\\overline{x}$ ." ], [ "Joinings", "Take $(X,\\mu ,T)$ and $(Y,\\nu ,S)$ from ${{F}}_{\\rm ec}$ and let $\\rho \\in J(T,S)$ be their joining.", "Let $\\rho =\\int _0^1 \\rho _t\\,dP(t),\\;\\mu =\\int _0^1 \\mu _t\\,dQ(t),\\;\\nu =\\int _0^1\\nu _t\\,dR(t)$ be the relevant ergodic decompositions.", "Then $\\int _0^1\\mu _t\\,dQ(t)=\\mu =\\rho |_X=\\int _0^1\\rho _t|_X\\,dP(t),$ so since $\\rho _t|X$ are also ergodic, these two decompositions are the same.", "So for a $P$ -“typical” $t\\in [0,1]$ , the projection of $\\rho _t$ on $X$ is an ergodic component of $T$ .", "The same argument applies on the coordinate $Y$ and we see that the ergodic components of $\\rho $ are joinings of ergodic components of $\\mu $ and $\\nu $ .", "It follows that $(X\\times Y,\\rho ,T\\times S)\\in {{F}}_{\\rm ec}$ .", "The argument extends to countable joinings." ], [ "ID, ZE, DISP and RIG$_{(q_n)}$", "Given a characteristic class ${{F}}$ , according to Proposition REF , each automorphism $R$ acting on $(Z,{\\cal D},\\kappa )$ has a largest ${{F}}$ -factor ${\\cal D}_{{{F}}}\\subset {\\cal D}$ .", "Often, its description is classical: the $\\sigma $ -algebra of invariant sets for ${{F}}={\\rm ID}$ , the Pinsker factor for ${{F}}={\\rm ZE}$ , the Kronecker factor for ${{F}}={\\rm DISP}$ , the largest factor for which $(q_n)$ is a rigidity time for ${{F}}={\\rm RIG}_{(q_n)}$ ." ], [ "DISP$_{\\rm ec}$", "We will comment now on ${\\cal D}_{{{F}}_{\\rm ec}}$ when ${{F}}={\\rm DISP}$ , cf.", "Proposition REF  (ii) and its connections with the theory of nil-factors.", "Most of the material presented below is known to aficionados but not necessarily the material is explicitly present in the literature.", "Our discussion is based on [18], [23], [30] and [31].", "We provide some details to explain clearly why the problem of whether $\\mu \\perp {\\rm DISP}_{\\rm ec}$ is open, cf.", "Corollary REF , Corollary REF and Remark REF .", "Recall that according to the Furstenberg-Zimmer theory [23], given $R$ on $(Z,{\\cal D},\\kappa )$ and a factor $\\mathcal {C}\\subset \\mathcal {D}$ , there exists a certain intermediate factor $\\mathcal {C}\\subset {\\cal K}={\\cal K}(\\mathcal {C})\\subset \\mathcal {D},$ called the relative Kronecker factor (with respect to $\\mathcal {C}$ ).", "It is the largest intermediate factor with the following property (see condition C$_2$ in [23], p. 131): $\\begin{split}\\parbox [t]{0.8}{there exists a dense set of functions F\\in L^2({\\cal K},\\kappa |_{{\\cal K}}) such thatfor each \\delta >0 there is a finite set g_1,\\ldots ,g_k\\in L^2({\\cal K},\\kappa |_{{\\cal K}}) such that for each h\\in {\\mathbb {Z}},\\min _{1\\le j\\le k}\\Vert F\\circ R^h-g_j\\Vert _{L^2(\\kappa _y)}<\\delta }\\end{split}$ for a.e.", "$y\\in Z/\\mathcal {C}$ , where $\\kappa |_{{\\cal K}}=\\int _{Z/\\mathcal {C}}\\kappa _y\\,d\\kappa (y).$ Whenever condition (REF ) holds, we speak of relative compactness or of relatively discrete spectrum of the intermediate factor over $\\mathcal {C}$ .", "A particular situation arises when ${\\cal C}=\\mathcal {I}_\\kappa $ , i.e.", "it is the $\\sigma $ -algebra of invariant sets.", "Then (REF ) is nothing but the ergodic decomposition of $\\kappa $ and the conditional measures $\\kappa _y$ are also $R$ -invariant.", "In this case condition (REF ) yields in a.e.", "fiber $\\pi ^{-1}(y)$ (where $\\pi \\colon Z/{\\cal K}\\rightarrow Z/\\mathcal {I}_\\kappa $ stands for the factor map) a dense set of functions $F|_{\\pi ^{-1}(y)}$ in $L^2({\\cal K},\\kappa _y)$ whose orbits under the unitary action of $R$ are relatively compact.", "It follows that the (ergodic) automorphism $(R,\\kappa _y)$ has discrete spectrum for a.e.", "$y\\in Z/\\mathcal {I}_\\kappa $ .", "In other words, ${\\cal K}(\\mathcal {I}_\\kappa )\\subset \\mathcal {D}_{{\\rm DISP_{ec}}}$ In fact, the opposite inclusion is also true, i.e.", "$\\mathcal {D}_{{\\rm DISP_{ec}}} = {\\cal K}(\\mathcal {I}_\\kappa ),$ that is, $\\mathcal {A}:=\\mathcal {D}_{{\\rm DISP_{ec}}}$ has relatively discrete spectrum over $\\mathcal {I}_\\kappa $ .", "Indeed, by the definition of $\\mathcal {A}$ , a.e.", "ergodic component of $R|_{\\mathcal {A}}$ has discrete spectrum.", "Fix $F\\in L^\\infty ({\\cal A},\\kappa |_{{\\cal A}})$ .", "Fix also $\\varepsilon ,\\delta >0$ and $k\\ge 1$ .", "Consider the set $W_k\\subset Z/\\mathcal {I}_\\kappa $ of those $y$ for which $\\min _{-k\\le j\\le k}\\left\\Vert F\\circ R^n-F\\circ R^j\\right\\Vert _{L^2(\\kappa _y)}<\\varepsilon $ for each $n\\in {\\mathbb {Z}}$ .", "Since on each fiber $R$ is an ergodic automorphism with discrete spectrum, the measure of $W_k$ goes to 1, when $k\\rightarrow \\infty $ , so it will be greater than $1-\\delta $ for $k$ large enough.", "It follows that the function $F$ is compact as it has been defined in the proof of Theorem 6.15 [23].", "Therefore, $F\\in L^2({\\cal K}(\\mathcal {I}_\\kappa ))$ , which (by [23]) concludes the proof of (REF ).", "Remark 2.10 As a matter of fact, in [5], the Furstenberg-Zimmer theory is developed without assuming ergodicity (cf.", "e.g.", "Proposition 5.7 therein to obtain the equality $\\mathcal {D}_{{\\rm DISP}_{\\rm ec}}=\\mathcal {K}(\\mathcal {I}_\\kappa )$ ).", "We will now see that $\\mathcal {D}_{\\rm DISP_{ec}}$ appears naturally in the classical theory of characteristic nil-factors [30], [31].We would like to thank Bryna Kra and Nikos Frantzikinakis for fruitful discussions and useful references on this subject.", "Recall that if $R$ acting on $(Z,{\\cal D},\\kappa )$ is ergodic then for a function $f\\in L^\\infty (Z,{\\cal D},\\kappa )$ its $u^s$ norms (in fact, seminorms) are defined in the following way: $\\Vert f\\Vert _{u^1}:=\\Big |\\int f\\,d\\kappa \\Big |,$ $\\Vert f\\Vert _{u^{s+1}}^{2^{s+1}}:=\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\Vert f\\circ R^h\\cdot \\overline{f}\\Vert _{u^s}^{2^s}.$ If $R$ is non-ergodic then instead of (REF ), we put $\\Vert f\\Vert ^2_{u^1}:=\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\int f\\circ R^h\\cdot \\overline{f}\\,d\\kappa $ and (REF ) remains unchanged.", "Then, by [30], [31], for each $s\\ge 1$ there is a special factor $\\mathcal {Z}_s=\\mathcal {Z}_s(R)\\subset {\\cal B}$ , namely, the largest factor whose $\\mbox{a.e.\\ ergodic component is an inverse limit of $s$-step nil-systems.", "}$ In other words, $\\mathcal {Z}_s(R)$ is the largest (characteristic) ${\\rm NIL}_s$ -factor of $R$ .", "Moreover (see Proposition 7 (page 138) and Proposition 13 (page 141) in [31]), $\\Vert f\\Vert _{u^{s+1}}=0 \\iff f\\perp L^2(\\mathcal {Z}_s) \\iff f\\perp L^2(\\mathcal {Z}_s(R,\\kappa _y)) \\text{ for $\\kappa $-a.e.", "}y.$ A special case arises when our measure-preserving systems are Furstenberg systems of a bounded $u\\colon {\\mathbb {N}}\\rightarrow .As in (for example) \\cite {Fr}, see Sections~2.4 and 2.5 therein, one can introduce the uniformity norms (along subsequences of intervals) for $ u$.", "The definitions are given in (\\ref {eq:1us}) and (\\ref {eq:2us}).", "They are very similar to those (in the non-ergodic case) to the definitions for functions.$ We will now show that $\\mathcal {Z}_1(R)=\\mathcal {K}(\\mathcal {I}_\\kappa ).$ If $R$ is ergodic then the above means just that $\\mathcal {Z}_1 \\text{ is the Kronecker factor of $R$}.$ To see that (REF ) indeed holds, notice that (REF ) for $s=1$ yields $\\Vert f\\Vert _{u^2}=0 \\iff f\\perp L^2(\\mathcal {Z}_1)$ and it remains to notice that (using the Wiener lemma) $\\Vert f\\Vert ^4_{u^2}=\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\Vert f\\circ R^h \\cdot \\overline{f} \\Vert ^2_{u^1}=$ $\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\left| \\int f\\circ R^h \\cdot \\overline{f}\\right|^2\\,d\\kappa \\rightarrow \\sum _{z\\in \\mathbb {T}}\\sigma _f(\\lbrace z\\rbrace )^2,$ where $\\sigma _f$ stands for the spectral measure of $f$ .", "Let us return to a possibly non-ergodic $R$ .", "The inclusion $\\mathcal {Z}_1(R)\\subset \\mathcal {K}:=\\mathcal {K}(\\mathcal {I}_\\kappa )$ follows directly by Theorem 5.2 in [18].", "To obtain the opposite inclusion, one can argue in the following way.", "Suppose that $f\\perp L^2(\\mathcal {Z}_1(R))$ and $|f|\\le 1$ .", "Take $g\\in L^2(\\mathcal {K})$ , cf.", "(REF ) with $F=g$ .", "We want to show that $\\int fg\\, d\\kappa =0$ .", "Notice that $\\int fg\\, d\\kappa =\\int \\left(\\frac{1}{N}\\sum _{n\\le N}\\int f\\circ T^n \\cdot g\\circ T^n\\, d\\kappa _y\\right)\\, d\\kappa (y).$ Let $g_j$ , $1\\le j\\le k$ , be as in (REF ).", "Then $\\left|\\frac{1}{N}\\sum _{n\\le N}\\int f\\circ T^n \\cdot g\\circ T^n\\, d\\kappa _y\\right|\\\\\\le \\left|\\sum _{1\\le j\\le k}\\frac{1}{N}\\sum _{n\\le N}\\int f\\circ T^n \\cdot g_j\\, d\\kappa _y \\right| + \\frac{1}{N}\\sum _{n\\le N}\\min _{1\\le i\\le k}\\int \\left|f\\circ T^n (g\\circ T^n-g_i) \\right|\\, d\\kappa _y.$ Each term in the average in the second summand is bounded by $\\delta $ .", "Moreover, $\\frac{1}{N}\\sum _{n\\le N}\\left|\\int f\\circ T^n \\cdot g_j\\, d\\kappa _y \\right|^2 \\rightarrow \\sum _{z\\in \\mathbb {T}}\\sigma _{f,g_j,\\kappa _y}(\\lbrace z\\rbrace )^2,$ where $\\sigma _{f,g_j,\\kappa _y}$ stands for the spectral measure of the pair $f,g_j$ (on the ergodic component $(\\pi ^{-1}(y),\\kappa _y)$ given by $y$ ).", "But by (REF ) and (REF ), we have $f\\perp L^2(\\mathcal {Z}_1) &\\iff f\\perp L^2(\\mathcal {Z}_1(R,\\kappa _y))\\text{ for a.e.", "}y\\\\& \\iff \\sigma _{f,\\kappa _y} \\text{ is continuous for a.e.", "}y$ ($\\sigma _{f,\\kappa _y}$ stands for the spectral measure of $f$ on the ergodic component $(\\pi ^{-1}(y),\\kappa _y)$ given by $y$ ).", "Since $f\\perp L^2(\\mathcal {Z}_1)$ and $\\sigma _{f,g_i,\\kappa _y}\\ll \\sigma _{f,\\kappa _y}$ , it remains to use the classical equivalence $\\frac{1}{N}\\sum _{n\\le N}a_n \\rightarrow 0 \\iff \\frac{1}{N}\\sum _{n\\le N}a_n^2\\rightarrow 0$ for any bounded sequence $(a_n)\\subset [0,\\infty )$ , to conclude that the limit in (REF ) is equal to zero.", "Thus $f\\perp L^2(\\mathcal {Z}_1) \\Rightarrow f\\perp L^2(\\mathcal {K})$ .", "Finally, let us compare the above with the notion of relative weak mixing.", "Recall that relative weak mixing over $\\mathcal {I}_\\kappa $ for $f$ means that $\\frac{1}{H}\\sum _{h\\le H}\\int \\left|{\\mathbb {E}}(f\\circ R^h\\cdot \\overline{f}|\\mathcal {I}_\\kappa )\\right|^2\\, d \\kappa \\rightarrow 0.$ Moreover, $\\frac{1}{H}\\sum _{h\\le H}\\int \\left|{\\mathbb {E}}(f\\circ R^h\\cdot \\overline{f}|\\mathcal {I}_\\kappa )\\right|^2\\, d \\kappa =\\int \\left(\\frac{1}{H}\\sum _{h\\le H}\\left|\\int f\\circ R^h\\cdot \\overline{f}\\,d\\kappa _y\\right|^2\\right)\\, d \\kappa ,$ and, once more by the Wiener lemma, $\\frac{1}{H}\\sum _{h\\le H}\\left| \\int f\\circ R^h\\cdot \\overline{f}\\,d\\kappa _y\\right|^2\\rightarrow \\sum _{z\\in \\sigma _{f,\\kappa _y}(\\lbrace z\\rbrace )^2.It follows immediately that \\sigma _{f,\\kappa _y} is continuous for a.e.\\ y if and only if f is relatively weakly mixing over \\mathcal {I}_\\kappa .", "}The above discussion can be summarized in the following statement.\\begin{Cor}Let (Z,{\\cal D},\\kappa ,R) be a measure-theoretic dynamical system and let f\\in L^2(Z,{\\cal D},\\kappa ).", "The following conditions are equivalent:\\begin{enumerate}\\item [(i)] f\\perp L^2(\\mathcal {Z}_1),\\item [(ii)] f\\perp L^2(\\mathcal {D}_{\\rm DISP_{ec}}),\\item [(iii)] f\\perp L^2(\\mathcal {K}(\\mathcal {I}_\\kappa )),\\item [(iv)] \\sigma _{f,\\kappa _y} is continuous for \\kappa -a.e.", "y,\\item [(v)] f is relatively weakly mixing over \\mathcal {I}_\\kappa .\\end{enumerate}\\end{Cor}$" ], [ "A class vs. its ec-class", "Let us continue our observations on the relations between characteristic classes and the corresponding ec-classes.", "Note that in general there are no relations between ${{F}}$ and ${{F}}_{\\rm ec}$ : Proposition 2.11 We have: (i) ${\\rm ZE}={\\rm ZE}_{\\rm ec}$ , ${\\rm ALL}={\\rm ALL}_{\\rm ec}$ , ${\\rm ID}={\\rm ID}_{\\rm ec}$ , ${\\rm NIL}_s=({\\rm NIL}_s)_{\\rm ec}$ , $\\lbrace \\ast \\rbrace \\subsetneq \\lbrace \\ast \\rbrace _{\\rm ec}$ ; (ii) ${\\rm DISP}\\subsetneq {\\rm DISP}_{\\rm ec}$ ; (iii) ${\\rm RDISP}={\\rm RDISP}_{\\rm ec}$ ; (iv) ${\\rm DIST}={\\rm DIST}_{\\rm ec}$ ; (v) $\\Big ({\\rm RIG}_{(q_n)}\\Big )_{\\rm ec}\\subsetneq {\\rm RIG}_{(q_n)}$ .", "[Proof of (i)-(iii)]  (i) The first claim follows from the fact that the entropy function is convex, the other claims are obvious.", "(ii) If an automorphism has discrete spectrum then its $L^2$ -space is generated by eigenfunctions.", "The restrictions (if non-zero) of these (global) eigenfunctions yield orthonormal bases in $L^2$ -spaces of ergodic components.", "The inclusion is strict since $(x,y)\\mapsto (x,x+y)$ (on 2, considered with Lebesgue measure) does not have discrete spectrum while the ergodic components do.", "(iii) We want to show that if each ergodic component has rational discrete spectrum then the whole automorphism has.", "Given $p/q\\in \\mathbb {Q}$ and an ergodic component $c$ , we choose $f_c$ a modulus 1 eigenfunction corresponding to the eigenvalue $e^{2\\pi ip/q}$ .", "Since $f_c$ is unique up to a constant of modulus 1, this choice can be done measurably.", "In this way, we will create global eigefunctions.The same argument works if we consider the characteristic class of automorphisms having discrete spectrum contained in a fixed countable subgroup of the circle.", "Before we give the proof of (iv), we need to recall some more notions and facts from the relative ergodic theory, e.g.", "[23], [25].", "Given an automorphism $T$ of $(X,{\\cal B},\\mu )$ and its factor $S$ on $(Y,{\\cal C},\\nu )$ with the factor map $\\pi \\colon X\\rightarrow Y$ , we say that this extension is relatively ergodic (rel.", "erg.)", "if each $f\\in L^1(X,{\\cal B},\\mu )$ satisfying $f\\circ T=f$ ($\\mu $ -a.e.)", "is $\\pi ^{-1}(\\mathcal {C})$ -measurable.", "It follows immediately from the definition that: any composition of relatively ergodic extensions remains relatively ergodic; an inverse limit of relatively ergodic extensions remains relatively ergodic (as the conditional expectation, with respect to a factor, of an invariant function remains invariant); $\\overline{\\pi }\\colon Y\\rightarrow \\overline{Y}:=Y/\\mathcal {I}_\\nu $ , where $(\\overline{Y},\\overline{\\nu })$ stands for the space of ergodic components (on which acts the identity map), is relatively ergodic.", "Let $\\mu =\\int _Y \\mu _y\\,d\\nu (y)$ stand for the disintegration of $\\mu $ over $\\nu $ and let $\\nu =\\int _{\\overline{Y}}\\overline{\\nu }_{\\overline{y}}\\,d\\overline{\\nu }$ denote the ergodic decomposition of $\\nu $ (which is precisely the disintegration of $\\nu $ over $\\overline{\\nu }$ ).", "Then the ergodic components of $S$ acting on $Y$ are of the form $(\\overline{\\pi }^{-1}(\\overline{y}),\\overline{\\nu }_{\\overline{y}}, S)$ (the measures $\\overline{\\nu }_{\\overline{y}}$ are $S$ -invariant).", "Therefore, we have the following lemma.", "Lemma 2.12 If $T$ is relatively ergodic over $S$ then the ergodic components of $T$ are of the form $\\left(\\pi ^{-1}(\\overline{\\pi }^{-1}(\\overline{y})),\\int _{\\overline{\\pi }^{-1}(\\overline{y})}\\mu _y\\, d \\overline{\\nu }_{\\overline{y}}(y)\\right).$ Note that it follows that the ergodic components of $T$ have as their factors (via the relevant restriction of $\\pi $ ) ergodic components of $S$ , and that the spaces of ergodic components of $T$ and $S$ are the same (i.e.", "$\\overline{X}=\\overline{Y}$ ).", "Lemma 2.13 Let $T$ be relatively ergodic over $S$ .", "Then the following are equivalent: $T$ over $S$ has relatively discrete spectrum.", "The ergodic components of $T$ have relatively discrete spectrum over the ergodic components of $S$ being their relevant factors.", "By Lemma REF , we see that the disintegration of an ergodic component $\\pi ^{-1}(\\overline{\\pi }^{-1}(\\overline{y}))$ over $\\overline{\\pi }^{-1}(\\overline{y})$ (which is its factor) consists of the same conditional measures $\\mu _y$ as the total disintegration of $\\mu $ over $\\nu $ .", "We proceed now as in the proof of the equality $\\mathcal {K}(I_\\kappa )=\\mathcal {D}_{\\rm DISP_{ec}}$ (page REF ), showing compactness.", "Recall that an automorphism $T$ is distal if it is a limit of a transfinite (indexed by countable ordinals) sequence of consecutive maximal Kronecker extensions (if an ordinal is not isolated, we pass to the corresponding inverse limit).", "Note that, by the very definition, the $\\sigma $ -algebra Inv is contained in the Kronecker factor of $T$ , so in this transfinite chain of consecutive extensions, all but (perhaps) the first one are relatively ergodic.", "By applying Lemma REF and transfinite induction, we obtain the following.", "Lemma 2.14 $T$ is distal if and only if all its ergodic components are distal.", "[Proof of (iv)-(v)]    (iv) This follows directly from Lemma REF .", "(v) It is clear that if $(q_n)$ is a rigidity time for an a.e.", "ergodic component, it is also a rigidity time for the whole automorphisms.", "Not vice versa however (for $(q_n)$ sufficiently sparsed).", "We will provide a relevant construction below." ], [ "${\\rm RIG}_{\\rm ec}$ is a proper subclass of {{formula:3509d1c9-0286-4aa1-af65-1e70a9710123}}", "Let us first notice that we only need to construct a continuous measure $\\sigma $ on the circle such that $e^{2\\pi i q_n\\cdot }\\rightarrow 1\\text{ in measure }\\sigma \\text{ but not } \\sigma -\\text{a.e.", "}$ Indeed, suppose (REF ) holds, and consider on 2 the automorphism $T(x,y)=(x,y+x)\\text{ with measure }\\sigma \\otimes \\,{\\rm Leb}.$ If $F(x,y)=f(x)e^{2\\pi i\\ell y}$ then by (REF ), $\\int |F(T^{q_n}(x,y))-F(x,y)|\\,d\\sigma (x)dy=\\int |f(x)||e^{2\\pi iq_n\\ell x}-1|\\,d\\sigma (x)\\rightarrow 0$ when $n\\rightarrow \\infty $ .", "On the other hand, the rotation by $x$ on an ergodic component $\\lbrace x\\rbrace \\times has $ (qn)$ as its rigidity time if and only if $ qnx0$ mod~1.", "This is not true for $$-a.e.\\ $ x in view of (REF ).", "We now sketch how to construct such a measure assuming that $(q_n)$ is sufficiently sparsed.", "Fix $0<p_n<1$ so that $p_n$ is decreasing to zero and $\\sum _{n\\ge 1}p_n=\\infty $ .", "Set $f_n(x)=\\lbrace q_nx\\rbrace $ .", "We intend to construct a Cantor set (together with a Cantor measure $\\sigma $ on it).", "Let $A_n:=f_n^{-1}([1/4,3/4]), \\;B_n=f_n^{-1}([0,p_n]).$ Our postulates are: $\\sigma (B_n)=1-p_n,\\; \\sigma (A_n)=p_n.$ In fact, we need to be more precise in description of the measure at stage $n$ to be able to continue its definition.", "So at stage $n$ the circle is divided into intervals of the form $[\\frac{j}{q_n},\\frac{j+1}{q_n})$ (many of such intervals are of measure $\\sigma $ equal to zero).", "We now require that the conditional measures satisfy: $\\sigma \\left(B_n|[\\frac{j}{q_n},\\frac{j+1}{q_n})\\right)=1-p_n,\\,\\sigma \\left(A_n|[\\frac{j}{q_n},\\frac{j+1}{q_n})\\right)=p_n$ for each $j=0,\\ldots ,q_n-1)$ .", "Passing to step $n+1$ , we require that all the intervals $[\\frac{j}{q_n},\\frac{j+1}{q_n}))$ contain at least two intervals of the form $[\\frac{k}{q_{n+1}},\\frac{k+1}{q_{n+1}})$ , we choose two of such (of course only in those $[\\frac{j}{q_n},\\frac{j+1}{q_n})$ which are of positive measure $\\sigma $ ) and apply the rule (REF ) to $A_{n+1}$ , $B_{n+1}$ with $p_n$ replaced with $p_{n+1}$ .", "Note that $\\int e^{2\\pi i q_nx}\\,d\\sigma (x)=1+O(p_n(1+p_n)+1\\cdot p_n)$ , so $e^{2\\pi i q_n\\cdot }\\rightarrow 1$ in measure $\\sigma $ .", "On the other hand $\\sigma (A_n)=p_n$ and the sets $A_n$ are almost independent.", "Since $\\sum _{n\\ge 1}p_n=\\infty $ , for $\\sigma $ -a.e.", "$x$ , we have $x\\in A_n$ for infinitely many $n$ (by the Borel-Cantelli lemma), so (REF ) holds." ], [ "Strong $u$ -MOMO property of systems whose visible measures yield systems in an ec-class", "While we have seen rather unclear relations between ${{F}}$ and ${{F}}_{\\rm ec}$ (cf.", "Proposition REF ), on the topological level we always have the following.", "Proposition 2.15 Let ${{F}}$ be a characteristic class.", "Then ${C}_{{{F}}}\\subset {C}_{{{F}}_{\\rm ec}}$ .", "This follows immediately from the fact that a homeomorphism $T$ (acting on a compact metric space $X$ ) belongs to ${C}_{{{F}}_{\\rm ec}}$ if and only if for each $\\kappa \\in M^e(X,T)$ , $(X,\\mathcal {B}(X),\\kappa ,T)\\in {{F}}$ .", "Note that in view of Proposition REF and Proposition REF , ${C}_{{\\rm RIG}_{(q_n)}}={C}_{({\\rm RIG}_{(q_n)})_{\\rm ec}}.$ The special role of ec-classes stands in the next proposition.", "Proposition 2.16 Let ${{F}}$ be a characteristic class.", "Then $u\\perp {C}_{{{F}}_{\\rm ec}}$ if and only if each element in ${C}_{{{F}}_{\\rm ec}}$ satisfies the strong $u$ -MOMO property.", "The below proof of Proposition REF is an adaptation of the proof of Corollary 9 in [3].", "It uses the following elementary result (see Lemma 18 in [3]).", "Lemma 2.17 Assume that $(c_n)\\subset and $ (mn)N$.", "Then if the sequence $ (cn)$ is contained in a closed convex cone which is not a half-plane then$$\\frac{1}{m_N}\\sum _{n\\le N}c_n \\rightarrow 0 \\iff \\frac{1}{m_N}\\sum _{n\\le N}|c_n|\\rightarrow 0 \\text{ as }N\\rightarrow \\infty .$$$ [Proof of Proposition REF ] Only one implication needs to be proved.", "Suppose that $u\\perp {C}_{{{F}}_{\\rm ec}}$ , and let $(Y,S)\\in {C}_{{{F}}_{\\rm ec}}$ .", "We fix $f\\in C(Y)$ , an increasing sequence $(b_k)$ in ${\\mathbb {N}}$ , with $b_1=1$ and $b_{k+1}-b_k\\rightarrow \\infty $ , and a sequence $(y_k)$ of points in $Y$ .", "We introduce the finite set $\\mathbb {A}:=\\lbrace 1,e^{2\\pi i/3},e^{4\\pi i /3}\\rbrace $ , and for each $k\\ge 1$ , we define $e_k\\in \\mathbb {A}$ such that the complex number $e_k\\Big (\\sum _{b_k\\le n<b_{k+1}}f(S^{n-b_k}y_k)u(n) \\Big )$ is in the closed convex cone $\\lbrace 0\\rbrace \\cup \\lbrace z\\in * : \\text{arg}(z)\\in [-\\pi /3,\\pi /3]\\rbrace $ .", "Then, by Lemma REF , the convergence that we need to prove, i.e.", "$\\frac{1}{b_K}\\sum _{k< K}\\Big |\\sum _{b_k \\le n <b_{k+1}}f(S^{n-b_k}y_k)u(n) \\Big |\\xrightarrow[K\\rightarrow \\infty ]{} 0$ is equivalent to the convergence $\\frac{1}{b_K}\\sum _{k< K}\\sum _{b_k\\le n<b_{k+1}}e_kf(S^{n-b_k}y_k)u(n)\\xrightarrow[K\\rightarrow \\infty ]{} 0.$ Consider the dynamical system $(X,T)$ , where $X:=(Y\\times \\mathbb {A})^{\\mathbb {Z}}$ , and $T$ is the left shift.", "Let $x\\in X$ be such that $x_n:=(S^{n-b_k}y_k,e_k),\\text{ whenever }b_k\\le n<b_{k+1},$ and $x_n=x_0$ is a fixed arbitrary point of $Y\\times \\mathbb {A}$ for $n\\le 0$ .", "By setting $F:=f\\otimes {\\rm Id}$ on $Y\\times \\mathbb {A}$ , it easily follows that (REF ) amounts to $\\frac{1}{b_K}\\sum _{k< K}\\sum _{b_k\\le n<b_{k+1}}F\\circ \\pi _0(T^nx)u(n)\\xrightarrow[K\\rightarrow \\infty ]{} 0.$ To prove the convergence above, we define the subspace $X_x$ as the closure of $\\lbrace T^nx : n\\in {\\mathbb {Z}}\\rbrace $ .", "By assumption on $u$ , we only have to check that the system $(X_x,T)$ is in ${C}_{{{F}}_{\\rm ec}}$ .", "So let $\\mu $ be a visible measure in $(X_x,T)$ , and we first consider the case where $x$ itself is generic for $\\mu $ , along a sequence $(N_\\ell )$ .", "Set $B:=\\bigl \\lbrace (v_j,a_j)_{j\\in {\\mathbb {Z}}}\\in X : (v_1,a_1) = (Sv_0,a_0) \\bigr \\rbrace $ Since $b_{k+1}-b_k\\rightarrow \\infty $ , we have $\\frac{1}{N_\\ell }\\sum _{n<N_\\ell }\\delta _{T^nx}(B)\\xrightarrow[\\ell \\rightarrow \\infty ]{} 1,$ and since the set $B$ is closed, by the Portmanteau theorem, it must be of full measure $\\mu $ in $(X,T)$ .", "Moreover, such a measure $\\mu $ must be $T$ -invariant, hence, $1=\\mu \\left(\\bigcap _{n\\in {\\mathbb {Z}}}T^nB\\right)=\\mu \\Big ( \\bigl \\lbrace (v_j,a_j)_{j\\in {\\mathbb {Z}}}\\in X : \\forall j,\\ (v_j,a_j) = (S^jv_0,a_0) \\bigr \\rbrace \\Big ).$ Denote by $\\mu ^{(0)}$ the restriction of $\\mu $ to the zero-coordinate (that is, with the above notation, the distribution of $(v_0,a_0)$ under $\\mu $ ).", "Since $\\mu $ is $T$ -invariant, it follows that $\\mu ^{(0)}$ is $(S\\times {\\rm Id}_{\\mathbb {A}})$ -invariant.", "Moreover, $Y\\times \\mathbb {A}$ consists of three copies of $Y$ , each of them is invariant under $S\\times {\\rm Id}_{\\mathbb {A}}$ .", "Thus, $\\mu ^{(0)}=\\alpha _0 \\mu ^{(0)}_1 + \\alpha _1 \\mu ^{(0)}_2 + \\alpha _2 \\mu ^{(0)}_3,$ where $\\alpha _0+\\alpha _1+\\alpha _2=1$ , $\\alpha _j\\ge 0$ and $\\mu ^{(0)}_j(Y\\times \\lbrace e^{2\\pi ij /3}\\rbrace )=1$ for $j=0,1,2$ .", "It follows that the ergodic components of $(Y\\times \\mathbb {A},\\mu ^{(0)},S\\times {\\rm Id}_{\\mathbb {A}})$ yield measure-theoretic systems isomorphic to ergodic measures on $(Y,S)$ , hence in ${{F}}$ since $(Y,S)\\in {C}_{{{F}}_{\\rm ec}}$ (this is the moment in our proof where we use the fact that we deal with a characteristic ec-class and not a general characteristic class ${{F}}$ ).", "Thus $(Y\\times \\mathbb {A},\\mu ^{(0)},S\\times {\\rm Id}_{\\mathbb {A}})\\in {{F}}_{\\rm ec}$ .", "Now, using (REF ), we see that $(X_x,\\mu ,T)$ is isomorphic to $(Y\\times \\mathbb {A},\\mu ^{(0)},S\\times {\\rm Id}_{\\mathbb {A}})$ , thus is also in ${{F}}_{\\rm ec}$ .", "Now, suppose that $\\mu \\in V_T(x^{\\prime })$ for some point $x^{\\prime }$ in the orbit closure of $x$ , say $x^{\\prime }=\\lim _{r\\rightarrow \\infty }T^{n_r}x$ .", "If $x^{\\prime }=T^nx$ for some $n\\in {\\mathbb {Z}}$ , then $\\mu \\in V_T(x)$ and we already know that $(X_x,\\mu ,T)\\in \\lbrace \\ast \\rbrace \\subset {{F}}_{\\rm ec}$ in this case.", "If $n_r\\rightarrow -\\infty $ , then $x^{\\prime }=(\\ldots ,x_0,x_0,x_0,\\ldots )$ is a fixed point, and $\\mu =\\delta _{x^{\\prime }}$ .", "In this case, $(X_x,\\mu ,T)\\in \\lbrace \\ast \\rbrace \\subset {{F}}_{\\rm ec}$ .", "If $n_r\\rightarrow +\\infty $ , and if we write $x^{\\prime }=(v_j,a_j)_{j\\in {\\mathbb {Z}}}=\\lim _{r\\rightarrow \\infty }T^{n_r}x$ , then as $b_{k+1}-b_k\\rightarrow \\infty $ , there exists at most one $j\\in {\\mathbb {Z}}$ such that $(v_{j+1},a_j)\\ne (Sv_j,a_j)$ .", "We can then use the same arguments as for $x$ to show that a measure $\\mu $ for which $x$ is quasi-generic satisfies $(X_x,\\mu ,T) \\in {{F}}_{\\rm ec}$ .", "We conclude that $(X_x,T)$ is in ${C}_{{{F}}_{\\rm ec}}$ .", "Remark 2.18 In general, when instead of ${{F}}_{\\rm ec}$ we consider ${{F}}$ , $u\\perp {C}_{{{F}}}$ implies the strong $u$ -MOMO property for each for $(Y,S)$ in which all invariant measures yield systems in ${{F}}$ (in particular, if $(Y,S)\\in {C}_{{{F}}}$ and each invariant measure is visible).", "Question 1 Is Proposition REF true for each characteristic class?", "Remark 2.19 A straightforward adaptation of the proof shows that the subsequence version of Proposition REF also holds: for each characteristic class ${{F}}$ and each increasing sequence of integers $(N_\\ell )$ , $u$ is $(N_\\ell )$ -orthogonal to ${C}_{{{F}}_{\\rm ec}}$ if and only if each element in ${C}_{{{F}}_{\\rm ec}}$ satisfies the strong $u$ -MOMO property along $(N_\\ell )$ .", "See Remarks REF and REF ." ], [ "Lifting lemma", "The purpose of this section is to prove Proposition REF , which is an alternative version of Conze-Downarowicz-Serafin lifting lemma from [9] and seems to be of independent interest.", "It may seem weaker than the original where the genericity was lifted to a single orbit, but the main advantage here is that we do not need assumptions on the nature of the second topological space: it does not have to be a full shift.", "The second advantage is that the result has its extension to the logarithmic case, see Appendix REF , while the lifting lemma of Conze-Downarowicz-Serafin and other results of that type so far have been proved for Cesàro averages.", "Proposition 3.1 Let $(Y,S)$ and $(X,T)$ be two topological systems and $u\\in Y$ generic along an increasing sequence $(N_m)$ for some $S$ -invariant measure $\\kappa $ on $Y$ .", "Let $\\rho $ be a joining of $\\kappa $ with a $T$ -invariant measure $\\nu $ on $X$ .", "Then there exist a sequence $(x_n)\\subset X$ and a subsequence $(N_{m_\\ell })$ such that: the sequence $(S^nu,x_n)$ is generic for $\\rho $ along $(N_{m_\\ell })$ : $ \\frac{1}{N_{m_\\ell }}\\sum _{0\\le n< N_{m_\\ell }}\\delta _{(S^nu,x_n)}\\xrightarrow[\\ell \\rightarrow \\infty ]{}\\rho , $ The sequence $(x_n)$ is constituted of longer and longer pieces of orbits.", "More precisely, $\\lbrace n\\ge 0:\\ x_{n+1}\\ne Tx_n\\rbrace $ is of the form $\\lbrace b_1<b_2<\\cdots <b_k<b_{k+1}<\\cdots \\rbrace $ , where $b_{k+1}-b_k\\rightarrow \\infty $ ." ], [ "Good sequences of partitions", "We need a convenient tool to estimate the weak*-convergence of a sequence of probability measures to a given measure.", "Definition 3.1 Let $(E,d)$ be a compact metric space, and let $\\nu $ be a Borel probability measure on $E$ , i.e.", "$\\nu \\in M(E)$ .", "We consider a sequence $({P}_\\ell )$ of finite partitions of $E$ into Borel subsets.", "The sequence $({P}_\\ell )$ is said to be good for $(E,\\nu )$ if the following conditions hold: for each $\\ell $ , ${P}_{\\ell +1}$ refines ${P}_\\ell $ , $\\mathop {\\mathrm {diam}}({P}_\\ell ):=\\max _{P\\text{ atom of }{P}_\\ell } \\mathop {\\mathrm {diam}}(P) \\xrightarrow[\\ell \\rightarrow \\infty ]{}0$ , for each $\\ell $ and each atom $P$ of ${P}_\\ell $ , $\\nu (\\partial P)=0$ .", "The motivation for introducing this definition comes from the following result.", "Lemma 3.2 If $({P}_\\ell )$ is a good sequence of partitions for $(E,\\nu )$ , then a sequence $(\\nu _n)\\subset M(E)$ converges to $\\nu $ in the weak*-topology if and only if, for each $\\ell $ and each atom $P$ of ${P}_\\ell $ , we have $\\nu _n(P) \\xrightarrow[n\\rightarrow \\infty ]{} \\nu (P).$ If $\\nu _n\\xrightarrow[n\\rightarrow \\infty ]{\\text{w}*} \\nu $ , then by the Portmanteau theorem, for each $P\\subset E$ such that $\\nu (\\partial P)=0$ , we have $\\nu _n(P)\\rightarrow \\nu (P)$ .", "Conversely, assume that for each $\\ell $ and each atom $P$ of ${P}_\\ell $ we have $\\nu _n(P)\\rightarrow \\nu (P)$ .", "Then any weak*-limit $\\mu $ of a subsequence of $(\\nu _n)$ satisfies (again by the Portmanteau theorem) $\\mu (P)=\\nu (P)$ for each atom $P$ of ${P}_\\ell $ .", "But since $\\mathop {\\mathrm {diam}}({P}_\\ell )\\rightarrow 0$ , the sequence $({P}_\\ell )$ separates points in $E$ , hence it generates the Borel $\\sigma $ -algebra of $E$ .", "Thus we have $\\mu =\\nu $ , and using the compactness of $M(E)$ for the weak* topology, we get that $\\nu _n\\xrightarrow[n\\rightarrow \\infty ]{\\text{w}*} \\nu $ .", "Lemma 3.3 For each $\\nu \\in M(E)$ of a compact metric space $(E,d)$ , there exists a good sequence of partitions for $(E,\\nu )$ .", "We first show that, for each $\\ell \\ge 1$ , there exists a finite partition ${Q}_\\ell $ in which each atom $Q$ satisfies $\\mathop {\\mathrm {diam}}(Q)<1/\\ell $ , $\\nu (\\partial Q)=0$ .", "Indeed, by compactness, there exists a finite set $\\lbrace x_1,\\ldots ,x_k\\rbrace \\subset E$ such that $ E\\subset \\bigcup _{1\\le i\\le k} B\\bigl (x_i,\\frac{1}{3\\ell }\\bigr ).", "$ Then, for each $1\\le i\\le k$ , there exist at most countably many $r>0$ such that $\\nu \\left(\\partial B(x_i,r)\\right) > 0.$ Therefore, we can find $r\\in \\left(\\frac{1}{3\\ell },\\frac{1}{2\\ell }\\right)$ such that $\\forall 1\\le i\\le k,\\quad \\nu \\left(\\partial B(x_i,r)\\right) = 0.$ Then the partition ${Q}_\\ell $ generated by the open balls $B(x_i,r)$ , $1\\le i\\le k$ , satisfies the required conditions.", "Once we have ${Q}_\\ell $ for each $\\ell \\ge 1$ , we set ${P}_\\ell :={Q}_1\\vee \\cdots \\vee {Q}_\\ell ,$ and we get a good sequence $({P}_\\ell )$ for $(E,\\nu )$ .", "Lemma 3.4 Let $({P}_\\ell )$ be a good sequence of partitions for $(E_1,\\nu _1)$ , and let $({Q}_\\ell )$ be a good sequence of partitions for $(E_2,\\nu _2)$ .", "Then for each coupling $\\rho $ of $\\nu _1$ and $\\nu _2$ , $({P}_\\ell \\times {Q}_\\ell )$ is a good sequence of partitions for $(E_1\\times E_2,\\rho )$ .", "This is obvious, since for each atom $P$ of ${P}_\\ell $ and each atom $Q$ of ${Q}_\\ell $ , $ \\partial (P\\times Q) \\subset (\\partial P\\times E_2) \\cup (E_1\\times \\partial Q), $ and the marginals of $\\rho $ are $\\nu _1$ and $\\nu _2$ ." ], [ "Proof of Proposition ", "Without loss of generality, we can (and we do) assume that the measure-theoretic dynamical system $(Y,\\kappa ,S)$ is aperiodic.", "Indeed, if this is not the case, we consider any uniquely ergodic topological system $(Y^{\\prime },S^{\\prime })$ whose unique invariant measure $\\kappa ^{\\prime }$ is such that $(Y^{\\prime },\\kappa ^{\\prime },S^{\\prime })$ is aperiodic.", "Then we take any point $u^{\\prime }\\in Y^{\\prime }$ , and we replace $Y$ by $Y\\times Y^{\\prime }$ , $S$ by $S\\times S^{\\prime }$ , and $u$ by $(u,u^{\\prime })$ .", "We also replace $(N_m)$ by a subsequence of $(N_m)$ along which $(u,u^{\\prime })$ is generic, for some measure $\\tilde{\\kappa }$ whose marginals have to be $\\kappa $ and $\\kappa ^{\\prime }$ .", "But then the system $(Y\\times Y^{\\prime },\\tilde{\\kappa },S\\times S^{\\prime })$ is aperiodic, because it is an extension of the aperiodic system $(Y^{\\prime },\\kappa ^{\\prime },S^{\\prime })$ .", "We fix a good sequence of partitions $({Q}_\\ell )$ for $(Y,\\kappa )$ and a good sequence of partitions $({P}_\\ell )$ for $(X,\\nu )$ .", "Then by Lemma REF , $({Q}_\\ell \\times {P}_\\ell )$ is a good sequence of partitions for $(Y\\times X,\\rho )$ .", "Definition 3.2 Let $M>0$ .", "A subset $E$ of ${\\mathbb {N}}$ is said to be $M$ -separated if for each integers $n\\ne m$ , $n,m\\in E\\Longrightarrow |n-m|\\ge M$ .", "The main argument to prove Proposition REF stands in the following proposition.", "Proposition 3.5 Under the assumptions of Proposition REF , and assuming also that $(Y,\\kappa ,S)$ is aperiodic (see above), given ${\\ell _0}\\ge 1$ and $\\varepsilon \\in (0,\\frac{1}{2})$ , there exists a sequence $(x_n)$ of points in $X$ such that: $\\lbrace n\\ge 0: x_{n+1}\\ne Tx_n\\rbrace $ is $\\frac{1}{\\varepsilon }$ -separated, for each atom $A$ of ${Q}_{\\ell _0}\\times {P}_{\\ell _0}$ , we have $\\rho (A)-\\varepsilon < \\liminf _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A }(S^nu,x_n),$ and $\\limsup _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A }(S^nu,x_n) < \\rho (A)+\\varepsilon .$ Let $h$ be a natural number such that $\\frac{1}{h}<\\varepsilon $ .", "We claim that for $\\ell $ large enough, we can find a set $B\\subset Y$ which is measurable with respect to $\\bigvee _{0\\le j\\le h-1}S^j{Q}_\\ell $ , and such that $B,SB,\\ldots ,S^{h-1}B$ are pairwise disjoint, $\\kappa \\left(\\bigcup _{0\\le j\\le h-1}S^jB\\right)>1-\\varepsilon $ .", "Indeed, since $(Y,\\kappa ,S)$ is assumed to be aperiodic, we can use the Rokhlin lemma to find a Borel subset $\\tilde{B}\\subset Y$ such that $\\tilde{B}, S\\tilde{B},\\ldots , S^{h-1}\\tilde{B}$ are pairwise disjoint, and such that $ \\kappa \\left(\\bigcup _{0\\le j\\le h-1}S^j\\tilde{B}\\right)>1-\\frac{\\varepsilon }{2}.", "$ Then we use the fact that the good sequence of partitions $({Q}_\\ell )$ generates the Borel $\\sigma $ -algebra: it follows that for $\\ell $ large enough, we can find a ${Q}_\\ell $ -measurable set $B^{\\prime }$ such that $ \\kappa (B^{\\prime }\\bigtriangleup \\tilde{B})< \\frac{\\varepsilon }{8h^2}.", "$ For each $1\\le j\\le h-1$ , we have $ B^{\\prime }\\cap S^jB^{\\prime } \\subset (B^{\\prime }\\setminus \\tilde{B})\\cup (S^jB^{\\prime }\\setminus S^j\\tilde{B}), $ hence $\\kappa (B^{\\prime }\\cap S^jB^{\\prime })\\le \\frac{\\varepsilon }{4h^2}.$ It remains to define $B$ by $ B:= B^{\\prime }\\setminus \\left(\\bigcup _{1\\le j\\le h-1} S^jB^{\\prime }\\right).", "$ Then, by construction, $B$ is disjoint from $S^jB$ for each $1\\le j\\le h-1$ , thus $B,SB,\\ldots ,S^{h-1}B$ are pairwise disjoint.", "Moreover, from (REF ), we have $\\kappa (B) \\ge \\kappa (B^{\\prime })-\\frac{\\varepsilon }{4h}\\ge \\kappa (\\tilde{B})-\\frac{\\varepsilon }{2h},$ which implies $ \\kappa \\left(\\bigcup _{0\\le j\\le h-1}S^j B\\right) = h\\kappa (B) \\ge h\\kappa (\\tilde{B})-\\frac{\\varepsilon }{2}>1-\\varepsilon , $ and our first claim is proved.", "Since $u$ is generic for $\\kappa $ along $(N_m)$ , and since the set $\\bigcup _{0\\le j\\le h-1}S^jB$ is measurable with respect to $\\bigvee _{0\\le j\\le 2h}S^j{Q}_\\ell $ (in particular, the $\\kappa $ -measure of its boundary vanishes), we have $\\frac{1}{N_m} \\sum _{0\\le n< N_m} {1}_{\\bigcup _{0\\le j\\le h-1}S^jB} (S^n u) \\xrightarrow[m\\rightarrow \\infty ]{} \\kappa \\left(\\bigcup _{0\\le j\\le h-1}S^jB\\right) > 1-\\varepsilon .$ This implies in particular that the set $P_B(u):=\\lbrace n\\ge 0:\\ S^nu\\in B\\rbrace $ is infinite.", "We number in order the elements of this set: $ P_B(u) = \\lbrace b_1<b_2<\\cdots <b_k<\\cdots \\rbrace \\; $ The integers $(b_k)$ will correspond to the times when we will be allowed to change the orbit of the desired sequence.", "As $B$ is disjoint from $S^jB$ for each $1\\le j\\le h-1$ , the set $P_B(u)$ is $h$ -separated, hence $\\frac{1}{\\varepsilon }$ -separated.", "We consider the partition $\\bigvee _{0\\le j\\le h-1}(S\\times T)^{-j}({Q}_{\\ell _0}\\times {P}_{\\ell _0})$ of $Y\\times X$ .", "Any atom of this partition is of the form ${\\bar{Q}}\\times {\\bar{P}}$ , where ${\\bar{Q}}$ (respectively ${\\bar{P}}$ ) is an atom of $\\bigvee _{0\\le j\\le h-1}S^{-j}{Q}_{\\ell _0}$ (respectively of $\\bigvee _{0\\le j\\le h-1}T^{-j}{P}_{\\ell _0}$ ).", "For such atoms ${\\bar{Q}}$ and ${\\bar{P}}$ , we can write ${\\bar{Q}}=Q_0\\cap S^{-1}Q_1\\cap \\cdots \\cap S^{-(h-1)}Q_{h-1},$ each $Q_j$ being an atom of ${Q}_{\\ell _0}$ , and ${\\bar{P}}=P_0\\cap S^{-1}P_1\\cap \\cdots \\cap S^{-(h-1)}P_{h-1},$ each $P_j$ being an atom of ${P}_{\\ell _0}$ .", "Since the $\\kappa $ -measure of the boundary of each involved set is always 0, we again have for each atom ${\\bar{Q}}$ of $\\bigvee _{0\\le j\\le h-1}S^{-j}{Q}_{\\ell _0}$ $\\frac{1}{N_m} \\sum _{b_k< N_m} {1}_{{\\bar{Q}}} (S^{b_k} u) =\\frac{1}{N_m} \\sum _{0\\le n< N_m} {1}_{B\\cap {\\bar{Q}}} (S^n u)\\xrightarrow[m\\rightarrow \\infty ]{} \\kappa (B\\cap {\\bar{Q}}).$ If $C$ is a measurable subset of $Y$ with $\\kappa (C)>0$ , we denote by $\\rho ^Y_C$ the marginal on $X$ of the conditional probability measure $\\rho (\\,\\cdot \\,|C\\times X)$ .", "Then, for each measurable $A\\subset X$ , we have $\\begin{split}\\rho (C\\times A) &=\\rho \\bigl ( (C\\times X) \\cap (Y\\times A) \\bigr ) \\\\&=\\rho (C\\times X) \\, \\rho \\bigl (Y\\times A|C\\times X\\bigr )\\\\&=\\kappa (C) \\, \\rho ^Y_C(A).\\end{split}$ On an appropriate probability space, we construct a sequence $(\\xi _k)$ of independent random variables, taking values in $X$ , such that for each $k$ , $\\xi _k$ is distributed according to $\\rho ^Y_{B\\cap {\\bar{Q}}}$ , where ${\\bar{Q}}$ is the atom of $\\bigvee _{0\\le j\\le h-1}S^{-j}{Q}_{\\ell _0}$ containing $S^{b_k}u$ .", "For each atom ${\\bar{Q}}$ of $\\bigvee _{0\\le j\\le h-1}S^{-j}{Q}_{\\ell _0}$ and each atom ${\\bar{P}}$ of $\\bigvee _{0\\le j\\le h-1}T^{-j}{P}_{\\ell _0}$ , by (REF ), the law of large numbers and (REF ), with probability 1, we have $\\frac{1}{N_m} \\sum _{b_k< N_m} {1}_{{\\bar{Q}}} (S^{b_k} u) {1}_{{\\bar{P}}} (\\xi _k) \\xrightarrow[m\\rightarrow \\infty ]{} \\kappa (B\\cap {\\bar{Q}}) \\rho ^Y_{B\\cap {\\bar{Q}}}({\\bar{P}}) = \\rho \\bigl ((B\\cap {\\bar{Q}})\\times {\\bar{P}}\\bigr ).$ Let us fix a realization of $(\\xi _k)$ which satisfies (REF ) for each atom ${\\bar{Q}}\\times {\\bar{P}}$ of $\\bigvee _{0\\le j\\le h-1}(S\\times T)^{-j}({Q}_{\\ell _0}\\times {P}_{\\ell _0})$ .", "Then, for each $n\\ge 0$ , we define the point $x_n\\in X$ as follows: $x_n := {\\left\\lbrace \\begin{array}{ll}T^{n-b_1}\\xi _1 &\\text{ if }n<b_1,\\\\T^{n-b_k}\\xi _k &\\text{ if }b_k\\le n<b_{k+1}\\text{ for some }k\\ge 1.\\end{array}\\right.", "}$ The set of integers $n$ such that $x_{n+1}\\ne Tx_n$ is contained in $P_B(u)$ , therefore, it is $\\frac{1}{\\varepsilon }$ -separated.", "Now, let $A=Q\\times P$ be a fixed atom of ${Q}_{\\ell _0}\\times {P}_{\\ell _0}$ .", "We set $R:=\\bigcup _{0\\le j\\le h-1}S^jB\\times X,$ and we observe that $\\rho (R) = \\kappa \\left(\\bigcup _{0\\le j\\le h-1}S^jB\\right) >1-\\varepsilon .$ We also note that for each $n\\ge b_1$ , $(S^nu,x_n)\\in R$ if and only if there exists $k$ and $0\\le j\\le h-1$ such that $n=b_k+j$ .", "In this case, $(S^nu,x_n)\\in A\\cap R$ if and only if the atom ${\\bar{Q}}\\times {\\bar{P}}$ of $\\bigvee _{0\\le j\\le h-1}(S\\times T)^{-j}({Q}_{\\ell _0}\\times {P}_{\\ell _0})$ containing $(S^{b_k}u,\\xi _k)$ satisfies $Q_j=Q$ and $P_j=P$ (using the notations given in (REF ) and (REF ), and remembering that $A=Q\\times P$ ).", "We can then use (REF ) to get $\\begin{split}&\\frac{1}{N_m} \\sum _{b_1\\le n<N_m} {1}_{A\\cap R}(S^nu,x_n) \\\\&= \\frac{1}{N_m} \\sum _{b_k<N_m} \\sum _{0\\le j\\le h-1} \\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} {1}_{{\\bar{Q}}\\times {\\bar{P}}}(S^{b_k}u,\\xi _k)\\\\&\\xrightarrow[m\\rightarrow \\infty ]{} \\sum _{0\\le j\\le h-1} \\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} \\rho \\bigl ((B\\cap {\\bar{Q}})\\times {\\bar{P}}\\bigr ).\\end{split}$ But, on the other hand, we can write $\\begin{split}\\rho (A\\cap R) &= \\sum _{0\\le j\\le h-1} \\rho \\bigl (A\\cap (S^jB\\times X)\\bigr )\\\\&= \\sum _{0\\le j\\le h-1} \\rho \\bigl ((S^{-j}Q\\times T^{-j}P)\\cap (B\\times X)\\bigr )\\\\&= \\sum _{0\\le j\\le h-1} \\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} \\rho \\bigl ((B\\cap {\\bar{Q}})\\times {\\bar{P}}\\bigr ).\\end{split}$ From (REF ) and (REF ), it follows that $\\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A\\cap R}(S^nu,x_n) \\xrightarrow[m\\rightarrow \\infty ]{} \\rho (A\\cap R).$ From (REF ), we get that $\\lim _{m\\rightarrow \\infty }\\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{(Y\\times X)\\setminus R}(S^nu,x_n)<\\varepsilon ,$ and since ${1}_{A}\\le {1}_{A\\cap R}+{1}_{Y\\times X\\setminus R}$ , this yields by (REF ), $\\limsup _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A }(S^nu,x_n)&< \\lim _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A\\cap R}(S^nu,x_n) + \\varepsilon \\\\&=\\rho (A\\cap R)+ \\varepsilon \\\\&\\le \\rho (A)+\\varepsilon ,$ and we have (REF ).", "On the other hand, using ${1}_{A}\\ge {1}_{A\\cap R}$ , we get by (REF ) $\\liminf _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A }(S^nu,x_n)&\\ge \\lim _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A\\cap R}(S^nu,x_n) \\\\&=\\rho (A\\cap R) \\\\&>\\rho (A)-\\varepsilon .$ and we have (REF ).", "We can now give the proof of Proposition REF , in which we use the following obvious fact: if we modify the sequence $(x_n)$ given by Proposition REF on a finite number of terms, we still get (REF ) and (REF ).", "[Proof of Proposition (REF )] We fix a sequence $(\\varepsilon _\\ell )_{\\ell \\ge 1}$ of numbers in $(0,\\frac{1}{2})$ , decreasing to 0, and we construct inductively the desired sequence $(x_n)$ and the subsequence $(N_{m_\\ell })$ by a repeated use of Proposition REF .", "We start by applying Proposition REF with $\\varepsilon :=\\varepsilon _1$ and ${\\ell _0}:=1$ .", "It provides us with an integer $m_1$ , and a finite sequence $(x_n)_{0\\le n<N_{m_1}}$ of points in $X$ such that the set of integers $n\\in \\lbrace 0,\\ldots ,N_{m_1}-2\\rbrace $ such that $x_{n+1}\\ne Tx_n$ is $\\frac{1}{\\varepsilon _1}$ -separated, for each atom $A$ of ${Q}_1\\times {P}_1$ , we have $\\rho (A)-\\varepsilon _1<\\frac{1}{N_{m_1}} \\sum _{0\\le n<N_{m_1}} {1}_{A}(S^nu,x_n) < \\rho (A)+\\varepsilon _1.$ Now, assume that for some $\\ell \\ge 1$ we have already constructed $m_1<\\cdots <m_\\ell $ and the sequence $(x_n)_{0\\le n<N_{m_\\ell }}$ of points in $X$ such that for each $1\\le j<\\ell $ , the set of integers $n\\in \\lbrace N_{m_{j-1}},\\ldots ,N_{m_j}-2\\rbrace $ such that $x_{n+1}\\ne Tx_n$ is $\\frac{1}{\\varepsilon _j}$ -separated (with the convention that $N_{m_0}=0$ ), for each atom $A$ of ${Q}_\\ell \\times {P}_\\ell $ , we have $\\rho (A)-\\varepsilon _\\ell <\\frac{1}{N_{m_\\ell }} \\sum _{0\\le n<N_{m_\\ell }} {1}_{A}(S^nu,x_n) < \\rho (A)+\\varepsilon _\\ell .$ Then we apply again Proposition REF , with $\\varepsilon :=\\varepsilon _{\\ell +1}$ and ${\\ell _0}:=\\ell +1$ .", "It provides us with an integer $m_{\\ell +1}$ and a finite sequence of points $(x_n)_{N_{m_\\ell }\\le n<N_{m_{\\ell +1}}}$ in $X$ which satisfy: the set of integers $n\\in \\lbrace N_{m_\\ell },\\ldots ,N_{m_{\\ell +1}}-2\\rbrace $ such that $x_{n+1}\\ne Tx_n$ is $\\frac{1}{\\varepsilon _{\\ell +1}}$ -separated, for each atom $A$ of ${Q}_{\\ell +1}\\times {P}_{\\ell +1}$ , we have $\\rho (A)-\\varepsilon _{\\ell +1}<\\frac{1}{N_{m_{\\ell +1}}} \\sum _{0\\le n<N_{m_{\\ell +1}}} {1}_{A}(S^nu,x_n) < \\rho (A)+\\varepsilon _{\\ell +1}.$ (We keep the points $(x_n)_{0\\le n<N_{m_\\ell }}$ already provided by the induction hypothesis, refering to the obvious fact stated before the proof.)", "Moreover, we can assume that the sequence $(\\varepsilon _\\ell )$ decreases sufficiently fast so that the validity of (REF ) for each atom $A$ of ${Q}_{\\ell +1}\\times {P}_{\\ell +1}$ ensures the validity of the analog inequalities for each $A$ which is a finite union of atoms of ${Q}_{\\ell +1}\\times {P}_{\\ell +1}$ (in particular, for each $A$ which is an atom of the previous partitions), but with $\\varepsilon _\\ell $ instead of $\\varepsilon _{\\ell +1}$ .", "The sequence $(x_n)_{n\\ge 0}$ of points in $X$ and the subsequence $(N_{m_\\ell })$ we construct with the above inductive procedure then satisfy the conditions announced in Proposition REF ." ], [ "Logarithmic case", "We would like to study the logarithmic version of Proposition REF , in which we replace each arithmetic average of the form $\\frac{1}{N_m}\\sum _{0\\le n< N_m} f(n)$ by the logarithmic average $\\frac{1}{L(N_m)}\\sum _{1\\le n\\le N_m} \\frac{1}{n}f(n).$ (Here we use the notation $L(N):=1+\\frac{1}{2}+\\cdots +\\frac{1}{N}$ .)", "In fact, this logarithmic version, whose statement is written below, is also valid, and the arguments to prove it are exactly the same as in the arithmetic average case.", "We just point out below the few technical changes that need to be made in the proof for the logarithmic case.", "Proposition 3.6 Let $(Y,S)$ and $(X,T)$ be two topological systems and $u\\in Y$ , logarithmically generic along an increasing sequence $(N_m)$ for some $S$ -invariant measure $\\kappa $ on $Y$ .", "Let $\\rho $ be a joining of $\\kappa $ with a $T$ -invariant measure $\\nu $ on $X$ .", "Then there exist a sequence $(x_n)\\subset X$ and a subsequence $(N_{m_\\ell })$ such that: the sequence $(S^nu,x_n)$ is logarithmically generic for $\\rho $ along $(N_{m_\\ell })$ : $ \\frac{1}{L(N_{m_\\ell })}\\sum _{1\\le n\\le N_{m_\\ell }}\\frac{1}{n}\\delta _{(S^nu,x_n)}\\xrightarrow[\\ell \\rightarrow \\infty ]{}\\rho , $ the set $\\lbrace n\\ge 0:\\ x_{n+1}\\ne Tx_n\\rbrace $ is of the form $\\lbrace b_1<b_2<\\cdots <b_k<b_{k+1}<\\cdots \\rbrace $ , where $b_{k+1}-b_k\\rightarrow \\infty $ .", "The changes that need to be made to the proof are almost all quite obvious, they consist in formally replacing the arithmetic average by the logarithmic average.", "One point maybe needs some explanations, namely when we arrive at the proof of the logarithmic analog of (REF ).", "We put these explanations in the form of a lemma, which we will apply in the following context: $(d_k)$ is the ordered sequence of positive integers $n$ such that $S^nu\\in B\\cap {\\bar{Q}}$ , and the sequence $(\\rho _k)$ is defined by $\\rho _k:={1}_{{\\bar{P}}}(\\xi _k)$ .", "Lemma 3.7 Let $(d_k)$ be an increasing sequence of positive integers such that $\\frac{1}{L(N_{m})}\\sum _{d_k\\le N_{m}}\\frac{1}{d_k} \\xrightarrow[m\\rightarrow \\infty ]{} \\kappa \\in [0,1],$ and let $(\\rho _k)$ be a sequence of real numbers in $[0,1]$ such that $\\frac{1}{K} \\sum _{1\\le k\\le K} \\rho _k \\xrightarrow[K\\rightarrow \\infty ]{} \\rho \\in [0,1].$ Then we have $\\frac{1}{L(N_{m})}\\sum _{d_k\\le N_{m}}\\frac{\\rho _k}{d_k} \\xrightarrow[m\\rightarrow \\infty ]{} \\kappa \\rho .$ For each $m$ , let us denote by $k_m$ the largest $k$ such that $d_k\\le N_m$ .", "We use the classical identity $\\sum _{d_k\\le N_{m}} \\frac{\\rho _k}{d_k} = \\sum _{1\\le k < k_m} \\left( \\frac{1}{d_k}-\\frac{1}{d_{k+1}} \\right) (\\rho _1+\\cdots +\\rho _k) + \\frac{1}{d_{k_m}}(\\rho _1+\\cdots +\\rho _{k_m}).$ Given $\\varepsilon >0$ , let $K_\\varepsilon $ be such that $ K\\ge K_\\varepsilon \\Longrightarrow \\left| \\frac{1}{K} \\sum _{1\\le k\\le K} \\rho _k - \\rho \\right| < \\varepsilon .", "$ We can then write $&\\left| \\frac{1}{L(N_{m})} \\sum _{d_k\\le N_{m}} \\frac{\\rho _k}{d_k} - \\frac{1}{L(N_{m})} \\sum _{d_k\\le N_{m}} \\frac{\\rho }{d_k}\\right|\\\\&= \\left| \\frac{1}{L(N_{m})} \\sum _{K_\\varepsilon \\le k < k_m} k \\left( \\frac{1}{d_k}-\\frac{1}{d_{k+1}} \\right) \\left( \\frac{1}{k} (\\rho _1+\\cdots +\\rho _k)-\\rho \\right) \\right| + O\\left(\\frac{1}{L(N_{m})}\\right)\\\\&< \\varepsilon + O\\left(\\frac{1}{L(N_{m})}\\right).$ But by assumption, we have $\\frac{1}{L(N_{m})} \\sum _{d_k\\le N_{m}} \\frac{\\rho }{d_k}\\xrightarrow[m\\rightarrow \\infty ]{} \\kappa \\rho ,$ hence we get $\\frac{1}{L(N_{m})} \\sum _{d_k\\le N_{m}} \\frac{\\rho _k}{d_k}\\xrightarrow[m\\rightarrow \\infty ]{} \\kappa \\rho .$ The last place in the proof where a (very easy) correction should be made in the logarithmic case is to get the analog of (REF ): at some point we have to replace some coefficients $\\frac{1}{b_k+j}$ by $\\frac{1}{b_k}$ , which is of no consequence since $j$ remains bounded between 0 and $h-1$ here.", "To be more precise, (REF ) becomes $\\begin{split}&\\frac{1}{L(N_m)} \\sum _{b_1\\le n<N_m} \\frac{1}{n}{1}_{A\\cap R}(S^nu,x_n) \\\\&= \\frac{1}{L(N_m)} \\sum _{b_k<N_m} \\sum _{0\\le j\\le h-1} \\frac{1}{b_k+j}\\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} {1}_{{\\bar{Q}}\\times {\\bar{P}}}(S^{b_k}u,\\xi _k)\\\\&= \\frac{1}{L(N_m)} \\sum _{b_k<N_m} \\sum _{0\\le j\\le h-1} \\frac{1}{b_k}\\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} {1}_{{\\bar{Q}}\\times {\\bar{P}}}(S^{b_k}u,\\xi _k) + o(1)\\\\&\\xrightarrow[m\\rightarrow \\infty ]{} \\sum _{0\\le j\\le h-1} \\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} \\rho \\bigl ((B\\cap {\\bar{Q}})\\times {\\bar{P}}\\bigr ).\\end{split}$" ], [ "Proof of Theorem ", "[Proof of Theorem REF ] Take any topological system $(X,T)\\in {C}_{{{F}}}$ and fix $f\\in C(X)$ , $x\\in X$ .", "Take any increasing sequence $(N_k)$ for which, with no loss of generality, we can assume that $\\frac{1}{N_k}\\sum _{n\\le N_k} \\delta _{(T^nx,S^nu)}\\rightarrow \\rho $ .", "It follows that $\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}f(T^nx)u(n)=\\int f\\otimes \\pi _0\\,d\\rho .$ But $\\rho $ is a joining of some $T$ -invariant measure $\\nu \\in V(X,T)$ for which $x$ is generic along $(N_k)$ , and some Furstenberg system $\\kappa $ of $u$ .", "Since $(X,T)\\in {C}_{{{F}}}$ , the system $(X,\\nu ,T)$ is in ${{F}}$ , and the integral on the right-hand side above vanishes by the Veech condition and Proposition REF ." ], [ "Proof of Theorem ", "Before we begin the proof, let us make the following remark concerning topological models.", "Given an automorphism $(Z,\\mathcal {D},\\kappa , R)$ , and a fixed subset of full measure of ergodic components of $\\kappa $ , recall that by a Hansel model of $R$ , we mean a topological system $(X,T)$ which has a $T$ -invariant measure $\\nu $ such that, as dynamical systems, $\\kappa $ and $\\nu $ are isomorphic and such that each point $x\\in X$ is generic for one of these chosen ergodic components.", "In [29], it is proved that each automorphism has a Hansel model.", "We assume that $u\\perp {C}_{{{F}}_{\\rm ec}}$ for some characteristic class ${{F}}$ .", "Take $\\kappa \\in V(u)$ and fix $(N_m)$ so that $\\frac{1}{N_m}\\sum _{n\\le N_m} \\delta _{S^nu}\\rightarrow \\kappa .$ Denote by ${\\cal A}(\\kappa )\\subset {\\cal B}(X_{u})$ the largest ${{F}}_{\\rm ec}$ -factor of $(X_{u},\\kappa ,S)$ , i.e.", "${\\cal A}(\\kappa )={\\cal B}(X_{u})_{{{F}}_{\\rm ec}}$ .", "Consider the factor $(X_{u}/{\\cal A}(\\kappa ), {\\cal A}(\\kappa ), \\kappa |_{{\\cal A}(\\kappa )},S)$ and take a Hansel model $(X,\\nu ,T)$ of it (by choosing in the ergodic decomposition of $\\kappa |_{{\\cal A}(\\kappa )}$ only ergodic measures in ${{F}}$ ).", "By definition, $(X,T)\\in {C}_{{{F}}_{\\rm ec}}.$ Fix a measure-theoretic factor map $J\\colon (X_{u},\\kappa ,S)\\rightarrow (X,\\nu ,T)$ such that $J^{-1}({\\cal B}(X))={\\cal A}(\\kappa )$ , and let $\\nu _J$ denote the corresponding graph joining (of $\\nu $ and $\\kappa |_{{\\cal A}(\\kappa )}$ ).", "Let $\\widehat{\\nu }_J$ be the relatively independent extension of $\\nu _J$ to a joining of $\\nu $ and $\\kappa $ : for $f\\in L^2(\\nu )$ and $g\\in L^2(\\kappa )$ , we have $\\int _{X_{u}\\times X} g\\otimes f \\, d\\widehat{\\nu }_J = \\int _{X_{u}} {\\mathbb {E}}^{\\kappa }\\bigl (g | {\\cal A}(\\kappa )\\bigr )\\,f\\circ J \\, d\\kappa .$ Now, by applying Proposition REF , we can find $(x_n)\\subset Y$ such that $((x_n),u) \\text{ is generic for }\\widehat{\\nu }_J\\text{ along some subsequence }(N_{m_\\ell }),$ and the set $\\lbrace n\\ge 0:\\ x_{n+1}\\ne Tx_n\\rbrace $ is of the form $\\lbrace b_1<b_2<\\cdots \\rbrace $ with $b_{k+1}-b_k\\rightarrow \\infty $ .", "Since $u\\perp {C}_{{{F}}_{\\rm ec}}$ , (REF ) and Proposition REF ensure that the system $(Y,S)$ satisfies the strong $u$ -MOMO property.", "Therefore, for each $f\\in C(Y)$ we have $\\lim _{m\\rightarrow \\infty }\\frac{1}{N_m}\\sum _{n\\le N_m}f(x_n)u(n)=$ $\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{b_{K_m}}\\sum _{k<K_m}\\left(\\sum _{b_k\\le n<b_{k+1}}f(T^{n-b_k}x_{b_k})u(n)\\right)=0,$ and it follows from (REF ) that $\\int f\\otimes \\pi _0\\, d \\widehat{\\nu }_J=0$ .", "Using (REF ), we get $\\int _{X_{u}} {\\mathbb {E}}^{\\kappa }\\bigl (\\pi _0 | {\\cal A}(\\kappa )\\bigr )\\,f\\circ J \\, d\\kappa =0.$ But $\\lbrace f\\circ J:\\ f\\in C(X)\\rbrace $ is dense in $L^2({\\cal A}(\\kappa ))$ and therefore $\\pi _0\\perp L^2({\\cal A}(\\kappa ))$ , which is the Veech condition for $u$ with respect to the characteristic class ${{F}}_{\\rm ec}$ ." ], [ "Cancellations. Proof of Corollaries ", "We need the following interpretation of the Veech condition in terms of relative uniform mixing (K-mixing) of the function $\\pi _0$ .", "For $n\\in {\\mathbb {N}}$ , let $\\pi _n:=\\pi _0\\circ S^n$ .", "Proposition 5.1 Let $\\kappa \\in V_S(u)$ and $\\int \\pi _0\\,d\\kappa =0$ .", "Then the following conditions are equivalent: $\\pi _0\\perp L^2(\\Pi (\\kappa ))$ , $\\pi _0$ is relatively K-mixing, i.e.", "for each $\\varepsilon >0$ , there exists $N$ such that $\\left|\\int \\pi _0 {1}_C\\,d\\kappa -\\int \\pi _0\\, d\\kappa \\int {1}_C d\\kappa \\right|=\\left| \\int \\pi _0 {1}_C\\,d\\kappa \\right|<\\varepsilon $ for each set $C\\in \\sigma (\\pi _n,\\pi _{n+1},...)$ and $n\\ge N$ .", "If we additionally assume that $u$ takes values in a finite set $E\\subset and $ (Mk)$ is a sequence along which we have a Furstenberg system $$ then the above conditions are equivalent to\\begin{enumerate}\\item [(c)]for each \\varepsilon >0 there exists N\\ge 1 such that for any s\\ge 1 and any function fdepending on coordinates N\\le n,n+1,\\ldots ,n+s, \\Vert f\\Vert _{C(X_{u})}\\le 1, we have\\limsup _{k\\rightarrow \\infty }\\left|\\frac{1}{M_k}\\sum _{m\\le M_k}u(m)f(S^mu)\\right|<\\varepsilon .\\end{enumerate}$ (a) $\\Rightarrow $ (b).", "Assume that ${\\mathbb {E}}(\\pi _0|\\Pi (\\kappa ))=0$ .", "Let $C\\in \\sigma (\\pi _n,\\pi _{n+1},\\ldots )$ .", "We have $\\left|\\int \\pi _0 {1}_C d\\kappa \\right|=\\left|\\int {1}_C {\\mathbb {E}}(\\pi _0|\\sigma (\\pi _n,\\pi _{n+1},...)\\, d\\kappa \\right|\\\\\\le \\int \\left|{\\mathbb {E}}(\\pi _0|\\sigma (\\pi _n,\\pi _{n+1},\\ldots ))\\right|\\, d\\kappa .$ Hence, we have an upper bound which does not depend on $C$ .", "Since ${\\mathbb {E}}(\\pi _0|\\sigma (\\pi _n,\\pi _{n+1},...))\\rightarrow {\\mathbb {E}}(\\pi _0|\\Pi (\\kappa ))=0$ $\\kappa $ -a.e.", "and thus also in $L^1$ , which is precisely the relative K-mixing for $\\pi _0$ .", "(b) $\\Rightarrow $ (a).", "Suppose that $\\pi _0$ is relatively K-mixing.", "Then, in particular, we have (REF ) for each $C\\in \\Pi (\\kappa )$ .", "In fact, since $\\varepsilon >0$ is arbitrary, $\\int \\pi _0 {1}_C\\,d\\kappa =0$ for each $C\\in \\Pi (\\kappa )$ .", "Whence ${\\mathbb {E}}(\\pi _0|\\Pi (\\kappa ))=0$ .", "(a) $\\Rightarrow $ (c).", "Since $\\frac{1}{M_k}\\sum _{m\\le M_k}u(m)f(S^mu)=\\frac{1}{M_k}\\sum _{m\\le M_k}(\\pi _0f)(S^mu)\\rightarrow \\int _{X_{u}} \\pi _0 f\\, d\\kappa ,$ we can repeat the same argument as was used to prove (a) $\\Rightarrow $ (b) (replacing ${1}_C$ by $f$ ).", "(c) $\\Rightarrow $ (b).", "Suppose that $\\left|\\int _{X_{u}}\\pi _0 f\\,d\\kappa \\right|<\\varepsilon $ , whenever $f$ depending on coordinates $n,n+1,\\ldots ,n+s$ with $n\\ge N$ is bounded by 1.", "Consider all blocks on coordinates $n,n+1,\\ldots ,n+s$ that is all $B=\\lbrace x\\in X_{u}:\\: x_n=b_0,\\ldots ,x_{n+s}=b_s\\rbrace $ with $b_j\\in E$ .", "Let $C$ be any union of such blocks.", "Then ${1}_C$ is a (continuous) function depending on coordinates $n,\\ldots ,n+s$ and is bounded by 1 and, by assumption, $\\left|\\int _{X_{u}}\\pi _0 {1}_C\\,d\\kappa \\right|<\\varepsilon .$ Note that with $N$ fixed and $s$ arbitrary, the family of $C$ defined above is dense in the $\\sigma $ -algebra $\\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ .", "Hence, given $D\\in \\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ and $\\varepsilon >0$ , we first find $s\\ge 0$ and then $C$ as above (a union of blocks “sitting” on coordinates $N,\\ldots ,N+s$ ) such that $\\kappa (C\\triangle D)<\\varepsilon $ and find that $\\left|\\int \\pi _0 {1}_D\\,d\\kappa \\right|\\le \\left|\\int \\pi _0 {1}_C\\,d\\kappa \\right|+\\kappa (C\\triangle D)<2\\varepsilon .$ Now, since each clopen set is a finite union of cylinders of a fixed length, Corollary REF follows directly by the above proposition.", "Corollary REF is a special case of Corollary REF ." ], [ "Conditional cancellations. Remark ", "The “cancellation law” of the values of $u$ along large shifts of the return times to a block (for most of the blocks) claimed in Remark REF is a consequence of a refinement of Proposition REF .", "Proposition 5.2 Let $\\kappa \\in V_S(u)$ and $\\int \\pi _0\\,d\\kappa =0$ .", "Then the following conditions are equivalent: $\\pi _0\\perp L^2(\\Pi (\\kappa ))$ , for each $\\varepsilon >0$ there exist $N\\ge 1$ and $L\\ge 1$ such that for each $\\ell \\ge L$ there is a family of “good” $\\ell $ -blocks $C\\in \\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ , i.e.", "of blocks satisfying $\\left|\\int {1}_C\\cdot \\pi _0\\,d\\kappa \\right|\\le \\kappa (C)\\varepsilon ,$ whose measure is $>1-\\varepsilon $ .", "In other words, for a “good” $\\ell $ -block $C$ , $\\left|\\int \\pi _0\\,d\\kappa _C\\right|<\\varepsilon $ , where $\\kappa _C$ stands for the conditional measure on $C$ .", "(a) $\\Rightarrow $ (d).", "Fix $\\varepsilon >0$ and note that ${\\mathbb {E}}^\\kappa (\\pi _0|\\sigma (\\pi _N,\\pi _{N+1},\\ldots ))\\rightarrow 0$ $\\kappa $ -a.e.", "This implies convergence in measure, i.e., we can find a set $A_\\varepsilon $ of measure at least $1-\\varepsilon $ such that for $N$ large enough, $|{\\mathbb {E}}^\\kappa (\\pi _0|\\sigma (\\pi _N,\\pi _{N+1},\\ldots ))(x)|<\\varepsilon \\text{ for all }x\\in A_{\\varepsilon }.$ Fix such an $N$ .", "There is $M\\ge 1$ large enough such that $\\kappa (A_{\\varepsilon }\\triangle A^{(M)}_{\\varepsilon })<\\varepsilon ,$ where $A^{(M)}_\\varepsilon \\in \\sigma (\\pi _{-M},\\pi _{-M+1},\\ldots )$ and note that $S^{N+M}A^{(M)}_\\varepsilon \\in \\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ .", "Now, for $\\ell $ large enough, we can approximate $S^{N+M}A^{(M)}_{\\varepsilon }$ by a (disjoint) union of $\\ell $ -blocks belonging to $\\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ , $\\kappa (S^{N+M}A^{(M)}_{\\varepsilon }\\triangle \\bigcup _{j\\in J} C^{(\\ell )}_j)<\\varepsilon .$ But $\\kappa (S^{N+M}A^{(M)}_{\\varepsilon }\\triangle A_{\\varepsilon })<2\\varepsilon $ , so $\\kappa (\\bigcup _{j\\in J} C^{(\\ell )}_j \\setminus A_{\\varepsilon })\\le \\kappa (A_{\\varepsilon }\\triangle \\bigcup _{j\\in J} C^{(\\ell )}_j)<3\\varepsilon .$ Consider those $j\\in J$ for which $\\kappa (C^{(\\ell )}_j\\setminus A_{\\varepsilon })\\ge \\sqrt{\\varepsilon }\\kappa (C_j^{(\\ell )})$ .", "Then the measure $m$ of the union of such blocks has to satisfy $\\sqrt{\\varepsilon } m<3\\varepsilon $ , so $m<3\\sqrt{\\varepsilon }$ .", "In other words “most” (in measure) of the $C_j^{(\\ell )}$ 's are “good”, i.e.", "they satisfy $\\kappa (C^{(\\ell )}_j\\cap A_{\\varepsilon })>(1-3\\sqrt{\\varepsilon })\\kappa (C^{(\\ell )}_j)$ .", "Take such a “good” $C$ .", "We have $\\left|\\int {1}_C\\cdot \\pi _0\\,d\\kappa \\right|\\le \\int {1}_{C}|{\\mathbb {E}}^\\kappa (\\pi _0|\\sigma (\\pi _N,\\pi _{N+1},\\ldots ))|\\,d\\kappa \\\\=\\int _{A_{\\varepsilon }}{1}_{C}|{\\mathbb {E}}^\\kappa (\\pi _0|\\sigma (\\pi _N,\\pi _{N+1},\\ldots ))|\\,d\\kappa +\\int _{A_{\\varepsilon }^c}{1}_{C}|{\\mathbb {E}}^\\kappa (\\pi _0|\\sigma (\\pi _N,\\pi _{N+1},\\ldots ))|\\,d\\kappa \\\\\\le \\varepsilon \\kappa (C)+3\\sqrt{\\varepsilon }\\kappa (C).$ (d) $\\Rightarrow $ (a).", "Fix $A\\in \\Pi (\\kappa )$ of positive measure $\\kappa $ .", "Then for $\\varepsilon >0$ , we can find $N$ such that for all $\\ell $ large enough “most” of the $\\ell $ -blocks in $\\sigma (\\pi _N,\\pi _{N+1}<\\ldots )$ is “good” in the sense that $\\left|\\int {1}_C\\cdot \\pi _0\\,d\\kappa \\right|\\le \\kappa (C)\\varepsilon .$ Since $A\\in \\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ , we can approximate it by unions of $\\ell $ -blocks (for $\\ell $ sufficiently large) and most of the used blocks is “good”.", "Whence $\\left|\\int {1}_A\\cdot \\pi _0\\,d\\kappa \\right|\\le 2\\varepsilon ,$ and since $\\varepsilon >0$ was arbitrary, $\\pi _0\\perp L^2(\\Pi (\\kappa ))$ ." ], [ "Orthogonality to ${C}_{\\rm {ID}}$ . Proof of Corollary ", "We recall that (Proposition REF ) ${C}_{\\rm ID}={C}_{{\\rm ID}_{\\rm ec}}.$ Since the characteristic factor is represented by the $\\sigma $ -algebra of invariant sets, by Theorems REF and REF , we obtain immediately that: Corollary 5.3 $u\\perp {C}_{\\rm ID}$ if and only if for each Furstenberg system $\\kappa $ of $u$ , $\\pi _0\\perp L^2(\\mathcal {I}_\\kappa )$ .", "Let us now pass to a combinatorial characterization of the Veech condition.", "Assume that $\\kappa $ is given as the limit of $\\frac{1}{N_k}\\sum _{n\\le N_k}\\delta _{S^nu}$ .", "In view of Corollary REF , we need to decipher ${\\mathbb {E}}(\\pi _0|\\mathcal {I}_\\kappa )=0$ .", "By the von Neumann theorem, we have $\\frac{1}{H}\\sum _{h\\le H}\\pi _0\\circ S^h\\rightarrow {\\mathbb {E}}(\\pi _0|\\mathcal {I}_\\kappa ) \\text{ in }L^2,$ i.e.", "$\\lim _{H\\rightarrow \\infty }\\int _{X_{u}}\\left|\\frac{1}{H}\\sum _{h\\le H}\\pi _0\\circ S^h\\right|^2\\,d\\kappa =0$ as ${\\mathbb {E}}(\\pi _0|\\mathcal {I}_\\kappa )=0$ .", "So, given $\\varepsilon >0$ , $\\int _{X_{u}}\\left| \\frac{1}{H}\\sum _{h\\le H}\\pi _0\\circ S^h\\right|^2\\,d\\kappa <\\varepsilon \\text{ for all }H\\ge H_{\\varepsilon }.$ The latter is equivalent to $\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}\\left|\\frac{1}{H}\\sum _{h\\le H}\\pi _0\\circ S^h\\right|^2(S^nu)<\\varepsilon ,$ that is, $\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}\\left|\\frac{1}{H}\\sum _{h\\le H}u(n+h)\\right|^2<\\varepsilon .$ The proof of Corollary REF follows immediately.$\\quad \\hbox{\\vrule \\vbox to 6pt {\\hrule width 4pt\\vfill \\hrule }\\vrule } $ Remark 5.4 Hence, the Matomäki-Radziwiłł theorem [41] on the behaviour of a strongly aperiodic multiplicative function $u$ on a typical short interval implies $u\\perp {C}_{\\rm ID}$ .", "However, as shown in [28], the aperiodic multiplicative functions defined in [42] do not satisfy the assertion of Corollary REF .", "In Corollary REF , the Veech condition (for $u$ ) equivalent to $u\\perp {C}_{\\rm ID}$ is written as $\\pi _0\\perp L^2(\\mathcal {I}_\\kappa )$ .", "If we look at it more spectrally, we obtain immediately that $u\\perp {C}_{\\rm ID}\\text{ if and only if }\\sigma _{\\pi _0,\\kappa }(\\lbrace 1\\rbrace )=0$ for all $\\kappa \\in V_S(u)$ , i.e.", "the spectral measure of $\\pi _0$ (with respect to each Furstenberg system) has no atom at 1.", "Classically (by a simple computation), we have: Lemma 5.5 If $\\sigma $ is a measure on the circle $\\mathbb {S}^1$ then $\\frac{1}{H}\\sum _{h=0}^{H-1} \\widehat{\\sigma }(h)\\rightarrow \\sigma (\\lbrace 1\\rbrace ).$ Hence, the Veech condition is equivalent to $\\frac{1}{H}\\sum _{h=0}^{H-1}\\int \\pi _0\\cdot \\overline{\\pi _0}\\circ S^h d\\kappa \\rightarrow 0.$ Combinatorially, we obtain $\\frac{1}{H}\\sum _{h=0}^{H-1}\\lim _{k\\rightarrow \\infty } \\frac{1}{N_k}\\sum _{n\\le N_k} u(n)\\overline{u}(n+h) \\rightarrow 0$ for each sequence $(N_k)$ defining a Furstenberg system $\\kappa $ .", "It follows that (REF ) is equivalent to the short interval behaviour ().", "In other words, condition $\\lim _{H\\rightarrow \\infty }\\left(\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}\\left|\\frac{1}{H}\\sum _{h\\le H} u(n+h)\\right|^2\\right)=0$ is equivalent to $\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\left(\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}u(n)\\overline{u}(n+h)=0\\right).$" ], [ "Orthogonality to ${C}_{\\rm DISP(G)}$ with {{formula:69a12a44-1485-4df7-8f5e-d208881bd559}} countable", "Let $G\\subset \\mathbb {S}^1$ be a countable subgroup and recall that ${{\\rm DISP}(G)}$ stands for the (characteristic) class of discrete spectrum automorphisms whose groups of eigenvalues are contained in $G$ .", "Since $z\\in \\mathbb {S}^1$ is an eigenvalue of $(Z,{\\cal D},\\kappa ,R)$ if and only if it is an eigenvalues of a subset of positive measure of ergodic components, it is not hard to see that ${{F}}_{{\\rm DISP}(G)}=({{F}}_{{\\rm DISP}(G)})_{\\rm ec}.$ It follows that $u\\perp {C}_{{{F}}_{{\\rm DISP}(G)}}\\text{ if and only if }\\sigma _{\\pi _0,\\kappa }(G)=0,$ i.e.", "the spectral measure of $\\pi _0$ has no atoms belonging to $G$ (for each Furstenberg system $\\kappa \\in V_S(u)$ ).", "Suppose that $e^{2\\pi i\\alpha }\\in G$ .", "Consider $v(n):=e^{2\\pi in\\alpha }u(n)$ for $n\\in {\\mathbb {N}}$ .", "Note that $\\frac{1}{N_k}\\sum _{n\\le N_k}v(n)\\overline{v}(n+h)=e^{-2\\pi ih\\alpha }\\frac{1}{N_k}\\sum _{n\\le N_k}u(n)\\overline{u}(n+h).$ So, if we have a subsequence $(N_k)$ along which both $\\frac{1}{N_k}\\sum _{n\\le N_k}\\delta _{S^nw}$ with $w=u,v$ converge to $\\kappa ,\\kappa ^{\\prime }$ respectively,Note that these common sequences yield all Furstenberg systems for both $u$ and $v$ .", "then $\\sigma _{\\pi _0,\\kappa }=\\delta _{e^{2\\pi i\\alpha }}\\ast \\sigma _{\\pi _0,\\kappa ^{\\prime }},$ whence $\\sigma _{\\pi _0,\\kappa }(\\lbrace e^{2\\pi i\\alpha }\\rbrace )=0 \\text{ if and only if }\\sigma _{\\pi _0,\\kappa ^{\\prime }}(\\lbrace 1\\rbrace ).$ By our previous subsection it follows that the latter condition is equivalent to: $\\lim _{H\\rightarrow \\infty }\\left(\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}\\left|\\frac{1}{H}\\sum _{h\\le H} v(n+h)\\right|^2\\right)=0,$ that is, $\\lim _{H\\rightarrow \\infty }\\left(\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}\\left|\\frac{1}{H}\\sum _{h\\le H}e^{2\\pi i(n+h)\\alpha } u(n+h)\\right|^2\\right)=0$ which is the strong $u$ -MOMO condition for the irrational rotation by $\\alpha $ .Note that if $f(t)=e^{2\\pi i t}$ then $\\frac{1}{b_K}\\sum _{k<K}\\left|\\sum _{b_k\\le n<b_{k+1}}f(R_\\alpha ^nx_k)u(n)\\right| =\\frac{1}{b_K}\\sum _{k<K}\\left|\\sum _{b_k\\le n<b_{k+1}}e^{2\\pi i n\\alpha }u(n)\\right|.$" ], [ "Furstenberg systems and the strong $u$ -MOMO property", "The following proposition helps us to exclude some measure-theoretic systems from the list of Furstenberg systems of an arithmetic function.", "Proposition 5.6 Let $u:{\\mathbb {N}}\\rightarrow be a bounded arithmetic function.", "Then no Furstenberg system $ VS(u)$ has a topological model which is strongly $ u$-MOMO.$ Suppose $(X_{u},\\kappa , S)$ has a topological model $(Z,\\nu ,R)$ which satisfies the strong $u$ -MOMO property.", "Let $J:Z\\rightarrow X_{u}$ settles a measure-theoretic isomorphism and let $\\nu _J$ be the corresponding graph joining.", "We assume that $\\frac{1}{N_j}\\sum _{n\\le N_j}\\delta _{S^nu}\\rightarrow \\kappa $ .", "From Proposition REF we can find a sequence $(z_n)\\subset Z$ consisting of pieces of orbits of different points: $\\lbrace n:\\: Rz_n\\ne z_{n+1}\\rbrace =\\lbrace b_k:\\:k\\ge 1\\rbrace $ with $b_{k+1}-b_k\\rightarrow \\infty $ , and a subsequence $(N_{j_\\ell })$ such that $\\frac{1}{N_{j_\\ell }}\\sum _{n\\le N_{j_\\ell }}\\delta _{(z_n,S^nu)}\\rightarrow \\nu _J.$ Then, by the strong $u$ -MOMO property of $(Z,R)$ , $\\int f\\otimes \\pi _0\\,d\\nu _J=\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{b_{K_\\ell }}\\sum _{k<K_\\ell }\\big (\\sum _{b_k\\le n<b_{k+1}}f(R^{n-b_k}z_{b_k})u(n)\\big )=0.$ Hence, $\\int {\\mathbb {E}}^{\\nu _J}(f|X_{u})\\pi _0\\,d\\kappa =0$ for each continuous $f$ on $Z$ , and we obtain a contradiction since ${\\mathbb {E}}^{\\nu _{J}}(L^2(\\nu )|X_{u})=L^2(\\kappa )$ .", "Corollary 5.7 Assume that for each $(b_k)$ with $b_{k+1}-b_k\\rightarrow \\infty $ , $\\lim _{K\\rightarrow \\infty }\\frac{1}{b_K}\\sum _{k<K}\\sup _{\\alpha \\in {\\mathbb {R}}}\\left|\\sum _{b_k\\le n< b_{k+1}}u(n)e^{2\\pi i\\alpha n}\\right|=0.$ Then the unipotent system $(x,y)\\mapsto (x,y+x)$ (on 2) is not a Furstenberg system of $u$ .", "Since condition (REF ) is the strong $u$ -MOMO property of the unipotent system, the result follows from Proposition REF .", "Remark 5.8 Corollary REF brings a better understanding of Problem 3.1 (due to Frantzikinakis) of the workshop [4]: The system $(x,y)\\mapsto (x,y+x)$ is not a Furstenberg system of the Liouville function (see also slide no 6 in [17]).", "We recall that in [42] (see Theorem 1.3 therein), it is proved that $\\lim _{M\\rightarrow \\infty }\\limsup _{N\\rightarrow \\infty }\\sup _{\\alpha \\in {\\mathbb {R}}}\\frac{1}{N}\\sum _{n\\le N}\\left|\\frac{1}{M}\\sum _{n\\le m<n+M}\\lambda (m)e^{2\\pi i m\\alpha }\\right|=0,$ so the sup has changed the place!", "The strong $\\lambda $ -MOMO property for the unipotent system remains hence open.", "For the equivalence of $\\lim _{M\\rightarrow \\infty }\\limsup _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n\\le N}\\sup _{\\alpha \\in {\\mathbb {R}}}\\left|\\frac{1}{M}\\sum _{n\\le m<n+M}u(m)e^{2\\pi i m\\alpha }\\right|=0,$ with (REF ) see the appendix in [34] - only in the arXiv version of the paper." ], [ "Orthogonality to ${C}_{{\\rm DISP}_{\\rm ec}}$ . Proof of Corollary ", "In view of Corollary REF (see also (REF )) and Theorem REF , in order to obtain $u\\perp \\mathcal {C}_{{\\rm DISP}_{ec}}$ it is sufficient and necessary to have $u\\perp L^2({\\cal K}(\\mathcal {I}_\\kappa ))$ for each Furstenberg system $\\kappa $ of $u$ .", "By our previous results, for the class of all topological systems whose all ergodic measures yield discrete spectra, Sarnak and Veech conditions are equivalent.", "We now write the Veech condition combinatorially, i.e., we provide the proof of Corollary REF .", "[Proof of Corollary REF ] By Corollary REF , we need to show that for each $\\kappa $ being a Furstenberg system of $u$ , we have $\\int \\frac{1}{H}\\sum _{h\\le H}|{\\mathbb {E}}(\\pi _0\\circ S^h\\cdot \\overline{\\pi _0}|\\mathcal {I}_\\kappa )|^2\\, d\\kappa \\rightarrow 0.$ By the von Neumann theorem, $\\int \\Big |{\\mathbb {E}}(\\pi _0\\circ S^h\\cdot \\overline{\\pi }_0|\\mathcal {I}_\\kappa )\\Big |^2\\,d\\kappa =\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n\\le N}\\int (\\pi _0\\circ S^h\\cdot \\overline{\\pi }_0)\\circ S^n\\cdot \\overline{\\pi _0\\circ S^h\\cdot \\overline{\\pi }_0}\\,d\\kappa .$ Therefore, (REF ) is equivalent to $\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n\\le N}\\int (\\pi _0\\circ S^h\\cdot \\overline{\\pi }_0)\\circ S^n\\cdot \\overline{\\pi _0\\circ S^h\\cdot \\overline{\\pi }_0}\\,d\\kappa =0.$ Let $(M_k)$ be such that $\\kappa =\\lim _{k\\rightarrow \\infty }\\frac{1}{M_k}\\sum _{m\\le M_k}\\delta _{S^mu}$ .", "It follows immediately that (REF ) is equivalent to $\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n\\le N}\\lim _{k\\rightarrow \\infty }\\frac{1}{M_k}\\sum _{m\\le M_k}u(m+n+h)\\overline{u(m+n)}\\overline{u(m+h)}u(m)=0$ which is precisely $\\Vert u\\Vert _{u^2((M_k))}=0$ .", "Now, it suffices to use (REF ).", "Remark 5.9 In fact, already Frantzikinakis [17] (see slide no 10) showed that if $u$ is generic then $\\Vert u\\Vert _{u^2}=0$ if and only if $\\lim _{M\\rightarrow \\infty }\\limsup _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n\\le N}\\sup _{\\alpha \\in {\\mathbb {R}}}\\left|\\frac{1}{M}\\sum _{n\\le m<n+M}u(m)e^{2\\pi i m\\alpha }\\right|=0.$ We recall that this condition is equivalent to the strong $u$ -MOMO property of the unipotent system $(x,y)\\mapsto (x, y+x)$ .", "Remark 5.10 Note that for each (bounded) $u\\colon {\\mathbb {N}}\\rightarrow satisfying $ uu2=0$ the system\\begin{equation}(x,y)\\mapsto (x,x+y)\\text{ on }(2,{\\rm Leb}\\,\\otimes \\, {\\rm Leb})\\end{equation}cannot appear (up to isomorphism) as a Furstenberg system of $ u$ (because $ 0$ is orthogonal to the $ L2(K(I))$ but for the unipotent system (\\ref {unip}) thewhole system is relative Kronecker over the $$-algebra of invariant sets).$ In particular, if $\\Vert \\lambda \\Vert _{u^2}=0$ holds for the Liouville function then () is not its Furstenberg system – this would answer a question by N. Frantzikinakis asked in 2016 (it is an official Problem 3.1 in [4]).", "However, the problem of whether $\\Vert \\lambda \\Vert _{u^2}=0$ (or more generally $\\Vert \\lambda \\Vert _{u^s}=0$ ) seems to be difficult.", "The best known results [43], [44] require a quantitative dependence between the parameters $M$ and $N$ , i.e.", "$M=N^{\\theta }$ , for arbitrary small, but fixed $\\theta >0$ .", "If $\\Vert \\lambda \\Vert _{u^2}=0$ holds then Sarnak's conjecture holds for all (zero entropy) systems whose ergodic measures yield discrete spectrum.", "So far it is only known that Sarnak's conjecture holds for systems whose all invariant measure yield discrete spectrum [33], [32], [15].", "Ruling out () (or, more generally, nilpotent type systems) from the list of potential Furstenberg systems of $\\lambda $ is important in view of Frantzikinakis and Host's results [20], [21] concerning the structure of Furstenberg systems of multiplicative functions (although, for the moment, this structure is known only for the logarithmic case).", "In the light of [42], it would be also interesting to know whether $\\Vert u\\Vert _{u^2}=0$ holds for some classical multiplicative functions.", "Note that this is not the case for the class of aperiodic multiplicative functions defined in [42] since as shown in [28] they have the unipotent system as a Furstenberg systemIn fact, for such functions $u$ we have already $\\Vert u\\Vert _{u^1((N_k))}>0$ (for some $(N_k)$ ), see Corollary 6.5 in [28].", "(see also Remark REF )." ], [ "Orthogonality to ${C}_{\\rm DISP}$ . Averaged Chowla property for multiplicative functions", "The assertion “iff” of Theorem REF cannot be applied to the class ${C}_{\\rm DISP}$ .", "In this section we will show however that the assertion of this theorem holds whenever $u\\colon {\\mathbb {N}}\\rightarrow satisfies the following additional property:\\begin{equation}\\mbox{all rotations on the circle satisfy the strong \\qquad \\mathrm {(\\ast )}$u$-MOMO property}.\\end{equation}We will need the following fact (see, e.g.,~\\cite {Ed}):\\begin{equation} \\mbox{each discrete spectrum automorphism is a factor of $R_\\alpha \\times {\\rm Id}_{[0,1]}$},\\end{equation}for some ergodic rotation by $ G$ on a compact (Abelian) metric group $ G$.", "Our key tool will be the following lemma.\\begin{Lemma}Suppose that (\\ast ) holds.", "Then R_\\alpha \\times {\\rm Id}_{[0,1]} satisfies the strong u-MOMO property.\\end{Lemma}\\begin{proof}It is enough to check the strong u-MOMO for functions F of the form \\chi \\otimes f, where \\chi \\in \\widehat{G} and f\\in C([0,1]).", "We have{\\begin{@align*}{1}{-1}\\frac{1}{b_K}\\sum _{k<K}&\\left| \\sum _{b_k\\le n<b_{k+1}}F((R_\\alpha \\times {\\rm Id})^n(h_k,u_k))u(n)\\right|\\\\&=\\frac{1}{b_K}\\sum _{k<K}\\left| \\sum _{b_k\\le n<b_{k+1}}\\chi ((R_\\alpha ^n(h_k))f(u_k)u(n)\\right|\\\\&={\\rm O}\\left( \\frac{1}{b_K}\\sum _{k<K}\\left| \\sum _{b_k\\le n<b_{k+1}}\\chi (n\\alpha )u(n)\\right| \\right).\\end{@align*}}Our claim follows from (\\ast ).\\end{proof}$ Theorem 5.11 Assume that $u$ enjoys the property $(\\ast )$ .", "Then $u\\perp {C}_{DISP}$ if and only if $\\pi _0\\perp L^2(\\mathcal {K}(\\kappa ))$ for each $\\kappa \\in V_S(u)$ (iff the spectral measure $\\sigma _{\\pi _0}$ is continuous for each Furstenberg system $\\kappa $ ).", "We only need to show that $u\\perp {C}_{DISP}$ implies $\\pi _0\\perp L^2(\\mathcal {K}(\\kappa ))$ for each $\\kappa \\in V_S(u)$ .$\\mathcal {K}(\\kappa )$ stands for the Kronecker factor of $(X_{u},\\kappa ,S)$ .", "Using (), let $p$ settle a factor map from $R_\\alpha \\times {\\rm Id}_{[0,1]}$ acting on $(G\\times [0,1],m_G \\otimes {\\rm Leb})$ and $(X_{u}/\\mathcal {K}(\\kappa ),\\mathcal {K}(\\kappa ),\\kappa |_{\\mathcal {K}(\\kappa )})$ .", "Let $(m_G\\otimes {\\rm Leb})_p$ stand for the corresponding graph joining and $\\rho $ for the relatively independent extension of it to a joining of $(G\\times [0,1], m_G\\otimes {\\rm Leb},R_\\alpha \\times {\\rm Id})$ with $(X_{u}, \\kappa ,S)$ .", "Now, by Proposition REF , the integral $\\int F\\otimes \\pi _0\\, d\\rho $ can be computed using a quasi-generic sequence $((g_n), (S^nu))$ .", "Since, by Lemma , our topological system $R_\\alpha \\times {\\rm Id}$ satisfies the strong $u$ -MOMO property, this integral vanishes.", "On the other hand, for each $F\\in C(G\\times [0,1])$ , $\\int F\\otimes \\pi _0\\,d\\rho =\\int {\\mathbb {E}}(F|X_{u})\\pi _0\\,d\\kappa $ and since ${\\mathbb {E}}^\\rho (C(G\\times [0,1])|X_{u})$ is dense in $L^2(\\mathcal {K},\\kappa |_{\\mathcal {K}})$ (in view of the definition of $\\rho $ ), it follows that $\\pi _0\\perp L^2(\\mathcal {K}(\\kappa ))$ .", "[Proof of Corollary REF ] Note that in the proof of Theorem REF , we have shown that our original assumption $(\\ast )$ already implies the Veech condition.", "In particular, the Sarnak and the Veech properties are equivalent.", "Condition () is just rewriting the Wiener condition combinatorially.", "Finally, the last part () is proved in Appendix .", "[Proof of Corollary REF ] By Corollary REF , we only need to show that irrational rotations satisfy the strong $u$ -MOMO property.", "This follows from the fact that irrational rotations satisfy the AOP property [1] and that the AOP property implies the strong $u$ -MOMO property [3]." ], [ "No strong $u$ -MOMO in positive entropy", "In this section we discuss the problem of orthogonality to ${C}_{\\rm ZE}$ and the reversed problem of the absence of orthogonality to an arbitrary positive entropy systems, following some ideas from [3].", "Recall that the following has been proved in [3].", "Proposition 6.1 ([3]) Let $u\\colon {\\mathbb {N}}\\rightarrow be a bounded arithmetic function.", "The following are equivalent:\\begin{enumerate}\\item [(a)] u\\perp {C}_{\\rm ZE}.\\item [(b)] For each (X,T) of zero entropy and f\\in C(X), (\\ref {ort1}) holds uniformly in x\\in X.\\item [(c)] Each zero entropy (X,T) satisfies the strong u-MOMO property.\\end{enumerate}$ On the other hand, it follows from the results of Downarowicz and Serafin [11], [12] that for each $u\\perp {C}_{\\rm ZE}$ there exists $(X,T)$ such that $u\\perp (X,T) \\text{ and }(X,T)\\notin {\\rm ZE}.$ In fact, one can get a positive entropy system $(X,T)$ in which for every $f\\in C(X)$ () holds uniformly in $x\\in X$ .", "We prove however that (REF ) fails if orthogonality is replaced by the strong $u$ -MOMO property.", "To avoid technical details, we restrict ourselves to the case of an arithmetic function $u$ taking finitely many values.", "Theorem D Let $u:{\\mathbb {N}}\\rightarrow be an arithmetic function taking finitely many values.Assume that $ uCZE$.", "Then no positive entropy topological dynamical system satisfies the strong $ u$-MOMO property.$" ], [ "Proof of Theorem ", "We fix a bounded arithmetic function $u\\colon {\\mathbb {N}}\\rightarrow .We need a series of results from \\cite {Ab-Ku-Le-Ru2} in some modified forms.", "In~\\cite {Ab-Ku-Le-Ru2}, the equivalence of certain three properties (P1), (P2) and (P3) of an ergodic measure-theoretic dynamical system $ (Z,B(Z),,R)$ was proved.", "Condition (P1) was nothing but the strong $ u$-MOMO for {\\bf some} topological system being a model of the system given by $$.", "Instead of recalling (P2), let us formulate red its subsequence version:\\begin{equation}\\begin{array}{l}\\mbox{Assume that \\qquad \\mathrm {(P2')}$(X,T)$ is any topological system and let $x\\in X$.", "}\\\\\\mbox{If $x$ is generic along $(N_k)$ for a measure which is isomorphic}\\\\\\mbox{(as dynamical systems) to $\\kappa $ then}\\\\\\mbox{$\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}f(T^nx)u(n)=0$for each $f\\in C(X)$}.\\end{array}\\end{equation}The proof of the implication (P1) $$ (P2^{\\prime }) is a repetition of the proof of (P1) implies (P2).", "In Lemma~17 in \\cite {Ab-Ku-Le-Ru2}, we need to consider the sequence $ (Nk)$ instead of $ N$ and start with $$ along this sequence.$ As a consequence of the above, we obtain the following version of Corollary 12 from [3].", "Corollary 6.2 Assume that $\\kappa $ is an ergodic shift-invariant measure on $L^{{\\mathbb {Z}}}$ , and that there exists $y\\in L^{{\\mathbb {Z}}}$ , generic along $(N_k)$ for $\\kappa $ , correlating with $u$ along $(N_k)$ , i.e.", "the sequence $(\\frac{1}{N_k}\\sum _{n\\le N_k}y(n)u(n))$ does not go to zero.", "Then the strong $u$ -MOMO property fails for any uniquely ergodic model of $(L^{{\\mathbb {Z}}},\\kappa ,S)$ .", "Then, by repeating the proof from [3], we obtain the following form of Corollary 14 in [3].", "Corollary 6.3 Assume that $y$ is generic along $(N_k)$ for a Bernoulli measure $\\nu $ , and that $y$ and $u$ correlate along $(N_k)$ .", "Then the strong $u$ -MOMO property fails for any $(X,T)$ with $h(X,T)>h(\\nu )$ .", "We also need the following crucial probabilistic lemma whose proof we postpone to the next subsection.", "Lemma 6.4 Assume that $X=(X_n)_{n\\in {\\mathbb {Z}}}$ is a a stationary process of positive entropy, taking finitely many complex values.", "Then for any non-trivial probability distribution $\\beta $ concentrated on a finite subset of ${\\mathbb {R}}$ , there exists a stationary coupling of $X$ with a Bernoulli process $Y=(Y_n)_{n\\in {\\mathbb {Z}}}$ of distribution $\\beta ^{\\otimes {\\mathbb {Z}}}$ such that ${\\mathbb {E}}[X_0 Y_0]\\ne {\\mathbb {E}}[X_0]{\\mathbb {E}}[Y_0]$ .", "We now assume that $u$ takes finitely many values and satisfies the Veech condition: $\\pi _0\\perp L^2(\\Pi (\\kappa ))$ for each Furstenberg system $\\kappa $ of $u$ .", "Lemma 6.5 For each $h>0$ there exists a sequence $y$ , generic for a Bernoulli measure of entropy $h$ along some increasing sequence $(N_k)$ , and correlating with $u$ along $(N_k)$ .", "Let $\\kappa $ be a Furstenberg system of $u$ , and $(M_\\ell )$ such that $u$ is generic for $\\kappa $ along $(M_\\ell )$ .", "By assumption, the entropy of the stationary process defined by $\\pi _0$ under $\\kappa $ is positive.", "Take a real-valued Bernoulli shift of entropy $h$ (Bernoulli measure denoted by $\\nu $ ).", "Using Lemma REF , find a joining of $\\kappa $ and $\\nu $ for which $\\pi _0$ (in $L^2(X_{u},\\kappa )$ ) is not orthogonal to $\\pi _0$ in $L^2(\\nu )$ : $\\int \\pi _0\\otimes \\pi _0\\,d\\rho \\ne 0$ .", "Now, use a subsequence version of the lifting lemma (Theorem 5.16 in [6]) to find $y$ in the subshift defining the Bernoulli automorphism such that $(u,y)$ is generic, along a subsequence $(N_k)=(M_{\\ell _k})$ , for $\\rho $ .", "Then $0\\ne \\int \\pi _0\\otimes \\pi _0\\,d\\rho =\\lim _{k\\rightarrow \\infty }\\frac{1}{N_{k}}\\sum _{n\\le N_{k}}\\pi _0(S^nu)\\pi _0(S^ny)=\\lim _{k\\rightarrow \\infty }\\frac{1}{N_{k}}\\sum _{n\\le N_{k}}u(n)y_n$ which means that $u$ and $y$ correlate along $(N_{k})$ .", "Now the proof of Theorem REF is a straightforward consequence of Lemma REF and Corollary REF ." ], [ "Proof of Lemma ", "Let $X=(X_n)_{n\\in {\\mathbb {Z}}}$ be a positive entropy stationary process as in the statement of the lemma.", "Without loss of generality (considering its real or imaginary part), we can assume that this process takes its values in a finite subset $\\lbrace x_1<x_2<\\cdots <x_r\\rbrace $ of ${\\mathbb {R}}$ .", "We also consider a given probability measure $\\beta $ supported on a possibly different finite subset of ${\\mathbb {R}}$ $\\lbrace y_1<y_2<\\cdots <y_s\\rbrace $ , which is supposed to be non trivial (i.e.", "not reduced to a Dirac measure).", "Thus we can assume that $s\\ge 2$ , and $\\beta (y_j)>0$ for each $1\\le j\\le s$ .", "The purpose of this section is to show how we can construct a stationary coupling of $X$ with a Bernoulli process $Y$ whose distribution is $\\beta ^{\\otimes {\\mathbb {Z}}}$ , in such a way that for each $n\\in {\\mathbb {Z}}$ , ${\\mathbb {E}}[X_n Y_n] > {\\mathbb {E}}[X_n]\\, {\\mathbb {E}}[Y_n].$ We observe that the validity of the preceding inequality is unchanged if we replace $Y_n$ by $Y_n+C$ for a fixed $C$ .", "Thus we can and we do assume without loss of generality that the probability $\\beta $ is such that ${\\mathbb {E}}[Y_n]=0$ .", "To construct the announced coupling, we just assume that, on the probability space where the process $X$ is defined, we also have an i.i.d.", "process $V=(V_n)_{n\\in {\\mathbb {Z}}}$ such that each $V_n$ is uniformly distributed on $[0,1]$ , $V$ is independent of $X$ .", "The construction will be divided into two steps: first we construct an auxiliary (uniform i.i.d.)", "process $U$ and then we use it to construct $Y$ which satisfies the assertion of Lemma REF ." ], [ "Step 1: uniform i.i.d. process $U$", "For $n\\in {\\mathbb {Z}}$ and $j\\in \\lbrace 1,\\ldots ,r\\rbrace $ , we consider the random variable $P_{j,n}$ defined by $P_{j,n} := {\\mathbb {P}}\\bigl ( X_n=x_j \\,|\\, (X_m)_{m\\le n-1}\\bigr ).$ When $j$ is fixed, $(P_{j,n})_{n\\in {\\mathbb {Z}}}$ is a stationary process.", "On the other hand, if we fix $n$ , then $(P_{1,n},\\ldots ,P_{r,n})$ is the conditional distribution of $X_n$ given $(X_m)_{m\\le n-1}$ , in particular we have almost surely $0\\le P_{j,n}\\le 1$ , and $\\sum _{j=1}^r P_{j,n} = 1.$ This allows us to define a random partition of $[0,1[$ into disjoint subintervals $I_{1,n},\\ldots , I_{r,n}$ where for each $j$ , $I_{j,n}$ is the interval of length $P_{j,n}$ defined by $I_{j,n}:=\\left[\\sum _{1\\le i\\le j-1} P_{i,n}\\ ; \\sum _{1\\le i\\le j} P_{i,n}\\right[.$ Then we can define the random variable $U_n$ by $U_n := \\sum _{j=1}^r {1}_{X_n=x_j}\\left(\\sum _{1\\le i\\le j-1} P_{i,n} + V_n P_{j,n}\\right).$ Figure: Definition of U n U_nInformally, if $X_n=x_j$ , we pick $U_n$ uniformly at random (using $V_n$ ) inside $I_{j,n}$ (see Figure REF ).", "Therefore, $\\mathcal {L}\\left(U_n\\,|\\,(X_m)_{m\\le n-1},(V_m)_{m\\le n-1}\\right)=\\mathcal {U}_{[0,1]},$ i.e., it is uniform on $[0,1]$ .", "But all $U_m$ , $m\\le n-1$ , are measurable with respect to $(X_m)_{m\\le n-1}$ and $(V_m)_{m\\le n-1}$ , thus we also have $\\mathcal {L}\\left(U_n\\,|\\,(U_m)_{m\\le n-1}\\right)=\\mathcal {U}_{[0,1]} \\text{ and } \\mathcal {L}\\left(U_n\\right)=\\mathcal {U}_{[0,1]}.$ Indeed, this is just the application of the tower property of conditional expectations: to obtain the left equality, notice that for any measurable $A\\subset [0,1]$ , we have $\\mathbb {P}\\bigl (U_n\\in A &\\,|\\, (U_m)_{m\\le n-1}\\bigl ) \\\\&=\\mathbb {E}\\Bigl [\\underbrace{\\mathbb {P}\\bigl (U_n\\in A\\,|\\, (X_m)_{m\\le n-1},(V_m)_{m\\le n-1}\\bigr )}_{{\\rm Leb}(A)}\\,|\\, (U_m)_{m\\le n-1}\\Bigr ]\\\\&={\\rm Leb}(A).$ Moreover, it also follows from (REF ) that $U$ is i.i.d.", "Note that by construction, $U_n$ is a measurable function of $V_n$ , $X_n$ and $(X_m)_{m\\le n-1}$ , which we abusively write as $U_n = U_n \\left(V_n,X_n,(X_m)_{m\\le n-1}\\right).$ Moreover, whenever we fix realizations $\\xi $ of $(X_m)_{m\\le n}$ and $v$ of $V_n$ then $U_n$ as a function of its second argument is increasing: $U_n(v,x_{j_1},\\xi ) < U_n(v,x_{j_2},\\xi ), \\text{ whenever }x_{j_1}<x_{j_2}.$ We want to define $Y_n$ for a given $n\\in {\\mathbb {Z}}$ .", "We use another partition of $[0,1[$ into subintervals, according to the probability distribution $\\beta $ intended for $Y_n$ : for $1\\le k\\le s$ , set $\\beta _k:=\\beta (y_k)$ and define the interval $J_k:=\\bigl [\\beta _1+\\cdots +\\beta _{k-1};\\beta _1+\\cdots +\\beta _k\\bigr [$ .", "Then we simply define $Y_n$ as a function of $U_n$ by setting $ Y_n := \\sum _{k=1}^s y_k\\,{1}_{J_k}(U_n).$ It follows by the choice of the intervals $J_k$ and by $\\mathcal {L}(U_n)=\\mathcal {U}_{[0,1]}$ that $Y_n$ is distributed according to $\\beta $ .", "Moreover, by the independence of $U$ , we have the independence of $Y$ .", "Thus, $Y$ is a Bernoulli process with distribution $\\beta ^{\\otimes \\mathbb {Z}}$ .", "It remains to prove the announced inequality (REF ).", "Observe that $Y_n$ is, like $U_n$ , constructed as a measurable function of $V_n$ , $X_n$ and $(X_m)_{m\\le n-1}$ , which we also abusively write as $ Y_n = Y_n \\left(V_n,X_n,(X_m)_{m\\le n-1}\\right).", "$ Since $Y_n$ is a non-decreasing function of $U_n$ , we get from (REF ) that for a fixed realization $\\xi $ of $(X_m)_{m\\le n-1}$ and $v$ of $V_n$ , we have for $1\\le j_1 < j_2 \\le r$ $Y_n \\left(v,x_{j_1},\\xi \\right) < Y_n \\left(v,x_{j_2},\\xi \\right)$ and it follows that the map $ x\\in A \\mapsto {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x\\bigr ] $ is non-decreasing.", "Moreover, by the construction of $Y$ , we have $ \\mathcal {L}\\bigl (Y_n \\,|\\, (X_m)_{m\\le n-1}\\bigr ) = \\mathcal {L}(Y_n)=\\beta , $ whence ${\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}\\bigr ] = {\\mathbb {E}}[Y_n] =0.$ Thus there exists $j_0\\in \\lbrace 1,\\ldots ,r\\rbrace $ (depending on $\\xi $ ) such that $\\begin{split}&{\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_j]\\le 0 \\text{ for }1\\le j\\le j_0,\\\\&{\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_j]> 0\\text{ for }j_0+1\\le j\\le r.\\end{split}$ We then have, using (REF ) and (REF ), ${\\mathbb {E}}\\bigl [ X_n Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] & = {\\mathbb {E}}\\bigl [ (X_n-x_{j_0}) Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] \\nonumber \\\\&= \\sum _{j=1}^{j_0} (x_j-x_{j_0})\\, {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_j] \\nonumber \\\\&\\quad + \\sum _{j=j_0+1}^{r} (x_j-x_{j_0})\\, {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_j] \\\\& \\ge 0.", "\\nonumber $ Now, we claim that the announced result is a consequence of the following lemma.", "Lemma 6.6 If the realization $\\xi $ of $(X_m)_{m\\le n-1}$ is such that the conditional distribution $\\mathcal {L}(X_n \\,|\\, (X_m)_{m\\le n-1}=\\xi )$ is non-trivial, then $ {\\mathbb {E}}\\bigl [ X_n Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] > 0.", "$ Indeed, since $X$ has positive entropy, $\\mathcal {L}(X_n \\,|\\, (X_m)_{m\\le n-1})$ is non-trivial with positive probability, and thus we can conclude that $ {\\mathbb {E}}\\bigl [ X_n Y_n\\bigr ] = {\\mathbb {E}}\\Bigl [{\\mathbb {E}}\\bigl [ X_n Y_n \\,|\\, (X_m)_{m\\le n-1}\\bigr ]\\Bigr ] >0.", "$ [Proof of Lemma REF ] We fix a realization $\\xi $ of $(X_m)_{m\\le n-1}$ such that the conditional distribution $\\mathcal {L}\\bigl (X_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr )$ is non-trivial.", "Then the random variables $P_{j,n}$ and the intervals $I_{j,n}$ are fixed, because their values only depend on $\\xi $ .", "Setting $j_1 &:= \\min \\bigl \\lbrace j\\in \\lbrace 1,\\ldots ,r\\rbrace :\\ P_{j,n}>0\\bigr \\rbrace ,\\\\\\text{and}\\quad j_2 &:= \\max \\bigl \\lbrace j\\in \\lbrace 1,\\ldots ,r\\rbrace :\\ P_{j,n}>0\\bigr \\rbrace ,$ we have $j_1<j_2$ .", "Moreover the intervals $I_{j_1,n}$ and $I_{j_2,n}$ are respectively of the form $[0,P_{j_1,n}[$ and $[1-P_{j_2,n},1[$ , with $0<P_{j_1,n}\\le 1-P_{j_2,n}<1$ .", "We now discuss according to the relative position of the interval $I_{j_2,n}$ with respect to the interval $J_1$ (used to define $Y_n$ ).", "Figure: Case 1 (J 1 ∩I j 2 ,n =∅J_1\\cap I_{j_2,n}=\\emptyset )" ], [ "Case 1:", "$J_1\\cap I_{j_2,n}=\\emptyset $ (see Figure REF ).", "Then we have ${\\mathbb {P}}\\bigl (Y_n=y_1 \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_2}\\bigr )=0,$ whereas ${\\mathbb {P}}\\bigl (Y_n=y_1 \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_1}\\bigr )>0.$ Moreover, notice that (REF ) is equivalent to ${\\mathbb {P}}\\bigl (Y_n>y_1 \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_2}\\bigr )=1,$ It follows from (REF ) and (REF ) that there exists a $V_n$ -measurable event $A$ of positive probability such that, on $A$ , $ Y_n\\bigl (V_n,x_{j_2},\\xi \\bigr ) > y_1 = Y_n\\bigl (V_n,x_{j_1},\\xi \\bigr ).", "$ Remembering (REF ), we get ${\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_{j_2}\\bigr ] > {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_{j_1}\\bigr ].$ Figure: Case 2 (J 1 ∩I j 2 ,n ≠∅J_1\\cap I_{j_2,n}\\ne \\emptyset )" ], [ "Case 2:", "$J_1\\cap I_{j_2,n}\\ne \\emptyset $ (see Figure REF ).", "Then $I_{j_1,n}\\subset J_1$ and $I_{j_1,n}\\cap J_s=\\emptyset $ .", "It follows that ${\\mathbb {P}}\\bigl (Y_n=y_s \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_1}\\bigr )=0,$ whereas ${\\mathbb {P}}\\bigl (Y_n=y_s \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_2}\\bigr )>0.$ In this case, we get a $V_n$ -measurable event $A$ of positive probability such that, on $A$ , $ Y_n\\bigl (V_n,x_{j_2},\\xi \\bigr ) = y_s > Y_n\\bigl (V_n,x_{j_1},\\xi \\bigr ), $ and as before we conclude that (REF ) holds.", "Now, since (REF ) always holds, and since $ {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] = 0 =\\sum _{j=1}^rP_{j,n}\\, {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_{j}\\bigr ], $ we deduce that $ {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_{j_2}\\bigr ]>0.", "$ It follows that in the sum (REF ), at least the term corresponding to $j=j_2$ is positive, and this yields $ {\\mathbb {E}}\\bigl [ X_n Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] > 0.", "$ Appendix" ], [ "From averaged double to averaged multiple correlations", "This section follows some arguments from [42].", "Remark A.1 In the proof below we will use the following standard fact: let $(x(n))$ be a sequence of complex number bounded by 1.", "Then $\\sum _{m\\le M}|x(m)|=o(M)$ is equivalent to $\\sum _{m\\le M}|x(m)|^2=o(M).$ The little “o” is uniform with respect to $M$ .", "If $\\varepsilon :=\\frac{1}{M}\\sum _{m\\le M}|x(m)|^2$ then by Markov's inequality $\\frac{1}{M}|\\lbrace m\\le M:\\: |c_m|^2\\ge \\varepsilon ^{1/2}\\rbrace |\\le \\frac{1}{\\varepsilon ^{1/2}}\\cdot \\varepsilon =\\varepsilon ^{1/2}$ and then $\\frac{1}{M}\\sum _{m\\le M}|x(m)|=\\frac{1}{M}\\sum _{m\\le M, |x(m)|\\ge e^{1/4}}|x(m)|+\\frac{1}{M}\\sum _{m\\le M, |x(m)|<\\varepsilon ^{1/4}}|x(m)|\\le \\varepsilon ^{1/2}+\\varepsilon ^{1/4}.$ We have the following general lemma: Lemma A.2 Let $(N_\\ell )_{\\ell \\in {\\mathbb {N}}}$ be a sequence of natural numbers.", "For $k\\in \\mathbb {N}$ let $a,b_1,\\ldots b_k\\colon \\mathbb {N} \\rightarrow \\mathbb {C}$ be sequences bounded by 1.", "Assume that $a$ satisfies $\\lim _{H\\rightarrow \\infty }\\frac{1}{H} \\sum _{h\\le H}\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n)a(n+h)\\Big |=0.$ Then $\\lim _{H\\rightarrow \\infty }\\frac{1}{H^k} \\sum _{h_1,\\ldots , h_k\\le H}\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n)\\prod _{i=1}^kb_i(n+h_i)\\Big |=0.$ Notice first that (REF ) can be rewritten as the following: for every $\\varepsilon >0$ , there exists $H_\\varepsilon $ such that for $H>H_\\varepsilon $ and all $\\ell $ sufficiently large (depending on $H$ ), we have $A:=\\frac{1}{H^k}\\sum _{h_1,\\ldots , h_k\\le H}\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n)\\prod _{i=1}^kb_i(n+h_i) \\Big |<\\varepsilon .$ Now, notice that for any $H,N_\\ell ,H^{\\prime }$ and any $h^{\\prime }\\le H^{\\prime }$ , by shifting the summation over $n\\le N_\\ell $ by $h^{\\prime }$ (for every fixed choice of $h_1,\\ldots h_k$ ), we have $\\sum _{h_1,\\ldots , h_k\\le H}\\Big | \\sum _{n\\le N_\\ell }a(n)\\prod _{i=1}^kb_i(n+h_i)\\Big |=$ $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{h^{\\prime }\\le n\\le N_\\ell +h^{\\prime }}a(n)\\prod _{i=1}^kb_i(n+h_i)\\Big |+{\\rm O}(H^k\\cdot h^{\\prime })=$ $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i+h^{\\prime })\\Big |+{\\rm O}(H^k\\cdot h^{\\prime })$ and ${\\rm O}(H^k\\cdot h^{\\prime })={\\rm O}(H^k\\cdot H^{\\prime })$ .", "Notice that as $h_i$ is taken from $[0,H]$ , then $h_i+h^{\\prime }$ is taken from $[h^{\\prime },H+h^{\\prime }]$ (which is a small shift of $[0,H]$ if $h^{\\prime }$ is much smaller than $H$ ).", "So putting $h^{\\prime }$ to the summation over $h_i$ , we get $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i+h^{\\prime })\\Big |=$ $\\sum _{h^{\\prime }\\le h_1,\\ldots , h_k\\le H+h^{\\prime }}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i)\\Big |=$ $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i)\\Big |+ {\\rm O}\\left((h^{\\prime } H^{k-1}+(h^{\\prime })^2H^{k-2}+\\ldots +(h^{\\prime })^k)N_\\ell \\right)$ and ${\\rm O}\\left((h^{\\prime } H^{k-1}+(h^{\\prime })^2H^{k-2}+\\ldots +(h^{\\prime })^k)N\\right)={\\rm O}((H^{\\prime })^kH^{k-1}N_\\ell )$ .", "Putting the two displayed equations together we get that for every $h^{\\prime }\\le H^{\\prime }$ , $A=\\frac{1}{H^k}\\sum _{h_1,\\ldots , h_k\\le H}\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i) \\Big |+{\\rm O}\\Big (\\frac{H^{\\prime }}{N_\\ell }\\Big )+{\\rm O}\\Big (\\frac{(H^{\\prime })^k}{H} \\Big ).$ Averaging the above equation over all $h^{\\prime }\\le H^{\\prime }$ , we get that $A=\\frac{1}{H^k}\\sum _{h_1,\\ldots , h_k\\le H}\\frac{1}{H^{\\prime }}\\sum _{h^{\\prime }\\le H^{\\prime }}\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })G(n)\\Big |+{\\rm O}\\Big (\\frac{H^{\\prime }}{N_\\ell }\\Big )+{\\rm O}\\Big (\\frac{(H^{\\prime })^k}{H} \\Big ),$ where $G(n)=\\prod _{i=1}^kb_i(n+h_i)$ .", "We will now estimate $\\frac{1}{H^{\\prime }}\\sum _{h^{\\prime }\\le H^{\\prime }}\\frac{1}{N_\\ell ^2}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })G(n) \\Big |^2=\\\\\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\Big (\\frac{1}{H^{\\prime }}\\sum _{h^{\\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })}\\Big )G(n)\\overline{G(n^{\\prime })},$ which will be easier to handle than the above expression for $A$ (and then use Remark REF to get rid of the squares).", "Clearly, to obtain an upper bound for (REF ), it suffices to obtain an upper bound for $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime }}\\Big |\\sum _{h^{\\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })} \\Big |.$ Again, it will be easier to deal with $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime 2}}\\Big |\\sum _{h^{\\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })} \\Big |^2$ (and use Remark REF to get rid of the squares).", "Expanding the square again we get that (REF ) is equal to $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })}\\overline{a(n+h^{\\prime \\prime })\\cdot \\overline{a(n^{\\prime }+h^{\\prime \\prime })}}=$ $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n+h^{\\prime \\prime })}a(n^{\\prime }+h^{\\prime \\prime })\\overline{a(n^{\\prime }+h^{\\prime })}.$ The sum in the last term by exchanging the order of summation is equal to $\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}\\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\overline{a(n+h^{\\prime \\prime })}\\Big |^2=$ $\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}\\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h^{\\prime \\prime }-h^{\\prime })}\\Big |^2+ {\\rm O}(\\frac{H^{\\prime 2}}{N_\\ell ^2}).$ Finally, grouping according to $h=h^{\\prime \\prime }-h^{\\prime }$ , we get that that the above is equal to $\\frac{1}{H^{\\prime 2}}\\sum _{|h|\\le H^{\\prime }}|H^{\\prime }-h|\\cdot \\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}\\Big |^2+{\\rm O}(\\frac{H^{\\prime 2}}{N_\\ell ^2})\\le $ $\\frac{1}{H^{\\prime }}\\cdot \\sum _{|h|\\le H^{\\prime }}\\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}\\Big |^2+{\\rm O}(\\frac{H^{\\prime 2}}{N_\\ell ^2}).$ That is, the expression from (REF ) equals $\\frac{1}{H^{\\prime }}\\sum _{|h|\\le H^{\\prime }}\\left|\\frac{\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}}{N_\\ell } \\right|^2 +{\\rm O}\\left(\\frac{(H^{\\prime })^2}{N_\\ell ^2} \\right).$ Now, by the assumption of our lemma, it follows that $\\frac{1}{H^{\\prime }}\\sum _{|h|\\le H^{\\prime }}\\left|\\frac{\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}}{N_\\ell } \\right|={\\rm o}(1),$ which, by Remark REF , is equivalent to $\\frac{1}{H^{\\prime }}\\sum _{|h|\\le H^{\\prime }}\\left|\\frac{\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}}{N_\\ell } \\right|^2={\\rm o}(1).$ Therefore,  (REF ) (and, thus, also (REF )) is of the order of ${\\rm o}(1)+{\\rm O}\\left(\\frac{(H^{\\prime })^2}{N_\\ell ^2} \\right)$ .", "Using again Remark REF , we conclude that also (REF ) is of the same order.", "It follows immediately that also the order of (REF ) is the same.", "Thus, we have proved that $A={\\rm o}\\left(1\\right)+{\\rm O}\\left(\\frac{(H^{\\prime })^2}{N_\\ell ^2} \\right)+{\\rm O}\\left(\\frac{H^{\\prime }}{N_\\ell } \\right)+{\\rm O}\\left(\\frac{(H^{\\prime })^k}{H}\\right).$ Acknowledgments: We would like to thank Tomasz Downarowicz, Nikos Frantzikinakis and Krzysztof Fra̧czek for useful discussions on the paper.", "Research of the second and third authors supported by Narodowe Centrum Nauki grant UMO-2019/33/B/ST1/00364.", "Department of Mathematics, The Maryland University [email protected] Faculty of Mathematics and Computer Science Nicolaus Copernicus University, Toruń, Poland [email protected], [email protected] Laboratoire de Mathématiques Raphaël Salem, CNRS – Université de Rouen Normandie Avenue de l’Université – 76801 Saint Étienne du Rouvray, France [email protected]" ], [ "Definition, examples, basic properties", "Recall that a class ${{F}}$ of measure-theoretic dynamical systems is characteristic if it is closed under taking isomorphisms, factors and (countable) joinings.", "Recall also the following classical result on such classes (see e.g.", "[45]).", "Proposition 2.1 Given a characteristic class ${{F}}$ , each automorphism $R$ on $(Z,\\mathcal {D},\\kappa )$ has a largest ${{F}}$ -factor, denoted by $\\mathcal {D}_{{{F}}}$ .", "The following result whose proof is based on a fundamental non-disjointness lemma from [39] will be crucial for us: Proposition 2.2 ([45]) Let $(X,{\\cal B},\\nu ,T)$ be a measure-theoretic dynamical system in the characteristic class ${{F}}$ , and let $(Z,{\\cal D},\\kappa ,R)$ be any measure-theoretic dynamical system.", "Then any joining of $R$ and $T$ is relatively independent over the largest ${{F}}$ -factor ${\\cal D}_{{{F}}}$ of $R$ .", "That is: if $g\\in L^2(Z,\\kappa )$ is such that ${\\mathbb {E}}^{\\kappa }[g|{\\cal D}_{{{F}}}]=0$ , and if $\\rho $ is a joining of $T$ and $R$ , then for any $f\\in L^2(X,\\nu )$ we have ${\\mathbb {E}}^\\rho (f\\otimes g)=0.$ Examples of characteristic classes (some acronyms are used for those which will be used in the sequel): ALL: all automorphisms of standard Borel probability spaces; $\\lbrace \\ast \\rbrace $ : the identity on the one-point space; ID: identity automorphisms (of all standard Borel probability spaces); DISP: discrete spectrum automorphisms; RDISP: rational discrete spectrum automorphisms; DISP($G$ ): discrete spectrum automorphisms whose group of eigenvalues is contained in fixed countable subgroup $G$ of the circle; ${\\rm NIL}_s$ : automorphisms whose a.a. ergodic components are inverse limits of $s$ -step nilautomorphisms.", "The fact that ergodic joinings of nilsystems remain nil, see Proposition 15, page 186 in the book [31], and the same holds for inverse limits (this is actually Lemma A.4 in [20]).", "Regarding factors of ergodic nilsystems, see Theorem 11 in page 230 [31].", "Here $s\\in {\\mathbb {N}}$ .", "DIST: distal automorphisms are those which are given as a transfinite (indexed by ordinals smaller than a fixed countable ordinal) sequence of consecutive extensions each of which either has relative discrete spectrum or (in case of a limit ordinal) is the corresponding inverse limit.", "The structural theorem Theorem 6.17 together with the concluding remark (for ${\\mathbb {Z}}$ -actions) on page 139 [22] tell us that each system has a largest distal factor, hence DIST is closed under countable joinings.", "In Lemma REF , we note that an automorphism is distal if and only if a.a. its ergodic components are distal.", "To see that this class is closed under taking factors, let us first recall that this fact holds for ergodic automorphisms (see Theorem 10.18 [25]).", "If now ${\\cal A}\\subset {\\cal D}$ is a factor of a distal automorphism $(Z,{\\cal D},\\kappa ,R)$ then ${\\cal A}$ (as an $R$ -invariant $\\sigma $ -algebra) is also a factor of a.e.", "of its ergodic components.", "So a double use of Lemma REF together with Theorem 10.18 from [25] gives that ${\\cal A}$ is also distal.", "ZE: zero entropy automorphisms; RIG$_{(q_n)}$ : automorphisms with a fixed sequence $(q_n)$ of rigidity; multipliers ${M}({D}^{\\perp })$ of a class ${D}^\\perp $ (${D}$ is any class of automorphisms and by ${D}^{\\perp }$ we mean the set of automorphisms disjoint from all systems from ${D}$ , and by ${M}({D}^{\\perp })$ we mean the set of systems whose all joinings with any element of ${D}^{\\perp }$ remain in ${D}^{\\perp }$ ); interesting classes of multipliers arise e.g.", "for ${D}$ =all weakly mixing (cf.", "Proposition 5.1 in [38]) or all mixing automorphisms; see [24], [38].", "the class of factors of all infinite self-joinings of a fixed automorphism $R$ (the smallest characteristic class containing $R$ ); especially in case of MSJ and simple automorphisms (cf.", "[25], Chapter 12).", "Characteristic classes of such type were used in [40].", "Note also that the intersection of any family of characteristic classes yields again a characteristic class.", "In Section REF , we will show that each characteristic class ${{F}}$ determines another characteristic class ${{F}}_{\\rm ec}$ consisting of those automorphisms whose ergodic components are in ${{F}}$ ." ], [ "The smallest nontrivial and the largest proper characteristic class", "An obvious observation has been made already in the introduction that the family ALL of all automorphisms is the largest characteristic class, while the one-element $\\lbrace \\ast \\rbrace $ family (which is the one-point space automorphism) is the smallest characteristic class.", "It is more interesting however that the smallest non-trivial and the largest proper characteristic classes exist.", "Proposition 2.3 ID is the smallest non-trivial characteristic class.", "Let us first notice that the system $([0,1],{\\rm Leb}, {\\rm Id})$ has all other identities as factors.", "Indeed, any standard Borel probability space is determined by a sequence $(t_i)_{i\\ge 0}$ of non-negative numbers such that $\\sum _{i\\ge 0}t_i=1$ and $t_0$ corresponds to the mass of the continuous part and $t_1,t_2,\\ldots $ correspond to the masses of atoms.", "Then, take the corresponding partition of $[0,1]$ into intervals $I_i$ of length $t_i$ and, for each $i\\ge 1$ , the factor map will glue points in $I_i$ .", "Now, notice that any non-trivial characteristic class ${{F}}$ contains a non-ergodic automorphism.", "Indeed, suppose that $T$ is ergodic, acting on a non-trivial space $(Y,\\nu )$ .", "Since $Y$ is non-trivial and so is $T$ , the graph joinings $\\Delta _{{\\rm Id}}$ and $\\Delta _T$ are ergodic and different, so any non-trivial convex combination of them yields a non-ergodic member of ${{F}}$ .", "It follows that by taking the factor $\\mathcal {I}_{\\nu }$ of (a.e.)", "$T$ -invariant sets (which belongs to ${{F}}$ ), we obtain the identity on a non-trivial standard Borel probability space $(\\overline{Y},\\overline{\\nu })$ .", "But then the infinite Cartesian product $(\\overline{Y}^{\\times \\infty },\\overline{\\nu }^{\\otimes \\infty })$ is also in ${{F}}$ and this infinite product is isomorphic to $([0,1],{\\rm Leb})$ , which completes the proof.", "In order to prove the existence of the largest characteristic (proper) class, we need to recall some results.", "Theorem 2.4 (non-ergodic Sinai's factor theorem [36], [48]) Assume that $R$ is an automorphism of $(Z,{\\cal D},\\rho )$ and let $\\rho =\\int _{X/\\mathcal {I}_{\\rho }}\\rho _{\\overline{x}}\\,dm(\\overline{x})$ stand for the ergodic decomposition of $\\rho $ .", "Assume that $m-{\\rm essinf}_{\\overline{x}\\in X/\\mathcal {I}_\\rho }h(\\rho _{\\overline{x}},R)\\ge \\alpha >0.$ Then a Bernoulli automorphism of entropy $\\alpha $ is a factor of $R$ .", "In [48] (see Theorem 4.3 therein) the above result is attributed to Kieffer and Rahe [36], see also [47] p.2 (The non-ergodic factor theorem).", "We also need the following well-known result (we include its proof for completeness).", "Proposition 2.5 Each automorphism $R$ is a factor of a self-joining of the (infinite entropy) Bernoulli automorphism $([0,1]^{{\\mathbb {Z}}},{\\rm Leb}^{\\otimes {\\mathbb {Z}}},S)$ .", "Remark 2.6 Before we prove the above result, let us notice that any automorphism $T$ of $(X,{\\cal B},\\mu )$ has an isomorphic copy in the space $([0,1]^{{\\mathbb {Z}}},\\kappa ,S)$ .The same arguments apply to $\\lbrace z\\in |z|\\le 1\\rbrace $ instead of $[0,1]$ .", "Consider first the aperiodic part of $T$ which is realized on a standard Borel space.", "This space is isomorphic to $[0,1]$ , via a Borel isomorphism $I$ .", "It follows that the distribution $\\mu ^{\\prime }$ of the process $(I\\circ S^k)_{k\\in {\\mathbb {Z}}}$ yields a realization of the aperiodic part.", "Now, $\\mu ^{\\prime }$ takes measure zero on the set of periodic points for the shift.", "Moreover, the set of periodic points of $S$ can be identified with a subset of $[0,1]$ , points of period 2 with a subset of $[0,1]^2$ etc., and we can easily settle an isomorphism of the fixed point subspace for $T$ with a subset of $[0,1]$ , period 2 points with a subset of $[0,1]^2$ , etc.", "Thus, it suffices to take $\\kappa $ equal to the sum of $\\mu ^{\\prime }$ and the relevant atomic measures corresponding to the periodic points.", "[Proof of Proposition REF ] Fix any automorphism $R$ of $(Z,{\\cal D},\\rho )$ and take its isomorphic copy in the space $([0,1]^{{\\mathbb {Z}}},\\kappa ,S)$ .", "Take the product space $([0,1]^{{\\mathbb {Z}}}\\times [0,1]^{{\\mathbb {Z}}},{\\rm Leb}^{\\otimes {\\mathbb {Z}}}\\otimes \\kappa )$ and consider the map $\\psi \\colon [0,1]^{{\\mathbb {Z}}}\\times [0,1]^{{\\mathbb {Z}}}\\rightarrow [0,1]^{{\\mathbb {Z}}}$ given by $(x_n,y_n)\\mapsto (x_n+y_n).$ Then $\\psi _\\ast ({\\rm Leb}^{\\otimes {\\mathbb {Z}}}\\otimes \\kappa )={\\rm Leb}^{\\otimes {\\mathbb {Z}}}$ and clearly the join of the $\\sigma $ -algebra of the first coordinate and of $\\psi ^{-1}(\\mathcal {B}([0,1]^{\\otimes {\\mathbb {Z}}}))$ is the product $\\sigma $ -algebra in $[0,1]^{\\otimes {\\mathbb {Z}}}\\times [0,1]^{\\otimes {\\mathbb {Z}}}$ .", "The result follows.", "We now prove the following.", "Lemma 2.7 Assume that ${{F}}$ is a characteristic class such that ${{F}}\\setminus {\\rm ZE}\\ne \\emptyset $ .", "Then ${{F}}={\\rm ALL}$ .", "Fix $T\\in {{F}}\\setminus {\\rm ZE}$ .", "Because of Proposition REF , we only need to prove that the infinite entropy Bernoulli automorphism is in ${{F}}$ .", "The first step is to consider the factor of $T$ that arises by gluing together the periodic part and the ergodic components from the aperiodic part whose entropy is smaller than $\\alpha =h(T)$ .", "Clearly, this factor remains in ${{F}}\\setminus {\\rm ZE}$ .", "Moreover, in its ergodic decomposition we have a single point and the remaining part (which may still be non-ergodic) consists of ergodic components of entropy at least $\\alpha $ .", "By Theorem REF , it follows that as a further factor $R\\in {{F}}\\setminus {\\rm ZE}$ we can obtain a non-ergodic automorphism with two ergodic components: one of them is a Bernoulli of entropy $\\alpha $ and the other one is a fixed point.", "Finally, we take the infinite Cartesian product $R^{\\times \\infty }$ .", "It is not hard to see that a.e.", "ergodic component of this automorphism is a Bernoulli with infinite entropy.", "Using once more Sinai's theorem (Theorem REF ), we obtain that a Bernoulli with infinite entropy belongs to ${{F}}$ which completes the proof.", "Now, using the lemma we obtain the following.", "Proposition 2.8 ZE is the largest proper characteristic class.$\\quad \\hbox{\\vrule \\vbox to 6pt {\\hrule width 4pt\\vfill \\hrule }\\vrule } $" ], [ "Characteristic classes given by ergodic components", "Assume that ${{F}}$ is a characteristic class.", "By ${{F}}_{\\rm ec}$ we denote the class of those automorphisms $R$ such that (a.e.)", "ergodic components of $R$ are in ${{F}}$ (or more precisely, in ${{F}}\\cap $  Erg, where Erg stands for the family of all ergodic automorphisms).", "Note that we have ${{F}}\\cap {\\rm Erg}={{F}}_{\\rm ec}\\cap {\\rm Erg}.$ Lemma 2.9 ${{F}}_{\\rm ec}$ is a characteristic class.", "The proof has two parts: we need to show that ${{F}}_{\\rm ec}$ is closed under taking factors and joinings." ], [ "Factors", "Let $R$ acting on $(Z,{\\cal D},\\kappa )$ belong to ${{F}}_{\\rm ec}$ and fix a factor ${\\cal A}\\subset {\\cal D}$ of $R$ .", "Let $\\kappa =\\int \\kappa _{\\overline{x}}\\,dP(\\overline{x})$ denote the ergodic decomposition of $\\kappa $ .", "Since the ergodic components $\\kappa _{\\overline{x}}$ are $R$ -invariant measures, ${\\cal A}$ (being an $R$ -invariant sub-$\\sigma $ -algebra) is also a factor of the automorphism $(Z,\\kappa _{\\overline{x}},R)$ and $\\kappa |_{{\\cal A}}=\\int \\kappa _{\\overline{x}}|_{{\\cal A}}\\,dP(\\overline{x})$ is the ergodic decomposition of $\\kappa |_{{\\cal A}}$ .", "It follows that the ergodic components of the factor are factors of ergodic components of $R$ , and since $R\\in {{F}}_{\\rm ec}$ , $(\\kappa _{\\overline{x}},R)\\in {{F}}$ , so also $(\\kappa _{\\overline{x}}|_{{\\cal A}},R|_{{\\cal A}})\\in {{F}}$ for $P$ -a.e.", "$\\overline{x}$ ." ], [ "Joinings", "Take $(X,\\mu ,T)$ and $(Y,\\nu ,S)$ from ${{F}}_{\\rm ec}$ and let $\\rho \\in J(T,S)$ be their joining.", "Let $\\rho =\\int _0^1 \\rho _t\\,dP(t),\\;\\mu =\\int _0^1 \\mu _t\\,dQ(t),\\;\\nu =\\int _0^1\\nu _t\\,dR(t)$ be the relevant ergodic decompositions.", "Then $\\int _0^1\\mu _t\\,dQ(t)=\\mu =\\rho |_X=\\int _0^1\\rho _t|_X\\,dP(t),$ so since $\\rho _t|X$ are also ergodic, these two decompositions are the same.", "So for a $P$ -“typical” $t\\in [0,1]$ , the projection of $\\rho _t$ on $X$ is an ergodic component of $T$ .", "The same argument applies on the coordinate $Y$ and we see that the ergodic components of $\\rho $ are joinings of ergodic components of $\\mu $ and $\\nu $ .", "It follows that $(X\\times Y,\\rho ,T\\times S)\\in {{F}}_{\\rm ec}$ .", "The argument extends to countable joinings." ], [ "ID, ZE, DISP and RIG$_{(q_n)}$", "Given a characteristic class ${{F}}$ , according to Proposition REF , each automorphism $R$ acting on $(Z,{\\cal D},\\kappa )$ has a largest ${{F}}$ -factor ${\\cal D}_{{{F}}}\\subset {\\cal D}$ .", "Often, its description is classical: the $\\sigma $ -algebra of invariant sets for ${{F}}={\\rm ID}$ , the Pinsker factor for ${{F}}={\\rm ZE}$ , the Kronecker factor for ${{F}}={\\rm DISP}$ , the largest factor for which $(q_n)$ is a rigidity time for ${{F}}={\\rm RIG}_{(q_n)}$ ." ], [ "DISP$_{\\rm ec}$", "We will comment now on ${\\cal D}_{{{F}}_{\\rm ec}}$ when ${{F}}={\\rm DISP}$ , cf.", "Proposition REF  (ii) and its connections with the theory of nil-factors.", "Most of the material presented below is known to aficionados but not necessarily the material is explicitly present in the literature.", "Our discussion is based on [18], [23], [30] and [31].", "We provide some details to explain clearly why the problem of whether $\\mu \\perp {\\rm DISP}_{\\rm ec}$ is open, cf.", "Corollary REF , Corollary REF and Remark REF .", "Recall that according to the Furstenberg-Zimmer theory [23], given $R$ on $(Z,{\\cal D},\\kappa )$ and a factor $\\mathcal {C}\\subset \\mathcal {D}$ , there exists a certain intermediate factor $\\mathcal {C}\\subset {\\cal K}={\\cal K}(\\mathcal {C})\\subset \\mathcal {D},$ called the relative Kronecker factor (with respect to $\\mathcal {C}$ ).", "It is the largest intermediate factor with the following property (see condition C$_2$ in [23], p. 131): $\\begin{split}\\parbox [t]{0.8}{there exists a dense set of functions F\\in L^2({\\cal K},\\kappa |_{{\\cal K}}) such thatfor each \\delta >0 there is a finite set g_1,\\ldots ,g_k\\in L^2({\\cal K},\\kappa |_{{\\cal K}}) such that for each h\\in {\\mathbb {Z}},\\min _{1\\le j\\le k}\\Vert F\\circ R^h-g_j\\Vert _{L^2(\\kappa _y)}<\\delta }\\end{split}$ for a.e.", "$y\\in Z/\\mathcal {C}$ , where $\\kappa |_{{\\cal K}}=\\int _{Z/\\mathcal {C}}\\kappa _y\\,d\\kappa (y).$ Whenever condition (REF ) holds, we speak of relative compactness or of relatively discrete spectrum of the intermediate factor over $\\mathcal {C}$ .", "A particular situation arises when ${\\cal C}=\\mathcal {I}_\\kappa $ , i.e.", "it is the $\\sigma $ -algebra of invariant sets.", "Then (REF ) is nothing but the ergodic decomposition of $\\kappa $ and the conditional measures $\\kappa _y$ are also $R$ -invariant.", "In this case condition (REF ) yields in a.e.", "fiber $\\pi ^{-1}(y)$ (where $\\pi \\colon Z/{\\cal K}\\rightarrow Z/\\mathcal {I}_\\kappa $ stands for the factor map) a dense set of functions $F|_{\\pi ^{-1}(y)}$ in $L^2({\\cal K},\\kappa _y)$ whose orbits under the unitary action of $R$ are relatively compact.", "It follows that the (ergodic) automorphism $(R,\\kappa _y)$ has discrete spectrum for a.e.", "$y\\in Z/\\mathcal {I}_\\kappa $ .", "In other words, ${\\cal K}(\\mathcal {I}_\\kappa )\\subset \\mathcal {D}_{{\\rm DISP_{ec}}}$ In fact, the opposite inclusion is also true, i.e.", "$\\mathcal {D}_{{\\rm DISP_{ec}}} = {\\cal K}(\\mathcal {I}_\\kappa ),$ that is, $\\mathcal {A}:=\\mathcal {D}_{{\\rm DISP_{ec}}}$ has relatively discrete spectrum over $\\mathcal {I}_\\kappa $ .", "Indeed, by the definition of $\\mathcal {A}$ , a.e.", "ergodic component of $R|_{\\mathcal {A}}$ has discrete spectrum.", "Fix $F\\in L^\\infty ({\\cal A},\\kappa |_{{\\cal A}})$ .", "Fix also $\\varepsilon ,\\delta >0$ and $k\\ge 1$ .", "Consider the set $W_k\\subset Z/\\mathcal {I}_\\kappa $ of those $y$ for which $\\min _{-k\\le j\\le k}\\left\\Vert F\\circ R^n-F\\circ R^j\\right\\Vert _{L^2(\\kappa _y)}<\\varepsilon $ for each $n\\in {\\mathbb {Z}}$ .", "Since on each fiber $R$ is an ergodic automorphism with discrete spectrum, the measure of $W_k$ goes to 1, when $k\\rightarrow \\infty $ , so it will be greater than $1-\\delta $ for $k$ large enough.", "It follows that the function $F$ is compact as it has been defined in the proof of Theorem 6.15 [23].", "Therefore, $F\\in L^2({\\cal K}(\\mathcal {I}_\\kappa ))$ , which (by [23]) concludes the proof of (REF ).", "Remark 2.10 As a matter of fact, in [5], the Furstenberg-Zimmer theory is developed without assuming ergodicity (cf.", "e.g.", "Proposition 5.7 therein to obtain the equality $\\mathcal {D}_{{\\rm DISP}_{\\rm ec}}=\\mathcal {K}(\\mathcal {I}_\\kappa )$ ).", "We will now see that $\\mathcal {D}_{\\rm DISP_{ec}}$ appears naturally in the classical theory of characteristic nil-factors [30], [31].We would like to thank Bryna Kra and Nikos Frantzikinakis for fruitful discussions and useful references on this subject.", "Recall that if $R$ acting on $(Z,{\\cal D},\\kappa )$ is ergodic then for a function $f\\in L^\\infty (Z,{\\cal D},\\kappa )$ its $u^s$ norms (in fact, seminorms) are defined in the following way: $\\Vert f\\Vert _{u^1}:=\\Big |\\int f\\,d\\kappa \\Big |,$ $\\Vert f\\Vert _{u^{s+1}}^{2^{s+1}}:=\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\Vert f\\circ R^h\\cdot \\overline{f}\\Vert _{u^s}^{2^s}.$ If $R$ is non-ergodic then instead of (REF ), we put $\\Vert f\\Vert ^2_{u^1}:=\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\int f\\circ R^h\\cdot \\overline{f}\\,d\\kappa $ and (REF ) remains unchanged.", "Then, by [30], [31], for each $s\\ge 1$ there is a special factor $\\mathcal {Z}_s=\\mathcal {Z}_s(R)\\subset {\\cal B}$ , namely, the largest factor whose $\\mbox{a.e.\\ ergodic component is an inverse limit of $s$-step nil-systems.", "}$ In other words, $\\mathcal {Z}_s(R)$ is the largest (characteristic) ${\\rm NIL}_s$ -factor of $R$ .", "Moreover (see Proposition 7 (page 138) and Proposition 13 (page 141) in [31]), $\\Vert f\\Vert _{u^{s+1}}=0 \\iff f\\perp L^2(\\mathcal {Z}_s) \\iff f\\perp L^2(\\mathcal {Z}_s(R,\\kappa _y)) \\text{ for $\\kappa $-a.e.", "}y.$ A special case arises when our measure-preserving systems are Furstenberg systems of a bounded $u\\colon {\\mathbb {N}}\\rightarrow .As in (for example) \\cite {Fr}, see Sections~2.4 and 2.5 therein, one can introduce the uniformity norms (along subsequences of intervals) for $ u$.", "The definitions are given in (\\ref {eq:1us}) and (\\ref {eq:2us}).", "They are very similar to those (in the non-ergodic case) to the definitions for functions.$ We will now show that $\\mathcal {Z}_1(R)=\\mathcal {K}(\\mathcal {I}_\\kappa ).$ If $R$ is ergodic then the above means just that $\\mathcal {Z}_1 \\text{ is the Kronecker factor of $R$}.$ To see that (REF ) indeed holds, notice that (REF ) for $s=1$ yields $\\Vert f\\Vert _{u^2}=0 \\iff f\\perp L^2(\\mathcal {Z}_1)$ and it remains to notice that (using the Wiener lemma) $\\Vert f\\Vert ^4_{u^2}=\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\Vert f\\circ R^h \\cdot \\overline{f} \\Vert ^2_{u^1}=$ $\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\left| \\int f\\circ R^h \\cdot \\overline{f}\\right|^2\\,d\\kappa \\rightarrow \\sum _{z\\in \\mathbb {T}}\\sigma _f(\\lbrace z\\rbrace )^2,$ where $\\sigma _f$ stands for the spectral measure of $f$ .", "Let us return to a possibly non-ergodic $R$ .", "The inclusion $\\mathcal {Z}_1(R)\\subset \\mathcal {K}:=\\mathcal {K}(\\mathcal {I}_\\kappa )$ follows directly by Theorem 5.2 in [18].", "To obtain the opposite inclusion, one can argue in the following way.", "Suppose that $f\\perp L^2(\\mathcal {Z}_1(R))$ and $|f|\\le 1$ .", "Take $g\\in L^2(\\mathcal {K})$ , cf.", "(REF ) with $F=g$ .", "We want to show that $\\int fg\\, d\\kappa =0$ .", "Notice that $\\int fg\\, d\\kappa =\\int \\left(\\frac{1}{N}\\sum _{n\\le N}\\int f\\circ T^n \\cdot g\\circ T^n\\, d\\kappa _y\\right)\\, d\\kappa (y).$ Let $g_j$ , $1\\le j\\le k$ , be as in (REF ).", "Then $\\left|\\frac{1}{N}\\sum _{n\\le N}\\int f\\circ T^n \\cdot g\\circ T^n\\, d\\kappa _y\\right|\\\\\\le \\left|\\sum _{1\\le j\\le k}\\frac{1}{N}\\sum _{n\\le N}\\int f\\circ T^n \\cdot g_j\\, d\\kappa _y \\right| + \\frac{1}{N}\\sum _{n\\le N}\\min _{1\\le i\\le k}\\int \\left|f\\circ T^n (g\\circ T^n-g_i) \\right|\\, d\\kappa _y.$ Each term in the average in the second summand is bounded by $\\delta $ .", "Moreover, $\\frac{1}{N}\\sum _{n\\le N}\\left|\\int f\\circ T^n \\cdot g_j\\, d\\kappa _y \\right|^2 \\rightarrow \\sum _{z\\in \\mathbb {T}}\\sigma _{f,g_j,\\kappa _y}(\\lbrace z\\rbrace )^2,$ where $\\sigma _{f,g_j,\\kappa _y}$ stands for the spectral measure of the pair $f,g_j$ (on the ergodic component $(\\pi ^{-1}(y),\\kappa _y)$ given by $y$ ).", "But by (REF ) and (REF ), we have $f\\perp L^2(\\mathcal {Z}_1) &\\iff f\\perp L^2(\\mathcal {Z}_1(R,\\kappa _y))\\text{ for a.e.", "}y\\\\& \\iff \\sigma _{f,\\kappa _y} \\text{ is continuous for a.e.", "}y$ ($\\sigma _{f,\\kappa _y}$ stands for the spectral measure of $f$ on the ergodic component $(\\pi ^{-1}(y),\\kappa _y)$ given by $y$ ).", "Since $f\\perp L^2(\\mathcal {Z}_1)$ and $\\sigma _{f,g_i,\\kappa _y}\\ll \\sigma _{f,\\kappa _y}$ , it remains to use the classical equivalence $\\frac{1}{N}\\sum _{n\\le N}a_n \\rightarrow 0 \\iff \\frac{1}{N}\\sum _{n\\le N}a_n^2\\rightarrow 0$ for any bounded sequence $(a_n)\\subset [0,\\infty )$ , to conclude that the limit in (REF ) is equal to zero.", "Thus $f\\perp L^2(\\mathcal {Z}_1) \\Rightarrow f\\perp L^2(\\mathcal {K})$ .", "Finally, let us compare the above with the notion of relative weak mixing.", "Recall that relative weak mixing over $\\mathcal {I}_\\kappa $ for $f$ means that $\\frac{1}{H}\\sum _{h\\le H}\\int \\left|{\\mathbb {E}}(f\\circ R^h\\cdot \\overline{f}|\\mathcal {I}_\\kappa )\\right|^2\\, d \\kappa \\rightarrow 0.$ Moreover, $\\frac{1}{H}\\sum _{h\\le H}\\int \\left|{\\mathbb {E}}(f\\circ R^h\\cdot \\overline{f}|\\mathcal {I}_\\kappa )\\right|^2\\, d \\kappa =\\int \\left(\\frac{1}{H}\\sum _{h\\le H}\\left|\\int f\\circ R^h\\cdot \\overline{f}\\,d\\kappa _y\\right|^2\\right)\\, d \\kappa ,$ and, once more by the Wiener lemma, $\\frac{1}{H}\\sum _{h\\le H}\\left| \\int f\\circ R^h\\cdot \\overline{f}\\,d\\kappa _y\\right|^2\\rightarrow \\sum _{z\\in \\sigma _{f,\\kappa _y}(\\lbrace z\\rbrace )^2.It follows immediately that \\sigma _{f,\\kappa _y} is continuous for a.e.\\ y if and only if f is relatively weakly mixing over \\mathcal {I}_\\kappa .", "}The above discussion can be summarized in the following statement.\\begin{Cor}Let (Z,{\\cal D},\\kappa ,R) be a measure-theoretic dynamical system and let f\\in L^2(Z,{\\cal D},\\kappa ).", "The following conditions are equivalent:\\begin{enumerate}\\item [(i)] f\\perp L^2(\\mathcal {Z}_1),\\item [(ii)] f\\perp L^2(\\mathcal {D}_{\\rm DISP_{ec}}),\\item [(iii)] f\\perp L^2(\\mathcal {K}(\\mathcal {I}_\\kappa )),\\item [(iv)] \\sigma _{f,\\kappa _y} is continuous for \\kappa -a.e.", "y,\\item [(v)] f is relatively weakly mixing over \\mathcal {I}_\\kappa .\\end{enumerate}\\end{Cor}$" ], [ "A class vs. its ec-class", "Let us continue our observations on the relations between characteristic classes and the corresponding ec-classes.", "Note that in general there are no relations between ${{F}}$ and ${{F}}_{\\rm ec}$ : Proposition 2.11 We have: (i) ${\\rm ZE}={\\rm ZE}_{\\rm ec}$ , ${\\rm ALL}={\\rm ALL}_{\\rm ec}$ , ${\\rm ID}={\\rm ID}_{\\rm ec}$ , ${\\rm NIL}_s=({\\rm NIL}_s)_{\\rm ec}$ , $\\lbrace \\ast \\rbrace \\subsetneq \\lbrace \\ast \\rbrace _{\\rm ec}$ ; (ii) ${\\rm DISP}\\subsetneq {\\rm DISP}_{\\rm ec}$ ; (iii) ${\\rm RDISP}={\\rm RDISP}_{\\rm ec}$ ; (iv) ${\\rm DIST}={\\rm DIST}_{\\rm ec}$ ; (v) $\\Big ({\\rm RIG}_{(q_n)}\\Big )_{\\rm ec}\\subsetneq {\\rm RIG}_{(q_n)}$ .", "[Proof of (i)-(iii)]  (i) The first claim follows from the fact that the entropy function is convex, the other claims are obvious.", "(ii) If an automorphism has discrete spectrum then its $L^2$ -space is generated by eigenfunctions.", "The restrictions (if non-zero) of these (global) eigenfunctions yield orthonormal bases in $L^2$ -spaces of ergodic components.", "The inclusion is strict since $(x,y)\\mapsto (x,x+y)$ (on 2, considered with Lebesgue measure) does not have discrete spectrum while the ergodic components do.", "(iii) We want to show that if each ergodic component has rational discrete spectrum then the whole automorphism has.", "Given $p/q\\in \\mathbb {Q}$ and an ergodic component $c$ , we choose $f_c$ a modulus 1 eigenfunction corresponding to the eigenvalue $e^{2\\pi ip/q}$ .", "Since $f_c$ is unique up to a constant of modulus 1, this choice can be done measurably.", "In this way, we will create global eigefunctions.The same argument works if we consider the characteristic class of automorphisms having discrete spectrum contained in a fixed countable subgroup of the circle.", "Before we give the proof of (iv), we need to recall some more notions and facts from the relative ergodic theory, e.g.", "[23], [25].", "Given an automorphism $T$ of $(X,{\\cal B},\\mu )$ and its factor $S$ on $(Y,{\\cal C},\\nu )$ with the factor map $\\pi \\colon X\\rightarrow Y$ , we say that this extension is relatively ergodic (rel.", "erg.)", "if each $f\\in L^1(X,{\\cal B},\\mu )$ satisfying $f\\circ T=f$ ($\\mu $ -a.e.)", "is $\\pi ^{-1}(\\mathcal {C})$ -measurable.", "It follows immediately from the definition that: any composition of relatively ergodic extensions remains relatively ergodic; an inverse limit of relatively ergodic extensions remains relatively ergodic (as the conditional expectation, with respect to a factor, of an invariant function remains invariant); $\\overline{\\pi }\\colon Y\\rightarrow \\overline{Y}:=Y/\\mathcal {I}_\\nu $ , where $(\\overline{Y},\\overline{\\nu })$ stands for the space of ergodic components (on which acts the identity map), is relatively ergodic.", "Let $\\mu =\\int _Y \\mu _y\\,d\\nu (y)$ stand for the disintegration of $\\mu $ over $\\nu $ and let $\\nu =\\int _{\\overline{Y}}\\overline{\\nu }_{\\overline{y}}\\,d\\overline{\\nu }$ denote the ergodic decomposition of $\\nu $ (which is precisely the disintegration of $\\nu $ over $\\overline{\\nu }$ ).", "Then the ergodic components of $S$ acting on $Y$ are of the form $(\\overline{\\pi }^{-1}(\\overline{y}),\\overline{\\nu }_{\\overline{y}}, S)$ (the measures $\\overline{\\nu }_{\\overline{y}}$ are $S$ -invariant).", "Therefore, we have the following lemma.", "Lemma 2.12 If $T$ is relatively ergodic over $S$ then the ergodic components of $T$ are of the form $\\left(\\pi ^{-1}(\\overline{\\pi }^{-1}(\\overline{y})),\\int _{\\overline{\\pi }^{-1}(\\overline{y})}\\mu _y\\, d \\overline{\\nu }_{\\overline{y}}(y)\\right).$ Note that it follows that the ergodic components of $T$ have as their factors (via the relevant restriction of $\\pi $ ) ergodic components of $S$ , and that the spaces of ergodic components of $T$ and $S$ are the same (i.e.", "$\\overline{X}=\\overline{Y}$ ).", "Lemma 2.13 Let $T$ be relatively ergodic over $S$ .", "Then the following are equivalent: $T$ over $S$ has relatively discrete spectrum.", "The ergodic components of $T$ have relatively discrete spectrum over the ergodic components of $S$ being their relevant factors.", "By Lemma REF , we see that the disintegration of an ergodic component $\\pi ^{-1}(\\overline{\\pi }^{-1}(\\overline{y}))$ over $\\overline{\\pi }^{-1}(\\overline{y})$ (which is its factor) consists of the same conditional measures $\\mu _y$ as the total disintegration of $\\mu $ over $\\nu $ .", "We proceed now as in the proof of the equality $\\mathcal {K}(I_\\kappa )=\\mathcal {D}_{\\rm DISP_{ec}}$ (page REF ), showing compactness.", "Recall that an automorphism $T$ is distal if it is a limit of a transfinite (indexed by countable ordinals) sequence of consecutive maximal Kronecker extensions (if an ordinal is not isolated, we pass to the corresponding inverse limit).", "Note that, by the very definition, the $\\sigma $ -algebra Inv is contained in the Kronecker factor of $T$ , so in this transfinite chain of consecutive extensions, all but (perhaps) the first one are relatively ergodic.", "By applying Lemma REF and transfinite induction, we obtain the following.", "Lemma 2.14 $T$ is distal if and only if all its ergodic components are distal.", "[Proof of (iv)-(v)]    (iv) This follows directly from Lemma REF .", "(v) It is clear that if $(q_n)$ is a rigidity time for an a.e.", "ergodic component, it is also a rigidity time for the whole automorphisms.", "Not vice versa however (for $(q_n)$ sufficiently sparsed).", "We will provide a relevant construction below." ], [ "${\\rm RIG}_{\\rm ec}$ is a proper subclass of {{formula:3509d1c9-0286-4aa1-af65-1e70a9710123}}", "Let us first notice that we only need to construct a continuous measure $\\sigma $ on the circle such that $e^{2\\pi i q_n\\cdot }\\rightarrow 1\\text{ in measure }\\sigma \\text{ but not } \\sigma -\\text{a.e.", "}$ Indeed, suppose (REF ) holds, and consider on 2 the automorphism $T(x,y)=(x,y+x)\\text{ with measure }\\sigma \\otimes \\,{\\rm Leb}.$ If $F(x,y)=f(x)e^{2\\pi i\\ell y}$ then by (REF ), $\\int |F(T^{q_n}(x,y))-F(x,y)|\\,d\\sigma (x)dy=\\int |f(x)||e^{2\\pi iq_n\\ell x}-1|\\,d\\sigma (x)\\rightarrow 0$ when $n\\rightarrow \\infty $ .", "On the other hand, the rotation by $x$ on an ergodic component $\\lbrace x\\rbrace \\times has $ (qn)$ as its rigidity time if and only if $ qnx0$ mod~1.", "This is not true for $$-a.e.\\ $ x in view of (REF ).", "We now sketch how to construct such a measure assuming that $(q_n)$ is sufficiently sparsed.", "Fix $0<p_n<1$ so that $p_n$ is decreasing to zero and $\\sum _{n\\ge 1}p_n=\\infty $ .", "Set $f_n(x)=\\lbrace q_nx\\rbrace $ .", "We intend to construct a Cantor set (together with a Cantor measure $\\sigma $ on it).", "Let $A_n:=f_n^{-1}([1/4,3/4]), \\;B_n=f_n^{-1}([0,p_n]).$ Our postulates are: $\\sigma (B_n)=1-p_n,\\; \\sigma (A_n)=p_n.$ In fact, we need to be more precise in description of the measure at stage $n$ to be able to continue its definition.", "So at stage $n$ the circle is divided into intervals of the form $[\\frac{j}{q_n},\\frac{j+1}{q_n})$ (many of such intervals are of measure $\\sigma $ equal to zero).", "We now require that the conditional measures satisfy: $\\sigma \\left(B_n|[\\frac{j}{q_n},\\frac{j+1}{q_n})\\right)=1-p_n,\\,\\sigma \\left(A_n|[\\frac{j}{q_n},\\frac{j+1}{q_n})\\right)=p_n$ for each $j=0,\\ldots ,q_n-1)$ .", "Passing to step $n+1$ , we require that all the intervals $[\\frac{j}{q_n},\\frac{j+1}{q_n}))$ contain at least two intervals of the form $[\\frac{k}{q_{n+1}},\\frac{k+1}{q_{n+1}})$ , we choose two of such (of course only in those $[\\frac{j}{q_n},\\frac{j+1}{q_n})$ which are of positive measure $\\sigma $ ) and apply the rule (REF ) to $A_{n+1}$ , $B_{n+1}$ with $p_n$ replaced with $p_{n+1}$ .", "Note that $\\int e^{2\\pi i q_nx}\\,d\\sigma (x)=1+O(p_n(1+p_n)+1\\cdot p_n)$ , so $e^{2\\pi i q_n\\cdot }\\rightarrow 1$ in measure $\\sigma $ .", "On the other hand $\\sigma (A_n)=p_n$ and the sets $A_n$ are almost independent.", "Since $\\sum _{n\\ge 1}p_n=\\infty $ , for $\\sigma $ -a.e.", "$x$ , we have $x\\in A_n$ for infinitely many $n$ (by the Borel-Cantelli lemma), so (REF ) holds." ], [ "Strong $u$ -MOMO property of systems whose visible measures yield systems in an ec-class", "While we have seen rather unclear relations between ${{F}}$ and ${{F}}_{\\rm ec}$ (cf.", "Proposition REF ), on the topological level we always have the following.", "Proposition 2.15 Let ${{F}}$ be a characteristic class.", "Then ${C}_{{{F}}}\\subset {C}_{{{F}}_{\\rm ec}}$ .", "This follows immediately from the fact that a homeomorphism $T$ (acting on a compact metric space $X$ ) belongs to ${C}_{{{F}}_{\\rm ec}}$ if and only if for each $\\kappa \\in M^e(X,T)$ , $(X,\\mathcal {B}(X),\\kappa ,T)\\in {{F}}$ .", "Note that in view of Proposition REF and Proposition REF , ${C}_{{\\rm RIG}_{(q_n)}}={C}_{({\\rm RIG}_{(q_n)})_{\\rm ec}}.$ The special role of ec-classes stands in the next proposition.", "Proposition 2.16 Let ${{F}}$ be a characteristic class.", "Then $u\\perp {C}_{{{F}}_{\\rm ec}}$ if and only if each element in ${C}_{{{F}}_{\\rm ec}}$ satisfies the strong $u$ -MOMO property.", "The below proof of Proposition REF is an adaptation of the proof of Corollary 9 in [3].", "It uses the following elementary result (see Lemma 18 in [3]).", "Lemma 2.17 Assume that $(c_n)\\subset and $ (mn)N$.", "Then if the sequence $ (cn)$ is contained in a closed convex cone which is not a half-plane then$$\\frac{1}{m_N}\\sum _{n\\le N}c_n \\rightarrow 0 \\iff \\frac{1}{m_N}\\sum _{n\\le N}|c_n|\\rightarrow 0 \\text{ as }N\\rightarrow \\infty .$$$ [Proof of Proposition REF ] Only one implication needs to be proved.", "Suppose that $u\\perp {C}_{{{F}}_{\\rm ec}}$ , and let $(Y,S)\\in {C}_{{{F}}_{\\rm ec}}$ .", "We fix $f\\in C(Y)$ , an increasing sequence $(b_k)$ in ${\\mathbb {N}}$ , with $b_1=1$ and $b_{k+1}-b_k\\rightarrow \\infty $ , and a sequence $(y_k)$ of points in $Y$ .", "We introduce the finite set $\\mathbb {A}:=\\lbrace 1,e^{2\\pi i/3},e^{4\\pi i /3}\\rbrace $ , and for each $k\\ge 1$ , we define $e_k\\in \\mathbb {A}$ such that the complex number $e_k\\Big (\\sum _{b_k\\le n<b_{k+1}}f(S^{n-b_k}y_k)u(n) \\Big )$ is in the closed convex cone $\\lbrace 0\\rbrace \\cup \\lbrace z\\in * : \\text{arg}(z)\\in [-\\pi /3,\\pi /3]\\rbrace $ .", "Then, by Lemma REF , the convergence that we need to prove, i.e.", "$\\frac{1}{b_K}\\sum _{k< K}\\Big |\\sum _{b_k \\le n <b_{k+1}}f(S^{n-b_k}y_k)u(n) \\Big |\\xrightarrow[K\\rightarrow \\infty ]{} 0$ is equivalent to the convergence $\\frac{1}{b_K}\\sum _{k< K}\\sum _{b_k\\le n<b_{k+1}}e_kf(S^{n-b_k}y_k)u(n)\\xrightarrow[K\\rightarrow \\infty ]{} 0.$ Consider the dynamical system $(X,T)$ , where $X:=(Y\\times \\mathbb {A})^{\\mathbb {Z}}$ , and $T$ is the left shift.", "Let $x\\in X$ be such that $x_n:=(S^{n-b_k}y_k,e_k),\\text{ whenever }b_k\\le n<b_{k+1},$ and $x_n=x_0$ is a fixed arbitrary point of $Y\\times \\mathbb {A}$ for $n\\le 0$ .", "By setting $F:=f\\otimes {\\rm Id}$ on $Y\\times \\mathbb {A}$ , it easily follows that (REF ) amounts to $\\frac{1}{b_K}\\sum _{k< K}\\sum _{b_k\\le n<b_{k+1}}F\\circ \\pi _0(T^nx)u(n)\\xrightarrow[K\\rightarrow \\infty ]{} 0.$ To prove the convergence above, we define the subspace $X_x$ as the closure of $\\lbrace T^nx : n\\in {\\mathbb {Z}}\\rbrace $ .", "By assumption on $u$ , we only have to check that the system $(X_x,T)$ is in ${C}_{{{F}}_{\\rm ec}}$ .", "So let $\\mu $ be a visible measure in $(X_x,T)$ , and we first consider the case where $x$ itself is generic for $\\mu $ , along a sequence $(N_\\ell )$ .", "Set $B:=\\bigl \\lbrace (v_j,a_j)_{j\\in {\\mathbb {Z}}}\\in X : (v_1,a_1) = (Sv_0,a_0) \\bigr \\rbrace $ Since $b_{k+1}-b_k\\rightarrow \\infty $ , we have $\\frac{1}{N_\\ell }\\sum _{n<N_\\ell }\\delta _{T^nx}(B)\\xrightarrow[\\ell \\rightarrow \\infty ]{} 1,$ and since the set $B$ is closed, by the Portmanteau theorem, it must be of full measure $\\mu $ in $(X,T)$ .", "Moreover, such a measure $\\mu $ must be $T$ -invariant, hence, $1=\\mu \\left(\\bigcap _{n\\in {\\mathbb {Z}}}T^nB\\right)=\\mu \\Big ( \\bigl \\lbrace (v_j,a_j)_{j\\in {\\mathbb {Z}}}\\in X : \\forall j,\\ (v_j,a_j) = (S^jv_0,a_0) \\bigr \\rbrace \\Big ).$ Denote by $\\mu ^{(0)}$ the restriction of $\\mu $ to the zero-coordinate (that is, with the above notation, the distribution of $(v_0,a_0)$ under $\\mu $ ).", "Since $\\mu $ is $T$ -invariant, it follows that $\\mu ^{(0)}$ is $(S\\times {\\rm Id}_{\\mathbb {A}})$ -invariant.", "Moreover, $Y\\times \\mathbb {A}$ consists of three copies of $Y$ , each of them is invariant under $S\\times {\\rm Id}_{\\mathbb {A}}$ .", "Thus, $\\mu ^{(0)}=\\alpha _0 \\mu ^{(0)}_1 + \\alpha _1 \\mu ^{(0)}_2 + \\alpha _2 \\mu ^{(0)}_3,$ where $\\alpha _0+\\alpha _1+\\alpha _2=1$ , $\\alpha _j\\ge 0$ and $\\mu ^{(0)}_j(Y\\times \\lbrace e^{2\\pi ij /3}\\rbrace )=1$ for $j=0,1,2$ .", "It follows that the ergodic components of $(Y\\times \\mathbb {A},\\mu ^{(0)},S\\times {\\rm Id}_{\\mathbb {A}})$ yield measure-theoretic systems isomorphic to ergodic measures on $(Y,S)$ , hence in ${{F}}$ since $(Y,S)\\in {C}_{{{F}}_{\\rm ec}}$ (this is the moment in our proof where we use the fact that we deal with a characteristic ec-class and not a general characteristic class ${{F}}$ ).", "Thus $(Y\\times \\mathbb {A},\\mu ^{(0)},S\\times {\\rm Id}_{\\mathbb {A}})\\in {{F}}_{\\rm ec}$ .", "Now, using (REF ), we see that $(X_x,\\mu ,T)$ is isomorphic to $(Y\\times \\mathbb {A},\\mu ^{(0)},S\\times {\\rm Id}_{\\mathbb {A}})$ , thus is also in ${{F}}_{\\rm ec}$ .", "Now, suppose that $\\mu \\in V_T(x^{\\prime })$ for some point $x^{\\prime }$ in the orbit closure of $x$ , say $x^{\\prime }=\\lim _{r\\rightarrow \\infty }T^{n_r}x$ .", "If $x^{\\prime }=T^nx$ for some $n\\in {\\mathbb {Z}}$ , then $\\mu \\in V_T(x)$ and we already know that $(X_x,\\mu ,T)\\in \\lbrace \\ast \\rbrace \\subset {{F}}_{\\rm ec}$ in this case.", "If $n_r\\rightarrow -\\infty $ , then $x^{\\prime }=(\\ldots ,x_0,x_0,x_0,\\ldots )$ is a fixed point, and $\\mu =\\delta _{x^{\\prime }}$ .", "In this case, $(X_x,\\mu ,T)\\in \\lbrace \\ast \\rbrace \\subset {{F}}_{\\rm ec}$ .", "If $n_r\\rightarrow +\\infty $ , and if we write $x^{\\prime }=(v_j,a_j)_{j\\in {\\mathbb {Z}}}=\\lim _{r\\rightarrow \\infty }T^{n_r}x$ , then as $b_{k+1}-b_k\\rightarrow \\infty $ , there exists at most one $j\\in {\\mathbb {Z}}$ such that $(v_{j+1},a_j)\\ne (Sv_j,a_j)$ .", "We can then use the same arguments as for $x$ to show that a measure $\\mu $ for which $x$ is quasi-generic satisfies $(X_x,\\mu ,T) \\in {{F}}_{\\rm ec}$ .", "We conclude that $(X_x,T)$ is in ${C}_{{{F}}_{\\rm ec}}$ .", "Remark 2.18 In general, when instead of ${{F}}_{\\rm ec}$ we consider ${{F}}$ , $u\\perp {C}_{{{F}}}$ implies the strong $u$ -MOMO property for each for $(Y,S)$ in which all invariant measures yield systems in ${{F}}$ (in particular, if $(Y,S)\\in {C}_{{{F}}}$ and each invariant measure is visible).", "Question 1 Is Proposition REF true for each characteristic class?", "Remark 2.19 A straightforward adaptation of the proof shows that the subsequence version of Proposition REF also holds: for each characteristic class ${{F}}$ and each increasing sequence of integers $(N_\\ell )$ , $u$ is $(N_\\ell )$ -orthogonal to ${C}_{{{F}}_{\\rm ec}}$ if and only if each element in ${C}_{{{F}}_{\\rm ec}}$ satisfies the strong $u$ -MOMO property along $(N_\\ell )$ .", "See Remarks REF and REF ." ], [ "Lifting lemma", "The purpose of this section is to prove Proposition REF , which is an alternative version of Conze-Downarowicz-Serafin lifting lemma from [9] and seems to be of independent interest.", "It may seem weaker than the original where the genericity was lifted to a single orbit, but the main advantage here is that we do not need assumptions on the nature of the second topological space: it does not have to be a full shift.", "The second advantage is that the result has its extension to the logarithmic case, see Appendix REF , while the lifting lemma of Conze-Downarowicz-Serafin and other results of that type so far have been proved for Cesàro averages.", "Proposition 3.1 Let $(Y,S)$ and $(X,T)$ be two topological systems and $u\\in Y$ generic along an increasing sequence $(N_m)$ for some $S$ -invariant measure $\\kappa $ on $Y$ .", "Let $\\rho $ be a joining of $\\kappa $ with a $T$ -invariant measure $\\nu $ on $X$ .", "Then there exist a sequence $(x_n)\\subset X$ and a subsequence $(N_{m_\\ell })$ such that: the sequence $(S^nu,x_n)$ is generic for $\\rho $ along $(N_{m_\\ell })$ : $ \\frac{1}{N_{m_\\ell }}\\sum _{0\\le n< N_{m_\\ell }}\\delta _{(S^nu,x_n)}\\xrightarrow[\\ell \\rightarrow \\infty ]{}\\rho , $ The sequence $(x_n)$ is constituted of longer and longer pieces of orbits.", "More precisely, $\\lbrace n\\ge 0:\\ x_{n+1}\\ne Tx_n\\rbrace $ is of the form $\\lbrace b_1<b_2<\\cdots <b_k<b_{k+1}<\\cdots \\rbrace $ , where $b_{k+1}-b_k\\rightarrow \\infty $ ." ], [ "Good sequences of partitions", "We need a convenient tool to estimate the weak*-convergence of a sequence of probability measures to a given measure.", "Definition 3.1 Let $(E,d)$ be a compact metric space, and let $\\nu $ be a Borel probability measure on $E$ , i.e.", "$\\nu \\in M(E)$ .", "We consider a sequence $({P}_\\ell )$ of finite partitions of $E$ into Borel subsets.", "The sequence $({P}_\\ell )$ is said to be good for $(E,\\nu )$ if the following conditions hold: for each $\\ell $ , ${P}_{\\ell +1}$ refines ${P}_\\ell $ , $\\mathop {\\mathrm {diam}}({P}_\\ell ):=\\max _{P\\text{ atom of }{P}_\\ell } \\mathop {\\mathrm {diam}}(P) \\xrightarrow[\\ell \\rightarrow \\infty ]{}0$ , for each $\\ell $ and each atom $P$ of ${P}_\\ell $ , $\\nu (\\partial P)=0$ .", "The motivation for introducing this definition comes from the following result.", "Lemma 3.2 If $({P}_\\ell )$ is a good sequence of partitions for $(E,\\nu )$ , then a sequence $(\\nu _n)\\subset M(E)$ converges to $\\nu $ in the weak*-topology if and only if, for each $\\ell $ and each atom $P$ of ${P}_\\ell $ , we have $\\nu _n(P) \\xrightarrow[n\\rightarrow \\infty ]{} \\nu (P).$ If $\\nu _n\\xrightarrow[n\\rightarrow \\infty ]{\\text{w}*} \\nu $ , then by the Portmanteau theorem, for each $P\\subset E$ such that $\\nu (\\partial P)=0$ , we have $\\nu _n(P)\\rightarrow \\nu (P)$ .", "Conversely, assume that for each $\\ell $ and each atom $P$ of ${P}_\\ell $ we have $\\nu _n(P)\\rightarrow \\nu (P)$ .", "Then any weak*-limit $\\mu $ of a subsequence of $(\\nu _n)$ satisfies (again by the Portmanteau theorem) $\\mu (P)=\\nu (P)$ for each atom $P$ of ${P}_\\ell $ .", "But since $\\mathop {\\mathrm {diam}}({P}_\\ell )\\rightarrow 0$ , the sequence $({P}_\\ell )$ separates points in $E$ , hence it generates the Borel $\\sigma $ -algebra of $E$ .", "Thus we have $\\mu =\\nu $ , and using the compactness of $M(E)$ for the weak* topology, we get that $\\nu _n\\xrightarrow[n\\rightarrow \\infty ]{\\text{w}*} \\nu $ .", "Lemma 3.3 For each $\\nu \\in M(E)$ of a compact metric space $(E,d)$ , there exists a good sequence of partitions for $(E,\\nu )$ .", "We first show that, for each $\\ell \\ge 1$ , there exists a finite partition ${Q}_\\ell $ in which each atom $Q$ satisfies $\\mathop {\\mathrm {diam}}(Q)<1/\\ell $ , $\\nu (\\partial Q)=0$ .", "Indeed, by compactness, there exists a finite set $\\lbrace x_1,\\ldots ,x_k\\rbrace \\subset E$ such that $ E\\subset \\bigcup _{1\\le i\\le k} B\\bigl (x_i,\\frac{1}{3\\ell }\\bigr ).", "$ Then, for each $1\\le i\\le k$ , there exist at most countably many $r>0$ such that $\\nu \\left(\\partial B(x_i,r)\\right) > 0.$ Therefore, we can find $r\\in \\left(\\frac{1}{3\\ell },\\frac{1}{2\\ell }\\right)$ such that $\\forall 1\\le i\\le k,\\quad \\nu \\left(\\partial B(x_i,r)\\right) = 0.$ Then the partition ${Q}_\\ell $ generated by the open balls $B(x_i,r)$ , $1\\le i\\le k$ , satisfies the required conditions.", "Once we have ${Q}_\\ell $ for each $\\ell \\ge 1$ , we set ${P}_\\ell :={Q}_1\\vee \\cdots \\vee {Q}_\\ell ,$ and we get a good sequence $({P}_\\ell )$ for $(E,\\nu )$ .", "Lemma 3.4 Let $({P}_\\ell )$ be a good sequence of partitions for $(E_1,\\nu _1)$ , and let $({Q}_\\ell )$ be a good sequence of partitions for $(E_2,\\nu _2)$ .", "Then for each coupling $\\rho $ of $\\nu _1$ and $\\nu _2$ , $({P}_\\ell \\times {Q}_\\ell )$ is a good sequence of partitions for $(E_1\\times E_2,\\rho )$ .", "This is obvious, since for each atom $P$ of ${P}_\\ell $ and each atom $Q$ of ${Q}_\\ell $ , $ \\partial (P\\times Q) \\subset (\\partial P\\times E_2) \\cup (E_1\\times \\partial Q), $ and the marginals of $\\rho $ are $\\nu _1$ and $\\nu _2$ ." ], [ "Proof of Proposition ", "Without loss of generality, we can (and we do) assume that the measure-theoretic dynamical system $(Y,\\kappa ,S)$ is aperiodic.", "Indeed, if this is not the case, we consider any uniquely ergodic topological system $(Y^{\\prime },S^{\\prime })$ whose unique invariant measure $\\kappa ^{\\prime }$ is such that $(Y^{\\prime },\\kappa ^{\\prime },S^{\\prime })$ is aperiodic.", "Then we take any point $u^{\\prime }\\in Y^{\\prime }$ , and we replace $Y$ by $Y\\times Y^{\\prime }$ , $S$ by $S\\times S^{\\prime }$ , and $u$ by $(u,u^{\\prime })$ .", "We also replace $(N_m)$ by a subsequence of $(N_m)$ along which $(u,u^{\\prime })$ is generic, for some measure $\\tilde{\\kappa }$ whose marginals have to be $\\kappa $ and $\\kappa ^{\\prime }$ .", "But then the system $(Y\\times Y^{\\prime },\\tilde{\\kappa },S\\times S^{\\prime })$ is aperiodic, because it is an extension of the aperiodic system $(Y^{\\prime },\\kappa ^{\\prime },S^{\\prime })$ .", "We fix a good sequence of partitions $({Q}_\\ell )$ for $(Y,\\kappa )$ and a good sequence of partitions $({P}_\\ell )$ for $(X,\\nu )$ .", "Then by Lemma REF , $({Q}_\\ell \\times {P}_\\ell )$ is a good sequence of partitions for $(Y\\times X,\\rho )$ .", "Definition 3.2 Let $M>0$ .", "A subset $E$ of ${\\mathbb {N}}$ is said to be $M$ -separated if for each integers $n\\ne m$ , $n,m\\in E\\Longrightarrow |n-m|\\ge M$ .", "The main argument to prove Proposition REF stands in the following proposition.", "Proposition 3.5 Under the assumptions of Proposition REF , and assuming also that $(Y,\\kappa ,S)$ is aperiodic (see above), given ${\\ell _0}\\ge 1$ and $\\varepsilon \\in (0,\\frac{1}{2})$ , there exists a sequence $(x_n)$ of points in $X$ such that: $\\lbrace n\\ge 0: x_{n+1}\\ne Tx_n\\rbrace $ is $\\frac{1}{\\varepsilon }$ -separated, for each atom $A$ of ${Q}_{\\ell _0}\\times {P}_{\\ell _0}$ , we have $\\rho (A)-\\varepsilon < \\liminf _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A }(S^nu,x_n),$ and $\\limsup _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A }(S^nu,x_n) < \\rho (A)+\\varepsilon .$ Let $h$ be a natural number such that $\\frac{1}{h}<\\varepsilon $ .", "We claim that for $\\ell $ large enough, we can find a set $B\\subset Y$ which is measurable with respect to $\\bigvee _{0\\le j\\le h-1}S^j{Q}_\\ell $ , and such that $B,SB,\\ldots ,S^{h-1}B$ are pairwise disjoint, $\\kappa \\left(\\bigcup _{0\\le j\\le h-1}S^jB\\right)>1-\\varepsilon $ .", "Indeed, since $(Y,\\kappa ,S)$ is assumed to be aperiodic, we can use the Rokhlin lemma to find a Borel subset $\\tilde{B}\\subset Y$ such that $\\tilde{B}, S\\tilde{B},\\ldots , S^{h-1}\\tilde{B}$ are pairwise disjoint, and such that $ \\kappa \\left(\\bigcup _{0\\le j\\le h-1}S^j\\tilde{B}\\right)>1-\\frac{\\varepsilon }{2}.", "$ Then we use the fact that the good sequence of partitions $({Q}_\\ell )$ generates the Borel $\\sigma $ -algebra: it follows that for $\\ell $ large enough, we can find a ${Q}_\\ell $ -measurable set $B^{\\prime }$ such that $ \\kappa (B^{\\prime }\\bigtriangleup \\tilde{B})< \\frac{\\varepsilon }{8h^2}.", "$ For each $1\\le j\\le h-1$ , we have $ B^{\\prime }\\cap S^jB^{\\prime } \\subset (B^{\\prime }\\setminus \\tilde{B})\\cup (S^jB^{\\prime }\\setminus S^j\\tilde{B}), $ hence $\\kappa (B^{\\prime }\\cap S^jB^{\\prime })\\le \\frac{\\varepsilon }{4h^2}.$ It remains to define $B$ by $ B:= B^{\\prime }\\setminus \\left(\\bigcup _{1\\le j\\le h-1} S^jB^{\\prime }\\right).", "$ Then, by construction, $B$ is disjoint from $S^jB$ for each $1\\le j\\le h-1$ , thus $B,SB,\\ldots ,S^{h-1}B$ are pairwise disjoint.", "Moreover, from (REF ), we have $\\kappa (B) \\ge \\kappa (B^{\\prime })-\\frac{\\varepsilon }{4h}\\ge \\kappa (\\tilde{B})-\\frac{\\varepsilon }{2h},$ which implies $ \\kappa \\left(\\bigcup _{0\\le j\\le h-1}S^j B\\right) = h\\kappa (B) \\ge h\\kappa (\\tilde{B})-\\frac{\\varepsilon }{2}>1-\\varepsilon , $ and our first claim is proved.", "Since $u$ is generic for $\\kappa $ along $(N_m)$ , and since the set $\\bigcup _{0\\le j\\le h-1}S^jB$ is measurable with respect to $\\bigvee _{0\\le j\\le 2h}S^j{Q}_\\ell $ (in particular, the $\\kappa $ -measure of its boundary vanishes), we have $\\frac{1}{N_m} \\sum _{0\\le n< N_m} {1}_{\\bigcup _{0\\le j\\le h-1}S^jB} (S^n u) \\xrightarrow[m\\rightarrow \\infty ]{} \\kappa \\left(\\bigcup _{0\\le j\\le h-1}S^jB\\right) > 1-\\varepsilon .$ This implies in particular that the set $P_B(u):=\\lbrace n\\ge 0:\\ S^nu\\in B\\rbrace $ is infinite.", "We number in order the elements of this set: $ P_B(u) = \\lbrace b_1<b_2<\\cdots <b_k<\\cdots \\rbrace \\; $ The integers $(b_k)$ will correspond to the times when we will be allowed to change the orbit of the desired sequence.", "As $B$ is disjoint from $S^jB$ for each $1\\le j\\le h-1$ , the set $P_B(u)$ is $h$ -separated, hence $\\frac{1}{\\varepsilon }$ -separated.", "We consider the partition $\\bigvee _{0\\le j\\le h-1}(S\\times T)^{-j}({Q}_{\\ell _0}\\times {P}_{\\ell _0})$ of $Y\\times X$ .", "Any atom of this partition is of the form ${\\bar{Q}}\\times {\\bar{P}}$ , where ${\\bar{Q}}$ (respectively ${\\bar{P}}$ ) is an atom of $\\bigvee _{0\\le j\\le h-1}S^{-j}{Q}_{\\ell _0}$ (respectively of $\\bigvee _{0\\le j\\le h-1}T^{-j}{P}_{\\ell _0}$ ).", "For such atoms ${\\bar{Q}}$ and ${\\bar{P}}$ , we can write ${\\bar{Q}}=Q_0\\cap S^{-1}Q_1\\cap \\cdots \\cap S^{-(h-1)}Q_{h-1},$ each $Q_j$ being an atom of ${Q}_{\\ell _0}$ , and ${\\bar{P}}=P_0\\cap S^{-1}P_1\\cap \\cdots \\cap S^{-(h-1)}P_{h-1},$ each $P_j$ being an atom of ${P}_{\\ell _0}$ .", "Since the $\\kappa $ -measure of the boundary of each involved set is always 0, we again have for each atom ${\\bar{Q}}$ of $\\bigvee _{0\\le j\\le h-1}S^{-j}{Q}_{\\ell _0}$ $\\frac{1}{N_m} \\sum _{b_k< N_m} {1}_{{\\bar{Q}}} (S^{b_k} u) =\\frac{1}{N_m} \\sum _{0\\le n< N_m} {1}_{B\\cap {\\bar{Q}}} (S^n u)\\xrightarrow[m\\rightarrow \\infty ]{} \\kappa (B\\cap {\\bar{Q}}).$ If $C$ is a measurable subset of $Y$ with $\\kappa (C)>0$ , we denote by $\\rho ^Y_C$ the marginal on $X$ of the conditional probability measure $\\rho (\\,\\cdot \\,|C\\times X)$ .", "Then, for each measurable $A\\subset X$ , we have $\\begin{split}\\rho (C\\times A) &=\\rho \\bigl ( (C\\times X) \\cap (Y\\times A) \\bigr ) \\\\&=\\rho (C\\times X) \\, \\rho \\bigl (Y\\times A|C\\times X\\bigr )\\\\&=\\kappa (C) \\, \\rho ^Y_C(A).\\end{split}$ On an appropriate probability space, we construct a sequence $(\\xi _k)$ of independent random variables, taking values in $X$ , such that for each $k$ , $\\xi _k$ is distributed according to $\\rho ^Y_{B\\cap {\\bar{Q}}}$ , where ${\\bar{Q}}$ is the atom of $\\bigvee _{0\\le j\\le h-1}S^{-j}{Q}_{\\ell _0}$ containing $S^{b_k}u$ .", "For each atom ${\\bar{Q}}$ of $\\bigvee _{0\\le j\\le h-1}S^{-j}{Q}_{\\ell _0}$ and each atom ${\\bar{P}}$ of $\\bigvee _{0\\le j\\le h-1}T^{-j}{P}_{\\ell _0}$ , by (REF ), the law of large numbers and (REF ), with probability 1, we have $\\frac{1}{N_m} \\sum _{b_k< N_m} {1}_{{\\bar{Q}}} (S^{b_k} u) {1}_{{\\bar{P}}} (\\xi _k) \\xrightarrow[m\\rightarrow \\infty ]{} \\kappa (B\\cap {\\bar{Q}}) \\rho ^Y_{B\\cap {\\bar{Q}}}({\\bar{P}}) = \\rho \\bigl ((B\\cap {\\bar{Q}})\\times {\\bar{P}}\\bigr ).$ Let us fix a realization of $(\\xi _k)$ which satisfies (REF ) for each atom ${\\bar{Q}}\\times {\\bar{P}}$ of $\\bigvee _{0\\le j\\le h-1}(S\\times T)^{-j}({Q}_{\\ell _0}\\times {P}_{\\ell _0})$ .", "Then, for each $n\\ge 0$ , we define the point $x_n\\in X$ as follows: $x_n := {\\left\\lbrace \\begin{array}{ll}T^{n-b_1}\\xi _1 &\\text{ if }n<b_1,\\\\T^{n-b_k}\\xi _k &\\text{ if }b_k\\le n<b_{k+1}\\text{ for some }k\\ge 1.\\end{array}\\right.", "}$ The set of integers $n$ such that $x_{n+1}\\ne Tx_n$ is contained in $P_B(u)$ , therefore, it is $\\frac{1}{\\varepsilon }$ -separated.", "Now, let $A=Q\\times P$ be a fixed atom of ${Q}_{\\ell _0}\\times {P}_{\\ell _0}$ .", "We set $R:=\\bigcup _{0\\le j\\le h-1}S^jB\\times X,$ and we observe that $\\rho (R) = \\kappa \\left(\\bigcup _{0\\le j\\le h-1}S^jB\\right) >1-\\varepsilon .$ We also note that for each $n\\ge b_1$ , $(S^nu,x_n)\\in R$ if and only if there exists $k$ and $0\\le j\\le h-1$ such that $n=b_k+j$ .", "In this case, $(S^nu,x_n)\\in A\\cap R$ if and only if the atom ${\\bar{Q}}\\times {\\bar{P}}$ of $\\bigvee _{0\\le j\\le h-1}(S\\times T)^{-j}({Q}_{\\ell _0}\\times {P}_{\\ell _0})$ containing $(S^{b_k}u,\\xi _k)$ satisfies $Q_j=Q$ and $P_j=P$ (using the notations given in (REF ) and (REF ), and remembering that $A=Q\\times P$ ).", "We can then use (REF ) to get $\\begin{split}&\\frac{1}{N_m} \\sum _{b_1\\le n<N_m} {1}_{A\\cap R}(S^nu,x_n) \\\\&= \\frac{1}{N_m} \\sum _{b_k<N_m} \\sum _{0\\le j\\le h-1} \\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} {1}_{{\\bar{Q}}\\times {\\bar{P}}}(S^{b_k}u,\\xi _k)\\\\&\\xrightarrow[m\\rightarrow \\infty ]{} \\sum _{0\\le j\\le h-1} \\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} \\rho \\bigl ((B\\cap {\\bar{Q}})\\times {\\bar{P}}\\bigr ).\\end{split}$ But, on the other hand, we can write $\\begin{split}\\rho (A\\cap R) &= \\sum _{0\\le j\\le h-1} \\rho \\bigl (A\\cap (S^jB\\times X)\\bigr )\\\\&= \\sum _{0\\le j\\le h-1} \\rho \\bigl ((S^{-j}Q\\times T^{-j}P)\\cap (B\\times X)\\bigr )\\\\&= \\sum _{0\\le j\\le h-1} \\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} \\rho \\bigl ((B\\cap {\\bar{Q}})\\times {\\bar{P}}\\bigr ).\\end{split}$ From (REF ) and (REF ), it follows that $\\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A\\cap R}(S^nu,x_n) \\xrightarrow[m\\rightarrow \\infty ]{} \\rho (A\\cap R).$ From (REF ), we get that $\\lim _{m\\rightarrow \\infty }\\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{(Y\\times X)\\setminus R}(S^nu,x_n)<\\varepsilon ,$ and since ${1}_{A}\\le {1}_{A\\cap R}+{1}_{Y\\times X\\setminus R}$ , this yields by (REF ), $\\limsup _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A }(S^nu,x_n)&< \\lim _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A\\cap R}(S^nu,x_n) + \\varepsilon \\\\&=\\rho (A\\cap R)+ \\varepsilon \\\\&\\le \\rho (A)+\\varepsilon ,$ and we have (REF ).", "On the other hand, using ${1}_{A}\\ge {1}_{A\\cap R}$ , we get by (REF ) $\\liminf _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A }(S^nu,x_n)&\\ge \\lim _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A\\cap R}(S^nu,x_n) \\\\&=\\rho (A\\cap R) \\\\&>\\rho (A)-\\varepsilon .$ and we have (REF ).", "We can now give the proof of Proposition REF , in which we use the following obvious fact: if we modify the sequence $(x_n)$ given by Proposition REF on a finite number of terms, we still get (REF ) and (REF ).", "[Proof of Proposition (REF )] We fix a sequence $(\\varepsilon _\\ell )_{\\ell \\ge 1}$ of numbers in $(0,\\frac{1}{2})$ , decreasing to 0, and we construct inductively the desired sequence $(x_n)$ and the subsequence $(N_{m_\\ell })$ by a repeated use of Proposition REF .", "We start by applying Proposition REF with $\\varepsilon :=\\varepsilon _1$ and ${\\ell _0}:=1$ .", "It provides us with an integer $m_1$ , and a finite sequence $(x_n)_{0\\le n<N_{m_1}}$ of points in $X$ such that the set of integers $n\\in \\lbrace 0,\\ldots ,N_{m_1}-2\\rbrace $ such that $x_{n+1}\\ne Tx_n$ is $\\frac{1}{\\varepsilon _1}$ -separated, for each atom $A$ of ${Q}_1\\times {P}_1$ , we have $\\rho (A)-\\varepsilon _1<\\frac{1}{N_{m_1}} \\sum _{0\\le n<N_{m_1}} {1}_{A}(S^nu,x_n) < \\rho (A)+\\varepsilon _1.$ Now, assume that for some $\\ell \\ge 1$ we have already constructed $m_1<\\cdots <m_\\ell $ and the sequence $(x_n)_{0\\le n<N_{m_\\ell }}$ of points in $X$ such that for each $1\\le j<\\ell $ , the set of integers $n\\in \\lbrace N_{m_{j-1}},\\ldots ,N_{m_j}-2\\rbrace $ such that $x_{n+1}\\ne Tx_n$ is $\\frac{1}{\\varepsilon _j}$ -separated (with the convention that $N_{m_0}=0$ ), for each atom $A$ of ${Q}_\\ell \\times {P}_\\ell $ , we have $\\rho (A)-\\varepsilon _\\ell <\\frac{1}{N_{m_\\ell }} \\sum _{0\\le n<N_{m_\\ell }} {1}_{A}(S^nu,x_n) < \\rho (A)+\\varepsilon _\\ell .$ Then we apply again Proposition REF , with $\\varepsilon :=\\varepsilon _{\\ell +1}$ and ${\\ell _0}:=\\ell +1$ .", "It provides us with an integer $m_{\\ell +1}$ and a finite sequence of points $(x_n)_{N_{m_\\ell }\\le n<N_{m_{\\ell +1}}}$ in $X$ which satisfy: the set of integers $n\\in \\lbrace N_{m_\\ell },\\ldots ,N_{m_{\\ell +1}}-2\\rbrace $ such that $x_{n+1}\\ne Tx_n$ is $\\frac{1}{\\varepsilon _{\\ell +1}}$ -separated, for each atom $A$ of ${Q}_{\\ell +1}\\times {P}_{\\ell +1}$ , we have $\\rho (A)-\\varepsilon _{\\ell +1}<\\frac{1}{N_{m_{\\ell +1}}} \\sum _{0\\le n<N_{m_{\\ell +1}}} {1}_{A}(S^nu,x_n) < \\rho (A)+\\varepsilon _{\\ell +1}.$ (We keep the points $(x_n)_{0\\le n<N_{m_\\ell }}$ already provided by the induction hypothesis, refering to the obvious fact stated before the proof.)", "Moreover, we can assume that the sequence $(\\varepsilon _\\ell )$ decreases sufficiently fast so that the validity of (REF ) for each atom $A$ of ${Q}_{\\ell +1}\\times {P}_{\\ell +1}$ ensures the validity of the analog inequalities for each $A$ which is a finite union of atoms of ${Q}_{\\ell +1}\\times {P}_{\\ell +1}$ (in particular, for each $A$ which is an atom of the previous partitions), but with $\\varepsilon _\\ell $ instead of $\\varepsilon _{\\ell +1}$ .", "The sequence $(x_n)_{n\\ge 0}$ of points in $X$ and the subsequence $(N_{m_\\ell })$ we construct with the above inductive procedure then satisfy the conditions announced in Proposition REF ." ], [ "Logarithmic case", "We would like to study the logarithmic version of Proposition REF , in which we replace each arithmetic average of the form $\\frac{1}{N_m}\\sum _{0\\le n< N_m} f(n)$ by the logarithmic average $\\frac{1}{L(N_m)}\\sum _{1\\le n\\le N_m} \\frac{1}{n}f(n).$ (Here we use the notation $L(N):=1+\\frac{1}{2}+\\cdots +\\frac{1}{N}$ .)", "In fact, this logarithmic version, whose statement is written below, is also valid, and the arguments to prove it are exactly the same as in the arithmetic average case.", "We just point out below the few technical changes that need to be made in the proof for the logarithmic case.", "Proposition 3.6 Let $(Y,S)$ and $(X,T)$ be two topological systems and $u\\in Y$ , logarithmically generic along an increasing sequence $(N_m)$ for some $S$ -invariant measure $\\kappa $ on $Y$ .", "Let $\\rho $ be a joining of $\\kappa $ with a $T$ -invariant measure $\\nu $ on $X$ .", "Then there exist a sequence $(x_n)\\subset X$ and a subsequence $(N_{m_\\ell })$ such that: the sequence $(S^nu,x_n)$ is logarithmically generic for $\\rho $ along $(N_{m_\\ell })$ : $ \\frac{1}{L(N_{m_\\ell })}\\sum _{1\\le n\\le N_{m_\\ell }}\\frac{1}{n}\\delta _{(S^nu,x_n)}\\xrightarrow[\\ell \\rightarrow \\infty ]{}\\rho , $ the set $\\lbrace n\\ge 0:\\ x_{n+1}\\ne Tx_n\\rbrace $ is of the form $\\lbrace b_1<b_2<\\cdots <b_k<b_{k+1}<\\cdots \\rbrace $ , where $b_{k+1}-b_k\\rightarrow \\infty $ .", "The changes that need to be made to the proof are almost all quite obvious, they consist in formally replacing the arithmetic average by the logarithmic average.", "One point maybe needs some explanations, namely when we arrive at the proof of the logarithmic analog of (REF ).", "We put these explanations in the form of a lemma, which we will apply in the following context: $(d_k)$ is the ordered sequence of positive integers $n$ such that $S^nu\\in B\\cap {\\bar{Q}}$ , and the sequence $(\\rho _k)$ is defined by $\\rho _k:={1}_{{\\bar{P}}}(\\xi _k)$ .", "Lemma 3.7 Let $(d_k)$ be an increasing sequence of positive integers such that $\\frac{1}{L(N_{m})}\\sum _{d_k\\le N_{m}}\\frac{1}{d_k} \\xrightarrow[m\\rightarrow \\infty ]{} \\kappa \\in [0,1],$ and let $(\\rho _k)$ be a sequence of real numbers in $[0,1]$ such that $\\frac{1}{K} \\sum _{1\\le k\\le K} \\rho _k \\xrightarrow[K\\rightarrow \\infty ]{} \\rho \\in [0,1].$ Then we have $\\frac{1}{L(N_{m})}\\sum _{d_k\\le N_{m}}\\frac{\\rho _k}{d_k} \\xrightarrow[m\\rightarrow \\infty ]{} \\kappa \\rho .$ For each $m$ , let us denote by $k_m$ the largest $k$ such that $d_k\\le N_m$ .", "We use the classical identity $\\sum _{d_k\\le N_{m}} \\frac{\\rho _k}{d_k} = \\sum _{1\\le k < k_m} \\left( \\frac{1}{d_k}-\\frac{1}{d_{k+1}} \\right) (\\rho _1+\\cdots +\\rho _k) + \\frac{1}{d_{k_m}}(\\rho _1+\\cdots +\\rho _{k_m}).$ Given $\\varepsilon >0$ , let $K_\\varepsilon $ be such that $ K\\ge K_\\varepsilon \\Longrightarrow \\left| \\frac{1}{K} \\sum _{1\\le k\\le K} \\rho _k - \\rho \\right| < \\varepsilon .", "$ We can then write $&\\left| \\frac{1}{L(N_{m})} \\sum _{d_k\\le N_{m}} \\frac{\\rho _k}{d_k} - \\frac{1}{L(N_{m})} \\sum _{d_k\\le N_{m}} \\frac{\\rho }{d_k}\\right|\\\\&= \\left| \\frac{1}{L(N_{m})} \\sum _{K_\\varepsilon \\le k < k_m} k \\left( \\frac{1}{d_k}-\\frac{1}{d_{k+1}} \\right) \\left( \\frac{1}{k} (\\rho _1+\\cdots +\\rho _k)-\\rho \\right) \\right| + O\\left(\\frac{1}{L(N_{m})}\\right)\\\\&< \\varepsilon + O\\left(\\frac{1}{L(N_{m})}\\right).$ But by assumption, we have $\\frac{1}{L(N_{m})} \\sum _{d_k\\le N_{m}} \\frac{\\rho }{d_k}\\xrightarrow[m\\rightarrow \\infty ]{} \\kappa \\rho ,$ hence we get $\\frac{1}{L(N_{m})} \\sum _{d_k\\le N_{m}} \\frac{\\rho _k}{d_k}\\xrightarrow[m\\rightarrow \\infty ]{} \\kappa \\rho .$ The last place in the proof where a (very easy) correction should be made in the logarithmic case is to get the analog of (REF ): at some point we have to replace some coefficients $\\frac{1}{b_k+j}$ by $\\frac{1}{b_k}$ , which is of no consequence since $j$ remains bounded between 0 and $h-1$ here.", "To be more precise, (REF ) becomes $\\begin{split}&\\frac{1}{L(N_m)} \\sum _{b_1\\le n<N_m} \\frac{1}{n}{1}_{A\\cap R}(S^nu,x_n) \\\\&= \\frac{1}{L(N_m)} \\sum _{b_k<N_m} \\sum _{0\\le j\\le h-1} \\frac{1}{b_k+j}\\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} {1}_{{\\bar{Q}}\\times {\\bar{P}}}(S^{b_k}u,\\xi _k)\\\\&= \\frac{1}{L(N_m)} \\sum _{b_k<N_m} \\sum _{0\\le j\\le h-1} \\frac{1}{b_k}\\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} {1}_{{\\bar{Q}}\\times {\\bar{P}}}(S^{b_k}u,\\xi _k) + o(1)\\\\&\\xrightarrow[m\\rightarrow \\infty ]{} \\sum _{0\\le j\\le h-1} \\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} \\rho \\bigl ((B\\cap {\\bar{Q}})\\times {\\bar{P}}\\bigr ).\\end{split}$" ], [ "Proof of Theorem ", "[Proof of Theorem REF ] Take any topological system $(X,T)\\in {C}_{{{F}}}$ and fix $f\\in C(X)$ , $x\\in X$ .", "Take any increasing sequence $(N_k)$ for which, with no loss of generality, we can assume that $\\frac{1}{N_k}\\sum _{n\\le N_k} \\delta _{(T^nx,S^nu)}\\rightarrow \\rho $ .", "It follows that $\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}f(T^nx)u(n)=\\int f\\otimes \\pi _0\\,d\\rho .$ But $\\rho $ is a joining of some $T$ -invariant measure $\\nu \\in V(X,T)$ for which $x$ is generic along $(N_k)$ , and some Furstenberg system $\\kappa $ of $u$ .", "Since $(X,T)\\in {C}_{{{F}}}$ , the system $(X,\\nu ,T)$ is in ${{F}}$ , and the integral on the right-hand side above vanishes by the Veech condition and Proposition REF ." ], [ "Proof of Theorem ", "Before we begin the proof, let us make the following remark concerning topological models.", "Given an automorphism $(Z,\\mathcal {D},\\kappa , R)$ , and a fixed subset of full measure of ergodic components of $\\kappa $ , recall that by a Hansel model of $R$ , we mean a topological system $(X,T)$ which has a $T$ -invariant measure $\\nu $ such that, as dynamical systems, $\\kappa $ and $\\nu $ are isomorphic and such that each point $x\\in X$ is generic for one of these chosen ergodic components.", "In [29], it is proved that each automorphism has a Hansel model.", "We assume that $u\\perp {C}_{{{F}}_{\\rm ec}}$ for some characteristic class ${{F}}$ .", "Take $\\kappa \\in V(u)$ and fix $(N_m)$ so that $\\frac{1}{N_m}\\sum _{n\\le N_m} \\delta _{S^nu}\\rightarrow \\kappa .$ Denote by ${\\cal A}(\\kappa )\\subset {\\cal B}(X_{u})$ the largest ${{F}}_{\\rm ec}$ -factor of $(X_{u},\\kappa ,S)$ , i.e.", "${\\cal A}(\\kappa )={\\cal B}(X_{u})_{{{F}}_{\\rm ec}}$ .", "Consider the factor $(X_{u}/{\\cal A}(\\kappa ), {\\cal A}(\\kappa ), \\kappa |_{{\\cal A}(\\kappa )},S)$ and take a Hansel model $(X,\\nu ,T)$ of it (by choosing in the ergodic decomposition of $\\kappa |_{{\\cal A}(\\kappa )}$ only ergodic measures in ${{F}}$ ).", "By definition, $(X,T)\\in {C}_{{{F}}_{\\rm ec}}.$ Fix a measure-theoretic factor map $J\\colon (X_{u},\\kappa ,S)\\rightarrow (X,\\nu ,T)$ such that $J^{-1}({\\cal B}(X))={\\cal A}(\\kappa )$ , and let $\\nu _J$ denote the corresponding graph joining (of $\\nu $ and $\\kappa |_{{\\cal A}(\\kappa )}$ ).", "Let $\\widehat{\\nu }_J$ be the relatively independent extension of $\\nu _J$ to a joining of $\\nu $ and $\\kappa $ : for $f\\in L^2(\\nu )$ and $g\\in L^2(\\kappa )$ , we have $\\int _{X_{u}\\times X} g\\otimes f \\, d\\widehat{\\nu }_J = \\int _{X_{u}} {\\mathbb {E}}^{\\kappa }\\bigl (g | {\\cal A}(\\kappa )\\bigr )\\,f\\circ J \\, d\\kappa .$ Now, by applying Proposition REF , we can find $(x_n)\\subset Y$ such that $((x_n),u) \\text{ is generic for }\\widehat{\\nu }_J\\text{ along some subsequence }(N_{m_\\ell }),$ and the set $\\lbrace n\\ge 0:\\ x_{n+1}\\ne Tx_n\\rbrace $ is of the form $\\lbrace b_1<b_2<\\cdots \\rbrace $ with $b_{k+1}-b_k\\rightarrow \\infty $ .", "Since $u\\perp {C}_{{{F}}_{\\rm ec}}$ , (REF ) and Proposition REF ensure that the system $(Y,S)$ satisfies the strong $u$ -MOMO property.", "Therefore, for each $f\\in C(Y)$ we have $\\lim _{m\\rightarrow \\infty }\\frac{1}{N_m}\\sum _{n\\le N_m}f(x_n)u(n)=$ $\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{b_{K_m}}\\sum _{k<K_m}\\left(\\sum _{b_k\\le n<b_{k+1}}f(T^{n-b_k}x_{b_k})u(n)\\right)=0,$ and it follows from (REF ) that $\\int f\\otimes \\pi _0\\, d \\widehat{\\nu }_J=0$ .", "Using (REF ), we get $\\int _{X_{u}} {\\mathbb {E}}^{\\kappa }\\bigl (\\pi _0 | {\\cal A}(\\kappa )\\bigr )\\,f\\circ J \\, d\\kappa =0.$ But $\\lbrace f\\circ J:\\ f\\in C(X)\\rbrace $ is dense in $L^2({\\cal A}(\\kappa ))$ and therefore $\\pi _0\\perp L^2({\\cal A}(\\kappa ))$ , which is the Veech condition for $u$ with respect to the characteristic class ${{F}}_{\\rm ec}$ ." ], [ "Cancellations. Proof of Corollaries ", "We need the following interpretation of the Veech condition in terms of relative uniform mixing (K-mixing) of the function $\\pi _0$ .", "For $n\\in {\\mathbb {N}}$ , let $\\pi _n:=\\pi _0\\circ S^n$ .", "Proposition 5.1 Let $\\kappa \\in V_S(u)$ and $\\int \\pi _0\\,d\\kappa =0$ .", "Then the following conditions are equivalent: $\\pi _0\\perp L^2(\\Pi (\\kappa ))$ , $\\pi _0$ is relatively K-mixing, i.e.", "for each $\\varepsilon >0$ , there exists $N$ such that $\\left|\\int \\pi _0 {1}_C\\,d\\kappa -\\int \\pi _0\\, d\\kappa \\int {1}_C d\\kappa \\right|=\\left| \\int \\pi _0 {1}_C\\,d\\kappa \\right|<\\varepsilon $ for each set $C\\in \\sigma (\\pi _n,\\pi _{n+1},...)$ and $n\\ge N$ .", "If we additionally assume that $u$ takes values in a finite set $E\\subset and $ (Mk)$ is a sequence along which we have a Furstenberg system $$ then the above conditions are equivalent to\\begin{enumerate}\\item [(c)]for each \\varepsilon >0 there exists N\\ge 1 such that for any s\\ge 1 and any function fdepending on coordinates N\\le n,n+1,\\ldots ,n+s, \\Vert f\\Vert _{C(X_{u})}\\le 1, we have\\limsup _{k\\rightarrow \\infty }\\left|\\frac{1}{M_k}\\sum _{m\\le M_k}u(m)f(S^mu)\\right|<\\varepsilon .\\end{enumerate}$ (a) $\\Rightarrow $ (b).", "Assume that ${\\mathbb {E}}(\\pi _0|\\Pi (\\kappa ))=0$ .", "Let $C\\in \\sigma (\\pi _n,\\pi _{n+1},\\ldots )$ .", "We have $\\left|\\int \\pi _0 {1}_C d\\kappa \\right|=\\left|\\int {1}_C {\\mathbb {E}}(\\pi _0|\\sigma (\\pi _n,\\pi _{n+1},...)\\, d\\kappa \\right|\\\\\\le \\int \\left|{\\mathbb {E}}(\\pi _0|\\sigma (\\pi _n,\\pi _{n+1},\\ldots ))\\right|\\, d\\kappa .$ Hence, we have an upper bound which does not depend on $C$ .", "Since ${\\mathbb {E}}(\\pi _0|\\sigma (\\pi _n,\\pi _{n+1},...))\\rightarrow {\\mathbb {E}}(\\pi _0|\\Pi (\\kappa ))=0$ $\\kappa $ -a.e.", "and thus also in $L^1$ , which is precisely the relative K-mixing for $\\pi _0$ .", "(b) $\\Rightarrow $ (a).", "Suppose that $\\pi _0$ is relatively K-mixing.", "Then, in particular, we have (REF ) for each $C\\in \\Pi (\\kappa )$ .", "In fact, since $\\varepsilon >0$ is arbitrary, $\\int \\pi _0 {1}_C\\,d\\kappa =0$ for each $C\\in \\Pi (\\kappa )$ .", "Whence ${\\mathbb {E}}(\\pi _0|\\Pi (\\kappa ))=0$ .", "(a) $\\Rightarrow $ (c).", "Since $\\frac{1}{M_k}\\sum _{m\\le M_k}u(m)f(S^mu)=\\frac{1}{M_k}\\sum _{m\\le M_k}(\\pi _0f)(S^mu)\\rightarrow \\int _{X_{u}} \\pi _0 f\\, d\\kappa ,$ we can repeat the same argument as was used to prove (a) $\\Rightarrow $ (b) (replacing ${1}_C$ by $f$ ).", "(c) $\\Rightarrow $ (b).", "Suppose that $\\left|\\int _{X_{u}}\\pi _0 f\\,d\\kappa \\right|<\\varepsilon $ , whenever $f$ depending on coordinates $n,n+1,\\ldots ,n+s$ with $n\\ge N$ is bounded by 1.", "Consider all blocks on coordinates $n,n+1,\\ldots ,n+s$ that is all $B=\\lbrace x\\in X_{u}:\\: x_n=b_0,\\ldots ,x_{n+s}=b_s\\rbrace $ with $b_j\\in E$ .", "Let $C$ be any union of such blocks.", "Then ${1}_C$ is a (continuous) function depending on coordinates $n,\\ldots ,n+s$ and is bounded by 1 and, by assumption, $\\left|\\int _{X_{u}}\\pi _0 {1}_C\\,d\\kappa \\right|<\\varepsilon .$ Note that with $N$ fixed and $s$ arbitrary, the family of $C$ defined above is dense in the $\\sigma $ -algebra $\\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ .", "Hence, given $D\\in \\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ and $\\varepsilon >0$ , we first find $s\\ge 0$ and then $C$ as above (a union of blocks “sitting” on coordinates $N,\\ldots ,N+s$ ) such that $\\kappa (C\\triangle D)<\\varepsilon $ and find that $\\left|\\int \\pi _0 {1}_D\\,d\\kappa \\right|\\le \\left|\\int \\pi _0 {1}_C\\,d\\kappa \\right|+\\kappa (C\\triangle D)<2\\varepsilon .$ Now, since each clopen set is a finite union of cylinders of a fixed length, Corollary REF follows directly by the above proposition.", "Corollary REF is a special case of Corollary REF ." ], [ "Conditional cancellations. Remark ", "The “cancellation law” of the values of $u$ along large shifts of the return times to a block (for most of the blocks) claimed in Remark REF is a consequence of a refinement of Proposition REF .", "Proposition 5.2 Let $\\kappa \\in V_S(u)$ and $\\int \\pi _0\\,d\\kappa =0$ .", "Then the following conditions are equivalent: $\\pi _0\\perp L^2(\\Pi (\\kappa ))$ , for each $\\varepsilon >0$ there exist $N\\ge 1$ and $L\\ge 1$ such that for each $\\ell \\ge L$ there is a family of “good” $\\ell $ -blocks $C\\in \\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ , i.e.", "of blocks satisfying $\\left|\\int {1}_C\\cdot \\pi _0\\,d\\kappa \\right|\\le \\kappa (C)\\varepsilon ,$ whose measure is $>1-\\varepsilon $ .", "In other words, for a “good” $\\ell $ -block $C$ , $\\left|\\int \\pi _0\\,d\\kappa _C\\right|<\\varepsilon $ , where $\\kappa _C$ stands for the conditional measure on $C$ .", "(a) $\\Rightarrow $ (d).", "Fix $\\varepsilon >0$ and note that ${\\mathbb {E}}^\\kappa (\\pi _0|\\sigma (\\pi _N,\\pi _{N+1},\\ldots ))\\rightarrow 0$ $\\kappa $ -a.e.", "This implies convergence in measure, i.e., we can find a set $A_\\varepsilon $ of measure at least $1-\\varepsilon $ such that for $N$ large enough, $|{\\mathbb {E}}^\\kappa (\\pi _0|\\sigma (\\pi _N,\\pi _{N+1},\\ldots ))(x)|<\\varepsilon \\text{ for all }x\\in A_{\\varepsilon }.$ Fix such an $N$ .", "There is $M\\ge 1$ large enough such that $\\kappa (A_{\\varepsilon }\\triangle A^{(M)}_{\\varepsilon })<\\varepsilon ,$ where $A^{(M)}_\\varepsilon \\in \\sigma (\\pi _{-M},\\pi _{-M+1},\\ldots )$ and note that $S^{N+M}A^{(M)}_\\varepsilon \\in \\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ .", "Now, for $\\ell $ large enough, we can approximate $S^{N+M}A^{(M)}_{\\varepsilon }$ by a (disjoint) union of $\\ell $ -blocks belonging to $\\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ , $\\kappa (S^{N+M}A^{(M)}_{\\varepsilon }\\triangle \\bigcup _{j\\in J} C^{(\\ell )}_j)<\\varepsilon .$ But $\\kappa (S^{N+M}A^{(M)}_{\\varepsilon }\\triangle A_{\\varepsilon })<2\\varepsilon $ , so $\\kappa (\\bigcup _{j\\in J} C^{(\\ell )}_j \\setminus A_{\\varepsilon })\\le \\kappa (A_{\\varepsilon }\\triangle \\bigcup _{j\\in J} C^{(\\ell )}_j)<3\\varepsilon .$ Consider those $j\\in J$ for which $\\kappa (C^{(\\ell )}_j\\setminus A_{\\varepsilon })\\ge \\sqrt{\\varepsilon }\\kappa (C_j^{(\\ell )})$ .", "Then the measure $m$ of the union of such blocks has to satisfy $\\sqrt{\\varepsilon } m<3\\varepsilon $ , so $m<3\\sqrt{\\varepsilon }$ .", "In other words “most” (in measure) of the $C_j^{(\\ell )}$ 's are “good”, i.e.", "they satisfy $\\kappa (C^{(\\ell )}_j\\cap A_{\\varepsilon })>(1-3\\sqrt{\\varepsilon })\\kappa (C^{(\\ell )}_j)$ .", "Take such a “good” $C$ .", "We have $\\left|\\int {1}_C\\cdot \\pi _0\\,d\\kappa \\right|\\le \\int {1}_{C}|{\\mathbb {E}}^\\kappa (\\pi _0|\\sigma (\\pi _N,\\pi _{N+1},\\ldots ))|\\,d\\kappa \\\\=\\int _{A_{\\varepsilon }}{1}_{C}|{\\mathbb {E}}^\\kappa (\\pi _0|\\sigma (\\pi _N,\\pi _{N+1},\\ldots ))|\\,d\\kappa +\\int _{A_{\\varepsilon }^c}{1}_{C}|{\\mathbb {E}}^\\kappa (\\pi _0|\\sigma (\\pi _N,\\pi _{N+1},\\ldots ))|\\,d\\kappa \\\\\\le \\varepsilon \\kappa (C)+3\\sqrt{\\varepsilon }\\kappa (C).$ (d) $\\Rightarrow $ (a).", "Fix $A\\in \\Pi (\\kappa )$ of positive measure $\\kappa $ .", "Then for $\\varepsilon >0$ , we can find $N$ such that for all $\\ell $ large enough “most” of the $\\ell $ -blocks in $\\sigma (\\pi _N,\\pi _{N+1}<\\ldots )$ is “good” in the sense that $\\left|\\int {1}_C\\cdot \\pi _0\\,d\\kappa \\right|\\le \\kappa (C)\\varepsilon .$ Since $A\\in \\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ , we can approximate it by unions of $\\ell $ -blocks (for $\\ell $ sufficiently large) and most of the used blocks is “good”.", "Whence $\\left|\\int {1}_A\\cdot \\pi _0\\,d\\kappa \\right|\\le 2\\varepsilon ,$ and since $\\varepsilon >0$ was arbitrary, $\\pi _0\\perp L^2(\\Pi (\\kappa ))$ ." ], [ "Orthogonality to ${C}_{\\rm {ID}}$ . Proof of Corollary ", "We recall that (Proposition REF ) ${C}_{\\rm ID}={C}_{{\\rm ID}_{\\rm ec}}.$ Since the characteristic factor is represented by the $\\sigma $ -algebra of invariant sets, by Theorems REF and REF , we obtain immediately that: Corollary 5.3 $u\\perp {C}_{\\rm ID}$ if and only if for each Furstenberg system $\\kappa $ of $u$ , $\\pi _0\\perp L^2(\\mathcal {I}_\\kappa )$ .", "Let us now pass to a combinatorial characterization of the Veech condition.", "Assume that $\\kappa $ is given as the limit of $\\frac{1}{N_k}\\sum _{n\\le N_k}\\delta _{S^nu}$ .", "In view of Corollary REF , we need to decipher ${\\mathbb {E}}(\\pi _0|\\mathcal {I}_\\kappa )=0$ .", "By the von Neumann theorem, we have $\\frac{1}{H}\\sum _{h\\le H}\\pi _0\\circ S^h\\rightarrow {\\mathbb {E}}(\\pi _0|\\mathcal {I}_\\kappa ) \\text{ in }L^2,$ i.e.", "$\\lim _{H\\rightarrow \\infty }\\int _{X_{u}}\\left|\\frac{1}{H}\\sum _{h\\le H}\\pi _0\\circ S^h\\right|^2\\,d\\kappa =0$ as ${\\mathbb {E}}(\\pi _0|\\mathcal {I}_\\kappa )=0$ .", "So, given $\\varepsilon >0$ , $\\int _{X_{u}}\\left| \\frac{1}{H}\\sum _{h\\le H}\\pi _0\\circ S^h\\right|^2\\,d\\kappa <\\varepsilon \\text{ for all }H\\ge H_{\\varepsilon }.$ The latter is equivalent to $\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}\\left|\\frac{1}{H}\\sum _{h\\le H}\\pi _0\\circ S^h\\right|^2(S^nu)<\\varepsilon ,$ that is, $\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}\\left|\\frac{1}{H}\\sum _{h\\le H}u(n+h)\\right|^2<\\varepsilon .$ The proof of Corollary REF follows immediately.$\\quad \\hbox{\\vrule \\vbox to 6pt {\\hrule width 4pt\\vfill \\hrule }\\vrule } $ Remark 5.4 Hence, the Matomäki-Radziwiłł theorem [41] on the behaviour of a strongly aperiodic multiplicative function $u$ on a typical short interval implies $u\\perp {C}_{\\rm ID}$ .", "However, as shown in [28], the aperiodic multiplicative functions defined in [42] do not satisfy the assertion of Corollary REF .", "In Corollary REF , the Veech condition (for $u$ ) equivalent to $u\\perp {C}_{\\rm ID}$ is written as $\\pi _0\\perp L^2(\\mathcal {I}_\\kappa )$ .", "If we look at it more spectrally, we obtain immediately that $u\\perp {C}_{\\rm ID}\\text{ if and only if }\\sigma _{\\pi _0,\\kappa }(\\lbrace 1\\rbrace )=0$ for all $\\kappa \\in V_S(u)$ , i.e.", "the spectral measure of $\\pi _0$ (with respect to each Furstenberg system) has no atom at 1.", "Classically (by a simple computation), we have: Lemma 5.5 If $\\sigma $ is a measure on the circle $\\mathbb {S}^1$ then $\\frac{1}{H}\\sum _{h=0}^{H-1} \\widehat{\\sigma }(h)\\rightarrow \\sigma (\\lbrace 1\\rbrace ).$ Hence, the Veech condition is equivalent to $\\frac{1}{H}\\sum _{h=0}^{H-1}\\int \\pi _0\\cdot \\overline{\\pi _0}\\circ S^h d\\kappa \\rightarrow 0.$ Combinatorially, we obtain $\\frac{1}{H}\\sum _{h=0}^{H-1}\\lim _{k\\rightarrow \\infty } \\frac{1}{N_k}\\sum _{n\\le N_k} u(n)\\overline{u}(n+h) \\rightarrow 0$ for each sequence $(N_k)$ defining a Furstenberg system $\\kappa $ .", "It follows that (REF ) is equivalent to the short interval behaviour ().", "In other words, condition $\\lim _{H\\rightarrow \\infty }\\left(\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}\\left|\\frac{1}{H}\\sum _{h\\le H} u(n+h)\\right|^2\\right)=0$ is equivalent to $\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\left(\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}u(n)\\overline{u}(n+h)=0\\right).$" ], [ "Orthogonality to ${C}_{\\rm DISP(G)}$ with {{formula:69a12a44-1485-4df7-8f5e-d208881bd559}} countable", "Let $G\\subset \\mathbb {S}^1$ be a countable subgroup and recall that ${{\\rm DISP}(G)}$ stands for the (characteristic) class of discrete spectrum automorphisms whose groups of eigenvalues are contained in $G$ .", "Since $z\\in \\mathbb {S}^1$ is an eigenvalue of $(Z,{\\cal D},\\kappa ,R)$ if and only if it is an eigenvalues of a subset of positive measure of ergodic components, it is not hard to see that ${{F}}_{{\\rm DISP}(G)}=({{F}}_{{\\rm DISP}(G)})_{\\rm ec}.$ It follows that $u\\perp {C}_{{{F}}_{{\\rm DISP}(G)}}\\text{ if and only if }\\sigma _{\\pi _0,\\kappa }(G)=0,$ i.e.", "the spectral measure of $\\pi _0$ has no atoms belonging to $G$ (for each Furstenberg system $\\kappa \\in V_S(u)$ ).", "Suppose that $e^{2\\pi i\\alpha }\\in G$ .", "Consider $v(n):=e^{2\\pi in\\alpha }u(n)$ for $n\\in {\\mathbb {N}}$ .", "Note that $\\frac{1}{N_k}\\sum _{n\\le N_k}v(n)\\overline{v}(n+h)=e^{-2\\pi ih\\alpha }\\frac{1}{N_k}\\sum _{n\\le N_k}u(n)\\overline{u}(n+h).$ So, if we have a subsequence $(N_k)$ along which both $\\frac{1}{N_k}\\sum _{n\\le N_k}\\delta _{S^nw}$ with $w=u,v$ converge to $\\kappa ,\\kappa ^{\\prime }$ respectively,Note that these common sequences yield all Furstenberg systems for both $u$ and $v$ .", "then $\\sigma _{\\pi _0,\\kappa }=\\delta _{e^{2\\pi i\\alpha }}\\ast \\sigma _{\\pi _0,\\kappa ^{\\prime }},$ whence $\\sigma _{\\pi _0,\\kappa }(\\lbrace e^{2\\pi i\\alpha }\\rbrace )=0 \\text{ if and only if }\\sigma _{\\pi _0,\\kappa ^{\\prime }}(\\lbrace 1\\rbrace ).$ By our previous subsection it follows that the latter condition is equivalent to: $\\lim _{H\\rightarrow \\infty }\\left(\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}\\left|\\frac{1}{H}\\sum _{h\\le H} v(n+h)\\right|^2\\right)=0,$ that is, $\\lim _{H\\rightarrow \\infty }\\left(\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}\\left|\\frac{1}{H}\\sum _{h\\le H}e^{2\\pi i(n+h)\\alpha } u(n+h)\\right|^2\\right)=0$ which is the strong $u$ -MOMO condition for the irrational rotation by $\\alpha $ .Note that if $f(t)=e^{2\\pi i t}$ then $\\frac{1}{b_K}\\sum _{k<K}\\left|\\sum _{b_k\\le n<b_{k+1}}f(R_\\alpha ^nx_k)u(n)\\right| =\\frac{1}{b_K}\\sum _{k<K}\\left|\\sum _{b_k\\le n<b_{k+1}}e^{2\\pi i n\\alpha }u(n)\\right|.$" ], [ "Furstenberg systems and the strong $u$ -MOMO property", "The following proposition helps us to exclude some measure-theoretic systems from the list of Furstenberg systems of an arithmetic function.", "Proposition 5.6 Let $u:{\\mathbb {N}}\\rightarrow be a bounded arithmetic function.", "Then no Furstenberg system $ VS(u)$ has a topological model which is strongly $ u$-MOMO.$ Suppose $(X_{u},\\kappa , S)$ has a topological model $(Z,\\nu ,R)$ which satisfies the strong $u$ -MOMO property.", "Let $J:Z\\rightarrow X_{u}$ settles a measure-theoretic isomorphism and let $\\nu _J$ be the corresponding graph joining.", "We assume that $\\frac{1}{N_j}\\sum _{n\\le N_j}\\delta _{S^nu}\\rightarrow \\kappa $ .", "From Proposition REF we can find a sequence $(z_n)\\subset Z$ consisting of pieces of orbits of different points: $\\lbrace n:\\: Rz_n\\ne z_{n+1}\\rbrace =\\lbrace b_k:\\:k\\ge 1\\rbrace $ with $b_{k+1}-b_k\\rightarrow \\infty $ , and a subsequence $(N_{j_\\ell })$ such that $\\frac{1}{N_{j_\\ell }}\\sum _{n\\le N_{j_\\ell }}\\delta _{(z_n,S^nu)}\\rightarrow \\nu _J.$ Then, by the strong $u$ -MOMO property of $(Z,R)$ , $\\int f\\otimes \\pi _0\\,d\\nu _J=\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{b_{K_\\ell }}\\sum _{k<K_\\ell }\\big (\\sum _{b_k\\le n<b_{k+1}}f(R^{n-b_k}z_{b_k})u(n)\\big )=0.$ Hence, $\\int {\\mathbb {E}}^{\\nu _J}(f|X_{u})\\pi _0\\,d\\kappa =0$ for each continuous $f$ on $Z$ , and we obtain a contradiction since ${\\mathbb {E}}^{\\nu _{J}}(L^2(\\nu )|X_{u})=L^2(\\kappa )$ .", "Corollary 5.7 Assume that for each $(b_k)$ with $b_{k+1}-b_k\\rightarrow \\infty $ , $\\lim _{K\\rightarrow \\infty }\\frac{1}{b_K}\\sum _{k<K}\\sup _{\\alpha \\in {\\mathbb {R}}}\\left|\\sum _{b_k\\le n< b_{k+1}}u(n)e^{2\\pi i\\alpha n}\\right|=0.$ Then the unipotent system $(x,y)\\mapsto (x,y+x)$ (on 2) is not a Furstenberg system of $u$ .", "Since condition (REF ) is the strong $u$ -MOMO property of the unipotent system, the result follows from Proposition REF .", "Remark 5.8 Corollary REF brings a better understanding of Problem 3.1 (due to Frantzikinakis) of the workshop [4]: The system $(x,y)\\mapsto (x,y+x)$ is not a Furstenberg system of the Liouville function (see also slide no 6 in [17]).", "We recall that in [42] (see Theorem 1.3 therein), it is proved that $\\lim _{M\\rightarrow \\infty }\\limsup _{N\\rightarrow \\infty }\\sup _{\\alpha \\in {\\mathbb {R}}}\\frac{1}{N}\\sum _{n\\le N}\\left|\\frac{1}{M}\\sum _{n\\le m<n+M}\\lambda (m)e^{2\\pi i m\\alpha }\\right|=0,$ so the sup has changed the place!", "The strong $\\lambda $ -MOMO property for the unipotent system remains hence open.", "For the equivalence of $\\lim _{M\\rightarrow \\infty }\\limsup _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n\\le N}\\sup _{\\alpha \\in {\\mathbb {R}}}\\left|\\frac{1}{M}\\sum _{n\\le m<n+M}u(m)e^{2\\pi i m\\alpha }\\right|=0,$ with (REF ) see the appendix in [34] - only in the arXiv version of the paper." ], [ "Orthogonality to ${C}_{{\\rm DISP}_{\\rm ec}}$ . Proof of Corollary ", "In view of Corollary REF (see also (REF )) and Theorem REF , in order to obtain $u\\perp \\mathcal {C}_{{\\rm DISP}_{ec}}$ it is sufficient and necessary to have $u\\perp L^2({\\cal K}(\\mathcal {I}_\\kappa ))$ for each Furstenberg system $\\kappa $ of $u$ .", "By our previous results, for the class of all topological systems whose all ergodic measures yield discrete spectra, Sarnak and Veech conditions are equivalent.", "We now write the Veech condition combinatorially, i.e., we provide the proof of Corollary REF .", "[Proof of Corollary REF ] By Corollary REF , we need to show that for each $\\kappa $ being a Furstenberg system of $u$ , we have $\\int \\frac{1}{H}\\sum _{h\\le H}|{\\mathbb {E}}(\\pi _0\\circ S^h\\cdot \\overline{\\pi _0}|\\mathcal {I}_\\kappa )|^2\\, d\\kappa \\rightarrow 0.$ By the von Neumann theorem, $\\int \\Big |{\\mathbb {E}}(\\pi _0\\circ S^h\\cdot \\overline{\\pi }_0|\\mathcal {I}_\\kappa )\\Big |^2\\,d\\kappa =\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n\\le N}\\int (\\pi _0\\circ S^h\\cdot \\overline{\\pi }_0)\\circ S^n\\cdot \\overline{\\pi _0\\circ S^h\\cdot \\overline{\\pi }_0}\\,d\\kappa .$ Therefore, (REF ) is equivalent to $\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n\\le N}\\int (\\pi _0\\circ S^h\\cdot \\overline{\\pi }_0)\\circ S^n\\cdot \\overline{\\pi _0\\circ S^h\\cdot \\overline{\\pi }_0}\\,d\\kappa =0.$ Let $(M_k)$ be such that $\\kappa =\\lim _{k\\rightarrow \\infty }\\frac{1}{M_k}\\sum _{m\\le M_k}\\delta _{S^mu}$ .", "It follows immediately that (REF ) is equivalent to $\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n\\le N}\\lim _{k\\rightarrow \\infty }\\frac{1}{M_k}\\sum _{m\\le M_k}u(m+n+h)\\overline{u(m+n)}\\overline{u(m+h)}u(m)=0$ which is precisely $\\Vert u\\Vert _{u^2((M_k))}=0$ .", "Now, it suffices to use (REF ).", "Remark 5.9 In fact, already Frantzikinakis [17] (see slide no 10) showed that if $u$ is generic then $\\Vert u\\Vert _{u^2}=0$ if and only if $\\lim _{M\\rightarrow \\infty }\\limsup _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n\\le N}\\sup _{\\alpha \\in {\\mathbb {R}}}\\left|\\frac{1}{M}\\sum _{n\\le m<n+M}u(m)e^{2\\pi i m\\alpha }\\right|=0.$ We recall that this condition is equivalent to the strong $u$ -MOMO property of the unipotent system $(x,y)\\mapsto (x, y+x)$ .", "Remark 5.10 Note that for each (bounded) $u\\colon {\\mathbb {N}}\\rightarrow satisfying $ uu2=0$ the system\\begin{equation}(x,y)\\mapsto (x,x+y)\\text{ on }(2,{\\rm Leb}\\,\\otimes \\, {\\rm Leb})\\end{equation}cannot appear (up to isomorphism) as a Furstenberg system of $ u$ (because $ 0$ is orthogonal to the $ L2(K(I))$ but for the unipotent system (\\ref {unip}) thewhole system is relative Kronecker over the $$-algebra of invariant sets).$ In particular, if $\\Vert \\lambda \\Vert _{u^2}=0$ holds for the Liouville function then () is not its Furstenberg system – this would answer a question by N. Frantzikinakis asked in 2016 (it is an official Problem 3.1 in [4]).", "However, the problem of whether $\\Vert \\lambda \\Vert _{u^2}=0$ (or more generally $\\Vert \\lambda \\Vert _{u^s}=0$ ) seems to be difficult.", "The best known results [43], [44] require a quantitative dependence between the parameters $M$ and $N$ , i.e.", "$M=N^{\\theta }$ , for arbitrary small, but fixed $\\theta >0$ .", "If $\\Vert \\lambda \\Vert _{u^2}=0$ holds then Sarnak's conjecture holds for all (zero entropy) systems whose ergodic measures yield discrete spectrum.", "So far it is only known that Sarnak's conjecture holds for systems whose all invariant measure yield discrete spectrum [33], [32], [15].", "Ruling out () (or, more generally, nilpotent type systems) from the list of potential Furstenberg systems of $\\lambda $ is important in view of Frantzikinakis and Host's results [20], [21] concerning the structure of Furstenberg systems of multiplicative functions (although, for the moment, this structure is known only for the logarithmic case).", "In the light of [42], it would be also interesting to know whether $\\Vert u\\Vert _{u^2}=0$ holds for some classical multiplicative functions.", "Note that this is not the case for the class of aperiodic multiplicative functions defined in [42] since as shown in [28] they have the unipotent system as a Furstenberg systemIn fact, for such functions $u$ we have already $\\Vert u\\Vert _{u^1((N_k))}>0$ (for some $(N_k)$ ), see Corollary 6.5 in [28].", "(see also Remark REF )." ], [ "Orthogonality to ${C}_{\\rm DISP}$ . Averaged Chowla property for multiplicative functions", "The assertion “iff” of Theorem REF cannot be applied to the class ${C}_{\\rm DISP}$ .", "In this section we will show however that the assertion of this theorem holds whenever $u\\colon {\\mathbb {N}}\\rightarrow satisfies the following additional property:\\begin{equation}\\mbox{all rotations on the circle satisfy the strong \\qquad \\mathrm {(\\ast )}$u$-MOMO property}.\\end{equation}We will need the following fact (see, e.g.,~\\cite {Ed}):\\begin{equation} \\mbox{each discrete spectrum automorphism is a factor of $R_\\alpha \\times {\\rm Id}_{[0,1]}$},\\end{equation}for some ergodic rotation by $ G$ on a compact (Abelian) metric group $ G$.", "Our key tool will be the following lemma.\\begin{Lemma}Suppose that (\\ast ) holds.", "Then R_\\alpha \\times {\\rm Id}_{[0,1]} satisfies the strong u-MOMO property.\\end{Lemma}\\begin{proof}It is enough to check the strong u-MOMO for functions F of the form \\chi \\otimes f, where \\chi \\in \\widehat{G} and f\\in C([0,1]).", "We have{\\begin{@align*}{1}{-1}\\frac{1}{b_K}\\sum _{k<K}&\\left| \\sum _{b_k\\le n<b_{k+1}}F((R_\\alpha \\times {\\rm Id})^n(h_k,u_k))u(n)\\right|\\\\&=\\frac{1}{b_K}\\sum _{k<K}\\left| \\sum _{b_k\\le n<b_{k+1}}\\chi ((R_\\alpha ^n(h_k))f(u_k)u(n)\\right|\\\\&={\\rm O}\\left( \\frac{1}{b_K}\\sum _{k<K}\\left| \\sum _{b_k\\le n<b_{k+1}}\\chi (n\\alpha )u(n)\\right| \\right).\\end{@align*}}Our claim follows from (\\ast ).\\end{proof}$ Theorem 5.11 Assume that $u$ enjoys the property $(\\ast )$ .", "Then $u\\perp {C}_{DISP}$ if and only if $\\pi _0\\perp L^2(\\mathcal {K}(\\kappa ))$ for each $\\kappa \\in V_S(u)$ (iff the spectral measure $\\sigma _{\\pi _0}$ is continuous for each Furstenberg system $\\kappa $ ).", "We only need to show that $u\\perp {C}_{DISP}$ implies $\\pi _0\\perp L^2(\\mathcal {K}(\\kappa ))$ for each $\\kappa \\in V_S(u)$ .$\\mathcal {K}(\\kappa )$ stands for the Kronecker factor of $(X_{u},\\kappa ,S)$ .", "Using (), let $p$ settle a factor map from $R_\\alpha \\times {\\rm Id}_{[0,1]}$ acting on $(G\\times [0,1],m_G \\otimes {\\rm Leb})$ and $(X_{u}/\\mathcal {K}(\\kappa ),\\mathcal {K}(\\kappa ),\\kappa |_{\\mathcal {K}(\\kappa )})$ .", "Let $(m_G\\otimes {\\rm Leb})_p$ stand for the corresponding graph joining and $\\rho $ for the relatively independent extension of it to a joining of $(G\\times [0,1], m_G\\otimes {\\rm Leb},R_\\alpha \\times {\\rm Id})$ with $(X_{u}, \\kappa ,S)$ .", "Now, by Proposition REF , the integral $\\int F\\otimes \\pi _0\\, d\\rho $ can be computed using a quasi-generic sequence $((g_n), (S^nu))$ .", "Since, by Lemma , our topological system $R_\\alpha \\times {\\rm Id}$ satisfies the strong $u$ -MOMO property, this integral vanishes.", "On the other hand, for each $F\\in C(G\\times [0,1])$ , $\\int F\\otimes \\pi _0\\,d\\rho =\\int {\\mathbb {E}}(F|X_{u})\\pi _0\\,d\\kappa $ and since ${\\mathbb {E}}^\\rho (C(G\\times [0,1])|X_{u})$ is dense in $L^2(\\mathcal {K},\\kappa |_{\\mathcal {K}})$ (in view of the definition of $\\rho $ ), it follows that $\\pi _0\\perp L^2(\\mathcal {K}(\\kappa ))$ .", "[Proof of Corollary REF ] Note that in the proof of Theorem REF , we have shown that our original assumption $(\\ast )$ already implies the Veech condition.", "In particular, the Sarnak and the Veech properties are equivalent.", "Condition () is just rewriting the Wiener condition combinatorially.", "Finally, the last part () is proved in Appendix .", "[Proof of Corollary REF ] By Corollary REF , we only need to show that irrational rotations satisfy the strong $u$ -MOMO property.", "This follows from the fact that irrational rotations satisfy the AOP property [1] and that the AOP property implies the strong $u$ -MOMO property [3]." ], [ "No strong $u$ -MOMO in positive entropy", "In this section we discuss the problem of orthogonality to ${C}_{\\rm ZE}$ and the reversed problem of the absence of orthogonality to an arbitrary positive entropy systems, following some ideas from [3].", "Recall that the following has been proved in [3].", "Proposition 6.1 ([3]) Let $u\\colon {\\mathbb {N}}\\rightarrow be a bounded arithmetic function.", "The following are equivalent:\\begin{enumerate}\\item [(a)] u\\perp {C}_{\\rm ZE}.\\item [(b)] For each (X,T) of zero entropy and f\\in C(X), (\\ref {ort1}) holds uniformly in x\\in X.\\item [(c)] Each zero entropy (X,T) satisfies the strong u-MOMO property.\\end{enumerate}$ On the other hand, it follows from the results of Downarowicz and Serafin [11], [12] that for each $u\\perp {C}_{\\rm ZE}$ there exists $(X,T)$ such that $u\\perp (X,T) \\text{ and }(X,T)\\notin {\\rm ZE}.$ In fact, one can get a positive entropy system $(X,T)$ in which for every $f\\in C(X)$ () holds uniformly in $x\\in X$ .", "We prove however that (REF ) fails if orthogonality is replaced by the strong $u$ -MOMO property.", "To avoid technical details, we restrict ourselves to the case of an arithmetic function $u$ taking finitely many values.", "Theorem D Let $u:{\\mathbb {N}}\\rightarrow be an arithmetic function taking finitely many values.Assume that $ uCZE$.", "Then no positive entropy topological dynamical system satisfies the strong $ u$-MOMO property.$" ], [ "Proof of Theorem ", "We fix a bounded arithmetic function $u\\colon {\\mathbb {N}}\\rightarrow .We need a series of results from \\cite {Ab-Ku-Le-Ru2} in some modified forms.", "In~\\cite {Ab-Ku-Le-Ru2}, the equivalence of certain three properties (P1), (P2) and (P3) of an ergodic measure-theoretic dynamical system $ (Z,B(Z),,R)$ was proved.", "Condition (P1) was nothing but the strong $ u$-MOMO for {\\bf some} topological system being a model of the system given by $$.", "Instead of recalling (P2), let us formulate red its subsequence version:\\begin{equation}\\begin{array}{l}\\mbox{Assume that \\qquad \\mathrm {(P2')}$(X,T)$ is any topological system and let $x\\in X$.", "}\\\\\\mbox{If $x$ is generic along $(N_k)$ for a measure which is isomorphic}\\\\\\mbox{(as dynamical systems) to $\\kappa $ then}\\\\\\mbox{$\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}f(T^nx)u(n)=0$for each $f\\in C(X)$}.\\end{array}\\end{equation}The proof of the implication (P1) $$ (P2^{\\prime }) is a repetition of the proof of (P1) implies (P2).", "In Lemma~17 in \\cite {Ab-Ku-Le-Ru2}, we need to consider the sequence $ (Nk)$ instead of $ N$ and start with $$ along this sequence.$ As a consequence of the above, we obtain the following version of Corollary 12 from [3].", "Corollary 6.2 Assume that $\\kappa $ is an ergodic shift-invariant measure on $L^{{\\mathbb {Z}}}$ , and that there exists $y\\in L^{{\\mathbb {Z}}}$ , generic along $(N_k)$ for $\\kappa $ , correlating with $u$ along $(N_k)$ , i.e.", "the sequence $(\\frac{1}{N_k}\\sum _{n\\le N_k}y(n)u(n))$ does not go to zero.", "Then the strong $u$ -MOMO property fails for any uniquely ergodic model of $(L^{{\\mathbb {Z}}},\\kappa ,S)$ .", "Then, by repeating the proof from [3], we obtain the following form of Corollary 14 in [3].", "Corollary 6.3 Assume that $y$ is generic along $(N_k)$ for a Bernoulli measure $\\nu $ , and that $y$ and $u$ correlate along $(N_k)$ .", "Then the strong $u$ -MOMO property fails for any $(X,T)$ with $h(X,T)>h(\\nu )$ .", "We also need the following crucial probabilistic lemma whose proof we postpone to the next subsection.", "Lemma 6.4 Assume that $X=(X_n)_{n\\in {\\mathbb {Z}}}$ is a a stationary process of positive entropy, taking finitely many complex values.", "Then for any non-trivial probability distribution $\\beta $ concentrated on a finite subset of ${\\mathbb {R}}$ , there exists a stationary coupling of $X$ with a Bernoulli process $Y=(Y_n)_{n\\in {\\mathbb {Z}}}$ of distribution $\\beta ^{\\otimes {\\mathbb {Z}}}$ such that ${\\mathbb {E}}[X_0 Y_0]\\ne {\\mathbb {E}}[X_0]{\\mathbb {E}}[Y_0]$ .", "We now assume that $u$ takes finitely many values and satisfies the Veech condition: $\\pi _0\\perp L^2(\\Pi (\\kappa ))$ for each Furstenberg system $\\kappa $ of $u$ .", "Lemma 6.5 For each $h>0$ there exists a sequence $y$ , generic for a Bernoulli measure of entropy $h$ along some increasing sequence $(N_k)$ , and correlating with $u$ along $(N_k)$ .", "Let $\\kappa $ be a Furstenberg system of $u$ , and $(M_\\ell )$ such that $u$ is generic for $\\kappa $ along $(M_\\ell )$ .", "By assumption, the entropy of the stationary process defined by $\\pi _0$ under $\\kappa $ is positive.", "Take a real-valued Bernoulli shift of entropy $h$ (Bernoulli measure denoted by $\\nu $ ).", "Using Lemma REF , find a joining of $\\kappa $ and $\\nu $ for which $\\pi _0$ (in $L^2(X_{u},\\kappa )$ ) is not orthogonal to $\\pi _0$ in $L^2(\\nu )$ : $\\int \\pi _0\\otimes \\pi _0\\,d\\rho \\ne 0$ .", "Now, use a subsequence version of the lifting lemma (Theorem 5.16 in [6]) to find $y$ in the subshift defining the Bernoulli automorphism such that $(u,y)$ is generic, along a subsequence $(N_k)=(M_{\\ell _k})$ , for $\\rho $ .", "Then $0\\ne \\int \\pi _0\\otimes \\pi _0\\,d\\rho =\\lim _{k\\rightarrow \\infty }\\frac{1}{N_{k}}\\sum _{n\\le N_{k}}\\pi _0(S^nu)\\pi _0(S^ny)=\\lim _{k\\rightarrow \\infty }\\frac{1}{N_{k}}\\sum _{n\\le N_{k}}u(n)y_n$ which means that $u$ and $y$ correlate along $(N_{k})$ .", "Now the proof of Theorem REF is a straightforward consequence of Lemma REF and Corollary REF ." ], [ "Proof of Lemma ", "Let $X=(X_n)_{n\\in {\\mathbb {Z}}}$ be a positive entropy stationary process as in the statement of the lemma.", "Without loss of generality (considering its real or imaginary part), we can assume that this process takes its values in a finite subset $\\lbrace x_1<x_2<\\cdots <x_r\\rbrace $ of ${\\mathbb {R}}$ .", "We also consider a given probability measure $\\beta $ supported on a possibly different finite subset of ${\\mathbb {R}}$ $\\lbrace y_1<y_2<\\cdots <y_s\\rbrace $ , which is supposed to be non trivial (i.e.", "not reduced to a Dirac measure).", "Thus we can assume that $s\\ge 2$ , and $\\beta (y_j)>0$ for each $1\\le j\\le s$ .", "The purpose of this section is to show how we can construct a stationary coupling of $X$ with a Bernoulli process $Y$ whose distribution is $\\beta ^{\\otimes {\\mathbb {Z}}}$ , in such a way that for each $n\\in {\\mathbb {Z}}$ , ${\\mathbb {E}}[X_n Y_n] > {\\mathbb {E}}[X_n]\\, {\\mathbb {E}}[Y_n].$ We observe that the validity of the preceding inequality is unchanged if we replace $Y_n$ by $Y_n+C$ for a fixed $C$ .", "Thus we can and we do assume without loss of generality that the probability $\\beta $ is such that ${\\mathbb {E}}[Y_n]=0$ .", "To construct the announced coupling, we just assume that, on the probability space where the process $X$ is defined, we also have an i.i.d.", "process $V=(V_n)_{n\\in {\\mathbb {Z}}}$ such that each $V_n$ is uniformly distributed on $[0,1]$ , $V$ is independent of $X$ .", "The construction will be divided into two steps: first we construct an auxiliary (uniform i.i.d.)", "process $U$ and then we use it to construct $Y$ which satisfies the assertion of Lemma REF ." ], [ "Step 1: uniform i.i.d. process $U$", "For $n\\in {\\mathbb {Z}}$ and $j\\in \\lbrace 1,\\ldots ,r\\rbrace $ , we consider the random variable $P_{j,n}$ defined by $P_{j,n} := {\\mathbb {P}}\\bigl ( X_n=x_j \\,|\\, (X_m)_{m\\le n-1}\\bigr ).$ When $j$ is fixed, $(P_{j,n})_{n\\in {\\mathbb {Z}}}$ is a stationary process.", "On the other hand, if we fix $n$ , then $(P_{1,n},\\ldots ,P_{r,n})$ is the conditional distribution of $X_n$ given $(X_m)_{m\\le n-1}$ , in particular we have almost surely $0\\le P_{j,n}\\le 1$ , and $\\sum _{j=1}^r P_{j,n} = 1.$ This allows us to define a random partition of $[0,1[$ into disjoint subintervals $I_{1,n},\\ldots , I_{r,n}$ where for each $j$ , $I_{j,n}$ is the interval of length $P_{j,n}$ defined by $I_{j,n}:=\\left[\\sum _{1\\le i\\le j-1} P_{i,n}\\ ; \\sum _{1\\le i\\le j} P_{i,n}\\right[.$ Then we can define the random variable $U_n$ by $U_n := \\sum _{j=1}^r {1}_{X_n=x_j}\\left(\\sum _{1\\le i\\le j-1} P_{i,n} + V_n P_{j,n}\\right).$ Figure: Definition of U n U_nInformally, if $X_n=x_j$ , we pick $U_n$ uniformly at random (using $V_n$ ) inside $I_{j,n}$ (see Figure REF ).", "Therefore, $\\mathcal {L}\\left(U_n\\,|\\,(X_m)_{m\\le n-1},(V_m)_{m\\le n-1}\\right)=\\mathcal {U}_{[0,1]},$ i.e., it is uniform on $[0,1]$ .", "But all $U_m$ , $m\\le n-1$ , are measurable with respect to $(X_m)_{m\\le n-1}$ and $(V_m)_{m\\le n-1}$ , thus we also have $\\mathcal {L}\\left(U_n\\,|\\,(U_m)_{m\\le n-1}\\right)=\\mathcal {U}_{[0,1]} \\text{ and } \\mathcal {L}\\left(U_n\\right)=\\mathcal {U}_{[0,1]}.$ Indeed, this is just the application of the tower property of conditional expectations: to obtain the left equality, notice that for any measurable $A\\subset [0,1]$ , we have $\\mathbb {P}\\bigl (U_n\\in A &\\,|\\, (U_m)_{m\\le n-1}\\bigl ) \\\\&=\\mathbb {E}\\Bigl [\\underbrace{\\mathbb {P}\\bigl (U_n\\in A\\,|\\, (X_m)_{m\\le n-1},(V_m)_{m\\le n-1}\\bigr )}_{{\\rm Leb}(A)}\\,|\\, (U_m)_{m\\le n-1}\\Bigr ]\\\\&={\\rm Leb}(A).$ Moreover, it also follows from (REF ) that $U$ is i.i.d.", "Note that by construction, $U_n$ is a measurable function of $V_n$ , $X_n$ and $(X_m)_{m\\le n-1}$ , which we abusively write as $U_n = U_n \\left(V_n,X_n,(X_m)_{m\\le n-1}\\right).$ Moreover, whenever we fix realizations $\\xi $ of $(X_m)_{m\\le n}$ and $v$ of $V_n$ then $U_n$ as a function of its second argument is increasing: $U_n(v,x_{j_1},\\xi ) < U_n(v,x_{j_2},\\xi ), \\text{ whenever }x_{j_1}<x_{j_2}.$ We want to define $Y_n$ for a given $n\\in {\\mathbb {Z}}$ .", "We use another partition of $[0,1[$ into subintervals, according to the probability distribution $\\beta $ intended for $Y_n$ : for $1\\le k\\le s$ , set $\\beta _k:=\\beta (y_k)$ and define the interval $J_k:=\\bigl [\\beta _1+\\cdots +\\beta _{k-1};\\beta _1+\\cdots +\\beta _k\\bigr [$ .", "Then we simply define $Y_n$ as a function of $U_n$ by setting $ Y_n := \\sum _{k=1}^s y_k\\,{1}_{J_k}(U_n).$ It follows by the choice of the intervals $J_k$ and by $\\mathcal {L}(U_n)=\\mathcal {U}_{[0,1]}$ that $Y_n$ is distributed according to $\\beta $ .", "Moreover, by the independence of $U$ , we have the independence of $Y$ .", "Thus, $Y$ is a Bernoulli process with distribution $\\beta ^{\\otimes \\mathbb {Z}}$ .", "It remains to prove the announced inequality (REF ).", "Observe that $Y_n$ is, like $U_n$ , constructed as a measurable function of $V_n$ , $X_n$ and $(X_m)_{m\\le n-1}$ , which we also abusively write as $ Y_n = Y_n \\left(V_n,X_n,(X_m)_{m\\le n-1}\\right).", "$ Since $Y_n$ is a non-decreasing function of $U_n$ , we get from (REF ) that for a fixed realization $\\xi $ of $(X_m)_{m\\le n-1}$ and $v$ of $V_n$ , we have for $1\\le j_1 < j_2 \\le r$ $Y_n \\left(v,x_{j_1},\\xi \\right) < Y_n \\left(v,x_{j_2},\\xi \\right)$ and it follows that the map $ x\\in A \\mapsto {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x\\bigr ] $ is non-decreasing.", "Moreover, by the construction of $Y$ , we have $ \\mathcal {L}\\bigl (Y_n \\,|\\, (X_m)_{m\\le n-1}\\bigr ) = \\mathcal {L}(Y_n)=\\beta , $ whence ${\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}\\bigr ] = {\\mathbb {E}}[Y_n] =0.$ Thus there exists $j_0\\in \\lbrace 1,\\ldots ,r\\rbrace $ (depending on $\\xi $ ) such that $\\begin{split}&{\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_j]\\le 0 \\text{ for }1\\le j\\le j_0,\\\\&{\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_j]> 0\\text{ for }j_0+1\\le j\\le r.\\end{split}$ We then have, using (REF ) and (REF ), ${\\mathbb {E}}\\bigl [ X_n Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] & = {\\mathbb {E}}\\bigl [ (X_n-x_{j_0}) Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] \\nonumber \\\\&= \\sum _{j=1}^{j_0} (x_j-x_{j_0})\\, {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_j] \\nonumber \\\\&\\quad + \\sum _{j=j_0+1}^{r} (x_j-x_{j_0})\\, {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_j] \\\\& \\ge 0.", "\\nonumber $ Now, we claim that the announced result is a consequence of the following lemma.", "Lemma 6.6 If the realization $\\xi $ of $(X_m)_{m\\le n-1}$ is such that the conditional distribution $\\mathcal {L}(X_n \\,|\\, (X_m)_{m\\le n-1}=\\xi )$ is non-trivial, then $ {\\mathbb {E}}\\bigl [ X_n Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] > 0.", "$ Indeed, since $X$ has positive entropy, $\\mathcal {L}(X_n \\,|\\, (X_m)_{m\\le n-1})$ is non-trivial with positive probability, and thus we can conclude that $ {\\mathbb {E}}\\bigl [ X_n Y_n\\bigr ] = {\\mathbb {E}}\\Bigl [{\\mathbb {E}}\\bigl [ X_n Y_n \\,|\\, (X_m)_{m\\le n-1}\\bigr ]\\Bigr ] >0.", "$ [Proof of Lemma REF ] We fix a realization $\\xi $ of $(X_m)_{m\\le n-1}$ such that the conditional distribution $\\mathcal {L}\\bigl (X_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr )$ is non-trivial.", "Then the random variables $P_{j,n}$ and the intervals $I_{j,n}$ are fixed, because their values only depend on $\\xi $ .", "Setting $j_1 &:= \\min \\bigl \\lbrace j\\in \\lbrace 1,\\ldots ,r\\rbrace :\\ P_{j,n}>0\\bigr \\rbrace ,\\\\\\text{and}\\quad j_2 &:= \\max \\bigl \\lbrace j\\in \\lbrace 1,\\ldots ,r\\rbrace :\\ P_{j,n}>0\\bigr \\rbrace ,$ we have $j_1<j_2$ .", "Moreover the intervals $I_{j_1,n}$ and $I_{j_2,n}$ are respectively of the form $[0,P_{j_1,n}[$ and $[1-P_{j_2,n},1[$ , with $0<P_{j_1,n}\\le 1-P_{j_2,n}<1$ .", "We now discuss according to the relative position of the interval $I_{j_2,n}$ with respect to the interval $J_1$ (used to define $Y_n$ ).", "Figure: Case 1 (J 1 ∩I j 2 ,n =∅J_1\\cap I_{j_2,n}=\\emptyset )" ], [ "Case 1:", "$J_1\\cap I_{j_2,n}=\\emptyset $ (see Figure REF ).", "Then we have ${\\mathbb {P}}\\bigl (Y_n=y_1 \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_2}\\bigr )=0,$ whereas ${\\mathbb {P}}\\bigl (Y_n=y_1 \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_1}\\bigr )>0.$ Moreover, notice that (REF ) is equivalent to ${\\mathbb {P}}\\bigl (Y_n>y_1 \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_2}\\bigr )=1,$ It follows from (REF ) and (REF ) that there exists a $V_n$ -measurable event $A$ of positive probability such that, on $A$ , $ Y_n\\bigl (V_n,x_{j_2},\\xi \\bigr ) > y_1 = Y_n\\bigl (V_n,x_{j_1},\\xi \\bigr ).", "$ Remembering (REF ), we get ${\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_{j_2}\\bigr ] > {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_{j_1}\\bigr ].$ Figure: Case 2 (J 1 ∩I j 2 ,n ≠∅J_1\\cap I_{j_2,n}\\ne \\emptyset )" ], [ "Case 2:", "$J_1\\cap I_{j_2,n}\\ne \\emptyset $ (see Figure REF ).", "Then $I_{j_1,n}\\subset J_1$ and $I_{j_1,n}\\cap J_s=\\emptyset $ .", "It follows that ${\\mathbb {P}}\\bigl (Y_n=y_s \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_1}\\bigr )=0,$ whereas ${\\mathbb {P}}\\bigl (Y_n=y_s \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_2}\\bigr )>0.$ In this case, we get a $V_n$ -measurable event $A$ of positive probability such that, on $A$ , $ Y_n\\bigl (V_n,x_{j_2},\\xi \\bigr ) = y_s > Y_n\\bigl (V_n,x_{j_1},\\xi \\bigr ), $ and as before we conclude that (REF ) holds.", "Now, since (REF ) always holds, and since $ {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] = 0 =\\sum _{j=1}^rP_{j,n}\\, {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_{j}\\bigr ], $ we deduce that $ {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_{j_2}\\bigr ]>0.", "$ It follows that in the sum (REF ), at least the term corresponding to $j=j_2$ is positive, and this yields $ {\\mathbb {E}}\\bigl [ X_n Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] > 0.", "$ Appendix" ], [ "From averaged double to averaged multiple correlations", "This section follows some arguments from [42].", "Remark A.1 In the proof below we will use the following standard fact: let $(x(n))$ be a sequence of complex number bounded by 1.", "Then $\\sum _{m\\le M}|x(m)|=o(M)$ is equivalent to $\\sum _{m\\le M}|x(m)|^2=o(M).$ The little “o” is uniform with respect to $M$ .", "If $\\varepsilon :=\\frac{1}{M}\\sum _{m\\le M}|x(m)|^2$ then by Markov's inequality $\\frac{1}{M}|\\lbrace m\\le M:\\: |c_m|^2\\ge \\varepsilon ^{1/2}\\rbrace |\\le \\frac{1}{\\varepsilon ^{1/2}}\\cdot \\varepsilon =\\varepsilon ^{1/2}$ and then $\\frac{1}{M}\\sum _{m\\le M}|x(m)|=\\frac{1}{M}\\sum _{m\\le M, |x(m)|\\ge e^{1/4}}|x(m)|+\\frac{1}{M}\\sum _{m\\le M, |x(m)|<\\varepsilon ^{1/4}}|x(m)|\\le \\varepsilon ^{1/2}+\\varepsilon ^{1/4}.$ We have the following general lemma: Lemma A.2 Let $(N_\\ell )_{\\ell \\in {\\mathbb {N}}}$ be a sequence of natural numbers.", "For $k\\in \\mathbb {N}$ let $a,b_1,\\ldots b_k\\colon \\mathbb {N} \\rightarrow \\mathbb {C}$ be sequences bounded by 1.", "Assume that $a$ satisfies $\\lim _{H\\rightarrow \\infty }\\frac{1}{H} \\sum _{h\\le H}\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n)a(n+h)\\Big |=0.$ Then $\\lim _{H\\rightarrow \\infty }\\frac{1}{H^k} \\sum _{h_1,\\ldots , h_k\\le H}\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n)\\prod _{i=1}^kb_i(n+h_i)\\Big |=0.$ Notice first that (REF ) can be rewritten as the following: for every $\\varepsilon >0$ , there exists $H_\\varepsilon $ such that for $H>H_\\varepsilon $ and all $\\ell $ sufficiently large (depending on $H$ ), we have $A:=\\frac{1}{H^k}\\sum _{h_1,\\ldots , h_k\\le H}\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n)\\prod _{i=1}^kb_i(n+h_i) \\Big |<\\varepsilon .$ Now, notice that for any $H,N_\\ell ,H^{\\prime }$ and any $h^{\\prime }\\le H^{\\prime }$ , by shifting the summation over $n\\le N_\\ell $ by $h^{\\prime }$ (for every fixed choice of $h_1,\\ldots h_k$ ), we have $\\sum _{h_1,\\ldots , h_k\\le H}\\Big | \\sum _{n\\le N_\\ell }a(n)\\prod _{i=1}^kb_i(n+h_i)\\Big |=$ $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{h^{\\prime }\\le n\\le N_\\ell +h^{\\prime }}a(n)\\prod _{i=1}^kb_i(n+h_i)\\Big |+{\\rm O}(H^k\\cdot h^{\\prime })=$ $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i+h^{\\prime })\\Big |+{\\rm O}(H^k\\cdot h^{\\prime })$ and ${\\rm O}(H^k\\cdot h^{\\prime })={\\rm O}(H^k\\cdot H^{\\prime })$ .", "Notice that as $h_i$ is taken from $[0,H]$ , then $h_i+h^{\\prime }$ is taken from $[h^{\\prime },H+h^{\\prime }]$ (which is a small shift of $[0,H]$ if $h^{\\prime }$ is much smaller than $H$ ).", "So putting $h^{\\prime }$ to the summation over $h_i$ , we get $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i+h^{\\prime })\\Big |=$ $\\sum _{h^{\\prime }\\le h_1,\\ldots , h_k\\le H+h^{\\prime }}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i)\\Big |=$ $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i)\\Big |+ {\\rm O}\\left((h^{\\prime } H^{k-1}+(h^{\\prime })^2H^{k-2}+\\ldots +(h^{\\prime })^k)N_\\ell \\right)$ and ${\\rm O}\\left((h^{\\prime } H^{k-1}+(h^{\\prime })^2H^{k-2}+\\ldots +(h^{\\prime })^k)N\\right)={\\rm O}((H^{\\prime })^kH^{k-1}N_\\ell )$ .", "Putting the two displayed equations together we get that for every $h^{\\prime }\\le H^{\\prime }$ , $A=\\frac{1}{H^k}\\sum _{h_1,\\ldots , h_k\\le H}\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i) \\Big |+{\\rm O}\\Big (\\frac{H^{\\prime }}{N_\\ell }\\Big )+{\\rm O}\\Big (\\frac{(H^{\\prime })^k}{H} \\Big ).$ Averaging the above equation over all $h^{\\prime }\\le H^{\\prime }$ , we get that $A=\\frac{1}{H^k}\\sum _{h_1,\\ldots , h_k\\le H}\\frac{1}{H^{\\prime }}\\sum _{h^{\\prime }\\le H^{\\prime }}\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })G(n)\\Big |+{\\rm O}\\Big (\\frac{H^{\\prime }}{N_\\ell }\\Big )+{\\rm O}\\Big (\\frac{(H^{\\prime })^k}{H} \\Big ),$ where $G(n)=\\prod _{i=1}^kb_i(n+h_i)$ .", "We will now estimate $\\frac{1}{H^{\\prime }}\\sum _{h^{\\prime }\\le H^{\\prime }}\\frac{1}{N_\\ell ^2}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })G(n) \\Big |^2=\\\\\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\Big (\\frac{1}{H^{\\prime }}\\sum _{h^{\\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })}\\Big )G(n)\\overline{G(n^{\\prime })},$ which will be easier to handle than the above expression for $A$ (and then use Remark REF to get rid of the squares).", "Clearly, to obtain an upper bound for (REF ), it suffices to obtain an upper bound for $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime }}\\Big |\\sum _{h^{\\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })} \\Big |.$ Again, it will be easier to deal with $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime 2}}\\Big |\\sum _{h^{\\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })} \\Big |^2$ (and use Remark REF to get rid of the squares).", "Expanding the square again we get that (REF ) is equal to $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })}\\overline{a(n+h^{\\prime \\prime })\\cdot \\overline{a(n^{\\prime }+h^{\\prime \\prime })}}=$ $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n+h^{\\prime \\prime })}a(n^{\\prime }+h^{\\prime \\prime })\\overline{a(n^{\\prime }+h^{\\prime })}.$ The sum in the last term by exchanging the order of summation is equal to $\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}\\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\overline{a(n+h^{\\prime \\prime })}\\Big |^2=$ $\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}\\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h^{\\prime \\prime }-h^{\\prime })}\\Big |^2+ {\\rm O}(\\frac{H^{\\prime 2}}{N_\\ell ^2}).$ Finally, grouping according to $h=h^{\\prime \\prime }-h^{\\prime }$ , we get that that the above is equal to $\\frac{1}{H^{\\prime 2}}\\sum _{|h|\\le H^{\\prime }}|H^{\\prime }-h|\\cdot \\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}\\Big |^2+{\\rm O}(\\frac{H^{\\prime 2}}{N_\\ell ^2})\\le $ $\\frac{1}{H^{\\prime }}\\cdot \\sum _{|h|\\le H^{\\prime }}\\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}\\Big |^2+{\\rm O}(\\frac{H^{\\prime 2}}{N_\\ell ^2}).$ That is, the expression from (REF ) equals $\\frac{1}{H^{\\prime }}\\sum _{|h|\\le H^{\\prime }}\\left|\\frac{\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}}{N_\\ell } \\right|^2 +{\\rm O}\\left(\\frac{(H^{\\prime })^2}{N_\\ell ^2} \\right).$ Now, by the assumption of our lemma, it follows that $\\frac{1}{H^{\\prime }}\\sum _{|h|\\le H^{\\prime }}\\left|\\frac{\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}}{N_\\ell } \\right|={\\rm o}(1),$ which, by Remark REF , is equivalent to $\\frac{1}{H^{\\prime }}\\sum _{|h|\\le H^{\\prime }}\\left|\\frac{\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}}{N_\\ell } \\right|^2={\\rm o}(1).$ Therefore,  (REF ) (and, thus, also (REF )) is of the order of ${\\rm o}(1)+{\\rm O}\\left(\\frac{(H^{\\prime })^2}{N_\\ell ^2} \\right)$ .", "Using again Remark REF , we conclude that also (REF ) is of the same order.", "It follows immediately that also the order of (REF ) is the same.", "Thus, we have proved that $A={\\rm o}\\left(1\\right)+{\\rm O}\\left(\\frac{(H^{\\prime })^2}{N_\\ell ^2} \\right)+{\\rm O}\\left(\\frac{H^{\\prime }}{N_\\ell } \\right)+{\\rm O}\\left(\\frac{(H^{\\prime })^k}{H}\\right).$ Acknowledgments: We would like to thank Tomasz Downarowicz, Nikos Frantzikinakis and Krzysztof Fra̧czek for useful discussions on the paper.", "Research of the second and third authors supported by Narodowe Centrum Nauki grant UMO-2019/33/B/ST1/00364.", "Department of Mathematics, The Maryland University [email protected] Faculty of Mathematics and Computer Science Nicolaus Copernicus University, Toruń, Poland [email protected], [email protected] Laboratoire de Mathématiques Raphaël Salem, CNRS – Université de Rouen Normandie Avenue de l’Université – 76801 Saint Étienne du Rouvray, France [email protected]" ], [ "Lifting lemma", "The purpose of this section is to prove Proposition REF , which is an alternative version of Conze-Downarowicz-Serafin lifting lemma from [9] and seems to be of independent interest.", "It may seem weaker than the original where the genericity was lifted to a single orbit, but the main advantage here is that we do not need assumptions on the nature of the second topological space: it does not have to be a full shift.", "The second advantage is that the result has its extension to the logarithmic case, see Appendix REF , while the lifting lemma of Conze-Downarowicz-Serafin and other results of that type so far have been proved for Cesàro averages.", "Proposition 3.1 Let $(Y,S)$ and $(X,T)$ be two topological systems and $u\\in Y$ generic along an increasing sequence $(N_m)$ for some $S$ -invariant measure $\\kappa $ on $Y$ .", "Let $\\rho $ be a joining of $\\kappa $ with a $T$ -invariant measure $\\nu $ on $X$ .", "Then there exist a sequence $(x_n)\\subset X$ and a subsequence $(N_{m_\\ell })$ such that: the sequence $(S^nu,x_n)$ is generic for $\\rho $ along $(N_{m_\\ell })$ : $ \\frac{1}{N_{m_\\ell }}\\sum _{0\\le n< N_{m_\\ell }}\\delta _{(S^nu,x_n)}\\xrightarrow[\\ell \\rightarrow \\infty ]{}\\rho , $ The sequence $(x_n)$ is constituted of longer and longer pieces of orbits.", "More precisely, $\\lbrace n\\ge 0:\\ x_{n+1}\\ne Tx_n\\rbrace $ is of the form $\\lbrace b_1<b_2<\\cdots <b_k<b_{k+1}<\\cdots \\rbrace $ , where $b_{k+1}-b_k\\rightarrow \\infty $ ." ], [ "Good sequences of partitions", "We need a convenient tool to estimate the weak*-convergence of a sequence of probability measures to a given measure.", "Definition 3.1 Let $(E,d)$ be a compact metric space, and let $\\nu $ be a Borel probability measure on $E$ , i.e.", "$\\nu \\in M(E)$ .", "We consider a sequence $({P}_\\ell )$ of finite partitions of $E$ into Borel subsets.", "The sequence $({P}_\\ell )$ is said to be good for $(E,\\nu )$ if the following conditions hold: for each $\\ell $ , ${P}_{\\ell +1}$ refines ${P}_\\ell $ , $\\mathop {\\mathrm {diam}}({P}_\\ell ):=\\max _{P\\text{ atom of }{P}_\\ell } \\mathop {\\mathrm {diam}}(P) \\xrightarrow[\\ell \\rightarrow \\infty ]{}0$ , for each $\\ell $ and each atom $P$ of ${P}_\\ell $ , $\\nu (\\partial P)=0$ .", "The motivation for introducing this definition comes from the following result.", "Lemma 3.2 If $({P}_\\ell )$ is a good sequence of partitions for $(E,\\nu )$ , then a sequence $(\\nu _n)\\subset M(E)$ converges to $\\nu $ in the weak*-topology if and only if, for each $\\ell $ and each atom $P$ of ${P}_\\ell $ , we have $\\nu _n(P) \\xrightarrow[n\\rightarrow \\infty ]{} \\nu (P).$ If $\\nu _n\\xrightarrow[n\\rightarrow \\infty ]{\\text{w}*} \\nu $ , then by the Portmanteau theorem, for each $P\\subset E$ such that $\\nu (\\partial P)=0$ , we have $\\nu _n(P)\\rightarrow \\nu (P)$ .", "Conversely, assume that for each $\\ell $ and each atom $P$ of ${P}_\\ell $ we have $\\nu _n(P)\\rightarrow \\nu (P)$ .", "Then any weak*-limit $\\mu $ of a subsequence of $(\\nu _n)$ satisfies (again by the Portmanteau theorem) $\\mu (P)=\\nu (P)$ for each atom $P$ of ${P}_\\ell $ .", "But since $\\mathop {\\mathrm {diam}}({P}_\\ell )\\rightarrow 0$ , the sequence $({P}_\\ell )$ separates points in $E$ , hence it generates the Borel $\\sigma $ -algebra of $E$ .", "Thus we have $\\mu =\\nu $ , and using the compactness of $M(E)$ for the weak* topology, we get that $\\nu _n\\xrightarrow[n\\rightarrow \\infty ]{\\text{w}*} \\nu $ .", "Lemma 3.3 For each $\\nu \\in M(E)$ of a compact metric space $(E,d)$ , there exists a good sequence of partitions for $(E,\\nu )$ .", "We first show that, for each $\\ell \\ge 1$ , there exists a finite partition ${Q}_\\ell $ in which each atom $Q$ satisfies $\\mathop {\\mathrm {diam}}(Q)<1/\\ell $ , $\\nu (\\partial Q)=0$ .", "Indeed, by compactness, there exists a finite set $\\lbrace x_1,\\ldots ,x_k\\rbrace \\subset E$ such that $ E\\subset \\bigcup _{1\\le i\\le k} B\\bigl (x_i,\\frac{1}{3\\ell }\\bigr ).", "$ Then, for each $1\\le i\\le k$ , there exist at most countably many $r>0$ such that $\\nu \\left(\\partial B(x_i,r)\\right) > 0.$ Therefore, we can find $r\\in \\left(\\frac{1}{3\\ell },\\frac{1}{2\\ell }\\right)$ such that $\\forall 1\\le i\\le k,\\quad \\nu \\left(\\partial B(x_i,r)\\right) = 0.$ Then the partition ${Q}_\\ell $ generated by the open balls $B(x_i,r)$ , $1\\le i\\le k$ , satisfies the required conditions.", "Once we have ${Q}_\\ell $ for each $\\ell \\ge 1$ , we set ${P}_\\ell :={Q}_1\\vee \\cdots \\vee {Q}_\\ell ,$ and we get a good sequence $({P}_\\ell )$ for $(E,\\nu )$ .", "Lemma 3.4 Let $({P}_\\ell )$ be a good sequence of partitions for $(E_1,\\nu _1)$ , and let $({Q}_\\ell )$ be a good sequence of partitions for $(E_2,\\nu _2)$ .", "Then for each coupling $\\rho $ of $\\nu _1$ and $\\nu _2$ , $({P}_\\ell \\times {Q}_\\ell )$ is a good sequence of partitions for $(E_1\\times E_2,\\rho )$ .", "This is obvious, since for each atom $P$ of ${P}_\\ell $ and each atom $Q$ of ${Q}_\\ell $ , $ \\partial (P\\times Q) \\subset (\\partial P\\times E_2) \\cup (E_1\\times \\partial Q), $ and the marginals of $\\rho $ are $\\nu _1$ and $\\nu _2$ ." ], [ "Proof of Proposition ", "Without loss of generality, we can (and we do) assume that the measure-theoretic dynamical system $(Y,\\kappa ,S)$ is aperiodic.", "Indeed, if this is not the case, we consider any uniquely ergodic topological system $(Y^{\\prime },S^{\\prime })$ whose unique invariant measure $\\kappa ^{\\prime }$ is such that $(Y^{\\prime },\\kappa ^{\\prime },S^{\\prime })$ is aperiodic.", "Then we take any point $u^{\\prime }\\in Y^{\\prime }$ , and we replace $Y$ by $Y\\times Y^{\\prime }$ , $S$ by $S\\times S^{\\prime }$ , and $u$ by $(u,u^{\\prime })$ .", "We also replace $(N_m)$ by a subsequence of $(N_m)$ along which $(u,u^{\\prime })$ is generic, for some measure $\\tilde{\\kappa }$ whose marginals have to be $\\kappa $ and $\\kappa ^{\\prime }$ .", "But then the system $(Y\\times Y^{\\prime },\\tilde{\\kappa },S\\times S^{\\prime })$ is aperiodic, because it is an extension of the aperiodic system $(Y^{\\prime },\\kappa ^{\\prime },S^{\\prime })$ .", "We fix a good sequence of partitions $({Q}_\\ell )$ for $(Y,\\kappa )$ and a good sequence of partitions $({P}_\\ell )$ for $(X,\\nu )$ .", "Then by Lemma REF , $({Q}_\\ell \\times {P}_\\ell )$ is a good sequence of partitions for $(Y\\times X,\\rho )$ .", "Definition 3.2 Let $M>0$ .", "A subset $E$ of ${\\mathbb {N}}$ is said to be $M$ -separated if for each integers $n\\ne m$ , $n,m\\in E\\Longrightarrow |n-m|\\ge M$ .", "The main argument to prove Proposition REF stands in the following proposition.", "Proposition 3.5 Under the assumptions of Proposition REF , and assuming also that $(Y,\\kappa ,S)$ is aperiodic (see above), given ${\\ell _0}\\ge 1$ and $\\varepsilon \\in (0,\\frac{1}{2})$ , there exists a sequence $(x_n)$ of points in $X$ such that: $\\lbrace n\\ge 0: x_{n+1}\\ne Tx_n\\rbrace $ is $\\frac{1}{\\varepsilon }$ -separated, for each atom $A$ of ${Q}_{\\ell _0}\\times {P}_{\\ell _0}$ , we have $\\rho (A)-\\varepsilon < \\liminf _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A }(S^nu,x_n),$ and $\\limsup _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A }(S^nu,x_n) < \\rho (A)+\\varepsilon .$ Let $h$ be a natural number such that $\\frac{1}{h}<\\varepsilon $ .", "We claim that for $\\ell $ large enough, we can find a set $B\\subset Y$ which is measurable with respect to $\\bigvee _{0\\le j\\le h-1}S^j{Q}_\\ell $ , and such that $B,SB,\\ldots ,S^{h-1}B$ are pairwise disjoint, $\\kappa \\left(\\bigcup _{0\\le j\\le h-1}S^jB\\right)>1-\\varepsilon $ .", "Indeed, since $(Y,\\kappa ,S)$ is assumed to be aperiodic, we can use the Rokhlin lemma to find a Borel subset $\\tilde{B}\\subset Y$ such that $\\tilde{B}, S\\tilde{B},\\ldots , S^{h-1}\\tilde{B}$ are pairwise disjoint, and such that $ \\kappa \\left(\\bigcup _{0\\le j\\le h-1}S^j\\tilde{B}\\right)>1-\\frac{\\varepsilon }{2}.", "$ Then we use the fact that the good sequence of partitions $({Q}_\\ell )$ generates the Borel $\\sigma $ -algebra: it follows that for $\\ell $ large enough, we can find a ${Q}_\\ell $ -measurable set $B^{\\prime }$ such that $ \\kappa (B^{\\prime }\\bigtriangleup \\tilde{B})< \\frac{\\varepsilon }{8h^2}.", "$ For each $1\\le j\\le h-1$ , we have $ B^{\\prime }\\cap S^jB^{\\prime } \\subset (B^{\\prime }\\setminus \\tilde{B})\\cup (S^jB^{\\prime }\\setminus S^j\\tilde{B}), $ hence $\\kappa (B^{\\prime }\\cap S^jB^{\\prime })\\le \\frac{\\varepsilon }{4h^2}.$ It remains to define $B$ by $ B:= B^{\\prime }\\setminus \\left(\\bigcup _{1\\le j\\le h-1} S^jB^{\\prime }\\right).", "$ Then, by construction, $B$ is disjoint from $S^jB$ for each $1\\le j\\le h-1$ , thus $B,SB,\\ldots ,S^{h-1}B$ are pairwise disjoint.", "Moreover, from (REF ), we have $\\kappa (B) \\ge \\kappa (B^{\\prime })-\\frac{\\varepsilon }{4h}\\ge \\kappa (\\tilde{B})-\\frac{\\varepsilon }{2h},$ which implies $ \\kappa \\left(\\bigcup _{0\\le j\\le h-1}S^j B\\right) = h\\kappa (B) \\ge h\\kappa (\\tilde{B})-\\frac{\\varepsilon }{2}>1-\\varepsilon , $ and our first claim is proved.", "Since $u$ is generic for $\\kappa $ along $(N_m)$ , and since the set $\\bigcup _{0\\le j\\le h-1}S^jB$ is measurable with respect to $\\bigvee _{0\\le j\\le 2h}S^j{Q}_\\ell $ (in particular, the $\\kappa $ -measure of its boundary vanishes), we have $\\frac{1}{N_m} \\sum _{0\\le n< N_m} {1}_{\\bigcup _{0\\le j\\le h-1}S^jB} (S^n u) \\xrightarrow[m\\rightarrow \\infty ]{} \\kappa \\left(\\bigcup _{0\\le j\\le h-1}S^jB\\right) > 1-\\varepsilon .$ This implies in particular that the set $P_B(u):=\\lbrace n\\ge 0:\\ S^nu\\in B\\rbrace $ is infinite.", "We number in order the elements of this set: $ P_B(u) = \\lbrace b_1<b_2<\\cdots <b_k<\\cdots \\rbrace \\; $ The integers $(b_k)$ will correspond to the times when we will be allowed to change the orbit of the desired sequence.", "As $B$ is disjoint from $S^jB$ for each $1\\le j\\le h-1$ , the set $P_B(u)$ is $h$ -separated, hence $\\frac{1}{\\varepsilon }$ -separated.", "We consider the partition $\\bigvee _{0\\le j\\le h-1}(S\\times T)^{-j}({Q}_{\\ell _0}\\times {P}_{\\ell _0})$ of $Y\\times X$ .", "Any atom of this partition is of the form ${\\bar{Q}}\\times {\\bar{P}}$ , where ${\\bar{Q}}$ (respectively ${\\bar{P}}$ ) is an atom of $\\bigvee _{0\\le j\\le h-1}S^{-j}{Q}_{\\ell _0}$ (respectively of $\\bigvee _{0\\le j\\le h-1}T^{-j}{P}_{\\ell _0}$ ).", "For such atoms ${\\bar{Q}}$ and ${\\bar{P}}$ , we can write ${\\bar{Q}}=Q_0\\cap S^{-1}Q_1\\cap \\cdots \\cap S^{-(h-1)}Q_{h-1},$ each $Q_j$ being an atom of ${Q}_{\\ell _0}$ , and ${\\bar{P}}=P_0\\cap S^{-1}P_1\\cap \\cdots \\cap S^{-(h-1)}P_{h-1},$ each $P_j$ being an atom of ${P}_{\\ell _0}$ .", "Since the $\\kappa $ -measure of the boundary of each involved set is always 0, we again have for each atom ${\\bar{Q}}$ of $\\bigvee _{0\\le j\\le h-1}S^{-j}{Q}_{\\ell _0}$ $\\frac{1}{N_m} \\sum _{b_k< N_m} {1}_{{\\bar{Q}}} (S^{b_k} u) =\\frac{1}{N_m} \\sum _{0\\le n< N_m} {1}_{B\\cap {\\bar{Q}}} (S^n u)\\xrightarrow[m\\rightarrow \\infty ]{} \\kappa (B\\cap {\\bar{Q}}).$ If $C$ is a measurable subset of $Y$ with $\\kappa (C)>0$ , we denote by $\\rho ^Y_C$ the marginal on $X$ of the conditional probability measure $\\rho (\\,\\cdot \\,|C\\times X)$ .", "Then, for each measurable $A\\subset X$ , we have $\\begin{split}\\rho (C\\times A) &=\\rho \\bigl ( (C\\times X) \\cap (Y\\times A) \\bigr ) \\\\&=\\rho (C\\times X) \\, \\rho \\bigl (Y\\times A|C\\times X\\bigr )\\\\&=\\kappa (C) \\, \\rho ^Y_C(A).\\end{split}$ On an appropriate probability space, we construct a sequence $(\\xi _k)$ of independent random variables, taking values in $X$ , such that for each $k$ , $\\xi _k$ is distributed according to $\\rho ^Y_{B\\cap {\\bar{Q}}}$ , where ${\\bar{Q}}$ is the atom of $\\bigvee _{0\\le j\\le h-1}S^{-j}{Q}_{\\ell _0}$ containing $S^{b_k}u$ .", "For each atom ${\\bar{Q}}$ of $\\bigvee _{0\\le j\\le h-1}S^{-j}{Q}_{\\ell _0}$ and each atom ${\\bar{P}}$ of $\\bigvee _{0\\le j\\le h-1}T^{-j}{P}_{\\ell _0}$ , by (REF ), the law of large numbers and (REF ), with probability 1, we have $\\frac{1}{N_m} \\sum _{b_k< N_m} {1}_{{\\bar{Q}}} (S^{b_k} u) {1}_{{\\bar{P}}} (\\xi _k) \\xrightarrow[m\\rightarrow \\infty ]{} \\kappa (B\\cap {\\bar{Q}}) \\rho ^Y_{B\\cap {\\bar{Q}}}({\\bar{P}}) = \\rho \\bigl ((B\\cap {\\bar{Q}})\\times {\\bar{P}}\\bigr ).$ Let us fix a realization of $(\\xi _k)$ which satisfies (REF ) for each atom ${\\bar{Q}}\\times {\\bar{P}}$ of $\\bigvee _{0\\le j\\le h-1}(S\\times T)^{-j}({Q}_{\\ell _0}\\times {P}_{\\ell _0})$ .", "Then, for each $n\\ge 0$ , we define the point $x_n\\in X$ as follows: $x_n := {\\left\\lbrace \\begin{array}{ll}T^{n-b_1}\\xi _1 &\\text{ if }n<b_1,\\\\T^{n-b_k}\\xi _k &\\text{ if }b_k\\le n<b_{k+1}\\text{ for some }k\\ge 1.\\end{array}\\right.", "}$ The set of integers $n$ such that $x_{n+1}\\ne Tx_n$ is contained in $P_B(u)$ , therefore, it is $\\frac{1}{\\varepsilon }$ -separated.", "Now, let $A=Q\\times P$ be a fixed atom of ${Q}_{\\ell _0}\\times {P}_{\\ell _0}$ .", "We set $R:=\\bigcup _{0\\le j\\le h-1}S^jB\\times X,$ and we observe that $\\rho (R) = \\kappa \\left(\\bigcup _{0\\le j\\le h-1}S^jB\\right) >1-\\varepsilon .$ We also note that for each $n\\ge b_1$ , $(S^nu,x_n)\\in R$ if and only if there exists $k$ and $0\\le j\\le h-1$ such that $n=b_k+j$ .", "In this case, $(S^nu,x_n)\\in A\\cap R$ if and only if the atom ${\\bar{Q}}\\times {\\bar{P}}$ of $\\bigvee _{0\\le j\\le h-1}(S\\times T)^{-j}({Q}_{\\ell _0}\\times {P}_{\\ell _0})$ containing $(S^{b_k}u,\\xi _k)$ satisfies $Q_j=Q$ and $P_j=P$ (using the notations given in (REF ) and (REF ), and remembering that $A=Q\\times P$ ).", "We can then use (REF ) to get $\\begin{split}&\\frac{1}{N_m} \\sum _{b_1\\le n<N_m} {1}_{A\\cap R}(S^nu,x_n) \\\\&= \\frac{1}{N_m} \\sum _{b_k<N_m} \\sum _{0\\le j\\le h-1} \\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} {1}_{{\\bar{Q}}\\times {\\bar{P}}}(S^{b_k}u,\\xi _k)\\\\&\\xrightarrow[m\\rightarrow \\infty ]{} \\sum _{0\\le j\\le h-1} \\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} \\rho \\bigl ((B\\cap {\\bar{Q}})\\times {\\bar{P}}\\bigr ).\\end{split}$ But, on the other hand, we can write $\\begin{split}\\rho (A\\cap R) &= \\sum _{0\\le j\\le h-1} \\rho \\bigl (A\\cap (S^jB\\times X)\\bigr )\\\\&= \\sum _{0\\le j\\le h-1} \\rho \\bigl ((S^{-j}Q\\times T^{-j}P)\\cap (B\\times X)\\bigr )\\\\&= \\sum _{0\\le j\\le h-1} \\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} \\rho \\bigl ((B\\cap {\\bar{Q}})\\times {\\bar{P}}\\bigr ).\\end{split}$ From (REF ) and (REF ), it follows that $\\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A\\cap R}(S^nu,x_n) \\xrightarrow[m\\rightarrow \\infty ]{} \\rho (A\\cap R).$ From (REF ), we get that $\\lim _{m\\rightarrow \\infty }\\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{(Y\\times X)\\setminus R}(S^nu,x_n)<\\varepsilon ,$ and since ${1}_{A}\\le {1}_{A\\cap R}+{1}_{Y\\times X\\setminus R}$ , this yields by (REF ), $\\limsup _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A }(S^nu,x_n)&< \\lim _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A\\cap R}(S^nu,x_n) + \\varepsilon \\\\&=\\rho (A\\cap R)+ \\varepsilon \\\\&\\le \\rho (A)+\\varepsilon ,$ and we have (REF ).", "On the other hand, using ${1}_{A}\\ge {1}_{A\\cap R}$ , we get by (REF ) $\\liminf _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A }(S^nu,x_n)&\\ge \\lim _{m\\rightarrow \\infty } \\frac{1}{N_m} \\sum _{0\\le n<N_m} {1}_{A\\cap R}(S^nu,x_n) \\\\&=\\rho (A\\cap R) \\\\&>\\rho (A)-\\varepsilon .$ and we have (REF ).", "We can now give the proof of Proposition REF , in which we use the following obvious fact: if we modify the sequence $(x_n)$ given by Proposition REF on a finite number of terms, we still get (REF ) and (REF ).", "[Proof of Proposition (REF )] We fix a sequence $(\\varepsilon _\\ell )_{\\ell \\ge 1}$ of numbers in $(0,\\frac{1}{2})$ , decreasing to 0, and we construct inductively the desired sequence $(x_n)$ and the subsequence $(N_{m_\\ell })$ by a repeated use of Proposition REF .", "We start by applying Proposition REF with $\\varepsilon :=\\varepsilon _1$ and ${\\ell _0}:=1$ .", "It provides us with an integer $m_1$ , and a finite sequence $(x_n)_{0\\le n<N_{m_1}}$ of points in $X$ such that the set of integers $n\\in \\lbrace 0,\\ldots ,N_{m_1}-2\\rbrace $ such that $x_{n+1}\\ne Tx_n$ is $\\frac{1}{\\varepsilon _1}$ -separated, for each atom $A$ of ${Q}_1\\times {P}_1$ , we have $\\rho (A)-\\varepsilon _1<\\frac{1}{N_{m_1}} \\sum _{0\\le n<N_{m_1}} {1}_{A}(S^nu,x_n) < \\rho (A)+\\varepsilon _1.$ Now, assume that for some $\\ell \\ge 1$ we have already constructed $m_1<\\cdots <m_\\ell $ and the sequence $(x_n)_{0\\le n<N_{m_\\ell }}$ of points in $X$ such that for each $1\\le j<\\ell $ , the set of integers $n\\in \\lbrace N_{m_{j-1}},\\ldots ,N_{m_j}-2\\rbrace $ such that $x_{n+1}\\ne Tx_n$ is $\\frac{1}{\\varepsilon _j}$ -separated (with the convention that $N_{m_0}=0$ ), for each atom $A$ of ${Q}_\\ell \\times {P}_\\ell $ , we have $\\rho (A)-\\varepsilon _\\ell <\\frac{1}{N_{m_\\ell }} \\sum _{0\\le n<N_{m_\\ell }} {1}_{A}(S^nu,x_n) < \\rho (A)+\\varepsilon _\\ell .$ Then we apply again Proposition REF , with $\\varepsilon :=\\varepsilon _{\\ell +1}$ and ${\\ell _0}:=\\ell +1$ .", "It provides us with an integer $m_{\\ell +1}$ and a finite sequence of points $(x_n)_{N_{m_\\ell }\\le n<N_{m_{\\ell +1}}}$ in $X$ which satisfy: the set of integers $n\\in \\lbrace N_{m_\\ell },\\ldots ,N_{m_{\\ell +1}}-2\\rbrace $ such that $x_{n+1}\\ne Tx_n$ is $\\frac{1}{\\varepsilon _{\\ell +1}}$ -separated, for each atom $A$ of ${Q}_{\\ell +1}\\times {P}_{\\ell +1}$ , we have $\\rho (A)-\\varepsilon _{\\ell +1}<\\frac{1}{N_{m_{\\ell +1}}} \\sum _{0\\le n<N_{m_{\\ell +1}}} {1}_{A}(S^nu,x_n) < \\rho (A)+\\varepsilon _{\\ell +1}.$ (We keep the points $(x_n)_{0\\le n<N_{m_\\ell }}$ already provided by the induction hypothesis, refering to the obvious fact stated before the proof.)", "Moreover, we can assume that the sequence $(\\varepsilon _\\ell )$ decreases sufficiently fast so that the validity of (REF ) for each atom $A$ of ${Q}_{\\ell +1}\\times {P}_{\\ell +1}$ ensures the validity of the analog inequalities for each $A$ which is a finite union of atoms of ${Q}_{\\ell +1}\\times {P}_{\\ell +1}$ (in particular, for each $A$ which is an atom of the previous partitions), but with $\\varepsilon _\\ell $ instead of $\\varepsilon _{\\ell +1}$ .", "The sequence $(x_n)_{n\\ge 0}$ of points in $X$ and the subsequence $(N_{m_\\ell })$ we construct with the above inductive procedure then satisfy the conditions announced in Proposition REF ." ], [ "Logarithmic case", "We would like to study the logarithmic version of Proposition REF , in which we replace each arithmetic average of the form $\\frac{1}{N_m}\\sum _{0\\le n< N_m} f(n)$ by the logarithmic average $\\frac{1}{L(N_m)}\\sum _{1\\le n\\le N_m} \\frac{1}{n}f(n).$ (Here we use the notation $L(N):=1+\\frac{1}{2}+\\cdots +\\frac{1}{N}$ .)", "In fact, this logarithmic version, whose statement is written below, is also valid, and the arguments to prove it are exactly the same as in the arithmetic average case.", "We just point out below the few technical changes that need to be made in the proof for the logarithmic case.", "Proposition 3.6 Let $(Y,S)$ and $(X,T)$ be two topological systems and $u\\in Y$ , logarithmically generic along an increasing sequence $(N_m)$ for some $S$ -invariant measure $\\kappa $ on $Y$ .", "Let $\\rho $ be a joining of $\\kappa $ with a $T$ -invariant measure $\\nu $ on $X$ .", "Then there exist a sequence $(x_n)\\subset X$ and a subsequence $(N_{m_\\ell })$ such that: the sequence $(S^nu,x_n)$ is logarithmically generic for $\\rho $ along $(N_{m_\\ell })$ : $ \\frac{1}{L(N_{m_\\ell })}\\sum _{1\\le n\\le N_{m_\\ell }}\\frac{1}{n}\\delta _{(S^nu,x_n)}\\xrightarrow[\\ell \\rightarrow \\infty ]{}\\rho , $ the set $\\lbrace n\\ge 0:\\ x_{n+1}\\ne Tx_n\\rbrace $ is of the form $\\lbrace b_1<b_2<\\cdots <b_k<b_{k+1}<\\cdots \\rbrace $ , where $b_{k+1}-b_k\\rightarrow \\infty $ .", "The changes that need to be made to the proof are almost all quite obvious, they consist in formally replacing the arithmetic average by the logarithmic average.", "One point maybe needs some explanations, namely when we arrive at the proof of the logarithmic analog of (REF ).", "We put these explanations in the form of a lemma, which we will apply in the following context: $(d_k)$ is the ordered sequence of positive integers $n$ such that $S^nu\\in B\\cap {\\bar{Q}}$ , and the sequence $(\\rho _k)$ is defined by $\\rho _k:={1}_{{\\bar{P}}}(\\xi _k)$ .", "Lemma 3.7 Let $(d_k)$ be an increasing sequence of positive integers such that $\\frac{1}{L(N_{m})}\\sum _{d_k\\le N_{m}}\\frac{1}{d_k} \\xrightarrow[m\\rightarrow \\infty ]{} \\kappa \\in [0,1],$ and let $(\\rho _k)$ be a sequence of real numbers in $[0,1]$ such that $\\frac{1}{K} \\sum _{1\\le k\\le K} \\rho _k \\xrightarrow[K\\rightarrow \\infty ]{} \\rho \\in [0,1].$ Then we have $\\frac{1}{L(N_{m})}\\sum _{d_k\\le N_{m}}\\frac{\\rho _k}{d_k} \\xrightarrow[m\\rightarrow \\infty ]{} \\kappa \\rho .$ For each $m$ , let us denote by $k_m$ the largest $k$ such that $d_k\\le N_m$ .", "We use the classical identity $\\sum _{d_k\\le N_{m}} \\frac{\\rho _k}{d_k} = \\sum _{1\\le k < k_m} \\left( \\frac{1}{d_k}-\\frac{1}{d_{k+1}} \\right) (\\rho _1+\\cdots +\\rho _k) + \\frac{1}{d_{k_m}}(\\rho _1+\\cdots +\\rho _{k_m}).$ Given $\\varepsilon >0$ , let $K_\\varepsilon $ be such that $ K\\ge K_\\varepsilon \\Longrightarrow \\left| \\frac{1}{K} \\sum _{1\\le k\\le K} \\rho _k - \\rho \\right| < \\varepsilon .", "$ We can then write $&\\left| \\frac{1}{L(N_{m})} \\sum _{d_k\\le N_{m}} \\frac{\\rho _k}{d_k} - \\frac{1}{L(N_{m})} \\sum _{d_k\\le N_{m}} \\frac{\\rho }{d_k}\\right|\\\\&= \\left| \\frac{1}{L(N_{m})} \\sum _{K_\\varepsilon \\le k < k_m} k \\left( \\frac{1}{d_k}-\\frac{1}{d_{k+1}} \\right) \\left( \\frac{1}{k} (\\rho _1+\\cdots +\\rho _k)-\\rho \\right) \\right| + O\\left(\\frac{1}{L(N_{m})}\\right)\\\\&< \\varepsilon + O\\left(\\frac{1}{L(N_{m})}\\right).$ But by assumption, we have $\\frac{1}{L(N_{m})} \\sum _{d_k\\le N_{m}} \\frac{\\rho }{d_k}\\xrightarrow[m\\rightarrow \\infty ]{} \\kappa \\rho ,$ hence we get $\\frac{1}{L(N_{m})} \\sum _{d_k\\le N_{m}} \\frac{\\rho _k}{d_k}\\xrightarrow[m\\rightarrow \\infty ]{} \\kappa \\rho .$ The last place in the proof where a (very easy) correction should be made in the logarithmic case is to get the analog of (REF ): at some point we have to replace some coefficients $\\frac{1}{b_k+j}$ by $\\frac{1}{b_k}$ , which is of no consequence since $j$ remains bounded between 0 and $h-1$ here.", "To be more precise, (REF ) becomes $\\begin{split}&\\frac{1}{L(N_m)} \\sum _{b_1\\le n<N_m} \\frac{1}{n}{1}_{A\\cap R}(S^nu,x_n) \\\\&= \\frac{1}{L(N_m)} \\sum _{b_k<N_m} \\sum _{0\\le j\\le h-1} \\frac{1}{b_k+j}\\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} {1}_{{\\bar{Q}}\\times {\\bar{P}}}(S^{b_k}u,\\xi _k)\\\\&= \\frac{1}{L(N_m)} \\sum _{b_k<N_m} \\sum _{0\\le j\\le h-1} \\frac{1}{b_k}\\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} {1}_{{\\bar{Q}}\\times {\\bar{P}}}(S^{b_k}u,\\xi _k) + o(1)\\\\&\\xrightarrow[m\\rightarrow \\infty ]{} \\sum _{0\\le j\\le h-1} \\sum _{({\\bar{Q}},{\\bar{P}}):\\ Q_j=Q\\text{ and }P_j=P} \\rho \\bigl ((B\\cap {\\bar{Q}})\\times {\\bar{P}}\\bigr ).\\end{split}$" ], [ "Proof of Theorem ", "[Proof of Theorem REF ] Take any topological system $(X,T)\\in {C}_{{{F}}}$ and fix $f\\in C(X)$ , $x\\in X$ .", "Take any increasing sequence $(N_k)$ for which, with no loss of generality, we can assume that $\\frac{1}{N_k}\\sum _{n\\le N_k} \\delta _{(T^nx,S^nu)}\\rightarrow \\rho $ .", "It follows that $\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}f(T^nx)u(n)=\\int f\\otimes \\pi _0\\,d\\rho .$ But $\\rho $ is a joining of some $T$ -invariant measure $\\nu \\in V(X,T)$ for which $x$ is generic along $(N_k)$ , and some Furstenberg system $\\kappa $ of $u$ .", "Since $(X,T)\\in {C}_{{{F}}}$ , the system $(X,\\nu ,T)$ is in ${{F}}$ , and the integral on the right-hand side above vanishes by the Veech condition and Proposition REF ." ], [ "Proof of Theorem ", "Before we begin the proof, let us make the following remark concerning topological models.", "Given an automorphism $(Z,\\mathcal {D},\\kappa , R)$ , and a fixed subset of full measure of ergodic components of $\\kappa $ , recall that by a Hansel model of $R$ , we mean a topological system $(X,T)$ which has a $T$ -invariant measure $\\nu $ such that, as dynamical systems, $\\kappa $ and $\\nu $ are isomorphic and such that each point $x\\in X$ is generic for one of these chosen ergodic components.", "In [29], it is proved that each automorphism has a Hansel model.", "We assume that $u\\perp {C}_{{{F}}_{\\rm ec}}$ for some characteristic class ${{F}}$ .", "Take $\\kappa \\in V(u)$ and fix $(N_m)$ so that $\\frac{1}{N_m}\\sum _{n\\le N_m} \\delta _{S^nu}\\rightarrow \\kappa .$ Denote by ${\\cal A}(\\kappa )\\subset {\\cal B}(X_{u})$ the largest ${{F}}_{\\rm ec}$ -factor of $(X_{u},\\kappa ,S)$ , i.e.", "${\\cal A}(\\kappa )={\\cal B}(X_{u})_{{{F}}_{\\rm ec}}$ .", "Consider the factor $(X_{u}/{\\cal A}(\\kappa ), {\\cal A}(\\kappa ), \\kappa |_{{\\cal A}(\\kappa )},S)$ and take a Hansel model $(X,\\nu ,T)$ of it (by choosing in the ergodic decomposition of $\\kappa |_{{\\cal A}(\\kappa )}$ only ergodic measures in ${{F}}$ ).", "By definition, $(X,T)\\in {C}_{{{F}}_{\\rm ec}}.$ Fix a measure-theoretic factor map $J\\colon (X_{u},\\kappa ,S)\\rightarrow (X,\\nu ,T)$ such that $J^{-1}({\\cal B}(X))={\\cal A}(\\kappa )$ , and let $\\nu _J$ denote the corresponding graph joining (of $\\nu $ and $\\kappa |_{{\\cal A}(\\kappa )}$ ).", "Let $\\widehat{\\nu }_J$ be the relatively independent extension of $\\nu _J$ to a joining of $\\nu $ and $\\kappa $ : for $f\\in L^2(\\nu )$ and $g\\in L^2(\\kappa )$ , we have $\\int _{X_{u}\\times X} g\\otimes f \\, d\\widehat{\\nu }_J = \\int _{X_{u}} {\\mathbb {E}}^{\\kappa }\\bigl (g | {\\cal A}(\\kappa )\\bigr )\\,f\\circ J \\, d\\kappa .$ Now, by applying Proposition REF , we can find $(x_n)\\subset Y$ such that $((x_n),u) \\text{ is generic for }\\widehat{\\nu }_J\\text{ along some subsequence }(N_{m_\\ell }),$ and the set $\\lbrace n\\ge 0:\\ x_{n+1}\\ne Tx_n\\rbrace $ is of the form $\\lbrace b_1<b_2<\\cdots \\rbrace $ with $b_{k+1}-b_k\\rightarrow \\infty $ .", "Since $u\\perp {C}_{{{F}}_{\\rm ec}}$ , (REF ) and Proposition REF ensure that the system $(Y,S)$ satisfies the strong $u$ -MOMO property.", "Therefore, for each $f\\in C(Y)$ we have $\\lim _{m\\rightarrow \\infty }\\frac{1}{N_m}\\sum _{n\\le N_m}f(x_n)u(n)=$ $\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{b_{K_m}}\\sum _{k<K_m}\\left(\\sum _{b_k\\le n<b_{k+1}}f(T^{n-b_k}x_{b_k})u(n)\\right)=0,$ and it follows from (REF ) that $\\int f\\otimes \\pi _0\\, d \\widehat{\\nu }_J=0$ .", "Using (REF ), we get $\\int _{X_{u}} {\\mathbb {E}}^{\\kappa }\\bigl (\\pi _0 | {\\cal A}(\\kappa )\\bigr )\\,f\\circ J \\, d\\kappa =0.$ But $\\lbrace f\\circ J:\\ f\\in C(X)\\rbrace $ is dense in $L^2({\\cal A}(\\kappa ))$ and therefore $\\pi _0\\perp L^2({\\cal A}(\\kappa ))$ , which is the Veech condition for $u$ with respect to the characteristic class ${{F}}_{\\rm ec}$ ." ], [ "Cancellations. Proof of Corollaries ", "We need the following interpretation of the Veech condition in terms of relative uniform mixing (K-mixing) of the function $\\pi _0$ .", "For $n\\in {\\mathbb {N}}$ , let $\\pi _n:=\\pi _0\\circ S^n$ .", "Proposition 5.1 Let $\\kappa \\in V_S(u)$ and $\\int \\pi _0\\,d\\kappa =0$ .", "Then the following conditions are equivalent: $\\pi _0\\perp L^2(\\Pi (\\kappa ))$ , $\\pi _0$ is relatively K-mixing, i.e.", "for each $\\varepsilon >0$ , there exists $N$ such that $\\left|\\int \\pi _0 {1}_C\\,d\\kappa -\\int \\pi _0\\, d\\kappa \\int {1}_C d\\kappa \\right|=\\left| \\int \\pi _0 {1}_C\\,d\\kappa \\right|<\\varepsilon $ for each set $C\\in \\sigma (\\pi _n,\\pi _{n+1},...)$ and $n\\ge N$ .", "If we additionally assume that $u$ takes values in a finite set $E\\subset and $ (Mk)$ is a sequence along which we have a Furstenberg system $$ then the above conditions are equivalent to\\begin{enumerate}\\item [(c)]for each \\varepsilon >0 there exists N\\ge 1 such that for any s\\ge 1 and any function fdepending on coordinates N\\le n,n+1,\\ldots ,n+s, \\Vert f\\Vert _{C(X_{u})}\\le 1, we have\\limsup _{k\\rightarrow \\infty }\\left|\\frac{1}{M_k}\\sum _{m\\le M_k}u(m)f(S^mu)\\right|<\\varepsilon .\\end{enumerate}$ (a) $\\Rightarrow $ (b).", "Assume that ${\\mathbb {E}}(\\pi _0|\\Pi (\\kappa ))=0$ .", "Let $C\\in \\sigma (\\pi _n,\\pi _{n+1},\\ldots )$ .", "We have $\\left|\\int \\pi _0 {1}_C d\\kappa \\right|=\\left|\\int {1}_C {\\mathbb {E}}(\\pi _0|\\sigma (\\pi _n,\\pi _{n+1},...)\\, d\\kappa \\right|\\\\\\le \\int \\left|{\\mathbb {E}}(\\pi _0|\\sigma (\\pi _n,\\pi _{n+1},\\ldots ))\\right|\\, d\\kappa .$ Hence, we have an upper bound which does not depend on $C$ .", "Since ${\\mathbb {E}}(\\pi _0|\\sigma (\\pi _n,\\pi _{n+1},...))\\rightarrow {\\mathbb {E}}(\\pi _0|\\Pi (\\kappa ))=0$ $\\kappa $ -a.e.", "and thus also in $L^1$ , which is precisely the relative K-mixing for $\\pi _0$ .", "(b) $\\Rightarrow $ (a).", "Suppose that $\\pi _0$ is relatively K-mixing.", "Then, in particular, we have (REF ) for each $C\\in \\Pi (\\kappa )$ .", "In fact, since $\\varepsilon >0$ is arbitrary, $\\int \\pi _0 {1}_C\\,d\\kappa =0$ for each $C\\in \\Pi (\\kappa )$ .", "Whence ${\\mathbb {E}}(\\pi _0|\\Pi (\\kappa ))=0$ .", "(a) $\\Rightarrow $ (c).", "Since $\\frac{1}{M_k}\\sum _{m\\le M_k}u(m)f(S^mu)=\\frac{1}{M_k}\\sum _{m\\le M_k}(\\pi _0f)(S^mu)\\rightarrow \\int _{X_{u}} \\pi _0 f\\, d\\kappa ,$ we can repeat the same argument as was used to prove (a) $\\Rightarrow $ (b) (replacing ${1}_C$ by $f$ ).", "(c) $\\Rightarrow $ (b).", "Suppose that $\\left|\\int _{X_{u}}\\pi _0 f\\,d\\kappa \\right|<\\varepsilon $ , whenever $f$ depending on coordinates $n,n+1,\\ldots ,n+s$ with $n\\ge N$ is bounded by 1.", "Consider all blocks on coordinates $n,n+1,\\ldots ,n+s$ that is all $B=\\lbrace x\\in X_{u}:\\: x_n=b_0,\\ldots ,x_{n+s}=b_s\\rbrace $ with $b_j\\in E$ .", "Let $C$ be any union of such blocks.", "Then ${1}_C$ is a (continuous) function depending on coordinates $n,\\ldots ,n+s$ and is bounded by 1 and, by assumption, $\\left|\\int _{X_{u}}\\pi _0 {1}_C\\,d\\kappa \\right|<\\varepsilon .$ Note that with $N$ fixed and $s$ arbitrary, the family of $C$ defined above is dense in the $\\sigma $ -algebra $\\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ .", "Hence, given $D\\in \\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ and $\\varepsilon >0$ , we first find $s\\ge 0$ and then $C$ as above (a union of blocks “sitting” on coordinates $N,\\ldots ,N+s$ ) such that $\\kappa (C\\triangle D)<\\varepsilon $ and find that $\\left|\\int \\pi _0 {1}_D\\,d\\kappa \\right|\\le \\left|\\int \\pi _0 {1}_C\\,d\\kappa \\right|+\\kappa (C\\triangle D)<2\\varepsilon .$ Now, since each clopen set is a finite union of cylinders of a fixed length, Corollary REF follows directly by the above proposition.", "Corollary REF is a special case of Corollary REF ." ], [ "Conditional cancellations. Remark ", "The “cancellation law” of the values of $u$ along large shifts of the return times to a block (for most of the blocks) claimed in Remark REF is a consequence of a refinement of Proposition REF .", "Proposition 5.2 Let $\\kappa \\in V_S(u)$ and $\\int \\pi _0\\,d\\kappa =0$ .", "Then the following conditions are equivalent: $\\pi _0\\perp L^2(\\Pi (\\kappa ))$ , for each $\\varepsilon >0$ there exist $N\\ge 1$ and $L\\ge 1$ such that for each $\\ell \\ge L$ there is a family of “good” $\\ell $ -blocks $C\\in \\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ , i.e.", "of blocks satisfying $\\left|\\int {1}_C\\cdot \\pi _0\\,d\\kappa \\right|\\le \\kappa (C)\\varepsilon ,$ whose measure is $>1-\\varepsilon $ .", "In other words, for a “good” $\\ell $ -block $C$ , $\\left|\\int \\pi _0\\,d\\kappa _C\\right|<\\varepsilon $ , where $\\kappa _C$ stands for the conditional measure on $C$ .", "(a) $\\Rightarrow $ (d).", "Fix $\\varepsilon >0$ and note that ${\\mathbb {E}}^\\kappa (\\pi _0|\\sigma (\\pi _N,\\pi _{N+1},\\ldots ))\\rightarrow 0$ $\\kappa $ -a.e.", "This implies convergence in measure, i.e., we can find a set $A_\\varepsilon $ of measure at least $1-\\varepsilon $ such that for $N$ large enough, $|{\\mathbb {E}}^\\kappa (\\pi _0|\\sigma (\\pi _N,\\pi _{N+1},\\ldots ))(x)|<\\varepsilon \\text{ for all }x\\in A_{\\varepsilon }.$ Fix such an $N$ .", "There is $M\\ge 1$ large enough such that $\\kappa (A_{\\varepsilon }\\triangle A^{(M)}_{\\varepsilon })<\\varepsilon ,$ where $A^{(M)}_\\varepsilon \\in \\sigma (\\pi _{-M},\\pi _{-M+1},\\ldots )$ and note that $S^{N+M}A^{(M)}_\\varepsilon \\in \\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ .", "Now, for $\\ell $ large enough, we can approximate $S^{N+M}A^{(M)}_{\\varepsilon }$ by a (disjoint) union of $\\ell $ -blocks belonging to $\\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ , $\\kappa (S^{N+M}A^{(M)}_{\\varepsilon }\\triangle \\bigcup _{j\\in J} C^{(\\ell )}_j)<\\varepsilon .$ But $\\kappa (S^{N+M}A^{(M)}_{\\varepsilon }\\triangle A_{\\varepsilon })<2\\varepsilon $ , so $\\kappa (\\bigcup _{j\\in J} C^{(\\ell )}_j \\setminus A_{\\varepsilon })\\le \\kappa (A_{\\varepsilon }\\triangle \\bigcup _{j\\in J} C^{(\\ell )}_j)<3\\varepsilon .$ Consider those $j\\in J$ for which $\\kappa (C^{(\\ell )}_j\\setminus A_{\\varepsilon })\\ge \\sqrt{\\varepsilon }\\kappa (C_j^{(\\ell )})$ .", "Then the measure $m$ of the union of such blocks has to satisfy $\\sqrt{\\varepsilon } m<3\\varepsilon $ , so $m<3\\sqrt{\\varepsilon }$ .", "In other words “most” (in measure) of the $C_j^{(\\ell )}$ 's are “good”, i.e.", "they satisfy $\\kappa (C^{(\\ell )}_j\\cap A_{\\varepsilon })>(1-3\\sqrt{\\varepsilon })\\kappa (C^{(\\ell )}_j)$ .", "Take such a “good” $C$ .", "We have $\\left|\\int {1}_C\\cdot \\pi _0\\,d\\kappa \\right|\\le \\int {1}_{C}|{\\mathbb {E}}^\\kappa (\\pi _0|\\sigma (\\pi _N,\\pi _{N+1},\\ldots ))|\\,d\\kappa \\\\=\\int _{A_{\\varepsilon }}{1}_{C}|{\\mathbb {E}}^\\kappa (\\pi _0|\\sigma (\\pi _N,\\pi _{N+1},\\ldots ))|\\,d\\kappa +\\int _{A_{\\varepsilon }^c}{1}_{C}|{\\mathbb {E}}^\\kappa (\\pi _0|\\sigma (\\pi _N,\\pi _{N+1},\\ldots ))|\\,d\\kappa \\\\\\le \\varepsilon \\kappa (C)+3\\sqrt{\\varepsilon }\\kappa (C).$ (d) $\\Rightarrow $ (a).", "Fix $A\\in \\Pi (\\kappa )$ of positive measure $\\kappa $ .", "Then for $\\varepsilon >0$ , we can find $N$ such that for all $\\ell $ large enough “most” of the $\\ell $ -blocks in $\\sigma (\\pi _N,\\pi _{N+1}<\\ldots )$ is “good” in the sense that $\\left|\\int {1}_C\\cdot \\pi _0\\,d\\kappa \\right|\\le \\kappa (C)\\varepsilon .$ Since $A\\in \\sigma (\\pi _N,\\pi _{N+1},\\ldots )$ , we can approximate it by unions of $\\ell $ -blocks (for $\\ell $ sufficiently large) and most of the used blocks is “good”.", "Whence $\\left|\\int {1}_A\\cdot \\pi _0\\,d\\kappa \\right|\\le 2\\varepsilon ,$ and since $\\varepsilon >0$ was arbitrary, $\\pi _0\\perp L^2(\\Pi (\\kappa ))$ ." ], [ "Orthogonality to ${C}_{\\rm {ID}}$ . Proof of Corollary ", "We recall that (Proposition REF ) ${C}_{\\rm ID}={C}_{{\\rm ID}_{\\rm ec}}.$ Since the characteristic factor is represented by the $\\sigma $ -algebra of invariant sets, by Theorems REF and REF , we obtain immediately that: Corollary 5.3 $u\\perp {C}_{\\rm ID}$ if and only if for each Furstenberg system $\\kappa $ of $u$ , $\\pi _0\\perp L^2(\\mathcal {I}_\\kappa )$ .", "Let us now pass to a combinatorial characterization of the Veech condition.", "Assume that $\\kappa $ is given as the limit of $\\frac{1}{N_k}\\sum _{n\\le N_k}\\delta _{S^nu}$ .", "In view of Corollary REF , we need to decipher ${\\mathbb {E}}(\\pi _0|\\mathcal {I}_\\kappa )=0$ .", "By the von Neumann theorem, we have $\\frac{1}{H}\\sum _{h\\le H}\\pi _0\\circ S^h\\rightarrow {\\mathbb {E}}(\\pi _0|\\mathcal {I}_\\kappa ) \\text{ in }L^2,$ i.e.", "$\\lim _{H\\rightarrow \\infty }\\int _{X_{u}}\\left|\\frac{1}{H}\\sum _{h\\le H}\\pi _0\\circ S^h\\right|^2\\,d\\kappa =0$ as ${\\mathbb {E}}(\\pi _0|\\mathcal {I}_\\kappa )=0$ .", "So, given $\\varepsilon >0$ , $\\int _{X_{u}}\\left| \\frac{1}{H}\\sum _{h\\le H}\\pi _0\\circ S^h\\right|^2\\,d\\kappa <\\varepsilon \\text{ for all }H\\ge H_{\\varepsilon }.$ The latter is equivalent to $\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}\\left|\\frac{1}{H}\\sum _{h\\le H}\\pi _0\\circ S^h\\right|^2(S^nu)<\\varepsilon ,$ that is, $\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}\\left|\\frac{1}{H}\\sum _{h\\le H}u(n+h)\\right|^2<\\varepsilon .$ The proof of Corollary REF follows immediately.$\\quad \\hbox{\\vrule \\vbox to 6pt {\\hrule width 4pt\\vfill \\hrule }\\vrule } $ Remark 5.4 Hence, the Matomäki-Radziwiłł theorem [41] on the behaviour of a strongly aperiodic multiplicative function $u$ on a typical short interval implies $u\\perp {C}_{\\rm ID}$ .", "However, as shown in [28], the aperiodic multiplicative functions defined in [42] do not satisfy the assertion of Corollary REF .", "In Corollary REF , the Veech condition (for $u$ ) equivalent to $u\\perp {C}_{\\rm ID}$ is written as $\\pi _0\\perp L^2(\\mathcal {I}_\\kappa )$ .", "If we look at it more spectrally, we obtain immediately that $u\\perp {C}_{\\rm ID}\\text{ if and only if }\\sigma _{\\pi _0,\\kappa }(\\lbrace 1\\rbrace )=0$ for all $\\kappa \\in V_S(u)$ , i.e.", "the spectral measure of $\\pi _0$ (with respect to each Furstenberg system) has no atom at 1.", "Classically (by a simple computation), we have: Lemma 5.5 If $\\sigma $ is a measure on the circle $\\mathbb {S}^1$ then $\\frac{1}{H}\\sum _{h=0}^{H-1} \\widehat{\\sigma }(h)\\rightarrow \\sigma (\\lbrace 1\\rbrace ).$ Hence, the Veech condition is equivalent to $\\frac{1}{H}\\sum _{h=0}^{H-1}\\int \\pi _0\\cdot \\overline{\\pi _0}\\circ S^h d\\kappa \\rightarrow 0.$ Combinatorially, we obtain $\\frac{1}{H}\\sum _{h=0}^{H-1}\\lim _{k\\rightarrow \\infty } \\frac{1}{N_k}\\sum _{n\\le N_k} u(n)\\overline{u}(n+h) \\rightarrow 0$ for each sequence $(N_k)$ defining a Furstenberg system $\\kappa $ .", "It follows that (REF ) is equivalent to the short interval behaviour ().", "In other words, condition $\\lim _{H\\rightarrow \\infty }\\left(\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}\\left|\\frac{1}{H}\\sum _{h\\le H} u(n+h)\\right|^2\\right)=0$ is equivalent to $\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\left(\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}u(n)\\overline{u}(n+h)=0\\right).$" ], [ "Orthogonality to ${C}_{\\rm DISP(G)}$ with {{formula:69a12a44-1485-4df7-8f5e-d208881bd559}} countable", "Let $G\\subset \\mathbb {S}^1$ be a countable subgroup and recall that ${{\\rm DISP}(G)}$ stands for the (characteristic) class of discrete spectrum automorphisms whose groups of eigenvalues are contained in $G$ .", "Since $z\\in \\mathbb {S}^1$ is an eigenvalue of $(Z,{\\cal D},\\kappa ,R)$ if and only if it is an eigenvalues of a subset of positive measure of ergodic components, it is not hard to see that ${{F}}_{{\\rm DISP}(G)}=({{F}}_{{\\rm DISP}(G)})_{\\rm ec}.$ It follows that $u\\perp {C}_{{{F}}_{{\\rm DISP}(G)}}\\text{ if and only if }\\sigma _{\\pi _0,\\kappa }(G)=0,$ i.e.", "the spectral measure of $\\pi _0$ has no atoms belonging to $G$ (for each Furstenberg system $\\kappa \\in V_S(u)$ ).", "Suppose that $e^{2\\pi i\\alpha }\\in G$ .", "Consider $v(n):=e^{2\\pi in\\alpha }u(n)$ for $n\\in {\\mathbb {N}}$ .", "Note that $\\frac{1}{N_k}\\sum _{n\\le N_k}v(n)\\overline{v}(n+h)=e^{-2\\pi ih\\alpha }\\frac{1}{N_k}\\sum _{n\\le N_k}u(n)\\overline{u}(n+h).$ So, if we have a subsequence $(N_k)$ along which both $\\frac{1}{N_k}\\sum _{n\\le N_k}\\delta _{S^nw}$ with $w=u,v$ converge to $\\kappa ,\\kappa ^{\\prime }$ respectively,Note that these common sequences yield all Furstenberg systems for both $u$ and $v$ .", "then $\\sigma _{\\pi _0,\\kappa }=\\delta _{e^{2\\pi i\\alpha }}\\ast \\sigma _{\\pi _0,\\kappa ^{\\prime }},$ whence $\\sigma _{\\pi _0,\\kappa }(\\lbrace e^{2\\pi i\\alpha }\\rbrace )=0 \\text{ if and only if }\\sigma _{\\pi _0,\\kappa ^{\\prime }}(\\lbrace 1\\rbrace ).$ By our previous subsection it follows that the latter condition is equivalent to: $\\lim _{H\\rightarrow \\infty }\\left(\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}\\left|\\frac{1}{H}\\sum _{h\\le H} v(n+h)\\right|^2\\right)=0,$ that is, $\\lim _{H\\rightarrow \\infty }\\left(\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}\\left|\\frac{1}{H}\\sum _{h\\le H}e^{2\\pi i(n+h)\\alpha } u(n+h)\\right|^2\\right)=0$ which is the strong $u$ -MOMO condition for the irrational rotation by $\\alpha $ .Note that if $f(t)=e^{2\\pi i t}$ then $\\frac{1}{b_K}\\sum _{k<K}\\left|\\sum _{b_k\\le n<b_{k+1}}f(R_\\alpha ^nx_k)u(n)\\right| =\\frac{1}{b_K}\\sum _{k<K}\\left|\\sum _{b_k\\le n<b_{k+1}}e^{2\\pi i n\\alpha }u(n)\\right|.$" ], [ "Furstenberg systems and the strong $u$ -MOMO property", "The following proposition helps us to exclude some measure-theoretic systems from the list of Furstenberg systems of an arithmetic function.", "Proposition 5.6 Let $u:{\\mathbb {N}}\\rightarrow be a bounded arithmetic function.", "Then no Furstenberg system $ VS(u)$ has a topological model which is strongly $ u$-MOMO.$ Suppose $(X_{u},\\kappa , S)$ has a topological model $(Z,\\nu ,R)$ which satisfies the strong $u$ -MOMO property.", "Let $J:Z\\rightarrow X_{u}$ settles a measure-theoretic isomorphism and let $\\nu _J$ be the corresponding graph joining.", "We assume that $\\frac{1}{N_j}\\sum _{n\\le N_j}\\delta _{S^nu}\\rightarrow \\kappa $ .", "From Proposition REF we can find a sequence $(z_n)\\subset Z$ consisting of pieces of orbits of different points: $\\lbrace n:\\: Rz_n\\ne z_{n+1}\\rbrace =\\lbrace b_k:\\:k\\ge 1\\rbrace $ with $b_{k+1}-b_k\\rightarrow \\infty $ , and a subsequence $(N_{j_\\ell })$ such that $\\frac{1}{N_{j_\\ell }}\\sum _{n\\le N_{j_\\ell }}\\delta _{(z_n,S^nu)}\\rightarrow \\nu _J.$ Then, by the strong $u$ -MOMO property of $(Z,R)$ , $\\int f\\otimes \\pi _0\\,d\\nu _J=\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{b_{K_\\ell }}\\sum _{k<K_\\ell }\\big (\\sum _{b_k\\le n<b_{k+1}}f(R^{n-b_k}z_{b_k})u(n)\\big )=0.$ Hence, $\\int {\\mathbb {E}}^{\\nu _J}(f|X_{u})\\pi _0\\,d\\kappa =0$ for each continuous $f$ on $Z$ , and we obtain a contradiction since ${\\mathbb {E}}^{\\nu _{J}}(L^2(\\nu )|X_{u})=L^2(\\kappa )$ .", "Corollary 5.7 Assume that for each $(b_k)$ with $b_{k+1}-b_k\\rightarrow \\infty $ , $\\lim _{K\\rightarrow \\infty }\\frac{1}{b_K}\\sum _{k<K}\\sup _{\\alpha \\in {\\mathbb {R}}}\\left|\\sum _{b_k\\le n< b_{k+1}}u(n)e^{2\\pi i\\alpha n}\\right|=0.$ Then the unipotent system $(x,y)\\mapsto (x,y+x)$ (on 2) is not a Furstenberg system of $u$ .", "Since condition (REF ) is the strong $u$ -MOMO property of the unipotent system, the result follows from Proposition REF .", "Remark 5.8 Corollary REF brings a better understanding of Problem 3.1 (due to Frantzikinakis) of the workshop [4]: The system $(x,y)\\mapsto (x,y+x)$ is not a Furstenberg system of the Liouville function (see also slide no 6 in [17]).", "We recall that in [42] (see Theorem 1.3 therein), it is proved that $\\lim _{M\\rightarrow \\infty }\\limsup _{N\\rightarrow \\infty }\\sup _{\\alpha \\in {\\mathbb {R}}}\\frac{1}{N}\\sum _{n\\le N}\\left|\\frac{1}{M}\\sum _{n\\le m<n+M}\\lambda (m)e^{2\\pi i m\\alpha }\\right|=0,$ so the sup has changed the place!", "The strong $\\lambda $ -MOMO property for the unipotent system remains hence open.", "For the equivalence of $\\lim _{M\\rightarrow \\infty }\\limsup _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n\\le N}\\sup _{\\alpha \\in {\\mathbb {R}}}\\left|\\frac{1}{M}\\sum _{n\\le m<n+M}u(m)e^{2\\pi i m\\alpha }\\right|=0,$ with (REF ) see the appendix in [34] - only in the arXiv version of the paper." ], [ "Orthogonality to ${C}_{{\\rm DISP}_{\\rm ec}}$ . Proof of Corollary ", "In view of Corollary REF (see also (REF )) and Theorem REF , in order to obtain $u\\perp \\mathcal {C}_{{\\rm DISP}_{ec}}$ it is sufficient and necessary to have $u\\perp L^2({\\cal K}(\\mathcal {I}_\\kappa ))$ for each Furstenberg system $\\kappa $ of $u$ .", "By our previous results, for the class of all topological systems whose all ergodic measures yield discrete spectra, Sarnak and Veech conditions are equivalent.", "We now write the Veech condition combinatorially, i.e., we provide the proof of Corollary REF .", "[Proof of Corollary REF ] By Corollary REF , we need to show that for each $\\kappa $ being a Furstenberg system of $u$ , we have $\\int \\frac{1}{H}\\sum _{h\\le H}|{\\mathbb {E}}(\\pi _0\\circ S^h\\cdot \\overline{\\pi _0}|\\mathcal {I}_\\kappa )|^2\\, d\\kappa \\rightarrow 0.$ By the von Neumann theorem, $\\int \\Big |{\\mathbb {E}}(\\pi _0\\circ S^h\\cdot \\overline{\\pi }_0|\\mathcal {I}_\\kappa )\\Big |^2\\,d\\kappa =\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n\\le N}\\int (\\pi _0\\circ S^h\\cdot \\overline{\\pi }_0)\\circ S^n\\cdot \\overline{\\pi _0\\circ S^h\\cdot \\overline{\\pi }_0}\\,d\\kappa .$ Therefore, (REF ) is equivalent to $\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n\\le N}\\int (\\pi _0\\circ S^h\\cdot \\overline{\\pi }_0)\\circ S^n\\cdot \\overline{\\pi _0\\circ S^h\\cdot \\overline{\\pi }_0}\\,d\\kappa =0.$ Let $(M_k)$ be such that $\\kappa =\\lim _{k\\rightarrow \\infty }\\frac{1}{M_k}\\sum _{m\\le M_k}\\delta _{S^mu}$ .", "It follows immediately that (REF ) is equivalent to $\\lim _{H\\rightarrow \\infty }\\frac{1}{H}\\sum _{h\\le H}\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n\\le N}\\lim _{k\\rightarrow \\infty }\\frac{1}{M_k}\\sum _{m\\le M_k}u(m+n+h)\\overline{u(m+n)}\\overline{u(m+h)}u(m)=0$ which is precisely $\\Vert u\\Vert _{u^2((M_k))}=0$ .", "Now, it suffices to use (REF ).", "Remark 5.9 In fact, already Frantzikinakis [17] (see slide no 10) showed that if $u$ is generic then $\\Vert u\\Vert _{u^2}=0$ if and only if $\\lim _{M\\rightarrow \\infty }\\limsup _{N\\rightarrow \\infty }\\frac{1}{N}\\sum _{n\\le N}\\sup _{\\alpha \\in {\\mathbb {R}}}\\left|\\frac{1}{M}\\sum _{n\\le m<n+M}u(m)e^{2\\pi i m\\alpha }\\right|=0.$ We recall that this condition is equivalent to the strong $u$ -MOMO property of the unipotent system $(x,y)\\mapsto (x, y+x)$ .", "Remark 5.10 Note that for each (bounded) $u\\colon {\\mathbb {N}}\\rightarrow satisfying $ uu2=0$ the system\\begin{equation}(x,y)\\mapsto (x,x+y)\\text{ on }(2,{\\rm Leb}\\,\\otimes \\, {\\rm Leb})\\end{equation}cannot appear (up to isomorphism) as a Furstenberg system of $ u$ (because $ 0$ is orthogonal to the $ L2(K(I))$ but for the unipotent system (\\ref {unip}) thewhole system is relative Kronecker over the $$-algebra of invariant sets).$ In particular, if $\\Vert \\lambda \\Vert _{u^2}=0$ holds for the Liouville function then () is not its Furstenberg system – this would answer a question by N. Frantzikinakis asked in 2016 (it is an official Problem 3.1 in [4]).", "However, the problem of whether $\\Vert \\lambda \\Vert _{u^2}=0$ (or more generally $\\Vert \\lambda \\Vert _{u^s}=0$ ) seems to be difficult.", "The best known results [43], [44] require a quantitative dependence between the parameters $M$ and $N$ , i.e.", "$M=N^{\\theta }$ , for arbitrary small, but fixed $\\theta >0$ .", "If $\\Vert \\lambda \\Vert _{u^2}=0$ holds then Sarnak's conjecture holds for all (zero entropy) systems whose ergodic measures yield discrete spectrum.", "So far it is only known that Sarnak's conjecture holds for systems whose all invariant measure yield discrete spectrum [33], [32], [15].", "Ruling out () (or, more generally, nilpotent type systems) from the list of potential Furstenberg systems of $\\lambda $ is important in view of Frantzikinakis and Host's results [20], [21] concerning the structure of Furstenberg systems of multiplicative functions (although, for the moment, this structure is known only for the logarithmic case).", "In the light of [42], it would be also interesting to know whether $\\Vert u\\Vert _{u^2}=0$ holds for some classical multiplicative functions.", "Note that this is not the case for the class of aperiodic multiplicative functions defined in [42] since as shown in [28] they have the unipotent system as a Furstenberg systemIn fact, for such functions $u$ we have already $\\Vert u\\Vert _{u^1((N_k))}>0$ (for some $(N_k)$ ), see Corollary 6.5 in [28].", "(see also Remark REF )." ], [ "Orthogonality to ${C}_{\\rm DISP}$ . Averaged Chowla property for multiplicative functions", "The assertion “iff” of Theorem REF cannot be applied to the class ${C}_{\\rm DISP}$ .", "In this section we will show however that the assertion of this theorem holds whenever $u\\colon {\\mathbb {N}}\\rightarrow satisfies the following additional property:\\begin{equation}\\mbox{all rotations on the circle satisfy the strong \\qquad \\mathrm {(\\ast )}$u$-MOMO property}.\\end{equation}We will need the following fact (see, e.g.,~\\cite {Ed}):\\begin{equation} \\mbox{each discrete spectrum automorphism is a factor of $R_\\alpha \\times {\\rm Id}_{[0,1]}$},\\end{equation}for some ergodic rotation by $ G$ on a compact (Abelian) metric group $ G$.", "Our key tool will be the following lemma.\\begin{Lemma}Suppose that (\\ast ) holds.", "Then R_\\alpha \\times {\\rm Id}_{[0,1]} satisfies the strong u-MOMO property.\\end{Lemma}\\begin{proof}It is enough to check the strong u-MOMO for functions F of the form \\chi \\otimes f, where \\chi \\in \\widehat{G} and f\\in C([0,1]).", "We have{\\begin{@align*}{1}{-1}\\frac{1}{b_K}\\sum _{k<K}&\\left| \\sum _{b_k\\le n<b_{k+1}}F((R_\\alpha \\times {\\rm Id})^n(h_k,u_k))u(n)\\right|\\\\&=\\frac{1}{b_K}\\sum _{k<K}\\left| \\sum _{b_k\\le n<b_{k+1}}\\chi ((R_\\alpha ^n(h_k))f(u_k)u(n)\\right|\\\\&={\\rm O}\\left( \\frac{1}{b_K}\\sum _{k<K}\\left| \\sum _{b_k\\le n<b_{k+1}}\\chi (n\\alpha )u(n)\\right| \\right).\\end{@align*}}Our claim follows from (\\ast ).\\end{proof}$ Theorem 5.11 Assume that $u$ enjoys the property $(\\ast )$ .", "Then $u\\perp {C}_{DISP}$ if and only if $\\pi _0\\perp L^2(\\mathcal {K}(\\kappa ))$ for each $\\kappa \\in V_S(u)$ (iff the spectral measure $\\sigma _{\\pi _0}$ is continuous for each Furstenberg system $\\kappa $ ).", "We only need to show that $u\\perp {C}_{DISP}$ implies $\\pi _0\\perp L^2(\\mathcal {K}(\\kappa ))$ for each $\\kappa \\in V_S(u)$ .$\\mathcal {K}(\\kappa )$ stands for the Kronecker factor of $(X_{u},\\kappa ,S)$ .", "Using (), let $p$ settle a factor map from $R_\\alpha \\times {\\rm Id}_{[0,1]}$ acting on $(G\\times [0,1],m_G \\otimes {\\rm Leb})$ and $(X_{u}/\\mathcal {K}(\\kappa ),\\mathcal {K}(\\kappa ),\\kappa |_{\\mathcal {K}(\\kappa )})$ .", "Let $(m_G\\otimes {\\rm Leb})_p$ stand for the corresponding graph joining and $\\rho $ for the relatively independent extension of it to a joining of $(G\\times [0,1], m_G\\otimes {\\rm Leb},R_\\alpha \\times {\\rm Id})$ with $(X_{u}, \\kappa ,S)$ .", "Now, by Proposition REF , the integral $\\int F\\otimes \\pi _0\\, d\\rho $ can be computed using a quasi-generic sequence $((g_n), (S^nu))$ .", "Since, by Lemma , our topological system $R_\\alpha \\times {\\rm Id}$ satisfies the strong $u$ -MOMO property, this integral vanishes.", "On the other hand, for each $F\\in C(G\\times [0,1])$ , $\\int F\\otimes \\pi _0\\,d\\rho =\\int {\\mathbb {E}}(F|X_{u})\\pi _0\\,d\\kappa $ and since ${\\mathbb {E}}^\\rho (C(G\\times [0,1])|X_{u})$ is dense in $L^2(\\mathcal {K},\\kappa |_{\\mathcal {K}})$ (in view of the definition of $\\rho $ ), it follows that $\\pi _0\\perp L^2(\\mathcal {K}(\\kappa ))$ .", "[Proof of Corollary REF ] Note that in the proof of Theorem REF , we have shown that our original assumption $(\\ast )$ already implies the Veech condition.", "In particular, the Sarnak and the Veech properties are equivalent.", "Condition () is just rewriting the Wiener condition combinatorially.", "Finally, the last part () is proved in Appendix .", "[Proof of Corollary REF ] By Corollary REF , we only need to show that irrational rotations satisfy the strong $u$ -MOMO property.", "This follows from the fact that irrational rotations satisfy the AOP property [1] and that the AOP property implies the strong $u$ -MOMO property [3]." ], [ "No strong $u$ -MOMO in positive entropy", "In this section we discuss the problem of orthogonality to ${C}_{\\rm ZE}$ and the reversed problem of the absence of orthogonality to an arbitrary positive entropy systems, following some ideas from [3].", "Recall that the following has been proved in [3].", "Proposition 6.1 ([3]) Let $u\\colon {\\mathbb {N}}\\rightarrow be a bounded arithmetic function.", "The following are equivalent:\\begin{enumerate}\\item [(a)] u\\perp {C}_{\\rm ZE}.\\item [(b)] For each (X,T) of zero entropy and f\\in C(X), (\\ref {ort1}) holds uniformly in x\\in X.\\item [(c)] Each zero entropy (X,T) satisfies the strong u-MOMO property.\\end{enumerate}$ On the other hand, it follows from the results of Downarowicz and Serafin [11], [12] that for each $u\\perp {C}_{\\rm ZE}$ there exists $(X,T)$ such that $u\\perp (X,T) \\text{ and }(X,T)\\notin {\\rm ZE}.$ In fact, one can get a positive entropy system $(X,T)$ in which for every $f\\in C(X)$ () holds uniformly in $x\\in X$ .", "We prove however that (REF ) fails if orthogonality is replaced by the strong $u$ -MOMO property.", "To avoid technical details, we restrict ourselves to the case of an arithmetic function $u$ taking finitely many values.", "Theorem D Let $u:{\\mathbb {N}}\\rightarrow be an arithmetic function taking finitely many values.Assume that $ uCZE$.", "Then no positive entropy topological dynamical system satisfies the strong $ u$-MOMO property.$" ], [ "Proof of Theorem ", "We fix a bounded arithmetic function $u\\colon {\\mathbb {N}}\\rightarrow .We need a series of results from \\cite {Ab-Ku-Le-Ru2} in some modified forms.", "In~\\cite {Ab-Ku-Le-Ru2}, the equivalence of certain three properties (P1), (P2) and (P3) of an ergodic measure-theoretic dynamical system $ (Z,B(Z),,R)$ was proved.", "Condition (P1) was nothing but the strong $ u$-MOMO for {\\bf some} topological system being a model of the system given by $$.", "Instead of recalling (P2), let us formulate red its subsequence version:\\begin{equation}\\begin{array}{l}\\mbox{Assume that \\qquad \\mathrm {(P2')}$(X,T)$ is any topological system and let $x\\in X$.", "}\\\\\\mbox{If $x$ is generic along $(N_k)$ for a measure which is isomorphic}\\\\\\mbox{(as dynamical systems) to $\\kappa $ then}\\\\\\mbox{$\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}f(T^nx)u(n)=0$for each $f\\in C(X)$}.\\end{array}\\end{equation}The proof of the implication (P1) $$ (P2^{\\prime }) is a repetition of the proof of (P1) implies (P2).", "In Lemma~17 in \\cite {Ab-Ku-Le-Ru2}, we need to consider the sequence $ (Nk)$ instead of $ N$ and start with $$ along this sequence.$ As a consequence of the above, we obtain the following version of Corollary 12 from [3].", "Corollary 6.2 Assume that $\\kappa $ is an ergodic shift-invariant measure on $L^{{\\mathbb {Z}}}$ , and that there exists $y\\in L^{{\\mathbb {Z}}}$ , generic along $(N_k)$ for $\\kappa $ , correlating with $u$ along $(N_k)$ , i.e.", "the sequence $(\\frac{1}{N_k}\\sum _{n\\le N_k}y(n)u(n))$ does not go to zero.", "Then the strong $u$ -MOMO property fails for any uniquely ergodic model of $(L^{{\\mathbb {Z}}},\\kappa ,S)$ .", "Then, by repeating the proof from [3], we obtain the following form of Corollary 14 in [3].", "Corollary 6.3 Assume that $y$ is generic along $(N_k)$ for a Bernoulli measure $\\nu $ , and that $y$ and $u$ correlate along $(N_k)$ .", "Then the strong $u$ -MOMO property fails for any $(X,T)$ with $h(X,T)>h(\\nu )$ .", "We also need the following crucial probabilistic lemma whose proof we postpone to the next subsection.", "Lemma 6.4 Assume that $X=(X_n)_{n\\in {\\mathbb {Z}}}$ is a a stationary process of positive entropy, taking finitely many complex values.", "Then for any non-trivial probability distribution $\\beta $ concentrated on a finite subset of ${\\mathbb {R}}$ , there exists a stationary coupling of $X$ with a Bernoulli process $Y=(Y_n)_{n\\in {\\mathbb {Z}}}$ of distribution $\\beta ^{\\otimes {\\mathbb {Z}}}$ such that ${\\mathbb {E}}[X_0 Y_0]\\ne {\\mathbb {E}}[X_0]{\\mathbb {E}}[Y_0]$ .", "We now assume that $u$ takes finitely many values and satisfies the Veech condition: $\\pi _0\\perp L^2(\\Pi (\\kappa ))$ for each Furstenberg system $\\kappa $ of $u$ .", "Lemma 6.5 For each $h>0$ there exists a sequence $y$ , generic for a Bernoulli measure of entropy $h$ along some increasing sequence $(N_k)$ , and correlating with $u$ along $(N_k)$ .", "Let $\\kappa $ be a Furstenberg system of $u$ , and $(M_\\ell )$ such that $u$ is generic for $\\kappa $ along $(M_\\ell )$ .", "By assumption, the entropy of the stationary process defined by $\\pi _0$ under $\\kappa $ is positive.", "Take a real-valued Bernoulli shift of entropy $h$ (Bernoulli measure denoted by $\\nu $ ).", "Using Lemma REF , find a joining of $\\kappa $ and $\\nu $ for which $\\pi _0$ (in $L^2(X_{u},\\kappa )$ ) is not orthogonal to $\\pi _0$ in $L^2(\\nu )$ : $\\int \\pi _0\\otimes \\pi _0\\,d\\rho \\ne 0$ .", "Now, use a subsequence version of the lifting lemma (Theorem 5.16 in [6]) to find $y$ in the subshift defining the Bernoulli automorphism such that $(u,y)$ is generic, along a subsequence $(N_k)=(M_{\\ell _k})$ , for $\\rho $ .", "Then $0\\ne \\int \\pi _0\\otimes \\pi _0\\,d\\rho =\\lim _{k\\rightarrow \\infty }\\frac{1}{N_{k}}\\sum _{n\\le N_{k}}\\pi _0(S^nu)\\pi _0(S^ny)=\\lim _{k\\rightarrow \\infty }\\frac{1}{N_{k}}\\sum _{n\\le N_{k}}u(n)y_n$ which means that $u$ and $y$ correlate along $(N_{k})$ .", "Now the proof of Theorem REF is a straightforward consequence of Lemma REF and Corollary REF ." ], [ "Proof of Lemma ", "Let $X=(X_n)_{n\\in {\\mathbb {Z}}}$ be a positive entropy stationary process as in the statement of the lemma.", "Without loss of generality (considering its real or imaginary part), we can assume that this process takes its values in a finite subset $\\lbrace x_1<x_2<\\cdots <x_r\\rbrace $ of ${\\mathbb {R}}$ .", "We also consider a given probability measure $\\beta $ supported on a possibly different finite subset of ${\\mathbb {R}}$ $\\lbrace y_1<y_2<\\cdots <y_s\\rbrace $ , which is supposed to be non trivial (i.e.", "not reduced to a Dirac measure).", "Thus we can assume that $s\\ge 2$ , and $\\beta (y_j)>0$ for each $1\\le j\\le s$ .", "The purpose of this section is to show how we can construct a stationary coupling of $X$ with a Bernoulli process $Y$ whose distribution is $\\beta ^{\\otimes {\\mathbb {Z}}}$ , in such a way that for each $n\\in {\\mathbb {Z}}$ , ${\\mathbb {E}}[X_n Y_n] > {\\mathbb {E}}[X_n]\\, {\\mathbb {E}}[Y_n].$ We observe that the validity of the preceding inequality is unchanged if we replace $Y_n$ by $Y_n+C$ for a fixed $C$ .", "Thus we can and we do assume without loss of generality that the probability $\\beta $ is such that ${\\mathbb {E}}[Y_n]=0$ .", "To construct the announced coupling, we just assume that, on the probability space where the process $X$ is defined, we also have an i.i.d.", "process $V=(V_n)_{n\\in {\\mathbb {Z}}}$ such that each $V_n$ is uniformly distributed on $[0,1]$ , $V$ is independent of $X$ .", "The construction will be divided into two steps: first we construct an auxiliary (uniform i.i.d.)", "process $U$ and then we use it to construct $Y$ which satisfies the assertion of Lemma REF ." ], [ "Step 1: uniform i.i.d. process $U$", "For $n\\in {\\mathbb {Z}}$ and $j\\in \\lbrace 1,\\ldots ,r\\rbrace $ , we consider the random variable $P_{j,n}$ defined by $P_{j,n} := {\\mathbb {P}}\\bigl ( X_n=x_j \\,|\\, (X_m)_{m\\le n-1}\\bigr ).$ When $j$ is fixed, $(P_{j,n})_{n\\in {\\mathbb {Z}}}$ is a stationary process.", "On the other hand, if we fix $n$ , then $(P_{1,n},\\ldots ,P_{r,n})$ is the conditional distribution of $X_n$ given $(X_m)_{m\\le n-1}$ , in particular we have almost surely $0\\le P_{j,n}\\le 1$ , and $\\sum _{j=1}^r P_{j,n} = 1.$ This allows us to define a random partition of $[0,1[$ into disjoint subintervals $I_{1,n},\\ldots , I_{r,n}$ where for each $j$ , $I_{j,n}$ is the interval of length $P_{j,n}$ defined by $I_{j,n}:=\\left[\\sum _{1\\le i\\le j-1} P_{i,n}\\ ; \\sum _{1\\le i\\le j} P_{i,n}\\right[.$ Then we can define the random variable $U_n$ by $U_n := \\sum _{j=1}^r {1}_{X_n=x_j}\\left(\\sum _{1\\le i\\le j-1} P_{i,n} + V_n P_{j,n}\\right).$ Figure: Definition of U n U_nInformally, if $X_n=x_j$ , we pick $U_n$ uniformly at random (using $V_n$ ) inside $I_{j,n}$ (see Figure REF ).", "Therefore, $\\mathcal {L}\\left(U_n\\,|\\,(X_m)_{m\\le n-1},(V_m)_{m\\le n-1}\\right)=\\mathcal {U}_{[0,1]},$ i.e., it is uniform on $[0,1]$ .", "But all $U_m$ , $m\\le n-1$ , are measurable with respect to $(X_m)_{m\\le n-1}$ and $(V_m)_{m\\le n-1}$ , thus we also have $\\mathcal {L}\\left(U_n\\,|\\,(U_m)_{m\\le n-1}\\right)=\\mathcal {U}_{[0,1]} \\text{ and } \\mathcal {L}\\left(U_n\\right)=\\mathcal {U}_{[0,1]}.$ Indeed, this is just the application of the tower property of conditional expectations: to obtain the left equality, notice that for any measurable $A\\subset [0,1]$ , we have $\\mathbb {P}\\bigl (U_n\\in A &\\,|\\, (U_m)_{m\\le n-1}\\bigl ) \\\\&=\\mathbb {E}\\Bigl [\\underbrace{\\mathbb {P}\\bigl (U_n\\in A\\,|\\, (X_m)_{m\\le n-1},(V_m)_{m\\le n-1}\\bigr )}_{{\\rm Leb}(A)}\\,|\\, (U_m)_{m\\le n-1}\\Bigr ]\\\\&={\\rm Leb}(A).$ Moreover, it also follows from (REF ) that $U$ is i.i.d.", "Note that by construction, $U_n$ is a measurable function of $V_n$ , $X_n$ and $(X_m)_{m\\le n-1}$ , which we abusively write as $U_n = U_n \\left(V_n,X_n,(X_m)_{m\\le n-1}\\right).$ Moreover, whenever we fix realizations $\\xi $ of $(X_m)_{m\\le n}$ and $v$ of $V_n$ then $U_n$ as a function of its second argument is increasing: $U_n(v,x_{j_1},\\xi ) < U_n(v,x_{j_2},\\xi ), \\text{ whenever }x_{j_1}<x_{j_2}.$ We want to define $Y_n$ for a given $n\\in {\\mathbb {Z}}$ .", "We use another partition of $[0,1[$ into subintervals, according to the probability distribution $\\beta $ intended for $Y_n$ : for $1\\le k\\le s$ , set $\\beta _k:=\\beta (y_k)$ and define the interval $J_k:=\\bigl [\\beta _1+\\cdots +\\beta _{k-1};\\beta _1+\\cdots +\\beta _k\\bigr [$ .", "Then we simply define $Y_n$ as a function of $U_n$ by setting $ Y_n := \\sum _{k=1}^s y_k\\,{1}_{J_k}(U_n).$ It follows by the choice of the intervals $J_k$ and by $\\mathcal {L}(U_n)=\\mathcal {U}_{[0,1]}$ that $Y_n$ is distributed according to $\\beta $ .", "Moreover, by the independence of $U$ , we have the independence of $Y$ .", "Thus, $Y$ is a Bernoulli process with distribution $\\beta ^{\\otimes \\mathbb {Z}}$ .", "It remains to prove the announced inequality (REF ).", "Observe that $Y_n$ is, like $U_n$ , constructed as a measurable function of $V_n$ , $X_n$ and $(X_m)_{m\\le n-1}$ , which we also abusively write as $ Y_n = Y_n \\left(V_n,X_n,(X_m)_{m\\le n-1}\\right).", "$ Since $Y_n$ is a non-decreasing function of $U_n$ , we get from (REF ) that for a fixed realization $\\xi $ of $(X_m)_{m\\le n-1}$ and $v$ of $V_n$ , we have for $1\\le j_1 < j_2 \\le r$ $Y_n \\left(v,x_{j_1},\\xi \\right) < Y_n \\left(v,x_{j_2},\\xi \\right)$ and it follows that the map $ x\\in A \\mapsto {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x\\bigr ] $ is non-decreasing.", "Moreover, by the construction of $Y$ , we have $ \\mathcal {L}\\bigl (Y_n \\,|\\, (X_m)_{m\\le n-1}\\bigr ) = \\mathcal {L}(Y_n)=\\beta , $ whence ${\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}\\bigr ] = {\\mathbb {E}}[Y_n] =0.$ Thus there exists $j_0\\in \\lbrace 1,\\ldots ,r\\rbrace $ (depending on $\\xi $ ) such that $\\begin{split}&{\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_j]\\le 0 \\text{ for }1\\le j\\le j_0,\\\\&{\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_j]> 0\\text{ for }j_0+1\\le j\\le r.\\end{split}$ We then have, using (REF ) and (REF ), ${\\mathbb {E}}\\bigl [ X_n Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] & = {\\mathbb {E}}\\bigl [ (X_n-x_{j_0}) Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] \\nonumber \\\\&= \\sum _{j=1}^{j_0} (x_j-x_{j_0})\\, {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_j] \\nonumber \\\\&\\quad + \\sum _{j=j_0+1}^{r} (x_j-x_{j_0})\\, {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_j] \\\\& \\ge 0.", "\\nonumber $ Now, we claim that the announced result is a consequence of the following lemma.", "Lemma 6.6 If the realization $\\xi $ of $(X_m)_{m\\le n-1}$ is such that the conditional distribution $\\mathcal {L}(X_n \\,|\\, (X_m)_{m\\le n-1}=\\xi )$ is non-trivial, then $ {\\mathbb {E}}\\bigl [ X_n Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] > 0.", "$ Indeed, since $X$ has positive entropy, $\\mathcal {L}(X_n \\,|\\, (X_m)_{m\\le n-1})$ is non-trivial with positive probability, and thus we can conclude that $ {\\mathbb {E}}\\bigl [ X_n Y_n\\bigr ] = {\\mathbb {E}}\\Bigl [{\\mathbb {E}}\\bigl [ X_n Y_n \\,|\\, (X_m)_{m\\le n-1}\\bigr ]\\Bigr ] >0.", "$ [Proof of Lemma REF ] We fix a realization $\\xi $ of $(X_m)_{m\\le n-1}$ such that the conditional distribution $\\mathcal {L}\\bigl (X_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr )$ is non-trivial.", "Then the random variables $P_{j,n}$ and the intervals $I_{j,n}$ are fixed, because their values only depend on $\\xi $ .", "Setting $j_1 &:= \\min \\bigl \\lbrace j\\in \\lbrace 1,\\ldots ,r\\rbrace :\\ P_{j,n}>0\\bigr \\rbrace ,\\\\\\text{and}\\quad j_2 &:= \\max \\bigl \\lbrace j\\in \\lbrace 1,\\ldots ,r\\rbrace :\\ P_{j,n}>0\\bigr \\rbrace ,$ we have $j_1<j_2$ .", "Moreover the intervals $I_{j_1,n}$ and $I_{j_2,n}$ are respectively of the form $[0,P_{j_1,n}[$ and $[1-P_{j_2,n},1[$ , with $0<P_{j_1,n}\\le 1-P_{j_2,n}<1$ .", "We now discuss according to the relative position of the interval $I_{j_2,n}$ with respect to the interval $J_1$ (used to define $Y_n$ ).", "Figure: Case 1 (J 1 ∩I j 2 ,n =∅J_1\\cap I_{j_2,n}=\\emptyset )" ], [ "Case 1:", "$J_1\\cap I_{j_2,n}=\\emptyset $ (see Figure REF ).", "Then we have ${\\mathbb {P}}\\bigl (Y_n=y_1 \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_2}\\bigr )=0,$ whereas ${\\mathbb {P}}\\bigl (Y_n=y_1 \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_1}\\bigr )>0.$ Moreover, notice that (REF ) is equivalent to ${\\mathbb {P}}\\bigl (Y_n>y_1 \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_2}\\bigr )=1,$ It follows from (REF ) and (REF ) that there exists a $V_n$ -measurable event $A$ of positive probability such that, on $A$ , $ Y_n\\bigl (V_n,x_{j_2},\\xi \\bigr ) > y_1 = Y_n\\bigl (V_n,x_{j_1},\\xi \\bigr ).", "$ Remembering (REF ), we get ${\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_{j_2}\\bigr ] > {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_{j_1}\\bigr ].$ Figure: Case 2 (J 1 ∩I j 2 ,n ≠∅J_1\\cap I_{j_2,n}\\ne \\emptyset )" ], [ "Case 2:", "$J_1\\cap I_{j_2,n}\\ne \\emptyset $ (see Figure REF ).", "Then $I_{j_1,n}\\subset J_1$ and $I_{j_1,n}\\cap J_s=\\emptyset $ .", "It follows that ${\\mathbb {P}}\\bigl (Y_n=y_s \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_1}\\bigr )=0,$ whereas ${\\mathbb {P}}\\bigl (Y_n=y_s \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_2}\\bigr )>0.$ In this case, we get a $V_n$ -measurable event $A$ of positive probability such that, on $A$ , $ Y_n\\bigl (V_n,x_{j_2},\\xi \\bigr ) = y_s > Y_n\\bigl (V_n,x_{j_1},\\xi \\bigr ), $ and as before we conclude that (REF ) holds.", "Now, since (REF ) always holds, and since $ {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] = 0 =\\sum _{j=1}^rP_{j,n}\\, {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_{j}\\bigr ], $ we deduce that $ {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_{j_2}\\bigr ]>0.", "$ It follows that in the sum (REF ), at least the term corresponding to $j=j_2$ is positive, and this yields $ {\\mathbb {E}}\\bigl [ X_n Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] > 0.", "$ Appendix" ], [ "From averaged double to averaged multiple correlations", "This section follows some arguments from [42].", "Remark A.1 In the proof below we will use the following standard fact: let $(x(n))$ be a sequence of complex number bounded by 1.", "Then $\\sum _{m\\le M}|x(m)|=o(M)$ is equivalent to $\\sum _{m\\le M}|x(m)|^2=o(M).$ The little “o” is uniform with respect to $M$ .", "If $\\varepsilon :=\\frac{1}{M}\\sum _{m\\le M}|x(m)|^2$ then by Markov's inequality $\\frac{1}{M}|\\lbrace m\\le M:\\: |c_m|^2\\ge \\varepsilon ^{1/2}\\rbrace |\\le \\frac{1}{\\varepsilon ^{1/2}}\\cdot \\varepsilon =\\varepsilon ^{1/2}$ and then $\\frac{1}{M}\\sum _{m\\le M}|x(m)|=\\frac{1}{M}\\sum _{m\\le M, |x(m)|\\ge e^{1/4}}|x(m)|+\\frac{1}{M}\\sum _{m\\le M, |x(m)|<\\varepsilon ^{1/4}}|x(m)|\\le \\varepsilon ^{1/2}+\\varepsilon ^{1/4}.$ We have the following general lemma: Lemma A.2 Let $(N_\\ell )_{\\ell \\in {\\mathbb {N}}}$ be a sequence of natural numbers.", "For $k\\in \\mathbb {N}$ let $a,b_1,\\ldots b_k\\colon \\mathbb {N} \\rightarrow \\mathbb {C}$ be sequences bounded by 1.", "Assume that $a$ satisfies $\\lim _{H\\rightarrow \\infty }\\frac{1}{H} \\sum _{h\\le H}\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n)a(n+h)\\Big |=0.$ Then $\\lim _{H\\rightarrow \\infty }\\frac{1}{H^k} \\sum _{h_1,\\ldots , h_k\\le H}\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n)\\prod _{i=1}^kb_i(n+h_i)\\Big |=0.$ Notice first that (REF ) can be rewritten as the following: for every $\\varepsilon >0$ , there exists $H_\\varepsilon $ such that for $H>H_\\varepsilon $ and all $\\ell $ sufficiently large (depending on $H$ ), we have $A:=\\frac{1}{H^k}\\sum _{h_1,\\ldots , h_k\\le H}\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n)\\prod _{i=1}^kb_i(n+h_i) \\Big |<\\varepsilon .$ Now, notice that for any $H,N_\\ell ,H^{\\prime }$ and any $h^{\\prime }\\le H^{\\prime }$ , by shifting the summation over $n\\le N_\\ell $ by $h^{\\prime }$ (for every fixed choice of $h_1,\\ldots h_k$ ), we have $\\sum _{h_1,\\ldots , h_k\\le H}\\Big | \\sum _{n\\le N_\\ell }a(n)\\prod _{i=1}^kb_i(n+h_i)\\Big |=$ $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{h^{\\prime }\\le n\\le N_\\ell +h^{\\prime }}a(n)\\prod _{i=1}^kb_i(n+h_i)\\Big |+{\\rm O}(H^k\\cdot h^{\\prime })=$ $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i+h^{\\prime })\\Big |+{\\rm O}(H^k\\cdot h^{\\prime })$ and ${\\rm O}(H^k\\cdot h^{\\prime })={\\rm O}(H^k\\cdot H^{\\prime })$ .", "Notice that as $h_i$ is taken from $[0,H]$ , then $h_i+h^{\\prime }$ is taken from $[h^{\\prime },H+h^{\\prime }]$ (which is a small shift of $[0,H]$ if $h^{\\prime }$ is much smaller than $H$ ).", "So putting $h^{\\prime }$ to the summation over $h_i$ , we get $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i+h^{\\prime })\\Big |=$ $\\sum _{h^{\\prime }\\le h_1,\\ldots , h_k\\le H+h^{\\prime }}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i)\\Big |=$ $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i)\\Big |+ {\\rm O}\\left((h^{\\prime } H^{k-1}+(h^{\\prime })^2H^{k-2}+\\ldots +(h^{\\prime })^k)N_\\ell \\right)$ and ${\\rm O}\\left((h^{\\prime } H^{k-1}+(h^{\\prime })^2H^{k-2}+\\ldots +(h^{\\prime })^k)N\\right)={\\rm O}((H^{\\prime })^kH^{k-1}N_\\ell )$ .", "Putting the two displayed equations together we get that for every $h^{\\prime }\\le H^{\\prime }$ , $A=\\frac{1}{H^k}\\sum _{h_1,\\ldots , h_k\\le H}\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i) \\Big |+{\\rm O}\\Big (\\frac{H^{\\prime }}{N_\\ell }\\Big )+{\\rm O}\\Big (\\frac{(H^{\\prime })^k}{H} \\Big ).$ Averaging the above equation over all $h^{\\prime }\\le H^{\\prime }$ , we get that $A=\\frac{1}{H^k}\\sum _{h_1,\\ldots , h_k\\le H}\\frac{1}{H^{\\prime }}\\sum _{h^{\\prime }\\le H^{\\prime }}\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })G(n)\\Big |+{\\rm O}\\Big (\\frac{H^{\\prime }}{N_\\ell }\\Big )+{\\rm O}\\Big (\\frac{(H^{\\prime })^k}{H} \\Big ),$ where $G(n)=\\prod _{i=1}^kb_i(n+h_i)$ .", "We will now estimate $\\frac{1}{H^{\\prime }}\\sum _{h^{\\prime }\\le H^{\\prime }}\\frac{1}{N_\\ell ^2}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })G(n) \\Big |^2=\\\\\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\Big (\\frac{1}{H^{\\prime }}\\sum _{h^{\\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })}\\Big )G(n)\\overline{G(n^{\\prime })},$ which will be easier to handle than the above expression for $A$ (and then use Remark REF to get rid of the squares).", "Clearly, to obtain an upper bound for (REF ), it suffices to obtain an upper bound for $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime }}\\Big |\\sum _{h^{\\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })} \\Big |.$ Again, it will be easier to deal with $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime 2}}\\Big |\\sum _{h^{\\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })} \\Big |^2$ (and use Remark REF to get rid of the squares).", "Expanding the square again we get that (REF ) is equal to $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })}\\overline{a(n+h^{\\prime \\prime })\\cdot \\overline{a(n^{\\prime }+h^{\\prime \\prime })}}=$ $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n+h^{\\prime \\prime })}a(n^{\\prime }+h^{\\prime \\prime })\\overline{a(n^{\\prime }+h^{\\prime })}.$ The sum in the last term by exchanging the order of summation is equal to $\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}\\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\overline{a(n+h^{\\prime \\prime })}\\Big |^2=$ $\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}\\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h^{\\prime \\prime }-h^{\\prime })}\\Big |^2+ {\\rm O}(\\frac{H^{\\prime 2}}{N_\\ell ^2}).$ Finally, grouping according to $h=h^{\\prime \\prime }-h^{\\prime }$ , we get that that the above is equal to $\\frac{1}{H^{\\prime 2}}\\sum _{|h|\\le H^{\\prime }}|H^{\\prime }-h|\\cdot \\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}\\Big |^2+{\\rm O}(\\frac{H^{\\prime 2}}{N_\\ell ^2})\\le $ $\\frac{1}{H^{\\prime }}\\cdot \\sum _{|h|\\le H^{\\prime }}\\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}\\Big |^2+{\\rm O}(\\frac{H^{\\prime 2}}{N_\\ell ^2}).$ That is, the expression from (REF ) equals $\\frac{1}{H^{\\prime }}\\sum _{|h|\\le H^{\\prime }}\\left|\\frac{\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}}{N_\\ell } \\right|^2 +{\\rm O}\\left(\\frac{(H^{\\prime })^2}{N_\\ell ^2} \\right).$ Now, by the assumption of our lemma, it follows that $\\frac{1}{H^{\\prime }}\\sum _{|h|\\le H^{\\prime }}\\left|\\frac{\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}}{N_\\ell } \\right|={\\rm o}(1),$ which, by Remark REF , is equivalent to $\\frac{1}{H^{\\prime }}\\sum _{|h|\\le H^{\\prime }}\\left|\\frac{\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}}{N_\\ell } \\right|^2={\\rm o}(1).$ Therefore,  (REF ) (and, thus, also (REF )) is of the order of ${\\rm o}(1)+{\\rm O}\\left(\\frac{(H^{\\prime })^2}{N_\\ell ^2} \\right)$ .", "Using again Remark REF , we conclude that also (REF ) is of the same order.", "It follows immediately that also the order of (REF ) is the same.", "Thus, we have proved that $A={\\rm o}\\left(1\\right)+{\\rm O}\\left(\\frac{(H^{\\prime })^2}{N_\\ell ^2} \\right)+{\\rm O}\\left(\\frac{H^{\\prime }}{N_\\ell } \\right)+{\\rm O}\\left(\\frac{(H^{\\prime })^k}{H}\\right).$ Acknowledgments: We would like to thank Tomasz Downarowicz, Nikos Frantzikinakis and Krzysztof Fra̧czek for useful discussions on the paper.", "Research of the second and third authors supported by Narodowe Centrum Nauki grant UMO-2019/33/B/ST1/00364.", "Department of Mathematics, The Maryland University [email protected] Faculty of Mathematics and Computer Science Nicolaus Copernicus University, Toruń, Poland [email protected], [email protected] Laboratoire de Mathématiques Raphaël Salem, CNRS – Université de Rouen Normandie Avenue de l’Université – 76801 Saint Étienne du Rouvray, France [email protected]" ], [ "No strong $u$ -MOMO in positive entropy", "In this section we discuss the problem of orthogonality to ${C}_{\\rm ZE}$ and the reversed problem of the absence of orthogonality to an arbitrary positive entropy systems, following some ideas from [3].", "Recall that the following has been proved in [3].", "Proposition 6.1 ([3]) Let $u\\colon {\\mathbb {N}}\\rightarrow be a bounded arithmetic function.", "The following are equivalent:\\begin{enumerate}\\item [(a)] u\\perp {C}_{\\rm ZE}.\\item [(b)] For each (X,T) of zero entropy and f\\in C(X), (\\ref {ort1}) holds uniformly in x\\in X.\\item [(c)] Each zero entropy (X,T) satisfies the strong u-MOMO property.\\end{enumerate}$ On the other hand, it follows from the results of Downarowicz and Serafin [11], [12] that for each $u\\perp {C}_{\\rm ZE}$ there exists $(X,T)$ such that $u\\perp (X,T) \\text{ and }(X,T)\\notin {\\rm ZE}.$ In fact, one can get a positive entropy system $(X,T)$ in which for every $f\\in C(X)$ () holds uniformly in $x\\in X$ .", "We prove however that (REF ) fails if orthogonality is replaced by the strong $u$ -MOMO property.", "To avoid technical details, we restrict ourselves to the case of an arithmetic function $u$ taking finitely many values.", "Theorem D Let $u:{\\mathbb {N}}\\rightarrow be an arithmetic function taking finitely many values.Assume that $ uCZE$.", "Then no positive entropy topological dynamical system satisfies the strong $ u$-MOMO property.$" ], [ "Proof of Theorem ", "We fix a bounded arithmetic function $u\\colon {\\mathbb {N}}\\rightarrow .We need a series of results from \\cite {Ab-Ku-Le-Ru2} in some modified forms.", "In~\\cite {Ab-Ku-Le-Ru2}, the equivalence of certain three properties (P1), (P2) and (P3) of an ergodic measure-theoretic dynamical system $ (Z,B(Z),,R)$ was proved.", "Condition (P1) was nothing but the strong $ u$-MOMO for {\\bf some} topological system being a model of the system given by $$.", "Instead of recalling (P2), let us formulate red its subsequence version:\\begin{equation}\\begin{array}{l}\\mbox{Assume that \\qquad \\mathrm {(P2')}$(X,T)$ is any topological system and let $x\\in X$.", "}\\\\\\mbox{If $x$ is generic along $(N_k)$ for a measure which is isomorphic}\\\\\\mbox{(as dynamical systems) to $\\kappa $ then}\\\\\\mbox{$\\lim _{k\\rightarrow \\infty }\\frac{1}{N_k}\\sum _{n\\le N_k}f(T^nx)u(n)=0$for each $f\\in C(X)$}.\\end{array}\\end{equation}The proof of the implication (P1) $$ (P2^{\\prime }) is a repetition of the proof of (P1) implies (P2).", "In Lemma~17 in \\cite {Ab-Ku-Le-Ru2}, we need to consider the sequence $ (Nk)$ instead of $ N$ and start with $$ along this sequence.$ As a consequence of the above, we obtain the following version of Corollary 12 from [3].", "Corollary 6.2 Assume that $\\kappa $ is an ergodic shift-invariant measure on $L^{{\\mathbb {Z}}}$ , and that there exists $y\\in L^{{\\mathbb {Z}}}$ , generic along $(N_k)$ for $\\kappa $ , correlating with $u$ along $(N_k)$ , i.e.", "the sequence $(\\frac{1}{N_k}\\sum _{n\\le N_k}y(n)u(n))$ does not go to zero.", "Then the strong $u$ -MOMO property fails for any uniquely ergodic model of $(L^{{\\mathbb {Z}}},\\kappa ,S)$ .", "Then, by repeating the proof from [3], we obtain the following form of Corollary 14 in [3].", "Corollary 6.3 Assume that $y$ is generic along $(N_k)$ for a Bernoulli measure $\\nu $ , and that $y$ and $u$ correlate along $(N_k)$ .", "Then the strong $u$ -MOMO property fails for any $(X,T)$ with $h(X,T)>h(\\nu )$ .", "We also need the following crucial probabilistic lemma whose proof we postpone to the next subsection.", "Lemma 6.4 Assume that $X=(X_n)_{n\\in {\\mathbb {Z}}}$ is a a stationary process of positive entropy, taking finitely many complex values.", "Then for any non-trivial probability distribution $\\beta $ concentrated on a finite subset of ${\\mathbb {R}}$ , there exists a stationary coupling of $X$ with a Bernoulli process $Y=(Y_n)_{n\\in {\\mathbb {Z}}}$ of distribution $\\beta ^{\\otimes {\\mathbb {Z}}}$ such that ${\\mathbb {E}}[X_0 Y_0]\\ne {\\mathbb {E}}[X_0]{\\mathbb {E}}[Y_0]$ .", "We now assume that $u$ takes finitely many values and satisfies the Veech condition: $\\pi _0\\perp L^2(\\Pi (\\kappa ))$ for each Furstenberg system $\\kappa $ of $u$ .", "Lemma 6.5 For each $h>0$ there exists a sequence $y$ , generic for a Bernoulli measure of entropy $h$ along some increasing sequence $(N_k)$ , and correlating with $u$ along $(N_k)$ .", "Let $\\kappa $ be a Furstenberg system of $u$ , and $(M_\\ell )$ such that $u$ is generic for $\\kappa $ along $(M_\\ell )$ .", "By assumption, the entropy of the stationary process defined by $\\pi _0$ under $\\kappa $ is positive.", "Take a real-valued Bernoulli shift of entropy $h$ (Bernoulli measure denoted by $\\nu $ ).", "Using Lemma REF , find a joining of $\\kappa $ and $\\nu $ for which $\\pi _0$ (in $L^2(X_{u},\\kappa )$ ) is not orthogonal to $\\pi _0$ in $L^2(\\nu )$ : $\\int \\pi _0\\otimes \\pi _0\\,d\\rho \\ne 0$ .", "Now, use a subsequence version of the lifting lemma (Theorem 5.16 in [6]) to find $y$ in the subshift defining the Bernoulli automorphism such that $(u,y)$ is generic, along a subsequence $(N_k)=(M_{\\ell _k})$ , for $\\rho $ .", "Then $0\\ne \\int \\pi _0\\otimes \\pi _0\\,d\\rho =\\lim _{k\\rightarrow \\infty }\\frac{1}{N_{k}}\\sum _{n\\le N_{k}}\\pi _0(S^nu)\\pi _0(S^ny)=\\lim _{k\\rightarrow \\infty }\\frac{1}{N_{k}}\\sum _{n\\le N_{k}}u(n)y_n$ which means that $u$ and $y$ correlate along $(N_{k})$ .", "Now the proof of Theorem REF is a straightforward consequence of Lemma REF and Corollary REF ." ], [ "Proof of Lemma ", "Let $X=(X_n)_{n\\in {\\mathbb {Z}}}$ be a positive entropy stationary process as in the statement of the lemma.", "Without loss of generality (considering its real or imaginary part), we can assume that this process takes its values in a finite subset $\\lbrace x_1<x_2<\\cdots <x_r\\rbrace $ of ${\\mathbb {R}}$ .", "We also consider a given probability measure $\\beta $ supported on a possibly different finite subset of ${\\mathbb {R}}$ $\\lbrace y_1<y_2<\\cdots <y_s\\rbrace $ , which is supposed to be non trivial (i.e.", "not reduced to a Dirac measure).", "Thus we can assume that $s\\ge 2$ , and $\\beta (y_j)>0$ for each $1\\le j\\le s$ .", "The purpose of this section is to show how we can construct a stationary coupling of $X$ with a Bernoulli process $Y$ whose distribution is $\\beta ^{\\otimes {\\mathbb {Z}}}$ , in such a way that for each $n\\in {\\mathbb {Z}}$ , ${\\mathbb {E}}[X_n Y_n] > {\\mathbb {E}}[X_n]\\, {\\mathbb {E}}[Y_n].$ We observe that the validity of the preceding inequality is unchanged if we replace $Y_n$ by $Y_n+C$ for a fixed $C$ .", "Thus we can and we do assume without loss of generality that the probability $\\beta $ is such that ${\\mathbb {E}}[Y_n]=0$ .", "To construct the announced coupling, we just assume that, on the probability space where the process $X$ is defined, we also have an i.i.d.", "process $V=(V_n)_{n\\in {\\mathbb {Z}}}$ such that each $V_n$ is uniformly distributed on $[0,1]$ , $V$ is independent of $X$ .", "The construction will be divided into two steps: first we construct an auxiliary (uniform i.i.d.)", "process $U$ and then we use it to construct $Y$ which satisfies the assertion of Lemma REF ." ], [ "Step 1: uniform i.i.d. process $U$", "For $n\\in {\\mathbb {Z}}$ and $j\\in \\lbrace 1,\\ldots ,r\\rbrace $ , we consider the random variable $P_{j,n}$ defined by $P_{j,n} := {\\mathbb {P}}\\bigl ( X_n=x_j \\,|\\, (X_m)_{m\\le n-1}\\bigr ).$ When $j$ is fixed, $(P_{j,n})_{n\\in {\\mathbb {Z}}}$ is a stationary process.", "On the other hand, if we fix $n$ , then $(P_{1,n},\\ldots ,P_{r,n})$ is the conditional distribution of $X_n$ given $(X_m)_{m\\le n-1}$ , in particular we have almost surely $0\\le P_{j,n}\\le 1$ , and $\\sum _{j=1}^r P_{j,n} = 1.$ This allows us to define a random partition of $[0,1[$ into disjoint subintervals $I_{1,n},\\ldots , I_{r,n}$ where for each $j$ , $I_{j,n}$ is the interval of length $P_{j,n}$ defined by $I_{j,n}:=\\left[\\sum _{1\\le i\\le j-1} P_{i,n}\\ ; \\sum _{1\\le i\\le j} P_{i,n}\\right[.$ Then we can define the random variable $U_n$ by $U_n := \\sum _{j=1}^r {1}_{X_n=x_j}\\left(\\sum _{1\\le i\\le j-1} P_{i,n} + V_n P_{j,n}\\right).$ Figure: Definition of U n U_nInformally, if $X_n=x_j$ , we pick $U_n$ uniformly at random (using $V_n$ ) inside $I_{j,n}$ (see Figure REF ).", "Therefore, $\\mathcal {L}\\left(U_n\\,|\\,(X_m)_{m\\le n-1},(V_m)_{m\\le n-1}\\right)=\\mathcal {U}_{[0,1]},$ i.e., it is uniform on $[0,1]$ .", "But all $U_m$ , $m\\le n-1$ , are measurable with respect to $(X_m)_{m\\le n-1}$ and $(V_m)_{m\\le n-1}$ , thus we also have $\\mathcal {L}\\left(U_n\\,|\\,(U_m)_{m\\le n-1}\\right)=\\mathcal {U}_{[0,1]} \\text{ and } \\mathcal {L}\\left(U_n\\right)=\\mathcal {U}_{[0,1]}.$ Indeed, this is just the application of the tower property of conditional expectations: to obtain the left equality, notice that for any measurable $A\\subset [0,1]$ , we have $\\mathbb {P}\\bigl (U_n\\in A &\\,|\\, (U_m)_{m\\le n-1}\\bigl ) \\\\&=\\mathbb {E}\\Bigl [\\underbrace{\\mathbb {P}\\bigl (U_n\\in A\\,|\\, (X_m)_{m\\le n-1},(V_m)_{m\\le n-1}\\bigr )}_{{\\rm Leb}(A)}\\,|\\, (U_m)_{m\\le n-1}\\Bigr ]\\\\&={\\rm Leb}(A).$ Moreover, it also follows from (REF ) that $U$ is i.i.d.", "Note that by construction, $U_n$ is a measurable function of $V_n$ , $X_n$ and $(X_m)_{m\\le n-1}$ , which we abusively write as $U_n = U_n \\left(V_n,X_n,(X_m)_{m\\le n-1}\\right).$ Moreover, whenever we fix realizations $\\xi $ of $(X_m)_{m\\le n}$ and $v$ of $V_n$ then $U_n$ as a function of its second argument is increasing: $U_n(v,x_{j_1},\\xi ) < U_n(v,x_{j_2},\\xi ), \\text{ whenever }x_{j_1}<x_{j_2}.$ We want to define $Y_n$ for a given $n\\in {\\mathbb {Z}}$ .", "We use another partition of $[0,1[$ into subintervals, according to the probability distribution $\\beta $ intended for $Y_n$ : for $1\\le k\\le s$ , set $\\beta _k:=\\beta (y_k)$ and define the interval $J_k:=\\bigl [\\beta _1+\\cdots +\\beta _{k-1};\\beta _1+\\cdots +\\beta _k\\bigr [$ .", "Then we simply define $Y_n$ as a function of $U_n$ by setting $ Y_n := \\sum _{k=1}^s y_k\\,{1}_{J_k}(U_n).$ It follows by the choice of the intervals $J_k$ and by $\\mathcal {L}(U_n)=\\mathcal {U}_{[0,1]}$ that $Y_n$ is distributed according to $\\beta $ .", "Moreover, by the independence of $U$ , we have the independence of $Y$ .", "Thus, $Y$ is a Bernoulli process with distribution $\\beta ^{\\otimes \\mathbb {Z}}$ .", "It remains to prove the announced inequality (REF ).", "Observe that $Y_n$ is, like $U_n$ , constructed as a measurable function of $V_n$ , $X_n$ and $(X_m)_{m\\le n-1}$ , which we also abusively write as $ Y_n = Y_n \\left(V_n,X_n,(X_m)_{m\\le n-1}\\right).", "$ Since $Y_n$ is a non-decreasing function of $U_n$ , we get from (REF ) that for a fixed realization $\\xi $ of $(X_m)_{m\\le n-1}$ and $v$ of $V_n$ , we have for $1\\le j_1 < j_2 \\le r$ $Y_n \\left(v,x_{j_1},\\xi \\right) < Y_n \\left(v,x_{j_2},\\xi \\right)$ and it follows that the map $ x\\in A \\mapsto {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x\\bigr ] $ is non-decreasing.", "Moreover, by the construction of $Y$ , we have $ \\mathcal {L}\\bigl (Y_n \\,|\\, (X_m)_{m\\le n-1}\\bigr ) = \\mathcal {L}(Y_n)=\\beta , $ whence ${\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}\\bigr ] = {\\mathbb {E}}[Y_n] =0.$ Thus there exists $j_0\\in \\lbrace 1,\\ldots ,r\\rbrace $ (depending on $\\xi $ ) such that $\\begin{split}&{\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_j]\\le 0 \\text{ for }1\\le j\\le j_0,\\\\&{\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_j]> 0\\text{ for }j_0+1\\le j\\le r.\\end{split}$ We then have, using (REF ) and (REF ), ${\\mathbb {E}}\\bigl [ X_n Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] & = {\\mathbb {E}}\\bigl [ (X_n-x_{j_0}) Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] \\nonumber \\\\&= \\sum _{j=1}^{j_0} (x_j-x_{j_0})\\, {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_j] \\nonumber \\\\&\\quad + \\sum _{j=j_0+1}^{r} (x_j-x_{j_0})\\, {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_j] \\\\& \\ge 0.", "\\nonumber $ Now, we claim that the announced result is a consequence of the following lemma.", "Lemma 6.6 If the realization $\\xi $ of $(X_m)_{m\\le n-1}$ is such that the conditional distribution $\\mathcal {L}(X_n \\,|\\, (X_m)_{m\\le n-1}=\\xi )$ is non-trivial, then $ {\\mathbb {E}}\\bigl [ X_n Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] > 0.", "$ Indeed, since $X$ has positive entropy, $\\mathcal {L}(X_n \\,|\\, (X_m)_{m\\le n-1})$ is non-trivial with positive probability, and thus we can conclude that $ {\\mathbb {E}}\\bigl [ X_n Y_n\\bigr ] = {\\mathbb {E}}\\Bigl [{\\mathbb {E}}\\bigl [ X_n Y_n \\,|\\, (X_m)_{m\\le n-1}\\bigr ]\\Bigr ] >0.", "$ [Proof of Lemma REF ] We fix a realization $\\xi $ of $(X_m)_{m\\le n-1}$ such that the conditional distribution $\\mathcal {L}\\bigl (X_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr )$ is non-trivial.", "Then the random variables $P_{j,n}$ and the intervals $I_{j,n}$ are fixed, because their values only depend on $\\xi $ .", "Setting $j_1 &:= \\min \\bigl \\lbrace j\\in \\lbrace 1,\\ldots ,r\\rbrace :\\ P_{j,n}>0\\bigr \\rbrace ,\\\\\\text{and}\\quad j_2 &:= \\max \\bigl \\lbrace j\\in \\lbrace 1,\\ldots ,r\\rbrace :\\ P_{j,n}>0\\bigr \\rbrace ,$ we have $j_1<j_2$ .", "Moreover the intervals $I_{j_1,n}$ and $I_{j_2,n}$ are respectively of the form $[0,P_{j_1,n}[$ and $[1-P_{j_2,n},1[$ , with $0<P_{j_1,n}\\le 1-P_{j_2,n}<1$ .", "We now discuss according to the relative position of the interval $I_{j_2,n}$ with respect to the interval $J_1$ (used to define $Y_n$ ).", "Figure: Case 1 (J 1 ∩I j 2 ,n =∅J_1\\cap I_{j_2,n}=\\emptyset )" ], [ "Case 1:", "$J_1\\cap I_{j_2,n}=\\emptyset $ (see Figure REF ).", "Then we have ${\\mathbb {P}}\\bigl (Y_n=y_1 \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_2}\\bigr )=0,$ whereas ${\\mathbb {P}}\\bigl (Y_n=y_1 \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_1}\\bigr )>0.$ Moreover, notice that (REF ) is equivalent to ${\\mathbb {P}}\\bigl (Y_n>y_1 \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_2}\\bigr )=1,$ It follows from (REF ) and (REF ) that there exists a $V_n$ -measurable event $A$ of positive probability such that, on $A$ , $ Y_n\\bigl (V_n,x_{j_2},\\xi \\bigr ) > y_1 = Y_n\\bigl (V_n,x_{j_1},\\xi \\bigr ).", "$ Remembering (REF ), we get ${\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_{j_2}\\bigr ] > {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_{j_1}\\bigr ].$ Figure: Case 2 (J 1 ∩I j 2 ,n ≠∅J_1\\cap I_{j_2,n}\\ne \\emptyset )" ], [ "Case 2:", "$J_1\\cap I_{j_2,n}\\ne \\emptyset $ (see Figure REF ).", "Then $I_{j_1,n}\\subset J_1$ and $I_{j_1,n}\\cap J_s=\\emptyset $ .", "It follows that ${\\mathbb {P}}\\bigl (Y_n=y_s \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_1}\\bigr )=0,$ whereas ${\\mathbb {P}}\\bigl (Y_n=y_s \\,|\\, (X_m)_{m\\le n-1}=\\xi ,X_n=x_{j_2}\\bigr )>0.$ In this case, we get a $V_n$ -measurable event $A$ of positive probability such that, on $A$ , $ Y_n\\bigl (V_n,x_{j_2},\\xi \\bigr ) = y_s > Y_n\\bigl (V_n,x_{j_1},\\xi \\bigr ), $ and as before we conclude that (REF ) holds.", "Now, since (REF ) always holds, and since $ {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] = 0 =\\sum _{j=1}^rP_{j,n}\\, {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_{j}\\bigr ], $ we deduce that $ {\\mathbb {E}}\\bigl [ Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi , X_n=x_{j_2}\\bigr ]>0.", "$ It follows that in the sum (REF ), at least the term corresponding to $j=j_2$ is positive, and this yields $ {\\mathbb {E}}\\bigl [ X_n Y_n \\,|\\, (X_m)_{m\\le n-1}=\\xi \\bigr ] > 0.", "$ Appendix" ], [ "From averaged double to averaged multiple correlations", "This section follows some arguments from [42].", "Remark A.1 In the proof below we will use the following standard fact: let $(x(n))$ be a sequence of complex number bounded by 1.", "Then $\\sum _{m\\le M}|x(m)|=o(M)$ is equivalent to $\\sum _{m\\le M}|x(m)|^2=o(M).$ The little “o” is uniform with respect to $M$ .", "If $\\varepsilon :=\\frac{1}{M}\\sum _{m\\le M}|x(m)|^2$ then by Markov's inequality $\\frac{1}{M}|\\lbrace m\\le M:\\: |c_m|^2\\ge \\varepsilon ^{1/2}\\rbrace |\\le \\frac{1}{\\varepsilon ^{1/2}}\\cdot \\varepsilon =\\varepsilon ^{1/2}$ and then $\\frac{1}{M}\\sum _{m\\le M}|x(m)|=\\frac{1}{M}\\sum _{m\\le M, |x(m)|\\ge e^{1/4}}|x(m)|+\\frac{1}{M}\\sum _{m\\le M, |x(m)|<\\varepsilon ^{1/4}}|x(m)|\\le \\varepsilon ^{1/2}+\\varepsilon ^{1/4}.$ We have the following general lemma: Lemma A.2 Let $(N_\\ell )_{\\ell \\in {\\mathbb {N}}}$ be a sequence of natural numbers.", "For $k\\in \\mathbb {N}$ let $a,b_1,\\ldots b_k\\colon \\mathbb {N} \\rightarrow \\mathbb {C}$ be sequences bounded by 1.", "Assume that $a$ satisfies $\\lim _{H\\rightarrow \\infty }\\frac{1}{H} \\sum _{h\\le H}\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n)a(n+h)\\Big |=0.$ Then $\\lim _{H\\rightarrow \\infty }\\frac{1}{H^k} \\sum _{h_1,\\ldots , h_k\\le H}\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n)\\prod _{i=1}^kb_i(n+h_i)\\Big |=0.$ Notice first that (REF ) can be rewritten as the following: for every $\\varepsilon >0$ , there exists $H_\\varepsilon $ such that for $H>H_\\varepsilon $ and all $\\ell $ sufficiently large (depending on $H$ ), we have $A:=\\frac{1}{H^k}\\sum _{h_1,\\ldots , h_k\\le H}\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n)\\prod _{i=1}^kb_i(n+h_i) \\Big |<\\varepsilon .$ Now, notice that for any $H,N_\\ell ,H^{\\prime }$ and any $h^{\\prime }\\le H^{\\prime }$ , by shifting the summation over $n\\le N_\\ell $ by $h^{\\prime }$ (for every fixed choice of $h_1,\\ldots h_k$ ), we have $\\sum _{h_1,\\ldots , h_k\\le H}\\Big | \\sum _{n\\le N_\\ell }a(n)\\prod _{i=1}^kb_i(n+h_i)\\Big |=$ $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{h^{\\prime }\\le n\\le N_\\ell +h^{\\prime }}a(n)\\prod _{i=1}^kb_i(n+h_i)\\Big |+{\\rm O}(H^k\\cdot h^{\\prime })=$ $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i+h^{\\prime })\\Big |+{\\rm O}(H^k\\cdot h^{\\prime })$ and ${\\rm O}(H^k\\cdot h^{\\prime })={\\rm O}(H^k\\cdot H^{\\prime })$ .", "Notice that as $h_i$ is taken from $[0,H]$ , then $h_i+h^{\\prime }$ is taken from $[h^{\\prime },H+h^{\\prime }]$ (which is a small shift of $[0,H]$ if $h^{\\prime }$ is much smaller than $H$ ).", "So putting $h^{\\prime }$ to the summation over $h_i$ , we get $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i+h^{\\prime })\\Big |=$ $\\sum _{h^{\\prime }\\le h_1,\\ldots , h_k\\le H+h^{\\prime }}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i)\\Big |=$ $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i)\\Big |+ {\\rm O}\\left((h^{\\prime } H^{k-1}+(h^{\\prime })^2H^{k-2}+\\ldots +(h^{\\prime })^k)N_\\ell \\right)$ and ${\\rm O}\\left((h^{\\prime } H^{k-1}+(h^{\\prime })^2H^{k-2}+\\ldots +(h^{\\prime })^k)N\\right)={\\rm O}((H^{\\prime })^kH^{k-1}N_\\ell )$ .", "Putting the two displayed equations together we get that for every $h^{\\prime }\\le H^{\\prime }$ , $A=\\frac{1}{H^k}\\sum _{h_1,\\ldots , h_k\\le H}\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i) \\Big |+{\\rm O}\\Big (\\frac{H^{\\prime }}{N_\\ell }\\Big )+{\\rm O}\\Big (\\frac{(H^{\\prime })^k}{H} \\Big ).$ Averaging the above equation over all $h^{\\prime }\\le H^{\\prime }$ , we get that $A=\\frac{1}{H^k}\\sum _{h_1,\\ldots , h_k\\le H}\\frac{1}{H^{\\prime }}\\sum _{h^{\\prime }\\le H^{\\prime }}\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })G(n)\\Big |+{\\rm O}\\Big (\\frac{H^{\\prime }}{N_\\ell }\\Big )+{\\rm O}\\Big (\\frac{(H^{\\prime })^k}{H} \\Big ),$ where $G(n)=\\prod _{i=1}^kb_i(n+h_i)$ .", "We will now estimate $\\frac{1}{H^{\\prime }}\\sum _{h^{\\prime }\\le H^{\\prime }}\\frac{1}{N_\\ell ^2}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })G(n) \\Big |^2=\\\\\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\Big (\\frac{1}{H^{\\prime }}\\sum _{h^{\\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })}\\Big )G(n)\\overline{G(n^{\\prime })},$ which will be easier to handle than the above expression for $A$ (and then use Remark REF to get rid of the squares).", "Clearly, to obtain an upper bound for (REF ), it suffices to obtain an upper bound for $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime }}\\Big |\\sum _{h^{\\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })} \\Big |.$ Again, it will be easier to deal with $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime 2}}\\Big |\\sum _{h^{\\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })} \\Big |^2$ (and use Remark REF to get rid of the squares).", "Expanding the square again we get that (REF ) is equal to $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })}\\overline{a(n+h^{\\prime \\prime })\\cdot \\overline{a(n^{\\prime }+h^{\\prime \\prime })}}=$ $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n+h^{\\prime \\prime })}a(n^{\\prime }+h^{\\prime \\prime })\\overline{a(n^{\\prime }+h^{\\prime })}.$ The sum in the last term by exchanging the order of summation is equal to $\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}\\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\overline{a(n+h^{\\prime \\prime })}\\Big |^2=$ $\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}\\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h^{\\prime \\prime }-h^{\\prime })}\\Big |^2+ {\\rm O}(\\frac{H^{\\prime 2}}{N_\\ell ^2}).$ Finally, grouping according to $h=h^{\\prime \\prime }-h^{\\prime }$ , we get that that the above is equal to $\\frac{1}{H^{\\prime 2}}\\sum _{|h|\\le H^{\\prime }}|H^{\\prime }-h|\\cdot \\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}\\Big |^2+{\\rm O}(\\frac{H^{\\prime 2}}{N_\\ell ^2})\\le $ $\\frac{1}{H^{\\prime }}\\cdot \\sum _{|h|\\le H^{\\prime }}\\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}\\Big |^2+{\\rm O}(\\frac{H^{\\prime 2}}{N_\\ell ^2}).$ That is, the expression from (REF ) equals $\\frac{1}{H^{\\prime }}\\sum _{|h|\\le H^{\\prime }}\\left|\\frac{\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}}{N_\\ell } \\right|^2 +{\\rm O}\\left(\\frac{(H^{\\prime })^2}{N_\\ell ^2} \\right).$ Now, by the assumption of our lemma, it follows that $\\frac{1}{H^{\\prime }}\\sum _{|h|\\le H^{\\prime }}\\left|\\frac{\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}}{N_\\ell } \\right|={\\rm o}(1),$ which, by Remark REF , is equivalent to $\\frac{1}{H^{\\prime }}\\sum _{|h|\\le H^{\\prime }}\\left|\\frac{\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}}{N_\\ell } \\right|^2={\\rm o}(1).$ Therefore,  (REF ) (and, thus, also (REF )) is of the order of ${\\rm o}(1)+{\\rm O}\\left(\\frac{(H^{\\prime })^2}{N_\\ell ^2} \\right)$ .", "Using again Remark REF , we conclude that also (REF ) is of the same order.", "It follows immediately that also the order of (REF ) is the same.", "Thus, we have proved that $A={\\rm o}\\left(1\\right)+{\\rm O}\\left(\\frac{(H^{\\prime })^2}{N_\\ell ^2} \\right)+{\\rm O}\\left(\\frac{H^{\\prime }}{N_\\ell } \\right)+{\\rm O}\\left(\\frac{(H^{\\prime })^k}{H}\\right).$ Acknowledgments: We would like to thank Tomasz Downarowicz, Nikos Frantzikinakis and Krzysztof Fra̧czek for useful discussions on the paper.", "Research of the second and third authors supported by Narodowe Centrum Nauki grant UMO-2019/33/B/ST1/00364.", "Department of Mathematics, The Maryland University [email protected] Faculty of Mathematics and Computer Science Nicolaus Copernicus University, Toruń, Poland [email protected], [email protected] Laboratoire de Mathématiques Raphaël Salem, CNRS – Université de Rouen Normandie Avenue de l’Université – 76801 Saint Étienne du Rouvray, France [email protected] This section follows some arguments from [42].", "Remark A.1 In the proof below we will use the following standard fact: let $(x(n))$ be a sequence of complex number bounded by 1.", "Then $\\sum _{m\\le M}|x(m)|=o(M)$ is equivalent to $\\sum _{m\\le M}|x(m)|^2=o(M).$ The little “o” is uniform with respect to $M$ .", "If $\\varepsilon :=\\frac{1}{M}\\sum _{m\\le M}|x(m)|^2$ then by Markov's inequality $\\frac{1}{M}|\\lbrace m\\le M:\\: |c_m|^2\\ge \\varepsilon ^{1/2}\\rbrace |\\le \\frac{1}{\\varepsilon ^{1/2}}\\cdot \\varepsilon =\\varepsilon ^{1/2}$ and then $\\frac{1}{M}\\sum _{m\\le M}|x(m)|=\\frac{1}{M}\\sum _{m\\le M, |x(m)|\\ge e^{1/4}}|x(m)|+\\frac{1}{M}\\sum _{m\\le M, |x(m)|<\\varepsilon ^{1/4}}|x(m)|\\le \\varepsilon ^{1/2}+\\varepsilon ^{1/4}.$ We have the following general lemma: Lemma A.2 Let $(N_\\ell )_{\\ell \\in {\\mathbb {N}}}$ be a sequence of natural numbers.", "For $k\\in \\mathbb {N}$ let $a,b_1,\\ldots b_k\\colon \\mathbb {N} \\rightarrow \\mathbb {C}$ be sequences bounded by 1.", "Assume that $a$ satisfies $\\lim _{H\\rightarrow \\infty }\\frac{1}{H} \\sum _{h\\le H}\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n)a(n+h)\\Big |=0.$ Then $\\lim _{H\\rightarrow \\infty }\\frac{1}{H^k} \\sum _{h_1,\\ldots , h_k\\le H}\\lim _{\\ell \\rightarrow \\infty }\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n)\\prod _{i=1}^kb_i(n+h_i)\\Big |=0.$ Notice first that (REF ) can be rewritten as the following: for every $\\varepsilon >0$ , there exists $H_\\varepsilon $ such that for $H>H_\\varepsilon $ and all $\\ell $ sufficiently large (depending on $H$ ), we have $A:=\\frac{1}{H^k}\\sum _{h_1,\\ldots , h_k\\le H}\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n)\\prod _{i=1}^kb_i(n+h_i) \\Big |<\\varepsilon .$ Now, notice that for any $H,N_\\ell ,H^{\\prime }$ and any $h^{\\prime }\\le H^{\\prime }$ , by shifting the summation over $n\\le N_\\ell $ by $h^{\\prime }$ (for every fixed choice of $h_1,\\ldots h_k$ ), we have $\\sum _{h_1,\\ldots , h_k\\le H}\\Big | \\sum _{n\\le N_\\ell }a(n)\\prod _{i=1}^kb_i(n+h_i)\\Big |=$ $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{h^{\\prime }\\le n\\le N_\\ell +h^{\\prime }}a(n)\\prod _{i=1}^kb_i(n+h_i)\\Big |+{\\rm O}(H^k\\cdot h^{\\prime })=$ $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i+h^{\\prime })\\Big |+{\\rm O}(H^k\\cdot h^{\\prime })$ and ${\\rm O}(H^k\\cdot h^{\\prime })={\\rm O}(H^k\\cdot H^{\\prime })$ .", "Notice that as $h_i$ is taken from $[0,H]$ , then $h_i+h^{\\prime }$ is taken from $[h^{\\prime },H+h^{\\prime }]$ (which is a small shift of $[0,H]$ if $h^{\\prime }$ is much smaller than $H$ ).", "So putting $h^{\\prime }$ to the summation over $h_i$ , we get $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i+h^{\\prime })\\Big |=$ $\\sum _{h^{\\prime }\\le h_1,\\ldots , h_k\\le H+h^{\\prime }}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i)\\Big |=$ $\\sum _{h_1,\\ldots , h_k\\le H}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i)\\Big |+ {\\rm O}\\left((h^{\\prime } H^{k-1}+(h^{\\prime })^2H^{k-2}+\\ldots +(h^{\\prime })^k)N_\\ell \\right)$ and ${\\rm O}\\left((h^{\\prime } H^{k-1}+(h^{\\prime })^2H^{k-2}+\\ldots +(h^{\\prime })^k)N\\right)={\\rm O}((H^{\\prime })^kH^{k-1}N_\\ell )$ .", "Putting the two displayed equations together we get that for every $h^{\\prime }\\le H^{\\prime }$ , $A=\\frac{1}{H^k}\\sum _{h_1,\\ldots , h_k\\le H}\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\prod _{i=1}^kb_i(n+h_i) \\Big |+{\\rm O}\\Big (\\frac{H^{\\prime }}{N_\\ell }\\Big )+{\\rm O}\\Big (\\frac{(H^{\\prime })^k}{H} \\Big ).$ Averaging the above equation over all $h^{\\prime }\\le H^{\\prime }$ , we get that $A=\\frac{1}{H^k}\\sum _{h_1,\\ldots , h_k\\le H}\\frac{1}{H^{\\prime }}\\sum _{h^{\\prime }\\le H^{\\prime }}\\frac{1}{N_\\ell }\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })G(n)\\Big |+{\\rm O}\\Big (\\frac{H^{\\prime }}{N_\\ell }\\Big )+{\\rm O}\\Big (\\frac{(H^{\\prime })^k}{H} \\Big ),$ where $G(n)=\\prod _{i=1}^kb_i(n+h_i)$ .", "We will now estimate $\\frac{1}{H^{\\prime }}\\sum _{h^{\\prime }\\le H^{\\prime }}\\frac{1}{N_\\ell ^2}\\Big |\\sum _{n\\le N_\\ell }a(n+h^{\\prime })G(n) \\Big |^2=\\\\\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\Big (\\frac{1}{H^{\\prime }}\\sum _{h^{\\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })}\\Big )G(n)\\overline{G(n^{\\prime })},$ which will be easier to handle than the above expression for $A$ (and then use Remark REF to get rid of the squares).", "Clearly, to obtain an upper bound for (REF ), it suffices to obtain an upper bound for $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime }}\\Big |\\sum _{h^{\\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })} \\Big |.$ Again, it will be easier to deal with $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime 2}}\\Big |\\sum _{h^{\\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })} \\Big |^2$ (and use Remark REF to get rid of the squares).", "Expanding the square again we get that (REF ) is equal to $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n^{\\prime }+h^{\\prime })}\\overline{a(n+h^{\\prime \\prime })\\cdot \\overline{a(n^{\\prime }+h^{\\prime \\prime })}}=$ $\\frac{1}{N_\\ell ^2}\\sum _{n,n^{\\prime }\\le N_\\ell }\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}a(n+h^{\\prime })\\overline{a(n+h^{\\prime \\prime })}a(n^{\\prime }+h^{\\prime \\prime })\\overline{a(n^{\\prime }+h^{\\prime })}.$ The sum in the last term by exchanging the order of summation is equal to $\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}\\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n+h^{\\prime })\\overline{a(n+h^{\\prime \\prime })}\\Big |^2=$ $\\frac{1}{H^{\\prime 2}}\\sum _{h^{\\prime },h^{\\prime \\prime }\\le H^{\\prime }}\\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h^{\\prime \\prime }-h^{\\prime })}\\Big |^2+ {\\rm O}(\\frac{H^{\\prime 2}}{N_\\ell ^2}).$ Finally, grouping according to $h=h^{\\prime \\prime }-h^{\\prime }$ , we get that that the above is equal to $\\frac{1}{H^{\\prime 2}}\\sum _{|h|\\le H^{\\prime }}|H^{\\prime }-h|\\cdot \\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}\\Big |^2+{\\rm O}(\\frac{H^{\\prime 2}}{N_\\ell ^2})\\le $ $\\frac{1}{H^{\\prime }}\\cdot \\sum _{|h|\\le H^{\\prime }}\\Big |\\frac{1}{N_\\ell }\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}\\Big |^2+{\\rm O}(\\frac{H^{\\prime 2}}{N_\\ell ^2}).$ That is, the expression from (REF ) equals $\\frac{1}{H^{\\prime }}\\sum _{|h|\\le H^{\\prime }}\\left|\\frac{\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}}{N_\\ell } \\right|^2 +{\\rm O}\\left(\\frac{(H^{\\prime })^2}{N_\\ell ^2} \\right).$ Now, by the assumption of our lemma, it follows that $\\frac{1}{H^{\\prime }}\\sum _{|h|\\le H^{\\prime }}\\left|\\frac{\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}}{N_\\ell } \\right|={\\rm o}(1),$ which, by Remark REF , is equivalent to $\\frac{1}{H^{\\prime }}\\sum _{|h|\\le H^{\\prime }}\\left|\\frac{\\sum _{n\\le N_\\ell }a(n)\\overline{a(n+h)}}{N_\\ell } \\right|^2={\\rm o}(1).$ Therefore,  (REF ) (and, thus, also (REF )) is of the order of ${\\rm o}(1)+{\\rm O}\\left(\\frac{(H^{\\prime })^2}{N_\\ell ^2} \\right)$ .", "Using again Remark REF , we conclude that also (REF ) is of the same order.", "It follows immediately that also the order of (REF ) is the same.", "Thus, we have proved that $A={\\rm o}\\left(1\\right)+{\\rm O}\\left(\\frac{(H^{\\prime })^2}{N_\\ell ^2} \\right)+{\\rm O}\\left(\\frac{H^{\\prime }}{N_\\ell } \\right)+{\\rm O}\\left(\\frac{(H^{\\prime })^k}{H}\\right).$ Acknowledgments: We would like to thank Tomasz Downarowicz, Nikos Frantzikinakis and Krzysztof Fra̧czek for useful discussions on the paper.", "Research of the second and third authors supported by Narodowe Centrum Nauki grant UMO-2019/33/B/ST1/00364.", "Department of Mathematics, The Maryland University [email protected] Faculty of Mathematics and Computer Science Nicolaus Copernicus University, Toruń, Poland [email protected], [email protected] Laboratoire de Mathématiques Raphaël Salem, CNRS – Université de Rouen Normandie Avenue de l’Université – 76801 Saint Étienne du Rouvray, France [email protected]" ] ]
2105.11737
[ [ "Extranatural Flux Inflation" ], [ "Abstract We propose a new inflation scenario in flux compactification, where a zero mode scalar field of extra components of the higher dimensional gauge field is identified with an inflaton.", "The scalar field is a pseudo Nambu-Goldstone boson of spontaneously broken translational symmetry in compactified spaces.", "The inflaton potential is non-local and finite, which is protected against the higher dimensional non-derivative local operators by quantum gravity corrections thanks to the gauge symmetry in higher dimensions and the shift symmetry originated from the translation in extra spaces.", "We give an explicit inflation model in a six dimensional scalar QED, which is shown to be consistent with Planck 2018 data." ], [ "Introduction", "The hierarchy problem has been considered as one of the guiding principles to study the physics beyond the Standard Model (SM) of particle physics.", "In the SM, the quantum corrections to the mass of Higgs field are sensitive to the square of the ultraviolet cutoff scale of the theory, which is typically Planck scale or the scale of grand unified theory.", "Since the cutoff scale is much larger than the 125 GeV Higgs mass, solving the hierarchy problem requires an unnatural fine-tuning of parameters or a new physics beyond the SM around TeV scale.", "Although the latter approaches have been mainly studied so far, no signature of new physics has been discovered, which is likely to increase a discrepancy between the new physics scale and the Higgs mass.", "Therefore, it seems to be desirable such that the Higgs mass vanishes at the classical level and is generated at the quantum effects by the dynamics different from the new physics one.", "As one of the approaches to the hierarchy problem, the higher dimensional theory with magnetic flux compactification recently attracts attention.", "Originally, the magnetic flux compactification has been studied in string theory [1], [2] and many attractive properties has been known even in the field theory: attempt to explain the number of the generations of the SM fermion [3], computation of Yukawa coupling hierarchy [4], [5], [6].", "Remarkably, it has recently been shown that the quantum corrections to the masses of the scalar zero-mode field originated from extra components of higher dimensional gauge field (called Wilson-line (WL) scalar field) are canceled [7], [8], [9], [10], [11], [12].", "The physical reason of the cancellation is that the WL scalar field becomes a Nambu-Goldstone boson of the spontaneously broken translational symmetry in compactified spaces and the shift symmetry from the translation in compactified spaces forbids the non-derivative terms as well as the mass term of the WL scalar field.", "Therefore, the massless nature of the WL scalar field is expected to be valid at any order of perturbation.", "In order to apply the scenario of the flux compactification where the WL scalar field is identified with Higgs field to the hierarchy problem, we need some mechanism to generate an explicit breaking term of the translational symmetry in compactified space and the WL scalar field must be a pseudo NG boson such as pion.", "In our previous paper [13], we have studied a possibility of realizing nonvanishing WL scalar mass in flux compactification.", "By generalizing loop integrals of the quantum correction to WL scalar mass at one-loop, we derived the conditions for the loop integral and mode sum to be finite.", "We further classified the four-point and three-point interaction terms explicitly breaking the translational symmetry in compactified space and generating the finite WL scalar mass at one-loop.", "Using the simplest interactions of them, we have explicitly shown the finite WL scalar mass at one-loop in a six dimensional scalar QED.", "Inflation is a very attractive scenario to solve many problems in the standard Big Bang cosmology and its existence is supported by observations of cosmological parameters[14].", "Inflation has been considered to happen by a scalar field called as inflaton.", "Although many models of inflation has been proposed so far, there is still no compelling model of inflation.", "In a slow-roll scenario of the inflation, the scalar potential is required to be flat and stable under quantum gravity corrections, which usually causes an unnatural fine-tuning of parameters of the theory unless we have some dynamics or symmetry to control the inflaton dynamics.", "For instance, the inflaton in natural inflation [15] is identified with the pseudo Nambu-Goldstone boson of some global symmetry.", "In extranatural inflation [16], the inflaton is identified with the WL scalar field of the gauge field in higher dimensions without magnetic flux.", "In [17], the inflaton and the curvaton are identified with the WL scalar fields in a six dimensional gauge theory.", "In this paper, we propose a new scenario of inflation in flux compactification where the WL scalar field is identified with an inflaton.", "Since this is a model of “extranatural inflation\" [16] with magnetic flux in compactified space, we refer to our scenario as “extranatural flux inflation\".The extranatural inflation is an inflation model where the idea of gauge-Higgs unification is applied.", "For the discussion of UV insensitivity for the WL scalar mass and potential, see [18] In constructing a slow-roll inflation model, it is important to realize a flat inflaton potential stable against quantum corrections and quantum gravity corrections since the slow-roll conditions typically require the large field value larger than the Planck scale for the inflaton.", "Unless this stability is guaranteed by some physical reasons, we cannot discuss the inflation by an effective field theory description because it is beyond the range of their applicability.", "Similar to the extranatural inflation [16], the inflaton potential in our scenario is controlled by the gauge symmetry in higher dimensions.", "The non-derivative local operators of the WL scalar fields are forbidden by the gauge symmetry, and by the shift symmetry originated from the translational symmetry in compactified spaces in a context of flux compactification.", "Therefore, the higher dimensional operators suppressed by the Planck scale, which is expected to be generated by the black holes, are forbidden.", "The quantum corrections to the local operators (by non-gravity interactions) are also forbidden.", "The remarkable nature is that the non-local Wlison line operators of the WL scalar field are generated by quantum effects and become finite inspite of the non-renormalizable theory.", "This is a great advantage which is absent in other scenarios.", "As a simplest model, we consider a six dimensional scalar QED which was discussed in [13].", "In this model, the WL scalar field is identified with the inflaton and the one-loop Coleman-Weinberg potential for the WL scalar field is calculated.", "The potential is found to be finite as expected.", "Then, the slow-roll parameters, the spectral index, the scalar-to-tensor ratio, and the e-folding are numerically computed and some viable parameter sets are found.", "This paper is organized as follows.", "In the next section, our model is introduced and one-loop effective potential is calculated.", "Identifying the WL scalar field with an inflaton, we discuss our inflation scenario by applying the one-loop effective potential to the inflaton potential in section 3.", "In section 4, our numerical analysis is found and some viable parameter sets consistent with Planck 2018 data are shown.", "A final section gives our summary." ], [ "Our setup", "In this section, we introduce our model and calculate the one-loop effective potential." ], [ "Flux compactifiaction", "We consider a six-dimensional U(1) gauge theory with a constant magnetic flux couples to a scalar field in the bulk.", "The six-dimensional spacetime is a product of four-dimensional Minkowski spacetime $M^4$ and two-dimensional torus $T^2$ .", "The Lagrangian we consider is $\\mathcal {L}=-\\frac{1}{4}F_{MN}F^{MN}-(D_M\\Phi )^*D^M\\Phi +\\kappa (\\bar{\\phi }\\overline{\\Phi }\\Phi +\\phi \\overline{\\Phi }\\Phi ),$ where the spacetime index is given by $M,N=0,1,2,3,5,6,~\\mu ,\\nu =0,1,2,3,~m,n=5,6$ respectively and we follow the metric convention as $\\eta _{MN}=(-1,+1,\\cdots ,+1)$ .", "The field strength and the covariant derivative of U(1) gauge field $A_M$ are defined by $F_{MN}=\\partial _M A_N-\\partial _N A_M,~D_M=\\partial _M-ig A_M$ with a gauge coupling constant $g$ .", "$\\Phi $ is a bulk scalar field.", "$\\phi $ is a scalar field related to the complex combination of $A_{5,6}$ and will be defined in detail later.", "$\\kappa $ is a dimensionless coupling constant.", "The third term in (REF ) is introduced to generates nonvanishing quantum corrections to the mass of $\\phi $ at one-loop [13].", "In the context of this paper, this term is also crucial to obtain a nonvanishing one-loop effective potential of $\\phi $ as well as the mass term.", "This is due to the property that the scalar field $\\phi $ is the Nambu-Goldstone boson of the translational symmetry in compactified spaces, which forbids non-derivative terms such as the mass and potential terms and then the explicit breaking terms for the translational symmetry are required.", "Let us introduce the magnetic flux in our model.", "The magnetic flux is given by the nontrivial background (or vacuum expectation value (VEV)) of fifth and sixth component of the gauge field $A_{5,6}$ , which must satisfy their classical equation of motion $\\partial ^m\\mathinner {\\langle {F_{mn}}\\rangle }=0$ .", "In our flux compactification, the background of $A_{5,6}$ is chosen as $\\mathinner {\\langle {A_5}\\rangle }=-\\frac{1}{2}fx_6,~~~\\mathinner {\\langle {A_5}\\rangle }=\\frac{1}{2}fx_5,$ which introduces a constant magnetic flux density $\\mathinner {\\langle {F_{56}}\\rangle }=f$ with a real number $f$ .", "Note that this solution breaks an extra-dimensional translational invariance spontaneously.", "Integrating over $T^2$ , the magnetic flux is quantized as follows $\\frac{g}{2 \\pi } \\int _{T^{2}} d x_{5} d x_{6}\\left\\langle F_{56}\\right\\rangle =\\frac{g}{2 \\pi } L^{2} f=N \\in \\mathbb {Z},$ where $L^2$ is an area of two-dimensional torus.", "It is useful to define $\\partial ,z,$ and $\\phi $ as $\\partial \\equiv \\partial _z=\\partial _5-i\\partial _6,~~~z=\\frac{1}{2}(x_5+ix_6),~~~\\phi =\\frac{1}{\\sqrt{2}}(A_6+iA_5).$ Note that the VEV of $\\phi $ is given by $\\mathinner {\\langle {\\phi }\\rangle }=(\\mathinner {\\langle {A_6}\\rangle }+i\\mathinner {\\langle {A_5}\\rangle })/\\sqrt{2}=f\\bar{z}/\\sqrt{2}$ .", "We expand $\\phi $ around the flux background $\\phi =\\mathinner {\\langle {\\phi }\\rangle }+\\varphi $ , where $\\varphi $ is a quantum fluctuation.", "To distinguish $\\varphi $ from an introduced bulk scalar $\\Phi $ , we call $\\varphi $ Wilson line (WL) scalar field." ], [ "Kaluza-Klein mass spectrum", "We need to derive Kaluza-Klein mass spectrum of the bulk scalar field $\\Phi $ for the calculation of effective potential.", "To begin, we define the covariant derivatives in the complex coordinates $D,\\bar{D}$ as $D&=D_5-iD_6=\\partial -\\sqrt{2}g\\phi =\\mathcal {D}-\\sqrt{2}g\\varphi , \\\\\\bar{D}&=D_5+iD_6=\\bar{\\partial }+\\sqrt{2}g\\bar{\\phi }=\\bar{\\mathcal {D}}+\\sqrt{2}g\\bar{\\varphi },\\\\\\mathcal {D}&=\\mathcal {D}_5-i\\mathcal {D}_6=\\partial -\\sqrt{2}g\\mathinner {\\langle {\\phi }\\rangle }, \\\\\\bar{\\mathcal {D}}&=\\bar{\\mathcal {D}}_5+i\\bar{\\mathcal {D}}_6=\\bar{\\partial }+\\sqrt{2}g\\mathinner {\\langle {\\bar{\\phi }}\\rangle }.$ Next, we regard $\\mathcal {D},\\bar{\\mathcal {D}}$ as creation and annihilation operators by $a=\\frac{1}{\\sqrt{2gf}}i\\bar{\\mathcal {D}},~~~a^\\dag =\\frac{1}{\\sqrt{2gf}}i\\mathcal {D},$ which satisfy the commutation relation $[a,a^\\dag ]=1$ .", "This correspondence is an analogy to the quantum mechanics in magnetic field.", "Hereafter, we denote $\\alpha =2gf=4\\pi N/L^2$ .", "We summarize the property of creation and annihilation operators.", "The ground state mode functions $\\xi _{0,j},~\\bar{\\xi }_{0,j}$ are determined by $a\\xi _{0,j}=0,~a^\\dag \\bar{\\xi }_{0,j}=0$ , where $j=0,\\cdots ,|N|-1$ accounts for the degeneracy of the ground state.", "Creation and annihilation operators act on mode functions as $a^\\dag \\xi _{n,j}=\\sqrt{n+1}\\xi _{n+1,j},~~~a\\xi _{n,j}=\\sqrt{n}\\xi _{n-1,j}, $ and we can construct the higher mode function $\\xi _{n,j}$ in the same way as the harmonic oscillator (in detail, see [19]) $\\xi _{n,j}=\\frac{1}{\\sqrt{n!", "}}(a^\\dag )^n\\xi _{0,j},~~~\\bar{\\xi }_{n,j}=\\frac{1}{\\sqrt{n!", "}}(a)^n\\bar{\\xi }_{0,j},$ where $n=0,1,2\\cdots $ is Landau level.", "The higher mode function satisfies an orthonormality condition $\\int _{T^2}d^2x \\bar{\\xi }_{n^{\\prime },j^{\\prime }}\\xi _{n,j}=\\delta _{n,n^{\\prime }}\\delta _{j,j^{\\prime }}.$ Finally, we extract the mass term from the second term in (REF ).", "Focusing on the extra-dimensional part of the second term in (REF ), we obtain $\\mathcal {L}_{scalar~mass}&=-\\mathcal {D}_m\\overline{\\Phi } \\mathcal {D}^m\\Phi \\nonumber \\\\&=-\\overline{\\Phi }\\alpha \\left(a^\\dag a+\\frac{1}{2}\\right)\\Phi .$ Noting that $a^\\dag a$ is a number operator, the KK mass of bulk scalar field is given by $m^2_{scalar}=\\alpha \\left(n+\\frac{1}{2}\\right).", "$" ], [ "Four-dimensional effective Lagrangian", "Using the expansion of $\\phi =\\mathinner {\\langle {\\phi }\\rangle }+\\varphi $ , the Lagrangian (REF ) is accordingly deformed as $\\mathcal {L}&\\supset -\\frac{1}{4} F^{\\mu \\nu } F_{\\mu \\nu }-D_{\\mu } \\overline{\\Phi } D^{\\mu } \\Phi -m^2_{scalar}\\bar{\\Phi }\\Phi \\nonumber \\\\&\\quad - i g \\sqrt{2\\alpha } \\bar{\\varphi } \\bar{\\Phi } a^{\\dagger } \\Phi + i g \\sqrt{2\\alpha } \\varphi \\bar{\\Phi } a \\Phi -2 g^{2} \\bar{\\varphi } \\varphi \\overline{\\Phi } \\Phi \\nonumber \\\\&\\quad +\\kappa (\\bar{\\varphi }\\overline{\\Phi }\\Phi +\\varphi \\overline{\\Phi }\\Phi )+\\kappa (\\mathinner {\\langle {\\bar{\\phi }}\\rangle }\\overline{\\Phi }\\Phi +\\mathinner {\\langle {\\phi }\\rangle }\\overline{\\Phi }\\Phi ),$ where we note that the unnecessary terms in our discussion are omitted.", "To derive a four-dimensional effective Lagrangian by KK reduction, we need to expand $\\Phi $ in terms of mode functions $\\xi _{n,j}$ $\\Phi =\\sum _{n,j}\\Phi _{n,j}\\xi _{n,j}.", "$ Integrating over $T^2$ , the four-dimensional effective Lagrangian is obtained by $\\mathcal {L}_{4D}&=-\\frac{1}{4} F^{\\mu \\nu } F_{\\mu \\nu }-\\partial ^{\\mu } \\bar{\\varphi } \\partial _{\\mu } \\varphi \\nonumber \\\\&\\quad +\\sum _{n, j}\\left(-D_{\\mu } \\overline{\\Phi }_{n, j} D^{\\mu } \\Phi _{n, j}-\\alpha \\left(n+\\frac{1}{2}\\right)\\overline{\\Phi }_{n, j} \\Phi _{n, j}\\right.", "\\nonumber \\\\&\\quad - i g \\sqrt{2\\alpha (n+1)} \\overline{\\varphi } \\overline{\\Phi }_{n+1, j} \\Phi _{n, j}+ i g \\sqrt{2\\alpha (n+1)}\\varphi \\overline{\\Phi }_{n, j} \\Phi _{n+1, j}-2g^2\\bar{\\varphi }\\varphi \\overline{\\Phi }_{n,j}\\Phi _{n,j} \\nonumber \\\\&\\quad +\\kappa \\bar{\\varphi }\\overline{\\Phi }_{n,j}\\Phi _{n,j}+\\kappa \\varphi \\overline{\\Phi }_{n,j}\\Phi _{n,j}+\\kappa \\mathinner {\\langle {\\phi }\\rangle }_I\\overline{\\Phi }_{n,j}\\Phi _{n,j}+\\kappa \\mathinner {\\langle {\\bar{\\phi }}\\rangle }_I\\overline{\\Phi }_{n,j}\\Phi _{n,j}\\Big ), $ where $\\mathinner {\\langle {\\phi }\\rangle }_I$ and $\\mathinner {\\langle {\\bar{\\phi }}\\rangle }_I$ are expressed by $\\mathinner {\\langle {\\phi }\\rangle }_I=\\int _{T^2}d^2 x \\mathinner {\\langle {\\phi }\\rangle }\\bar{\\xi }_{n,j}\\xi _{n^{\\prime },j^{\\prime }},~~~\\mathinner {\\langle {\\bar{\\phi }}\\rangle }_I=\\int _{T^2}d^2 x \\mathinner {\\langle {\\bar{\\phi }}\\rangle }\\bar{\\xi }_{n,j}\\xi _{n^{\\prime },j^{\\prime }}.$ When $\\mathinner {\\langle {\\phi }\\rangle }=f\\bar{z}/\\sqrt{2}$ , $\\mathinner {\\langle {\\phi }\\rangle }_I$ and $\\mathinner {\\langle {\\bar{\\phi }}\\rangle }_I$ lead to zero because of odd function with respect to integral variables $z$ or $\\bar{z}$ .", "In the following, we omit the third and fourth terms in the fourth line of (REF )." ], [ "One-loop effective potential", "One-loop effective potential is described as $V(\\varphi ,\\bar{\\varphi })=N \\sum _{n=0}^\\infty \\int \\frac{d^{4} k}{(2 \\pi )^{4}}\\ln \\left(k^2+\\alpha \\left(n+\\frac{1}{2}\\right) + M^2(\\varphi ,\\bar{\\varphi })\\right),$ where we have taken into account loop contributions from the bulk scalar field $\\Phi $ .", "$N$ is a number of the degeneracy.", "$M^2(\\varphi ,\\bar{\\varphi })$ is a field-dependent mass for the bulk scalar field $\\Phi $ .", "As for this $M^2(\\varphi ,\\bar{\\varphi })$ , we consider two limiting cases for a free parameter $U(1)$ gauge coupling, namely $g \\ll 1$ and $g \\gg 1$ .", "For that purpose, we read $M^2(\\varphi ,\\bar{\\varphi })$ from (REF ) as $M^2(\\varphi ,\\bar{\\varphi })&=-\\kappa \\bar{\\varphi }-\\kappa \\varphi + 2g^2\\bar{\\varphi }\\varphi .$ While only the first two terms in (REF ) are considered in the $g \\ll 1$ case, the last term proportional to $g^2$ in (REF ) is also considered in the $g \\gg 1$ case in addition to the first two terms.", "In the case of $g \\simeq {\\cal O}(1)$ , the terms linear in $g$ in (REF ) should be also taken into account in $M^2(\\varphi ,\\bar{\\varphi })$ .", "However, the obtained eigenvalues of $M^2(\\varphi ,\\bar{\\varphi })$ become complicated and makes the computation of the effective potential hard.", "Therefore, we do not discuss this case in this paper.", "We can express the effective potential by using Schwinger's proper time as $V&=- N \\sum _{n=0}^\\infty \\int \\frac{d^{4} k}{(2 \\pi )^{4}}\\int _{0}^{\\infty }\\frac{dt}{t}e^{-k^2 t-\\alpha \\left(n+\\frac{1}{2}\\right)t}e^{-M^2(\\varphi ,\\bar{\\varphi })t} \\nonumber \\\\&=- N \\frac{1}{16\\pi ^2}\\int _{0}^{\\infty }\\frac{dt}{t^3}\\frac{e^{-\\frac{\\alpha }{2}t}}{1-e^{-\\alpha t}}e^{-M^2(\\varphi ,\\bar{\\varphi })t}.", "$ To proceed a calculation of the effective potential further, we notice an integral representation of Hurwitz zeta function $\\zeta [s,a]=\\frac{1}{\\Gamma (s)}\\int _{0}^{\\infty }dt\\frac{t^{s-1}e^{-at}}{1-e^{-t}},~~~\\mathrm {Re}~s>1.$ Then, the effective potential and its derivatives by $\\varphi $ can be expressed by $V&=- N \\frac{\\alpha ^2}{16\\pi ^2}\\lim _{\\epsilon \\rightarrow 0}\\Gamma (\\epsilon -2)\\zeta \\left[\\epsilon -2,\\frac{1}{2}+\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi })\\right], \\\\V_{\\varphi }&=- N \\frac{\\alpha \\kappa }{16\\pi ^2}\\lim _{\\epsilon \\rightarrow 0} \\Gamma (\\epsilon -1)\\zeta \\left[\\epsilon -1,\\frac{1}{2}+\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi })\\right], \\\\V_{\\varphi \\bar{\\varphi }}&= - N \\frac{\\kappa ^2}{16\\pi ^2}\\lim _{\\epsilon \\rightarrow 0}\\Gamma (\\epsilon )\\zeta \\left[\\epsilon ,\\frac{1}{2}+\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi })\\right],$ where a parameter $\\epsilon $ is introduced to regularize the integral of $t$ .", "In particular, we can check that the $\\epsilon \\rightarrow 0$ limit indeed agrees with the results in case of $M^2(\\varphi ,\\bar{\\varphi })=0$ obtained in[13] by diagrammatic calculations using the dimensional regularization.", "In the $g\\ll 1$ case, we ignore $2g^2\\bar{\\varphi }\\varphi $ in (REF ) as mentioned above.", "For convenience, we define the dimensionless variables in a four dimensional sense as $z=\\frac{\\varphi }{M_P},~~~y=M_P\\frac{\\kappa }{\\alpha },$ $M^2(\\varphi ,\\bar{\\varphi })/\\alpha $ is then expressed by $\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi })=-(z+\\bar{z})y=-2xy, \\quad \\mathrm {Re}~z=x.$ Thus, the effective potential is rewritten by $V=- N \\frac{\\alpha ^2}{16\\pi ^2}\\lim _{\\epsilon \\rightarrow 0}\\Gamma (\\epsilon -2)\\zeta \\left[\\epsilon -2,\\frac{1}{2}-2xy\\right],$ and the effective potential is shown in figure REF .", "Figure: Schematic picture of the effective potential in the case of g≪1g\\ll 1.The blue and yellow lines shows y=1.0×10 0 y=1.0\\times 10^{0}, y=1.0×10 -1 y=1.0\\times 10^{-1} respectively.If $L\\sim M^{-1}_P$ , the effective potential is close to flat as $y$ (or $\\kappa $ ) takes smaller value.", "Taking into account for the consistency with the original theory [7], [9], the small value of $y$ is favored.", "If $y\\ll 1$ , $\\kappa $ is small, which is independent of $g$ .", "This implies that linear terms in $g$ can be neglected because we can always take $g\\ll \\kappa \\ll 1$ .", "In the $g\\gg 1$ case, $M^2(\\varphi ,\\bar{\\varphi })/\\alpha $ is expressed by $\\frac{1}{\\alpha } M^2(\\varphi ,\\bar{\\varphi })&=-(z+\\bar{z})y+2\\frac{g^2M^2_P}{\\alpha }|z|^2\\nonumber \\\\&=-2uy+2G(u^2+v^2), $ where $z\\equiv u+i v$ and $G \\equiv g^2 M^2_P/\\alpha $ are defined in the second equality.", "Note that $G$ is almost an order of $g^2$ because $\\alpha $ is independent of $g$ .", "Setting $u=v$ for simplicity, the effective potential is given by $V=- N \\frac{\\alpha ^2}{16\\pi ^2}\\lim _{\\epsilon \\rightarrow 0}\\Gamma (\\epsilon -2)\\zeta \\left[\\epsilon -2,\\frac{1}{2}-2uy+4Gu^2\\right],$ which is shown in figure REF .", "Figure: Schematic picture of the effective potential in the case of g≫1g\\gg 1.We take y=1y=1 for simplicity.The yellow and blue lines shows G=1.0×10 2 G=1.0\\times 10^{2}, G=1.0×10 3 G=1.0\\times 10^{3} respectively.This effective potential in the case of $g\\gg 1$ behaves as $V\\propto \\Gamma [\\epsilon -2]\\zeta [\\epsilon -2,4Gu^2]$ .", "Comparing with the potential in figure REF , it seems difficult to apply the potential in figure REF to an inflation model." ], [ "Inflationary parameters", "Using the four-dimensional effective potential for the WL scalar field (REF ), we propose a cosmological inflation model in flux compactifiaction, where the WL scalar field is identified with an inflaton.", "Slow-roll parameters $\\epsilon $ and $\\eta $ in our model are given by $\\epsilon &= \\frac{M^2_P}{2}\\left(\\frac{V_\\varphi }{V}\\right)^2=\\frac{M^2_P}{2}\\left(\\frac{\\kappa }{\\alpha }\\lim _{\\epsilon \\rightarrow 0}\\frac{\\Gamma (\\epsilon -1)}{\\Gamma (\\epsilon -2)}\\frac{\\zeta \\left[\\epsilon -1,\\frac{1}{2}+\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi })\\right]}{\\zeta \\left[\\epsilon -2,\\frac{1}{2}+\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi })\\right]}\\right)^2, \\\\\\eta &=M^2_P\\frac{V_{\\varphi \\varphi }}{V}=M^2_P\\left(\\frac{\\kappa ^2}{\\alpha ^2}\\lim _{\\epsilon \\rightarrow 0}\\frac{\\Gamma (\\epsilon )}{\\Gamma (\\epsilon -2)}\\frac{\\zeta \\left[\\epsilon ,\\frac{1}{2}+\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi })\\right]}{\\zeta \\left[\\epsilon -2,\\frac{1}{2}+\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi })\\right]}\\right).$ Noting that Hurwitz zeta function can be expressed by Bernoulli polynomials $B_n(x)$ as follows $\\zeta [-n,x]=-\\frac{B_{n+1}(x)}{n+1},$ we can further simplify (REF ) and (), $\\epsilon &=\\frac{y^2}{2}\\left(-2\\frac{\\zeta \\left[-1,\\frac{1}{2}+\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi })\\right]}{\\zeta \\left[-2,\\frac{1}{2}+\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi })\\right]}\\right)^2=\\frac{9y^2}{2}\\left(\\frac{B_2(\\frac{1}{2}+\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi }))}{B_3(\\frac{1}{2}+\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi }))}\\right)^2, \\\\\\eta &=y^2\\left((-1)(-2)\\frac{\\zeta \\left[0,\\frac{1}{2}+\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi })\\right]}{\\zeta \\left[-2,\\frac{1}{2}+\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi })\\right]}\\right)=6y^2\\frac{B_1(\\frac{1}{2}+\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi }))}{B_3(\\frac{1}{2}+\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi }))}.$ Slow-roll conditions to realize inflation require $\\epsilon \\ll 1, |\\eta | \\ll 1$ .", "The number of e-folding before the end of inflation is $N_*=\\frac{1}{M^2_P}\\int _{\\varphi _{f}}^{\\varphi _*}\\frac{V}{V_{\\varphi }}d\\varphi =\\frac{2}{3y}\\int _{\\varphi _*}^{\\varphi _f}\\frac{B_3\\left(\\frac{1}{2}+\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi })\\right)}{B_2\\left(\\frac{1}{2}+\\frac{1}{\\alpha }M^2(\\varphi ,\\bar{\\varphi })\\right)} d\\varphi .$ To solve the horizon and flatness problems, the number of e-folding $N_*$ has to be at least $50<N_*<60$ .", "$\\varphi _f$ is the value of the end of inflation determined by $\\epsilon (\\varphi _f)=1$ , which violates the slow-roll conditions.", "$\\varphi _*$ is determined so that the e-folding can satisfy $50<N_*<60$ .", "The spectral index and the tensor-to-scalar ratio are given in a slow-roll approximation as $n_s=1-6\\epsilon +2\\eta ,~~~r=16\\epsilon .$ Planck 2018 data [14] gives constraints on $n_s=0.9649 \\pm 0.0042$ and $r < 0.10$ ." ], [ "Numerical results", "In this section, our numerical results are shown." ], [ "$g\\ll $ 1 case", "In this case, $M^2(\\varphi ,\\bar{\\varphi })/\\alpha $ corresponds to (REF ), where the slow-roll parameters $\\epsilon $ and $\\eta $ are provided by $\\epsilon &=\\frac{9y^2}{2}\\left(\\frac{B_2(\\frac{1}{2}-2xy)}{B_3(\\frac{1}{2}-2xy)}\\right)^2,~~~\\eta =6y^2\\frac{B_1(\\frac{1}{2}-2xy)}{B_3(\\frac{1}{2}-2xy)}.$ To compute the e-folding $N_*$ , we need to know the value of end of inflation $x_f=\\varphi _f/M_P$ , which is determined by the condition of the end of inflation $\\epsilon (x_f)=1$ .", "The number of e-folding is $N_*=\\frac{2}{3y}\\int _{x_i}^{x_f}\\frac{B_3(\\frac{1}{2}-2xy)}{B_2(\\frac{1}{2}-2xy)}dx,$ where $x_i=\\mathrm {Re}\\varphi _*/M_P$ .", "Sample of our numerical solutions $x_i, x_f, N_*$ at some points of $y$ are shown in Table REF , where the e-folding $N_*=50, 60$ are taken.", "One might think that our results are not reliable since the WL scalar field value is quite larger than the Planck scale, which is beyond an applicability of the effective field theory.", "As mentioned in introduction, the gauge symmetry in our theory is not broken by quantum gravity effects and forbids any dangerous higher dimensional local operators suppressed by the Planck scale as well as the non-derivative local operators of the WL scalar field.", "Therefore, our results are reliable.", "Table: Sample of our numerical solutions x i ,x f ,N * x_i, x_f, N_* at some points of yy.Using the numerical solutions in Table REF , the slow-roll parameters $\\epsilon , \\eta $ , the spectral index $n_s$ , and the scalar-to-tensor ratio $r$ are calculated and shown in Table REF .", "Comparing our results in Table REF with $n_s$ and $r$ in Planck 2018 data, our results are found to be relatively good agreement with the data.", "If $y$ is taken to be a large value such as $y=1.0\\times 10^{2}$ , $n_s$ and $r$ cannot be satisified with Planck 2018 data.", "Table: Inflation parameters ϵ,η,n s ,r\\epsilon , \\eta , n_s, r obtained from our model.Figure: Our results (table ) in the n s n_s-rr plot from Planck 2018 data .Orange circles are our results,where the small and large ones represent N * =50N_*=50 and N * =60N_*=60, respectively.Our results are shown in $(n_s, r)$ plot of Figure REF from Planck 2018 data [14].", "Orange circles are our results where small and large ones corresponds to $N_*=50$ and $N_*=60$ , respectively.", "As the parameter $y$ is decreased, our results in $(n_s, r)$ plot go downward.", "Our results are within a parameter region indicating the combining data of Planck TT, TE, EE+lowE+lensing at CL95%." ], [ "$g\\gg $ 1 case", "In this case, $M^2(\\varphi ,\\bar{\\varphi })$ corresponds to (REF ).", "Under $u=v$ , $\\epsilon $ and $\\eta $ are expressed by $\\epsilon =\\frac{9y^2}{2}\\left(\\frac{B_2(\\frac{1}{2}+4Gu^2)}{B_3(\\frac{1}{2}+4Gu^2)}\\right)^2,~~~\\eta =6y^2\\frac{B_1(\\frac{1}{2}+4Gu^2)}{B_3(\\frac{1}{2}+4Gu^2)},$ where we ignore $-2uy$ because $y$ is small.", "The number of e-folding is $N_*=\\frac{2}{3y}\\int _{u_i}^{u_f}\\frac{B_3(\\frac{1}{2}+4Gu^2)}{B_2(\\frac{1}{2}+4Gu^2)}du.$ As in the $g\\ll 1$ case, We obtain the value of $u_i$ and $u_f$ for a value of $G$ .", "Taking $G=1,0\\times 10^{3}$ as an example, we find $u_i=-3.8403$ and $u_f=-0.7282$ .", "Using these values, $n_s$ and $r$ are $n_s=0.99569$ and $r=0.0206896$ .", "$r$ is consistent with Planck 2018 constraint, but $n_s$ is not.", "Thus, comparing with the potential in $g\\ll 1$ case, the potential in the $g\\gg 1$ case is not suitable for the inflation." ], [ "Summary", "In this paper, we have proposed a new inflation model in flux compactification, which is refered to as “extranatural flux inflation\".", "In this model, the WL scalar field, which is originated from extra components of the gauge field in higher dimensions, is identified with an inflaton field.", "The great advantage of our model is that the inflation potential is protected by the gauge symmetry of the theory from the dangerous higher dimensional local operators by quantum gravity effects.", "Our inflation potential is nonlocal and finite, which is generated by quantum corrections.", "Therefore, the extranatural inflation with flux is very predictable regardless of the non-renormalizable theory.", "We have considered a model of six dimensional scalar QED in flux compactification and calculated an inflation potential.", "We have shown that our model in the weak gauge coupling case is consistent with Planck 2018 data." ], [ "Acknowledgments", "We would like to thank Kazumasa Okabayashi for useful discussions and comments." ] ]
2105.11782
[ [ "Towards Compact Single Image Super-Resolution via Contrastive\n Self-distillation" ], [ "Abstract Convolutional neural networks (CNNs) are highly successful for super-resolution (SR) but often require sophisticated architectures with heavy memory cost and computational overhead, significantly restricts their practical deployments on resource-limited devices.", "In this paper, we proposed a novel contrastive self-distillation (CSD) framework to simultaneously compress and accelerate various off-the-shelf SR models.", "In particular, a channel-splitting super-resolution network can first be constructed from a target teacher network as a compact student network.", "Then, we propose a novel contrastive loss to improve the quality of SR images and PSNR/SSIM via explicit knowledge transfer.", "Extensive experiments demonstrate that the proposed CSD scheme effectively compresses and accelerates several standard SR models such as EDSR, RCAN and CARN.", "Code is available at https://github.com/Booooooooooo/CSD." ], [ "Introduction", "Single Image Super-Resolution (SISR) aims to reconstruct a high-resolution (HR) image given a low-resolution (LR) image.", "Recently, convolutional neural networks (CNNs) [12], [15] have dominated SR approaches by directly learning a mapping from LR input to HR output with a deep neural network.", "However, the existing methods [34], [27] tend to explore complex and delicate network architectures for recovering edge structures and missing texture details.", "These networks consume significant amounts of memory and computational cost and are therefore impractical to apply in resource-limited devices such as wearables and IoT.", "Several SR model compression methods have been proposed to remove the redundant parameters.", "Recursive models [30], [13] share the SR network's major blocks and reduces the model size and number of parameters.", "However, the recursive nature still results in time-consuming inference procedures as the model cost up to 15s per 2K image with 4$\\times $ SR on one Intel i9-10980XE CPU [13].", "Parameter quantization [18], [14] can reduce memory storage by converting the parameters into lower bits.", "However, the computational complexity is still high as the models are rarely fully quantized.", "For example, [18] quantize parameters but still use full-precision activations.", "Similarly, [14] uses additional float scales and low-bit weights/activations to approximate the original full-precision weights/activations, still requiring float-based convolutional computation.", "Figure: Visual comparison of CSSR-Net with different losses.We aim to simultaneously compress and accelerate SR models.", "We propose a simple self-distillation framework inspired by [32], in which a student network is split from a teacher (target) network by using parts of teacher's channels in each layer.", "We term this student network as Channel-Splitting Super-Resolution network (CSSR-Net).", "The teacher and student networks are trained jointly to form two SR models with different computation.", "According to different computation resources in devices, we can dynamically allocate these two models, i.e.", "select CSSR-Net if exceeding the required computation overhead in the limited-resource devices, and the teacher model otherwise.", "Training CSSR-Net and its teacher network jointly using a reconstruction loss implicitly transfers knowledge from teacher to CSSR-Net.", "Implicit knowledge transfer considers to transfer the knowledge from the teacher to CSSR-Net by weight sharing and joint training, which only provides limited internal knowledge.", "This will result in low PSNR/SSIM and low quality HR images on CSSR-Net.", "Recently, perceptual loss [11] have been widely used as additional knowledge to improve the quality, but the results of CSSR-Net still suffer from blurred lines and edges, as shown in Fig.", "REF .", "This is due to limited external knowledge of the perceptual loss that uses only ground-truth HR images as the upper bound of CSSR-Net and teacher.", "This begs our rethinking: Why not select multiple negative samples as the lower bounds to reduce the optimization space and provide more explicit knowledge to improve the student's performance?", "Figure: The framework of our proposed CSD.", "Here we choose EDSR as our backbone for example.", "The darker green parts of the CSSR-Net (student) are shared from its teacher.", "The student and teacher separately produce a high-resolution image constrained by the reconstruction loss.", "The rich knowledge is constructed by contrastive loss and explicitly transferred from teacher to student, where the output O S O_S is pulled to closer to the O T O_T and pushed far away from the negative samples in the embedding space of VGG network.To answer the above question, we propose a contrastive self-distillation (CSD) scheme to explicitly transfer the knowledge from teacher to CSSR-Net using contrastive loss (CL), which is inspired by contrastive learning [21], [3], [6].", "As shown in Fig.", "REF , SR model compression and acceleration is accomplished by CSSR-Net as a student.", "Meanwhile, CSD receives rich internal and external knowledge from reconstruction error and CL, respectively.", "There are two “opposing forces” on CL; One pulls the output of CSSR-Net closer to its teacher, the other one pushes the output of CSSR-Net farther away from the negative images in the latent feature space.", "CL constrains the output of CSSR-Net into the closed upper and lower bounds, which shows better quality images (see Fig.", "REF ).", "Our main contributions could be summarized as follows: The proposed CSD scheme as a universal method can simultaneously compress and accelerate various SR networks which is also runtime friendly for practical use.", "Self-distillation is introduced to compress and accelerate SR models, while contrastive loss is proposed to further improve the performance of CSSR-Net by effective knowledge transferring.", "Extensive experiments demonstrate the effectiveness of our CSD scheme .", "For example, on Urban100, the compressed EDSR+ achieves $4\\times $ compression rate and $1.77\\times $ speedup, with only a minor loss of 0.13db PSNR and 0.0039 SSIM at the resolution scale of $\\times $ 4." ], [ "Single Image Super-Resolution", "DCNNs based Super-resolution methods [15], [34], [1] has shown impressive performance for SR in recent years.", "To improve the visual effect of the reconstructed images, the perceptual loss [11] and the adversarial loss [27] were introduced.", "Both losses are effective as regularizers, but cannot prevent blurry areas as they are limited by the information of the ground truth images as an upper bound.", "Moreover, they require heavy parameters and computations to improve the quality of HR images, making them impractical to be employed on resource-limited embedded devices.", "Several approaches have been proposed to reduce parameters for SR; they can be divided into recursive-based SR [30], [13], quantization-based SR[18], [14] and compact architecture-based SR[36], [10].", "Recently, AdderSR [25] utilize adder neural networks to avoid massive energy consumptions while GhostSR [20] reduced parameters by generating ghost features.", "For recursive-based SR and quantization-based SR, they are difficult to accelerate SR models in practical applications, In contrast, Our CSD scheme can simultaneously compress and accelerate various off-the-shelf SR models.", "Our CSD can also be integrated with compact architecture based SR models, which are orthogonal to our core contribution." ], [ "Knowledge Distillation", "Knowledge Distillation (KD) [7] is a teacher-student framework that transfers information from a full teacher network to a compact student network.", "[16] achieve accelerating and compressing via low-rank decomposition with knowledge transfer.", "Recently, self-distillation has been proposed [35], [23], where knowledge is distilled within the network itself.", "[17] proposes a stage-based self-distillation method termed as MetaDistiller by transferring the knowledge from deeper stages to earlier stages.", "However, few self-distillation methods are exploited for SR task.", "Different from MetaDistiller, our teacher and student have the same number of stages and knowledge is constructed explicitly via the contrastive loss rather than using a label generator to produce soft targets.", "In line with our work, [32] proposed slimmable neural networks executing different widths on various classification models for dynamic resource adjustment.", "Training with different widths and switchable batch normalization is a class of implicit knowledge distillation which works well for image classification.", "However, it will leads to sub-optimal results [31] when directly applying these methods into SR task.", "In contrast, we introduce a contrastive loss to explicitly transfer the knowledge from teacher to CSSR-Net via the closed upper and lower bound constraints, which significantly improve the performance of CSSR-Net." ], [ "Contrastive Learning", "Contrastive losses are widely used in self-supervised learning [21], [6], [3] and aim to pull the anchor close to positive points while pushing it away from negative points in the representation space.", "Previous works [3], [29] have applied contrastive learning in high-level tasks.", "[26] presents a contrastive distillation framework to capture correlations of structured representations for image classification.", "Our work on the other hand uses contrastive learning to provide external knowledge with upper and lower bounds for SR tasks.", "Instead of using two separate networks, we adopt a self-distillation framework, facilitating dynamic resource adjustment by using different channel rates in the teacher model.", "Recently, [22] employed contrastive learning to improve unpaired image-to-image translation quality.", "[28] proposed a contrastive regularization to improve the performance of various SOTA dehazing networks.", "Inspired by [28], we introduce a contrastive loss for the SR task; contrastive learning is unexplored for SR and different from [22] and [28], we also introduce a new way to generate negative samples." ], [ "Method", "Our CSD contains two parts: CSSR-Net and contrastive loss (CL).", "First, we describe the CSSR-Net.", "Then, we present our CL to construct the upper and lower bound for CSSR-Net.", "Finally, the overall loss function of CSD scheme is presented and solved by a new optimization strategy." ], [ "Channel-Splitting Super-Resolution Networks", "The channel-splitting super-resolution network (CSSR-Net) is a network constructed by splitting any CNN-based super-resolution network in the channel dimension.", "The CSSR-Net can be considered as a student network while the original network from which it is derived is a teacher network.", "The pair can be used to construct a self-distillation framework and implicitly transferring the knowledge from the teacher to the student.", "CSSR-Net is entangled with the teacher network as it shares a portion of weights from the teacher, as illustrated in the left panel of Fig.", "REF .", "The width of the CSSR-Net is controlled by a manually set scale factor $r_w$ multiplied by the teacher's width uniformly at all layers.", "For example, $r_w = 0.5$ , corresponds to the CSSR-Net retaining half of the width or number of channels as the teacher in all layers.", "Inspired by [32], we can jointly minimize the reconstruction loss for both CSSR-Net and the teacher as: ${\\leavevmode \\xbox {resizebox}{\\XMLaddatt {width}{388.57156pt}\\begin{aligned}L_{Rec} &= \\sum _i^N L_1(O_S^{(i)}, I_{GT}^{(i)})+\\lambda _TL_1(O_T^{(i)}, I_{GT}^{(i)}\\big ), \\\\&=\\sum _i^N L_1\\big (f^S(I^{(i)},\\theta _S), I_{GT}^{(i)}\\big )+\\lambda _TL_1\\big (g^T(I^{(i)},\\theta _T), I_{GT}^{(i)}\\big ),\\end{aligned} }}$ where $O_S^{(i)}=f^S(I^{(i)},\\theta _S)$ and $O_T^{(i)}=g^T(I^{(i)},\\theta _T)$ are the output of the CSSR-Net $f^S$ and the teacher network $g^T$ on the LR input $I^{(i)}$ with parameters $\\theta _S$ and $\\theta _T$ respectively.", "$\\theta _S$ is shared from $\\theta _T$ and satisfy $\\theta _S\\subset \\theta _T$ .", "$I_{GT}^{(i)}$ is the ground-truth HR image and $N$ is the number of training images.", "Directly minimizing Eq.", "REF via stochastic gradient descent (SGD) leads to sub-optimal results in SR task (See the results in Table REF ).", "As a result, CSSR-Net and the teacher network converge with worse results than the corresponding individual training.", "Moreover, the generated SR images have blurry parts, e.g.", "the area on dense straight lines (see Fig.", "REF ).", "We speculate the following reason: implicit KD is not strong enough to provide insightful information by two independent loss terms.", "We expect to add explicit knowledge, which can provide richer internal and external knowledge.", "Therefore, we introduce contrastive learning to explicitly construct a relationship between the student and the teacher, also providing the closed upper and lower bound to improve performances of both CSSR-Net and the teacher.", "Upper bound is constructed to pull the output of CSSR-Net to teacher's, while lower bound is to constrain CSSR-Net's output to be far away from negative samples (e.g.", "bicubic upsampled images)." ], [ "Contrastive Loss", "Contrastive learning has improved representation learning by pulling anchors close to positive samples while pushing away negative samples [21], [22], [3], [6].", "We propose a novel contrastive loss (CL) to explicitly represent knowledge for training both the teacher network and CSSR-Net.", "For contrastive learning, we need to consider two aspects: One is to construct the “positive” and “negative” samples, the other is to find the latent feature space to compare the samples.", "For the former, we construct the output $O_S^{(i)}$ of CSSR-Net and the output $O_T^{(i)}$ of its teacher as the anchor and the positive sample, respectively.", "More negative samples may better cover the undesired distribution.", "We thus sample $K$ images (other than the anchor) from the same mini-batch to $I^{(i)}$ as negative samples.", "Then upsample them to the same resolution as $O_S^{(i)}$ via bicubic interpolation.", "Each of them is denoted by $O_{Neg}^{(k)},k=1,2\\cdots ,K$ .", "More detailed experimental results on numbers of negative samples are shown in experiments.", "For the latent feature space, we use intermediate features of a pre-trained model $\\phi $ (e.g.", "VGG [24]).", "Given positive and negative samples, we can construct the contrastive loss as: $L_{CL}=\\sum _{i}^N\\sum _{j}^M \\lambda _j\\frac{d\\big (\\phi _j(O_S^{(i)}),\\phi _j(O_T^{(i)})\\big )}{\\sum _{k}^K d\\big (\\phi _j(O_S^{(i)}),\\phi _j(O_{Neg}^{(k)})\\big )},$ where $\\phi _j, j=1,2,\\cdots M$ is the intermediate features from the $j$ -th layer of pre-trained model.", "$M$ is the number of total hidden layers.", "$d(x,y)$ is the L1-distance loss between $x$ and $y$ .", "$\\lambda _j$ is the balancing weight for each layer.", "We do not update the parameters of pre-trained model $\\phi $ during training.", "The contrastive loss is presented in Fig.", "REF right panel; the loss introduces opposing forces pulling the output of CSSR-Net $O_S^{(i)}$ to the output of its teacher $O_T^{(i)}$ and pushing $O_S^{(i)}$ to the negative samples $O_{Neg}^{(k)}$ .", "Note that our contrastive loss is different from InfoNCE [21], which uses a dot product-based similarity.", "Instead of this similarity, our L1-distance loss achieves better performance (See experiments for more details).", "Additionally related to our CL is the perceptual loss [11], which minimizes the distance loss between the student and the ground-truth from multi-layer features of the pre-trained VGG.", "This is an upper bound to constrain the student network.", "Unlike the perceptual loss, however, we also adopt multiple negative samples as a lower bound to reduce the solution space and further improve the performance of CSSR-Net and its teacher." ], [ "Overall Loss.", "The overall loss of our CSD scheme is constructed by leveraging contrastive loss Eq.", "REF into reconstruction loss Eq.", "REF , which can be formulated as: $L(\\theta _S, \\theta _T)=L_{Rec}+\\lambda _{C}L_{CL},$ where $\\lambda _{C}$ is a hyper-parameter for balancing $L_{Rec}$ and $L_{CL}$ ." ], [ "Solver.", "Since our CSSR-Net and its teacher are entangled, we need to update both gradients from them.", "One naive solver is to update all parameters (i.e., $\\theta _S$ and $\\theta _T$ ) by directly minimize Eq.", "REF based on SGD.", "However, the teacher simply becomes weaker rather than maintain good performance.", "As a solution, we detach teacher's gradients from the contrastive loss and only update gradients from reconstruction loss.", "For student, we take normal gradient updating of Eq.", "REF .", "Pseudocode of CSD scheme is summarized in Algorithm .", "0mm-2.5mmfalseflexibleblackdkgreenmauvetrue # f_t: Pre-trained teacher network # width_mult: the width of network f_s # lt, lc: lambda_t, lambda_c in Eq. 1, Eq.", "3 initialize() for lr, hr in loader: # load a minibatch     neg = bic(generate()) # negative samples     o_s = f_s.forward(lr) # anchor     o_t = f_t.forward(lr) # positive samples     # reconstruction loss, Eq.", "1     loss = L1(o_s, hr) + lt * L1(o_t, hr)     # No gradient to o_t from contrastive loss     o_t.detach()     # contrastive loss, Eq.", "2     vgg_s, vgg_t, vgg_n = VGG19(o_s, o_t, neg)     loss += lc * CL(vgg_s, vgg_t, vgg_n)     loss.backward() # update" ], [ "Implementation Details.", "Our CSD scheme is implemented by PyTorch 1.2.0 and MindSpore 1.2.0[9] with one NVIDIA TITAN RTX GPU.", "The models are trained with ADAM optimizer by setting $\\beta _1 = 0.9, \\beta _2 = 0.999$ , and $\\epsilon = 10^{-8}$ .", "The batch size and total epochs are set to 16 and 300 epochs, respectively.", "The initial learning rate is $10^{-4}$ and decayed by 10$\\times $ at every $2\\times 10^5$ iterations.", "For the latent features in Eq.", "REF , We extract the features from the 1st, 3rd, 5th, 9th and 13th layers of the pre-trained VGG-19 while with the corresponding coefficients $\\lambda _i, i=1,\\cdots 5 $ to $\\frac{1}{32}, \\frac{1}{16}, \\frac{1}{8}, \\frac{1}{4}$ and 1, respectively.", "We set the hyper-parameters $\\lambda _T$ and $ \\lambda _C$ in Eq.", "REF to 1 and 200, respectively.", "The scale factor for the width of CSSR-Net is default set to 0.25 if not specially stated.", "The input is randomly cropped into patches and augmented with random horizontal flip and $90^{\\circ }$ rotation to produce the fixed $192\\times 192$ HR patches during training." ], [ "Datasets.", "We train all SR models with 800 training images on DIV2K and evaluate on the 100 validation images.", "We additionally test on four SR benchmarks: Set5[2], Set14[33], BSD100[19] and Urban100[8]." ], [ "Evaluation Metric.", "We calculate PSNR and SSIM on the Y channel, also evaluate the compression rate and real speedup on one NVIDIA TITANX RTX GPU." ], [ "Teacher Backbones.", "To validate our CSD scheme, we choose models with different sizes and structures (EDSR+[15], RCAN+[34] and CARN+[1]“+” means self-ensemble technique [15]) is applied into the networks) as teacher backbones, upon which we construct their corresponding CSSR-Nets as students.", "Note that CARN+ is the most compact network with only 1.1M parameters, compared to RCAN+ (15.6M) and EDSR+ (43.1M)." ], [ "Baselines.", "We select individually trained CSSR-Net and its teacher as our baseline.", "We also compare the proposed CSD with CSSR-Net, which only optimizes Eq.", "REF via joint training, denoted as J-T1." ], [ "Quantitative Results", "We first compress EDSR+, as shown in Table REF .", "EDSR+ 1.0$\\times $ denotes the original EDSR+ model as a teacher, while EDSR+ 0.25$\\times $ presents a student with 2.7M parameters by setting $r_w = 0.25$ .", "Compared to the teacher, EDSR+ 0.25$\\times $ achieves 16.0$\\times $ compression rate.", "We can observe that (1) Joint training (J-T1) achieves the consistent PSNR/SSIM with baseline (i.e.", "individual training) both in student and teacher at the same SR scale.", "For dynamic inference, the baseline needs to store two individual models with different parameters , while J-T1 only requires the parameter of the teacher; (2) Compared to baseline and J-T1, CSD achieves the best performance both in student and teacher at the same SR scale, except the SSIM metric of teacher at SR scale of 2 on DIV2K.", "For example, on Urban100, EDSR+ 0.25$\\times $ using CSD achieves the PSNR gains over baseline by 0.26dB and 0.24dB at the SR scale of 2 and 4, respectively; (3) Using CSD, our EDSR+ 0.25$\\times $ model achieves 16$\\times $ compression rate, only with the loss of 0.2db PSNR at the SR scale of 4 (i.e.", "29.13dB vs. 29.33dB in baseline) on DIV2K.", "We further compress RCAN+ and CARN+, which are more compact.", "Additionally, we set $r_w = 0.5$ .", "As shown in Fig.", "4, our CSD achieves higher PSNR at the SR scale of 4 on Urban100, compared to baseline.", "For example, on Urban100 $\\times 4$ , our CSD-RCAN+ 0.5$\\times $ achieves 0.12dB PSNR gains over B-RCAN+ 0.5$\\times $ .", "We also found that CARN+ compression is relatively difficult, which is due to the smallest number of redundant parameters.", "However, our CSD still achieves higher PSNR, compared to the baseline based on individual training.", "Moreover, compared to SOTA SR quantization methods (i.e.", "PAMS-4bit [14] and PACT-4bit [5]), our CSD achieves 0.19dB and 0.37dB PSNR gains over PAMS and PACT, while with the smallest parameter number of 0.3M (vs. 0.48MThis number has been converted to 32-bit float number.", "both in PAMS and PACT).", "For fair comparison to evaluate the speedup, we report the total inference time across Urban100 dataset with SR scalar of 4 (See Fig.", "REF ).", "Our CSD achieves $3.9\\times $ compression rate and $1.38\\times $ GPU speedup rate only with 0.2dB PSNR loss, compared to the original model." ], [ "Qualitative Results", "As shown in Fig.", "REF , we compare our CSD with other training methods on the quality of the enhanced images at $4\\times $ SR scale on Urban100, BSD100 and DIV2K.", "For simplicity, we select the student model EDSR+ $0.25\\times $ for visual comparison.", "We can see that both outputs images based on individual training (baseline) and joint training (CSSR-Net) are blurry, especially in the areas where straight lines are very dense.", "In contrast, our CSD can effectively alleviate the blurry problem, which means that high frequency information is effectively recovered.", "More examples are presented in Supplementary.", "Figure: PSNR-Speed" ], [ "Ablation Study", "To demonstrate the effectiveness of the proposed CSD scheme, we conduct ablation study to analyze the effect of contrastive loss, S-T distillation, updating strategies and the number of negative samples.", "EDSR+ is selected as our backbone in ablation study." ], [ "Effect of InfoNCE Loss, Perceptual Loss and CL.", "InfoNCE loss [21], [6] has been widely used in contrastive learning.", "It uses the dot-product operation to measure the distance between two vectors.", "Instead of InfoNCE loss, we use contrastive loss in Eq.", "REF based on L1-distance measurement.", "As shown in Tab.", "REF , our CL achieves the best results, compared to InfoNCE and J-T with perceptual loss.", "For example, Our CSD based on L1-distance achieves 0.0014 and 0.0041 SSIM gains over InfoNCE based CSD on DIV2K and Urban100, respectively.", "Table: Comparison of different contrastive losses and S-T distillations on EDSR+ 0.25×\\times model on Urban100 ×\\times 4." ], [ "Effect of S-T Distillation", "We further evaluate the effectiveness of our CSSR-Net.", "We consider three S-T distillation strategies: (1) W/O T (GT Pos.)", "only use ground-truth HR images as positive samples, which remove the teacher's branch; (2) CSD (GT Pos.)", "uses ground-truth HR images as postive samples in CSD scheme; (3) T-S Separate where the student does not share weights with the teacher.", "As shown in Table REF , our CSSR-Net enables self-distillation, which receives richer information from its entangled teacher.", "It is worth noticing that the performance of CSD is better than CSD (GT Pos.).", "We speculate that HR images provide a stronger upper bound which is more difficult for the limited capacity S to fully exploit.", "This is consistent with findings in [4], where better teachers do not necessarily help to train better students." ], [ "Effect of Updating Strategy.", "Fig.", "REF presents a comparison of different updating strategies on CL.", "CSD-A removes the reconstruction loss for $T$ in Eq.", "REF ; CSD-B directly minimize Eq.", "REF by both updating weights of $S$ and $T$ from the gradient of CL; CSD detaches $T$ 's gradient from CL.", "Obviously, our CSD achieves significantly higher performance, compared to CSD-A and CSD-B.", "In CSD-A, $T$ collapses quickly in the early stage during training and is unable to provide valid information to $S$ , resulting in much worse performance.", "In CSD-B, $T$ tends to learn the student such that $T$ become weaker during training." ], [ "Effect of the Number of Negative Samples.", "Finally, we explore the effect of the number of negative samples.", "As shown in Fig.", "REF , adding more negative samples achieves better performance.", "However, training memory and time grows as the number of negative samples increases.", "For the performance-efficiency trade-off, we choose the number of 10 negative samples in our all experiments." ], [ "Conclusion", "In this paper, we propose a novel CSD scheme for simultaneously compressing and accelerating SR models, which consists of CSSR-Net and contrastive loss.", "Constructed from the target (teacher) network, CSSR-Net shares the part weights, directly compressing the target network.", "To mine for rich knowledge from the teacher, a novel contrastive loss is proposed for explicit knowledge transfer, which ensures that the output of CSSR-Net is pulled closer to the teacher's and pushed far away from the blurry images.", "We have comprehensively evaluated the performance of CSD scheme for compressing and accelerating various SR models on standard benchmark datasets with superior performance.", "We believe the proposed CSD scheme can be generalized to other low-level vision tasks (e.g.", "dehazing, denoising and debluring), which will be explored in future work." ], [ "Acknowledgements", "This work is sponsored by National Natural Science Foundation of China (61972157, 61772524, 61876161, 61902129), Natural Science Foundation of Shanghai (20ZR1417700), CAAI-Huawei MindSpore Open Fund, National Key Research and Development Program of China (2019YFC1521104), Zhejiang Lab (2019KD0AC02, 2020NB0AB01), the Fundamental Research Funds for the Central Universities, Shanghai Sailing Program (21YF1411200).", "We thank MindSpore[9] for their partial support." ] ]
2105.11683
[ [ "Minimizing the Number of Wireless Charging PAD for UAV-Based Wireless\n Rechargeable Sensor Networks" ], [ "Abstract In wireless rechargeable sensor networks (WRSNs), most of researches address energy scarcity by introducing one or multiple ground mobile vehicles to recharge energy-hungry sensor nodes.", "The charging efficiency is limited by the moving speed of ground chargers and rough environments, especially in large-scale scenarios or challenging scenarios such as separate islands.", "To address the limitations, some researchers consider replacing ground mobile chargers with lightweight unmanned aerial vehicles (UAVs) to support extremely large-scale scenarios, because of the UAV moving at higher speed without geographical limitation.", "Moreover, multiple automatic landing wireless charging PADs are deployed in the network to recharge UAVs automatically.", "In this work, we investigate the problem of introducing the minimal number of PADs in UAV-based WRSNs.", "We propose a novel and adaptive PAD deployment scheme named CDC & DSC that can adapt to arbitrary locations of the base station, arbitrary geographic distributions of sensor nodes, and arbitrary sizes of network areas.", "In the proposed scheme, we first obtain an initial PAD deployment solution by clustering nodes in geographic locations.", "Then, we propose a center shift combining algorithm to optimize this solution by shifting the location of PADs and attempting to merge the adjacent PADs.", "The simulation results show that compared to existing algorithms, our proposed scheme can use fewer PADs to charge the whole network." ], [ "Introduction", "Wireless Rechargeable Sensor Networks (WRSNs) provide a promising approach to prolong the lifetime of sensor networks by introducing mobile chargers to recharge energy-hungry nodes [1], [2], [3].", "In the most of researches, mobile chargers are generally ground vehicles, such as mobile cars and intelligent robots.", "However, there are some limitations on using ground vehicle chargers.", "It is common that sensor nodes may be deployed in complex and cross-distribution geographical conditions such as rivers, land, islands, and rugged mountains.", "In these challenging scenarios, the complex terrain may hinder the movement of ground vehicles from one node to another.", "On the other hand, the lightweight Unmanned Aerial Vehicle (UAV) with high speed can adapt to large-scale scenarios or challenging scenarios regardless of geographical constraints [4], [5].", "A few studies have considered introducing the UAV as a mobile charger in WRSNs [4], [5], [6], [7].", "In large-scale scenarios or challenging scenarios, the UAV works better than ground chargers for charging nodes but it consumes more energy in flight.", "It is impracticable to directly increase the battery size because of the negative impacts on cost and performance [8].", "Therefore, [9], [10] developed a Wireless Charging Station (PAD) to recharge the UAV automatically based on some special Wireless Power Transmission (WPT) systems [11], [12].", "By introducing PADs, the UAV can get energy supplement during charging flights in large-scale scenarios or challenging scenarios.", "[13] first introduced PADs into UAV-based WRSNs and proposed four heuristic algorithms to minimize the number of PADs.", "However, these algorithms can only work in the network with uniform node distribution and the central base station(BS).", "Obviously, they are not practicable and adaptable for typical application scenarios of UAV-based WRSNs.", "In this work, inspired by [13], we investigate the problem of introducing the minimal number of PADs in a UAV-based WRSN (Minimizing the Number of Deployed PADs, MNDP).", "Unlike the work in [13], our work aims to minimize the number of PADs in scenarios with arbitrary node distribution and arbitrary BS location to exploit the advantages of the UAV.", "We first define and formulize this problem and then propose a novel PAD deployment scheme Clustering-with-double-constraints and Disks-shift-combining (CDC${\\&}$ DSC) to address this problem.", "CDC${\\&}$ DSC scheme works in two phases.", "In the first phase, we propose the CDC (Clustering-with-double-constraints) algorithm, to generate an initial solution to our problem.", "In the second phase, we propose the DSC (Disks-shift-combining) algorithm to optimize the initial solution by shifting the locations of PADs to their nearest neighbors and trying to merge the adjacent PADs.", "Simulations show that our scheme can adaptively deploy fewer PADs in UAV-based WRSNs than any scheme in [13]." ], [ "Literature Reveiw", "Most of the existing studies in WRSNs focused on using ground vehicles recharging nodes.", "[1] designed an algorithm to maximize charging throughput.", "[2] focused on the joint optimization of data gathering and energy replenishment.", "Moreover, [3] proposed a multi-chargers collaborative charging scheme to improve the shortages of a single charger in large-scale WRSNs.", "However, the speed limitations and moving obstructive challenges of ground chargers cannot adapt to large-scale scenarios or challenging scenarios.", "Recent work applies the lightweight UAV in WRSNs to adapt to large-scale scenarios or challenging scenarios to overcome geographical barriers.", "[14] developed a specialized WPT system that supports the UAV to transfer power to ground sensor nodes.", "[4] proposed a spatial discretization scheme to obtain a UAV charging spot set and maximize the overall charging energy.", "[5] considered the UAV hovering energy consumption based on the model by [15] to minimize the number of hovering points.", "Considering the shortages of a single UAV, [7] discussed the multi-UAVs charging problem to minimize the number of UAVs.", "Although these above schemes have improved the energy efficiency of UAVs during charging tasks, it is still necessary to recharge UAVs to extend the working time and the working range of UAVs, especially in large-scale or challenging WRSNs [8].", "[11] designed a lightweight and efficient WPT system to charge UAV and [12] achieved the larger charging area by multiple extended coils.", "Further, some researchers developed a small-scale charging station that can be deployed flexibly.", "[9] devised an automatic landing pad (PAD) to transfer energy wirelessly through a pair of lightweight induction coils when the UAV is hovering.", "[10] introduced two automatic charging platforms with direct contact way and wireless way to adapt to different kinds of UAVs.", "Based on the above studies, [13] introduced PADs into UAV-based WRSNs, so that the UAV with low residual energy can fly to a nearby PAD and replenish its energy.", "To minimize the number of PADs, [13] proposed four heuristic algorithms for PAD deployment, namely MSC, TNC, GNC, and DC algorithms.", "However, these algorithms only consider the central BS and the uniform distribution of nodes, and cannot verify the superiority of UAVs in large-scale scenarios or challenging scenarios." ], [ "Preliminaries", "In this section, we first introduce the network model and the UAV consumption model, and then define and formulize the MNDP problem." ], [ "Network Model", "We consider a large monitoring area with a BS, N static sensor nodes, a UAV as the mobile charger, and several PADs.", "Let $S = \\lbrace {s_1},{s_2},...,{s_N}\\rbrace $ denotes the set of sensor nodes and $P = \\lbrace {p_1},{p_2},...,{p_M}\\rbrace $ denotes the set of PADs.", "Since BS and PADs work similarly when charging the UAV, for convenience of description, they are sometimes collectively referred to as the charging stations in this work.", "Let ${p_0}$ denotes the BS and ${P^{^{\\prime }}} = \\lbrace {p_0},{p_1},{p_2},...,{p_M}\\rbrace $ denotes the set of charging stations.", "We also use ${p_i} \\in P^{\\prime }$ to denote the location of the PAD/BS deployed.", "To simplify the problem, the energy supply of each charging station (the BS or a PAD) is unlimited.", "Each sensor node ${s_i} \\in S$ is powered by a rechargeable battery with energy capacity $e$ and deployed statically on a given location $l({s_i})$ that is known by the BS and the UAV.", "However, the distribution of all nodes deployment is unknown.", "The BS is also deployed arbitrarily according to the network requirement and it is responsible for data gathering, charging schedule, and serving the UAV." ], [ "UAV Consumption Model", "The UAV is powered by a rechargeable battery with high capacity ${E_{max}}$ .", "The UAV charges sensor nodes in the point-to-point charging pattern, which means the UAV has to hover over and fully charge only one node.", "The UAV flies at a constant speed ${V_U}$ between nodes, PADs and the BS.", "The UAV has to fly to the nearest charging station before its energy runs off.", "During the charging process, the UAV can only get recharging from the charging stations.", "The UAV departs from a charging station with full energy to charge sensor nodes.", "We call the duration between the fully charged UAV leaving a charging station and landing at the next charging station for recharging as one charging flight.", "The energy consumption of the UAV comprises of two parts, namely the energy consumed by traveling ${E_{travel}}$ and the energy consumed by charging ${E_{charge}}$ .", "Particularly, the traveling energy consumption can be divided into two parts: one is the energy consumption of flying between moving targets (nodes/BS/PADs) ${E_{mov}}$ ; the other is the energy consumption of hovering over nodes to perform the energy transferring ${E_{hov}}$ .", "We have the UAV consumed energy ${E_{consume}}$ as follows: $\\begin{aligned}{E_{consume}} & = {E_{travel}} + {E_{charge}}\\\\& = {E_{mov}} + {E_{hov}} + {E_{charge}}\\end{aligned}$ Let’s assume the energy transfer efficiency between the UAV and the charged node is $\\eta $ .", "We have ${E_{charge}} = {E_{rec}}/\\eta $ where ${E_{rec}}$ is the total amount of energy that charged nodes need to be replenished.", "We calculate the moving power ${P_{mov}}$ and the hovering power ${P_{hov}}$ of the rotary-wing UAV according to the propulsion power formula derived by [15].", "The moving energy ${E_{mov}}$ and the hovering energy ${E_{hov}}$ for one charging flight can be calculated as follows: ${E_{hov}} = {P_{hov}}{t_h} = ({P_0} + {P_i}){t_h}$ ${E_{mov}} = {P_{mov}}{t_m}$ where ${t_m}$ and ${t_h}$ are the flying time and the hovering time in the current charging flight, respectively.", "Since $e \\ll {E_{max}}$ , the time that one sensor node get fully charged can be approximate as a constant $\\Delta $ .", "Therefore, we have ${t_h} = n\\Delta $ where ${n}$ is the number of charged nodes in the current charging flight.", "According to pre-defined ${V_U}$ and ${E_{max}}$ , we calculate the maximum flight distance of the UAV ${d_{max}}$ as follows: ${d_{max}} = \\frac{{{E_{max}}}}{{{P_{mov}}}} \\times {V_U}$" ], [ "The Problem of Definition", "Let’s first consider the case without introducing PADs.", "If no PADs in the network, the UAV only can get charged at the BS.", "According to the last subsection, the flight radius of the UAV is $\\frac{1}{2}{d_{\\max }}$ .", "Considering the UAV has to have enough energy to return to the BS after charging one node, the distance ${d_{cover}}$ between the farthest rechargeable node and the BS has to satisfy the following equation: $\\frac{{\\rm {1}}}{{\\rm {2}}}{d_{max}} > {d_{cover}} \\ge \\frac{{\\rm {1}}}{{\\rm {2}}}\\frac{{({E_{max}} - {P_{hov}}\\Delta - e/\\eta )}}{{{P_{mov}}}} \\times {V_U}$ To be simplified, we can use equation (REF ) in our work.", "${d_{cover}}{\\rm { = }}\\frac{{\\rm {1}}}{{\\rm {2}}}\\frac{{({E_{max}} - {P_{hov}}\\Delta - e/\\eta )}}{{{P_{mov}}}} \\times {V_U}$ We call ${d_{cover}}$ as the charging coverage range of the UAV.", "Obviously, in the case without PADs, the UAV only can charge nodes in a cycle area with the location of BS as the center and ${d_{cover}}$ as the radius.", "Therefore, to charge the nodes outside this cycle area of BS, we have to introduce PADs to replenish the energy of the UAV.", "In our work, to easily depict, given a node ${s_i}$ and a charging station ${p_j}$ , if the distance between ${s_i}$ and ${p_j}$ is less than ${d_{cover}}$ , we call ${s_i}$ is covered by ${p_j}$ and denote this relationship as ${s_i} \\in C({p_j})$ , where $C({p_j})$ is the set of all the nodes covered by ${p_j}$ .", "Let $A({p_j})$ denotes the cycle area ${p_j}$ covered, we also call it as the coverage disk of ${p_j}$ .", "Once PADs are deployed, to ensure each node is chargeable, each node needs to be covered by at least one charging station.", "We call this requirement for PAD deployment as the coverage constraint.", "On the other hand, since a single UAV is used in our case, to ensure each node is chargeable, the UAV needs to fly between two charging stations.", "In this way, the UAV can charge the nodes in a new charging coverage disk after charging all the nodes in the current charging coverage disk.", "This requirement can be represented as the following: Let $dis({p_i},{p_j})$ denotes the distance between ${p_i}$ and ${p_j}$ .", "Given a graph $G = (V,E)$ , where $V = P^{\\prime }$ and $E{\\rm { = \\lbrace (}}{p_i},{p_j})|dis({p_i},{p_j}) < {d_{max}}\\rbrace $ .", "The $G$ must be a connected graph.", "We refer to this requirement for PAD deployment as the connectivity constraint.", "Based on the above analysis, we can define the problem of minimizing the number of deployed PADs in the UAV-based WRSN below: Definition 1 The problem of minimizing the number of deployed PADs Given a monitoring area $\\Omega $ , a BS ${p_0}$ and a set of nodes $S = \\lbrace {s_1},{s_2},...,{s_N}\\rbrace $ , a UAV, the objective of the MNDP problem is to find some locations to deploy PADs $P = \\lbrace {p_1},{p_2},...,{p_M}\\rbrace $ , which minimizes the number of PADs ${M}$ , under the coverage constraint and the connectivity constraint.", "i.e.", "$Min{\\rm { }}\\quad {M}$ $dis\\left( {{s_i},{p_j}} \\right) \\le {d_{cover}},\\forall {s_i} \\in S,\\exists {p_j} \\in P^{\\prime }$ ${\\rho _{i,j}} = \\left\\lbrace {\\begin{array}{*{20}{l}}{1,{\\rm { if\\;}}dis\\left( {{p_i},{p_j}} \\right) \\le {d_{max}}}\\\\{{\\rm {0, otherwise }}}\\end{array}} \\right.$ $\\begin{split} \\sum \\limits _{|\\Omega |} {\\left( {{\\rho _i}{,_{\\Omega (1)}} \\times \\left( {\\prod \\limits _{k = 1}^{|\\Omega | - 1} {{\\rho _{\\Omega (k),\\Omega (k + 1)}}} } \\right) \\times {\\rho _{\\Omega (|\\Omega |),j}}} \\right)} \\ge 1,\\\\\\forall {p_i},{p_j} \\in P^{\\prime }\\end{split}$ Formula (REF ) ensures the coverage for all nodes with PADs and the coverage radius is different from [13].", "Formula (REF ) and (REF ) demonstrate the connectivity for PADs, in (REF ) $\\Omega $ is a permutation of a subset of $P$ which means the UAV can fly from $p_i$ to $p_j$ through at least one path combined by other PADs.", "The problem is a NP-Hard problem deduced by [13]." ], [ "Clustering-with-double-constraints ${\\&}$ Disks-shift-combine Algorithm (CDC{{formula:0a39c7cb-eb97-47a0-8fab-6fbcd5bc03a8}} DSC)", "We propose a PAD placement scheme, named CDC&DSC (Clustering-with-double-constraints & Disks-shift-combining), to address the MNDP problem, which can automatically adapt to network scenarios with arbitrary node distribution and arbitrary BS location.", "The CDC&DSC scheme works in two phases and we propose two algorithms, CDC (Clustering-with-double-constraints) algorithm and DSC (Disks-shift-combining) algorithm, to accomplish the task of each phase separately.", "The task of phase 1 is to generate the initial solution of the MNDP problem, which we achieve by using the CDC algorithm.", "In the CDC algorithm, we first cluster all nodes based on geographic locations, then deploy a PAD at the center of each cluster, and then obtain an initial solution to our problem by adding new PADs to satisfy the coverage constraint and the connectivity constraint.", "The task of phase 2 is to optimize the initial solution from phase 1, which we complete by using the DSC algorithm.", "In the DSC algorithm, we decrease the number of PADs by combing the PADs based on the principle of triangle circumcircle after trying to shift each PAD to its nearest neighbor.", "In the rest of this section, we depict the details of the two algorithms one by one." ], [ "CDC Algorithm", "The goal of the CDC algorithm is to generate an initial solution to the MNDP problem.", "Essentially, the output of CDC should be a set of PADs with their coverage disks.", "CDC Algorithm Input: A sensor node set $S = \\lbrace {s_1},{s_2},...,{s_N}\\rbrace $ , a BS Output: An initial PAD set ${P}$ [1] ${P_c} \\leftarrow \\emptyset $ $I = \\lbrace {s_i}|{d_i} > \\sum \\limits _{{s_j} \\in S^{\\prime }} {{d_j}} /\\left| {S^{\\prime }} \\right|\\rbrace , K = \\left| I \\right|*\\alpha $ Calculate $P$ by K-means algorithm with ${S^{^{\\prime }}}$ in which $K = \\left| I \\right|*\\alpha $ , the initial points Update $C({p_i})$ for each ${p_i} \\in P$ $S^{\\prime } = S - \\bigcup \\limits _{{p_i} \\in P} {C({p_i})} $ [/*Check coverage constraint*/]$S^{\\prime } \\ne \\emptyset $ Deploy a PAD ${p^{^{\\prime }}}$ on $l({s_i})$ with the largest ${d_i}$ $S^{\\prime } = S - C({p^{^{\\prime }}}),P \\leftarrow {\\rm {\\lbrace }}{p^{^{\\prime }}}{\\rm {\\rbrace }}$ $G = (V,E),V \\leftarrow {\\rm {\\lbrace }}BS{\\rm {\\rbrace }},E \\leftarrow \\emptyset $ [/*Check connectivity constraint*/]$P \\ne \\emptyset $ Select ${p_i} \\in P$ with the minimal $d({p_i},{p_j}),{p_j} \\in V$ $d({p_i},{p_j}) > {d_{max}}$ Deploy a PAD ${p^{^{\\prime }}}$ on the line connecting ${p_i}$ and ${p_j}$ $V \\leftarrow \\lbrace {p^{^{\\prime }}}\\rbrace ,E \\leftarrow \\lbrace ({p^{^{\\prime }}},{p_j})\\rbrace $ $V \\leftarrow \\lbrace {p_i}\\rbrace ,E \\leftarrow \\lbrace ({p_i},{p_j})\\rbrace ,P - \\lbrace {p_i}\\rbrace $ $P = V$ Considering the MNDP problem, we must cover all the nodes with as few PADs as possible.", "Intuitively, the ideal situation is that each node is covered by one and only one PAD.", "Furthermore, considering the coverage constraint, the geographical locations of nodes covered by the same PAD should be close.", "Based on the above considerations, we first use a clustering algorithm to cluster the nodes and deploy a PAD in the center of each cluster.", "Then we can obtain an initial solution to the MNDP problem by adding some PADs to meet the coverage constraint and the connectivity constraint.", "Figure: after combiningThe goal of clustering nodes is to determine a series of locations to place PADs to cover the nodes in the network.", "The node ${s_i} \\in C({p_0})$ is surely covered by the BS.", "Thus, we only cluster the set of nodes $S^{^{\\prime }} = S - C({p_0})$ .", "In our case, we use the K-means algorithm to cluster the nodes.", "One problem with using the K-means algorithm is that we need to determine the parameter ${K}$ .", "We notice that isolated nodes may appear in the network scenarios since our goal is to be able to handle scenarios with arbitrary node distribution and arbitrary BS locations.", "The isolated nodes are more difficult to cover than the other nodes.", "Based on this consideration, we determine the value of $K$ by the number of isolated nodes.", "Let ${d_i}$ denotes the distance between node ${s_i}$ and its nearest neighbor node ${n_i}$ , and ${I}$ is the set of isolate nodes.", "Besides, we denote $\\alpha $ as a parameter to adjust $K$ .", "We have $I = \\lbrace {s_i}|{n_i} > \\sum \\limits _{{s_i} \\in S^{\\prime }} {{d_i}} /\\left| {S^{\\prime }} \\right|\\rbrace $ $K = \\left\\lfloor {\\alpha \\left| I \\right|} \\right\\rfloor $ We use the locations of isolate nodes as the initial points of K-means in Fig.", "REF .", "After clustering, we place a PAD in the center of each cluster and generate a coverage disk for each PAD.", "Then, we check if all nodes are covered.", "If some nodes are not covered, we select the node farthest from its nearest neighbor node and deploy a PAD in its place $l({s_i})$ .", "We repeat the process until all nodes are covered.", "For the connectivity constraint, we construct a connected graph from the BS with edge length not exceeding ${d_{max}}$ .", "The PADs closest to the graph are added to the connected graph in turn.", "If the distance between a PAD to be added and the nearest vertex in the graph exceeds ${d_{max}}$ , a new PAD is generated on the line connecting the PAD to the corresponding vertex at a distance ${d_{max}}$ from the vertex, until all PADs are added to the connected graph.", "The details of the CDC algorithm are shown in algorithm REF ." ], [ "Disks-shift-combine", "After obtaining a feasible solution ${P_c}$ , and we decrease the number of PADs in the DSC algorithm.", "To reduce computational complexity, we first delete some redundant PADs in ${P_c}$ .", "The PAD ${p_i}$ is redundant if the ${P_c}$ still satisfies two constraints after removing ${p_i}$ .", "We can observe some redundant PADs in Fig.", "REF and they are deleted in Fig.", "REF .", "After deleting the redundant PAD, we merge the adjacent PADs to decrease the number of PADs.", "Before merging the PADs, we first try to reduce the distance between the PADs by a nearest neighbor shift method to facilitate the next merging operation.", "Let ${Effect}\\_C({p_i})$ denotes the set of nodes only covered by the PAD ${p_i}$ .", "We select the PAD with the minimal ${Effect}\\_C({p_i}),i \\ne 0$ to perform the following operations.", "We first shift the location of the PAD with the step length ${d_\\Delta }$ toward its nearest neighbor PAD, then generate a coverage disk at the new location, and then check the coverage constraint and the connectivity constraint.", "If these two constraints are not satisfied, we delete the new location and begin the other loop for the next PAD.", "Otherwise, we continue to shift this PAD with the step length ${d_\\Delta }$ .", "We repeat the above process until all PADs have been shifted.", "Fig.", "REF shows the result of shifting.", "After all the PADs are shifted, we try to re-delete the redundant PADs.", "After the shifting operations, we can merge the adjacent PADs to get the minimal set of PADs.", "We attempt to perform the adjacent PAD merge operation in the following way.", "For each neighbor PAD ${p_{i}}$ of ${p_{j}}$ , $i \\ne 0,j \\ne 0,i \\ne j$ , we calculate the discrete point set $C({p_i},{p_j})$ , which is the set of nodes only covered by ${p_{i}}$ and ${p_{j}}$ : $C({p_i},{p_j}) = S - \\bigcup \\limits _{{p_k} \\in P,k \\ne i,k \\ne j} {C({p_k})}$ We select the two most distant nodes in $C({p_i},{p_j})$ and try to merge ${p_{i}}$ and ${p_{j}}$ to a new PAD $p^{\\prime }$ at the midpoint between these two selected nodes.", "If the merging result can not meet both the coverage constraint and the connectivity constraint, we check whether there is an enclosing circle of radius ${d_{cover}}$ covering $C({p_i},{p_j})$ .", "If there is, we merge ${p_{i}}$ and ${p_{j}}$ to a new PAD $p^{\\prime \\prime }$ at the center of this enclosing circle [16], [17].", "We repeat the above steps until all the ${p_i},i \\ne 0$ have been tried the merging operation.", "We show the details in algorithm REF .", "We give a demonstration on how the DSC algorithm works in Fig.", "REF .", "Fig.", "REF presents the result of deleting redundant operation on the deployment result in Fig.", "REF , while Fig.", "REF and Fig.", "REF show the results of shifting operations and merging operations, respectively.", "DSC algorithm Input: An initial PAD set ${P}$ Output: A final PAD set ${P}$ [1] Check redundant PADs from ${P}$ and delete them ${P^{^{\\prime }}} = \\emptyset $ [/*Shifting PADs*/]$P \\ne \\emptyset $ Select ${p_i}$ with the minimal $Effect\\_C({p_i}),i \\ne 0$ Two constraints are satisfied Shift ${p_i}$ toward its nearest neighbor PAD with ${d_\\Delta }$ $P - \\lbrace {p_i}\\rbrace ,{P^{^{\\prime }}} \\leftarrow \\lbrace {p_i}\\rbrace $ $P = {P^{^{\\prime }}}$ , Check redundant again [/*Combining PADs*/]$P \\ne \\emptyset $ ${N_i} = \\lbrace {p_i}\\left|\\;{d({p_i},{p_j}) \\le {d_{max}},{p_j} \\in {P^{^{\\prime }}}} \\right.\\rbrace $ ${N_i} \\ne \\emptyset $ Select ${s_a},{s_b}$ with the minimal $d({s_a},{s_b})$ ${s_a},{s_b} \\in C({p_i},{p_j})$ Calculate the midpoint ${p^{^{\\prime }}}$ of ${s_a}$ and ${s_b}$ ${P^{^{\\prime \\prime }}} = {P^{^{\\prime }}} - \\lbrace {p_i},{p_j}\\rbrace ,{P^{^{\\prime \\prime }}} \\leftarrow {\\rm {\\lbrace }}{p^{^{\\prime }}}{\\rm {\\rbrace }}$ ${P^{^{\\prime \\prime }}}$ satisfies two constraints $P - \\lbrace {p_i}\\rbrace ,P - \\lbrace {p_j}\\rbrace ,{P^{^{\\prime }}} = {P^{^{\\prime \\prime }}}$ break Select ${s_c}$ with the minimal $d({s_c},{p^{^{\\prime }}})$ ${s_c} \\in C({p_i},{p_j}),c \\ne a,b$ Calculate circumcenter ${p^{^{\\prime \\prime }}}$ of triangle with three vertexes ${s_a}$ , ${s_b}$ and ${s_c}$ ${P^{^{\\prime \\prime }}} - \\lbrace {p^{^{\\prime }}}\\rbrace ,{P^{^{\\prime \\prime }}} \\leftarrow \\lbrace {p^{^{\\prime \\prime }}}\\rbrace $ ${P^{^{\\prime \\prime }}}$ satisfies two constraints $P \\leftarrow {v_{cir}},P - \\lbrace {p_i}\\rbrace ,P - \\lbrace {p_j}\\rbrace $ break ${N_i} - \\lbrace {p_j}\\rbrace $ $P - \\lbrace {p_i}\\rbrace $ $P = {P^{^{\\prime }}}$" ], [ "Simulation Result", "In this section, we evaluate the performance of the proposed scheme.", "To illuminate the effectiveness of our proposed scheme, we compare it with the four algorithms in [13].", "To the best of our knowledge, the work in [13] and our scheme are the only two works investigating the PADs deployment in a UAV-based WRSN.", "Figure: The result of the region size.Figure: The result of the number of nodes.We carry out four groups of simulations by varying the size of the network region, the number of nodes, the battery capacity of UAV, and the different node distributions, respectively.", "To verify the adaptability of the proposed scheme to arbitrary BS locations, we transform two BS locations for each group of simulations: the BS-center scenario, where the BS is surrounded by nodes in the center of the network area, and the BS-isolated scenario, where the BS is outside of the network area and the distance to any node is greater than ${d_{max}}$ .", "Due to the algorithms in [13] cannot work in the scenarios of BS-isolated, we compare our scheme with the four algorithms in the BS-center scenario first and then we compare the performance of our scheme in these two scenarios.", "We take the average results with 10 sets of data to smooth data in each simulation.", "The default parameters are shown in Table REF .", "Table: The default parameters." ], [ "The impacts of the region size", "Fig.", "REF presents the simulation results with varying the size of the network region from $1000 \\times 1000{m^2}$ to $25000 \\times 25000{m^2}$ .", "As expected, for all the algorithms, the number of PADs increases as the size of the network region increases.", "The reason is that as the network size increases, the distance between nodes increases and more PADs are needed to satisfy the coverage constraint.", "Fig.", "REF shows the BS-isolated scenario always requires more PADs than the BS-center scenario.", "This is because we need to introduce additional PADs to ensure the connectivity constraint in the BS-isolated scenario.", "It is important to notice from Fig.", "REF that the number of PADs deployed by the proposed scheme is always less than four comparison algorithms, and this advantage continues to grow as the network area increases.", "This changing trend demonstrates the proposed algorithm has advantages in WRSNs with large network region." ], [ "The impacts of the number of nodes", "Fig.", "REF depicts the simulation results for varying the number of nodes from 100 to 1000.", "Comparing with Fig.", "REF , for all the algorithms, the number of PADs is growing slowly as the node number increases in Fig.", "REF .", "It is because the PAD deployment problem is essentially a coverage problem, and the number of PADs depends mainly on the coverage radius of the UAV and the size of the network, while it is less affected by the node density.", "We also notice that the result of the DC algorithm is always 16 when the node number is over 200.", "One reason is that the DC algorithm always deploys PADs at the center of each cell after celling the network area and then removes the PADs of the cells with no node.", "Therefore, when the node density is above the threshold value (all cells have nodes), the result of the DC algorithm will not change.", "The other approaches determine the locations of PADs from the geographic distributions of nodes, so an increase in node density has a slight effect on their results.", "Still, we can find that the proposed scheme achieves the best performance in Fig.", "REF .", "From Fig.", "REF , we conclude the similar conclusion within Fig.", "REF .", "In the BS-isolated scenario, it takes more PADs to ensure the connectivity constraint than in the BS-center scenario.", "Figure: The result of UAV energy with the uniform nodes.Figure: The result of UAV energy with the mixed Gaussian distribution nodes." ], [ "The impacts of the battery capacity of UAV", "We also vary the battery capacity of UAV with results presented in Fig.", "REF .", "As the battery capacity increases, the number of PADs of all the schemes decreases.", "The more energy the UAV itself carries, the larger ${d_{max}}$ and ${d_{cover}}$ become.", "As a result, each PAD has a larger coverage disk, while the connectivity constraint can be satisfied by fewer PADs.", "The number of PADs in the proposed scheme is obviously 8 to 10 less than all the comparing algorithms.", "One reason is that the four comparison algorithms always deploy PADs at fixed locations: TNC, GNC, and MSC deploy PADs at nodal locations, while DC deploys PADs at the centers of some cells after celling the network area.", "Also, none of the four comparison algorithms attempted to merge adjacent PADs." ], [ "The impacts of the triple Gaussian Mixture distribution", "We further carry out some simulations to verify the adaptability of the proposed scheme to the arbitrary node distribution.", "We build a scenario with 300 nodes with a triple Gaussian Mixture distribution.", "We divide 300 nodes into three groups and each group of one hundred nodes is deployed according to a Gaussian distribution.", "Since the expectation and variance of the three Gaussian distributions are random, the deployment areas of the three groups of nodes may or may not overlap.", "We vary the battery capacity of UAV in this scenario with results presented in Fig.", "REF .", "For the DC approach and our proposed scheme, the results are the average of 10 sets of data.", "Since the TNC, GNC, and MSC may not work in the scenarios with three groups of nodes isolate with each other, for these algorithms we only counted the results in scenarios where they could work.", "According to Fig.", "REF , the number of PADs decreases as the battery capacity increases.", "The reason is the same as the previous simulation results in Fig.", "REF .", "Comparing Fig.", "REF , the numbers of PADs in this type of scenario are much smaller than in the scenarios with a uniform distribution of nodes.", "It can be explained as follows, when the mixed Gaussian distribution is obeyed, the nodes are clustered more closely than when the uniform distribution is obeyed.", "Consequently, the actual network area size becomes smaller, the performances of all the approaches are improved in this type of scenario.", "Our proposed algorithm still achieves the minimum number of PADs in this type of scenario.", "In general, the CDC${\\&}$ DSC algorithm outperforms all the comparing algorithms.", "It can adapt to the scenarios with arbitrary node distribution and arbitrary BS location." ], [ "Conclution", "After years of research, the introduction of PADs for UAV energy replenishment has become a promising approach to improve the performance of UAV-based WRSNs.", "In this paper, we investigated the PADs deployment problem for the UAV-based WRSNs.", "We proposed a novel PADs deployment scheme, named CDC&DSC, to adapt to the scenarios with arbitrary node distribution and arbitrary BS location.", "We proposed the CDC algorithm to generate an initial deployment of PADs, and then proposed the DSC algorithm to optimize this initial solution in an attempt to merge adjacent PADs and remove redundant PADs by shifting the locations of PADs.", "Finally, we compared the proposed scheme with four existing PADs deployment approaches through simulations.", "The results showed that our proposed scheme outperforms the existing methods in all aspects.", "However, our proposed algorithm deals with the deployment of PADs only from the perspective of minimizing the number of PADs, without considering the charging scheduling.", "In the future, we are planning to solve the PADs deployment problem to improve the global charging efficiency by taking the charging demand of nodes and the scheduling of UAVs into account." ] ]
2105.11701
[ [ "An evolving objective function for improved variational quantum\n optimisation" ], [ "Abstract A promising approach to useful computational quantum advantage is to use variational quantum algorithms for optimisation problems.", "Crucial for the performance of these algorithms is to ensure that the algorithm converges with high probability to a near-optimal solution in a small time.", "In Barkoutsos et al (Quantum 2020) an alternative class of objective functions, called Conditional Value-at-Risk (CVaR), was introduced and it was shown that they perform better than standard objective functions.", "Here we extend that work by introducing an evolving objective function, which we call Ascending-CVaR and that can be used for any optimisation problem.", "We test our proposed objective function, in an emulation environment, using as case-studies three different optimisation problems: Max-Cut, Number Partitioning and Portfolio Optimisation.", "We examine multiple instances of different sizes and analyse the performance using the Variational Quantum Eigensolver (VQE) with hardware-efficient ansatz and the Quantum Approximate Optimization Algorithm (QAOA).", "We show that Ascending-CVaR in all cases performs better than standard objective functions or the \"constant\" CVaR of Barkoutsos et al (Quantum 2020) and that it can be used as a heuristic for avoiding sub-optimal minima.", "Our proposal achieves higher overlap with the ideal state in all problems, whether we consider easy or hard instances -- on average it gives up to ten times greater overlap at Portfolio Optimisation and Number Partitioning, while it gives an 80% improvement at Max-Cut.", "In the hard instances we consider, for the number partitioning problem, standard objective functions fail to find the correct solution in almost all cases, CVaR finds the correct solution at 60% of the cases, while Ascending-CVaR finds the correct solution in 95% of the cases." ], [ "Introduction", "We have recently entered the era where quantum computers have scaled up, from small proof-of-principle devices to devices that are beyond the classical simulation limit opening the prospect for providing computational speed-ups.", "However, we are still very far from the point that large fault tolerant quantum computers are developed.", "Our period has been termed as Noisy Intermediate Scale Quantum (NISQ) device era [2] and refers to the time that the existing devices vary from $\\approx 50$ qubits of Google's quantum-advantageAlso known as “quantum computational supremacy”.", "experiment [3] to devices with $O(1000)$ qubits that are anticipated in a horizon of five to ten years.", "There are two paths forward for quantum computing.", "The “long-term” path requires to intensify the efforts (theoretical and experimental) to overcome existing barriers and truly scale up these devices to the large fault-tolerant regime.", "The “near-term” one, is to determine if and how these NISQ devices can be used, directly, and offer advantage for problems of practical importance.", "A promising approach in the latter path, is the use of hybrid quantum-classical algorithms.", "A leading class of candidate algorithms, both due the possible importance of the applications and the promise it shows, is the class of variational quantum algorithms for optimisation problems.", "One can divide variational quantum algorithms (see more details [preliminaries]II]) into three main steps.", "The first step is to map the targeted problem to the mathematical task that these algorithms are designed to solve, which is the search for the ground state energy of a HamiltonianMathematically this is simply evaluating the smallest eigenvalue of a Hermitian matrix..", "The second step, is a method to estimate the energy of a quantum state, given a (polynomial in the size of the input) number of copies.", "Finally, the third step consists of a parameterised family of quantum states (“ansatz”) and a classical optimiser that given the above tools, outputs efficiently an approximation of the ground state energy.", "This is done by finding the choice of parameters that lead to the quantum state that has the smallest energy.", "The success of the algorithms depend on all those steps and extensive research on improving each of them exists, indicatively, [4] used warm-starting to improve QAOA on low depth, [5] improved QAOA by introducing a non-local version which outperformed classical QAOA on 3-regular graphs, [6], [7] introduced different procedures on how to optimise the variational parameters and [8] used reinforcement learning to assist the classical optimisation.", "What we focus in this contribution is the third part, and specifically on how to use the measurement outcomes performed in estimating the energy of a quantum state to (i) accelerate the speed and (ii) improve the accuracy that the classical optimiser finds an (approximation of the) ground state and thus solves the problem optimally.", "Prior to our work, [9] inspired by statistical physics, used a Gibbs objective function to improve the performance.", "Minimising the infidelity between the parameterised state and a target state [10], [11] appears to be another promising approach.", "For classical optimisation problems, the solution (ground state) is one of the computational basis quantum states.", "Preparing a quantum state that has big overlap with that state is sufficient to give a good and quick approximation of the ground state.", "For example, if one can achieve a constant, but possibly small, overlap with the correct solution, they are guaranteed after sampling this state a constant number of times to obtain at least one sample of the true ground state.", "In [1], the authors used this idea, and instead of evaluating the proximity of a quantum state to the desired (ground state) by minimising the (overall) energy, aimed to minimise the energy of the lowest tail of a quantum state.", "This, intuitively, would succeed quicker in finding a quantum state that has a non-negligible overlap with the solution (but not necessarily very high overlap).", "This state, however, suffices to solve the problem.", "This intuition was also confirmed with numerical simulations.", "In other words, the cost function used in the classical optimiser, in order to find the optimal parameters, was not the energy of the quantum state, but the tail of the corresponding distribution.", "Inspired by this idea but also by adiabatic quantum computing [12], we consider here an evolving cost function.", "In our proposal the way that the cost of a quantum state is computed, dynamically changes during the classical optimisation process.", "We start with a cost function as in [1] focusing on a small tail, but during the optimisation process we gradually increase the tail (fraction of the distribution we “count”) until we reach a point that all the distribution is included i.e.", "we measure the full expectation value of the energy (as in “standard” cost functions).", "Our contributions.", "We introduce an evolving objective function that starts with the CVaR defined in [1] and gradually in the optimisation process becomes the full energy of the quantum state.", "Alternative forms of this Ascending-CVaR objective functions are considered and a linear and a sigmoid functions (that appear to perform better) are selected.", "We test our proposal, with classical numerical simulations (using up to 20 qubits), both in the setting of VQE with hardware efficient ansatz and in QAOA.", "Our results suggest that our proposal leads to faster convergence with bigger overlap with the ideal solution than prior works, while crucially, succeeds in obtaining the solution in (many) instances that other techniques fail altogether (see Section for statistics and comparisons).", "Our analysis is done for three very different combinatorial optimisation problems, namely Max-Cut, Number Partitioning and Portfolio Optimisation.", "We consider many different instances and problem sizes where the conclusions persist in all cases.", "This has importance in its own right, since these problems are important by themselves, and our proposal gives an approach to improve the performance and bring closer achieving “useful” quantum advantage.", "Interestingly, our method offered greater advantage in “hard instances” of the problems, where the other methods frequently failed to find the solutions altogether.", "Structure.", "In Section we give the essential background: We introduce the variational quantum algorithms and specifically, the variational quantum eigensolver and the quantum approximate optimisation algorithm.", "We then introduce the CVaR objective function of [1] and finally analyse the three different combinatorial optimisation problems that we use as case-studies.", "In Section we introduce our novel method called Ascending-CVaR and we discuss the hyperparameters of our model.", "In Section we discuss our methodology.", "In section we present the results of our method compared to existing objective functions.", "We conclude in Section with a general discussion of our method and future work." ], [ "Preliminaries", "We introduce two of the main Variational Quantum Algorithms [13], the CVaR objective function [1] and the three types of combinatorial optimisation problems that we use.", "Figure: General framework of a variational quantum algorithm.", "The optimisation problem, described by a cost function C(x)C(x) is mapped to an interacting qubit Hamiltonian H C H_C.", "A parameterised family of states (“ansatz”) U(θ)U(\\theta ) with random initial parameters is chosen and a classical feedback loop iteratively updates the parameters θ\\theta .", "The optimisation ends when the stopping condition is met, and the optimal parameters θ * \\theta ^* are outputted." ], [ "Variational Quantum Algorithms", "Here we revise the methods with focus on optimisation problems.", "The general framework of a variational quantum algorithm is outlined in Figure REF .", "The first step is to map the classical cost function $C(x)$ that describes the optimisation problem, into an interacting qubit Hamiltonian $H_C$ whose ground state gives the solution we are seeking.", "The second step is to choose an ansatz family of unitary operators $U(\\theta )$ .", "This family is both efficiently expressible and trainable, parameterised by a $\\mu $ -parameter vector $\\theta = (\\theta _1, ..., \\theta _\\mu )$ where $\\mu = \\mathcal {O}(poly(n))$ and $n$ is the system size.", "In general, the parameters are initialized at randomThere are cases that a “clever” initialization could lead to faster convergence [14].", "The third step is to evaluate some objective function, usually taken to be the expectation value of the problem's Hamiltonian on the state considered $\\mathinner {\\langle {\\psi (\\theta )}|}H_C\\mathinner {|{\\psi (\\theta )}\\rangle }$ .", "This is done by preparing the state (applying the unitary $U(\\theta )$ on the initial state) and then measuring the output in the computational basis and repeating this procedure for a given number of times (typically called “shots”).", "This number determines the accuracy the objective function is evaluated.", "The fourth step is to update the parameters and repeat step three, iteratively using some classical optimiser until a stopping condition is satisfied.", "We then say that the parameters are optimal, i.e.", "$\\theta ^* = \\arg \\min _{\\theta } O \\left(\\theta , H_C\\right)$ The state produced by these parameters, $\\mathinner {|{\\psi (\\theta ^*)}\\rangle }=U(\\theta ^*)\\mathinner {|{0}\\rangle }^{\\otimes n}$ , can be used to give an estimate of the ground state energy of the Hamiltonian $H_C$ and thus an approximate solution to the desired optimisation problem.", "The objective function used during this process, as stated above, typically coincides with the expectation value of the problem's Hamiltonian.", "However, we note here that other choices may also be possible, especially if we realise that the true target of the optimisation algorithm is to sample, at least once, the optimal solution.", "This can efficiently be produced if the output state has a sufficiently large (or more precisely simply non-vanishing) overlap with the optimal solution $\\vert \\mathinner {\\langle {\\psi (\\theta ^*)}|}\\mathinner {|{\\psi _{opt}}\\rangle }\\vert ^2$ .", "The Variational Quantum Eigensolver, as proposed by [15], is a hybrid quantum-classical algorithm, originally designed to solve quantum chemistry problems, but it can be used to tackle optimisation problems [16].", "The main idea is to a map the optimisation problem into a cost function that is translated into a interacting qubit Hamiltonian [17], whose ground state corresponds to the solution of the optimisation problem.", "The encoded qubit Hamiltonian, $H_C$ , is decomposed into a linear combination of Pauli strings $P_a$ , consisted of tensor products of Pauli Matrices $\\hat{\\sigma }^x, \\hat{\\sigma }^y, \\hat{\\sigma }^z$ : $H_C = \\sum _{k=1}^Mc_kP_k$ where $M=\\mathcal {O}(poly(n))$ , $n$ is the system size and $c_k$ is the complex coefficient of the $P_k$ Pauli string.", "For combinatorial optimisation problems however, where the Hamiltonian is a diagonal matrix, the $H_C$ is decomposed only on Pauli strings consisting of $\\sigma _i^z$ operators.", "VQE is initialised by creating a parameterised state $\\mathinner {|{\\psi (\\theta )}\\rangle }$ whose parameters are iteratively updated by a classical optimiser so as to minimize an objective function, usually the expectation value of (REF ).", "The parameterised state is created by choosing a variational form $U(\\theta )$ which acts on the initial state $\\mathinner {|{0}\\rangle }^{\\otimes n}$ and produces $\\mathinner {|{\\psi (\\theta )}\\rangle }$ .", "Our choice of variational form $U(\\theta )$ is a hardware efficient ansatz [1], [18], where the qubits are initialised in the $\\mathinner {|{0}\\rangle }$ state and $R_y(\\theta _i)$ -rotations are applied in each qubit along with control-Z operators.", "Each layer of the variational form consists of Control-$Z_{ij}$ operations with $i$ the control qubit and $j$ the target qubit, as long as the condition $i<j$ holds, and $R_y(\\theta _i)$ rotations for every qubit.", "If $p$ is the number of layers, then the number of parameters is linear, $\\mu = n\\left(1+p\\right)$ , in the number of qubits and the variational form spans every basis state already within the first layer.", "Figure: Single Layer Hardware Efficient Ansatz for 3 qubits.The hardware efficient ansatz falls in the more general category of problem-agnostic ansatze, meaning that the structure of the ansatz carries no information about the problem itself and is mostly suited for optimisation problems.", "Other problems use different ansatz families, like the Unitary Coupled Cluster which is widely used in chemistry to obtain the ground state of a molecule [19] or the Variational Hamiltonian Ansatz which encodes the problem's Hamiltonian [20]." ], [ "Quantum Approximate Optimisation Algorithm", "The Quantum Approximate Optimisation Algorithm (QAOA) [21] is, a variational quantum algorithm mostly used in combinatorial optimisation problems, and while in shallow depths it is analytically and numerically explored for some problems [22], [23], its performance in intermediate depths is still unknown.", "The QAOA algorithm applies an alternation of two unitary transformations, one encoding the cost function $H_C$ , $U(H_C)=e^{-i\\gamma H_C}$ , and the other a mixer Hamiltonian $H_B = \\sum \\sigma _i^x$ , $U(H_B)=e^{-i\\beta H_B}$ , where $\\gamma $ and $\\beta $ are variational angles specifying the “time” for which the unitary transformations are applied.", "The system is initialised at the ground state of $H_B$ and the alternating ansatz of $U(H_B)$ $U(H_C)$ is applied $p$ -times, where $p$ defines the layers of the algorithm (see Figure REF ), producing the state : $\\mathinner {|{\\beta , \\gamma }\\rangle } = e^{-i\\beta _p H_B}e^{-i\\gamma _p H_C}\\ldots e^{-i\\beta _1 H_B}e^{-i\\gamma _1 H_C}\\mathinner {|{+}\\rangle }$ where $\\mathinner {|{+}\\rangle }$ is the uniform superposition state, $\\gamma = \\left(\\gamma _1\\ldots ,\\gamma _p\\right)$ , $\\beta = \\left(\\beta _1\\ldots ,\\beta _p\\right)$ .", "Figure: General framework of a pp-layer QAOA consisting of 2p2p variational angles.With sufficient repetitions of the algorithm, the expectation value is calculated as: $F_p(\\beta , \\gamma ) = \\mathinner {\\langle {\\beta ,\\gamma }|}H_C\\mathinner {|{\\beta , \\gamma }\\rangle }$ until the $2p$ optimal parameters $(\\beta ^*,\\gamma ^*)$ are found.", "If $C_{opt}$ is the optimal cost function, then the target of the algorithm is to maximise the approximation ratio, defined as: $r^* = \\frac{F_p(\\beta ^*, \\gamma ^*)}{C_{opt}}$ Finding the optimal parameters is far from trivial since the expectation value landscape is highly non-convex, filled with local minima where a classical optimiser could easily get stuck.", "The hardest part of QAOA, and in general of a variational quantum algorithm, is finding the optimal parameters that will lead in a high overlap with the optimal (or near-optimal) bit-string or high expectation value.", "Recently, [24] proved that training the optimization parameters is NP-Hard and that the landscape of the objective function is filled with far-from-optimal local minima.", "One way to avoid “getting stuck” in a local minima is using multi-start methods [25] or heuristic methods like using the global optimum of one layer, in QAOA, as a starting point for the next [23].", "Barkoutsos et al.", "[1] used an alternative objective function.", "They demonstrated that their proposal performed better than minimising the expectation value.", "The key observation is that for optimisation problems the optimal solution is a computational basis state.", "For computational basis states, one can compute their energy (efficiently).", "For a general quantum state $\\mathinner {|{\\psi (\\theta )}\\rangle }$ one can prepare and measure it (multiple times) in the computational basis, and the expectation value of the energy is simply the average of the individual computational basis states energies.", "To find the overlap of this state with the optimal solution (ground state) one can simply observe the frequency of the computational basis state with the smallest energy.", "Naturally, if that overlap is too small (or even zero), it is possible that none of the measurements outcomes will give the solution.", "On the other hand it is also clear that the overlap of this state with computational basis vectors with high energy are irrelevant for finding the ground state.", "The idea of [1] was to use this observation and instead of using all the measurement outcomes and compute the expectation value, they used as objective function the lower tail of the distribution of energies obtained, i.e.", "ignored all but a small fraction (with smallest energy) of their measurement outcomes.", "They then demonstrated that their technique succeeded in getting quicker a quantum state that has a sufficiently large overlap with the ground state.", "This in turn, is sufficient to actually find this ground state, since as a final step, once the optimal $\\theta ^*$ is found, one can keep the computational vector that has the smallest energy only.", "Specifically, let $H_k$ be the energy corresponding to a computational basis vector, and let us order them in such a way that larger $k$ corresponds to larger energy.", "For each state, one repeats the measurement $K$ -times, so there are (up to) $K$ distinct values $H_k$ .", "In [1] a new parameter $\\alpha $ was introduced.", "Let $\\alpha \\in (0,1]$ be the fraction (part of the tail) that we want to keep.", "This fraction, typically, needs to be non-negligible (we can assume for simplicity, that is constant).", "Then the objective function that was used, was the average of the smallest $\\alpha K$ samples, i.e.", "$CVaR_\\alpha = \\frac{1}{*{\\alpha K}}\\sum _{k=0}^{*{\\alpha K}}H_k$ In order to achieve the same accuracy when evaluating this objective function, as the accuracy achieved when computing the expectation value using $K$ shots, it is clear that the number of runs of the preparation circuit need to be increased to $K/\\alpha $ .", "As it was proven by [1], the angles $\\theta ^*$ that minimise CVaR$_\\alpha $ do not (in general) correspond to minima of the expectation value.", "As a result, the angles that lead to the smallest possible $\\alpha $ -tail differ from the angles that minimise the average of the samples.", "This fact motivates to introduce a lower $\\alpha $ -tail optimisation so as to achieve an overlap with the optimal state of at least $\\alpha $ , i.e find optimal $\\theta ^*$ that satisfy: $|\\mathinner {\\langle {\\psi (\\theta ^*)}|}\\mathinner {|{\\psi _{opt}}\\rangle }|^2 \\ge \\alpha $" ], [ "Combinatorial Optimisation Problems", "We test our proposed method in various instances of three different combinatorial optimisation problems.", "These are all important problems in their own right, so improving the performance of variational quantum algorithms for these problems is of independent interest.", "Moreover, testing our proposed objective function on different types of combinatorial optimisation problems demonstrates that improvements observed are generic and motivates further use for different applications.", "Given that our proposal's starting point is the work of [1], we included the problems that they tested their proposal to allow for more direct comparison.", "The easiest way to use variational quantum algorithms for an optimisation problem is to first map the problem to a Quadratic Unconstrained Binary Optimization (QUBO) problem.", "This is what we will do for all our examples.", "QUBO problems seek to solve (find the $x$ that minimises the expression): $\\min _{x} \\left(b^{T}x + x^TAx\\right)$ where $b\\in \\mathbb {R}^n$ and $A \\in \\mathbb {R}^{n \\times n}$ .", "These cost functions can easily be mapped to an Ising Hamiltonian [26] by first transforming the binary variables $x_i\\in \\lbrace 0,1\\rbrace $ according to: $x_i = \\frac{1-z_i}{2}$ where $z_i\\in \\lbrace -1,+1\\rbrace $ are spin variables, and then turning the cost function to a Hamiltonian by promoting these variables to Pauli $\\sigma ^z_i$ operators, one for each qubit $i$ .", "The QUBO problem then transforms to $\\min _z c^{T}z + z^TQz$ where the new $c\\in \\mathbb {R}^n$ and $Q \\in \\mathbb {R}^{n\\times n}$ are easily computable.", "Then, by replacing the spin variable $z_i$ with the Pauli $\\sigma _i^z$ operator with corresponding eigenvalues $\\lbrace -1, +1\\rbrace $ , the problem translates into finding the ground state, i.e.", "the spin configuration, of an $n$ -qubit system interacting with the Hamiltonian: $H = \\sum _{i=1}^n c_i \\sigma _i^z + \\sum _{i=1}^n Q_{ij} \\sigma _i^z \\sigma _j^z$" ], [ "Max-Cut Problem", "The first problem is Max-Cut.", "It is one of the most studied combinatorial problems in the context of variational quantum algorithms, due to the simplicity and guaranteed performance at least for some instances [21], [22].", "Let $G(V,E)$ be a non-directed $n$ -vertex graph, where $V$ is the set of vertices, $E$ is the set of edges, and $w_{ij}$ are the weights of the edges.", "A cut is defined as a bipartion of the set $V$ into two disjoint subsets $P,Q$ , i.e.", "$P\\cup Q = V$ and $ P\\cap Q=\\emptyset $ .", "Equivalently, we label every vertex with either 0 or 1, where it is understood that the vertex belongs to set $P$ if it takes the value 0 and to set $Q$ if it takes the value 1.", "The aim is to maximise the following cost function: $C(x) = \\sum _{i,j = 1}^n w_{ij}x_i\\left(1-x_j\\right)$ This intuitively corresponds to finding a partition of the vertices into two disjoint sets that “cuts” the maximum number of edges.", "By applying the transformation, Eq.", "(REF ), the cost function transforms into: $C(z) = \\sum _{\\left<i,j\\right>\\in E} \\frac{w_{ij}}{2}\\left(1-z_iz_j\\right)$ Maximising the cost function above corresponds into finding the ground state of the HamiltonianNote the overall minus sign that turns the maximisation of the cost function to finding the minimum energy for the Hamiltonian.", ": $H_{C} = -\\sum _{\\left<i,j\\right>\\in E}\\frac{w_{ij}}{2}\\left( 1- \\sigma _{i}^z \\sigma _j^z \\right)$ Max-Cut is known to be NP-Hard.", "The best classical approximation algorithm is that of Goemans and Williamson which uses semi-definite programming to achieve an approximation ratio, Eq.", "(REF ), $r^* \\approx 0.87856$ for all graphs.", "Note, that being NP-Hard implies that we do not expect to have an efficient quantum algorithm (poly-time) to solve the problem for its hardest instancesNP is strongly believed to not be included in BQP, but we could definitely get improvements using quantum algorithms (either by smaller speed-ups or by heuristics that could solve more instances than classical heuristics).", "Although it was proven that constant-depth QAOA does not outperform GW for certain class of problems [5], there are instances where the approximation ratio of the former is larger than the latter [27].", "Note here that QAOA beats random guessing even at $p=1$ , while Machine Learning techniques have been used to classify for which graph types is better to use QAOA instead of GW [28].", "In general, however, the performance of variational quantum algorithms in intermediate depths is still highly unexplored." ], [ "Number Partitioning", "The second problem is Number Partitioning and is stated as follows.", "Given a set of $N$ positive integers $S = \\lbrace n_1, n_2, ..., n_N\\rbrace $ , the target is to find a bipartion of the set $S$ into two disjoint subsets $P,Q$ , where $P \\cup Q = S$ and $P\\cap Q=\\emptyset $ so that the difference between the sum of the elements on the set $P$ and the set $Q$ is minimized.", "We thus want to minimize the cost function: $C(x) = \\left(\\sum _{i=1}^N(2x_i-1)n_i\\right)^2$ The binary string $x=x_1x_2\\dots x_n$ corresponds to one configuration where a number $n_i$ is placed in the $P$ set ($x_i=0$ ) or in the $Q$ set ($x_i = 1)$ .", "The cost function can easily be mapped to the Ising Hamiltonian: $H_C = \\left(\\sum _{i=1}^N\\sigma _i^zn_i\\right)^2$ Although the problem is known to be NP-Hard, it is also known as the “easiest hard problem”.", "That is, because there exists a “hard-easy” phase transition [29] where instances belonging in the easy-phase can be efficiently tackled using heuristics [30].", "Interestingly, it appears that one may be able to tackle some of the instances in the “hard phase” using variational quantum algorithms." ], [ "Portfolio Optimisation", "The third problem is Portfolio Optimisation [31], [32] and is stated as follows.", "Given a set of $n$ assets $\\lbrace 0,\\cdots ,n\\rbrace $ , corresponding expected returns $\\mu _i$ and covariances $\\Sigma _{ij}$ , a risk factor $q > 0$ and a budget $B \\in \\lbrace 1,\\ldots ,n\\rbrace $ , the considered portfolio optimisation problem tries to find a subset of assets $P\\subset \\lbrace 1,\\ldots ,n\\rbrace $ with $|P|=B$ such that the resulting q-weighted-mean-variance, i.e.", "$\\sum _{i\\in P} \\mu _i - q\\sum _{i,j\\in P}\\Sigma _{ij}$ , is maximised.", "In other words, we want to maximise the cost function: $C(x) = \\sum _{i=1}^n \\mu _i x_i - q\\sum _{i,j=1}^n\\Sigma _{ij}x_ix_j$ along with the constraint $\\sum _{i=1}^nx_i = B$ The portfolio vector $x\\in \\lbrace 0,1\\rbrace ^n$ , consisting of $n$ binary decision variables, indicates whether an asset is picked ($x_i = 1$ ) or not ($x_i = 0$ ).", "The constraint in (REF ) is translated as an extra penalty term in the Hamiltonian $(\\sum _{i=1}^nx_i-B)^2$ .", "The problem is known to be NP-complete [33].", "We apply the transformation, Eq.", "(REF ), so the cost function transforms into: $\\begin{aligned}C(z) &= - q\\sum _{i,j=1}^n \\frac{\\Sigma _{ij}}{4}z_i z_j + \\sum _{i=1}^n\\left(\\sum _{j=1}^n \\frac{q\\Sigma _{ij}z_i}{2}-\\frac{\\mu _iz_i}{2}\\right) \\\\&+\\sum _{i=1}^n\\left(\\frac{\\mu _i}{2} - \\sum _{j=1}^n\\frac{q\\Sigma _{ij}}{4}\\right)\\end{aligned}$ which, along with the extra penalty term, corresponds to minimising the Hamiltonian : $\\begin{aligned}H_C &= \\sum _{i,j=1}^n \\frac{q\\Sigma _{ij}}{4}\\sigma _i^z\\sigma _j^z - \\sum _{i=1}^n\\left(\\sum _{j=1}^n \\frac{q\\Sigma _{ij}\\sigma _i^z}{2}-\\frac{\\mu _i\\sigma _i^z}{2}\\right) \\\\& -\\sum _{i=1}^n\\left(\\frac{\\mu _i}{2} - \\sum _{j=1}^n\\frac{q\\Sigma _{ij}}{4}\\right) + \\left(\\sum _{i=1}^n\\sigma _i^z + \\frac{n}{2}-B\\right)^2\\end{aligned}$ Portfolio optimisation as given in Eq.", "(REF ) was recently tackled using variational quantum algorithms [34], using warm-starting QAOA [4] and on D-wave systems using quantum annealing [35].", "Prior to our work, [36] considered a more general setting of portfolio optimisation, called dynamic portfolio optimisation, where one has to allocate weights to a number of assets in a period of time in order to maximise the overall return.", "The CVaR cost function of [1] was shown to perform better in general, than the “standard” expectation value.", "There are three observations, however, that motivates our proposal.", "First, as noted in [1], the choice of $\\alpha $ is somehow random, and importantly, for different problems and even for different instances of the same class of problems, the optimal choice of $\\alpha $ varies in a non-obvious (e.g.", "monotonic) way.", "The performance of the algorithm's speed, but also if it finds the solution at all, depends on that choice.", "The second point is that optimising with a fixed small $\\alpha $ has further disadvantages: (i) it “finds” parameters $\\theta $ that result to a state that does not have the greatest overlap with the solution and (ii) the true running time of the algorithm to achieve same accuracy is larger, in other words for each iteration one requires $1/\\alpha $ times more measurements to achieve the same accuracy in estimating the cost function (since only the lower $\\alpha $ fraction of the measurements are used).", "Finally, the third observation is that CVaR with different $\\alpha $ 's all agree in the ground state (it has minimum energy for all of them) but the rest energy landscape is different.", "For any fixed choice of $\\alpha $ the optimiser could “get stuck” at a local minimum.", "Interestingly, if one varies the $\\alpha $ during the optimisation, while we still ensure that if the algorithm finds the true ground state it remains there, we also avoid getting stuck at local minima since those are different for different choices of $\\alpha $ .", "Therefore if the optimiser reaches a point that has a local minimum for one value of $\\alpha $ , when $\\alpha $ changes this point (may) no longer be a local minimum and thus could continue “moving” towards the true global minimum (ground state).", "The cost functions used in variational quantum algorithms, to our knowledge, are “constant in time”, meaning that the whole optimisation is run with a fixed cost function.", "To solve the issue of “selecting the best $\\alpha $ ”, and the other reasons listed above, we propose to use a dynamically evolving cost function, that essentially passes through all the choices of $\\alpha $ , starting from a very small value and ending with $\\alpha =1$ that is the standard expectation value of the Hamiltonian.", "We call all these cost functions Ascending-CVaR.", "This has also a great(er) number of free choices, since we can now freely choose the (ascending) function.", "However, all choices we tried for the ascending function performed (in general) better than fixed $\\alpha $ , which indicates that the evolving cost function is a promising approach.", "For the remaining of the paper we focused on two functions that performed better: The linear ascending in which the parameter $\\alpha _t$ is iteratively and discretely increased by the rule: $\\begin{aligned}\\alpha _{t+1} &= \\alpha _t + \\lambda \\\\CVaR_{\\alpha _t} &= \\frac{1}{*{\\alpha _t K}}\\sum _{k=0}^{*{\\alpha _tK}}H_k\\end{aligned}$ where $\\lambda \\in [0.025, 0.045]$ is the ascending factor and $0< \\alpha _t \\le 1$ .", "The sigmoid ascending in which the parameter $\\alpha $ is discretely increased according to the function : $\\begin{aligned}\\alpha _t &= \\frac{1}{1+e^{5 - \\lambda t}}\\end{aligned}$ where $\\lambda \\in [0.3, 0.4]$ is again the ascending factor and $0 < \\alpha _t \\le 1$ .", "To reach this conclusion we tested four different functions, a sigmoid, a linear, an exponential and a logarithmic (see Figure REF ) on VQE-CVaR${_{\\alpha _t}}$ with various different ascending rates.", "All functions were tested on all three problems.", "The metrics used were the magnitude of the overlap with the optimal solution, the success rate (i.e.", "the number of times where it succeeds to achieve a non-negligible overlap) as well as the average time taken to achieve at least $10\\%$ overlap (for details see ).", "Figure: Different choices for the ascending function.", "All functions start from the same initial point, α 0 =0.01\\alpha _0=0.01 and ascend until α f =1\\alpha _f=1 is reached.The linear ascending, Eq.", "(REF ), and the sigmoid ascending, Eq.", "(REF ), functions have the most steady behavior, outperforming the other two types on the majority of instances.", "The sigmoid was slightly slower in terms of speed, which is why we mainly used the linear one.", "However, as we will discuss in the next section, it appears to be better in some classes of problems especially on harder instances with $\\sim 50$ qubits.", "It seems that in those cases, the optimiser is doing better spending the majority of its iterations on low $\\alpha $ values and thus the sigmoid performs better.", "Figure: Portfolio optimisation instance for 18 assets and different ascending functions.", "The blue line indicates the linear ascending and always achieves a high overlap with the optimal solution in contrast to the orange line, the exponential ascending, which fails in almost any instance." ], [ "Methods", "One common metric used, especially in QAOA, is the approximation ratio as given in Eq.", "(REF ).", "However, as we noted earlier, the true aim of variational quantum algorithms for combinatorial optimisation is to obtain quickly a sufficiently high (but not necessarily close to unity) overlap with the optimal solution.", "The CVaR method, for example, is constructed in a way that the maximum overlap achieved is not unity but determined by the risk $\\alpha $ .", "While our approach does achieve high approximation ratio, to make a fair and more complete comparison with prior works and importantly with [1], we use different metrics.", "Specifically, to benchmark and test our proposed method, we used three different types of metrics.", "The first is the overlap with the optimal solution.", "If $\\mathinner {|{\\psi _{opt,i}}\\rangle }$ is a $d$ -degenerate ground state of the problem Hamiltonian, then the overlap is defined as: $\\sum _{i=1}^d|\\mathinner {\\langle {\\psi (\\theta )}|}\\mathinner {|{\\psi _{opt,i}}\\rangle }|^2$ i.e.", "the probability of obtaining the optimal solution, given the parameters $\\theta $ .", "It follows that the parameterised state with the highest overlap with the optimal solution leads to sampling that optimal solution with the least number of circuit executions.", "The second metric we want to test is the time taken to reach a given fixed overlap.", "We set a threshold of $10\\%$ probability of obtaining the optimal solution and we tested which method achieves at least that probability faster.", "We note however that, in order to test which method converges to a $10\\%$ overlap faster, we have to use $\\alpha \\ge 0.1$ because all $\\alpha <0.1$ are not guaranteed to converge in an overlap of $10\\%$ since the parameters $\\theta $ than minimise $\\alpha $ lead in an overlap smaller than 0.1.", "To summarise the results and compare better the different approaches, for each cost function we divided the problem instances to those that the cost function is successful and to those that it fails.", "The meaning of what constitutes a “successful” run or a “failed” run cannot be unambiguously defined.", "For our work we consider that an optimiser is successful at a given instance of a problem if it achieves at least $10\\%$ overlap with the optimal solution.", "It is clear that as the size of the problem instances increase, achieving a fixed $10\\%$ overlap becomes harderWe should note that even a much smaller overlap is sufficient to find at least once the solution, provided that the number of “shots” is sufficiently large..", "In our analysis we chose $10\\%$ since this leads to interesting behaviour where the methods analysed differ in their performance.", "In our experiment, for comparing with fixed $\\alpha $ we used four different choices: $\\alpha =0.1, 0.2, 0.5, 1$ .", "The $\\alpha =1$ choice corresponds to the expectation value (it includes all the measurement outcomes) and it is this objective function that has been used in all existing literature apart from [1].", "All these were compared with our proposed $\\alpha _t$ .", "We also note that ascending factors $\\lambda \\in [0.025, 0.045]$ and $\\lambda \\in [0.3, 0.4]$ were found to be a good choice for the three different problems on instances with 15 to 20 qubits for the linear and sigmoid ascending respectively.", "In the QAOA algorithm we tested instances using depth $p=1$ to $p=6$ while on VQE we worked only on the depth $p=1$ , since this depth was sufficient to get very good accuracy.", "In near term devices for the QAOA algorithm, increasing the depth even more becomes impractical due noise and decoherence.", "For this reason we did not consider greater depth, despite the fact that theoretically this could lead to better performance.", "This means that the variational ansatz for QAOA has only 2 to 12 parameters, i.e.", "only a fraction of the total parameters present in hardware efficient ansatz used for VQE in depth-1.", "To account for the different sizes of problem instances, and to make a fair comparison for the speed of convergence, we used the normalised optimiser iterations [37].", "Note that this choice is made in order to be able to compare the performance of the algorithm among instances that involve different number of qubits, and see how the improvement offered by Ascending-CVaR is independent of the instance size.", "Concretely, the normalised optimiser iterations is defined as the number of times the optimiser evaluates the objective function divided by the function's number of parameters, i.e.", "the number of parameters of the ansatz.", "In the case of the VQE the number of parameters are $n(1 + p)$ while on QAOA are $2p$ .", "We note however, that the real time of convergence could be used as seen in Appendix , where we compare the performance with respect to the total number of circuit repetitions.", "However, as we show below, there are instances where the constant CVaR does not achieve even a small overlap with the optimal solution and in those cases the time taken becomes irrelevant.", "We ran our experiments on IBM's Qiskit Aer simulator, allowing noiseless multi-shot executions of our circuit.", "We set the number of executions of our circuit to $K=1000$ , which scaled up as $K/\\alpha $ with the choice of $\\alpha $ .", "All instances were given a maximum of $\\left(66\\times parameters\\right)$ optimiser iterations.", "They were initialised with a random choice of parameters, but the same for all different choices of $\\alpha $ .", "We used the same gradient-free optimiser, COBYLA [38], for all different problems and instances as it was shown to outperform other classical optimisers [16]." ], [ "Results", "We will analyse the results for each of the three combinatorial optimisation problems separately.", "For each of them we will present the results for VQE with hardware-efficient ansatz first and then the results for QAOA.", "We note that for all three combinatorial optimisation problems and for all methods used (Ascending-CVaR, constant CVaR, expectation value), VQE performs (much) better than QAOA, at least for the sufficiently shallow circuits that we consider.", "Our method improves the performance in both cases (VQE, QAOA) but since VQE gives much better results for these problems, in the comparison and discussion we will focus on VQE instances only." ], [ "Max-Cut", "For the Max-Cut problem we worked on unweighted graphs with 15-19 vertices, drawn from different graph classes and sampled them using the NetworkX library [39].", "For regular graphs, CVaR$_{\\alpha _t}$ -VQE behaved equally well with constant-$\\alpha $ 's optimisation as well as with the expectation value.", "All of the methods reached the chosen threshold of $10\\%$ overlap with the optimal state at almost equal times without any difficulty.", "For that reason we focused on harder non-regular instances where our method outperformed the latter methods.", "In Table REF we give a summary of the results for 100 random non-regular unweighted graph instances with 15 to 19 vertices.", "We can see that our method succeeds in more instances while the overlap achieved is also much higher.", "There are many reasons why non-regular random graphs are “harder” than regular graphs.", "The first is that the ground state of a regular graph, due to its symmetry, is highly degenerate where the optimiser could easily reach without “stucking” in a sub-optimal minimum.", "The second is that the Hamiltonian corresponding to a random graph has more distinct eigenvalues and as it was shown numerically by [16], the number of distinct eigenvalues correlates inversely with the performance of hardware efficient ansatz.", "Indicatively, in Figure REF we plot the probability of sampling the optimal solution over the normalised number of iterations for two random graphs with 17 vertices.", "For the left figure, we can see how the optimiser for the Ascending-CVaR optimisation is able to to find the optimal solution in under 10 normalised iterations which by the end of the optimisation is able to increase the probability up to $70\\%$ .", "Notably the expectation value or constant CVaR completely fail.", "The right part of the figure, gives another example where our approach performs better.", "This instance constitutes an example where smaller $\\alpha $ 's do not lead to better performance for constant CVaRIn most cases, small $\\alpha $ gives better performance, but one cannot know a-priori which is the suitable $\\alpha $ in the constant CVaR case.. We can see in the figure that while $\\alpha = 0.1$ failed, $\\alpha = 0.2$ was able to achieve a high quality parameterised state.", "This is another indication why our approach is more flexible.", "Figure: Max-Cut instances with 17 vertices for random non-regular unweighted graphs.", "Ascending-CVaR, drawn with a blue line, results in a fast and high overlap with the optimal solution in contrast to constant CVaR." ], [ "CVaR$_{\\alpha _t}$ -QAOA", "Solving the Max-Cut problem using QAOA, with small depth circuits, does not seem a very promising approach in any of the methods considered (constant CVaR or Ascending-CVaR).", "In terms of speed, all methods converged equally fast but in states with small overlap with the solution (with relatively small differences within different approaches).", "Having said that, as explained below, our method still gives improved performance.", "While CVaR$_{\\alpha _t}$ -VQE optimisation results in high overlap states, CVaR$_{\\alpha _t}$ -QAOA produces “flat” states, a behaviour also observed in [1].", "These states have almost equal probability amplitudes to the majority of the computational basis states.", "For the Max-Cut problem, as noted in [21], it seems that the states produced with QAOA with small $p$ result to states with energy close to the (random) initialisation point.", "The spread of the energies does increase with $p$ , possibly leading to a state close to the ground state, but in our analysis we focused on small $p\\le 6$ .", "Intuitively, the main reason why QAOA cannot achieve the same probability amplitudes as VQE, in the same depth, is due to having a smaller number of parameters as well as the architecture of the ansatz [40].", "Note that the parameter space is filled with sub-optimal local minima.", "Constant CVaR objective functions with different confidence level $\\alpha $ 's lead to different energy landscape.", "This means that a local minimum for a confidence level $\\alpha _1$ does not, in general, correspond to a local minimum for a confidence level $\\alpha _2$ if $\\alpha _1 \\ne \\alpha _2$ .", "This is probably the reason that we get improved performance.", "For example, in Figure REF we see how Ascending-CVaR can avoid local minima.", "In this example all constant CVaR achieve less than $3\\%$ overlap with the ground state, while the Ascending-CVaR gives $7\\%$ .", "Figure: CVaR α t _{\\alpha _t}-QAOA optimisation with linear ascending for a Max-Cut instance of 17 qubits.", "The blue line, indicating the ascending optimisation, results in more than a 100% increase in the overlap with the optimal solution in contrast to the expectation value or constant CVaR optimisation.On Number Partitioning we tested instances with 17 to 20 integers, on both VQE and QAOA." ], [ "CVaR$_{\\alpha _t}$ -VQE", "On CVaR$_{\\alpha _t}$ -VQE we tested 300 instances with 17 to 20 integers, sampled randomly from three sets; $N_1 = \\lbrace 0, \\ldots , 200\\rbrace $ , $N_2 = \\lbrace 0, \\ldots , 500\\rbrace $ and $N_3 = \\lbrace 0, \\ldots , 750\\rbrace $ .", "We highlight that the smaller the set that the numbers are uniformly drawn from, the easier the optimiser succeeds in finding the optimal solution.", "A summary of the results is given at Table REF .", "Table: Results table for the Number Partitioning problem (VQE) for the three different sets N 1 N_1, N 2 N_2 and N 3 * N_3^*, where the star at the last set indicates that we used the sigmoid ascending function.For the first two sets, we used a linear ascending function with an ascending factor $\\lambda = 0.03$ .", "Further optimisation of the parameter may lead in either faster convergence or more successful instances.", "Either way, the Ascending-CVaR method outperforms constant CVaR and the expectation value objective function on the aforementioned metrics (e.g.", "see typical performance on Figure REF ).", "Figure: Probability of sampling the optimal solution for Number Partitioning instances with 17-20 integers uniformly drawn from the sets N 1 ={0,...,200}N_1 = \\lbrace 0,\\ldots , 200\\rbrace (on the left) and N 2 ={0,...,500}N_2 = \\lbrace 0,\\ldots , 500\\rbrace (on the right).", "The blue line, indicating Ascending-CVaR outperforms constant CVaR in terms of speed and overlap with the optimal solution.For the last set $N_3$ , constant CVaR and the expectation value as objective functions struggled to achieve even a small overlap with the optimal solution.", "Indicatively, at $40\\%$ of the cases none of the constant CVaR objective functions could be “successful”Recall, that successful in our convention, means to achieve overlap of at least $10\\%$ with the optimal solution.. We found that by choosing a sigmoid ascending function, the optimiser is able to attain a high quality parameterised state and succeed in the majority of instances ($95\\%$ ).", "The trade-off is that using the sigmoid ascending function, in contrast to linear ascending, comes with some cost of more circuit shots in order to achieve the same accuracy.", "Note also, that the linear ascending function, while performing worse than the sigmoid, it was still more successful than the constant CVaR objective functions." ], [ "CVaR$_{\\alpha _t}$ -QAOA", "While CVaR$_{\\alpha _t}$ -VQE optimisation efficiently achieved a high overlap state already within the first layer for instances drawn from the two sets $N_1$ and $N_2$ , CVaR$_{\\alpha _t}$ -QAOA failed to achieve a high overlap on small depths.", "To address this issue without having to increase the depth of the ansatz we chose to work on instances drawn from the smaller set $M = \\lbrace 0, \\ldots , 50\\rbrace $ .", "Our method succeeds in finding quantum states with higher overlap, unreachable with constant CVaR optimisation, possibly because it avoids the high amount of local minima.", "Indicatively in Figure REF we see an example where Ascending-CVaR achieves more than double overlap with the optimal solution than other methods, but is still below the threshold of $10\\%$ required to classify this as a “successful run”.", "Figure: CVaR α t _{\\alpha _t}-QAOA for an 18-integer instance Number Partitioning problem with p=4p=4.", "The blue line, indicating Ascending-CVaR optimisation, is able to achieve 100% increase in the overlap with the optimal solution in respect to the other objective functions." ], [ "Portfolio Optimisation", "On Portfolio Optimisation we tested instances with 16 to 20 assets, on both VQE and QAOA, with a budget drawn uniformly at random from the set $B = \\lbrace 0, ... n\\rbrace $ where $n$ is the number of assets and many different risk factors $q$ ." ], [ "CVaR$_{\\alpha _t}$ -VQE", "We used linear ascending with an ascending factor $\\lambda = 0.045$ and the confidence level was initialised on $\\alpha _0 = 0.01$ .", "The results are summarised in Table REF .", "Table: Results table for the Portfolio Optimisation problem (VQE) for 100 random Portfolios with 16 to 20 assets.In Figure REF we see the typical performance of two different instances where we plotted the probability of obtaining the optimal solution over the normalised number of optimiser iterations for the CVaR$_{\\alpha _t}$ -VQE.", "Figure: Portfolio Optimisation problem for 18 and 20 asset instances with linear ascending and λ=0.045\\lambda = 0.045.", "The blue line indicates the CVaR α t _{\\alpha _t}-VQE optimisation which already within the first 10 optimiser iterations has achieved over 40%40\\% overlap, compared to constant α\\alpha 's where either fail (α=0.5,1\\alpha = 0.5, 1), or lead to slower and sub-optimal convergence (α=0.1,0.2\\alpha = 0.1, 0.2).We highlight the fact that Ascending-CVaR and constant CVaR with $\\alpha = 0.1, 0.2$ succeed in achieving at least $10\\%$ overlap on all instances tested (see results on Table REF ), while the expectation value ($\\alpha =1)$ failed in almost all cases.", "Moreover, it is worth noting that our method offers a significant improvement in comparison with all the other approaches in the speed that this overlap was achieved (in terms of normalised optimiser iterations and circuit repetitions) and in the overall magnitude of the overlap achieved (see also Table REF )." ], [ "CVaR$_{\\alpha _t}$ -QAOA", "CVaR$_{\\alpha _t}$ -QAOA, similarly with [1], underperforms significantly in terms of overlap with the optimal state, compared to CVaR$_{\\alpha _t}$ -VQE.", "Specifically, keeping the depth as in previous parts, and without increasing the shots each circuit is implemented, all methods fail achieving overlap with the optimal solution well below $1\\%$ .", "There are several reasons for this failure, including the Reachability Deficits [41], the large problem density [16] and Barren Plateaus [42].", "This, however, goes beyond the focus of this paper that is to find a way to improve the performance of previously used objective functions.", "To illustrate the improvement, we could have used (significantly) larger number of shots, where Ascending-CVaR would start showing better performance.", "This would make the comparison with other problems unfair (where in all cases we used the same “normalised” number of shots), and it would still not present a practical way to solve the Portfolio Optimisation problem (VQE is much better), so we omitted it." ], [ "Conclusions", "We introduced a novel type of objective function, Ascending-CVaR, to be used in variational quantum algorithms for any combinatorial optimisation problem.", "The starting point is the (constant) CVaR objective function of [1], where they illustrated that for any choice of risk $\\alpha $ the true ground state is a minimum, and that with (typically small values of) $\\alpha $ one can improve the performance compared to the “standard” expectation-value objective function.", "Our idea was to use an evolving objective function that “passes” through all the different values of $\\alpha $ to finish at the expectation value.", "This, intuitively, avoids getting stuck at local minima since the energy landscape for different $\\alpha $ 's differs apart from the global minimum.", "We tested numerically the proposal on three combinatorial optimisation problems (Max-Cut, Number Partitioning and Portfolio Optimisation), where in agreement with prior works we found that for these problems VQE seems more promising than QAOA with small depth.", "The improvement that Ascending-CVaR provides to VQE and QAOA are similar but we focus on VQE here since this was the overall more promising approach to solve the corresponding optimisation problems.", "We observed that Ascending-CVaR gave much greater on average overlap with the optimal solution (see Table REF ).", "In Portfolio Optimisation and Number Partitioning we got 10 times greater overlap than the expectation value (while we got at least double overlap than the best constant CVaR choice).", "In Max-Cut we got smaller improvement ($80\\%$ ) compared to the expectation value, but note that the constant CVaR actually gave much smaller overlap.", "Perhaps the most important feature is that in the Number Partitioning and Max-Cut, Ascending-CVaR succeeded in finding the solution in many instances that no other approach achieved more than the small chosen threshold of $10\\%$ overlap.", "This indicates that not only the approach improves the quality of the results, but is plausible that instances that are believed to be “hard” with the other methods, will become “easy” and thus solvable.", "Table: Overview of our methodBeyond the accuracy of the result, another factor to evaluate the performance of variational quantum algorithms is the speed, that can be counted with respect to the (average) number of iterations the optimisation needs to run until the algorithm outputs a (candidate) solution.", "Since our proposal “passes” through several choices of $\\alpha $ , one could expect that the “trade-off” for better overlap would be slower speed and thus more optimisation iterations.", "Interestingly, not only we do not get any cost in speed, but in most cases we see an improvement, i.e.", "our method requires fewer iterations to reach the threshold of $10\\%$ overlap with the solution (see Table REF ).", "The only case that our method required slightly more iterations than the $\\alpha =0.1$ was for the case that we actually observed the greater improvement in overlap.", "This was the Number Partitioning from the set $N_3$ , where the overlap was seven times better than the “next best” case, and 350 times greater overlap than the expectation value (see Table REF ).", "Table: Average Normalised Optimiser Iterations to achieve at least 10%10\\% overlap with the optimal solution for the three different combinatorial problems.Our work, not only offers a generic method to improve the performance of variational quantum algorithms for combinatorial optimisation problems, it also suggests a new direction of research where dynamic objective functions can be used to boost the performance in terms of quality and speed of near-term quantum algorithms.", "An immediate follow up to the proposal suggested here is to generalise our approach.", "Concretely, our method introduces two extra degrees of freedom.", "The hyperparameter $\\lambda $ and the function according to which the parameter $\\alpha $ increases.", "It is worth exploring a more systematic rule on how to fix these degrees of freedom according to the problem considered and the features of the specific instance.", "Finally, considering other dynamic objective functions is another direction that is worth pursuing." ], [ "Acknowledgements", "P.W.", "acknowledges support by the UK Hub in Quantum Computing and Simulation, part of the UK National Quantum Technologies Programme with funding from UKRI EPSRC grant EP/T001062/1 and the Collaborative Computational Project - Quantum Computing (CCP-QC) with funding from UKRI EPSRC grant EP/T026715/1." ], [ "Circuit Repetitions", "In this section, we demonstrate how our method outperforms, in terms of real circuit repetitions and quality of the output state, the previously used objective functions.", "We set our “default” circuit repetitions to $K=1000$ which we then scale it up, along the discretely increasing $\\alpha $ using the expression $K/\\alpha _t$ , for each given time.", "While one may think that this would weaken our results, as illustrated below, it seems that in terms of circuit repetitions our method converges to the chosen threshold of $10\\%$ faster than the best of constant CVaR or the expectation value approaches.", "Figure: Probability of sampling an optimal solution over the circuit repetitions for a Number-Partitioning instance." ] ]
2105.11766
[ [ "The Cynicism of Modern Cybercrime: Automating the Analysis of Surface\n Web Marketplaces" ], [ "Abstract Cybercrime is continuously growing in numbers and becoming more sophisticated.", "Currently, there are various monetisation and money laundering methods, creating a huge, underground economy worldwide.", "A clear indicator of these activities is online marketplaces which allow cybercriminals to trade their stolen assets and services.", "While traditionally these marketplaces are available through the dark web, several of them have emerged in the surface web.", "In this work, we perform a longitudinal analysis of a surface web marketplace.", "The information was collected through targeted web scrapping that allowed us to identify hundreds of merchants' profiles for the most widely used surface web marketplaces.", "In this regard, we discuss the products traded in these markets, their prices, their availability, and the exchange currency.", "This analysis is performed in an automated way through a machine learning-based pipeline, allowing us to quickly and accurately extract the needed information.", "The outcomes of our analysis evince that illegal practices are leveraged in surface marketplaces and that there are not effective mechanisms towards their takedown at the time of writing." ], [ "Introduction", "With the continuous digitisation of procedures, services, and products, crime has shifted towards the same direction.", "As a result, almost every aspect of a modern crime is facilitated by digital means, and consequently, almost every criminal investigation involves some sort of digital evidence.", "The above are the primary reasons why cybercrime has evolved into a multi-billion underground economy.", "Its economic impact is devastating [50], with FBI estimating the losses to $3.5 billion only within USA [21].", "In fact, this continuous rise is so threatening that it has become the second most-concerning risk for global commerce over the next decade, according to the World Economic Forum [16].", "Notably, the recent COVID-19 pandemic resulted in a huge spike in cybercrime activitieshttps://www.europol.europa.eu/newsroom/news/covid-19-sparks-upward-trend-in-cybercrime https://www.interpol.int/en/News-and-Events/News/2020/INTERPOL-report-shows-alarming-rate-of-cyberattacks-during-COVID-19 making the problem even more thorny.", "It is indisputable that the introduction of cryptocurrencies has significantly facilitated illegal money flows.", "The transactions of many cryptocurrencies offer high inherent privacy guarantees; e.g.", "Monero, ZCash, or they are difficult to be traced due to the wallet anonymity; e.g.", "Bitcoins.", "The rise of the dark web has also boosted the underground economy as it provides perpetrators with more advanced privacy guarantees.", "While dark web markets are still the key stakeholders when it comes to illegal trading, several surface web marketplaces have recently been repeatedly reported for trading leaked datahttps://www.forbes.com/sites/zakdoffman/2020/04/20/facebook-users-beware-hackers-just-sold-267-million-of-your-profiles-for-540/ https://www.zdnet.com/article/hacker-selling-data-of-538-million-weibo-users/.", "A very interesting characteristic, in this case, is that despite the trading of illicit products, these marketplaces have a very open form, e.g.", "they do not require any registration to access them, and the “loot” that is traded is advertised openly across the web." ], [ "Motivation and Contribution", "Delinquent behaviour is unarguably a characteristic that can be observed in every human society.", "The extent of this behaviour, in terms of how many people exhibit it, and the harm that is caused by it define the ethics of the society and its limits.", "The ethics on the Internet follow different rules [47].", "Moreover, the dark web is considered the default place on the Internet in which such behaviours and actions flourish.", "Nonetheless, they are promoted in closed circles so that there is some “control” on who can access this information and to retain the anonymity of the perpetrators.", "However, should this information be openly disseminated in public channels, it implies that the promoted behaviour is widely practised and is considered a norm by some groups.", "In the past few years, there has been a significant increase in reported data leaks, online extortion schemes and credential trading.", "One of our initial research questions was whether such actions are so widely performed that they can be observed on the surface web.", "In this regard, we wanted to check whether the perpetrators were using platforms of the surface web to advertise their “loot” and the existence of marketplaces in the surface web.", "Currently, there are several such marketplaces operating with similar functionality; however, this work is mainly focused on Shoppy (https://shoppy.gg/) which appears to have the most users and products at the time of writing.", "Nonetheless, similar illicit trends have been found in the rest of the surface web marketplaces.", "The goal of this work is to provide an overview of what is actually being sold in such a marketplace, and leverage methods (e.g.", "machine learning) to automatically determine which are the illegal products and the main organisations affected.", "The main limitation in the automation of such a task is the lack of text.", "These sellers do not need to add a lot of text about what they trade in these marketplaces, and in many occasions there are typos, abbreviations, and slang, posing even more issues in the analysis of the derived text.", "Further to the analysis of the traded products, we discuss the modus operandi of the sellers and some insight regarding the pricing of “big leaked data”.", "The rest of this work is structured as follows.", "First, we provide an overview of the related work and a brief discussion of these marketplaces.", "Then, we detail our data collection methodology.", "In Section , we analyse the collected data to extract actual knowledge out of the short descriptions of the shops in an automated way.", "Finally, the article concludes, summarising our contributions and highlighting some ideas for future work." ], [ "Related work", "In recent years, there have been multiple incidents of massive data breaches affecting a broad spectrum of online services and service providers, including retailers, payment processors, and government entities [12].", "Malicious actors gain internal access to sensitive data sources, and then acquire millions of credit and debit card details, user credentials, as well as sensitive data which can be used to identify individuals uniquely.", "The sheer quantity of data that can be acquired has given rise to a burgeoning market for actors who sell the information that they obtain, through, e.g.", "hacking and other forms of data theft, to other users.", "Participants in these illegally acquired data markets leverage various communication and networking methods, enabling them to freely form communities and interaction mechanisms.", "The most prevalent forms of such marketplaces, as identified in the literature, are Internet forums and Internet-Relay-Chat (IRC) channels [37], [3].", "In particular, forums have been shown to comprise the principal medium for cybercriminals to network, form communities, and operate online stolen data markets, despite numerous successful infiltrations by law enforcement agencies [55].", "To a large extent, these marketplaces reside in the dark web, commonly behind Tor [15], and are referred to as “Darknet markets”.", "Darknet markets are popular among criminals since they enable them to anonymously trade illegal goods and services, extending well beyond stolen data.", "The latter was discussed by Thomas et al.", "[51] by pointing out the complex value chain of the underground market economy at scale.", "These marketplaces comprise the essential pillars of this global-scale cybercrime economy and thus have become the key information source for investigating the cybercriminal ecosystem.", "An extensive body of literature has explored the darknet marketplaces [54], the involved stakeholders and their communication patterns [20], and their modus operandi [53].", "Of particular interest to researchers, are the marketplaces dedicated to the sale of stolen personal and financial information, known as “carding forums”, where cybercriminals sell the artefacts of large scale data breaches, often containing stolen financial information [37], [25], [18].", "Subsequently, the compromised credit and debit card information enables malicious actors to commit crimes such as identity theft, financial fraud, and most importantly, online money laundering [12], [31].", "Moreover, due to their illicit and underground nature, carding forums and marketplaces are characterised by unique trading dynamics between vendors and sellers, since the quality of merchandise and the identities of traders are unknown to potential buyers.", "In this regard, a number of works focus on untangling the mechanics of transactions in this particular kind of underground marketplaces [56], [17], [19], [48].", "Identifying key players is essential when investigating emergent threats and developing efficient disruption strategies [34], in particular, considering the fact that members of such communities are characterised by cross-forum posting activity, which can be used to identify user roles based on the type of posts and their frequencies.", "The indicators of trustworthiness and reputation of a seller play a pivotal role for the sale of illicit services and stolen data through underground hacking forums and markets, as users are more likely to conduct business with sellers who hold reputable standing.", "As such, it has been shown that the status of reputation can be used to identify prominent players in illicit online marketplaces [56].", "Apart from the complexity of its dynamics, the ecosystem of illegal online markets is characterised by an equally wide range of offerings, services and products relevant for a variety of illicit topics, such as underground drug economies, data breaches, and cyber warfare.", "For instance, the work proposed in [44] focuses on malicious assets traded in hacker forums, such as hacking tools, rootkits and exploits, from which cyber threat intelligence can be distilled, and analyses the extracted data for predicting and mitigating cyber attacks.", "In [29], the authors provide both quantitative and qualitative categorisation of offerings in 17 different marketplaces.", "Their findings indicate the existence of both highly specialised products with respect to particular vendors and markets, as well as the cross-listing of products on multiple sites and nearly identical products for sale by multiple vendors.", "Nonetheless, the most prevalent categories are related to stolen credentials and information, extending beyond financial accounts.", "In this direction, Madarie et al.", "[27] examined how, a diverse set of outlets such as stolen credentials, are disseminated by malicious actors as “account dumps’’.", "Their analysis revealed that the illicit dissemination of stolen account credentials covers a broad spectrum of online services, and highlighted Pastebinhttps://pastebin.com/ as one of the main sites used to spread the information.", "Pastebin is a paste website intended for sharing plain text snippets, which is radically different compared to the typical marketplaces and forums, due to the complete lack of structure.", "Interestingly enough, while stolen combinations of usernames and passwords for various online services were posted, data thieves used advertisements (embedded within the dumps) for establishing communication with potential customers seeking access to financial or more sensitive data." ], [ "Methodology", "After thorough literature research, we observed a literature gap focusing on surface web marketplaces and an analysis of some of their activities, including automated methodologies for the extraction and processing of information to discover potentially malicious behaviour.", "First, we explored two well-known hacking-related forums, namely blackhatworldhttps://www.blackhatworld.com/ and crackedhttps://cracked.to/ looking for indications of the emergence of marketplaces supporting illegal activities, similar to the deep web forums/marketplaces.", "In this regard, we observed that Shoppyhttps://shoppy.gg/ was widely used in such platforms to monetise some of the reported activities.", "In fact, such activities were advertised.", "Therefore, we established a methodology to analyse what types of activities and products were being sold in Shoppy.", "Shoppy is a shop hosting service that provides the opportunity to individual vendors to sell their products, allows payments in different forms, and a set of APIs to, e.g.", "advertise one's products in forums etc.", "A crucial difference between Shoppy and the underground marketplaces studied in the literature is that the former does not offer a centralised listing of the sold vendors and products.", "Each vendor obtains a unique URL where they can host their shop, without providing any means for a user to look for similar shops of products offered by different vendors, a common feature in e-commerce platforms [45].", "The decentralised architecture of Shoppy hinders the extraction of knowledge, and thus, our proposed data collection methodology specifically aims to discover shops associated with illicit offerings and services, given the context established by focusing on hacking-related forums.", "Figure: Workflow of the steps followed in our research.The methodology that we adopted to address this challenge consists of several steps, as depicted in Figure REF .", "First, we crawled the blackhatworld and cracked forums, collecting usernames, as well as references to Shoppy accounts in post signatures.", "Given the size of these two communities, we specifically focused our crawling only to the “Marketplace” forums.", "To his end, we adopted the architecture of the Structure-driven Incremental Forum crawler (SInFo) [36], which enabled us to crawl data from the aforementioned forums.", "Nevertheless, we did not leverage user accounts that could potentially allow us to access even more content, restricted to authenticated users [49].", "Next, we examined the extent to which the collected usernames and Shoppy account data could be correlated with existing shops in the Shoppy ecosystem.", "The data collection process lasted from March to April of 2020.", "We collected a total of 68,045 usernames, and Shoppy links from forum post signatures, 2,906 of which were linked to existing Shoppy shops at the time of crawling.", "The results are summarised in Table REF .", "Notably, a large fraction of the links to Shoppy accounts found in post signatures, that did not resolve to existing shops, indicating that accounts in Shoppy may be banned, deleted, or renamed.", "Table: Collected usernames and Shoppy links.With the collected data, we used the open Shoppy API to retrieve all the information associated with these shops, including products, prices, and their corresponding metadata to create a curated dataset." ], [ "Data Exploration", "In the following sections, we explore the Shoppy data in different steps.", "First, we provide a quantitative review of the collected dataset.", "Next, we detail our topic modelling approach and, finally, we leverage an exploratory analysis of a subset of the surface data." ], [ "Shoppy in Numbers", "In this section, we provide a quantitative analysis of the collected Shoppy shops and advertised products, as well as highlight the particular behaviours of vendors.", "In total, our dataset contains 64,726 products advertised by 2,906 vendors.", "Shoppy provides vendors with the ability to categorise their products as accounts, services or files.", "The distribution of product categories in our dataset is provided in Table REF .", "“Account” is the default category, which evidently dominates the other two by a large margin.", "Table: Shoppy products per categoryFig REF describes the cumulative distribution functions (CDFs) of the number of items per shop (Fig REF ) and the product prices, in USD (Fig REF ).", "We can observe that, while around 40% of the shops have less than ten items listed, there exist shops with thousands of items.", "As seen in Fig REF , the price distribution of products is remarkably well described by a lognormal distribution ($\\mu =1.6, \\sigma =1.58$ ), highlighting that the prices of approximately 62% of the products fall within a small range comprised between 1 to 10$.", "Moreover, the median price of Shoppy products is 5 USD, and, as observed in our dataset, the prices can reach up to 10,000 USD.", "Figure: CDFsTo get deeper insight on how different types of products are priced, focusing on possible outliers priced well above the median of 5 USD, we bin the products based on their price and we calculate the fractions of each product type in each bin in Figure REF .", "We can observe that while the lowest price bin is dominated by accounts, the fractions of services per bin follow a consistently increasing trend as the prices increase.", "In contrast, the relative representation of accounts is inversely proportional to the price, ultimately making services the predominant product category (approx.", "70% of total) for the last bin reflecting the highest-priced offerings ($\\ge 500 \\$$ ).", "The fractions of the file type products, which as previously shown comprise only a small fraction of the total offerings, are generally sustained, accounting for less than 10% of the products in each bin.", "It is worth to note that our initial observation related with the use of default categories is reflected in Figure REF , which shows that, for instance, account products are well represented within all the range of possible prices.", "The latter behaviour seems quite unrealistic in a real and competitive market scenario and is further supported by the experiments performed in the next sections.", "Figure: Fractional price bins of Shoppy products.To investigate the high priced services dominating the upper price bracket, we manually examined them and provided some illustrative examples in Table REF .", "Evidently, these items are false products and rather contain information such as merchants' terms of service, notes regarding provided shop feedback, support information, and links to Discord servers and Telegram channels maintained by the merchants.", "This behaviour has been highlighted in recent literature by arguing that the unregulated and anonymous nature of platforms such as Telegram and Discord, makes them the perfect habitats for scammers and cybercriminals [33], [52].", "Table: Some illustrative false “services”, priced ≥500$\\ge 500 \\$." ], [ "Topic Modelling", "In this section, we analyse the Shoppy stores and elaborate a topic-based characterisation of the offered products by analysing their titles.", "To this end, we consider a statistical model, namely “topic model”, which is a method well suited to the study of high-level relationships between text documents.", "Specifically, we leverage Latent Dirichlet Allocation (LDA), a generative probabilistic model proposed in [4].", "It comprises an endogenous NLP technique, which as highlighted in [5] “involves the use of machine-learning techniques to perform semantic analysis of a corpus by building structures that approximate concepts from a large set of documents” without relying on any external knowledge base.", "LDA, as the name implies, is a latent variable model in which each item in a collection (e.g., each text document in a corpus) is modelled as a finite mixture over an underlying set of topics.", "Each of these topics is characterised by a distribution over item properties (e.g.", "words).", "LDA assumes that these properties are exchangeable (i.e.", "ordering of words is ignored, as in many other “bag of words” approaches in text modelling), and that the properties of each document are observable (e.g.", "the words in each document are known).", "The word distribution for each topic and the topic distribution for each document are unobserved; they are learned from the data.", "Since LDA is an unsupervised topic modelling method, there is no direct measure to identify the optimal number of topics to include in a model.", "In this sense, LDA assigns documents to different clusters of topics with certain probabilities (i.e.", "the number of clusters is defined with an integer number $k$ provided by the user), where these probabilities depend on the occurrence of words which are assumed to co-occur in documents belonging to the same topic (Dirichlet prior assumption).", "This exemplifies the main idea behind all unsupervised topic models, that language is organised by latent dimensions that actors may not even be aware of [30].", "Researchers have recommended various approaches to establish the optimal $k$ (e.g.", "[6], [1], [13], [42], [57]).", "These approaches provide a good range of possible $k$ values that are mathematically plausible.", "However, according to [14], when topic modelling is used to identify themes and assist in interpretation (like in the present study), rather than to predict a knowable state or quantity, there is no statistical test for the optimal number of topics or the quality of a solution.", "A simple way to evaluate topic models is to look at the qualities of each topic and discern whether they are reasonable [30].", "To the best of our knowledge, the topic coherence measure with the largest correlation to human interpretability is the $C_v$ score defined in [42], which we also adopt in this study to establish the optimal number of topics.", "In our setting, we consider as a document the aggregate titles of the offered products in each of the 2906 shops in our dataset.", "For training LDA models on the generated documents, we employed the implementation provided by Machine Learning for Language Toolkit (MALLET) http://mallet.cs.umass.edu/.", "To obtain the most coherent topic model for our data, we considered the number of topics $(k)$ within the range from 5 to 50 with a step of 5 and trained the LDA models with 1,000 Gibbs sampling iterations and priors $\\alpha = 5/k$ , $\\beta = 0.01$ .", "For each trained model, we compute the $C_v(k)$ metric .", "This metric combines the indirect cosine measure with the normalised pointwise mutual information (PMI) and the boolean sliding window technique, to determine the number of optimal topic classes according to data distribution [43].", "According to Figure REF , the value yielding the highest $C_v$ corresponds to $C_v(20)=0.621$ and thus, we set the number of topics $k$ to 20.", "Figure: C v C_v metric according to the number of topics.In Table REF , we present the topics learned by our best LDA model, including the most relevant terms describing each topic and the number of shops where each topic is dominant.", "To obtain the most descriptive terms for topic interpretation, we adopted the approach of ranking individual terms within topics presented in [46].", "Table: Different topic classes and their corresponding key terms, sorted according to the number of documents found.To provide an insight on the products sold by the shops classified in each topic, Table REF includes some indicative examples per topic, with respect to the number of topic-relevant terms contained in their titles.", "The latter further allows us to characterise each one of the learned topics in a qualitative manner.", "Topics #1 and #2 describe “premium” accounts for a variety of online services and software products including streaming and VPN services.", "Topic #3 describes accounts associated with popular restaurants and fast food companies.", "Topic #4 reflects accounts associated with in-game items and collectables for the popular online game Fortnite.", "This topic is found to be dominant in most shops, in comparison to the other topics, with 466 occurrences (i.e.", "16% of all shops).", "Although selling game accounts can be perceived as an innocuous activity, provided the context of our data collection, these selling activities could be linked with money laundering schemes, based on the idea of converting stolen money to virtual currencies which are used to purchase in-game items [10], [32].", "Topic #5 focuses on OpenBullet configurations.", "OpenBullet is a brute-forcing tool used for performing credential stuffing attacks against online services [24], which are described by configuration files “configs”, offering features such as checking multiple credentials simultaneously (advertised by metrics such as CPM, standing for “Checks Per Minute”) and bypassing rate-limiting.", "Topic #6 contains several classes associated with a broad spectrum of products ranging from game accounts to hacking and reconnaissance tools such as dorks.", "Topic #7 includes mainly subscriptions to various sports and video streaming services.", "Topic #8 highlights accounts, hacking tools and in-game items for the popular video game Minecraft.", "Topic #9 models the false products previously described (cf Table REF ), containing information regarding vendor's terms of service and links to external Discord servers, Telegram channels, and etc.", "Topic #10 includes product licences and keys for a variety of software packages, games and operating systems.", "Topic #11 describes subscription plans for streaming services, similar to Topic #7.", "Topic #12 involves accounts for the popular game League of Legends.", "Topic #13 describes selling leaked user data from security breaches, in the form of combo lists, i.e.", "combinations of usernames/emails and passwords [28], which can be used for compromising accounts with the same credentials in other services, by means of credential stuffing attacks, as seen in Topic #5.", "Topic #14 involves mostly guides and e-books regarding carding and other methods of financial fraud.", "Topic #15 contains discount codes and accounts containing redeemable credits for various online shops and e-commerce platforms.", "Topic #16 is closely related to topics #14 and #15 and includes vouchers for online purchases in various venues as well as methods to perform a fraud or to scam sellers.", "Topic #17 mainly includes subscriptions for online services and products with a focus on mobile apps.", "Topic #18 is related to serial numbers for computer peripherals such as monitors, keyboards, etc.", "Topic #19 provides assorted “random” accounts for various social media and sites.", "Finally, Topic #20 is related to products such as redeemable gift cards, mainly for restaurants and food suppliers." ], [ "Use case analysis of surface data", "In this section, we focus on the Topics #5 and #13, which as highlighted above, model products related with cybercriminal activities such as selling breached credential dumps and using tools for automating the compromise of accounts in different online services.", "To this end, we leverage the term-salience metric defined in [9], which given the set of representative terms per topic, ranks them according to their distinctiveness, i.e.", "how informative a specific term is for determining the generating topic, versus a randomly-selected term.", "Subsequently, we select the top-3 most salient terms for topics #5 (config, openbullet, capture) and #13 (combo, database, records), and we use them to query product titles, in order to identify the most prevalent products modelled by these topics.", "For Topic #13, we additionally include the term db which is a common abbreviation for the term database.", "As previously reported (Table REF ), Topic #13 models leaked data from online data breaches, which are sold in the form of username/email and password combinations, along with other personal information.", "Such listings usually advertise the number of the breached records, as well as the source of the leak.", "In Table REF we present some of the largest account dumps found in our Shoppy dataset, along with their prices.", "Indeed, we discovered that popular password breaches checker platforms, such as https://haveibeenpwned.com, list the majority of the account database dumps sold on Shoppy.", "Moreover, this could explain the relatively low price tag for leaks, including up to millions of records, as the respective breaches have already been made public.", "In Table REF , we list some illustrative products with titles including at least one of the selected salient terms for Topic #5.", "We observe that these products represent configurations for software such as OpenBullethttps://github.com/openbullet/openbullet/BlackBullethttps://redskyalliance.org/xindustry/blackbullet-credential-stuffing/Stormhttps://www.netacea.com/blog/storm-cracker-tool/.", "As previously stated, such tools can be used to automate credential stuffing attacks [41], versus various online services, as shown from the product titles.", "Sellers of such “configs” often advertise features such as CPM (checks per minute) and capturing functionality offered, i.e.", "the ability to capture specific information associated with a compromised account, such as saved credit cards and payment methods, reward points, etc.", "From the above, we can largely infer the modus operandi of the account sellers of Shoppy and other cybercriminal markets: One is able to purchase massive quantities of breached credentials, and by exploiting the password reuse behaviour exhibited by many users [40], she could compromise users accounts with same credentials in other online services by using credential stuffing tools with different configurations.", "Table: Known breaches sold through Shoppy, as identified by Topic #13's most salient terms.Table: Indicative products modelled by Topic #5 in descending price order." ], [ "Discussion and final remarks", "There are several conclusions that can be extracted from the analysis and the outcomes obtained in the previous sections.", "First and foremost, we found evidence of malicious activities which are usually taking place in the dark web, yet this time arising on the surface web.", "In this sense, the cynicism of malicious actors, who are perpetrating these activities, is covered by a lack of methodologies and takedown mechanisms, due to several factors such as, e.g.", "the decentralised nature of the marketplaces.", "To the best of our knowledge, this is the first work that provides a solid and automated methodology to find, quantify and classify in a comprehensive way such activities.", "Nevertheless, despite the promising analysis leveraged in this article, malicious actors always find a way to circumvent analysis, since, e.g.", "they only use such platforms as a contact point, redirecting all of their activities to other external channels such as Telegram or Discord.", "Moreover, the use of technologies such as IPFS can augment the possibilities and resilience of such malicious practices [35], [39].", "Another dimension to be yet explored is the underlying connection between the activities reported in this article and further criminal campaigns.", "Therefore, despite the fact that most of the sold products can be classified as `soft' cybercrime (i.e.", "passwords, credit card credentials, personal data) they can pose significant damage to individuals and businesses, and they may be just the tip of the iceberg.", "More concretely, money laundering and the financing of other, probably more dangerous activities, can be just happening in front of our eyes [22], [31], [11].", "As previously stated, there exist several challenges for the analysis and takedown of the illegal activities being hosted on such platforms.", "The decentralised nature of, e.g.", "Shoppy avoids crawling mechanisms that could be used to collect all the stored information.", "Moreover, Shoppy is a resilient platform, as well as Sellix and Selly.", "The latter is supported by the fact that users can just have back their shops easily.", "As a matter of fact, the activities reported in this article are taking place at the moment of writing without restrictions.", "The possibility of linking these activities by using novel blockchain platforms is a further issue that needs to be thoroughly explored.", "First, the immutable nature of blockchain may permit the development of shopping platforms which offer private and permanent selling services [23], [26], [7].", "The latter fact is a critical issue due to the lack of efficient erasure mechanisms [2], [38], [8].", "We argue that more effort should be devoted to the development of robust AI methods as well as data collection procedures such as the one proposed in this article to locate and quantify the extent of such activities.", "Moreover, robust investigation protocols and more support from law enforcement towards the prosecution of these activities, as well as legislation related to this phenomena are mandatory.", "Finally, proactive measures, including strategies such as abnormal behaviour detection and the corresponding mitigation actions should be implemented by design, especially in the cases in which a platform is using immutable architectures.", "In this article, we showed that most of the activities that are leveraged in the dark web are also taking place on the surface web and yet, no effective mechanisms or takedown measures are taking place.", "This claim is supported by our thorough analysis of a marketplace, namely Shoppy.", "First, we collected credentials from two well-known forums, namely cracked and blackhatworld.", "Next, due to decentralised and anonymous nature of Shoppy, we used such credentials to crawl and retrieve data regarding shops, products and descriptions.", "Subsequently, we used topic modelling-based analysis to categorise and further explore the collected data by reporting several qualitative and quantitative features.", "Our findings evince the cybercriminal nature of a myriad of shops and users in the Shoppy ecosystem, supporting our initial claim.", "Finally, to raise awareness and highlight the relevance of our findings, we discussed the implications of our research, the current challenges and limitations, and proposed some measures to overcome them.", "Future work will focus on exploring similar marketplaces and trying to find correlations between different platforms in an automated way.", "Moreover, we plan to analyse the possible links between the activities leveraged in such marketplaces and cryptocurrencies, as well as other widely used financial platforms." ], [ "Acknowledgements", "This work was supported by the European Commission under the Horizon 2020 Programme (H2020), as part of the projects CyberSec4Europe (Grant Agreement no.", "830929) and LOCARD (Grant Agreement no.", "832735).", "The content of this article does not reflect the official opinion of the European Union.", "Responsibility for the information and views expressed therein lies entirely with the authors." ] ]
2105.11805
[ [ "Fractal generation in a two-dimensional active-nematic fluid" ], [ "Abstract Active fluids, composed of individual self-propelled agents, can generate complex large-scale coherent flows.", "A particularly important laboratory realization of such an active fluid is a system composed of microtubules, aligned in a quasi-two-dimensional (2D) nematic phase, and driven by ATP-fueled kinesin motor proteins.", "This system exhibits robust chaotic advection and gives rise to a pronounced fractal structure in the nematic contours.", "We characterize such experimentally derived fractals using the power spectrum and discover that the power spectrum decays as $k^{-\\beta}$ for large wavenumbers $k$.", "The parameter $\\beta$ is measured for several experimental realizations.", "Though $\\beta$ is effectively constant in time, it does vary with experimental parameters, indicating differences in the scale-free behavior of the microtubule-based active nematic.", "Though the fractal patterns generated in this active system are reminiscent of passively advected dye in 2D chaotic flows, the underlying mechanism for fractal generation is more subtle.", "We provide a simple, physically inspired mathematical model of fractal generation in this system that relies on the material being locally compressible, though the total area of the material is conserved globally.", "The model also requires that large-scale density variations be injected into the material periodically.", "The model reproduces the power spectrum decay $k^{-\\beta}$ seen in experiments.", "Linearizing the model of fractal generation about the equilibrium density, we derive an analytic relationship between $\\beta$ and a single dimensionless quantity $r$, which characterizes the compressibility." ], [ "Introduction", "We examine a quasi-two-dimensional (2D) synthetic active nematic fluid, composed of biologically derived subunits [10], in which the rod-like microtubule subunits are driven relative to each other by the action of kinesin motor proteins.", "In recent years, this system has become an important prototype for active fluids, with numerous experimental and theoretical studies [9], [11], [12], [13], [14], [15], [16], [17], [18], [19].", "A key characteristic of this fluid is the emergence of mobile topological defects; points where the orientational order of the nematic phase breaks down.", "Positive and negative defects occur spontaneously as a result of the active flows, separate from each other and ultimately annihilate with other defects of their opposite charge in the fluid.", "In recent work, our group analyzed the motion of these defects in the context of chaotic mixing [19], revealing that the defects can be treated as virtual stirring rods, which braid around each other, driving advective flows that stretch and fold the material.", "One of the most striking features of 2D microtubule-based active nematics is the visually distinctive patterns of folded dark and light bands of varying intensity (Fig.", "REF a).", "These folded patterns appear fractal, with finer structure apparent at smaller length scales.", "These fractal patterns are the subject of the current paper.", "The fractal analysis of images has a rich history across numerous disciplines, with many quantitative measures of fractal behavior introduced and utilized [20], [21], [22], [23], [24].", "Fractal dimension is a well known tool and various definitions are commonly used.", "Here we choose to investigate the closely related decay of the power spectrum at large wavenumbers, i.e.", "small length scales.", "We analyze experimental images of active nematics, using data sets first reported in Ref. Tan19.", "We find that power, i.e.", "the norm-squared of the Fourier coefficients, scales as a power-law $k^{-\\beta }$ in the wavenumber $k$ .", "This power-law behavior begins at the active length scale of the system, roughly the spacing between topological defects, and continues down to smaller scales, eventually being dominated by pixel noise.", "In a log-log plot, power versus $k$ displays a strikingly linear behavior over a wide range of $k$ values, from which we extract $\\beta $ .", "The value of $\\beta $ is roughly constant in time.", "Fractal patterns are well known to occur within the passive advection of material in a traditional 2D fluid [25], [26], [27], [28], [29], [30].", "This is evident on large planetary scales, e.g.", "plankton blooms in the ocean [25], [31], or on small laboratory scales, e.g.", "microfluidic devices [32], [33].", "However, in a confined, incompressible flow, such fractal behavior is transient; eventually the system becomes well mixed and the fractal pattern washes out [29].", "Persistent fractal behavior can arise in an open incompressible flow, when a constant stream of impurity enters and then exits a chaotic mixing region.", "However, the fractal structure witnessed in the active microtubule-based nematic is both persistent and confined.", "Furthermore, the microtubule system is also incompressible, or area-preserving, at least when averaged over a large enough domain.", "On smaller length scales, laboratory videos do show local compression of microtubule bundles and the creation and expansion of small voids.", "(See the online supplemental video.)", "Despite this, most continuum-based simulations assume local incompressibility from the start [12], [15], [17].", "Figure: a) A fluorescence microscopy image of the microtubulesystem.", "b) The intensity across the top row of panel a.We present a simple model for the creation of fractal structure that does not assume incompressibility on small length scales.", "Area is, however, preserved at the active length scale.", "Similarly, density is not constant; the patterns seen in the active nematic are, after all, due to local variations in density.", "Our model contains three distinct steps: 1) extension of the material in the direction of the director field, i.e.", "along the microtubule bundles, and compression in the perpendicular direction; 2) folding of the material back over itself in a horseshoe pattern; and 3) creation of new large-scale density fluctuations at the active length scale.", "A critical aspect of this model is that the compressibility in Step 1 varies with density.", "We find that this simple model generates $k^{-\\beta }$ scaling in the power spectrum with a well defined $\\beta $ value.", "To better understand the model's consequences, we conduct an analysis of the fractal formation to first order in the density fluctuations.", "Within this linear analysis, we analytically compute $\\beta $ and find that it depends solely on a dimensionless parameter $r$ , that characterizes the compressibility.", "The fractal behavior is lost if the system is incompressible.", "This model is not intended to be a quantitatively faithful simulation of all the physics of a microtubule-based active nematic.", "Rather it is a reduced model designed to highlight the key physical processes necessary to create fractal structure.", "The lessons learned here could be incorporated into more quantitatively accurate numerical simulations in the future.", "Note that our work should be distinguished from prior studies of scale-free behavior in active nematics [12], [34].", "These studies assume uniform density, so are not sensitive to the density fluctuations considered here.", "Instead, they consider the decay in the energy and enstrophy spectra, which have a universal power-law behavior at small scales.", "This paper is organized as follows.", "Section  presents our results on the spectral decay of experimental images of the active nematic material.", "Section  explains our three-step model of fractal generation.", "Section  linearizes the model in the density fluctuations.", "Section  discusses the special case in which Step 1 of our model is incompressible but Step 3 is not.", "Conclusions are in Sect.", "." ], [ "Experimentally measured power-spectrum decay of microtubule-based\nactive nematics", "The analysis here is based on the experimental data presented in Ref.", "Tan19, where a complete description of the experimental technique is given.", "In brief, a quasi-2D layer of microtubules is suspended at an oil-water boundary.", "The microtubules are at high density and form bundles due to the use of a depletion agent (polyethylene glycol).", "Kinesin molecular motors, joined together in clusters, serve to cross-link the microtubules.", "ATP is added to the solution to power the molecular motors, which walk along the microtubules from their negative end to their positive end.", "Neighboring microtubules of opposite polarity are pushed in opposing directions by the motor clusters to produce the active stress in the material.", "This local stress produces large-scale dynamics in the 2D microtubule layer.", "The system is imaged using fluorescence microscopy.", "See Fig.", "REF a and the online supplemental video.", "Experiments were performed with six different ATP concentrations.", "The higher the ATP concentration, the more activity is induced in the system and the faster the system evolves, assuming all other factors are held constant.", "Figure: a) Power spectrum (black) of Fig.", "b.The blue curve is the power spectrum averaged in logk\\log k spaceusing a Gaussian of width σ=0.4\\sigma = 0.4.", "(See Appendix.)", "The redline is a linear fit to this average.", "The vertical dashed linedenotes the kk value corresponding to the velocity autocorrelationlength.", "Note that the scaling is chosen so that k=L 0 /λk = L_0/\\lambda for wavelength λ\\lambda , where L 0 L_0 is the width of the image; b)The 1D power spectrum averaged over all rows of the image inFig.", "a; c) The angle-averaged 2D power spectrum of theimage in Fig.", "a; d) The 1D (black) and 2D (green) β\\beta values as a function of time.", "The red curve is the green curveshifted down by 1.", "All data in these figures were taken at an ATPconcentration of 50μ50 \\mu M.We analyse the power spectrum of the resulting experimental images $f(x,y)$ in two ways.", "In the first technique, we compute the 1D Fourier transform $\\tilde{f}(k_x;y)$ of each row; here $y$ labels the row and $k_x$ is the $x$ wave vector.", "The power spectrum of row $y$ is then the norm-squared of the Fourier coefficients versus $k = |k_x|$ , i.e.", "$P(k;y) = |\\tilde{f}(k;y)|^2$ .", "For example, Fig.", "REF b shows the intensity of the top row in Fig.", "REF a; Fig.", "REF a shows a log-log plot of the resulting power spectrum (black).", "To better see the spectral decay, we smooth the power spectrum in $\\log k$ as discussed in the Appendix.", "This average is shown as the smooth blue curve.", "Finally, the blue curve is fit to a linear decay, shown in red, with slope $-1.33$ .", "The vertical dashed line marks the value $k = L_0/\\ell $ , where $L_0$ is the width of the image and $\\ell $ is the active length scale, defined here as the velocity autocorrelation length [18], [35] of the active nematic, computed in Ref. Tan19.", "Note that the linear fall off begins at $k$ values larger than approximately $L_0/\\ell $ .", "The initial (black) power spectrum in Fig.", "REF a has large fluctuations, which we can reduce by averaging $P(k;y)$ over all the rows $y$ of the image, producing the averaged 1D power spectrum $P_1(k)$ .", "This is shown as the black power spectrum in Fig.", "REF b, where the fluctuations have diminished dramatically.", "The red fit line is obtained as in Fig.", "REF a, i.e.", "the power spectrum is first smoothed in $\\log k$ (not shown in Fig.", "REF b), and then the red line is fit to this average.", "The resulting slope is $-1.37$ , comparable to that of Fig.", "REF a.", "The second technique for computing the spectral decay is to first take the 2D Fourier transform $\\tilde{f}(k_x,k_y)$ of the entire image (in practice, a square subset of the image).", "Then $P(k_x,k_y) = |\\tilde{f}(k_x,k_y)|^2$ is averaged over angle to produce the 2D power spectrum $P_2(k) = \\frac{1}{2 \\pi } \\int _0^{2 \\pi } P(k \\cos (\\phi ), k \\sin (\\phi ))\\, d\\phi ,$ which is shown in Fig.", "REF c. Its fluctuations are roughly comparable to that of averaging over the rows (Fig.", "REF b).", "The data is again smoothed in $\\log k$ and subsequently fit to the red line, with slope $-2.37$ .", "Note that these power spectra show that there is a single active length scale, which can be defined as the velocity autocorrelation length $\\ell $ , as done here.", "This is consistent with what is known from prior publications, e.g.", "Ref. Lemma19.", "On scales smaller than the active length scale, the density distribution appears scale free.", "Figure: The β\\beta value averaged over all 500 framesof the active nematic video for experiments performed at sixdifferent ATP concentrations: 50, 75, 100, 250, 500, 1000 μ\\mu M.The black data is computed from P 1 P_1 and the red data from P 2 P_2,shifted down by 1.", "Error bars are the standard deviations ofβ\\beta over all frames.We now investigate the time-dependence of the $\\beta $ 's.", "Figure REF d shows $\\beta $ computed from $P_1$ (black) and $P_2$ (green) for each of 500 experimental frames.", "There is modest variation in each graph.", "Note that the two curves differ by about 1.", "This is made clearer by the red curve in Fig.", "REF d, which is the green curve shifted down by 1.", "The reason for this correspondence can be understood as follows.", "Suppose the original image $f(x,y)$ were independent of $y$ , consisting only of vertical stripes.", "Then $\\tilde{f}(k_x, k_y)$ could be written as $\\tilde{f}(k_x; y) \\delta _{k_y 0}$ , i.e.", "the 2D Fourier transform depends on $k_x$ just as the 1D Fourier transform of any (equivalent) row of the image and is nonzero only when $k_y = 0$ , since the image is constant in $y$ .", "If $P(k_x;y)$ falls off like $k^{-\\beta }$ , then the angular average of $P(k_x, k_y)$ is one factor of $k$ smaller due to dividing by the circumference $2 \\pi k$ .", "The close agreement between the red and black curves confirms the validity of our analysis.", "Separate experiments were run for six different ATP concentrations.", "Figure REF shows the $\\beta $ value averaged over all 500 frames for each ATP concentration.", "The black data is computed from $P_1$ and the red data from $P_2$ , shifted down by one.", "Error bars are the standard deviation of $\\beta $ over the 500 frames.", "Note that the $P_1$ and $P_2$ data are consistent across all six experimental runs.", "However, there are significant variations in $\\beta $ from one experiment to the next, indicating that the nature of the scale-free behavior is not universal." ], [ "Step 1: One-dimensional model of compression dynamics", "We consider a model of the material in which the microtubule bundles, and hence the directors, point along the $y$ direction everywhere, and in which the density $\\rho $ depends solely on $x$ .", "See Fig.", "REF a.", "We assume that this material is under a constant stress perpendicular to the directors and that the response of the material depends solely on the density $\\rho (x)$ .", "Specifically, the relative change in length $d L/L$ of any interval $[x_0, x_1]$ , where $L = x_1-x_0$ , over time $dt$ depends solely on the average density $\\bar{\\rho }$ of the interval, i.e.", "$\\frac{d L}{dt} & = g(\\bar{\\rho }) L, \\\\\\bar{\\rho } &= \\frac{1}{L} \\int _{x_0}^{x_1} \\rho (x) dx,$ where $g$ is the compression rate for a given average density.", "The assumption that Eq.", "(REF ) applies to all intervals is equivalent to $g$ being an affine function, which we write as $g(\\bar{\\rho }) = -\\alpha (1 - \\bar{\\rho }/\\rho _m),$ where $\\alpha $ and $\\rho _m$ are constants.", "We take $\\rho _m > 0$ and $\\alpha > 0$ , so that the material becomes less compressible at higher density, becoming incompressible for the given stress when the average density is $\\rho _m$ .", "Using $v(x,t)$ for the velocity of the material at position $x$ and time $t$ , Eqs.", "(REF )–(REF ) imply $\\frac{1}{L} \\int _{x_0}^{x_1} \\frac{\\partial v}{\\partial x} dx& = \\frac{1}{L}[v(x_1) - v(x_0)] = \\frac{1}{L} \\frac{d L}{dt} =g(\\bar{\\rho }) \\nonumber \\\\& = -\\alpha + \\frac{\\alpha }{\\rho _m}\\frac{1}{L} \\int _{x_0}^{x_1} \\rho (x) dx \\nonumber \\\\& = \\frac{1}{L} \\int _{x_0}^{x_1} -\\alpha \\left(1 - \\frac{\\rho (x)}{\\rho _m}\\right) dx,$ from which we find $\\frac{\\partial v}{\\partial x} = -\\alpha \\left( 1 - \\frac{\\rho (x)}{\\rho _m} \\right).$ Applying the continuity equation $\\frac{\\partial }{\\partial t} \\rho +\\frac{\\partial }{\\partial x} (v \\rho ) = 0,$ we find $\\frac{D}{Dt} \\rho (x,t) = \\frac{\\partial }{\\partial t} \\rho + v\\frac{\\partial }{\\partial x} \\rho = -\\rho \\frac{\\partial }{\\partial x} v= \\alpha \\rho \\left( 1 - \\frac{\\rho }{\\rho _m}\\right).$ Here $D/Dt$ is the advective derivative.", "Note that the quadratic nature of Eq.", "(REF ) produces two fixed points for $\\rho $ in the Lagrangian frame, one at $\\rho = 0$ and one at $\\rho = \\rho _m$ ; these are linearly unstable and stable, respectively.", "See Fig.", "REF b.", "Thus, a typical parcel of material will be driven toward the maximum density $\\rho _m$ as it is compressed; a block of material already at density $\\rho _m$ is incompressible.", "Note that the combination of Eqs.", "(REF ) and (REF ) yields an integro-differential equation for the density $\\rho (x,t)$ , which can be solved numerically.", "Figure: a) The microtubules (green) and directors(black double arrow) are aligned in the yy direction.", "Themicrotubule density ρ\\rho depends only on xx.", "The microtubulesare compressed in the xx direction.", "b) The time-rate-of-change ofthe density in the Lagrangian frame.", "The density is driven towardthe stable fixed point ρ m \\rho _m.", "c) Stretching in the yy directionis added to the model.", "d) The density is driven toward the newstable fixed point ρ a <ρ m \\rho _a < \\rho _m.We next incorporate into the dynamics stretching in the $y$ -direction, given by the rate $\\gamma >0$ , which is taken to be constant in position and time and independent of density.", "See Fig.", "REF c. This does not change Eq.", "(REF ), but modifies Eq.", "(REF ) to be $\\frac{D}{Dt} \\rho (x,t) &= \\frac{\\partial }{\\partial t} \\rho + v\\frac{\\partial }{\\partial x} \\rho = \\rho ( \\alpha - \\gamma - \\alpha \\frac{\\rho }{\\rho _m}) \\nonumber \\\\& = \\alpha \\rho \\left( \\frac{ \\rho _a - \\rho }{\\rho _m} \\right),$ where $\\rho _a = \\rho _m (\\alpha - \\gamma )/\\alpha < \\rho _m.$ See Fig.", "REF d. With this modification, a parcel of material is driven toward the steady state density $\\rho _a$ instead of $\\rho _m$ .", "Note that the limit $\\rho _m = \\infty $ , with $\\alpha $ and $\\gamma $ constant, removes the density dependence from Eqs.", "(REF ) and (REF ).", "Density then increases at the uniform rate of $\\alpha - \\gamma $ everywhere.", "This behavior is identical to the compression-stretching step for both the baker and Lozi maps, and it is well known that both of these maps have a fractal attractor, with a singular density distribution, but only when the system is contracting, i.e.", "$\\alpha - \\gamma > 0$ .", "This is not an accurate model for the microtubule-based system, which must be area-preserving on average.", "One could instead consider the incompressible limit of Eq.", "(REF ) by setting $\\alpha = \\gamma $ .", "Though completely area-preserving dynamics will not produce fractal structure, it is possible to produce fractal structure if Step 1 is area preserving and it is combined with density fluctuations introduced in Step 3.", "(See Sect.", "REF below.)", "This special case is considered in Sect. .", "Until then, we take $\\rho _m$ to be finite.", "Though area is not conserved locally in the microtubule system, we do require that area be conserved across the entirety of the material, as noted above.", "This means that the stretching rate must be balanced by the compression rate, i.e.", "$\\gamma = -g(\\bar{\\rho }_T)$ , where $\\bar{\\rho }_T$ is the average density of the entire block of material.", "Since $\\gamma $ is assumed constant in time, $\\bar{\\rho }_T$ must also be constant in time.", "Equation (REF ) then implies that $\\bar{\\rho }_T = \\rho _a.$ The “a” subscript stands for “average”.", "By an appropriate choice of length and time scales, we can set $\\alpha = \\rho _m = 1$ so that Eqs.", "(REF ) and (REF ) become $\\frac{D}{Dt} \\rho (x,t) & = \\frac{\\partial }{\\partial t} \\rho + v\\frac{\\partial }{\\partial x} \\rho = \\rho ( r - \\rho ), \\\\\\frac{\\partial v}{\\partial x} & = \\rho - 1, $ where $r = \\rho _a/\\rho _m \\le 1$ is dimensionless.", "Equation (REF ) for the density averaged over the entire material becomes $\\bar{\\rho }_T = r.$ Note that there is no incompressible limit of Eqs.", "(REF ) and (), since the rescaling assumes $\\rho _m$ is finite." ], [ "Step 2: Folding dynamics", "After the system has been compressed to one-half its original width, we introduce a fold.", "In the one-dimensional model, this means that the system returns to its original width $L_0$ , with density $\\rho (x,t) =\\left\\lbrace \\begin{array}{ll}\\rho (x,t) & 0 < x < L_0/2, \\\\\\rho (L_0/2-x,t) & L_0/2 < x < L_0.\\end{array}\\right.$ An alternative perspective here is to extend $\\rho (x,t)$ to an $x$ -periodic function with wavelength $L_0$ that is also an even function in $x$ .", "In this case, the function $\\rho (x,t)$ will naturally evolve into the form of Eq.", "(REF ) once the original interval is compressed to half its length.", "Thus the fold step is naturally incorporated into the compression step.", "To implement this step, we compute the time $T$ to compress the entire block of material from length $L_0$ to an arbitrary length $L$ .", "First, assuming without loss of generality that the position of the left edge of the block is fixed at $x=0$ , the velocity at the right edge $x = L$ is computed as $\\frac{dL}{dt} & = \\int _0^L \\frac{\\partial v}{\\partial x} dx= - \\int _0^L ( 1 - \\rho ) dx = -( 1 - r) L,$ using Eqs.", "() and (REF ).", "Hence, the time to compress the block from $L_0$ to $L$ is $T = \\frac{ \\log (L_0/L)}{1 - r} = \\frac{ \\log (2)}{ 1 - r },$ where in the final step we have set $L = L_0/2$ ." ], [ "Step 3: Imposing large-scale density variations", "Immediately after the folding step we impose a multiplicate density variation $R(x)$ and normalize the density so that the average density remains $r$ .", "Explicitly, $\\rho (x) \\mapsto \\frac{R(x) \\rho (x)}{\\langle R(x) \\rho (x) \\rangle } r,$ where the angled brackets denote the average over $x$ .", "Consistent with the viewpoint that $\\rho (x)$ is an even, $L_0$ -periodic function, we similarly extend $R(x)$ to an even, $L_0$ -periodic function." ], [ "Fractal generation", "The sequence of compression by a factor of two, folding, and large-scale density variations generates a mapping $M$ of an original density $\\rho _0(x)$ to a density $\\rho (x)$ ; though the folding step can be ignored if $\\rho (x)$ is even and $L_0$ -periodic as discussed above.", "This mapping depends on the form $R(x)$ of the density variations imposed and the value of $r$ in the compression step.", "The mapping $M$ can then be repeatedly applied to the density.", "We numerically observe that it converges to a stationary fractal function, as seen in Fig.", "REF for the density variation $R(x) = 1 + B \\cos (x),$ with $B = 0.2$ .", "Figure REF shows the power spectrum (black) of the fractal image.", "The blue line is the power spectrum averaged over log(k), and the red line is a linear fit to this average.", "For comparison, Fig.", "REF shows the stationary density distribution and power spectrum for larger amplitude density variations with $B = 0.7$ .", "Figure: The density ρ\\rho after each iterate nn of themap MM with r=0.25r= 0.25, L 0 =2πL_0 = 2 \\pi , and density variation givenby Eq.", "() with B=0.2B = 0.2.", "More and more fine-scalestructure develops at each iterate.", "By n=15n = 15 the density hasvisually converged to an invariant state.Figure: The power spectrum (black) of the final graph inFig.", "(n=15n = 15).", "The blue curve is the power spectrumsmoothed according to Eq. ().", "The smoothed spectrum is fitto a line (red) with slope -1.71." ], [ "Linearization of compression dynamics", "The constant density $\\rho _0(x,t) = r$ and time-invariant velocity $v_0(x,t) = (r-1)x$ are steady state solutions of Eqs.", "(REF ) and ().", "Note that we have chosen a frame in which $v = 0$ at the origin $x=0$ .", "Expanding about this solution using $\\rho & = \\rho _0 + \\epsilon g, \\\\v & = v_0 + \\epsilon f,$ where $\\langle g \\rangle = 0$ , we find to first order in $\\epsilon $ $\\frac{\\partial }{\\partial t} g &= (1-r)x\\frac{\\partial }{\\partial x} g -r g, \\\\\\frac{\\partial f}{\\partial x} & = g.$ Figure: a) The stationary density for the sameparameters as Fig.", ", except B=0.7B = 0.7. b) Thecorresponding power spectrum.", "The fit slope -1.60-1.60 differsslightly from Fig.", ".Note that the density perturbation $g$ decouples from $f$ .", "Direct substitution shows that the solution to Eq.", "(REF ) for a given initial condition $g(x,0)$ has the form $g(x,t) = h(t)g(x/\\ell (t), 0),$ where $\\ell (t) &= \\exp ( (r-1) t), \\\\h(t) &= \\exp (-rt).$ This implies that the Fourier transform $\\tilde{g}(k,t)$ evolves in time as $\\tilde{g}(k,t) = h(t) \\tilde{g}(k \\ell (t), 0 ).$ Thus as time evolves, the graph of the power spectrum in log-log space shifts to larger wavenumbers and smaller amplitudes by translating along a line of slope $-2r/(r-1)$ .", "After time $T$ given by Eq.", "(REF ), we find $\\tilde{g}(k,T) = 2^{- r/(1-r) } \\tilde{g}(k/2, 0).$ This equation is valid when the Fourier transform is smooth, but must be treated carefully when the Fourier transform is singular, as when $\\rho $ is $L_0$ -periodic at $t = 0$ and $t = T$ .", "In this case $\\tilde{g}(k,t)$ is zero except when $k = 2\\pi n/L_0$ for $n = 0, 1, 2, 3, ...$ , at which value it is a delta function whose amplitude we denote $\\tilde{g}_n(t)$ .", "These coefficients satisfy $\\tilde{g}_n(T) =\\left\\lbrace \\begin{array}{ll}2^{- r / (1-r) } \\tilde{g}_{ n/2 }(0), & n \\text{even} \\\\0, & n \\text{ odd.", "}\\end{array}\\right.$ Thus, the spacing between nonzero values of $\\tilde{g}_n$ has increased by a factor of 2 over time $T$ .", "Thus, when smoothed over $n$ , as discussed in the Appendix, $|\\tilde{g}_n|^2$ scales as $\\langle |\\tilde{g}_n(T)|^2 \\rangle & \\approx 0.5 \\times 2^{- 2r/(1-r) }\\langle |\\tilde{g}_{\\lfloor n/2 \\rfloor } (0)|^2 \\rangle \\nonumber \\\\& \\approx 2^{- (1 + r)/(1-r) }\\langle |\\tilde{g}_{\\lfloor n/2 \\rfloor } (0)|^2 \\rangle ,$ where the angled brackets denote the average over $n$ and $\\lfloor \\quad \\rfloor $ denotes the floor function.", "From this we see that the smoothed power spectrum $\\langle |\\tilde{g}_n(t)|^2 \\rangle $ slides down a line of slope $(1+r)/(1-r)$ in log-log space." ], [ "Linearization of new density variations", "We assume the density variations introduced by Eq.", "(REF ) scale as $\\epsilon $ , i.e.", "$R(x) = 1 + \\epsilon p(x),$ with $\\langle p(x) \\rangle = 0$ .", "Then, to first order in $\\epsilon $ and using the expansion Eq.", "(REF ), Eq.", "(REF ) becomes $g(x) \\mapsto g(x) + p(x),$ which in terms of the Fourier coefficients, we write as $\\tilde{g}_n \\mapsto \\tilde{g}_n + \\tilde{p}_n.$" ], [ "Linearized fractal generation", "We denote by $A$ the linear operator that evolves $g(x,0)$ forward for time $T$ according to Eq.", "(REF ).", "Then the total linearized dynamics is $g \\mapsto A g+ p.$ Repeated application of this map, beginning with $g = 0$ and using $p(x) = 0.2 \\cos (x),$ converges to the stationary function (blue) in Fig.", "REF , which is consistent with the nonlinear stationary density from Fig.", "REF and reproduced in Fig.", "REF (black).", "It is straightforward to see that $P = \\sum _{m = 0}^\\infty A^m p,$ is the unique invariant function of Eq.", "(REF ) and that any initial function will converge to it.", "Figure: The invariant function (blue) obtained fromrepeated application of the linearized dynamics Eq.", "(),with r=0.25r = 0.25 and L 0 =2πL_0 = 2\\pi .", "For comparison is the invariantfunction (black) obtained from the nonlinear dynamics and shownpreviously in Fig.", ".It was noted to us by S. Berman that when $p(x)$ is a cosine, as in Eq.", "(REF ), the invariant function $P(x)$ is the Weierstrass function, which was the first published example of a continuous function that is nowhere differentiable.", "Furthermore, fixed-point equations with the form of Eq.", "(REF ) were introduced by de Rham to characterize such singular functions.", "Such techniques have a history of applications to dynamical systems, e.g.", "Ref. Tasaki98.", "In Fourier space, the linear operator $\\tilde{A}$ that maps coefficients $\\tilde{g}_n(0)$ forward to $\\tilde{g}_n(T)$ is given by Eq.", "(REF ), and the invariant function is $\\tilde{P} = \\sum _{m = 0}^\\infty \\tilde{A}^m \\tilde{p},$ with coefficients $\\tilde{P}_n = \\sum _{m = 0}^\\infty 2^{- m r/(1-r)}\\tilde{p}_{n/2^m},$ where it is understood that $\\tilde{p}_{n/2^m} = 0$ if $n/2^m$ is not an integer.", "Figure: The power spectrum (black) of the blue curve inFig. .", "The blue curve above is the power spectrumsmoothed according to Eq. ().", "The smoothed spectrum is fitto a line (red) with slope -1.67.Assuming that the density variation function $p(x)$ is analytic, its Fourier coefficients $\\tilde{p}_n$ decay exponentially in $n$ , meaning that the sum in Eq.", "(REF ) is dominated by the first nonzero term.", "This term will have an $m$ value satisfying $m \\approx \\log (n)/\\log (2)$ , and hence $\\tilde{P}_n \\propto n^{-r/(1-r)},$ from which follows $|\\tilde{P}_n|^2 \\propto n^{-2r/(1-r)},$ and finally $\\langle |\\tilde{P}_n|^2 \\rangle \\propto \\frac{1}{n} n^{-2r/(1-r)} = n^{-(1+r)/(1-r)},$ where the factor of $1/n$ that appears upon averaging is due to the diminishing density of nonzero coefficients, as discussed prior to Eq.", "(REF ).", "Thus in the linearized model, we find $\\beta = \\frac{1+r}{1-r}.$ This is the same scaling that we previously identified for the temporal behavior in Eq.", "(REF ), highlighting the origin of this scaling in the compression dynamics.", "To check this formula numerically, we plot in Fig.", "REF the power spectrum of the blue curve in Fig.", "REF , which has $r = 0.25$ .", "The linear fit yields a slope of $-1.67$ , which agrees with the value $\\beta = 1.67$ obtained from Eq.", "(REF ).", "Figure REF provides a more comprehensive comparison of the nonlinear analysis, the linear analysis, and Eq.", "(REF ).", "The black and red dots show $\\beta $ versus $r$ for the nonlinear and linearized systems, respectively, using $R$ and $p$ given by Eqs.", "(REF ) (with $B = 0.2$ ) and (REF ).", "The data are quite consistent with each other.", "The blue curve is the graph of Eq.", "(REF ), and it nicely tracks the red data points.", "Figure: The variation of β\\beta with rr.", "The blackdots are computed numerically from the nonlinear analysis withB=0.2B = 0.2.", "The red dots are computed numerically from the linearanalysis.", "The blue curve is the graph of Eq.", "()." ], [ "The fluctuation injection scale", "The value of $\\beta $ does not depend on the details of the density fluctuations introduced by $p(x)$ , except that its Fourier coefficients should fall off sufficiently rapidly.", "However, the $k^{-\\beta }$ scaling only begins at length scales below the smallest length scale of $p(x)$ .", "Figure REF illustrates this by plotting the smoothed power spectrum for $p(x)$ with wavelengths equal to the width of the block of material divided by 1 [$p(x) = \\cos (x)$ , black], 10 [$p(x) = \\cos (10 x)$ , blue], and 100 [$p(x) = \\cos (100 x)$ , red].", "These wavelengths are the scales at which density fluctuations are injected into the system.", "The $k$ value for each wavelength is denoted by the dashed vertical line of the corresponding color.", "The linear behavior only begins to the right of each line, i.e.", "at length scales smaller than the injection scale." ], [ "The special case of incompressibility in Step 1", "Returning to Eq.", "(REF ) and taking the limit $\\rho _m = \\infty $ and setting $\\alpha = \\gamma $ , density is conserved in the Lagrangian frame, $\\frac{D}{Dt} \\rho (x,t) = 0,$ so that the material is incompressible, regardless of its density.", "Furthermore, Eq.", "(REF ) reduces to $\\frac{\\partial v}{\\partial x} = -\\alpha ,$ so that the system is uniformly compressed in the $x$ direction.", "Combining this with the density variations in Step 3, Eq.", "(REF ), we obtain a steady state fractal density.", "(We ignore subtle points about the convergence of this function, which is technically a distribution.)", "This dynamics is admittedly physically inconsistent, as we cannot imagine a situation in which density fluctuations could arrise via folding on the active scale (Step 3) without the material also being subject to compression or expansion in Step 1.", "Nevertheless, it is instructive to compute the resulting value of $\\beta $ , which is straightforward when linearizing the density fluctuations, i.e.", "using Eq.", "(REF ).", "Repeating the analysis in Sec.", "REF , one finds $\\tilde{P}_n = \\sum _{m = 0}^\\infty \\tilde{p}_{n/2^m},$ which simply reflects the fact that as a cosine density fluctuation is squeezed in $x$ , its amplitude remains constant.", "This is the same result as Eq.", "(REF ) with $r = 0$ and therefore gives the same power spectrum decay with $\\beta = 1$ .", "Thus, even if one were to assume incompressibility in Step 1, the resulting value of $\\beta $ disagrees with the experimental measurements in Fig.", "REF (at least within the linearized analysis.)" ], [ "Conclusions", "By using a simple model of fractal generation, we have highlighted the necessity both of introducing density fluctuations on large scales and of a variable compressibility in the material.", "At least within the linearized analysis, the details of the density fluctuations are irrelevant to the value of $\\beta $ .", "The density fluctuations can take any functional form, so long as they have a minimum length scale.", "The $k^{-\\beta }$ scaling then manifests at scales below this smallest length scale.", "In the case of the microtubule-based active nematic material, the injection scale of the density fluctuations is the active length scale of the system, i.e.", "the average defect spacing.", "Physically, the injected density variations are most naturally explained by the creation of defect pairs and the associated fracturing.", "The $\\beta $ parameter depends on the compressibility via $r$ , and in the linearized analysis we derived an explicit relationship, Eq.", "(REF ).", "It is tempting to thus translate the experimentally measured values of $\\beta $ (Fig.", "REF ) into a material parameter $r$ via $r = \\frac{\\beta - 1}{\\beta + 1},$ yielding values of $r$ in the range $0.1$ – $0.3$ .", "But what is the physical meaning of $r$ ?", "Within the context of the simple model, we can combine Eqs.", "(REF ) and (REF ), to rewrite $r$ as $r = 1 - \\gamma /\\alpha .$ Recall $\\gamma $ is the (uniform and constant) extension rate and $\\alpha $ can be interpreted as the transverse compression rate in the low-density limit [Eq.", "(REF )].", "Whether this interpretation from the simple fractal model will hold for the actual laboratory system is not clear.", "This issue could be addressed via a realistic 2D hydrodynamic model with a density-dependent compressibility.", "Specifically, the dependence of $\\beta $ on the material properties and system parameters would make an interesting and informative study.", "Finally, it would be interesting to identify whether other physical systems exhibit a similar mechanism for fractal formation.", "Such systems would exhibit fractal structure not in the patterning of impurities mixed into the system, but in morphological properties such as density or surface texture.", "The classic example of chaos in a taffy puller comes to mind [37], in which large-scale surface irregularities are introduced periodically by the folding of the taffy and then pushed down to smaller length scales by the stretching and compression dynamics.", "The authors acknowledge generous funding from the National Science Foundation, through several awards: DMR-1808926, NSF-CREST: Center for Cellular and Biomolecular Machines at UC Merced (HRD-1547848), and from the Brandeis Biomaterials facility MRSEC-1420382, which provided materials.", "We would also like to thank Simon Berman for critiquing the manuscript, and in particular for pointing out the work in Ref.", "Tasaki98." ], [ "DATA AVAILABILITY", "The data that support the findings of this study are available from the corresponding author upon reasonable request." ], [ "Spectral averaging", "We smooth out local variations in a function $f(k)$ using a Gaussian function of width $\\sigma $ in log space $\\langle f \\rangle (k) = \\int _0^\\infty f(k^{\\prime }) \\frac{1}{N(k)}\\exp \\left[ - \\frac{1}{2 \\sigma ^2}(\\log k^{\\prime } - \\log k)^2 \\right] dk^{\\prime },$ with normalization $N(k) & = \\int _0^\\infty \\exp \\left[ - \\frac{1}{2 \\sigma ^2}(\\log k^{\\prime } - \\log k)^2 \\right] dk^{\\prime } \\nonumber \\\\& = \\sqrt{2 \\pi } \\sigma e^{\\sigma ^2/2} k.$ Note that this smoothing preserves the $L^1$ -norm, i.e.", "$\\int _0^\\infty |\\langle f \\rangle (k)| dk = \\int _0^\\infty | f(k) |dk$ .", "Thus, spectral averaging of the power function preserves the total power.", "For all computations in this paper, we use $\\sigma = 0.4$ ." ] ]
2105.11673
[ [ "Efficiently Explaining CSPs with Unsatisfiable Subset Optimization" ], [ "Abstract We build on a recently proposed method for explaining solutions of constraint satisfaction problems.", "An explanation here is a sequence of simple inference steps, where the simplicity of an inference step is measured by the number and types of constraints and facts used, and where the sequence explains all logical consequences of the problem.", "We build on these formal foundations and tackle two emerging questions, namely how to generate explanations that are provably optimal (with respect to the given cost metric) and how to generate them efficiently.", "To answer these questions, we develop 1) an implicit hitting set algorithm for finding optimal unsatisfiable subsets; 2) a method to reduce multiple calls for (optimal) unsatisfiable subsets to a single call that takes constraints on the subset into account, and 3) a method for re-using relevant information over multiple calls to these algorithms.", "The method is also applicable to other problems that require finding cost-optimal unsatiable subsets.", "We specifically show that this approach can be used to effectively find sequences of optimal explanation steps for constraint satisfaction problems like logic grid puzzles." ], [ "Introduction", "Building on old ideas to explain domain-specific propagations performed by constraint solvers [36], [10], we recently introduced a method that takes as input a satisfiable constraint program and explains the solution-finding process in a human-understandable way [2].", "Explanations in that work are sequences of simple inference steps, involving as few constraints and facts as possible.", "The explanation-generation algorithms presented in that work rely heavily on calls for Minimal Unsatisfiable Subsets (MUS) [29] of a derived program, exploiting a one-to-one correspondence between so-called non-redundant explanations and MUSs.", "The explanation steps in the seminal work are heuristically optimized with respect to a given cost function that should approximate human-understandability, e.g., taking the number of constraints and facts into account, as well as a valuation of their complexity (or priority).", "The algorithm developed in that work has two main weaknesses: first, it provides no guarantees on the quality of the produced explanations due to internally relying on the computation of $\\subseteq $ -minimal unsatisfiable subsets, which are often suboptimal with respect to the given cost function.", "Secondly, it suffers from performance problems: the lack of optimality is partly overcome by calling a MUS algorithm on increasingly larger subsets of constraints for each candidate implied fact.", "However, using multiple MUS calls per literal in each iterations quickly causes efficiency problems, causing the explanation generation process to take several hours.", "Motivated by these observations, we develop algorithms that aid explaining CSPs and improve the state-of-the-art in the following ways: We develop algorithms that compute (cost-)Optimal Unsatisfiable Subsets (from now on called OUSs) based on the well-known hitting-set duality that is also used for computing cardinality-minimal MUSs [18], [35].", "We observe that many of the individual calls for MUSs (or OUSs) can actually be replaced by a single call that searches for an optimal unsatisfiable subset among subsets satisfying certain structural constraints.", "In other words, we introduce the Optimal Constrained Unsatisfiable Subsets (OCUS) problem and we show how $O(n^2)$ calls to MUS/OUS can be replaced by $O(n)$ calls to an OCUS oracle, where $n$ denotes the number of facts to explain.", "Finally, we develop techniques for optimizing the O(C)US algorithms further, exploiting domain-specific information coming from the fact that we are in the explanation-generation context.", "One such optimization is the development of methods for information re-use between consecutive OCUS calls.", "In this paper, we apply our OCUS algorithms to generate step-wise explanations of satisfaction problems.", "However, MUSs have been used in a variety of contexts, and in particular lie at the foundations of several explanation techniques [21], [20], [7].", "We conjecture that OCUS can also prove useful in those settings, to take more fine-grained control over which MUSs, and eventually, which explanations are produced.", "The rest of this paper is structured as follows.", "We discuss background on the hitting-set duality in sec:background.", "sec:motviation motivates our work, while sec:ocus introduces the OCUS problem and a generic hitting set–based algorithm for computing OCUSs.", "In sec:ocusEx we show how to optimize this computation in the context of explanations and in sec:experiments we experimentally validate the approach.", "We discuss related work in sec:related and conclude in sec:conclusion." ], [ "Background", "We present all methods using propositional logic but our results easily generalize to richer languages, such as constraint languages, as long as the semantics is given in terms of a satisfaction relation between expressions in the language and possible states of affairs (assignments of values to variables).", "Let $\\Sigma $ be a set of propositional symbols, also called atoms; this set is implicit in the rest of the paper.", "A literal is an atom $p$ or its negation $\\lnot p$ .", "A clause is a disjunction of literals.", "A formula $\\mathcal {F} $ is a conjunction of clauses.", "Slightly abusing notation, a clause is also viewed as a set of literals and a formula as a set of clauses.", "We use the term clause and constraint interchangeably.", "A (partial) interpretation is a consistent (not containing both $p$ and $\\lnot p$ ) set of literals.", "Satisfaction of a formula $\\mathcal {F} $ by an interpretation is defined as usual [1].", "A model of $\\mathcal {F} $ is an interpretation that satisfies $\\mathcal {F} $ ; $\\mathcal {F} $ is said to be unsatisfiable if it has no models.", "A literal $l$ is a consequence of a formula $\\mathcal {F} $ if $l$ holds in all $\\mathcal {F} $ 's models.", "If $I$ is a set of literals, we write $\\overline{I}$ for the set of literals $\\lbrace \\lnot l\\mid l\\in I\\rbrace $ .", "Definition 1 A Minimal Unsatisfiable Subset (MUS) of $\\mathcal {F} $ is an unsatisfiable subset $\\mathcal {S}$ of $\\mathcal {F} $ for which every strict subset of $\\mathcal {S} $ is satisfiable.", "$\\mathit {MUSs}(\\mathcal {F} )$ denotes the set of MUSs of $\\mathcal {F} $ .", "Definition 2 A set $\\mathcal {S} \\subseteq \\mathcal {F} $ is a Maximal Satisfiable Subset (MSS) of $ \\mathcal {F} $ if $\\mathcal {S}$ is satisfiable and for all $\\mathcal {S}^{\\prime }$ with $\\mathcal {S} \\subsetneq \\mathcal {S}^{\\prime }\\subseteq \\mathcal {F} $ , $\\mathcal {S}^{\\prime }$ is unsatisfiable.", "Definition 3 A set $\\mathcal {S} \\subseteq \\mathcal {F} $ is a correction subset of $\\mathcal {F} $ if $\\mathcal {F} \\setminus \\mathcal {S}$ is satisfiable.", "Such a $\\mathcal {S}$ is a minimal correction subset (MCS) of $\\mathcal {F} $ if no strict subset of $\\mathcal {S}$ is also a correction subset.", "$\\mathit {MCSs}(\\mathcal {F} )$ denotes the set of MCSs of $\\mathcal {F} $ .", "Each MCS of $\\mathcal {F} $ is the complement of an MSS of $\\mathcal {F} $ and vice versa.", "Definition 4 Given a collection of sets $\\mathcal {H}$ , a hitting set of $\\mathcal {H}$ is a set $h$ such that $h \\cap C \\ne \\emptyset $ for every $C \\in \\mathcal {H}$ .", "A hitting set is minimal if no strict subset of it is also a hitting set.", "The next proposition is the well-known hitting set duality [24], [33] between MCSs and MUSs that forms the basis of our algorithms, as well as algorithms to compute MSSs [5] and cardinality-minimal MUSs [18].", "Proposition 5 A set $\\mathcal {S} \\subseteq \\mathcal {F} $ is an MCS of $ \\mathcal {F} $ iff it is a minimal hitting set of $\\mathit {MUSs}(\\mathcal {F} )$ .", "A set $\\mathcal {S} \\subseteq \\mathcal {F} $ is a MUS of $ \\mathcal {F} $ iff it is a minimal hitting set of $\\mathit {MCSs}(\\mathcal {F} )$ ." ], [ "Motivation", "Our work is motivated by the problem of explaining satisfaction problems through a sequence of simple explanation steps.", "This can be used to teach people problem-solving skills, to compare the difficulty of related satisfaction problems (through the number and complexity of steps needed), and in human-computer solving assistants.", "Our original explanation generation algorithm [2] starts from a formula $\\mathcal {C} $ (in the application coming from a high level CSP), a partial interpretation $I$ (here also viewed as a conjunction of literals) and a cost function $f$ quantifying the difficulty of an explanation step, by means of a weight for every clause and literal in $\\mathcal {F} $ .", "[t] $\\textsc {explain-One-Step} (\\mathcal {C} ,f,I,I_\\mathit {end} )$ $X_{best} \\leftarrow \\mathit {nil}$ $l \\in \\lbrace I_\\mathit {end} \\setminus I\\rbrace $ $X \\leftarrow \\textsc {MUS} {(\\mathcal {C} \\wedge I \\wedge \\lnot l)}$ $f(X)<f(X_{best})$ $X_{best} \\leftarrow X$ $X_{best}$ The goal is to find a sequence of simple explanation steps, where the simplicity of a step is measured by the total cost of the elements used in the explanation.", "An explanation step is an implication $I^{\\prime } \\wedge \\mathcal {C} ^{\\prime } \\Rightarrow N$ where $I^{\\prime }$ is a subset of already derived literals, $\\mathcal {C} ^{\\prime }$ is a subset of constraints of the input formula $\\mathcal {C} $ , and $N$ is a set of literals entailed by $I^{\\prime }$ and $\\mathcal {C} ^{\\prime }$ which are not yet explained.", "The key part of the algorithm is the search for the next best explanation, given an interpretation $I$ derived so far.", "alg:oneStep shows the gist of how this was done.", "It takes as input the formula $\\mathcal {C} $ , a cost function $f$ quantifying the quality of explanations, an interpretation $I$ containing all already derived literals in the sequence so far, and the interpretation-to-explain $I_\\mathit {end} $ .", "To compute an explanation, this procedure iterates over the literals that are still to explain, computes for each of them an associated MUS and subsequently selects the lowest cost one from found MUSs.", "The reason this works is because there is a one-to-one correspondence between MUSs of $\\mathcal {C} \\wedge I \\wedge \\lnot l$ and so-called non-redundant explanation of $l$ in terms of (subsets of) $\\mathcal {C} $ and $I$  [2].", "Experiments have shown that such a MUS-based approach can easily take hours, especially when multiple MUS calls are performed to increase the chance of finding a good MUS, and hence that algorithmic improvements are needed to make it more practical.", "We see three main points of improvement, all of which will be tackled by our generic OCUS algorithm presented in the next section.", "First of all, since the algorithm is based on $\\textsc {MUS}$ calls, there is no guarantee that the explanation found is indeed optimal (with respect to the given cost function).", "Performing multiple MUS calls is only a heuristic that is used to circumvent the restriction that there are no algorithms for cost-based unsatisfiable subset optimization.", "Second, this algorithm uses $\\textsc {MUS}$ calls for every literal to explain separately.", "The goal of all these calls is to find a single unsatisfiable subset of $\\mathcal {C} \\wedge I \\wedge \\overline{(I_\\mathit {end} \\setminus I)}$ that contains exactly one literal from $\\overline{(I_\\mathit {end} \\setminus I)}$ .", "This begs the questions whether it is possible to compute a single (optimal) unsatisfiable subset subject to constraints, where in our case, the constraint is to include exactly one literal from $\\overline{(I_\\mathit {end} \\setminus I)}$ .", "Finally, the algorithm that computes an entire explanation sequence makes use of repeated calls to $\\textsc {explain-One-Step} $ and hence will solve many similar problems.", "This raises the issue of incrementality: can we re-use the computed data structures to achieve speed-ups in later calls?" ], [ "Optimal Constrained Unsatisfiable Subsets", "The first two considerations from the previous section lead to the following definition.", "Definition 6 Let $\\mathcal {F} $ be a formula, $f:2^{\\mathcal {F} } \\rightarrow \\mathbb {N} $ a cost function and $p$ a predicate $p: 2^{\\mathcal {F} }\\rightarrow \\lbrace true,false\\rbrace $ .", "We call $\\mathcal {S} \\subseteq \\mathcal {F} $ an OCUS of $\\mathcal {F} $ (with respect to $f$ and $p$ ) if $\\mathcal {S}$ is unsatisfiable, $p(\\mathcal {S})$ is true all other unsatisfiable $\\mathcal {S}^{\\prime }\\subseteq \\mathcal {F} $ for which $p(\\mathcal {S}^{\\prime })$ is true satisfy $f(\\mathcal {S}^{\\prime })\\ge f(\\mathcal {S})$ .", "If we assume that the predicate $p$ is specified itself as a CNF over (meta)-variables indicating inclusion of clauses of $\\mathcal {F}$ , and $f$ is obtained by assigning a weight to each such meta-variable, then the complexity of the problem of finding an OCUS is the same as that of the SMUS (cardinality-minimal MUS) problem [18]: the associated decision problem is $\\Sigma ^P_2$ -complete.", "Hardness follows from the fact that SMUS is a special case of OCUS, containment follows - intuitively - from the fact that this can be encoded as an $\\exists \\forall $ -QBF using a Boolean circuit encoding of the costs.", "When considering the procedure $\\textsc {explain-One-Step} $ from the perspective of OCUS defined above, the task of the procedure is to compute an OCUS of the formula $\\mathcal {F} := \\mathcal {C} \\wedge I\\wedge \\overline{I_\\mathit {end} \\setminus I}$ with $p$ the predicate that holds for subsets that contain exactly one literal of $\\overline{I_\\mathit {end} \\setminus I}$ , see alg:oneStepOCUS.", "In order to compute an OCUS of a given formula, we propose to build on the hitting set duality of prop:MCS-MUS-hittingset.", "For this, we will assume to have access to a solver $\\textsc {CondOptHittingSet} $ that can compute hitting sets of a given collection of sets that are optimal (w.r.t.", "a given cost function $f$ ) among all hitting sets satisfying a condition $p$.", "The choice of the underlying hitting set solver will thus determine which types of cost functions and constraints are possible.", "In our implementation, we use a cost function $f$ as well as a condition $p$ that can easily be encoded as linear constraints, thus allowing the use of highly optimized mixed integer programming (MIP) solvers.", "The $\\textsc {CondOptHittingSet} $ formulation is as follows: $\\small minimize_S \\quad & f(S) \\\\s.t.", "\\quad & p(S) \\\\& sum(H) \\ge 1, \\quad &&\\forall H \\in \\mathcal {H} \\\\& s \\in \\lbrace 0,1\\rbrace , \\quad &&\\forall s \\in S$ where $S$ is a set of MIP decision variables, one for every clause in $\\mathcal {F} $ .", "In our case, $p$ is expressed as $\\sum _{s \\in \\overline{I_\\mathit {end} \\setminus I}} s = 1$ .", "$f$ is a weighted sum over the variables in $S$ .", "For example, (unit) clauses representing previously derived facts can be given small weights and regular constraints can be given large weights, such that explanations are penalized for including constraints when previously derived facts can be used instead.", "[t] $\\textsc {explain-One-Step-ocus} (\\mathcal {C} ,f,I,I_\\mathit {end} )$ $p \\leftarrow $ exactly one of $\\overline{I_\\mathit {end} \\setminus I}$ $\\textsc {OCUS} (\\mathcal {C} \\wedge I\\wedge \\overline{I_\\mathit {end} \\setminus I}, f, p)$ [t] $\\mathcal {H} \\leftarrow \\emptyset $ true $\\mathcal {S} \\leftarrow \\textsc {CondOptHittingSet} (\\mathcal {H},f,p) $ $\\lnot \\textsc {sat} (\\mathcal {S})$ $\\mathcal {S}$ $\\mathcal {S} \\leftarrow \\textsc {Grow} (\\mathcal {S},\\mathcal {F} ) $ $\\mathcal {H} \\leftarrow \\mathcal {H} \\cup \\lbrace \\mathcal {F} \\setminus \\mathcal {S}\\rbrace $ $\\textsc {OCUS} (\\mathcal {F} ,f,p)$ Our generic algorithm for computing OCUSs is depicted in alg:comus.", "It combines the hitting set-based approach for MUSs of [18] with the use of a MIP solver for (weighted) hitting sets as proposed for maximum satisfiability [5].", "The key novelty is the ability to add structural constraints to the hitting set solver, without impacting the duality principles of prop:MCS-MUS-hittingset, as we will show.", "Ignoring line:grow for a moment, the algorithm alternates calls to a hitting set solver with calls to a $\\textsc {sat} $ oracle on a subset $\\mathcal {S}$ of $\\mathcal {F} $ .", "In case the $\\textsc {sat} $ oracle returns true, i.e., the subset $\\mathcal {S}$ is satisfiable, the complement of $\\mathcal {S}$ is a correction subset of $\\mathcal {F}$ and is added to $\\mathcal {H}$ .", "As in the SMUS algorithm of [18], our algorithm contains an (optional) call to $\\textsc {Grow} $ .", "The purpose of the $\\textsc {Grow} $ is to expand a satisfiable subset of $\\mathcal {F}$ further, to find a smaller correction subset and as such find stronger constraints on the hitting sets.", "In our case, the calls for hitting sets will also take into account the cost ($f$ ), as well as the meta-level constraints ($p$ ); as such, it is not clear a priori which properties a good $\\textsc {Grow} $ function should have here.", "We discuss the different possible implementations of $\\textsc {Grow} $ later and evaluate their performance in sec:experiments.", "For correctness of the algorithm, all we need to know is that it returns a satisfiable subset $\\mathcal {S}^{\\prime }$ of $\\mathcal {F}$ with $\\mathcal {S}\\subseteq \\mathcal {S}^{\\prime }$ .", "Soundness and completeness of the proposal follow from the fact that all sets added to $\\mathcal {H}$ are correction subsets, and thm:soundcomplete, which states that what is returned is indeed a solution and that a solution will be found if it exists.", "Theorem 7 Let $\\mathcal {H}$ be a set of correction subsets of $\\mathcal {F} $ .", "If $\\mathcal {S}$ is a hitting set of $\\mathcal {H}$ that is $f$ -optimal among the hitting sets of $\\mathcal {H}$ satisfying a predicate $p$ , and $\\mathcal {S}$ is unsatisfiable, then $\\mathcal {S}$ is an OCUS of $\\mathcal {F} $ .", "If $\\mathcal {H}$ has no hitting sets satisfying $p$ , then $\\mathcal {F} $ has no OCUSs.", "For the first claim, it is clear that $\\mathcal {S}$ is unsatisfiable and satisfies $p$ .", "Hence all we need to show is $f$ -optimality of $\\mathcal {S}$ .", "If there would exist some other unsatisfiable subset $\\mathcal {S}^{\\prime }$ that satisfies $p$ with $f(\\mathcal {S}^{\\prime })\\le f(\\mathcal {S})$ , we know that $\\mathcal {S}^{\\prime }$ would hit every minimal correction set of $\\mathcal {F}$ , and hence also every set in $\\mathcal {H}$ (since every correction set is the superset of a minimal correction set).", "Since $\\mathcal {S}$ is $f$ -optimal among hitting sets of $\\mathcal {H}$ satisfying $p$ and $\\mathcal {S}^{\\prime }$ also hits $\\mathcal {H}$ and satisfies $p$ , it must thus be that $f(\\mathcal {S})=f(\\mathcal {S}^{\\prime })$ .", "The second claim immediately follows from prop:MCS-MUS-hittingset and the fact that an OCUS is an unsatisfiable subset of $\\mathcal {F} $ .", "Perhaps surprisingly, correctness of the proposed algorithm does not depend on monotonicity properties of $f$ nor $p$ .", "In principle, any (computable) cost function and condition on the unsatisfiable subsets can be used.", "In practice however, one is bound by limitations of the chosen hitting set solver.", "As an illustration, we now provide an example of one call to $\\textsc {explain-One-Step-ocus} $ (Algorithm ) and the corresponding $\\textsc {OCUS} $ -call (Algorithm ) in detail: Example 1 Let $\\mathcal {C} $ be a CNF formula over variables $x_1, x_2, x_3$ with the following four clauses: $ c_1 := \\lnot x_1 \\vee \\lnot x_2 \\vee x_3 \\qquad c_2 := \\lnot x_1 \\vee x_2 \\vee x_3$ $ c_3 := x_1 \\qquad c_4 := \\lnot x_2 \\vee \\lnot x_3 $ The final interpretation $I_\\mathit {end} $ is $\\lbrace x_1, \\lnot x_2, x_3\\rbrace $ .", "Let the current interpretation $I$ be $\\lbrace x_1\\rbrace $ , then $\\overline{I_\\mathit {end} \\setminus I} = \\lbrace x_2, \\lnot x_3\\rbrace $ .", "To define the input for the OCUS call, we add new clauses representing the already known facts $I$ and the to-be-derived facts $\\overline{I_\\mathit {end} \\setminus I}$ : $ c_5 := \\lbrace x_1\\rbrace \\qquad c_6:=\\lbrace x_2\\rbrace \\qquad c_7 := \\lbrace \\lnot x_3\\rbrace $ The formula $\\mathcal {F} $ in the $\\textsc {OCUS} $ -call is thus: $\\mathcal {F} = \\mathcal {C} \\wedge I \\wedge \\overline{(I_\\mathit {end} \\setminus I)} = \\lbrace c_1 \\wedge c_2 \\wedge c_3\\wedge c_4\\wedge c_5\\wedge c_6\\wedge c_7\\rbrace $ We define $p\\triangleq $ exactly-one$(c_6, c_7)$ and $f = \\sum w_ic_i$ with clause weights $w_1 = 60, w_2=60, w_3=100, w_4=100, w_5=1, w_6=1, w_7=1$ .", "$\\mathcal {H}$ is initialized as the empty set.", "At each iteration, the hitting set solver will search for a cost-minimal assignment that hits all sets in $\\mathcal {H}$ and that furthermore contains exactly one of $c_6$ and $c_7$ (due to $p$ ).", "Table REF shows the computed steps in the different iterations of Algorithm  given the above input.", "Table: Example of an OCUS-explanation computation.In this example, the $\\textsc {Grow} $ we used is the one called Max-Actual-Unif in sec:experiments." ], [ "Efficient OCUS Computation for Explanations", "Algorithm  is generic and can also be used to find (unconstrained) $\\textsc {OUS} $ s, namely with a trivially true $p$ .", "However, its constrainedness property allows to remove the need to compute a MUS/$\\textsc {OUS} $ for every literal.", "This decreases the complexity of explanation sequence generation from $O(n^2)$ calls to MUS to $O(n)$ calls to OCUS, namely, once for every step in the sequence.", "We now discuss optimizations to the OCUS algorithm that are specific to explanation sequence generation, though they can also be used when other forms of domain knowledge are present." ], [ "Incremental OCUS Computation.", "Inherently, generating a sequence of explanations still requires as many OCUS calls as there are literals to explain.", "Indeed, a greedy sequence construction algorithm calls $\\textsc {explain-One-Step-ocus} $ iteratively with a growing interpretation $I$ until $I=I_\\mathit {end} $ .", "All of these calls to $\\textsc {explain-One-Step-ocus} $ , and hence OCUS, are done with very similar input (the set of constraints does not change, and the $I$ slowly grows between two calls).", "For this reason, it makes sense that information computed during one of the earlier stages can be useful in later stages as well.", "The main question is, suppose two $\\textsc {OCUS} $ calls are done, first with inputs $\\mathcal {F} _1$ , $f_1$ , and $p_1$ , and later with $\\mathcal {F} _2$ , $f_2$ , and $p_2$ ; how can we make use as much as possible of the data computations of the first call to speed-up the second call?", "The answer is surprisingly elegant.", "The most important data $\\textsc {OCUS} $ keeps track of is the collection $\\mathcal {H}$ of correction subsets that need to be hit.", "This collection in itself is not useful for transfer between two calls, since – unless we assume that $\\mathcal {F} _2$ is a subset of $\\mathcal {F} _1$ , there is no reason to assume that a set in $\\mathcal {H} _1$ should also be hit in the second call.", "However, each set $H$ in $\\mathcal {H} $ is the complement (with respect to the formula at hand) of a satisfiable subset of constraints, and this satisfiability remains true.", "Thus, instead of storing $\\mathcal {H} $ , we can keep track of a set $\\mathbf {SSs}$ of satisfiable subsets (the sets $\\mathcal {S}$ in the $\\textsc {OCUS} $ algorithm).", "When a second call to $\\textsc {OCUS} $ is performed, we can then initialize $\\mathcal {H} $ as the complement of each of these satisfiable subsets with respect to $\\mathcal {F} _2$ , i.e., $\\mathcal {H} \\leftarrow \\lbrace \\mathcal {F} _2\\setminus \\mathcal {S}\\mid \\mathcal {S}\\in \\mathbf {SSs} \\rbrace .$ The effect of this is that we bootstrap the hitting set solver with an initial set $\\mathcal {H} $ .", "For hitting set solvers that natively implement incrementality, we can generalize this idea further: we know that all calls to $\\textsc {OCUS} (\\mathcal {F} ,f,p)$ will be cast with $\\mathcal {F} \\subseteq \\mathcal {C}\\cup I_\\mathit {end} \\cup \\overline{I_\\mathit {end} \\setminus I_0}$ , where $I_0$ is the start interpretation.", "Since our implementation uses a MIP solver for computing hitting sets (see Section ), and we have this upper bound on the set of formulas to be used, we can initialize all relevant decision variables once.", "To compute the conditional hitting set for a specific $\\mathcal {C} \\cup I\\cup \\overline{I_\\mathit {end} \\setminus I} \\subseteq \\mathcal {C}\\cup I_\\mathit {end} \\cup \\overline{I_\\mathit {end} \\setminus I_0}$ we need to ensure that the MIP solver only uses literals in $\\mathcal {C} \\cup I\\cup \\overline{I_\\mathit {end} \\setminus I}$ , for example by giving all other literals infinite weight in the cost function.", "In this way, the MIP solver will automatically maintain and reuse previously found sets-to-hit in each of its computations." ], [ "Explanations with Bounded OUS.", "Instead of working $\\textsc {OCUS} $ -based, we can now also generate optimal explanations by replacing the MUS call by an $\\textsc {OUS} $ call in Algorithm  (where OUS is computed as in Algorithm , but with a trivially true $p$ ).", "When doing this, we know that every $\\textsc {OCUS} $ of cost greater than or equal to $f(X_{\\mathit {best}})$ will be discarded by the check on Line 4 of Algorithm .", "As such, a next optimization is to, instead of searching for an OUS, perform a bounded OUS check, which only computes an OUS in case one of cost smaller than a given bound $\\mathit {ub}$ exists.", "In our specific implementation, bounded $\\textsc {OUS} $ is performed by interrupting this $\\textsc {OUS} $ -call (after Line 3 Algorithm ) if $f(\\mathcal {S}) > \\mathit {ub}$ .", "Since the bounding on the $\\textsc {OUS} $ cost has the most effect if cheap $\\textsc {OUS} $ s are found early in the loop across the different literals, we keep track of an upper bound of the cost of an OUS for each literal to explain.", "This is initialized to a value greater than any $\\textsc {OUS} $ , e.g., as $f(\\mathcal {C} \\wedge I_0 \\wedge \\overline{I_\\mathit {end} \\setminus I_0})$ , and is updated every time an $\\textsc {OUS} $ explaining that literal is found; when going through the loop in Line 2 of Algorithm , we then handle literals in order of increasing upper bounds." ], [ "Domain-Specific Implementations of $\\textsc {Grow} $ .", "The goal of the $\\textsc {Grow} $ procedure is to turn $\\mathcal {S}$ into a larger subformula of $\\mathcal {F} $ .", "The effect of this is that the complement added to $\\mathcal {H}$ will be smaller, and hence, a stronger restriction for the hitting set solver is found.", "Choosing an effective $\\textsc {Grow} $ procedure requires finding a difficult balance: on the one hand, we want our subformula to be as large as possible (which ultimately would correspond to computing the maximum satisfiable subformula), but on the other hand we also want the procedure to be very efficient as it is called in every iteration.", "For the case of explanations we are in, we make the following observations: Our formula at hand (using the notation from the $\\textsc {explain-One-Step-ocus} $ algorithm) consists of three types of clauses: (translations of) the problem constraints (this is $\\mathcal {C} $ ) literals representing the assignment found thus far (this is $I$ ), and the negations of literals not-yet-derived (this is $\\overline{I_\\mathit {end} \\setminus I}$ ).", "$\\mathcal {C} $ and $I$ together are satisfiable, with assignment $I_{end}$ , and mutually supportive, by this we mean that making more constraints in $\\mathcal {C} $ true, more literals in $I$ will automatically become true and vice versa.", "The constraint $p$ enforces that each hitting set will contain exactly one literal of $\\overline{I_\\mathit {end} \\setminus I}$ Since the restriction on the third type of elements of $\\mathcal {F} $ are already strong, it makes sense to use the $\\textsc {Grow} $ ($\\mathcal {S}$ ,$\\mathcal {F} $ ) procedure to search for a maximal satisfiable subset of $\\mathcal {C} \\cup I$ with hard constraints that $\\mathcal {S}$ should be satisfied, using a call to an efficient (partial) MaxSAT solver.", "Furthermore, we can initialize this call as well as any call to a $\\textsc {sat} $ solver with the polarities for all variables set to the value they take in $I_\\mathit {end} $ .", "We evaluate different grow strategies in the experiments section, including the use of partial MaxSAT as well as weighted partial MaxSAT based on the weights in the cost function $f$ ." ], [ "Example 1 (cont.)", "Consider line 0 in table REF .", "During the $\\textsc {Grow} $ procedure, the MaxSAT solver Max-Actual-Unif with polarities set to $I_\\mathit {end} $ branches when multiple assignment to a literal are possible.", "By hinting the polarities of the literals, we guide the solver and it assigns all values according to the end interpretation and neither $c_6$ nor $c_7$ is taken." ], [ "Experiments", "We now experimentally validate the performance of the different versions of our algorithm.", "Our benchmarks were run on a compute cluster, where each explanation sequence generation was assigned a single core on a 10-core INTEL Xeon Gold 61482 (Skylake) processor, a timelimit of 120 minutes and a memory-limit of 4GB.", "Everything was implemented in Python on top of PySAThttps://pysathq.github.io and is available at https://github.com/ML-KULeuven/ocus-explain.", "For MIP calls, we used Gurobi 9.0, for SAT calls MiniSat 2.2 and for MaxSAT calls RC2 as bundled with PySAT (version 0.1.6.dev11).", "In the MUS-based approach we used PySAT's deletion-based MUS extractor MUSX [29].", "All of our experiments were run on a direct translation to PySAT of the 10 puzzles of [2]In one of the puzzles, an error in the automatic translation of the natural language constraints was found and fixed.. We used a cost of 60 for puzzle-agnostic constraints; 100 for puzzle-specific constraints; and cost 1 for facts.", "When generating an explanation sequence for such puzzles, the unsatisfiable subset identifies which constraints and which previously derived facts should be combined to derive new information.", "Our experiments are designed to answer the following research questions: Q1 What is the effect of requiring optimality of the generated MUSs on the quality of the generated explanations?", "Q2 Which domain-specific $\\textsc {Grow} $ methods perform best?", "Q3 What is the effect of the use of constrainedness on the time required to compute an explanation sequence?", "Q4 Does re-use of information across the different iterations improve efficiency?", "Figure: Q1 - Explanation quality comparison of optimal versus subset-minimal explanations in the generated puzzle explanation sequences." ], [ "Explanation quality.", "To evaluate the effect of optimality on the quality of the generated explanations, we reimplemented a MUS-based explanation generator based on alg:oneStep.", "Before presenting the results, we want to stress that this is not a fair comparison with the implementation of [2], since there – in order to avoid the quality problems we will illustrate below – an extra inner loop was used that employs even more calls to $\\textsc {MUS}$ for a selected set of subsets of $\\mathcal {C} $ of increasing size.", "While this yields better explanations, it comes at the expense of computation time, thereby leading to several hours to generate the explanation of a single puzzle.", "To answer Q1, we ran the $\\textsc {MUS}$ -based algorithm as described in alg:oneStep and compared at every step the cost of the produced explanation with the cost of the optimal explanation.", "These costs are plotted on a heatmap in Figure REF , where the darkness represents the number of occurrences of the combination at hand.", "We see that the difference in quality is striking in many cases, with the MUS-based solution often missing very cheap explanations (as seen by the two dark squares in the column around cost 60), thereby confirming the need for a cost-based $\\textsc {OUS} $ /$\\textsc {OCUS} $ approach." ], [ "Domain-specific $\\textsc {Grow} $ .", "In our OCUS algorithm, we do not just aim to find any satisfiable subsets, but we prefer high quality satisfiable subsets: subsets that impose strong constraints on the assignments the optimal hitting set solver can find.", "This induces a trade-off between efficiency of the $\\textsc {Grow} $ strategy and quality of the produced satisfiable subset.", "Thus, to answer Q2, we compared variations of $\\textsc {OCUS} $ that only differ in which $\\textsc {Grow} $ strategy they use.", "Figure REF depicts the average (over all the puzzles) cumulative explanation time to derive a number of literals.", "Note that most puzzles only contain 150 literals, except for 2, which contain 96 and 250 literals respectively.", "When a method times out for a puzzle at one step, a runtime value of 7200 is used in computing the averages for all future steps.", "The configurations used are as follows: Max refers to growing with a MaxSAT solver and Greedy to growing using a heuristic method implemented by repeated sat calls, while no-grow refers to skipping the $\\textsc {Grow} $ step.", "Full refers to using the full unsatisfiable formula $\\mathcal {F}$ while Actual refers to using only the constraints that hold in the final interpretation (see Section ).", "For instance, for the MaxSAT-based calls, Actual means that only the previously derived facts and the original constraints are taken into account when computing optimality.", "The MaxSAT solver (Max) is combined with different weighing schemes: uniform weights (unif), cost-function weights (pos) (equal to the weights in $f$ ), or the inverse of these costs (inv) defined as $\\max _j(w_j) + 1 - w_i$ .", "We can observe that not using a grow strategy performs badly, as do weighted MaxSAT grows with costs $w_i$ (-Pos).", "Greedy grow strategies improve on not using a grow strategy, but not substantially.", "The two approaches that work best use the domain-specific knowledge of doing a MaxSAT grow on $\\mathcal {C} \\cup I$ , with the unweighted variant the only one that never times out.", "Figure: Q2 - Explanation specific Grow\\textsc {Grow} strategies for OCUS\\textsc {OCUS} ." ], [ "Constrainedness and incrementality.", "To answer Q3 and Q4, we compare the effect of constrainedness in the search for explanations (C) and incrementality.", "Next to OCUS, we also include the bounded OUS approach (OUSb), where we call the OUS algorithm for every literal in every step, but we reuse information by giving it the current best bound $f(X_{best})$ and iterating over the literals that performed best in the previous call first.", "Based on the previous experiment, we always use (both for OUSb and OCUS) Max-Actual-Unif as grow strategy.", "For OCUS, incrementality (+Incr.", "HS) is achieved by reusing the same incremental MIP hitting set solver throughout the explanation calls, as explained in Section .", "To have a better view on how incrementality also affects OUSb, we add incrementality to it in the following ways: SS.", "caching keeps track of the satisfiable subsets, which are used to initialize $\\mathcal {H}$ for a fresh hitting set solver instance each time.", "Lit. Incr.", "HS uses a separate incremental hitting set solver for every literal to explain, throughout the explanation calls.", "Once the literal is explained, the hitting set solver is discarded.", "Figure REF shows the results, where In this figure, the configurations are compared in a similar fashion to Figure REF .", "Figure: Q3 - Cumulative runtime evolution enhancements on incrementality and constrainedness.When comparing the configurations, we see that the plain MUS-based implementation is faster than the O(C)US implementations, as it solves a simpler problem (with worse quality results as shown in Q1).", "Replacing MUS by bounded OUS calls (orange line) leads to a much larger computation cost.", "The generic SS caching technique adds additional overhead.", "The OCUS variants significantly improve runtime compared to those two bounded OUS approaches, by reducing the number of OUS calls.", "For OCUS, using an incremental hitting set solver across all steps seems to be slightly faster for deriving literals earlier in the sequence, while inducing a small overhead for the literals at the end of the sequence.", "When looking at the runtime to explain the entire sequence, best results are obtained with OUSb + Lit.", "Incr.", "HS, that is, using an incremental hitting set solver for every individual literal to explain.", "However, for the first literals, we can see that it takes much more computation time and that reducing the number of OUS calls from $n$ to 1 per explanation step improves runtime (OCUS).", "However, making each of the $n$ calls incremental and bounded across the entire explanation sequence generation process leads to an even faster process overall (OUSb+Lit.", "Incr.", "HS)." ], [ "Related Work", "In the last few years, driven by the increasingly many successes of Artificial Intelligence (AI), there is a growing need for eXplainable Artificial Intelligence (XAI) [31].", "In the research community, this need manifests itself through the emergence of (interdisciplinary) workshops and conferences on this topic [30], [16] and American and European incentives to stimulate research in the area [14], [15], [8].", "While the main focus of XAI research has been on explaining black-box machine learning systems [26], [13], [20], also model-based systems, which are typically considered more transparent, are in need of explanation mechanisms.", "Indeed, by advances in solving methods in research fields such as constraint programming [34] and SAT [1], as well as by hardware improvement, such systems now easily consider millions of alternatives in short amounts of time.", "Because of this complexity, the question arises how to generate human-interpretable explanations of the conclusions they make.", "Explanations for model-based systems have been considered mostly for explain unsatisfiable problem instances [21], and have recently seen a rejuvenation in various subdomains of constraint reasoning [9], [4], [3], [2].", "In this context, we recently introduced step-wise explanations [2] and applied them to Zebra puzzles; similar explanations, but for a wider range of puzzles, have been investigated by [7].", "Our current work is motivated by a concrete algorithmic need: to generate these explanations efficiently, we need algorithms that can find optimal MUSs with respect to a given cost function, where the cost function approximates human-understandability of the corresponding explanation step.", "The closest related works can be found in the literature on generating or enumerating MUSs [27], [25].", "Different techniques are employed to find MUSs, including manipulating resolution proofs produced by SAT solvers [12], [11], [6], incremental solving to enable/disable clauses and branch-and-bound search [32], or by BDD-manipulation methods [17].", "Other methods work by means of translation into a so-called Quantified MaxSAT [19], a field that combines the expressivity of Quantified Boolean Formulas (QBF) [22] with optimization as known from MaxSAT [23], or by exploiting the so-called hitting set duality [18] bootstrapped using MCS-enumeration [28].", "An abstract framework for describing hitting set–based algorithms, including optimization was developed by [35].", "While our approach can be seen to fit the framework, the terminology is focused on MaxSAT rather than MUS and would complicate our exposition.", "To the best of our knowledge, only few have considered optimizing MUSs: the only criterion considered yet is cardinality-minimality [27], [18]." ], [ "Conclusion, Challenges and Future work", "We presented a hitting set–based algorithm for finding optimal constrained unsatisfiable subsets, with an application in generating explanation sequence for constraint satisfaction problems.", "We extended our methods with incrementality, as well as with a domain-specific method for extending satisfiable subsets ($\\textsc {Grow} $ ).", "This domain-specific $\\textsc {Grow} $ method was key to generating explanation sequences in a reasonable amount of time.", "We noticed that, independently, incrementality and constrainedness have major benefits on explanation-generation time.", "The best method on the tested puzzles was the incremental, bounded but non-constrained variant.", "It remains an open question how to make constrainedness and incrementality work together more effectively, as well as how to further optimize O(C)US-based explanations, for instance using disjoint MCS enumeration [28].", "With the observed impact of different `$\\textsc {Grow} $ ' methods, an open question remains whether we can quantify precisely and in a generic way what a good or even the best set-to-hit is in a hitting set approach.", "The synergies of our approach with the more general problem of QMaxSAT [19] is another open question.", "The concept of bounded (incremental) OUS and OCUS are not limited to explanations of satisfaction problems and we are keen to explore other applications too.", "A general direction here are explanations of optimisation problems and the role of the objective function in explanations." ], [ "Acknowledgments", "This research received partial funding from the Flemish Government (AI Research Program); the FWO Flanders project G070521N; and funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant No.", "101002802, CHAT-Opt)." ] ]
2105.11763
[ [ "Reversibility of Hermitian Isometries" ], [ "Abstract An element $g$ in a group $G$ is called reversible (or real) if it is conjugate to $g^{-1}$ in $G$, i.e., there exists $h$ in $G$ such that $g^{-1}=hgh^{-1}$.", "The element $g$ is called strongly reversible if the conjugating element $h$ is an involution (i.e., element of order at most two) in $G$.", "In this paper, we classify reversible and strongly reversible elements in the isometry groups of $\\mathbb{F}$-Hermitian spaces, where $\\mathbb{F}=\\mathbb{C}$ or $\\mathbb{H}$.", "More precisely, we classify reversible and strongly reversible elements in the groups $ \\mathrm{Sp}(n) \\ltimes \\mathbb{H}^n$, $\\mathrm{U}(n) \\ltimes \\mathbb{C}^n$ and $\\mathrm{SU}(n) \\ltimes \\mathbb{C}^n$.", "We also give a new proof of the classification of strongly reversible elements in $\\mathrm{Sp}(n)$." ], [ " Introduction", "Let $G$ be a group.", "An element $g$ in $G$ is called reversible if it is conjugate to $g^{-1}$ in $G$ , i.e.", "there is a $h$ in $G$ such that $g^{-1}=hgh^{-1}$ .", "The element $g$ in $G$ is called strongly reversible if $g$ is a product of two involutions (i.e.", "order one or two elements) in $G$ .", "Equivalently, $g$ in $G$ is strongly reversible if it is conjugate to its inverse by an involution in $G$ .", "Reversible elements are also known as real elements in the literature, and strongly reversible elements are known as `strongly real' and `bireflectional', e.g.", "[3], [5], [8], [9].", "A strongly reversible element is reversible, but the converse is generally not true.", "It has been a problem of wide interest to investigate the reversibility in groups, see the monograph [2] for an elaborate exposition of this theme.", "In [7], also see [2], Short proved that every element in the Euclidean isometry group $\\mathrm {O}(n) \\ltimes \\mathbb {R}^n$ is strongly reversible.", "The situation for the orientation-preserving isometries $\\mathrm {SO}(n) \\ltimes \\mathbb {R}^n$ is subtle and Short classified the strongly reversible elements in this group as well.", "In this paper we aim to investigate the reversibilities in the isometry group of the ${\\rm F}$ -Hermitian space where ${\\rm F}$ is either the complex numbers $\\mathbb {C}$ or the division ring of the Hamilton's quaternions $\\mathbb {H}$ .", "Before stating our main results, we briefly recall the basic objects." ], [ "The Hermitian Space", "Let $\\mathrm {V}$ be an $n$ dimensional right vector space over ${\\rm F}$ .", "Given a right linear transformation $T: \\mathrm {V}\\rightarrow \\mathrm {V}$ , it can be represented by an $n \\times n$ matrix over ${\\rm F}$ after choosing a basis.", "The linear algebra over $\\mathbb {C}$ is well-known.", "For theory of linear transformations over the quaternions, see [6].", "In particular, well-defined notions like characteristic or minimal polynomials are not available over the quaternions as eigenvalues appear in similarity classes: Let $T: \\mathrm {V}\\rightarrow \\mathrm {V}$ be a linear transformation and $v \\in \\mathrm {V}, v \\ne o $ , $\\lambda \\in \\mathbb {H}$ , be such that $T(v)=v \\lambda $ , then for $\\mu \\in {\\mathbb {H}}^{\\times }$ we have $T(v \\mu )=(v \\mu ) \\mu ^{-1} \\lambda \\mu .$ Therefore eigenvalues of $T$ occur in similarity classes and if $v$ is a $\\lambda $ -eigenvector, then $v \\mu \\in v \\mathbb {H}$ is a $\\mu ^{-1} \\lambda \\mu $ -eigenvector.", "Each similarity class of eigenvalues contains a unique pair of complex conjugate numbers.", "Often we shall refer to them as `eigenvalues', though it should be understood that our reference is towards their similarity classes.", "In places where we need to distinguish between the similarity class and a representative, we shall denote the similarity class of an eigenvalue representative $\\lambda $ by $[\\lambda ]$ .", "We shall mostly choose the complex representative of a similarity class where the argument lies in $[0, \\pi ]$ .", "Let $\\mathrm {V}:={\\rm F}^{n}$ be equipped with the ${\\rm F}$ -Hermitian form $\\Phi (z,w)=\\bar{z}_1w_1+\\cdots +\\bar{z}_nw_n,$ where $z=( z_1, \\ldots , z_n), \\; w=( w_1, \\ldots , w_n)\\in {\\rm F}^{n}$ .", "The group of linear transformations $g$ that preserves this form, i.e.", "for all $z, w \\in \\mathrm {V}$ , $\\Phi (gz, gw)=\\Phi (z, w)$ , is the unitary group $\\mathrm {U}(n, {\\rm F})$ .", "Following usual notations, we denote $\\mathrm {U}(n, \\mathbb {C}) := \\mathrm {U}(n)$ and $\\mathrm {U}(n, \\mathbb {H}):= \\mathrm {Sp}(n)$ .", "In matrix notations, $\\mathrm {U}(n, {\\rm F}):=\\lbrace g \\in \\mathrm {GL}(n,{\\rm F}) \\mid {}^{t}\\bar{g}g=I_n\\rbrace ,$ which is a compact subgroup of $\\mathrm {GL}(n,{\\rm F})$ .", "We denote ${}^{t}\\bar{g}=g^{\\bigstar }$ in the sequel.", "The following definition will be used.", "Definition 1.1 Let $g\\in \\mathrm {Sp}(n)$ .", "Let distinct eigenvalues of $g$ be represented by $e^{i\\theta _1}, e^{i\\theta _2}, \\ldots , e^{i\\theta _m}$ , $m \\le n$ .", "The right vector space $\\mathbb {H}^n$ has the following orthogonal decomposition into eigenspaces: $ \\mathbb {H}^n=\\mathrm {V}_{\\theta _1}\\oplus \\mathrm {V}_{\\theta _2}\\oplus \\cdots \\oplus \\mathrm {V}_{\\theta _m},$ where $\\mathrm {V}_{\\theta _l}=\\lbrace v\\in \\mathbb {H}^n\\mid gv=ve^{i\\theta _l}\\rbrace $ for $1\\le l\\le m$ .", "We define multiplicity of $e^{i\\theta _l}:=\\mathrm {dim}\\;(\\mathrm {V}_{\\theta _l})$ .", "Equivalently, it is the repetition of the eigenvalue $e^{i\\theta _l}$ in a diagonal form, up to conjugacy, of $g$ .", "Definition 1.2 Let $g \\in \\mathrm {U}(n)$ .", "The characteristic polynomial $\\chi _g(x)$ of $g$ is called self-dual if whenever $\\lambda \\ne \\pm 1$ is a root of $\\chi _g(x)$ , so is $\\lambda ^{-1}$ with the same multiplicity.", "The Hermitian form $\\Phi $ gives a natural metric $d(z, w)=\\Phi (z-w, z-w)^{\\frac{1}{2}}$ on $\\mathrm {V}$ .", "We call $(\\mathrm {V}, d)$ as the Hermitian space.", "The isometries of $(\\mathrm {V}, d)$ are affine transformations of the form $T: z \\mapsto Az+v$ , where $A \\in \\mathrm {U}(n, {\\rm F})$ and $v \\in {\\rm F}^n$ .", "The group $\\textnormal {Isom}(\\mathrm {V}, d)$ of the affine isometries of $(\\mathrm {V}, d)$ may be identified with $\\mathrm {U}(n, {\\rm F}) \\ltimes {\\rm F}^n$ ." ], [ "Main results", "In this paper, we classify the reversible and strongly reversible elements in the group $\\mathrm {U}(n, {\\rm F}) \\ltimes {\\rm F}^n$ .", "We prove that every element in $\\mathrm {Sp}(n) \\ltimes \\mathbb {H}^n$ is reversible, see Theorem REF .", "We also show that an element $g$ in $\\mathrm {U}(n) \\ltimes \\mathbb {C}^n$ is reversible if and only if it is strongly reversible.", "However, not every element in $\\mathrm {U}(n, {\\rm F}) \\ltimes {\\rm F}^n$ is strongly reversible.", "We have the following classification.", "Theorem 1.3 Let $g = ( L(g) ,w )$ be an element in $\\mathrm {U}(n, {\\rm F}) \\ltimes {\\rm F}^n$ .", "Then $g$ is strongly reversible in $\\mathrm {U}(n, {\\rm F}) \\ltimes {\\rm F}^n$ if and only if $L(g)$ is strongly reversible in $\\mathrm {U}(n, {\\rm F})$ .", "The proof has been divided over two sections, for $\\mathrm {Sp}(n) \\ltimes \\mathbb {H}^n$ , see Theorem REF and for $\\mathrm {U}(n) \\ltimes \\mathbb {C}^n$ , see Theorem REF .", "The situation is more tricky in $\\mathrm {SU}(n) \\ltimes \\mathbb {C}^n$ .", "When $g=(L(g), w)$ be such that $L(g)$ has an eigenvalue $-1$ , then Theorem REF holds for $\\mathrm {SU}(n) \\ltimes \\mathbb {C}^n$ as well.", "However, if $L(g)$ has no eigenvalue $-1$ , then $g$ may not be strongly reversible even if $L(g)$ is so.", "We classify elements in $\\mathrm {SU}(n) \\ltimes \\mathbb {C}^n$ which are not strongly reversible.", "Theorem 1.4 Let $ g\\in G = \\mathrm {SU}(n,\\mathbb {C})\\ltimes \\mathbb {C}^n $ be such that up to conjugacy in $G$ , $g$ = (A,v ) where $A= \\textnormal {diag}(e^ {i\\theta _1}, e^ {-i\\theta _1} , e^ {i\\theta _2},e^ {-i\\theta _2} ,~\\cdots ,~ e^ {i\\theta _r}, e^ {-i\\theta _r}, I_t)$ and $v = [0,0, \\cdots , 0,v_{r+1}, v_{r+2},\\cdots , v_n ]$ , $ t \\in \\mathbb {N}, \\theta _k \\in (0 ,\\pi ), ~k=1, \\ldots , r$ .", "Then the following are equivalent.", "$g$ is not strongly reversible.", "$v_\\ell \\ne 0$ for all $\\ell =r+1, \\ldots , n$ , and one of the following holds.", "(i) $n \\equiv 0 \\mod {2}$ , $r \\equiv 1 \\mod {2}$ .", "(ii) $n \\equiv 1 \\mod {2}$ , $r \\equiv 0 \\mod {2}$ .", "Note that the classification of the reversible and the strongly reversible elements in $\\mathrm {U}(n, {\\rm F})$ is intricately related to the classification of reversibility in $\\mathrm {U}(n, {\\rm F}) \\ltimes {\\rm F}^n$ .", "Complete classification of reversibility in $\\mathrm {U}(n)$ and $\\mathrm {SU}(n)$ are well-known, see [2], [4].", "However, complete classification of the strongly reversible elements in $\\mathrm {Sp}(n)$ was not known until very recently.", "It was raised as an open problem in [2] to classify the strongly reversible elements in $\\mathrm {Sp}(n)$ , see [2].", "Bhunia and Gongopadhyay have given a solution to this problem in [1].", "However, the proof in [1] is geometric and uses the notion of `projective points'.", "In this paper, we revisit this problem and have given a different proof to the classification of strongly reversible elements in $\\mathrm {Sp}(n)$ .", "Our proof uses only simple quaternionic linear algebra and may be of independent interest." ], [ "Structure of the paper", "We classify the strongly reversible elements in $\\mathrm {Sp}(n)$ in Section .", "In Section , we note down a few lemmas related to the conjugacy in $\\mathrm {U}(n, {\\rm F}) \\ltimes {\\rm F}^n$ .", "These lemmas are used in the proofs of the main theorems.", "In Section , we classify reversible and strongly reversible elements in $\\mathrm {Sp}(n) \\ltimes \\mathbb {H}^n$ .", "In Section , complete classification of reversible and strongly reversible elements in $\\mathrm {U}(n) \\ltimes \\mathbb {C}^n$ has been obtained.", "The Theorem REF follows by combining results of these later two sections.", "Further, we investigate reversibility in $\\mathrm {SU}(n) \\ltimes \\mathbb {C}^n$ and prove Theorem REF in Section ." ], [ "Reversibility in $\\mathrm {Sp}(n)$ ", "First note the following well-known lemma.", "Lemma 2.1 If ${A} \\in \\mathrm {Sp}(n)$ , then there exist ${U} \\in \\mathrm {Sp}(n)$ such that $ {UA}{U}^{ -1} = \\textnormal {diag} ( e^{ i\\theta _{1} } ,e^{ i\\theta _{2} } , \\cdots \\cdots , e^{ i\\theta _{n} } )$ where $ \\theta _{s} \\in [0 ,\\pi ]$ for all $ s \\in \\lbrace 1,2 , \\cdots \\, n \\rbrace $ .", "Next we observe the following about quaternionic numbers.", "Lemma 2.2 Let $ \\alpha \\in (0,\\pi )$ and $\\beta \\in [0,\\pi ]$ .", "Let $x \\in \\mathbb {H}$ be such that $x e^{i \\alpha }= e^{i \\beta } x$ , then for some real numbers $a$ and $b$ , x = $ {\\left\\lbrace \\begin{array}{ll}a + i b & \\text{if $ \\alpha = \\beta $ }\\\\( a +ib ) j & \\text{if $ \\alpha = -\\beta $ }\\\\0 & \\text{if $ \\alpha \\ne \\pm \\beta $}.\\end{array}\\right.}", "$ Let $ x = x_{0} + x_{1} i + x_{2} j + x_{3} k \\in \\mathbb {H}$ be such that $ x e^{i \\alpha }= e^{i \\beta } x$ .", "this implies the following set of equations: $x_{0} \\cos \\alpha - x_{1} \\sin \\alpha = x_{0} \\cos \\beta - x_{1} \\sin \\beta $ , $ x_{0} \\sin \\alpha + x_{1} \\cos \\alpha = x_{0} \\sin \\beta + x_{1} \\cos \\beta $ , $ x_{2} \\cos \\alpha + x_{3} \\sin \\alpha = x_{2} \\cos \\beta - x_{3} \\sin \\beta $ , and $ - x_{2} \\sin \\alpha + x_{3} \\cos \\alpha = x_{2} \\sin \\beta + x_{3} \\cos \\beta $ .", "Using matrix forms, we have $A_{1} X{_1} = O $ and $ A_{2} X{_2} = O$ , where $ X{_1} =\\begin{pmatrix}x_{0} \\\\x_{1}\\\\\\end{pmatrix} $ , $ X{_2}=\\begin{pmatrix}x_{2} \\\\x_{3}\\\\\\end{pmatrix}$ ,    $A_1 =\\begin{pmatrix}\\cos \\alpha - \\cos \\beta & - ( \\sin \\alpha - \\sin \\beta ) \\\\\\sin \\alpha - \\sin \\beta & \\cos \\alpha - \\cos \\beta \\\\\\end{pmatrix}$ and $A_2 =\\begin{pmatrix}\\cos \\alpha - \\cos \\beta & \\sin \\alpha + \\sin \\beta \\\\- ( \\sin \\alpha + \\sin \\beta ) & \\cos \\alpha - \\cos \\beta \\\\\\end{pmatrix}$ .", "The desired result will obtain by solving this system of homogeneous linear equations.", "Note that $\\det A_1 = (\\cos \\alpha - \\cos \\beta )^{2} + (\\sin \\alpha - \\sin \\beta )^{2} = 2 (1 - \\cos ( \\alpha - \\beta ) )$ , and $\\det (A_2 ) = (\\cos \\alpha - \\cos \\beta )^{2} + (\\sin \\alpha + \\sin \\beta )^{2} = 2 (1 - \\cos (\\alpha + \\beta ) )$ .", "As $ \\alpha \\in (0,\\pi ) \\beta \\in [0,\\pi ] $ , we observe: $\\det (A_1 ) = 0 \\Longleftrightarrow \\alpha =\\beta \\Longleftrightarrow A{_1} = 0, $ $ \\det (A_2 ) = 0 \\Longleftrightarrow \\alpha = - \\beta \\Longleftrightarrow A{_2} = 0.$ From these observations, we see that if $ \\alpha \\ne \\beta $ then $A{_1}$ has only trivial solution $X{_1} =O$ , and if $\\alpha = \\beta $ then $A{_1} = 0$ , so, $X{_1}$ may be any element from $\\mathbb {R}^{2}$ .", "If $ \\alpha \\ne - \\beta $ then $A{_2}$ has only trivial solution $ X{_2} $ =O, and if $\\alpha = - \\beta $ then $A_2 = 0$ , so $X{_2}$ may be any element from $\\mathbb {R}^{2}$ .", "This proves the lemma.", "Theorem 2.3 (1) Every element in $\\mathrm {Sp}(n)$ is reversible.", "(2) Every element in $\\mathrm {PSp}(n)$ is strongly reversible.", "Let ${A} \\in \\mathrm {Sp}(n)$ .", "By using Lemma REF , upto conjugacy in $\\mathrm {Sp}(n)$ assume: ${A}= \\textnormal {diag} ( e^{ i\\theta _{1} } ,e^{ i\\theta _{2} } , \\cdots \\cdots , e^{ i\\theta _{n} } )$ , where $\\theta _{s} \\in [0 ,\\pi ] $ for all $ s \\in \\lbrace 1,2 , \\cdots \\, n \\rbrace $ .", "Now consider $ B = \\textnormal {diag} ( j ,j, \\cdots \\cdots , j ) \\in \\mathrm {Sp}(n)$ .", "Then $ {BA}{B}^{ -1} = { A}^{ -1}$ .", "Hence every element of $\\mathrm {Sp}(n)$ is reversible.", "Now note that $ {B}^{2} = - $ $ I_{n}$ and $\\mathrm {PSp}(n) := \\mathrm {Sp}(n) / \\lbrace \\pm {I}_{n} \\rbrace , $ where ${I}_{n}$ is identity matrix in $\\mathrm {Sp}(n)$ .", "Hence every element of $\\mathrm {PSp}(n)$ is strongly reversible.", "Lemma 2.4 Let ${A} \\in \\mathrm {Sp}(n)$ be such that eigenvalue classes of ${A }$ are either $ [ \\pm 1 ] $ or have even multiplicity.", "Then ${A}$ is strongly reversible in $\\mathrm {Sp}(n)$ .", "Let $ {A} \\in \\mathrm {Sp}(n)$ .", "So by using Lemma REF , upto conjugacy in $\\mathrm {Sp}(n)$ , we can assume that: $ {A}= e^{ i\\theta _{1} } {I}_{2k_1} \\oplus \\ e^{ i\\theta _{2} } {I}_{2k_2} \\oplus \\cdots \\cdots e^{ i\\theta _{r} } {I}_{2k_r} \\oplus - {I}_{s } \\oplus {I}_{t }, $ where ${2k_1} + {2k_2} + \\cdots \\cdots + {2k_r} + s + t = n$ , $ 2k_m \\in \\mathbb {N}$ , $ \\theta _{m} \\in (0 ,\\pi )$ for all $ m \\in \\lbrace 1,2 , \\cdots \\, r \\rbrace $ .", "Let $K= \\begin{pmatrix}0 & j\\\\-j & 0 \\\\\\end{pmatrix} \\in \\mathrm {Sp}(2)$ .", "Then $K^2 = I_{2}$ .", "Consider ${B} = \\textnormal {diag} (K,~ K, ~K,~\\cdots ,~ K,~ I_{s+t}) \\in \\mathrm {Sp}(n)$ .", "Then ${B} {A} {B}^{-1} = {A}^{-1}$ and ${B}^2 = {I}_{n}$ , where ${I}_{n}$ is the identity element in $\\mathrm {Sp}(n)$ .", "Hence ${A}$ is strongly reversible in $\\mathrm {Sp}(n)$ .", "Lemma 2.5 Let ${A} \\in \\mathrm {Sp}(n)$ have an eigenvalue class $[\\lambda ]$ of odd multiplicity $k$ such that $ \\lambda \\ne \\pm 1$ .", "Define $R({A})= \\lbrace {B} \\in \\mathrm {Sp}(n): {BAB}^{-1} = {A}^{-1} \\rbrace $ .", "Then up to conjugacy, an element ${B}$ in $R({A})$ is of the form $ \\begin{pmatrix}B_{1} j & O_{k \\times n-k}\\\\O_{n-k \\times k} & B_{2} \\\\\\end{pmatrix},$ for some $B_{2} \\in {\\mathrm {M}}(n-k,\\mathbb {H}), ~ B_{1} \\in {\\mathrm {M}}(k,\\mathbb {C})$ , where $O$ denotes zero matrix and subscripts of $O$ denotes its order.", "Let $\\theta _{1} \\in (0 ,\\pi ) $ be such that $e^ { i\\theta _{1} } $ is a chosen representative of the eigenvalue-class $[\\lambda ]$ of odd multiplicity $k$ .", "Upto conjugacy in $\\mathrm {Sp}(n)$ let $A$ be given by: A $ = [ a_{s,t}]_{1\\le s,t \\le n} $ where $ a_{s,t} $ = $ {\\left\\lbrace \\begin{array}{ll}e^{i\\theta _{1} } & \\text{if $s = t$ , $ 1\\le s \\le k$ }\\\\e^{i\\theta _{s} } & \\text{if $s = t$, $ k +1\\le s \\le n$ }\\\\0 & \\text{if $s \\ne t$ }\\\\\\end{array}\\right.}", "$ $\\Bigg \\rbrace $ .", "Let ${B} = [b_{s,t}] \\in R({A})$ .", "From $ {BA}{B}^{ -1} = { A}^{ -1} $ we get : ${BA} & = &{ A}^{ -1} {B} \\\\ &\\Longleftrightarrow & \\sum _{\\ell =1}^n ( b_{s,\\ell })( a_{\\ell ,t}) = \\sum _{\\ell =1}^n (a_{s,\\ell } )^{-1} ( b_{\\ell ,t}), \\hbox{for all} \\ \\ 1\\le s,t \\le n.\\\\&\\Longleftrightarrow & ( b_{s,t})( a_{t,t}) = (a_{s,s} )^{-1} ( b_{s,t}), \\hbox{for all } 1\\le s,t \\le n.$ This implies $( b_{s,t})e^{i\\theta _{1} } = e^{- i\\theta _{1} } ( b_{s,t}) ~ \\text{if } 1\\le s,t \\le k;$ $( b_{s,t}) e^{i\\theta _{t} } = e^{- i\\theta _{1} } ( b_{s,t}) ~ \\text{if } k +1\\le t \\le n, 1\\le s \\le k;$ $( b_{s,t}) e^{i\\theta _{1} } = e^{- i\\theta _{s} } ( b_{s,t}) ~\\text{if } k +1\\le s \\le n, 1\\le t \\le k.$ As $\\theta _{1} \\in (0,\\pi ), ~\\theta _{s} \\in [0,\\pi ]$ and $\\theta _{1} \\ne \\pm \\theta _{s}$ for all $k +1\\le s \\le n, s \\in \\mathbb {N}$ , so by using Lemma REF , we get the following: ${\\left\\lbrace \\begin{array}{ll}( b_{s,t}) = ( w_{s,t}) j \\hbox{ for some} \\ w_{s,t} \\in \\mathbb {C}& \\text{if $ 1\\le s,t \\le k$}, \\\\b_{s,t} =0 & \\text{if $ k +1\\le t \\le n , 1\\le s \\le k$ }, \\\\b_{s,t} =0 & \\text{if $ k +1\\le s \\le n , 1\\le t \\le k $}.\\end{array}\\right.", "}$ This gives us the matrix $B$ of the form $ {B} =\\begin{pmatrix}B_{1} j& O_{k \\times n-k}\\\\O_{n-k \\times k} & B_{2} \\\\\\end{pmatrix}$ for some $ B_{2} \\in {\\mathrm {M}}(n-k,\\mathbb {H}) $ and $ B_{1} = [ w_{s,t}]_{1\\le s,t \\le k} \\in {\\mathrm {M}}(k,\\mathbb {C})$ .", "Lemma 2.6 If ${A} \\in \\mathrm {Sp}(n)$ has an eigenvalue class $[\\lambda ]$ , $\\lambda \\ne \\pm 1$ , of odd multiplicity, then ${A}$ is not be strongly reversible in $\\mathrm {Sp}(n)$ .", "If possible suppose that ${A}$ is strongly reversible in $\\mathrm {Sp}(n)$ .", "Then there exists $B$ in $\\mathrm {Sp}(n)$ such that ${BA}{B}^{ -1} = { A}^{ -1}$ , ${B}^2 = {I_{n}}$ and ${B}{B}^{\\bigstar } = {I_{n}}$ .", "Since $B \\in R(A)$ , by using Lemma REF , we can write B in following form with respect to the basis consisting of the eigenvectors of $A$ : $ {B} =\\begin{pmatrix}B_{1} j& O_{k \\times n-k}\\\\O_{n-k \\times k} & B_{2} \\\\\\end{pmatrix} $ for some $ B_{2} \\in {\\mathrm {M}}(n-k,\\mathbb {H}) $ and $ B_{1} \\in {\\mathrm {M}}(k,\\mathbb {C}) $ .", "Now, we see that the relations ${B}^{2} = {I_{n}}$     and     ${B}{B}^{\\bigstar } = {I_{n}}$ implies $(B_{1} j ) ( B_{1} j) = {I_{n}} ~~~~~~ \\hbox{and } ~~~~~~ (B_{1} j ) ( B_{1} j) ^{\\bigstar } = {I_{n}}.$ Thus, $(B_{1}) ( \\bar{ B_{1}} ) = - {I_{n}}$ and $(B_{1}) ( B_{1}^{\\bigstar }) = {I_{n}}$ , using $wj = j\\bar{w}$ for $w \\in \\mathbb {C}$ .", "In particular, $B_{1} \\in {\\mathrm {M}}(k,\\mathbb {C})$ is invertible such that $ B_{1} ^{-1} = - \\bar{B_{1}} = B_{1}^{\\bigstar }$ .", "On taking transpose both sides we have $B_{1}^{\\top } = - B_{1}$ .", "So $ B_{1} $ is skew-symmetric matrix in $ {\\mathrm {M}}(k,\\mathbb {C})$ of odd order k. This is a contradiction since a skew symmetric matrix of odd order $k$ in $ {\\mathrm {M}}(k,\\mathbb {C})$ is not invertible.", "Therefore, our initial assumption that $A$ is strongly reversible can not be true.", "This proves the lemma.", "Theorem 2.7 An element $A$ in $\\mathrm {Sp}(n)$ is strongly reversible if and only if every eigenvalue-class of ${A}$ is either $ \\pm 1 $ or of even multiplicity.", "The proof follows from Lemma REF and Lemma REF ." ], [ "Conjugacy in $\\mathrm {U}(n, {\\rm F}) \\ltimes {\\rm F}^n$", "Let $G$ denote either of the groups $\\mathrm {Sp}(n) \\ltimes \\mathbb {H}^n$ , $\\mathrm {U}(n) \\ltimes \\mathbb {C}^n$ or $\\mathrm {SU}(n) \\ltimes \\mathbb {C}^n$ .", "Let $L(G)$ denote the linear part of $G$ .", "Let ${\\rm F}$ denote $\\mathbb {H}$ or $\\mathbb {C}$ .", "Lemma 3.1 Up to conjugacy, each element of $g$ in $G$ can be written as $g = ({A},v)$ , where ${A}(v) = v$ .", "After conjugating $g$ by a suitable element $(B,o)$ in $G$ we can assume that $g = (A ,w)$ where $A= \\textnormal {diag} (e^ {i\\theta _1}, e^ {i\\theta _2},~\\cdots ,~ e^ {i\\theta _r}, -I_s, I_t)$ where $\\theta _k \\in (0 ,\\pi )$ and $r+s+t = n$ , $ k = 1,2 , \\cdots \\, r$ , where $r,~s,~ t \\in \\mathbb {N}\\cup \\lbrace 0\\rbrace $ .", "Case (1): 1 is not an eigenvalue of $A$ .", "So the linear transformation ${A}- {I_n}$ is invertible.", "Therefore we can choose $x_o =({A } - {I_n} )^{-1} (w) \\in {\\rm F}^n$ .", "Consider $h = ({I_n}, x_o) \\in G$ , i.e $h(x) = x + x_o $ for all x $\\in {\\rm F}^n$ .", "Now $ hgh^{-1} (x) = hg(x- x_o) = h({A}x - {A}x_o + w ) = {A}x - {A}x_o + w + x_o ={A}x + w - ({A}-I_n)x_o.$ This implies, $hgh^{-1} (x)= {A}x + o$ for all $x \\in {\\rm F}^n$ , since $x_o =({A} - {I_n} )^{-1} (w) \\in {\\rm F}^n$ .", "By taking $v = o$ , $hgh^{-1} =({A} , o) = ({A}, v) \\in G \\ \\hbox{ and }\\ {A}(v) = v = o,$ where o is zero element in ${\\rm F}^{n}.$ Hence the lemma is proved for this case.", "Case (2) : Let 1 is an eigenvalue of ${A}$ .", "In this case $t >0$ and ${A} - {I_n}$ has rank $r+s = n-t < n$ such that we can choose a vector $u \\in {\\rm F}^n$ having last $n -(r+s)$ coordinates zero such that $ [({A} - {I_n} )(u)]_\\ell = w_\\ell \\ for \\ 1 \\le \\ell \\le r+s, \\hbox{ where }\\ w = [w_\\ell ]_{1 \\le \\ell \\le n}$ .", "Let $v = w{ - }({A} {-} {I_n} )(u)$ .", "Then $v = [0,0, \\cdots , 0,w_{r+s+1}, w_{r+s+2},\\cdots , w_n ] \\ \\hbox{ and }\\ {A}(v) =v $ .", "Now consider $h = ( {I_n},u) \\in G$ , i.e $h(x) = x+u$ .", "Note that for $x \\in {\\rm F}^n$ , $hgh^{-1}(x) = hg(x-u) = h({A}x - {A}u +w) = {A}x + w-({A} - {I_n})(u) = {A}x +v.$ Therefore we have found a $v \\in {\\rm F}^n$ and $h \\in G$ such that $hgh^{-1} = ({A},v) \\in G$ and ${A}(v)= v$ .", "This proves the lemma.", "Corollary 3.2 Every element $g$ in $G$ , upto conjugacy, can be written as $g = ({A},v)$ such that ${A}(v) =v$ and ${ A} = \\textnormal {diag} (e^ {i\\theta _1} , e^ {i\\theta _2},~ \\cdots , ~e^ {i\\theta _r}, - I_s , I_t)$ , $\\theta _k \\in (0 ,\\pi )$ , $r+s+t = n$ , $ k = 1,2 , \\cdots \\, r$ , where $r,~s,~ t \\in \\mathbb {N}\\cup \\lbrace 0\\rbrace $ , and $v$ is of the form $v = [0,0, \\cdots , 0,v_{r+s+1}, v_{r+s+2},\\cdots , v_n ]$ .", "Remark 1 If $A$ is such that the eigenvalue classes $[\\lambda ]$ , $\\lambda \\ne \\pm 1$ , occur in pairs, e.g.", "when $A \\in \\mathrm {U}(n)$ and has self-dual characteristic polynomial, then in the above lemma we can choose $A$ up to conjugacy as: $A= \\textnormal {diag} (e^ {i\\theta _1} , e^ {-i\\theta _1},~ \\cdots , ~e^ {i\\theta _r}, e^{-i \\theta _r}, - I_s , I_t)$ .", "Lemma 3.3 Let $g = ({A},v)$ in $G$ be such that ${A}(v) = v$ .", "If there exist an element ${B}$ in $L(G)$ such that ${BA}{B}^{-1} ={ A}^{-1} $ and ${B}(v) = - v$ , then $g$ is reversible in $G$ .", "Consider $ h = ({B }, o) \\in G$ , where o is zero element in ${\\rm F}^n$ .", "Then $ hgh^{-1} = g^{-1} $ .", "This proves the claim.", "Corollary 3.4 Let $g = ({A},v)$ in $G$ be such that ${A}(v) = v$ .", "If there exist an element ${B}$ in $L(G)$ such that ${BA}{B}^{-1} ={ A}^{-1} $ , ${B}(v) = - v$ and ${ B}^{2} = I_{n}$ , then $g$ is strongly reversible in $G$ .", "The element $h = ({B }, o)$ is also an involution if $B$ is an involution.", "Lemma 3.5 If $g = ({A} , v )$ is strongly reversible in $G$ , then ${ A }$ is strongly reversible in $L(G)$ .", "In other words, if ${A}$ is not strongly reversible in $L(G)$ then $g$ is not strongly reversible in $G$ .", "Let $g$ be strongly reversible in $G$ .", "Then there exist $h = ({B},u)$ in $G$ such that $hgh^{-1} = g^{-1} $ and $ h^{2} = I$ , where I is the identity in $G$ .", "This implies, ${BAB}^{-1}(x) -{ BAB}^{-1} (u) + {B}(v) + u = {A}^{-1}(x) -{ A}^{-1}(v) $ and ${ B}^2 (x) + {B}(u) + u = I_{n} (x) $ for all $ x \\in {\\rm F}^n $ .", "This implies $ {BAB}^{-1} = {A}^{-1}$ and ${B}^2 = I_{n} $ .", "So A is strongly reversible in $L(G)$ .", "Corollary 3.6 If $g = ({A} , v )$ is reversible in $G$ , then ${ A }$ is reversible in $L(G)$ .", "In other words, if ${A}$ is not reversible in $L(G)$ then $g$ is not reversible in $G$ ." ], [ "Reversibility in $\\mathrm {Sp}(n) \\ltimes \\mathbb {H}^n$", "Theorem 4.1 Let $g$ be an element in $G = \\mathrm {Sp}(n) \\ltimes \\mathbb {H}^n$ .", "Then $g$ is reversible in $G$ .", "Let $g \\in G$ .", "By Corollary REF , upto conjugacy in $G$ , we can write $g= ({A},v)$ such that ${A}(v) =v$ , where ${A }= \\textnormal {diag} (e^ {i\\theta _1} , e^ {i\\theta _2} , \\cdots ,e^ {i\\theta _r}, - I_s, I_t)$ , $\\theta _k \\in (0 ,\\pi )$ , $r+s+t = n$ , $v = [0,0, \\cdots , 0,v_{r+s+1}, v_{r+s+2},\\cdots , v_n ] \\in \\mathbb {H}^{n}$ .", "If t =0 i.e.", "1 is not an eigenvalue of $L(g)$ then up to conjugacy, $g = ({A},o )$ , where A $ \\in Sp(n)$ .", "By Theorem REF , ${ A}$ is reversible in $\\mathrm {Sp}(n)$ , so $g$ is reversible in $G$ .", "If $t > 0$ , consider ${B} = \\textnormal {diag} ( j,~ j,~j, ~\\cdots ,~j,~ - I_{s+t})$ in $\\mathrm {Sp}(n)$ .", "Then $ {BA}{B}^{-1} ={ A}^{-1}$ , ${B}(v) = - v$ .", "Using Lemma REF , the result follows.", "Since $B^2$ projects to the identity element in $\\mathrm {PSp}(n)$ , as an imediate corollary we have the following.", "Corollary 4.2 Every element in $\\mathrm {PSp}(n) \\ltimes \\mathbb {H}^n $ is strongly reversible.", "Lemma 4.3 Let $g = (L(g), w )$ in $G = \\mathrm {Sp}(n) \\ltimes \\mathbb {H}^n$ .", "Let every eigenvalue class of $L(g)$ is either $ \\pm 1$ or of even multiplicity.", "Then $g$ is strongly reversible in $G$ .", "Upto conjugacy in $G$ we write $g$ as $g = ({A}, v )$ as in Corollary REF .", "Let the eigenvalues $\\ne \\pm 1$ appear in $r$ pairs.", "Let $K = \\begin{pmatrix}0 & j\\\\-j & 0 \\\\\\end{pmatrix}$ in $\\mathrm {Sp}(2) $ .", "Then $K^2 = I_{2}$ in $\\mathrm {Sp}(2)$ .", "Consider $ {B} = \\begin{pmatrix}K & 0 & \\ldots & 0 & 0 & 0 \\\\0 & K & \\ldots & 0 & 0& 0\\\\ & & \\ddots & & & \\\\ 0 & 0 & \\ldots & K & 0&0 \\\\0 & 0 &\\cdots &0 & I_{s}& 0 \\\\ 0 & 0 &\\cdots &0 & 0 & -I_t \\end{pmatrix}$ in $\\mathrm {Sp}(n)$ .", "Then ${BAB}^{-1} ={ A}^{-1}$ , ${B}(v)= - v$ and ${B}^{2} =I_n$ .", "The theorem now follows from Corollary REF .", "Combining the above lemma with Lemma REF and Theorem REF we have the following classification of strongly reversible elements in $\\mathrm {Sp}(n) \\ltimes \\mathbb {H}^n$ .", "Theorem 4.4 Let $g = ( L(g) ,w )$ be an element in $\\mathrm {Sp}(n) \\ltimes \\mathbb {H}^n$ .", "Then the following are equivalent.", "$g$ is strongly reversible in $\\mathrm {Sp}(n) \\ltimes \\mathbb {H}^n$ .", "Every eigenvalue class of $L(g)$ is either $ \\pm 1$ or of even multiplicity.", "$L(g)$ is strongly reversible.", "in $\\mathrm {Sp}(n)$ ." ], [ "Reversibility in $\\mathrm {U}(n) \\ltimes \\mathbb {C}^n$", "Theorem 5.1 Let $g=(L(g), w)$ be an element in $\\mathrm {U}(n) \\ltimes \\mathbb {C}^n$ .", "Then the following are equivalent.", "$g$ is reversible in $\\mathrm {U}(n) \\ltimes \\mathbb {C}^n$ .", "$g$ is strongly reversible in $\\mathrm {U}(n) \\ltimes \\mathbb {C}^n$ .", "The characteristic polynomial of $L(g)$ is self-dual.", "$L(g)$ is strongly reversible in $\\mathrm {U}(n)$ .", "(1) $\\Rightarrow $ (3) follow from Lemma REF and noting the fact that an element $L(g)$ in $\\mathrm {U}(n)$ is reversible if and only if it has self-dual characteristic polynomial.", "(3) $\\Leftrightarrow (2)$ : The proof is similar to the proof of Theorem REF in the previous section, except that in Lemma REF , one choose $K=\\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}$ and $B$ as in (4.1) .", "This shows that an element $g = ( L(g) ,w )$ is strongly reversible in $\\mathrm {U}(n) \\ltimes \\mathbb {C}^n$ if and only if $L(g)$ is strongly reversible in $\\mathrm {U}(n)$ .", "Now the result follow from the fact that an element in $\\mathrm {U}(n)$ is strongly reversible if and only if it has self-dual characteristic polynomial.", "(3) $\\Leftrightarrow $ (4) is classical, e.g.", "see [4].", "As a corollary, we obtain the following.", "Corollary 5.2 Let $g = ( L(g) ,w )$ be an element in $\\mathrm {SU}(n) \\ltimes \\mathbb {C}^n$ .", "Then the following are equivalent.", "$g$ is reversible in $\\mathrm {SU}(n) \\ltimes \\mathbb {C}^n$ .", "The characteristic polynomial of $L(g)$ is self-dual.", "$L(g)$ is reversible in $\\mathrm {SU}(n)$ .", "We follow the proof of Lemma REF .", "Note that $\\det K=-1$ , and $\\det B=(-1)^{r+t} $ where $K=\\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}$ and $B$ as in (4.1).", "If $\\det B=1$ , the same proof works.", "If not, i.e.", "if $\\det ( -I_{t})=-1$ , and $\\det K^r=1$ , or, $\\det K^r=-1$ and $\\det (-I_t) =1$ , we choose $B$ as $ {B} = \\begin{pmatrix}J & 0 & \\ldots & 0 & 0 & 0 \\\\0 & K & \\ldots & 0 & 0 & 0 \\\\ & & \\ddots & & & \\\\ 0 & 0 & \\ldots & K &0 & 0 \\\\0 & 0 &\\cdots &0 & I_{s}& 0 \\\\ 0 & 0 & \\cdots & 0 & 0 & -I_t \\end{pmatrix}$ where $J=\\begin{pmatrix} 0 & -1\\\\1& 0 \\end{pmatrix}$ in  (REF ).", "Then $\\det B=1$ , though note that $B$ is no more an involution.", "If $s \\ne 0$ , instead of replacing $K$ by $J$ , we may replace an 1 in $I_s$ by $-1$ , that would give $\\det B=1$ and also $B$ would be an involution.", "Corollary 5.3 Let $g=(L(g), w)$ in $\\mathrm {SU}(n) \\ltimes \\mathbb {C}^n$ .", "Assume that either one of the following holds: (i) $L(g)$ has an eigenvalue $-1$ .", "(ii) $n \\equiv 0 \\mod {4}$ and $L(g)$ does not have any eigenvalue 1 or $-1$ .", "Then the following are equivalent.", "$g$ is strongly reversible in $\\mathrm {SU}(n) \\ltimes \\mathbb {C}^n$ .", "The characteristic polynomial of $L(g)$ is self-dual.", "$L(g)$ is strongly reversible in $\\mathrm {SU}(n)$ .", "Note that the Corollary REF is not true in general.", "This is demonstrated by the following example.", "Example 5.4 Consider $g = (A ,v )$ in $G = \\mathrm {SU}(4,\\mathbb {C})\\ltimes \\mathbb {C}^4$ , where $ A = \\textnormal {diag} ( e^{i\\theta } ,e^{-i\\theta } ,1,1 ) $ and $v = (0,0,1,1)$ , $ \\theta \\in (0,\\pi )$ .", "If possible suppose that $g$ is strongly reversible in $G$ .", "Then there exist $ h = (B ,u ) \\in \\mathrm {SU}(4,\\mathbb {C})\\ltimes \\mathbb {C}^4$ such that $ hgh^{-1} =g^{-1} $ and $ h^2 = I_4$ .", "Let $ B = [b_{p,q}]_{1\\leqslant p,q,\\leqslant 4} \\in SU(4,\\mathbb {C})$ be such that $ BAB^{-1} =A^{-1}$ .", "Then $ \\begin{pmatrix}e^{i\\theta } b_{1,1} & e^{-i\\theta } b_{1,2} & b_{1,3} & b_{1,4} \\\\e^{i\\theta } b_{2,1} &e^{-i\\theta } b_{2,2} & b_{2,3} & b_{2,4} \\\\e^{i\\theta } b_{3,1} & e^{-i\\theta } b_{3,2} & b_{3,3} & b_{3,4} \\\\e^{i\\theta } b_{4,1} & e^{-i\\theta } b_{4,2} & b_{4,3} & b_{4,4} \\\\\\end{pmatrix} = \\begin{pmatrix}e^{-i\\theta } b_{1,1} & e^{-i\\theta } b_{1,2} & e^{-i\\theta } b_{1,3} & e^{-i\\theta } b_{1,4} \\\\e^{i\\theta } b_{2,1} & e^{i\\theta } b_{2,2} &e^{i\\theta } b_{2,3} & e^{i\\theta } b_{2,4} \\\\b_{3,1} & b_{3,2} & b_{3,3} & b_{3,4} \\\\b_{4,1} & b_{4,2} & b_{4,3} & b_{4,4} \\\\\\end{pmatrix}.", "$ From this matrix equation we can say that matrix $B \\in \\mathrm {SU}(4, \\mathbb {C})$ has block diagonal form: $ B = \\begin{pmatrix}B_1 & O_ {2 \\times 2 }\\\\O_ {2 \\times 2 } & B_2 \\\\\\end{pmatrix} $ where $ B _1 = \\begin{pmatrix}0 & a\\\\b & 0\\\\\\end{pmatrix}$ for some a, b $\\in \\mathbb {C}\\setminus \\lbrace 0\\rbrace $ and for some $ B_2 \\in GL (2,\\mathbb {C})$ .", "Using $hgh^{-1} =g^{-1} $ and $ h^2 = {I} $ where ${I}$ is identity in $G$ , we have for all $x \\in \\mathbb {C}^4$ , $ BAB^{-1}(x) - BAB^{-1} (u) + B(v) + u = A^{-1}(x) - A^{-1}(v) $ $ B^2 (x) + B(u) + u = I_{4}(x) $ This implies, $ BAB^{-1} =A^{-1}, ( A^{-1} - I_4)(v-u) = - (B + I_4 )(v), \\hbox{ and} \\ B ^2 = I_4 , (B + I_4 )(u) =0.", "$ Using the above equations with  (REF ) and noting that $v=(0,0,1,1)$ we obtain $B_2=-I_2$ .", "Now, $ B ^2 = I_4 $ implies $ab=1$ .", "Thus we have $B = \\begin{pmatrix}0 & a &0 & 0 \\\\b & 0 &0 & 0 \\\\0& 0 & -1 & 0 \\\\0 & 0 &0 & -1\\end{pmatrix}, \\hbox{ where } a, b \\in \\mathbb {C}\\hbox{ such that } ab = 1.$ Thus $ \\det (B ) = -1$ .", "This contradicts that $B \\in \\mathrm {SU}(n,\\mathbb {C})$ .", "Hence $g$ is not strongly reversible in $\\mathrm {SU}(4) \\ltimes \\mathbb {C}^4$ .", "The following theorem generalises the above example to classify, up to conjugacy, all such elements $ g = (A ,v ) \\in \\mathrm {SU}(n,\\mathbb {C})\\ltimes \\mathbb {C}^n$ which are not strongly reversible , although L(g) is strongly reversible in $ SU(n,\\mathbb {C}) $ .", "Before that we note the following lemma.", "Lemma 5.5 Let A $ \\in \\mathrm {SU}(2n,\\mathbb {C}) $ have self-dual characteristic polynomial that has no eigenvalue equal to 1 or $-1$ .", "Let $B$ in $\\mathrm {GL}(2n, \\mathbb {C})$ be such that $BAB^{-1} = A^{-1}$ and $B^2 = I_{2n}$ .", "Then $\\det (B ) = (-1) ^n $ .", "First consider $n = 1$ .", "Then upto conjugacy in $ \\mathrm {SU}(2,\\mathbb {C}) $ we can write A as : $A =\\begin{pmatrix} e^ {i\\theta } & 0 \\\\ 0 & e^ {- i\\theta } \\end{pmatrix}$ where $ \\theta \\in (0,\\pi )$ .", "Let $B = [b_{p,q}]_{1\\leqslant p,q,\\leqslant 2} \\in \\mathrm {GL}(2, \\mathbb {C})$ be such that $BAB^{-1} = A^{-1}$ and $B^2 = I_{2}$ .", "Now $ BAB^{-1} = A^{-1}$ implies $ \\begin{pmatrix}e^{i\\theta } b_{1,1} & e^{-i\\theta } b_{1,2} \\\\e^{i\\theta }b_{2,1} &e^{-i\\theta } b_{2,2} \\\\\\end{pmatrix}\\ = \\begin{pmatrix}e^{-i\\theta } b_{1,1} & e^{-i\\theta } b_{1,2} \\\\e^{i\\theta }b_{2,1} & e^{i\\theta } b_{2,2}\\end{pmatrix}.$ Thus $b_{1,1} = b_{2,2} = 0 $ and $ b_{1,2}$ and $b_{2,1} $ non-zero complex numbers.", "Now $B^2=I_2$ implies $ b_{1,2} b_{2,1} = 1$ .", "Thus, $\\det (B) = - b_{1,2} b_{2,1} = -1 = (-1)^{1}$ .", "Now the lemma follows by induction.", "The following well-known result follows from the above lemma.", "Corollary 5.6 Suppose $ A \\in \\mathrm {SU}(n,\\mathbb {C})$ has self reciprocal characteristic polynomial such that no eigenvalue of $A$ is equal to 1 or $-1$ .", "If $ n \\equiv 2 \\mod {4}$ then $A$ is not be strongly reversible in $\\mathrm {SU}(n,\\mathbb {C}) $ ." ], [ "Proof of Theorem ", "If possible assume that $g$ is strongly reversible in $G=\\mathrm {SU}(n, \\mathbb {C}) \\ltimes \\mathbb {C}^n$ .", "Then there exist $ h = (B ,u ) \\in G $ such that $ hgh^{-1} =g^{-1} $ and $ h^2 = {I} $ where ${I}$ is identity in $G$ .", "From this we have  (REF )– (REF ) as in the previous example.", "Further, using $BAB^{-1}= A^{-1}$ , we have the following form for $B$ for some $ B _1 \\in {\\mathrm {M}}(n-t,\\mathbb {C})$ and $ B_2 \\in {\\mathrm {M}}(t,\\mathbb {C})$ : $B = \\begin{pmatrix}B_1 & O_ {(n-t) \\times t }\\\\O_ {t \\times (n-t) } & B_2 \\\\\\end{pmatrix}.", "$ Using the above equations, we get $B_2 = -I_{t}$ .", "So , $B _1 \\in {\\mathrm {M}}(2r,\\mathbb {C})$ such that $ B_1 A_1 B_1^{-1} = A_1 ^{-1}$ , $B_{1} ^2 = I_ {n-t }$ , where $ {A_1} = \\begin{pmatrix}e^{i \\theta _1} & 0 & \\ldots & 0 & 0 \\\\0 & e^{-i \\theta _1}& \\ldots & 0 & 0 \\\\ & & \\ddots & &\\\\ 0 & 0 & \\ldots & e^{i \\theta _r}& 0 \\\\ 0 & 0 & \\ldots & 0 & e^{-i \\theta _r} \\end{pmatrix}.$ Now note that $B_1A_1B_1^{-1} = A_1^{-1}$ and $B_1^2 = I_{2r}$ .", "From Lemma REF we have $ \\det (B_1 ) = (-1)^r$ .", "Hence $\\det ( B ) = \\det (B_1) \\det ( -I_t) = (-1)^{r+t}.$ It follows that $\\det B=-1$ if and only if either $n \\equiv 0 \\mod {2}$ , $r \\equiv 1 \\mod {2}$ , or, $n \\equiv 1 \\mod {2}$ , $r \\equiv 0 \\mod {2}$ .", "If $v_{\\ell }=0$ for some $\\ell \\in \\lbrace r+1, \\ldots , n\\rbrace $ , we can choose $B$ such that the diagonal entry corresponding to $v_{\\ell }$ is 1, and that would make $\\det B=1$ .", "This proves the theorem.", "Acknowledgement The authors thank Ian Short, Sushil Bhunia and Chandan Maity for their comments on a first draft of this paper.", "Tejbir acknowledges full support from the CSIR SRF grant, file No : 09/947(0113)/2019-EMR-I, during the course of this work.", "Gongopadhyay acknowledges SERB MATRICS grant MTR/2017/000355." ] ]
2105.11707
[ [ "Algebraic structures in the family of non-Lebesgue measurable sets" ], [ "Abstract In the additive topological group $(\\mathbb{R},+)$ of real numbers, we construct families of sets for which elements are not measurable in the Lebesgue sense.", "The constructed families of sets have algebraic structures of being semigroups (i.e.", "closed under finite unions of sets), and they are invariant under the action of the group $\\Phi(\\mathbb{R})$ of all translations of $\\mathbb{R}$ onto itself.", "Those semigroups are constructed by using Vitali selectors and Bernstein sets on $\\mathbb{R}$ taken simultaneously differently to what exists in the literature." ], [ "1 1em" ], [ "1.1 1em colorlinks, linkcolor=blue, citecolor=magenta, urlcolor=blue Algebraic structures in the family of non-Lebesgue measurable sets Venuste NYAGAHAKWAGratien HAGUMA Department of Mathematics, University of RWANDA, PO.", "Box 3900, Kigali, [email protected] Department of Mathematics, University of RWANDA, PO.", "Box 3900, Kigali, [email protected] Primary 28A05; Secondary 28C10 Lebesgue measurability, Bernstein sets, Vitali selectors, Non-Lebesgue measurable sets, Baire property In the additive topological group $(\\mathbb {R},+)$ of real numbers, we construct families of sets for which elements are not measurable in the Lebesgue sense.", "The constructed families of sets have algebraic structures of being semigroups (i.e.", "closed under finite unions of sets), and they are invariant under the action of the group $\\Phi (\\mathbb {R})$ of all translations of $\\mathbb {R}$ onto itself.", "Those semigroups are constructed by using Vitali selectors and Bernstein sets on $\\mathbb {R}$ taken simultaneously differently to what exists in the literature." ], [ "Introduction", "Let $(\\mathbb {R},+)$ be the additive group of real numbers endowed with the Euclidean topology, and let $\\mathcal {P}(\\mathbb {R})$ be the collection of all subsets of $\\mathbb {R}$ .", "It is well-known that there exist subsets of $\\mathbb {R}$ which are not measurable in the Lebesgue sense [8], [2]; for instance, Vitali selectors of $\\mathbb {R}$ , Bernstein sets of $\\mathbb {R}$ , as well as non-Lebesgue measurable subsets of $\\mathbb {R}$ associated with Hamel basis.", "Accordingly, the family $\\mathcal {P}(\\mathbb {R})$ can be decomposed into two disjoint non-empty families; namely, the family $\\mathcal {L}(\\mathbb {R})$ of all Lebesgue measurable subsets of $\\mathbb {R}$ , and the family $\\mathcal {L}^c(\\mathbb {R})=\\mathcal {P}(\\mathbb {R})\\setminus \\mathcal {L}(\\mathbb {R})$ of all non-Lebesgue measurable subsets of $\\mathbb {R}$ .", "The algebraic structure; in the set-theoretic point of view, of the family $\\mathcal {L}(\\mathbb {R})$ is well-known.", "The family $\\mathcal {L}(\\mathbb {R})$ is a $\\sigma $ -algebra of sets on $\\mathbb {R}$ , and hence it is closed under all basic set-operations.", "It contains the collection $\\mathcal {B}_O(\\mathbb {R})$ of all Borel subsets of $\\mathbb {R}$ , as well as, the collection $\\mathcal {N}_0(\\mathbb {R})$ of all subsets of $\\mathbb {R}$ having the Lebesgue measure zero.", "In addition, the family $\\mathcal {L}(\\mathbb {R})$ is invariant under the action of the group $\\Phi (\\mathbb {R})$ of all translations of $\\mathbb {R}$ onto itself; it means that if $A\\subseteq \\mathbb {R}$ is such that $A\\in \\mathcal {L}(\\mathbb {R})$ and $h\\in \\Phi (\\mathbb {R})$ then $h(A)\\in \\mathcal {L}(\\mathbb {R})$ .", "On the other hand, the family $\\mathcal {L}^c(\\mathbb {R})$ does not have a well-defined structure in the set-theoretic point of view.", "Indeed, the union (resp.", "intersection, difference, and symmetric difference) of two elements in the family $\\mathcal {L}^c(\\mathbb {R})$ can be inside or outside of $\\mathcal {L}^c(\\mathbb {R})$ .", "However, like $\\mathcal {L}(\\mathbb {R})$ , the family $\\mathcal {L}^c(\\mathbb {R})$ is invariant under the action of the group $\\Phi (\\mathbb {R})$ .", "Given a countable dense subgroup $Q$ of $(\\mathbb {R},+)$ , let $\\mathcal {V}(Q)$ be the collection of all Vitali selectors related to $Q$ .", "The construction of Vitali selectors is discussed in Subsection REF and more facts about them can be found in [2].", "The following question constitutes a motivating key of this paper.", "Question 2.1 ([14]) Could we find in $\\mathcal {L}^c(\\mathbb {R})$ subfamilies of $\\mathcal {P}(\\mathbb {R})$ containing $\\mathcal {V}(Q)$ and have some algebraic structures in the set-theoretic point of view?", "In [11], it was shown that each element of the family $\\mathcal {V}_1(Q)=\\lbrace \\bigcup _{i=1}^n V_i: V_i\\in \\mathcal {V}(Q), n\\in \\mathbb {N}\\rbrace $ of all finite unions of Vitali selectors related to $Q$ , is a semigroup of sets with respect to the operation of union of sets, and that it is invariant under the action of the group $\\Phi (\\mathbb {R})$ such that $\\mathcal {V}(Q)\\subsetneq \\mathcal {V}_1(Q)\\subsetneq \\mathcal {L}^c(\\mathbb {R})$ .", "In addition, the family $\\mathcal {V}_2(Q)=\\mathcal {V}_1(Q)*\\mathcal {N}_0:=\\lbrace (U\\setminus M)\\cup N: U\\in \\mathcal {V}_1(Q), M, N\\in \\mathcal {N}_0\\rbrace $ was shown to be a semigroup of sets, invariant under the action of the group $\\Phi (\\mathbb {R})$ and that $\\mathcal {V}_1(Q)\\subsetneq \\mathcal {V}_2(Q)\\subsetneq \\mathcal {L}^c(\\mathbb {R})$ .", "A variety of semigroups satisfying these properties were constructed in [12] and [13] by using the collection $\\mathcal {V}_1(Q)$ and different ideal of sets on $\\mathbb {R}$ .", "More general statements related to Vitali selectors generalizing those facts are proved in [16], where in particular, it is shown that each topological group isomorphism of $(\\mathbb {R},+)$ onto itself preserves the non-Lebesgue measurability of Vitali selectors of $\\mathbb {R}$ .", "Let $\\mathcal {V}$ be the family of all Vitali selectors of $\\mathbb {R}$ , and let $\\mathcal {S}(\\mathcal {V})$ be the collection of all finite unions of elements of $\\mathcal {V}$ .", "In [16], it is shown that the family $\\mathcal {S}(\\mathcal {V})*\\mathcal {N}_0:=\\lbrace (U\\setminus M)\\cup N: U\\in \\mathcal {S}(\\mathcal {V}), M, N\\in \\mathcal {N}_0\\rbrace $ is an abelian semigroup of sets for which elements are not measurable in the Lebesgue sense, and that $\\mathcal {V}_2(Q)\\subsetneq \\mathcal {S}(\\mathcal {V})$ for every countable dense subgroup $Q$ of $(\\mathbb {R},+)$ .", "We note that the non-Lebesgue measurability of elements of the family $\\mathcal {S}(\\mathcal {V})*\\mathcal {N}_0$ was also proved in [15], together with other interesting facts about the semigroups generated by Vitali selectors and Bernstein sets of $\\mathbb {R}$ .", "In addition, it is proved in [16] that the abelian semigroup $\\mathcal {S}(\\mathcal {V})*\\mathcal {N}_0$ has an algebraic structure of being invariant under the action of the group $\\Pi (\\mathbb {R})$ of all affine transformations of $\\mathbb {R}$ onto itself.", "The families of sets that are not measurable in the Lebesgue sense that are discussed in the literature are constructed by using mostly Vitali selectors of $\\mathbb {R}$ .", "In this paper, we consider a more general setting, by looking away for extending Question REF .", "Accordingly, we consider a Bernstein set $B$ which has an algebraic structure of being a subgroup of $(\\mathbb {R},+)$ .", "Such a set exists as it is shown in [2].", "Furthermore, we consider the collection $\\mathbb {R}/B$ of all cosets (translates) of $B$ in $(\\mathbb {R},+)$ , that we denote by $\\mathcal {B}$ for simplicity.", "Since the family of Bernstein sets is preserved by homeomorphisms [2], it follows that each element of $\\mathcal {B}$ is also a Bernstein set.", "Question 2.2 Could we find in $\\mathcal {L}^c(\\mathbb {R})$ subfamilies of $\\mathcal {P}(\\mathbb {R})$ containing $\\mathcal {B}$ and have some algebraic structures in the set-theoretic point of view?", "Through the paper, different families of sets answering Question REF are constructed.", "In particular, the family $\\mathcal {S}(\\mathcal {B})=\\lbrace \\bigcup _{i=1}^n B_i: B_i\\in \\mathcal {B}, n\\in \\mathbb {N}\\rbrace $ of all finite unions of elements of $\\mathcal {B}$ , and its extension $\\mathcal {S}(\\mathcal {B})*\\mathcal {N}_0$ by the $\\mathcal {N}_0$ , constitute an answer to Question REF .", "Consider a new family of sets $\\mathcal {V}\\vee \\mathcal {B}:=\\lbrace V\\cup B: V\\in \\mathcal {V} \\text{ and } B\\in \\mathcal {B}\\rbrace $ .", "Note that $\\mathcal {V}\\subseteq \\mathcal {V}\\vee \\mathcal {B}_{\\emptyset }$ and $\\mathcal {B}\\subseteq \\mathcal {V}_{\\emptyset }\\vee \\mathcal {B}$ , where $\\mathcal {V}_{\\emptyset }=\\mathcal {V}\\cup \\lbrace \\emptyset \\rbrace $ and $\\mathcal {B}_{\\emptyset }=\\mathcal {B}\\cup \\lbrace \\emptyset \\rbrace $ .", "The main aim of the paper is to construct families of sets that constitute answers to the following question, which generalizes in some sense Question REF and Question REF .", "Question 2.3 Could we find in $\\mathcal {L}^{c}(\\mathbb {R})$ subfamilies of $\\mathcal {P}(\\mathbb {R})$ containing the family $\\mathcal {V}\\vee \\mathcal {B}$ and have some algebraic structures in the set-theoretic point of view?", "In this paper, different families of sets having the algebraic structure of being semigroups with respect to the operation of the union of sets, are constructed through the use of $\\mathcal {V}$ and $\\mathcal {B}$ .", "The constructed semigroups extend the existing ones, and in particular, some contains the family $\\mathcal {S}(\\mathcal {V})*\\mathcal {N}_0$ .", "The constructed families also consist of sets that are not measurable in the Lebesgue sense, and they are invariant under the action of the group $\\Phi (\\mathbb {R})$ .", "In particular, it is proved that, for any Bernstein sets $B_1$ and $B_2$ having the algebraic structures of being subgroups of $(\\mathbb {R},+)$ , the family $[\\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)\\vee \\mathcal {S}(\\mathcal {V})]*\\mathcal {N}_0:=\\lbrace [(U_1\\cup U_2\\cup U_3)\\setminus N]\\cup M: U_1\\in \\mathcal {S}(\\mathcal {B}_1), U_2\\in \\mathcal {S}(\\mathcal {B}_2), U_3\\in \\mathcal {S}(\\mathcal {V}), N,M\\in \\mathcal {N}_0\\rbrace $ is an abelian semigroup of sets for which elements are not measurable in the Lebesgue sense, and that it is invariant under the action of the group $\\Phi (\\mathbb {R})$ .", "We point out that $[\\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)\\vee \\mathcal {S}(\\mathcal {V})]*\\mathcal {N}_0=(\\mathcal {S}(\\mathcal {B}_1)*\\mathcal {N}_0)\\vee (\\mathcal {S}(\\mathcal {B}_2)*\\mathcal {N}_0)\\vee (\\mathcal {S}(\\mathcal {V})*\\mathcal {N}_0)$ and that $\\mathcal {S}(\\mathcal {V})*\\mathcal {N}_0 \\subseteq [\\mathcal {S}_{\\emptyset }(\\mathcal {B}_1)\\vee \\mathcal {S}_{\\emptyset }(\\mathcal {B}_2)\\vee \\mathcal {S}(\\mathcal {V})]*\\mathcal {N}_0$ , where $\\mathcal {S}_{\\emptyset }(\\mathcal {B}_i)=\\mathcal {S}(\\mathcal {B}_i)\\cup \\lbrace \\emptyset \\rbrace $ for $i=1,2$ .", "The same result holds also when $\\mathcal {S}(\\mathcal {V})$ is replaced by $\\mathcal {V}_1(Q)$ for any countable dense subgroup of $(\\mathbb {R},+)$ .", "This paper uses standard notation and facts from Set Theory and Real Analysis, and it is organized as follows: After an introductory section, where motivating ideas and the problem under investigation are developed, the second section deals with the theory of semigroups, the theory of Vitali selectors, and different facts about Bernstein sets.", "The third section concerns the semigroups of non-Lebesgue measurable sets constructed by using Bernstein sets.", "The fourth section complements the part about Vitali selections developed in the preliminary section, and it is about semigroups of non-Lebesgue measurable sets generated by Vitali selectors.", "The fifth section is devoted to the semigroups of non-Lebesgue measurable sets that are constructed by using Vitali selectors and Bernstein sets simultaneously." ], [ "Theory of semigroups and ideals of sets", "Let $\\mathcal {S}$ be a non-empty set.", "The set $\\mathcal {S}$ is called a semigroup of sets if there is a binary operation $\\ast : \\mathcal {S}\\times \\mathcal {S}\\longrightarrow \\mathcal {S}$ for which the associativity law is satisfied; i.e.", "$(x\\ast y)\\ast z =x\\ast (y\\ast z)$ for all $x,y,z\\in \\mathcal {S}$ .", "The semigroup $\\mathcal {S}$ is said to be abelian if $x\\ast y=y\\ast x$ for all $x,y\\in \\mathcal {S}$ .", "For a non-empty set $X$ , let $\\mathcal {P}(X$ be the collection of all subsets of $X$ .", "Consider a non-empty family of sets $\\mathcal {S}\\subseteq \\mathcal {P}(X)$ such that for each pair of elements $A,B\\in \\mathcal {S}$ we have $A\\cup B\\in \\mathcal {S}$ .", "Since the union of sets is both commutative and associative, such a family of sets is an abelian semigroup with respect to the operation of union of sets.", "Definition 3.1 A non-empty collection of sets $\\mathcal {S}\\subseteq \\mathcal {P}(X)$ is called a semigroup of sets on $X$ if it is closed under finite unions.", "If $\\mathcal {S}$ is closed under countable unions then it is said to be a $\\sigma $ -semigroup of sets on $X$ .", "It is evident that if $\\mathcal {S}$ is a semigroup of sets on $X$ with respect to the operation of union of sets then the collection $\\lbrace X\\setminus S: S\\in \\mathcal {S}\\rbrace $ of all complements of elements of $\\mathcal {S}$ in $X$ , is closed under finite intersection of sets, and thus, it is a semigroup of sets with respect to the set-theoretic operation of intersection of sets on $X$ .", "Recall [7] that a non-empty collection $\\mathcal {I}\\subseteq \\mathcal {P}(X)$ of sets is called an ideal of sets on $X$ if it satisfies the following conditions: If $A\\in \\mathcal {I}$ and $B\\in \\mathcal {I}$ then $A\\cup B\\in \\mathcal {I}$ .", "If $A\\in \\mathcal {I}$ and $B\\subseteq A$ then $B\\in \\mathcal {I}$ .", "If the ideal of sets $\\mathcal {I}$ is closed under countable unions of sets then it is called a $\\sigma $ -ideal of sets on $X$ .", "Clearly, each ideal of sets is a semigroup of sets which is closed under taking subsets.", "Example 3.2 ([12]) If $\\mathcal {A}\\subseteq \\mathcal {P}(X)$ is a non-empty family of sets, consider the collection $\\mathcal {S}({\\mathcal {A}})=\\left\\lbrace \\bigcup _{i=1}^nA_i: A_i\\in \\mathcal {A}, n\\in \\mathbb {N}\\right\\rbrace $ of all finite unions of elements of $\\mathcal {A}$ , and consider the collection $\\mathcal {I}({\\mathcal {A}})=\\lbrace B\\in \\mathcal {P}(X): \\text{there is }A\\in \\mathcal {S}({\\mathcal {A}}) \\text{ such that }B\\subseteq A\\rbrace $ .", "It is clear that family $\\mathcal {S}({\\mathcal {A}})$ is a semigroup of sets, while the collection $\\mathcal {I}({\\mathcal {A}})$ is an ideal of sets on $X$ .", "The family $\\mathcal {S}({\\mathcal {A}})$ is called the semigroup of sets generated by $\\mathcal {A}$ while $\\mathcal {I}({\\mathcal {A}})$ is called the ideal of sets generated by $\\mathcal {A}$.", "Evidently the inclusion $\\mathcal {S}(\\mathcal {A})\\subseteq \\mathcal {I}({\\mathcal {A}})$ holds, and if $\\mathcal {A}$ is a semigroup of sets then $\\mathcal {S}(\\mathcal {A})=\\mathcal {A}$ .", "Example 3.3 Let $X$ be the set $\\mathbb {R}$ of real numbers.", "The family $\\mathcal {I}_f$ of all finite subsets of $X$ is an ideal of sets.", "The family $\\mathcal {I}_c$ of all countable subsets of $X$ is a $\\sigma $ -ideal of sets.", "The family $\\mathcal {I}_b$ of all bounded subsets of $X$ is an ideal of sets on $X$ .", "It is clear all these three families are semigroups of sets on $X$ .", "Remark 3.4 If $\\mathcal {A}$ is a family of sets on $X$ and $Y\\subseteq X$ then collection $\\mathcal {A}\\cup \\lbrace Y\\rbrace $ is denoted by $\\mathcal {A}_Y$ , and the collection $\\mathcal {S}(\\mathcal {A})\\cup \\lbrace Y\\rbrace $ is denoted by $\\mathcal {S}_Y(\\mathcal {A})$ .", "It is clear that if $Y\\in \\mathcal {A}$ then $\\mathcal {A}_Y=\\mathcal {A}$ and $\\mathcal {S}_Y(\\mathcal {A})=\\mathcal {S}(\\mathcal {A})$ .", "If $Y=\\emptyset $ or $Y=X$ then $\\mathcal {S}_Y(\\mathcal {A})$ is a semigroup of sets.", "Let $\\mathcal {A}$ and $\\mathcal {B}$ be any families of subsets of $X$ .", "Define a new collection of sets on $X$ by setting $\\mathcal {A}* \\mathcal {B}=\\lbrace (A\\setminus B_1)\\cup B_2: A\\in \\mathcal {A}, B_1\\in \\mathcal {B}, B_2\\in \\mathcal {B}\\rbrace $ .", "It is clear that if $\\mathcal {A}, \\mathcal {B}, \\mathcal {C}$ and $\\mathcal {D}$ are families of sets on $X$ such that $\\mathcal {A}\\subseteq \\mathcal {B}, \\mathcal {C}\\subseteq \\mathcal {D}$ then $\\mathcal {A}* \\mathcal {C} \\subseteq \\mathcal {B}* \\mathcal {D}$ .", "For any family of sets $\\mathcal {A}$ on $X$ , the inclusion $\\mathcal {A}\\subseteq \\mathcal {A}* \\mathcal {A}$ always hold.", "If $\\mathcal {S}_1$ and $\\mathcal {S}_2$ are semigroups of sets then the family $\\mathcal {S}_1*\\mathcal {S}_2$ does not need to be a semigroup of sets, unless one of them is an ideal of sets.", "This fact is shown in the following statement presenting a way of extending a given semigroup by the use of ideals of sets.", "Proposition 3.5 ([12]) Let $\\mathcal {S}$ be a semigroup of sets on $X$ and let $\\mathcal {I}$ be an ideal of sets on $X$ .", "Then the families $\\mathcal {I}* \\mathcal {S}$ and $\\mathcal {S}* \\mathcal {I}$ are semigroups of sets on $X$ such that $\\mathcal {S}\\subseteq \\mathcal {I}* \\mathcal {S}\\subseteq \\mathcal {S}* \\mathcal {I}$ .", "Moreover, $\\mathcal {I}* (\\mathcal {I}* \\mathcal {S})=\\mathcal {I}* \\mathcal {S}$ and $(\\mathcal {S}* \\mathcal {I})* \\mathcal {I}=\\mathcal {S}* \\mathcal {I}$ .", "Let $\\mathcal {A}$ and $\\mathcal {B}$ be any families of subsets of $X$ .", "Define a new family of sets on $X$ by setting $\\mathcal {A}\\vee \\mathcal {B}=\\lbrace A\\cup B: A\\in \\mathcal {A}, B\\in \\mathcal {B}\\rbrace $ .", "It is clear that if $\\mathcal {A}, \\mathcal {B}, \\mathcal {C}$ and $\\mathcal {D}$ are families of sets on $X$ such that $\\mathcal {A}\\subseteq \\mathcal {B}, \\mathcal {C}\\subseteq \\mathcal {D}$ then $\\mathcal {A}\\vee \\mathcal {C} \\subseteq \\mathcal {B}\\vee \\mathcal {D}$ .", "For any family of sets $\\mathcal {A}$ on $X$ , the inclusion $\\mathcal {A}\\subseteq \\mathcal {A}\\vee \\mathcal {A}$ always hold.", "If $\\mathcal {A}$ is a semigroup of sets on $X$ then $\\mathcal {A}\\vee \\mathcal {A}=\\mathcal {A}$ .", "The inclusions $\\mathcal {A}\\subseteq \\mathcal {A}\\vee \\mathcal {B}$ and $\\mathcal {B}\\subseteq \\mathcal {A}\\vee \\mathcal {B}$ do not need to hold for any families $\\mathcal {A}$ and $\\mathcal {B}$ with or without the assumption of being semigroups.", "However, the inclusions $\\mathcal {A}\\subseteq \\mathcal {A}\\vee \\mathcal {B}_{\\emptyset }, \\mathcal {B}\\subseteq \\mathcal {A}_{\\emptyset }\\vee \\mathcal {B}$ and $\\mathcal {A}\\subseteq \\mathcal {A}\\vee \\mathcal {A}_{\\emptyset }$ always hold.", "If $\\mathcal {S}_1$ and $\\mathcal {S}_2$ are semigroups of sets then the union $\\mathcal {S}_1\\cup \\mathcal {S}_2$ does not need to be a semigroup of sets, however the following lemma holds.", "Lemma 3.6 If $\\mathcal {S}_1$ and $\\mathcal {S}_2$ are semigroups of sets on $X$ then the family $\\mathcal {S}_1\\vee \\mathcal {S}_2$ is also a semigroup of sets on $X$ .", "We observe that for any sets $A$ and $B$ we have $A\\cup B=(A\\setminus B)\\cup B=(B\\setminus A)\\cup A$ , then we have $\\mathcal {A}\\vee \\mathcal {B}\\subseteq \\mathcal {A}*\\mathcal {B}$ and $\\mathcal {A}\\vee \\mathcal {B}\\subseteq \\mathcal {B}*\\mathcal {A}$ for any families of sets $\\mathcal {A}$ and $\\mathcal {B}$ on $X$ .", "In addition, the equality $\\mathcal {S}(\\mathcal {A})\\cup \\mathcal {S}(\\mathcal {B})=\\mathcal {S}(\\mathcal {A} \\cup \\mathcal {B})$ does not need to hold.", "Example 3.7 On the set $X=\\lbrace a,b,c,d\\rbrace $ , let $A=\\lbrace a,b\\rbrace , B=\\lbrace b,c\\rbrace $ and $D=\\lbrace c,d\\rbrace $ , and consider the families $\\mathcal {A}=\\lbrace A\\rbrace $ and $\\mathcal {B}=\\lbrace B,D\\rbrace $ .", "Note that $\\mathcal {A}\\cup \\mathcal {B}=\\lbrace A,B,D\\rbrace $ , $\\mathcal {S}(\\mathcal {A})=\\mathcal {A}$ and $\\mathcal {S}(\\mathcal {B})=\\lbrace B, D, B\\cup D\\rbrace $ .", "It is clear that $\\mathcal {S}(\\mathcal {A})\\cup \\mathcal {S}(\\mathcal {B})=\\lbrace A,B,D,B\\cup D\\rbrace $ while $\\mathcal {S}(\\mathcal {A}\\cup \\mathcal {B})=\\lbrace A,B,D,A\\cup B, A\\cup D, B\\cup D, A\\cup B\\cup D\\rbrace $ .", "Since $\\mathcal {A}\\vee \\mathcal {B}=\\lbrace A\\cup B, A\\cup D\\rbrace $ , we further observe that $\\mathcal {S}(\\mathcal {A})\\vee \\mathcal {S}(\\mathcal {B})=\\mathcal {S}(\\mathcal {A}\\vee \\mathcal {B})=\\lbrace A\\cup B, A\\cup D, A\\cup B\\cup D\\rbrace $ , and this observation motivates the following statement.", "Lemma 3.8 If $\\mathcal {A}$ and $\\mathcal {B}$ are non-empty families of sets on $X$ , then the equality $\\mathcal {S}(\\mathcal {A \\vee \\mathcal {B}})= \\mathcal {S}(\\mathcal {A}) \\vee \\mathcal {S}(\\mathcal {B})$ always holds.", "Assume that $Y\\in \\mathcal {S}(\\mathcal {A}\\vee \\mathcal {B})$ .", "Then $Y=\\bigcup _{i=1}^n Y_i$ where $Y_i\\in \\mathcal {A}\\vee \\mathcal {B}$ , i.e.", "$Y_i=A_i\\cup B_i$ with $A_i\\in \\mathcal {A}$ , $B_i\\in \\mathcal {B}$ and $n\\in \\mathbb {N}$ .", "Hence $Y=\\bigcup _{i=1}^n \\left(A_i\\cup B_i\\right)=\\left( \\bigcup _{i=1}^n A_i\\right) \\cup \\left( \\bigcup _{i=1}^n B_i\\right)$ .", "Put $A= \\bigcup _{i=1}^n A_i$ and $B=\\bigcup _{i=1}^n B_i$ .", "Clearly, $A\\in \\mathcal {S}(\\mathcal {A})$ and $B\\in \\mathcal {S}(\\mathcal {B})$ .", "Hence $Y=A\\cup B\\in \\mathcal {S}(\\mathcal {A})\\vee \\mathcal {S}(\\mathcal {B})$ .", "Assume that $Y\\in \\mathcal {S}(\\mathcal {A})\\vee \\mathcal {S}(\\mathcal {B})$ .", "Then $Y=A\\cup B$ where $A\\in \\mathcal {S}(\\mathcal {A})$ and $B\\in \\mathcal {S}(\\mathcal {B})$ , i.e.", "that $A=\\bigcup _{i=1}^n A_i$ and $B=\\bigcup _{i=1}^m B_i$ where $A_i\\in \\mathcal {A}$ and $B_i\\in \\mathcal {B}_i$ for some $n$ and $m$ in $\\mathbb {N}$ .", "If $n=m$ then $Y=\\bigcup _{i=1}^n\\left(A_i\\cup B_i\\right)$ and hence $Y\\in \\mathcal {S}(\\mathcal {A}\\vee \\mathcal {B})$ .", "Assume that $n\\ne m$ .", "Without loosing of generality, we may assume that $n<m$ .", "Then $Y=\\left[\\bigcup _{i=1}^n \\left(A_i\\cup B_i\\right)\\right] \\cup \\left( \\bigcup _{i=n+1}^m B_i\\right)$ .", "For $i=n+1,n+2,\\cdots , m$ , put $A_i=A_k$ , where $k$ is some fixed integer in the set $\\lbrace 1,2,\\cdots , n\\rbrace $ .", "It follows that $Y=\\left[\\bigcup _{i=1}^n \\left(A_i\\cup B_i\\right)\\right]\\cup \\left[\\bigcup _{i=n+1}^m(A_i\\cup B_i)\\right]=\\bigcup _{i=1}^m \\left(A_i\\cup B_i\\right)$ .", "Since $A_i\\cup B_i\\in \\mathcal {A}\\vee \\mathcal {B}$ for $i=1,2, \\cdots ,m$ , it follows that $Y\\in \\mathcal {S}(\\mathcal {A}\\vee \\mathcal {B})$ .", "The following proposition is a generalization of Lemma REF for any finite collection of families of sets.", "Proposition 3.9 Let $\\mathcal {A}_i$ be a non-empty family of sets on $X$ , where $i=1,2, \\cdots , n$ , for some $n\\in \\mathbb {N}$ .", "Then the equality $\\mathcal {S}\\left( \\bigvee _{i=1}^n \\mathcal {A}_i\\right)= \\bigvee _{i=1}^n \\mathcal {S}(\\mathcal {A}_i)$ always holds.", "It follows from Proposition REF that the families $(\\mathcal {S}_1\\vee \\mathcal {S}_2)*\\mathcal {I}$ and $\\mathcal {I}*(\\mathcal {S}_1\\vee \\mathcal {S}_2)$ are semigroups of sets for any ideal of sets $\\mathcal {I}$ on $X$ .", "However, no one of the inclusions $\\mathcal {S}_1*\\mathcal {I}\\subseteq (\\mathcal {S}_1\\vee \\mathcal {S}_2)*\\mathcal {I}$ , $\\mathcal {S}_2*\\mathcal {I}\\subseteq (\\mathcal {S}_1\\vee \\mathcal {S}_2)*\\mathcal {I}$ , $\\mathcal {I}*\\mathcal {S}_1\\subseteq \\mathcal {I}*(\\mathcal {S}_1\\vee \\mathcal {S}_2)$ and $\\mathcal {I}*\\mathcal {S}_2\\subseteq \\mathcal {I}*(\\mathcal {S}_1\\vee \\mathcal {S}_2)$ needs to hold.", "Example 3.10 Let $X$ be a non-empty set such $\\operatorname{\\textrm {Card}}(X)\\ge 2$ , and let $A$ be a non-empty proper subset of $X$ .", "Let $B=X\\setminus A$ .", "Consider $\\mathcal {S}_1=\\lbrace A, X\\rbrace $ and $\\mathcal {S}_2=\\lbrace B, X\\rbrace $ .", "Consider the ideals of $\\mathcal {I}=\\mathcal {P}(A)$ and $\\mathcal {J}=\\mathcal {P}(B)$ .", "It is clear that $\\mathcal {S}_1\\vee \\mathcal {S}_2=\\lbrace X\\rbrace $ , and $\\emptyset $ and $A$ cannot be elements of $ (\\mathcal {S}_1\\vee \\mathcal {S}_2)*\\mathcal {I}$ , but $\\emptyset $ and $A$ are elements of $\\mathcal {S}_1*\\mathcal {I}$ .", "Similarly, the collection $\\mathcal {S}_2*\\mathcal {J}$ contains the elements $\\emptyset $ and $B$ , but the family $ (\\mathcal {S}_1\\vee \\mathcal {S}_2)*\\mathcal {J}$ cannot contain $\\emptyset $ and $B$ .", "Hence $\\mathcal {S}_1*\\mathcal {I}\\nsubseteq (\\mathcal {S}_1\\vee \\mathcal {S}_2)*\\mathcal {I}$ , and $\\mathcal {S}_2*\\mathcal {J}\\nsubseteq (\\mathcal {S}_1\\vee \\mathcal {S}_2)*\\mathcal {J}$ .", "Note that $\\mathcal {I}*(\\mathcal {S}_1\\vee \\mathcal {S}_2)=\\lbrace X\\rbrace =\\mathcal {J}*(\\mathcal {S}_1\\vee \\mathcal {S}_2)$ .", "Observe that the collection $\\mathcal {I}*\\mathcal {S}_2$ contains the set $B$ and the collection $\\mathcal {J}*\\mathcal {S}_2$ contains the set $A$ .", "Hence $\\mathcal {I}*\\mathcal {S}_2\\nsubseteq \\mathcal {I}*(\\mathcal {S}_1\\vee \\mathcal {S}_2)$ and $\\mathcal {J}*\\mathcal {S}_2\\nsubseteq \\mathcal {J}*(\\mathcal {S}_1\\vee \\mathcal {S}_2)$ .", "Proposition 3.11 Let $\\mathcal {S}_1$ and $\\mathcal {S}_2$ be semigroups of sets on $X$ .", "If $\\mathcal {I}$ is an ideal of sets on $X$ then $(\\mathcal {S}_1\\vee \\mathcal {S}_2)*\\mathcal {I}=(\\mathcal {S}_1*\\mathcal {I})\\vee (\\mathcal {S}_2*\\mathcal {I})$ and $ \\mathcal {I}*(\\mathcal {S}_1\\vee \\mathcal {S}_2)=(\\mathcal {I}*\\mathcal {S}_1)\\vee (\\mathcal {I}*\\mathcal {S}_2)$ .", "Assume that $A\\in (\\mathcal {S}_1\\vee \\mathcal {S}_2)*\\mathcal {I}$ .", "Then $A=[(S_1\\cup S_2)\\setminus I]\\cup K $ where $S_1\\in \\mathcal {S}_1, S_2\\in \\mathcal {S}_2$ and $I, K\\in \\mathcal {I}$ .", "It is clear that $A=(S_1\\setminus I)\\cup (S_2\\setminus I)\\cup K=[(S_1\\setminus I)\\cup K]\\cup [(S_2\\setminus I)\\cup K]\\in (\\mathcal {S}_1*\\mathcal {I})\\vee (\\mathcal {S}_2*\\mathcal {I})$ .", "Assume that $A\\in (\\mathcal {S}_1*\\mathcal {I})\\vee (\\mathcal {S}_2*\\mathcal {I})$ .", "Then $A=[(S_1\\setminus N)\\cup L)\\cup ((S_2\\setminus P)\\cup R]$ , where $S_1\\in \\mathcal {S}_1, S_2\\in \\mathcal {S}_1$ and $N,L,P,R \\in \\mathcal {I}, i=1,2$ .", "It is clear that $A=[(S_1\\setminus N)\\cup L]\\cup [(S_2\\setminus P)\\cup R]=(S_1\\setminus N)\\cup (S_2\\setminus P)\\cup (L\\cup R)$ .", "Putting $I=L\\cup R\\in \\mathcal {I}$ it follows that $A=\\left[(S_1\\cap N^{c})\\cup (S_2\\cap P^{c})\\right]^{cc}\\cup I= \\left[(S_1\\cap N^{c})^{c}\\cap (S_2\\cap P^{c})^{c}\\right]^{c}\\cup I= \\left[(S_1^{c}\\cup N)\\cap (S_2^{c}\\cup P)\\right]^{c}\\cup I= [(S_1^{c}\\cap S_2^{c})\\cup (S_1^{c}\\cap P)\\cup (S_2^{c}\\cap N)\\cup (N\\cap P)]^{c}\\cup I $ .", "Put $J=(S_1^{c}\\cap P)\\cup (S_2^{c}\\cap N)\\cup (N\\cap P)\\in \\mathcal {I} $ and note that $A=[(S_1^{c}\\cap S_2^{c})^{c}\\cap J^{c}]\\cup I=[(S_1\\cup S_2)\\cap J^{c}]\\cup I=((S_1\\cup S_2)\\setminus J)\\cup I$ .", "Since $S_1\\in \\mathcal {S}_1, S_2 \\in \\mathcal {S}_2 $ and $J,I \\in \\mathcal {I}$ then we have $A\\in (\\mathcal {S}_1\\vee \\mathcal {S}_2)*\\mathcal {I}$ .", "Assume that $A\\in (\\mathcal {I}*\\mathcal {S}_1)\\vee (\\mathcal {I}*\\mathcal {S}_2)$ .", "Then $A=\\left[I_1\\setminus U_1)\\cup W_1\\right] \\cup \\left[ (I_2\\setminus U_2)\\cup W_2\\right]$ where $I_1, I_2\\in \\mathcal {I}, U_1, W_1\\in \\mathcal {S}_1$ and $U_2, W_2\\in \\mathcal {S}_2$ .", "It is clear that $A=(I_1\\setminus U_1)\\cup (I_2\\setminus U_2)\\cup (W_1\\cup W_2)=I\\cup (W_1\\cup W_2)$ , where $I=(I_1\\setminus U_1)\\cup (I_2\\setminus U_2)$ .", "Since $I\\in \\mathcal {I}$ it follows that $A=\\left[I\\setminus (W_1\\cup W_2)\\right]\\cup (W_1\\cup W_2) \\in \\mathcal {I}*(\\mathcal {S}_1\\vee \\mathcal {S}_2)$ .", "Assume that $A\\in \\mathcal {I}*\\left(\\mathcal {S}_1\\vee \\mathcal {S}_2\\right)$ .", "It follows that $A=\\left[I\\setminus (U_1\\cup U_2)\\right] \\cup (W_1\\cup W_2)$ where $I\\in \\mathcal {I}, U_1, W_1\\in \\mathcal {S}_1$ and $U_2, W_2\\in \\mathcal {S}_2$ .", "Note that $A=\\left[(I\\setminus U_2)\\setminus U_1\\right]\\cup \\left[(I\\setminus U_1)\\setminus U_2\\right]\\cup (W_1\\cup W_2)=\\left[\\left((I\\setminus U_2)\\setminus U_1\\right) \\cup W_1\\right]\\cup \\left[\\left((I\\setminus U_1)\\setminus U_2\\right)\\cup W_2\\right]$ .", "Since the sets $I\\setminus U_1$ and $I\\setminus U_2$ are elements of $I$ then we have $A\\in (\\mathcal {I}*\\mathcal {S}_1)\\vee (\\mathcal {I}*\\mathcal {S}_2)$ .", "Corollary 3.12 Let $\\mathcal {S}_1, \\mathcal {S}_2, \\cdots , \\mathcal {S}_n$ be a finite collection of semigroups of sets on $X$ .", "If $\\mathcal {I}$ is an ideal of sets on $X$ then $\\left(\\bigvee _{i=1}^n \\mathcal {S}_i\\right)*\\mathcal {I}=\\bigvee _{i=1}^{n} \\left( \\mathcal {S}_i*\\mathcal {I} \\right)$ and $\\mathcal {I}*\\left(\\bigvee _{i=1}^n \\mathcal {S}_i\\right)=\\bigvee _{i=1}^{n} \\left( \\mathcal {I}*\\mathcal {S}_i \\right)$ .", "It follows from Proposition REF that if $\\mathcal {A}$ and $\\mathcal {B}$ are families of sets on $X$ , and $\\mathcal {I}$ is an ideal of sets on $X$ then $\\mathcal {S}[(\\mathcal {A}\\vee \\mathcal {B})*\\mathcal {I}]=\\mathcal {S}\\left[(\\mathcal {A}*\\mathcal {I})\\vee (\\mathcal {B}*\\mathcal {I})\\right]$ .", "Question 3.13 Let $\\mathcal {A}$ and $\\mathcal {B}$ be families of sets, and let $\\mathcal {I}$ be an ideal of sets.", "What is the relationship, in the sense of inclusion, between the semigroups of sets $\\mathcal {S}(\\mathcal {A}\\vee \\mathcal {B})*\\mathcal {I}$ and $\\mathcal {S}[(\\mathcal {A}\\vee \\mathcal {B})*\\mathcal {I}]$ ?", "It is clear that if $\\mathcal {A}$ and $\\mathcal {B}$ are semigroups of sets, then $\\mathcal {S}(\\mathcal {A}\\vee \\mathcal {B})*\\mathcal {I}=\\mathcal {S}[(\\mathcal {A}\\vee \\mathcal {B})*\\mathcal {I}]$ .", "The following statement can be used in the extension of semigroup of sets.", "Its proof is based on the properties of ideals of sets goes in the same line as Proposition REF .", "Proposition 3.14 Let $\\mathcal {I}_1$ and $\\mathcal {I}_2$ be ideals of sets on $X$ and let $\\mathcal {S}$ be a semigroup of sets on $X$ .", "Then $\\mathcal {S}*\\mathcal {I}_i\\subseteq \\mathcal {S}*(\\mathcal {I}_1\\vee \\mathcal {I}_2)=(\\mathcal {S}*\\mathcal {I}_1)\\vee (\\mathcal {S}* \\mathcal {I}_2)$ and $\\mathcal {I}_i*\\mathcal {S}\\subseteq (\\mathcal {I}_1\\vee \\mathcal {I}_2)*\\mathcal {S}=(\\mathcal {I}_1*\\mathcal {S})\\vee (\\mathcal {I}_2*\\mathcal {S})$ for $i=1,2$ .", "Corollary 3.15 Let $\\mathcal {I}_1, \\mathcal {I}_2,\\cdots , \\mathcal {I}_n$ be a finite collection of ideals of sets on $X$ .", "If $\\mathcal {S}$ is a semigroup of sets on $X$ then $\\mathcal {S}*\\left( \\bigvee _{i=1}^n \\mathcal {I}_i\\right)=\\bigvee _{i=1}^n \\left(\\mathcal {S}*\\mathcal {I}_i\\right)$ , and $\\left( \\bigvee _{i=1}^n \\mathcal {I}_i\\right)*\\mathcal {S}=\\bigvee _{i=1}^n \\left( \\mathcal {I}_i*\\mathcal {S}\\right)$ ." ], [ "Lebesgue measurability and the Baire property", "Recall that the Lebesgue outer measure of a set $E \\subseteq \\mathbb {R}$ , denoted by $\\mu ^{*}(E)$ , is meant the number $\\mu ^{*}(E)=\\inf \\left\\lbrace \\sum _{n=1}^{\\infty } \\ell (I_n): E\\subseteq \\bigcup _{n=1}^{\\infty }I_n\\right\\rbrace $ , where $\\inf $ is taken over all sequences $\\lbrace I_n\\rbrace _{n=1}^{\\infty }$ consisting of open intervals covering the set $E$ .", "The Lebesgue outer measure is defined for all subsets of $\\mathbb {R}$ but it is not countably additive.", "A subset $E$ of $\\mathbb {R}$ is said to be Lebesgue measurable if for each $A\\subseteq \\mathbb {R}$ , the equality $\\mu ^{*}(E)=\\mu ^{*}(A\\cap E)+\\mu ^{*}(A\\cap E^c)$ is satisfied.", "If $E$ is a Lebesgue measurable set, then the Lebesgue measure of $E$ is its outer measure, and it is denoted by $\\mu (E)$ .", "Let $\\mathcal {N}_0$ be the collection of all subsets of $\\mathbb {R}$ having the Lebesgue measure zero (null subsets of $\\mathbb {R}$ ).", "Note that $\\mathcal {N}_0$ is a $\\sigma $ -ideal of sets on $\\mathbb {R}$ .", "The family $\\mathcal {L}(\\mathbb {R})$ of all Lebesgue measurable sets on $\\mathbb {R}$ is a $\\sigma $ -algebra of sets on $\\mathbb {R}$ , containing the collection $\\mathcal {B}_O(\\mathbb {R})$ of all Borel sets on $\\mathbb {R}$ as well as the collection $\\mathcal {N}_0$ .", "Let us note that the family $\\mathcal {L}^c(\\mathbb {R})=\\mathcal {P}(\\mathbb {R})\\setminus \\mathcal {L}(\\mathbb {R})$ is not empty [5], [2].", "Recall [7] that if $\\mathcal {O}(\\mathbb {R})\\subseteq \\mathcal {P}(\\mathbb {R})$ and $\\Psi (\\mathbb {R})$ is a group of homeomorphisms of $\\mathbb {R}$ onto itself, then the family $\\mathcal {O}(\\mathbb {R})$ is said to be invariant under the action of $\\Psi (\\mathbb {R})$ , if for each $A\\in \\mathcal {O}(\\mathbb {R})$ and for each $h\\in \\Psi (\\mathbb {R})$ , we have $h(A)\\in \\mathcal {O}(\\mathbb {R})$ .", "The collection $\\mathcal {L}(\\mathbb {R})$ is invariant the action of the group $\\Phi (\\mathbb {R})$ of all translations of $\\mathbb {R}$ ; i.e., if $E\\in \\mathcal {L}(\\mathbb {R})$ and $t\\in \\mathbb {R}$ then $E+t=\\lbrace e+t: e\\in E\\rbrace \\in \\mathcal {L}(\\mathbb {R})$ .", "Furthermore, $\\mu (E+t)=\\mu (E)$ , and if $E\\in \\mathcal {L}(\\mathbb {R}),t\\in \\mathbb {R}$ then $tE=\\lbrace te: e\\in E\\rbrace \\in \\mathcal {L}(\\mathbb {R})$ and $\\mu (tE)=|t|\\mu (E)$ , where $|t|$ is the absolute value of $t$ .", "Hence the families $\\mathcal {L}(\\mathbb {R})$ and $\\mathcal {L}^c(\\mathbb {R})$ are invariant under the action of the group $\\Pi (\\mathbb {R})$ for which elements are of the form $h(x)=ax+b$ with $a,b\\in \\mathbb {R}$ and $a\\ne 0$ .", "Lemma 3.16 ([10]) Let $A$ and $B$ be subsets of $\\mathbb {R}$ .", "If $A\\in \\mathcal {L}(\\mathbb {R})$ and $\\mu (A\\Delta B)=0$ then $B\\in \\mathcal {L}(\\mathbb {R})$ and $\\mu (A)=\\mu (B)$ .", "If $A$ a subset of $\\mathbb {R}$ then $\\operatorname{\\textrm {Int}}(A)$ and $\\operatorname{\\textrm {Cl}}(A)$ are used to denote the interior and the closure of $A$ in $\\mathbb {R}$ , respectively.", "A subset $M$ of $\\mathbb {R}$ is said to be meager (or of first category) if it can be represented as a countable union of nowhere dense; i.e., $M=\\bigcup _{i=1}^{\\infty } M_i$ with $\\operatorname{\\textrm {Int}}\\operatorname{\\textrm {Cl}}(M_i)=\\emptyset $ for each $i=1,2, \\cdots $ .", "It is well known that the collection $\\mathcal {I}_m$ of all first category subsets of $\\mathbb {R}$ is a $\\sigma $ -ideal of sets on $\\mathbb {R}$ and that $\\mathcal {I}_f \\subseteq \\mathcal {I}_c\\subseteq \\mathcal {I}_m$ and $\\mathcal {I}_f \\subseteq \\mathcal {I}_c\\subseteq \\mathcal {N}_0$ .", "A subset $A$ of $\\mathbb {R}$ is said to have the Baire property in $\\mathbb {R}$ if $A$ can be represented as $A=O\\Delta M$ , where $O$ is open in $\\mathbb {R}$ , $M$ is a first category set on $\\mathbb {R}$ , and $\\Delta $ is the usual operation of standard symmetric difference of sets [8].", "The family $\\mathcal {B}_P(\\mathbb {R})$ of sets having the Baire property in $\\mathbb {R}$ is a $\\sigma $ -algebra, containing the collections $\\mathcal {B}_O(\\mathbb {R})$ and $\\mathcal {I}_m$ , and it is invariant under the action of the group $\\mathcal {H}(\\mathbb {R})$ of all homeomorphisms of $\\mathbb {R}$ onto itself." ], [ "Vitali selectors in the additive topological group of real numbers", "The Vitali selectors of $\\mathbb {R}$ constitute an example of subsets of $\\mathbb {R}$ which are not Lebesgue measurable and without the Baire property in $\\mathbb {R}$ .", "To define Vitali selectors, we follow [2] and [5], and we emphasize that their existence is granted by the Axiom of Choice.", "Consider a countable dense subgroup $Q$ of the additive topological group $(\\mathbb {R},+)$ of real numbers.", "Define a relation $\\mathcal {R}$ on $\\mathbb {R}$ as follows: for $x,y\\in \\mathbb {R}$ , let $x \\mathcal {R}y$ if and only if $x-y\\in {Q}$ .", "Clearly, $\\mathcal {R}$ is an equivalence relation on $\\mathbb {R}$ , and hence it divides $\\mathbb {R}$ into equivalence classes.", "Let $\\mathbb {R}/{Q}=\\lbrace E_{\\alpha }(Q): \\alpha \\in I\\rbrace $ be the collection of all equivalence classes, where $I$ is some indexing set.", "Accordingly, we have the following decomposition of $\\mathbb {R}$ : $\\mathbb {R}=\\bigcup \\lbrace E_\\alpha (Q): \\alpha \\in I\\rbrace .$ Equality (REF ) implies that $\\operatorname{\\textrm {Card}}(I)= \\mathfrak {c}$ , where $\\mathfrak {c}$ is the continuum.", "It follows from the definition of $\\mathcal {R}$ that the set $\\mathbb {R}/{Q}$ consists of disjoint translated copies of $Q$ by elements of $\\mathbb {R}$ .", "Namely, if $t\\in E_\\alpha (Q)$ and $E_\\alpha (Q) \\in \\mathbb {R}/Q$ then $E_\\alpha (Q)= Q+t=\\lbrace q+t: q\\in Q\\rbrace $ .", "Hence each equivalence class $E_\\alpha (Q)$ is a dense subset of $\\mathbb {R}$ .", "Example 3.17 The set $\\mathbb {Q}$ of rational numbers, the set $\\mathbb {D}=\\lbrace a+b\\sqrt{2}: a, b\\in 2\\mathbb {N}\\rbrace $ , the set $\\mathbb {Q}(\\gamma )=\\lbrace a+b \\gamma : a,b \\in \\mathbb {Q}\\rbrace $ for each irrational number $\\gamma $ , and the set $\\sqrt{2}\\mathbb {Q}=\\lbrace \\sqrt{2}q: q\\in \\mathbb {Q}\\rbrace $ are some examples of countable dense subgroups of $(\\mathbb {R},+)$ .", "Definition 3.18 ([5], [2]) A Vitali selector of $\\mathbb {R}$ related to $Q$ is any subset $V$ of $\\mathbb {R}$ containing one element for each equivalence class; i.e.", "any subset $V$ of $\\mathbb {R}$ for which $\\operatorname{\\textrm {Card}}(V\\cap E_\\alpha (Q))=1$ for each $\\alpha \\in I$ .", "A Vitali selector is called a Vitali set whenever the subgroup ${Q}$ coincides with the additive group $\\mathbb {Q}$ of rational numbers.", "Proposition 3.19 ([5], [2]) Let $Q$ be a countable dense subgroup of $(\\mathbb {R},+)$ and let $V$ be a Vitali selector related to $Q$ .", "Then the following statements hold: If $q_1,q_2\\in Q$ and $q_1\\ne q_2$ then $(V+q_1)\\cap (V+q_2)=\\emptyset $ .", "Any two sets in the collection $\\lbrace V+q: q\\in Q\\rbrace $ are homeomorphic, and $\\mathbb {R}=\\bigcup \\lbrace V+q: q\\in Q\\rbrace .$ The set $V$ is not of the first category in $\\mathbb {R}$ , and it is not a null set.", "The following theorem shows that the collection of all Vitali selectors of $\\mathbb {R}$ is invariant under the action of the group $\\Phi (\\mathbb {R})$ of all translations of $\\mathbb {R}$ .", "Theorem 3.20 ([11], [16]) If $U=\\bigcup _{i=1}^n V_i$ , where each $V_i$ is a Vitali selector related to $Q_i$ and $t\\in \\mathbb {R}$ then the set $U+t:=\\left(\\bigcup _{i=1}^n V_i\\right)+t=\\bigcup _{i=1}^n (V_i+t)$ is a union of Vitali selectors, where each $V_i+t$ is related to $Q_i$ , for $i=1,2, \\cdots ,n$ .", "It is possible to define bounded and unbounded Vitali selectors of $\\mathbb {R}$ .", "If $O$ is a non-empty open set of $\\mathbb {R}$ then one can define Vitali selectors which are dense in $O$ .", "This implies that there exist Vitali selectors which are dense in $\\mathbb {R}$ .", "We further point out that there exists a Vitali selector which contains a perfect set [7], and it can be easily observed that for any Vitali selector $V$ of $\\mathbb {R}$ the set $\\mathbb {R}\\setminus V$ is dense in $\\mathbb {R}$ .", "Theorem 3.21 ([2], [5]) Any Vitali selector $V$ of $\\mathbb {R}$ is not measurable in the Lebesgue sense and does not have the Baire property in $\\mathbb {R}$ .", "It follows from Theorem REF that if $V$ be a Vitali selector of $\\mathbb {R}$ , then every Lebesgue measurable subset of $V$ has the Lebesgue measure zero, and every subset of $V$ with the Baire property is of the first category.", "The next theorem is a more general result on the Baire property than Theorem REF .", "It was stated for Vitali sets, but it is valid even for Vitali selectors of $\\mathbb {R}$ .", "Theorem 3.22 ([14]) Let $V_i$ be a Vitali set for each $i\\le n$ , where $n$ is some integer such that $n\\ge 1$ .", "Then the set $U=\\bigcup _{i=1}^n V_i$ does not contain the difference $O\\setminus M$ , where $O$ is a non-empty open set and $M$ is a meager.", "In particular, the set $U$ does not possess the Baire property in $\\mathbb {R}$ .", "A similar result to Theorem REF about non-Lebesgue measurability of finite unions of Vitali sets, which is valid also for a finite union of Vitali selectors of $\\mathbb {R}$ , was proved by A.B.", "Kharazishvili.", "Theorem 3.23 ([1]) If $\\lbrace V_\\alpha : 1\\le \\alpha \\le m\\rbrace $ is a non-empty finite family of Vitali sets, then the union $\\bigcup \\lbrace V_\\alpha : 1\\le \\alpha \\le m\\rbrace $ is not measurable in the Lebesgue sense.", "Below, we recall the classical Banach Theorem, and two important lemmas, which were used in the proof of Theorem REF , and they will be very useful in the sequel.", "For, we let $\\mathcal {B}_b (\\mathbb {R})$ denotes the family of all bounded subsets of $\\mathbb {R}$ .", "Theorem 3.24 (Banach Theorem [1]) Let $\\mathcal {S}$ be a translation invariant ring of subsets of $\\mathbb {R}$ , satisfying the relations $\\mathcal {S} \\subseteq \\mathcal {B}_b (\\mathbb {R})$ and $[0, 1) \\in \\mathcal {S}$ , and let $\\vartheta : \\mathcal {S}\\longrightarrow [0,+\\infty )$ be a finitely additive translation invariant functional such that $\\vartheta ([0,1))=1$ .", "Then there exists a finitely additive translation invariant functional $\\eta : \\mathcal {B}_b (\\mathbb {R})\\longrightarrow [0, +\\infty )$ such that $\\eta $ is an extension of $\\vartheta $ .", "Lemma 3.25 ([1]) Let $\\vartheta $ be as in Theorem REF , and let $X\\in \\mathcal {B}_b(\\mathbb {R})$ have the following property: There exists a bounded infinite sequence $\\lbrace h_k: k\\in \\mathbb {N}\\rbrace $ of elements of $\\mathbb {R}$ such that the family $\\lbrace X+h_k: k\\in \\mathbb {N}\\rbrace $ is disjoint.", "If $X\\in \\operatorname{\\textrm {dom}}(\\vartheta )$ then necessarily $\\vartheta (X)=0$ .", "Lemma 3.26 ([1]) Let $X$ be a bounded subset of a Vitali selector $V$ .", "Then $X$ has the property indicated in Lemma REF .", "It follows from Equality REF in Proposition REF that the results of Theorem REF and Theorem REF are not valid for infinite countable unions of Vitali selectors of $\\mathbb {R}$ .", "However, the following theorem provides examples of infinite countable unions of Vitali selectors without the Baire property in $\\mathbb {R}$ .", "Theorem 3.27 ([9]) If $V$ is a Vitali selector of $\\mathbb {R}$ related to $Q$ and $\\Gamma $ is a non-empty proper subset of ${Q} $ then the set $U=\\bigcup \\lbrace V+q: q\\in \\Gamma \\rbrace $ does not possess the Baire property in $\\mathbb {R}$ .", "Question 3.28 Let $V$ be a Vitali selector of $\\mathbb {R}$ related to $Q$ and let $\\Gamma $ be an infinite countable proper subset of $Q$ .", "Under what conditions the set $U=\\bigcup \\lbrace V+q: q\\in \\Gamma \\rbrace $ is not measurable in the Lebesgue sense?", "It is clear that if $Q\\setminus \\Gamma $ is finite, then by Theorem REF , the set $W=\\bigcup \\lbrace V+q: q\\in Q\\setminus \\Gamma \\rbrace $ is not measurable in the Lebesgue sense.", "Consequently, by Equality REF , the set $U=\\bigcup \\lbrace V+q: q\\in \\Gamma \\rbrace $ is also not measurable in the Lebesgue sense.", "We also note [13] that if $O$ is a non-empty open subset of $\\mathbb {R}$ then there exists a sequence $\\lbrace V_1, V_2, \\cdots \\rbrace $ of (disjoint) Vitali selectors of $\\mathbb {R}$ such that $O=\\bigcup _{i=1}^{\\infty } V_i$ .", "It is clear that such a union is Lebesgue measurable and has the Baire property in $\\mathbb {R}$ ." ], [ "Bernstein sets in the additive topological group of real numbers", "The Bernstein sets on $\\mathbb {R}$ constitute also an example of elements belonging to the family $\\mathcal {L}^c(\\mathbb {R})$ .", "According to [2] and [8], a subset $B$ of $\\mathbb {R}$ is called a Bernstein set if $B\\cap F\\ne \\emptyset $ and $(\\mathbb {R}\\setminus B)\\cap F\\ne \\emptyset $ for each uncountable closed subset of $\\mathbb {R}$ .", "The existence and the construction of Bernstein sets on $\\mathbb {R}$ is based on the Method of Transfinite Recursion.", "We point out that the same method is used in the construction of Bernstein sets in higher-dimensional Euclidean spaces $\\mathbb {R}^n$ for $n\\ge 2$ .", "Proposition 3.29 ([8], [6]) Let $B$ be a Bernstein subset of $\\mathbb {R}$ .", "Then the following statements hold.", "The complement $\\mathbb {R}\\setminus B$ of $B$ is also a Bernstein set, and $\\operatorname{\\textrm {Int}}(B)=\\operatorname{\\textrm {Int}}(\\mathbb {R}\\setminus B)=\\emptyset $ .", "Both sets $B$ and $\\mathbb {R}\\setminus B$ are dense in $\\mathbb {R}$ and $\\operatorname{\\textrm {Card}}(B)=\\operatorname{\\textrm {Card}}(\\mathbb {R}\\setminus B)=\\mathfrak {c}$ .", "The family $\\mathcal {B}_E(\\mathbb {R})$ of all Bernstein subsets of $\\mathbb {R}$ is invariant under the action of the group $\\mathcal {H}(\\mathbb {R})$ of all homeomorphisms of $\\mathbb {R}$ onto itself, i.e.", "if $B\\in \\mathcal {B}_E(\\mathbb {R})$ and $h\\in \\mathcal {H}(\\mathbb {R})$ then $h(B)\\in \\mathcal {B}_E(\\mathbb {R})$ .", "Lemma 3.30 ([8], [6]) Let $A$ be a subset of $\\mathbb {R}$ .", "Then $A$ is a Bernstein set if and only if $F\\cap A\\ne \\emptyset $ and $F\\setminus A\\ne \\emptyset $ for every uncountable closed subset of $\\mathbb {R}$ .", "Theorem 3.31 ([8]) Any Bernstein set $B$ on $\\mathbb {R}$ is not measurable in the Lebesgue sense and does not have the Baire property.", "Indeed, every Lebesgue measurable subset of either $B$ or $\\mathbb {R}\\setminus B$ has the Lebesgue measure zero, and every subset of either $B$ or $\\mathbb {R}\\setminus B$ with the Baire property is of the first category.", "Corollary 3.32 ([4]) If $A$ is a Lebesgue measurable set with positive measure then the set $A\\cap B$ and $A\\setminus B$ are not measurable in the Lebesgue sense.", "If $A$ is set with the Baire property which is not of the first category then the sets $A\\cap B$ and $A\\setminus B$ do not have the Baire property.", "We point out that there exist Bernstein subsets of $\\mathbb {R}$ which have some additional algebraic structures for subgroups of the additive group $(\\mathbb {R},+)$ as it is indicated in the following statements.", "Lemma 3.33 ([2]) There exists a subgroup $B$ of $(\\mathbb {R},+)$ such that the factor group $\\mathbb {R}/B$ is isomorphic to the group $(\\mathbb {R},+)$ and $B$ is a Bernstein set in $\\mathbb {R}$ .", "Theorem 3.34 ([2]) There exists two subgroups $G_1$ and $G_2$ of the additive group $(\\mathbb {R},+)$ such that $G_1\\cap G_2=\\lbrace 0\\rbrace $ , and both $G_1$ and $G_2$ are Bernstein sets in $\\mathbb {R}$ .", "For other notions and facts we refer the reader to [12], [17] and [8]." ], [ "Semigroups of non-Lebesgue measurable sets generated by Bernstein sets", "Let $B$ be a Bernstein subset of $\\mathbb {R}$ which has an algebraic structure of being a subgroup of $(\\mathbb {R},+)$ as in Lemma REF .", "Consider the collection $\\mathbb {R}/B=\\lbrace B+x: x\\in \\mathbb {R}\\rbrace $ of all cosets of $B$ .", "Without loosing of generality, we may assume that the collection of $\\mathbb {R}/B$ consists of pairwise disjoint sets and we simply denote it by $\\mathcal {B}$ .", "It is clear that $\\mathcal {B}$ is the collection of all pairwise disjoint translates of $B$ by real numbers.", "It follows from [3] that $\\operatorname{\\textrm {Card}}(\\mathcal {B}) \\ge \\aleph _0$ , where $\\aleph _0=\\operatorname{\\textrm {Card}}(\\mathbb {N})$ , and $\\operatorname{\\textrm {Card}}(\\mathcal {B})$ is the same as the cardinality of the set $\\lbrace \\mathbb {R}\\setminus Y: Y\\in \\mathcal {B}\\rbrace $ .", "Since the family $\\mathcal {B}_E(\\mathbb {R})$ is invariant under the action of the group $\\mathcal {H}(\\mathbb {R})$ (in particular, invariant under translations), it follows that each element of $\\mathcal {B}$ is also a Bernstein set on $\\mathbb {R}$ .", "Let $\\mathcal {S}(\\mathcal {B})=\\lbrace \\bigcup _{i=1}^n B_i: B_i\\in \\mathcal {B}, n\\in \\mathbb {N}\\rbrace $ be the collection of all finite unions of elements of $\\mathcal {B}$ , i.e.", "$\\mathcal {S}(\\mathcal {B})$ is the semigroup of sets generated by $\\mathcal {B}$ .", "Evidently, the family $\\mathcal {S}(\\mathcal {B})$ is invariant under the action of the group $\\Phi (\\mathbb {R})$ .", "Lemma 4.1 Each element $U$ in the collection $\\mathcal {S}(\\mathcal {B})$ is a Bernstein set in $\\mathbb {R}$ .", "Assume that $U\\in \\mathcal {S}(\\mathcal {B})$ .", "Then $U=\\bigcup _{i=1}^n B_i$ , where $B_i\\in \\mathcal {B}$ for $i=1,2, \\cdots , n$ and $n\\in \\mathbb {N}$ .", "Let $F$ be an uncountable closed subset of $\\mathbb {R}$ .", "Since each $B_i$ is a Bernstein set of $\\mathbb {R}$ it is evident that $F\\cap B_i\\ne \\emptyset $ for each $i=1,2,\\cdots , n$ .", "Hence $F\\cap U=F\\cap \\left( \\bigcup _{i=1}^n B_i\\right)=\\bigcup _{i=1}^n (F\\cap B_i)\\ne \\emptyset $ .", "Now, we show that $F\\cap (\\mathbb {R}\\setminus \\bigcup _{i=1}^n B_i)\\ne \\emptyset $ .", "Accordingly, assume that $F\\cap (\\mathbb {R}\\setminus \\bigcup _{i=1}^n B_i)=\\emptyset $ .", "So $F\\subseteq \\bigcup _{i=1}^n B_i$ .", "Let $B_k$ be an element of $\\mathcal {B}$ for some $k\\notin \\lbrace 1,2, \\cdots , n\\rbrace $ .", "Such an element exists, since $\\operatorname{\\textrm {Card}}(\\mathcal {B})\\ge \\aleph _0$ .", "Since each element of $\\mathcal {B}$ is a Bernstein set then we have $F\\cap B_k\\ne \\emptyset $ .", "By construction $B_k\\cap (\\bigcup _{i=1}^n B_i)=\\emptyset $ and thus the inclusion $F\\subseteq \\bigcup _{i=1}^n B_i$ is impossible.", "Hence $U$ is a Bernstein set on $\\mathbb {R}$ .", "This implies that $\\mathbb {R}\\setminus U$ is also a Bernstein set.", "Corollary 4.2 Each element $U$ of the family $\\mathcal {S}(\\mathcal {B})$ , as well as its complement $\\mathbb {R}\\setminus U$ , is not measurable in the Lebesgue sense, and it does not possess the Baire property in $\\mathbb {R}$ .", "Proposition 4.3 The families $\\mathcal {S}(\\mathcal {B})*\\mathcal {N}_0$ and $\\mathcal {N}_0*\\mathcal {S}(\\mathcal {B})$ are semigroups of sets on $\\mathbb {R}$ such that $\\mathcal {S}(\\mathcal {B})\\subseteq \\mathcal {N}_0*\\mathcal {S}(\\mathcal {B})\\subseteq \\mathcal {S}(\\mathcal {B})*\\mathcal {N}_0$ .", "They are invariant under the action of the group $\\Phi (\\mathbb {R})$ and they consist of sets which are not measurable in the Lebesgue sense.", "The families are semigroups of sets by Proposition REF and the inclusions follow by the same proposition.", "Let $A\\in \\mathcal {S}(\\mathcal {B})*\\mathcal {N}_0$ and assume that $A\\in \\mathcal {L}(\\mathbb {R})$ .", "Then $A=(U\\setminus M)\\cup N$ where $U\\in \\mathcal {S}(\\mathcal {B})$ and $M,N\\in \\mathcal {N}_0$ .", "Note that $A\\setminus U\\subseteq N$ and $U\\setminus A\\subseteq M$ and hence $A\\Delta U \\subseteq M\\cup N$ .", "It follows that $\\mu (A\\Delta U)\\le \\mu (M\\cup N)=0$ and thus $\\mu (A\\Delta U)=0$ .", "Lemma REF indicates that the set $U$ must be measurable in the Lebesgue sense.", "However, $U$ is a Bernstein set on $\\mathbb {R}$ and thus it is not measurable in the Lebesgue sense, hence a contradiction.", "The family $\\mathcal {S}(\\mathcal {B})*\\mathcal {N}_0$ is invariant under the action of the group $\\Phi (\\mathbb {R})$ since both families $\\mathcal {S}(\\mathcal {B})$ and $\\mathcal {N}_0$ are invariant under the action of the group $\\Phi (\\mathbb {R})$ , and this ends the proof.", "Proposition 4.4 Let $\\mathcal {I}_{m}^n$ be the family of all meager subsets of $\\mathbb {R}^n$ for $n\\ge 1$ .", "For each element $A\\in \\mathcal {S}(\\mathcal {B})*\\mathcal {I}_{m}^n$ we have $0\\le \\dim A\\le n-1$ , where $\\dim $ is the Lebesgue covering dimension.", "Assume that $\\dim A=n$ .", "Then by the Brouwer Dimension Theorem, we must have a non-empty open set $O$ in $\\mathbb {R}^n$ such that $O\\subseteq A=(U\\setminus M)\\cup N$ , where $U\\in \\mathcal {S}(\\mathcal {B})$ and $M,N$ meager sets.", "Since $\\mathbb {R}^n$ is a Baire space (i.e.", "every meager subset of $\\mathbb {R}^n$ has an empty interior) then we have $\\operatorname{\\textrm {Int}}(M)=\\operatorname{\\textrm {Int}}(N)=\\emptyset $ .", "It follows from Proposition REF (i) and Lemma REF that $\\emptyset \\ne O\\subseteq \\operatorname{\\textrm {Int}}(A)\\subseteq \\operatorname{\\textrm {Int}}(U\\setminus M)\\cup \\operatorname{\\textrm {Int}}(N)\\subseteq \\operatorname{\\textrm {Int}}(U)\\cup \\operatorname{\\textrm {Int}}(N)=\\emptyset \\cup \\emptyset =\\emptyset $ , which is a contradiction.", "Lemma 4.5 Let $Y$ be a bounded subset of a Bernstein set $A$ in the collection $\\mathcal {B}$ .", "Then $Y$ has the property indicated in Lemma REF .", "Let $Y$ be an element of $\\mathcal {B}_b(\\mathbb {R})$ such that $Y$ is a subset of a Bernstein set $A\\in \\mathcal {B}=\\lbrace B+x: x\\in \\mathbb {R}\\rbrace $ , where $B$ is a Bernstein subset of $\\mathbb {R}$ having the property of being a subgroup of $(\\mathbb {R}, +)$ .", "This means that $A=B+x$ for some $x\\in \\mathbb {R}$ .", "Since the family $\\mathcal {B}$ is pairwise disjoint, it follows that the family $\\lbrace Y+x: x\\in \\mathbb {R}\\rbrace $ is also pairwise disjoint.", "Since every infinite set contains an infinitely countable set [18], let $\\Lambda $ be an infinitely countable bounded subset of $\\mathbb {R}$ .", "The family $\\lbrace x_k: x_k \\in \\Lambda , k=1,2,\\cdots \\rbrace $ can play the role of $\\lbrace h_k: k\\in \\mathbb {N}\\rbrace $ in Lemma REF .", "It follows that if $Y\\in \\operatorname{\\textrm {dom}}(\\vartheta )$ then $\\vartheta (Y)=0$ , and this ends the proof.", "Proposition 4.6 Let $B$ a Bernstein set of $\\mathbb {R}$ which has a structure of being a subgroup of $(\\mathbb {R},+)$ .", "Any element $U$ of the family $\\mathcal {S}(\\mathcal {B})$ cannot contain any set of positive Lebesgue measure.", "Suppose that there exists a Lebesgue measurable subset $Y$ of $\\mathbb {R}$ such that $\\mu (Y)>0$ and $Y\\subseteq U$ .", "Since $U\\in \\mathcal {S}(\\mathcal {B})$ then $U=\\bigcup _{i=1}^n B_i$ with $B_i\\in \\mathcal {B}$ for each $i=1,2,\\cdots , n$ .", "Since $Y=\\bigcup _{n=-\\infty }^{\\infty }\\left(Y\\cap [r,r+1)\\right)$ and $\\mu (Y)>0$ implies that $\\mu \\left( Y\\cap [r,r+1) \\right)>0$ for some $r$ , without loss of generality, we may assume that the set $Y$ is bounded.", "Let $\\vartheta $ be the restriction of $\\mu $ to $\\mathcal {B}_b(\\mathbb {R})\\cap \\operatorname{\\textrm {dom}}(\\mu )$ .", "For this $\\vartheta $ , there exists a functional $\\eta $ as in Theorem REF .", "Then $0<\\vartheta (Y)=\\eta (Y)=\\eta (Y\\cap U)=\\eta \\left[Y\\cap \\left(\\bigcup _{i=1}^n B_i\\right)\\right]=\\eta \\left[\\bigcup _{i=1}^n \\left(Y\\cap B_i\\right)\\right]$ Inequality REF implies that $\\eta (Y\\cap B_i)>0$ for some integer $i\\in \\lbrace 1,2,\\cdots ,n\\rbrace $ .", "Since $Y\\cap B_i$ is a bounded subset of the Bernstein set $B_i$ then it has the property described in Lemma REF .", "According to Lemma REF , we must have the equality $\\eta (Y\\cap B_i)=0$ , and this is a contradiction.", "Corollary 4.7 Let $B$ a Bernstein set of $\\mathbb {R}$ which has an algebraic structure of being a subgroup of $(\\mathbb {R},+)$ .", "Any element of the family $\\mathcal {S}(\\mathcal {B})*\\mathcal {N}_0$ cannot contain any set of positive measure.", "Theorem 4.8 Let $U_k$ be an element of $\\mathcal {S}(\\mathcal {B})$ and $h_k$ be an element of $\\Phi (\\mathbb {R})$ for $k=1,2,\\cdots ,n$ where $n\\in \\mathbb {N}$ .", "Then the set $U=\\bigcup _{k=1}^n h_k(U_k)$ is not measurable in the Lebesgue sense and it does not possess the Baire property in $\\mathbb {R}$ .", "It is enough to show that the set $U=\\bigcup _{k=1}^nh_k(U_k)$ is a Bernstein set on $\\mathbb {R}$ .", "Accordingly, we will show that $U\\in \\mathcal {S}(\\mathcal {B})$ .", "Since $U_k\\in \\mathcal {S}(\\mathcal {B})$ i.e.", "$U_k=\\bigcup _{i=1}^m B_{ki}$ where $B_{ki}\\in \\mathcal {B}$ then $U_k$ is a Bernstein set by Lemma REF .", "So by the invariance of the family $\\mathcal {S}(\\mathcal {B})$ under the action of $\\Phi (\\mathbb {R})$ , each set $h_k(U_k)$ is also a Bernstein set for each $k=1,2,\\cdots , n$ .", "Note that $U=\\bigcup _{k=1}^n h_k(U_k)=\\bigcup _{k=1}^n h_k\\left(\\bigcup _{i=1}^m B_{ki}\\right)=\\bigcup _{k=1}^n\\left[\\bigcup _{i=1}^m h_k(B_{ki})\\right]$ Put $B_k= \\bigcup _{i=1}^m h_k(B_{ki})$ .", "Due to the fact that the family $\\mathcal {S}(\\mathcal {B})$ is invariant under the action of $\\Phi (\\mathbb {R})$ , it follows that $B_k$ is an element of $\\mathcal {S}(\\mathcal {B})$ .", "Again, by the invariance of the family of $\\mathcal {S}(\\mathcal {B})$ under the action of $\\Phi (\\mathbb {R})$ , it follows that the set $U=\\bigcup _{k=1}^n B_k$ is a finite union of elements of $\\mathcal {S}(\\mathcal {B})$ and hence $U\\in \\mathcal {S}(\\mathcal {B})$ .", "Lemma REF implies that $U$ is a Bernstein set on $\\mathbb {R}$ and thus it is not measurable in the Lebesgue sense, and it does not possess the Baire property in $\\mathbb {R}$ .", "Question 4.9 Let $A_k$ be an element of $\\mathcal {S}(\\mathcal {B})*\\mathcal {N}_0$ and $h_k$ be a homeomorphism of $\\mathbb {R}$ for each $k=1,2, \\cdots n$ .", "Is the union $\\bigcup _{k=1}^nh_k(A_k)$ non-measurable in the Lebesgue sense and without the Baire property in $\\mathbb {R}$ ?", "Let $B_1$ and $B_2$ be Bernstein sets having an algebraic structure of being subgroups of $(\\mathbb {R},+)$ as in Theorem REF , and consider the families $\\mathcal {B}_1=\\lbrace B_1+x: x\\in \\mathbb {R}\\rbrace $ and $\\mathcal {B}_2=\\lbrace B_2+x: x\\in \\mathbb {R} \\rbrace $ of all disjoint translates (cosets) of $B_1$ and $B_2$ , respectively.", "Let $\\mathcal {S}(\\mathcal {B}_1)$ and $\\mathcal {S}(\\mathcal {B}_2)$ be the semigroups of sets generated by $\\mathcal {B}_1$ and $\\mathcal {B}_2$ , respectively.", "Lemma 4.10 Let $B_1$ and $B_2$ be Bernstein sets having an algebraic structure of being subgroups of $(\\mathbb {R},+)$ , and consider the sets $U_1\\in \\mathcal {S}(\\mathcal {B}_1)$ and $U_2\\in \\mathcal {S}(\\mathcal {B}_2)$ .", "Then the set $U=U_1\\cup U_2$ cannot contain any subset of positive measure.", "Assume that there exists a Lebesgue measurable set $Y$ such that $\\mu (Y)>0$ and $Y\\subseteq U_1\\cup U_2$ , where $U_1=\\bigcup _{i=1}^n B_{1i}$ and $U_2=\\bigcup _{k=1}^m B_{2k}$ with $B_{1i}\\in \\mathcal {B}_1$ and $B_{2k}\\in \\mathcal {B}_2$ .", "Without loss of generality we may assume, we may assume that the set $Y$ is bounded.", "It follows from Proposition REF that the set $Y$ cannot lie entirely in $U_1$ nor in $U_2$ .", "Write $U=\\bigcup _{i=1}^{n+m} X_i$ where $X_i=B_{1i}$ for $i=1,2, \\cdots , n$ and $X_{n+k}=B_{2k}$ for $k=1,2,\\cdots ,m$ .", "Accordingly, we have $0<\\mu (Y)=\\mu (Y\\cap U)=\\mu \\left[\\bigcup _{i=1}^{n+m} (Y\\cap X_i)\\right]\\le \\sum _{i=1}^{n+m}\\mu (Y\\cap X_i).$ Inequality REF implies that that $\\mu (Y\\cap X_i) >0$ for some index $i \\in \\lbrace 1,2, \\cdots , n+m\\rbrace $ .", "Since $Y\\cap X_i \\in \\mathcal {B}_b(\\mathbb {R})$ , let $\\vartheta $ be the restriction of $\\mu $ to $\\mathcal {B}_b(\\mathbb {R})\\cap \\operatorname{\\textrm {dom}}(\\mu )$ .", "For this $\\vartheta $ , there exists a functional $\\eta $ as in Theorem REF which is an extension of $\\vartheta $ .", "It follows that $0<\\mu (Y\\cap X_i)=\\vartheta (Y\\cap X_i)=\\eta (Y\\cap X_i)$ .", "If $X_i\\in \\mathcal {B}_1$ then we must have $\\eta (Y\\cap X_i)=0$ by Lemma REF .", "Similarly, if $X_i\\in \\mathcal {B}_2$ we must have $\\eta (Y\\cap X_i)=0$ by Lemma REF .", "We conclude that the set $Y$ cannot exists and this ends the proof.", "Theorem 4.11 The family $\\mathcal {S}(\\mathcal {B}_1) \\vee \\mathcal {S}(\\mathcal {B}_2)$ is a semigroup of sets on $\\mathbb {R}$ which consists of non-Lebesgue measurable sets and it is invariant under the action of the group $\\Phi (\\mathbb {R})$ .", "The family $\\mathcal {S}(\\mathcal {B}_1) \\vee \\mathcal {S}(\\mathcal {B}_2)$ is a semigroup by Lemma REF .", "Let $U\\in \\mathcal {S}(\\mathcal {B}_1) \\vee \\mathcal {S}(\\mathcal {B}_2)$ and assume that $U$ is a Lebesgue measurable set.", "Then $U=U_1\\cup U_2$ where $U_1\\in \\mathcal {S}(\\mathcal {B}_1)$ and $U_2\\in \\mathcal {S}(\\mathcal {B}_2)$ .", "Since $U_1$ and $U_2$ are Bernstein sets by Lemma REF then $\\mu (U)\\ne 0$ .", "But the inequality $\\mu (U)>0$ is also impossible by Lemma REF .", "It follows that the set $U$ is not Lebesgue measurable.", "It is evident that for any $h\\in \\Phi (\\mathbb {R})$ we have $h(U)=h(U_1\\cup U_2)=h(U_1)\\cup h(U_2)\\in \\mathcal {S}(\\mathcal {B}_1) \\vee \\mathcal {S}(\\mathcal {B}_2)$ , due to the fact that both families $\\mathcal {S}(\\mathcal {B}_1)$ and $\\mathcal {S}(\\mathcal {B}_2)$ are invariant under the action of the group $\\Phi (\\mathbb {R})$ .", "Corollary 4.12 If $ A\\in \\mathcal {S}(\\mathcal {B}_1) \\vee \\mathcal {S}(\\mathcal {B}_2)$ then $\\dim A=0$ , where $\\dim $ is the Lebesgue covering dimension.", "Assume that there is an element $A$ in $\\mathcal {S}(\\mathcal {B}_1) \\vee \\mathcal {S}(\\mathcal {B}_2)$ such that $\\dim A=1$ .", "It follows from the Brouwer Dimension Theorem that there exists an open set $O\\ne \\emptyset $ in $\\mathbb {R}$ such that $O\\subseteq A$ .", "For such an open set, it must have a positive Lebesgue measure, which contradicts Lemma REF .", "Theorem 4.13 The families $\\mathcal {N}_0*\\left(\\mathcal {S}(\\mathcal {B}_1) \\vee \\mathcal {S}(\\mathcal {B}_2)\\right) $ and $\\left(\\mathcal {S}(\\mathcal {B}_1) \\vee \\mathcal {S}(\\mathcal {B}_2)\\right) *\\mathcal {N}_0$ are semigroups of sets on $\\mathbb {R}$ such that $\\mathcal {S}(\\mathcal {B}_1) \\vee \\mathcal {S}(\\mathcal {B}_2) \\subseteq \\mathcal {N}_0*\\left(\\mathcal {S}(\\mathcal {B}_1) \\vee \\mathcal {S}(\\mathcal {B}_2)\\right) \\subseteq \\left(\\mathcal {S}(\\mathcal {B}_1) \\vee \\mathcal {S}(\\mathcal {B}_2)\\right)*\\mathcal {N}_0$ .", "They are invariant under the action of the group $\\Phi (\\mathbb {R})$ and they consist of sets which are not measurable in the Lebesgue sense.", "The families are semigroups by Proposition REF .", "The inclusions follow from the same proposition.", "The invariance of the family $\\left(\\mathcal {S}(\\mathcal {B}_1) \\vee \\mathcal {S}(\\mathcal {B}_2)\\right) *\\mathcal {N}_0$ under the action of the group $\\Phi (\\mathbb {R})$ follows from Theorem REF and the fact that the collection $\\mathcal {N}_0$ is invariant under the action of the group $\\Phi (\\mathbb {R})$ .", "The proof that each element of the family $\\left(\\mathcal {S}(\\mathcal {B}_1) \\vee \\mathcal {S}(\\mathcal {B}_2)\\right)*\\mathcal {N}_0$ is not measurable in the Lebesgue sense goes in the same line as in Proposition REF taking into consideration Theorem REF .", "It follows from Proposition REF and Lemma REF that $\\mathcal {S}(\\mathcal {B}_1\\vee \\mathcal {B}_2)*\\mathcal {N}_0 =\\left(\\mathcal {S}(\\mathcal {B}_1) \\vee \\mathcal {S}(\\mathcal {B}_2)\\right)*\\mathcal {N}_0=\\left(\\mathcal {S}(\\mathcal {B}_1)*\\mathcal {N}_0\\right) \\vee \\left(\\mathcal {S}(\\mathcal {B}_2)*\\mathcal {N}_0\\right)$ .", "We also note that $\\mathcal {S}(\\mathcal {B}_1)\\subseteq \\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}_{\\emptyset }(\\mathcal {B}_2)$ and $\\mathcal {S}(\\mathcal {B}_2)\\subseteq \\mathcal {S}_{\\emptyset }(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)$ and $\\mathcal {S}(\\mathcal {B}_i)*\\mathcal {N}_0\\subseteq (\\mathcal {S}(\\mathcal {B}_i)*\\mathcal {N}_0)\\vee (\\mathcal {S}_{\\emptyset }(\\mathcal {B}_j)*\\mathcal {N}_0)$ for $i,j=1,2$ with $i\\ne j$ .", "Both of these semigroups consist also non-Lebesgue measurable subsets of $\\mathbb {R}$ .", "The family $\\mathcal {S}(\\mathcal {B}_1)*\\mathcal {S}(\\mathcal {B}_1)$ does not need to be a semigroup of sets, but the following statement shows that it consists of elements which are not measurable in the Lebesgue sense.", "Corollary 4.14 Each element of the family $\\mathcal {S}(\\mathcal {B}_1) * \\mathcal {S}(\\mathcal {B}_2)$ is not measurable in the Lebesgue sense.", "Let $A\\in \\mathcal {S}(\\mathcal {B}_1) * \\mathcal {S}(\\mathcal {B}_2)$ and assume that $A$ is a Lebesgue measurable set.", "Note that $A=(U_1\\setminus U_2)\\cup U_3$ for some $U_1 \\in \\mathcal {S}(\\mathcal {B}_1)$ and $U_2, U_3\\in \\mathcal {S}(\\mathcal {B}_2)$ .", "Since $U_3$ is Bernstein set and $U_3\\subseteq A$ then the set $A$ cannot have the Lebesgue measure zero.", "Assume that $\\mu (A)>0$ .", "It follows that $A=(U_1\\setminus U_2)\\cup U_3\\subseteq U_1\\cup U_3\\in \\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)$ , which is in contradiction with Lemma REF .", "Corollary 4.15 The families $\\mathcal {N}_0*\\left(\\mathcal {S}(\\mathcal {B}_1) * \\mathcal {S}(\\mathcal {B}_2) \\right)$ and $\\left(\\mathcal {S}(\\mathcal {B}_1) * \\mathcal {S}(\\mathcal {B}_2)\\right)*\\mathcal {N}_0$ consist of elements which are not measurable in the Lebesgue sense.", "Note that $\\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)\\subseteq \\mathcal {S}(\\mathcal {B}_1)* \\mathcal {S}(\\mathcal {B}_2)$ and $\\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)\\subseteq \\mathcal {S}(\\mathcal {B}_2)* \\mathcal {S}(\\mathcal {B}_1)$ .", "Furthermore $\\mathcal {N}_0* \\left(\\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)\\right)\\subseteq \\mathcal {N}_0*\\left(\\mathcal {S}(\\mathcal {B}_1)* \\mathcal {S}(\\mathcal {B}_2)\\right)$ and $ \\left(\\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)\\right)*\\mathcal {N}_0\\subseteq \\left(\\mathcal {S}(\\mathcal {B}_1)* \\mathcal {S}(\\mathcal {B}_2)\\right)*\\mathcal {N}_0$ .", "Question 4.16 Is each element $U$ in the families $\\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)$ and $\\mathcal {S}(\\mathcal {B}_1)* \\mathcal {S}(\\mathcal {B}_2)$ without the Baire property in $\\mathbb {R}$ ?" ], [ "Semigroups of non-Lebesgue measurable sets generated by Vitali selectors", "Let $\\mathcal {C}$ be the family of all countable dense subgroups of $(\\mathbb {R}, +)$ .", "The following statement shows that each finite union of Vitali selectors is not measurable in the Lebesgue sense and it generalizes Theorem REF .", "Theorem 5.1 ([16]) Let $U=\\bigcup _{i=1}^n V_i$ be a finite union of Vitali selectors of $\\mathbb {R}$ , where $V_i\\in \\mathcal {V}(Q_i)$ and each $Q_i$ is a an element of $\\mathcal {C}$ for $i=1,2,\\cdots ,n$ .", "Then the set $U$ is not measurable in the Lebesgue sense.", "For $n=2$ , Theorem REF implies the following statement.", "Corollary 5.2 Suppose that $V_1$ and $V_2$ are Vitali selectors related to elements $Q_1$ and $Q_2$ respectively in $\\mathcal {C}$ .", "Then atleast one of the sets $V_1\\setminus V_2, V_2\\setminus V_1$ and $V_1\\cap V_2$ must be a non measurable set in the Lebesgue sense.", "It follows from Theorem REF that the set $V_1\\cup V_2$ is not measurable in the Lebesgue sense.", "Note that $V_1\\cup V_2=(V_1\\setminus V_2)\\cup (V_2\\setminus V_1)\\cup (V_1\\cap V_2)$ and sets in this union are disjoint.", "If all the sets in this union are Lebesgue measurable, then the set $V_1\\cup V_2$ will be a Lebesgue measurable set, and this will be a contradiction.", "A result similar to Theorem REF also holds in the case of the Baire property as it can be found in [12].", "If $Q\\in \\mathcal {C}$ then we denote by $\\mathcal {V}(Q)$ the family of all Vitali selectors related to $Q$ , and $\\mathcal {V}_1(Q)$ the semigroup generated by $\\mathcal {V}(Q)$ .", "The following statement shows that each topological group isomorphism maps Vitali selectors of $\\mathbb {R}$ to Vitali selectors of $\\mathbb {R}$ , not necessarily related to the same subgroups of $(\\mathbb {R}, +)$ .", "Theorem 5.3 ([16]) Let $Q$ be a countable dense subgroup $(\\mathbb {R},+)$ and let $V\\in \\mathcal {V}(Q)$ .", "If $h: (\\mathbb {R}, +) \\longrightarrow (\\mathbb {R}, +)$ is a topological group isomorphism then $P=h(Q)\\in \\mathcal {C}$ and $W=h(V) \\in \\mathcal {V}(P)$ .", "Let $Q_1$ and $Q_2$ be elements of $\\mathcal {C}$ such that $Q_1\\subseteq Q_2$ and $Q_1\\ne Q_2$ .", "It was shown in [13] that if $\\operatorname{\\textrm {Card}}(Q_2/Q_1)<\\infty $ then $\\mathcal {V}_1(Q_1)\\subseteq \\mathcal {V}_1(Q_2)$ and $\\mathcal {V}_1(Q_1)\\ne \\mathcal {V}_1(Q_2)$ , and if $\\operatorname{\\textrm {Card}}(Q_2/Q_1)=\\aleph _0$ then $\\mathcal {V}_1(Q_1)\\cap \\mathcal {V}_1(Q_2)=\\emptyset $ .", "From here, we consider the collection $\\mathcal {V}=\\lbrace V: V\\in \\mathcal {V}(Q), Q\\in \\mathcal {C}\\rbrace $ of all Vitali selectors of $\\mathbb {R}$ , and we define the semigroup $\\mathcal {S}(\\mathcal {V})=\\lbrace \\bigcup _{i=1}^n V_i: V_i\\in \\mathcal {V}, n\\in \\mathbb {N} \\rbrace $ generated by the collection $\\mathcal {V}$ of all Vitali selectors of $(\\mathbb {R}, +)$ .", "Clearly, $\\mathcal {V}(Q)\\subsetneq \\mathcal {V}$ and $\\mathcal {V}_1(Q)\\subsetneq \\mathcal {S}(\\mathcal {V})$ for each $Q\\in \\mathcal {C}$ .", "It is well known [12] that the families $\\mathcal {S}(\\mathcal {V})$ , $\\mathcal {I}_m*\\mathcal {S}(\\mathcal {V})$ and $\\mathcal {S}(\\mathcal {V})*\\mathcal {I}_m$ consist of sets without the Baire property and they are invariant under the action of $\\Phi (\\mathbb {R})$ .", "Theorem 5.4 ([16]) The families $\\mathcal {N}_0*\\mathcal {S}(\\mathcal {V})$ and $\\mathcal {S}(\\mathcal {V})*\\mathcal {N}_0$ are semigroups of sets on $\\mathbb {R}$ , for which elements are not measurable in the Lebesgue sense, such that $\\mathcal {S}(\\mathcal {V})\\subsetneq \\mathcal {N}_0*\\mathcal {S}(\\mathcal {V})\\subsetneq \\mathcal {S}(\\mathcal {V})*\\mathcal {N}_0$ , and they are invariant under the action of the group $\\Pi (\\mathbb {R})$ of all affine transformations of $\\mathbb {R}$ onto itself.", "The following theorem is a more general result than Theorem REF .", "Theorem 5.5 Let $U=\\bigcup _{i=1}^n V_i$ be a finite union of Vitali selectors of $\\mathbb {R}$ , where $V_i\\in \\mathcal {V}(Q_i)$ and each $Q_i$ is an element of $\\mathcal {C}$ for $i=1,2,\\cdots ,n$ .", "Then the set $U$ cannot contain any subset of positive Lebesgue measure.", "Suppose that there exists a Lebesgue measurable subset $Y$ of $\\mathbb {R}$ such that $\\mu (Y)>0$ and $Y\\subseteq U$ .", "Without loss of generality, we may assume that the set $Y$ is bounded.", "Let $\\vartheta $ be the restriction of $\\mu $ to $\\mathcal {B}_b(\\mathbb {R})\\cap \\operatorname{\\textrm {dom}}(\\mu )$ .", "For this $\\vartheta $ , there exists a functional $\\eta $ as in Theorem REF .", "Clearly, we have $0<\\vartheta (Y)=\\eta (Y)=\\eta (Y\\cap U)=\\eta \\left[Y\\cap \\left(\\bigcup _{i=1}^n V_i\\right)\\right]=\\eta \\left[\\bigcup _{i=1}^n \\left(Y\\cap V_i\\right)\\right]$ Inequality REF implies that $\\eta (Y\\cap V_i)>0$ for some index $i\\in \\lbrace 1,2,\\cdots ,n\\rbrace $ .", "Since $Y\\cap V_i$ is a bounded subset of the Vitali selector $V_i$ , it follows from Lemma REF that it has the property described in Lemma REF .", "According to Lemma REF , we must have the equality $\\eta (Y\\cap V_i)=0$ , but this a contradiction.", "We conclude that the set $U$ cannot contain any Lebesgue measurable set with positive measure.", "It is important to point out that Theorem REF is not valid in the case of countable unions of Vitali selectors.", "A simple way to observe this fact, is to consider Equality REF , but a more general result was proved in [13], where it was shown that if $O$ is a non-empty open subset of $\\mathbb {R}$ then there exist a sequence of Vitali selectors $V_1, V_2, \\cdots $ such that $O=\\bigcup _{i=1}^{\\infty } V_i$ .", "Such a union contains a set of positive measure.", "Corollary 5.6 No element of the family $\\mathcal {S}(\\mathcal {V})*\\mathcal {N}_0$ can contain any set of positive Lebesgue measure.", "With the help of Theorem REF , we can prove the following statement.", "Corollary 5.7 Let $U_k=\\bigcup _{i=1}^n V_{i k}$ be a finite union of Vitali selectors of $\\mathbb {R}$ , where $V_{ik}\\in \\mathcal {V}(Q_k)$ and $Q_k$ is an element of $\\mathcal {C}$ for each $k$ , and let $h_k$ be a topological group isomorphism of $(\\mathbb {R},+)$ onto itself, for each $k=1,2, \\cdots ,n$ .", "Then the union $U=\\bigcup _{k=1}^m h_k(U_k)$ cannot contain any Lebesgue measurable set of positive measure.", "By Theorem REF , it is enough to show that $U$ is a finite union of Vitali selectors of $\\mathbb {R}$ .", "Note that $h_k(U_k)=h_k\\left(\\bigcup _{i=1}^n V_{ik}\\right)=\\bigcup _{i=1}^n h_k\\left(V_{ik}\\right)$ .", "By Theorem REF , the set $h_k(V_{ik})$ is a Vitali selector related to the group $h_k(Q_k)\\in \\mathcal {C}$ .", "Hence $h_k(U_k)$ is a finite union of Vitali selectors.", "It follows that $U=\\bigcup _{k=1}^m h_k(U_k)= \\bigcup _{k=1}^m \\bigcup _{i=1}^n h_k(V_{ik})$ is also a finite union of Vitali selectors of $\\mathbb {R}$ ." ], [ "Semigroups of non-Lebesgue measurable sets generated by Bernstein sets and Vitali selectors simultaneously", "We now combine Bernstein sets and Vitali selectors of $\\mathbb {R}$ , to construct families of sets for which elements are not measurable in the Lebesgue sense.", "Theorem 6.1 Let $B$ a Bernstein set of $\\mathbb {R}$ which has an algebraic structure of being a subgroup of $(\\mathbb {R},+)$ , and let $\\mathcal {S}(\\mathcal {V})$ be the semigroup generated by the collection $\\mathcal {V}$ all Vitali selectors of $\\mathbb {R}$ .", "Any union $U=U_1\\cup U_2$ , where $U_1\\in \\mathcal {S}(\\mathcal {B})$ and $U_2\\in \\mathcal {S}(\\mathcal {V})$ , cannot contain any subset of positive Lebesgue measure.", "Assume that there exists a Lebesgue measurable set $Y$ such that $\\mu (Y)>0$ and $Y\\subseteq U=U_1\\cup U_2$ .", "Without loss of generality we may assume, we may assume that the set $Y$ is bounded.", "It follows from Proposition REF and Theorem REF that the set $Y$ cannot lie entirely in $U_1$ nor in $U_2$ .", "Write $U=\\bigcup _{i=1}^{n+m} X_i$ where $X_i=B_i \\in \\mathcal {B}$ for $i=1,2, \\cdots , n$ and $X_i=V_i\\in \\mathcal {V}$ for $i=n+1,n+2,\\cdots , n+m$ .", "Then $0<\\mu (Y)=\\mu (Y\\cap U)=\\mu \\left[\\bigcup _{i=1}^{n+m} (Y\\cap X_i)\\right]\\le \\sum _{i=1}^{n+m}\\mu (Y\\cap X_i).$ It follows from Inequality REF that $\\mu (Y\\cap X_i) >0$ for some index $i \\in \\lbrace 1,2, \\cdots , n+m\\rbrace $ .", "Since $Y\\cap X_i\\in \\mathcal {B}_b(\\mathbb {R})$ , let $\\vartheta $ be the restriction of $\\mu $ on $\\mathcal {B}_b(\\mathbb {R})\\cap \\operatorname{\\textrm {dom}}(\\mu )$ .", "For this $\\vartheta $ there exists a functional $\\eta $ as in Theorem REF which is an extension of $\\vartheta $ .", "So we have $0<\\mu (Y\\cap X_i=\\vartheta (Y\\cap X_i)=\\eta (Y\\cap X_i)$ .", "If $X_i$ is an element of $\\mathcal {V}$ then $\\eta (Y\\cap X_i)=0$ by Lemma REF .", "If $X_i$ is an element of $\\mathcal {B}$ then $\\eta (Y\\cap X_i)=0$ by Lemma REF .", "As a conclusion the set $Y$ cannot set exist.", "Corollary 6.2 Let $B$ a Bernstein set of $\\mathbb {R}$ which has an algebraic structure of being a subgroup of $(\\mathbb {R},+)$ , and let $\\mathcal {S}(\\mathcal {V})$ be the semigroup generated by the collection $\\mathcal {V}$ all Vitali selectors of $\\mathbb {R}$ .", "Then the semigroup $\\mathcal {S}(\\mathcal {B})\\vee \\mathcal {S}(\\mathcal {V})$ consists of sets which are not measurable in the Lebesgue sense, and it is invariant under the action of the group $\\Phi (\\mathbb {R})$ .", "Assume that there exists a Lebesgue measurable set $U$ in $\\mathcal {S}(\\mathcal {B})\\vee \\mathcal {S}(\\mathcal {V})$ .", "Then $U=U_1\\cup U_2$ where $U_1\\in \\mathcal {S}(\\mathcal {B})$ and $U_2\\in \\mathcal {S}(\\mathcal {V})$ .", "Since $U_2=\\bigcup _{i=1}^n V_i$ where $V_i\\in \\mathcal {V}(Q_i)$ , let $V_k$ be a fixed Vitali selector in this union such that $V_k\\in \\mathcal {V}(Q_k)$ .", "Since $\\mathbb {R}=\\bigcup \\lbrace V_k+q: q\\in Q_k\\rbrace $ and $V_k\\subseteq U_2\\subseteq U$ then we have $\\mathbb {R}=\\bigcup \\lbrace U+q: q\\in Q_k\\rbrace $ .", "Given that $\\mu (U+q)=\\mu (U)$ and $\\mu (\\mathbb {R})>0$ then we must have $\\mu (U)>0$ , and this contradicts Theorem REF .", "Since the family $\\mathcal {S}(\\mathcal {V})$ is invariant under the action of the group $\\Pi (\\mathbb {R})$ and the family $\\mathcal {S}(\\mathcal {B})$ invariant under the action of the group $\\Phi (\\mathbb {R})$ , it follows that the family $\\mathcal {S}(\\mathcal {B})\\vee \\mathcal {S}(\\mathcal {V})$ is invariant under the action of the group $\\Phi (\\mathbb {R})$ .", "Corollary 6.3 Each element of the family $\\mathcal {S}(\\mathcal {B}) * \\mathcal {S}(\\mathcal {V})$ is not measurable in the Lebesgue sense.", "Let $A\\in \\mathcal {S}(\\mathcal {B}) * \\mathcal {S}(\\mathcal {V})$ and assume that $A$ is a Lebesgue measurable set.Then $A=(U_1\\setminus U_2)\\cup U_3$ for some $U_1 \\in \\mathcal {S}(\\mathcal {B})$ and $U_2, U_3\\in \\mathcal {S}(\\mathcal {V})$ .", "Since $U_3$ is a finite union of Vitali selectors and $U_3\\subseteq A$ the set $A$ cannot have the Lebesgue measure zero.", "Assume that $\\mu (A)>0$ .", "It follows that $A=(U_1\\setminus U_2)\\cup U_3\\subseteq U_1\\cup U_3\\in \\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {V})$ , which contradicts Theorem REF .", "Theorem 6.4 The families $\\mathcal {S}(\\mathcal {V})\\vee \\mathcal {S}(\\mathcal {B}), \\mathcal {N}_0*\\left(\\mathcal {S}(\\mathcal {V})\\vee \\mathcal {S}(\\mathcal {B})\\right)$ and $\\left(\\mathcal {S}(\\mathcal {V})\\vee \\mathcal {S}(\\mathcal {B})\\right)*\\mathcal {N}_0$ are semigroups of sets on $\\mathbb {R}$ satisfying the inclusions $\\mathcal {S}(\\mathcal {V})\\vee \\mathcal {S}(\\mathcal {B}) \\subseteq \\mathcal {N}_0*\\left(\\mathcal {S}(\\mathcal {V})\\vee \\mathcal {S}(\\mathcal {B})\\right) \\subseteq \\left(\\mathcal {S}(\\mathcal {V})\\vee \\mathcal {S}(\\mathcal {B})\\right)*\\mathcal {N}_0$ .", "They are invariant under the action of the group $\\Phi (\\mathbb {R})$ and they consist of sets which are not measurable in the Lebesgue sense.", "The given families are semigroups of sets by Proposition REF , and the inclusions follow from the same statement.", "Let $A\\in \\left(\\mathcal {S}(\\mathcal {V})\\vee \\mathcal {S}(\\mathcal {B})\\right)*\\mathcal {N}_0$ and assume that $A$ is measurable in the Lebesgue sense.", "Then $A=((U_1\\cup U_2)\\setminus M)\\cup N$ , where $U_1\\in \\mathcal {S}(\\mathcal {V}), U_2\\in \\mathcal {S}(\\mathcal {B})$ and $M,N\\in \\mathcal {N}_0$ .", "Note that $A\\setminus (U_1\\cup U_2)\\subseteq N$ and $(U_1\\cup U_2)\\setminus A\\subseteq M$ and hence $A\\Delta (U_1\\cup U_2) \\subseteq M\\cup N$ .", "It follows that $\\mu (A\\Delta (U_1\\cup U_2)\\le \\mu (M\\cup N)=0$ and thus $\\mu (A\\Delta (U_1\\cup U_2)=0$ .", "It follows from Lemma REF that the set $U_1\\cup U_2$ must be measurable in the Lebesgue sense.", "But the set $U_1\\cup U_1$ is not measurable in the Lebesgue sense by Corollary REF , and this is a contradiction.", "Let us note that $\\mathcal {S}(\\mathcal {B})\\vee \\mathcal {S}(\\mathcal {V})\\subseteq \\mathcal {S}(\\mathcal {B}) * \\mathcal {S}(\\mathcal {V})$ and $\\left(\\mathcal {S}(\\mathcal {B})\\vee \\mathcal {S}(\\mathcal {V})\\right)*\\mathcal {N}_0\\subseteq \\left(\\mathcal {S}(\\mathcal {B}) * \\mathcal {S}(\\mathcal {V})\\right)*\\mathcal {N}_0$ .", "It follows from Proposition REF and Lemma REF that $\\mathcal {S}(\\mathcal {B}\\vee \\mathcal {V})*\\mathcal {N}_0=\\left(\\mathcal {S}(\\mathcal {V})\\vee \\mathcal {S}(\\mathcal {B})\\right)*\\mathcal {N}_0=(\\mathcal {S}(\\mathcal {V})*\\mathcal {N}_0)\\vee (\\mathcal {S}(\\mathcal {B})*\\mathcal {N}_0)$ .", "We also note that $\\mathcal {S}(\\mathcal {B})\\subseteq \\mathcal {S}(\\mathcal {B})\\vee \\mathcal {S}_{\\emptyset }(\\mathcal {V})$ , $\\mathcal {S}(\\mathcal {V})\\subseteq \\mathcal {S}_{\\emptyset }(\\mathcal {B})\\vee \\mathcal {S}(\\mathcal {V})$ , $\\mathcal {S}(\\mathcal {B})*\\mathcal {N}_0\\subseteq (\\mathcal {S}(\\mathcal {B})*\\mathcal {N}_0)\\vee (\\mathcal {S}_{\\emptyset }(\\mathcal {V})*\\mathcal {N}_0)$ and $\\mathcal {S}(\\mathcal {V})*\\mathcal {N}_0\\subseteq (\\mathcal {S}(\\mathcal {B}_{\\emptyset })*\\mathcal {N}_0)\\vee (\\mathcal {S}(\\mathcal {V})*\\mathcal {N}_0)$ .", "These semigroups consist of sets which are not measurable in the Lebesgue sense.", "Corollary 6.5 The families $\\mathcal {N}_0*\\left(\\mathcal {S}(\\mathcal {B}) * \\mathcal {S}(\\mathcal {V}) \\right)$ and $\\left(\\mathcal {S}(\\mathcal {B}) * \\mathcal {S}(\\mathcal {V})\\right)*\\mathcal {N}_0$ consist of elements which are not measurable in the Lebesgue sense.", "Question 6.6 Is each element of the families $\\mathcal {S}(\\mathcal {B}) \\vee \\mathcal {S}(\\mathcal {V})$ and $\\mathcal {S}(\\mathcal {B}) * \\mathcal {S}(\\mathcal {V})$ without the Baire property in $\\mathbb {R}$ ?", "The positive answer to Question REF will imply that the semigroups of sets $\\mathcal {S}(\\mathcal {V})\\vee \\mathcal {S}(\\mathcal {B}), \\mathcal {I}_m*\\left(\\mathcal {S}(\\mathcal {V})\\vee \\mathcal {S}(\\mathcal {B})\\right)$ and $\\left(\\mathcal {S}(\\mathcal {V})\\vee \\mathcal {S}(\\mathcal {B})\\right)*\\mathcal {I}_m$ , which are invariant under the action of the group $\\Phi (\\mathbb {R})$ , consist of sets without the Baire property in $\\mathbb {R}$ .", "Lemma 6.7 Let $B_1$ and $B_2$ be Bernstein of $\\mathbb {R}$ having the algebraic structures of being subgroups of $(\\mathbb {R},+)$ , and let $\\mathcal {S}(\\mathcal {V})$ be the semigroup generated by all Vitali selectors of $\\mathbb {R}$ .", "Then any union $U=U_1\\cup U_2\\cup U_3$ , where $U_1\\in \\mathcal {S}(\\mathcal {B}_1), U_2\\in \\mathcal {S}(\\mathcal {B}_3)$ and $U_3\\in \\mathcal {S}(\\mathcal {V})$ cannot contains any set of positive measure.", "In particular, the family $\\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)\\vee \\mathcal {S}(\\mathcal {V})$ is a semigroup of sets for which elements are not measurable in the Lebesgue sense, and it is invariant under the action of the group $\\Phi (\\mathbb {R})$ .", "To prove that the element $U$ cannot contain any set of positive measure, we proceed as in Theorem REF .", "It is clear that the family $\\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)\\vee \\mathcal {S}(\\mathcal {V})$ is a semigroup of sets by Lemma REF .", "The family is invariant under the action of the group $\\Phi (\\mathbb {R})$ , since both families $\\mathcal {S}(\\mathcal {B}_1),\\mathcal {S}(\\mathcal {B}_2)$ and $\\mathcal {S}(\\mathcal {V})$ are invariant under the action of the group $\\Phi (\\mathbb {R})$ .", "The following statement can be proved in a similar way as Theorem REF by taking into account Lemma REF .", "Theorem 6.8 Let $B_1$ and $B_2$ be Bernstein sets of $\\mathbb {R}$ having an algebraic structures of being subgroups of $(\\mathbb {R},+)$ , and let $\\mathcal {S}(\\mathcal {V})$ be the semigroup generated by all Vitali selectors of $\\mathbb {R}$ .", "Then the families $\\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)\\vee \\mathcal {S}(\\mathcal {V})$ , $\\mathcal {N}_0*(\\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)\\vee \\mathcal {S}(\\mathcal {V}))$ and $(\\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)\\vee \\mathcal {S}(\\mathcal {V}))*\\mathcal {N}_0$ are semigroups of sets on $\\mathbb {R}$ such that $\\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)\\vee \\mathcal {S}(\\mathcal {V})\\subseteq \\mathcal {N}_0*(\\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)\\vee \\mathcal {S}(\\mathcal {V}))\\subseteq (\\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)\\vee \\mathcal {S}(\\mathcal {V}))*\\mathcal {N}_0$ .", "They are invariant under the action of the group $\\Phi (\\mathbb {R})$ and they consist of sets that are not measurable in the Lebesgue sense.", "It follows from Proposition REF and Lemma REF that $\\mathcal {S}(\\mathcal {B}_1\\vee \\mathcal {B}_2\\vee \\mathcal {V})*\\mathcal {N}_0=(\\mathcal {S}(\\mathcal {B}_1)\\vee \\mathcal {S}(\\mathcal {B}_2)\\vee \\mathcal {S}(\\mathcal {V}))*\\mathcal {N}_0=(\\mathcal {S}(\\mathcal {B}_1) * \\mathcal {N}_0)\\vee (\\mathcal {S}(\\mathcal {B}_2)*\\mathcal {N}_0)\\vee (\\mathcal {S}(\\mathcal {V})*\\mathcal {N}_0)$ .", "We also note that $\\mathcal {S}(\\mathcal {B}_i)\\subseteq \\mathcal {S}(\\mathcal {B}_i) \\vee \\mathcal {S}_{\\emptyset }(\\mathcal {B}_j) \\vee \\mathcal {S}_{\\emptyset }(\\mathcal {V})$ for $i,j=1,2$ with $i\\ne j$ .", "Furthermore, $\\mathcal {S}(\\mathcal {B}_i)*\\mathcal {N}_0\\subseteq (\\mathcal {S}(\\mathcal {B}_i)*\\mathcal {N}_0)\\vee (\\mathcal {S}_{\\emptyset }(\\mathcal {B}_j)*\\mathcal {N}_0)\\vee (\\mathcal {S}_{\\emptyset }(\\mathcal {V})*\\mathcal {N}_0)$ and $\\mathcal {S}(\\mathcal {V})*\\mathcal {N}_0\\subseteq (\\mathcal {S}_{\\emptyset }(\\mathcal {B}_i)*\\mathcal {N}_0)\\vee (\\mathcal {S}_{\\emptyset }(\\mathcal {B}_j)*\\mathcal {N}_0)\\vee (\\mathcal {S}(\\mathcal {V})*\\mathcal {N}_0)$ .", "All these semigroups consist of sets which are not measurable in the Lebesgue sense.", "All the statements proved in which the $\\sigma $ -ideal $\\mathcal {N}_0$ is involved, they are even valid for any subideal $\\mathcal {I}$ of $\\mathcal {N}_0$ .", "We also note that those statements are valid when the family $\\mathcal {S}(\\mathcal {V})$ is replaced by $\\mathcal {V}_1(Q)$ for any countable dense subset $Q$ of $(\\mathbb {R},+)$ ." ], [ "Acknowledgements", "We would like to thank Prof. A.", "B Kharazishvili for the helpful information about Vitali selectors that he provided.", "We would like also to thank the referee for his (her) valuable comments and advice." ] ]
2105.11810
[ [ "Mechanical response of packings of non-spherical particles: A case study\n of 2D packings of circulo-lines" ], [ "Abstract We investigate the mechanical response of jammed packings of circulo-lines, interacting via purely repulsive, linear spring forces, as a function of pressure $P$ during athermal, quasistatic isotropic compression.", "Prior work has shown that the ensemble-averaged shear modulus for jammed disk packings scales as a power-law, $\\langle G(P) \\rangle \\sim P^{\\beta}$, with $\\beta \\sim 0.5$, over a wide range of pressure.", "For packings of circulo-lines, we also find robust power-law scaling of $\\langle G(P)\\rangle$ over the same range of pressure for aspect ratios ${\\cal R} \\gtrsim 1.2$.", "However, the power-law scaling exponent $\\beta \\sim 0.8$-$0.9$ is much larger than that for jammed disk packings.", "To understand the origin of this behavior, we decompose $\\langle G\\rangle$ into separate contributions from geometrical families, $G_f$, and from changes in the interparticle contact network, $G_r$, such that $\\langle G \\rangle = \\langle G_f\\rangle + \\langle G_r \\rangle$.", "We show that the shear modulus for low-pressure geometrical families for jammed packings of circulo-lines can both increase {\\it and} decrease with pressure, whereas the shear modulus for low-pressure geometrical families for jammed disk packings only decreases with pressure.", "For this reason, the geometrical family contribution $\\langle G_f \\rangle$ is much larger for jammed packings of circulo-lines than for jammed disk packings at finite pressure, causing the increase in the power-law scaling exponent." ], [ "Introduction", "Granular materials represent fascinating examples of nonequilibrium physical systems that display complex, collective behavior [1], [2], such as stick-slip motion [3], shear banding [4], and segregation [5].", "They are composed of discrete, macroscopic grains that interact via dissipative, frictional contact interactions.", "Granular materials occur in many important contexts, including numerous geological processes [6], [7], food and consumer product processing [8], [9], and robotics applications [10], [11].", "Because granular media are highly dissipative, the grains do not move unless they are driven by gravity, fluid flow, or the system boundaries.", "When granular systems are compressed to sufficiently large packing fractions, they become jammed and possess a nonzero static shear modulus and other soid-like properties [12].", "While most dry granular materials are composed of frictional, non-spherical grains, numerous computational and theoretical studies of dry granular packings have focused on the soft-particle model in which frictionless, spherical particles interact via the pairwise, purely repulsive potential energy [13]: $U(r_{ij}) \\propto (1-r_{ij}/\\sigma _{ij})^{\\alpha } \\Theta (1-r_{ij}/\\sigma _{ij})$ , where $r_{ij}$ is the separation between the centers of mass of particles $i$ and $j$ , $\\sigma _{ij}$ is their average diameter, the exponents $\\alpha =2$ and $5/2$ are set for purely repulsive linear and Hertzian spring interactions [14], and $\\Theta (.", ")$ is the Heaviside step function that ensures the potential energy is nonzero only when the particles are in contact.", "For frictionless, spherical particles, the jamming transition occurs when the number of interparticle contacts $N_c$ equals or exceeds the isostatic value $N_c^0=dN - d+1$ , where $d$ is the spatial dimension and $N$ is the number of non-rattler particles [15].", "When the system is compressed above jamming onset, the pressure increases from zero and the jammed packing develops nonzero bulk and shear moduli.", "A hallmark of the jamming transition in static packings of frictionless, spherical particles is that the ensemble-averaged contact number and shear modulus $\\langle G \\rangle $ scale as a power-law in the pressure $P$  [12], [16], above a characteristic pressure $P^{**}$ that decreases with increasing system size, as the packings are isotropically compressed above jamming onset [17].", "For example, $\\langle G \\rangle \\sim P^{1/2}$ for $P > P^{**}$ for packings of spherical particles with purely repulsive, linear spring interactions.", "Several previous experimental studies of compressed emulsions [18], [19] and packings of thin granular cylinders [20] have also found power-law scaling of the contact number and shear modulus with pressure during isotropic compression.", "In previous computational studies of packings of frictionless, spherical particles [21], we showed that there are two important contributions to the ensemble-averaged shear modulus: $G_f$ from geometrical families and $G_r$ from changes in the interparticle contact network during compression.", "For isotropic compression, jammed packings within a geometrical family are mechanically stable packings with different pressures that are related to each other by continuous, quasistatic changes in packing fraction with no changes in the interparticle contact network.", "$G_r$ includes discontinuities in the shear modulus that arise from point and jump changes in the contact network [22], [23].", "Point changes involve the addition or removal of a single interparticle contact (or multiple contacts when a rattler particle is added or removed from the contact network) without significant particle motion.", "Jump changes are caused by mechanical instabilities and typically involve multiple changes in the contact network and collective particle motion.", "For frictionless, spherical particles, the shear modulus along a geometrical family decreases with increasing pressure during isotropic compression (and no shearing of the boundaries).", "For systems with purely repulsive linear spring interactions, $G_f$ decreases roughly linearly with pressure, $G_f(P)/G_0 \\sim 1 - P/P_0$ , where $G_0$ is the shear modulus at $P=0$ and $P_0$ is the pressure at which $G_f(P)=0$ , although there are deviations from this simple form as the system approaches contact network changes [24], [21].", "Thus, the increase in the ensemble-averaged shear modulus, $\\langle G\\rangle $ , with pressure for packings of spherical particles arises from $G_r$ , where changes in the contact network cause discontinuous upward jumps in the shear modulus.", "We will show below that the sign of the second, pressure-dependent term in $G_f$ (i.e.", "whether $G_f$ increases or decreases with pressure) is determined by the curvature of geometrical families in the strain direction.", "Several prior computational studies have shown that the form of $\\langle G(P) \\rangle $ for jammed packings depends on the particle shape [25], [26].", "For example, the pressure-depenent shear modulus for packings of frictionless ellipse-shaped particles with repulsive linear spring interactions scales as $\\langle G(P) \\rangle \\sim P^{\\beta }$ , where $\\beta \\sim 1$ over a wide range of pressure, $P^{**} < P < P^*$ , $P^{**} \\sim N^{-2}$ , and $P^*$ does not depend strongly on system size.", "For $P>P^*$ , the power-law scaling crosses over to $\\langle G \\rangle \\sim P^{1/2}$ , as found for jammed packings of spherical particles [17].", "However, the ensemble-averaged shear modulus for jammed packings of dimer-shaped particles (and other composite particles formed from bonded spherical particles) scales as $\\langle G \\rangle \\sim P^{1/2}$ over the same range of pressures and for the same aspect ratios as those studied for packings of smooth, ellipse-shaped particles [26].", "Since packings of frictionless ellipse-shaped particles are hypostatic at jamming onset, while packings of dimer-shaped particles are isostatic at jamming onset, it is possible that the change in the power-law scaling exponent from $\\beta =0.5$ to $\\sim 1$ is related to the presence of low-frequency quartic modes in the vibrational response for packings of ellipse-shaped and other non-spherical particles [27].", "In this work, we address the question of what determines the power-law scaling exponent $\\beta $ for packings of non-spherical particles.", "Does $G_f$ decrease with pressure for packings of non-spherical particles?", "Are the frequency of contact network changes and the magnitude of the discontinuities in the shear modulus at contact changes different from those for packings of spherical particles?", "As a case study, we investigate the pressure-dependent mechanical response of jammed packings of circulo-lines interacting via purely repulsive, linear spring interactions in two spatial dimensions (2D) as a function of aspect ratio.", "We find several key results for the mechanical response of jammed packings of circulo-lines.", "First, the curvature of the variation of packing fraction with strain for geometrical families can be either negative or positive, and thus the shear modulus of geometrical families can either increase or decrease with pressure: $G_f/G_0 \\sim 1 \\pm P/P_0$ to linear order in pressure.", "We derive an exact expression for the pressure-dependent shear modulus of jammed packings and show that near jamming onset it can be approximated as $G_f \\sim -\\frac{1}{\\phi }\\left(\\frac{dP}{d\\gamma }\\right)_\\phi \\left(\\frac{d\\phi }{d\\gamma } \\right)_P - \\frac{P}{\\phi }\\left(\\frac{\\left(\\frac{d\\phi }{d\\gamma }\\right)_P}{d\\gamma } \\right)_\\phi $  [28], where $\\gamma $ is the shear strain.", "The first term tends to a constant, $G_0 >0$ , in the zero-pressure limit and the sign of the coefficient of the second term (that is roughly linear in $P$ ) is determined by the curvature of geometric families in the $\\phi $ -$\\gamma $ plane.", "Second, we decompose $G_f = G_a - G_{na}$ for each first, low-pressure geometrical family into its affine and non-affine contributions.", "$G_a$ gives the response of the system to a globally affine change of the particle positions and system boundaries, while $G_{na}$ also includes particle motion in response to potential energy minimization.", "We find that the non-affine term plays an important role in determining $G_f$ .", "In particular, the non-affine contribution can cause $G_f$ to increase with pressure, which does not occur in jammed packings of spherical particles.", "We also calculate the ensemble-averaged shear modulus $\\langle G\\rangle $ versus $P$ for jammed packings of circulo-lines over a range of aspect ratios ${\\cal R}$ and system sizes.", "For packings of circulo-lines, we find that $\\langle G \\rangle \\sim P^{\\beta }$ , where $\\beta \\sim 0.8$ -$0.9$ , over a range of pressures $P^{**} < P < P^*$ , where $P^{**} \\sim N^{-2}$ and $P^*$ decreases as ${\\cal R} \\rightarrow 1$ and does not depend strongly on system size.", "We find that the finite fraction of geometrical families with negative curvature in the $\\phi $ -$\\gamma $ plane causes the power-law exponent $\\beta $ for $\\langle G\\rangle $ versus $P$ to increase compared to that for jammed packings of spherical particles.", "The remainder of the article is organized as follows.", "In Sec.", ", we describe the interparticle potential and types of contacts that occur between pairs of circulo-lines, the numerical methods used to generate jammed packings of circulo-lines, and formulas for calculating the pressure, shear stress, and shear modulus.", "In Sec.", ", we describe the results concerning geometrical families, non-affine contributions to the shear modulus, point and jump changes, and the ensemble-averaged shear modulus for jammed packings of circulo-lines.", "Finally, in Sec.", ", we provide the conclusions and point to promising future research directions.", "We also include an appendix that gives explicit expressions for the affine shear modulus for packings of circulo-lines with purely repulsive, linear spring interactions." ], [ "Methods", "A circulo-line is the set of points that are equally distant from a line segment; it is thus a 2D shape composed of a rectangular middle region capped by two semi-circles on both ends.", "Fig.", "REF (a) shows a circulo-line (labeled $i$ ) with diameter of the semi-circles, $\\sigma _i$ , and length of the middle line segment, $2 l_i$ .", "We will refer to the end points of the middle line segment as the foci.", "The aspect ratio of a circulo-line is $\\mathcal {R}=(\\sigma _i+2 l_i)/\\sigma _i$ , and ${\\cal R} =1$ in the limit that the particle becomes coincident semi-circles.", "The asphericity of a circulo-line is $\\mathcal {A}=p_i^2/4\\pi a_i=\\frac{(2\\pi +4(\\mathcal {R}_i-1))^2}{4\\pi (\\pi +4(\\mathcal {R}_i-1))}$ , where $p_i$ is the perimeter and $a_i$ is the area of the circulo-line.", "In these studies, we vary ${\\cal R}-1$ and ${\\cal A}-1$ over the ranges $10^{-2}$ to 2 and $10^{-5}$ to $0.5$ , respectively.", "We study bidisperse packings of circulo-lines to inhibit positional and orientational order.", "We consider packings in rhombic boxes with edge length $L$ , periodic boundary conditions in the $x$ - and $y$ -directions, and $N/2$ large and $N/2$ small particles with end cap diameter ratio $\\sigma _l/\\sigma _s=1.4$ , but the same mass $m$ and aspect ratio ${\\cal R}$ .", "To investigate the effect of system size, we studied packings with $N=64$ , 128, 256, and 512.", "In Fig.", "REF (b)-(d), we show that there are three ways in which two circulo-lines can make contact (or make small overlaps) with each other: end-end, end-middle, and middle-middle contacts.", "Fig.", "REF (b) shows an end-end contact between circulo-lines $i$ and $j$ .", "In this case, the relevant separation $r_{ij}$ between the circulo-lines is the separation between the two closest foci on $i$ and $j$ .", "Another important distance is the separation $\\lambda _i$ between the center of circulo-line $i$ and the point at which the line from the closest foci on $j$ that is perpendicular to the axis of circulo-line $i$ intersects the axis of $i$ .", "For an end-end contact, the following three conditions must be satisfied: $r_{ij} < \\sigma _{ij} = (\\sigma _i+\\sigma _j)/2$ , $\\lambda _i > l_i$ , and $\\lambda _j > l_j$ .", "Fig.", "REF (c) shows an end-middle contact where the end of circulo-line $j$ is in contact with the middle region of circulo-line $i$ .", "For an end-middle contact, the relevant separation is the distance $r_{ji}$ between the closest focus on $i$ and the axis of $j$ .", "Similarly, we can define the separation $r_{ij}$ , but $r_{ji} \\ne r_{ij}$ .", "For an end-middle contact, the following four conditions must be satisfied: $r_{ij}< \\sigma _{ij}$ , $r_{ji} > \\sigma _{ij}$ , $\\lambda _i <l_i$ , and $\\lambda _{j} > l_j$ .", "Fig.", "REF (d) shows two circulo-lines for which the two middle regions are in contact.", "We model middle-middle contacts as two end-middle contacts, so that when a middle-middle contact becomes an end-end contact (from Fig.", "REF (d) to (b)) or end-middle contact (from Fig.", "REF (d) to (c)) due to small changes in the positions and orientations of the circulo-lines, the interparticle potential energy and forces are continuous.", "Thus, for a middle-middle contact, the following four conditions must be satisfied: $r_{ij}< \\sigma _{ij}$ , $r_{ji} < \\sigma _{ij}$ , $\\lambda _i <l_i$ , and $\\lambda _{j} < l_j$ .", "Figure: (a) A single circulo-line with middle segment length 2l i 2l_i and end cap diameter σ i \\sigma _i.", "The three types of contacts between a pair of circulo-lines ii and jj: (b) an end-end contact, (c) an end-middle contact, and (d) a middle-middle contact.", "Note that σ ij =(σ i +σ j )/2\\sigma _{ij} = (\\sigma _i+\\sigma _j)/2 and the distances r ij r_{ij}, r ji r_{ji}, λ i \\lambda _i, and λ j \\lambda _j are defined in the main text.", "(e) The effective point of contact C ij C_{ij} between two overlapping circulo-lines ii and jj is located at the midpoint of r ij r_{ij}.", "The vector ρ → ij {\\vec{\\rho }}_{ij} points from C ij C_{ij} to the center of circulo-line jj.For all three types of contacts, we use the pairwise, purely repulsive, linear spring potential $U(r_{ij}) = \\frac{\\epsilon }{2} \\left(1-\\frac{r_{ij}}{\\sigma _{ij}} \\right)^{2} \\Theta \\left( 1-\\frac{r_{ij}}{\\sigma _{ij}} \\right),$ where $\\epsilon $ is the characteristic energy scale of the interaction and $\\Theta (.", ")$ is the Heaviside step function, ensuring that pairs of circulo-lines do not interact when then are not in contact.", "The total potential energy, $U=\\sum _{i>j} U(r_{ij})$ , and pair force, ${\\vec{f}}_{ij} = (dU/dr_{ij}) {\\hat{r}}_{ij}$ , are continuous as a function of the coordinates ${\\vec{r}}_i$ of the centers of mass and orientations $\\theta _i$ of the circulo-lines.", "We use $\\epsilon $ , $\\epsilon /\\sigma _s$ , and $\\epsilon /\\sigma ^2_s$ for the units of energy, force, and stress.", "We employ the Love expression [29] to calculate the stress tensor: $\\Sigma _{\\alpha \\beta } = \\frac{1}{2L^2} \\sum ^N_{i,j=1} (f_{ij\\alpha }\\rho _{ij\\beta }+f_{ij\\beta }\\rho _{ij\\alpha }),$ where $f_{ij\\alpha }$ is the $\\alpha $ -component of the force on particle $i$ due to particle $j$ and $\\rho _{ij\\beta }$ is the $\\beta $ -component of the vector pointing from the effective contact point between two overlapping circulo-lines $i$ and $j$ to the center of $j$ .", "The effective contact point $C_{ij}$ is the midpoint of $r_{ij}$ as shown in Fig.", "REF (e).", "We define the pressure as $P=(\\Sigma _{xx}+\\Sigma _{yy})/2$ and the shear stress as $\\Sigma =-\\Sigma _{xy}$ .", "To calculate the shear modulus, we first apply an affine simple shear strain step $\\delta \\gamma $ , which changes the circulo-line center of mass positions and orientations, $x^{\\prime }_i = x_i + \\delta \\gamma y_i,$ and $\\theta ^{\\prime }_i = \\cot ^{-1}(\\cot \\theta _i+\\delta \\gamma ),$ together with Lees-Edwards periodic boundary conditions.", "($x_i$ ,$y_i$ ) gives the position of the center of mass of the $i$ th circulo-line and $\\theta _i$ gives the angle that the axis of the circulo-line makes with the $x$ -axis.", "After each simple shear strain step, we minimize the total potential energy using the FIRE algorithm and measure the shear stress $\\Sigma $ .", "We can then determine the shear modulus by calculating $G=d\\Sigma /d\\gamma $ .", "We employ an athermal, quasistatic isotropic compression protocol to generate packings of circulo-lines at jamming onset.", "We initialize the system with random positions and orientations of the circulo-lines in the dilute limit with packing fraction $\\phi < 10^{-2}$ .", "We then compress the system by $\\Delta \\phi /\\phi = 2 \\times 10^{-3}$ , minimize the total potential energy using the FIRE algorithm, and measure the pressure $P$ .", "If $P < P_t$ , where $P_t$ is the target pressure, we again compress the system by $\\Delta \\phi $ and minimize the total potential energy.", "If $P > P_t$ , we return to the previous configuration and decrease the packing fraction increment by a factor of two.", "We continue this compression process until $|P-P_t|/P_t < 10^{-5}$ .", "For $P_t < 10^{-7}$ , we find that the number of interparticle contacts satisfies the effective isostatic condition: $N_c=N_c^0-N_q$ , where $N_c^0=d_fN^{\\prime }-d+1$ , $d=2$ is the spatial dimension, the number of degrees of freedom per particle, $d_f =3$ , $N^{\\prime } = N - N_r - N_s/3$ , $N_r$ is the number of rattler particles with too few contacts to constrain the $d_f$ degrees of freedom, $N_s$ is the number of slider particles with one unconstrained translational degree of freedom, and $N_q$ is the number of quartic modes of the dynamical matrix [27].", "To investigate the pressure-dependent mechanical properties, we start with systems at $P_t =10^{-7}$ and then successively increase the target pressure over the range $10^{-7} < P_t < 10^{-2}$ .", "Figure: Pressure PP for jammed packings of N=6N=6 bidisperse disks as a function of packing fraction φ\\phi and shear strain γ\\gamma .", "White regions correspond to unjammed states.", "Points A and B correspond to the beginning and end of a geometrical family in the P→0P \\rightarrow 0 limit.", "At point B (γ∼0.17\\gamma \\sim 0.17), the system undergoes a jump change to the next geometrical family at point C. Point D indicates a point change from one geometrical family to another.Figure: (a) Pressure PP for jammed packings of N=6N=6 bidisperse circulo-lines with aspect ratio ℛ=2.0{\\cal R} = 2.0 as a function of packing fraction φ\\phi and shear strain γ\\gamma .", "White regions correspond to unjammed states.", "Points (b) and (c) correspond to the beginning and end of an upward geometrical family.", "At point (c) (γ∼0.19\\gamma \\sim 0.19), the system undergoes a jump change to point (d).", "A second upward geometrical family occurs between points (d) and (e).", "The system undergoes a point change from points (e) to (f) (γ∼0.205\\gamma \\sim 0.205).", "We also show a downward geometrical family in the P→0P\\rightarrow 0 limit (dotted line) that extends from γ∼0.82\\gamma \\sim 0.82 to 1.", "The packings in panels (b)-(f) correspond to points (b)-(f) in the φ\\phi -γ\\gamma plane in panel (a).", "The highlighted contacts in (c) do not occur in (d), and the highlighted contacts in (d) do not occur in (c).", "The highlighted contact in (e) does not occur in (f).", "EM (or ME) indicates an end-middle (or middle-end) contact between circulo-lines 3 and 4." ], [ "Results", "Here, we describe our main results in five subsections.", "In Sec.", "REF , we generalize the concept of geometrical families of jammed packings in the $\\phi $ -$\\gamma $ plane to packings of circulo-lines and show that the curvature $d^2\\phi /d\\gamma ^2$ can be both positive and negative for packings of circulo-lines.", "In Sec.", "REF , we derive a general expression for the pressure-dependent shear modulus $G$ in terms of derivatives of $U$ and $\\phi $ with respect to $\\gamma $ at fixed packing fraction and at fixed pressure.", "Using this expression, we find that the first geometrical family in the $P \\rightarrow 0$ limit scales as $G_f/G_0 \\sim 1 \\pm P/P_0$ to linear order in $P$ , where the sign of $d^2\\phi /d\\gamma ^2$ determines the sign of the second term in $G_f$ .", "In Sec.", "REF , we decompose the shear modulus for each low-pressure geometrical family, $G_f = G_a - G_{na}$ , into the affine and non-affine contributions, respectively.", "We show that the non-affine contribution to the shear modulus of the low-pressure geometrical families is larger for packings of circulo-lines compared to that for disk packings, and that the pressure dependence of $G_{na}$ can cause $G_f$ to increase with pressure, which does not occur for packings of spherical particles.", "In Sec.", "REF , we characterize point and jump changes in the contact network during isotropic compression in packings of circulo-lines (interacting via repulsive linear springs) and show that jump changes give rise to discontinuous changes in potential energy, shear stress, and shear modulus, whereas point changes give rise to discontinuous changes only in the shear modulus.", "We find that point changes are more frequent for packings of circulo-lines (compared to disk packings), but the resulting jumps in the shear modulus are smaller, over the same range of pressure as for disk packings.", "In Sec.", "REF , we discuss the results for the ensemble-averaged shear modulus $\\langle G\\rangle $ as a function of pressure.", "We show that, for large aspect ratios ${\\cal R} \\gtrsim 1.2$ , $\\langle G\\rangle $ scales as a power-law at large pressures with an exponent that is nearly a factor of 2 larger than that for disk packings.", "For small aspect ratios, ${\\cal R} \\lesssim 1.2$ , $\\langle G\\rangle $ versus pressure does not possess a single power-law scaling exponent.", "We further decompose $\\langle G\\rangle = \\langle G_f \\rangle + \\langle G_r\\rangle $ into contributions from geometrical families $G_f$ and changes in the contact network $G_r$ .", "We show that $\\langle G_r \\rangle $ is smaller for packings of circulo-lines compared to that for spherical particles, and thus the increase in the power-law exponent is caused by the fact that the shear modulus can increase in pressure during geometrical families for packings of circulo-lines." ], [ "Geometrical families of circulo-lines", "Jammed packings within a geometrical family are mechanically stable packings with different values of the pressure (and shear stress) that are related to each other by continuous, quasistatic changes in packing fraction and shear strain with no changes in the interparticle contact network.", "We showed in previous studies of jammed disk packings that near-isostatic geometrical families form upward parabolic segments in the $\\phi $ -$\\gamma $ plane [30], [28].", "An example geometrical family of $N=6$ jammed disk packings in the $P \\rightarrow 0$ limit runs from point A to B in Fig.", "REF .", "Similarly shaped geometrical families occur at higher pressures, as shown by the upward parabolic contours in Fig.", "REF .", "Geometrical families begin and end at point or jump changes in the interparticle contact network [22].", "For example, after making an infinitesimal increase in shear strain at point B ($\\gamma \\sim 0.17$ ) in Fig.", "REF , the system undergoes a jump change to point C. At a jump change, the packing becomes unstable, particles rearrange, and the packing fraction, shear stress, and shear modulus change discontinuously.", "At point D ($\\gamma \\sim 0.40$ ) in Fig.", "REF , the system undergoes a point change.", "For point changes, a single contact is added or removed from the contact network and the packing fraction and shear stress are continuous.", "For packings with purely repulsive linear spring interactions, the shear modulus is discontinuous at point changes [22].", "Figure: (a) Comparison of the shear modulus GG versus pressure PP for N=16N=16 jammed packings of bidisperse circulo-lines with ℛ=2{\\cal R}=2 obtained by calculating G=dΣ/dγG=d\\Sigma /d\\gamma numerically (circles) and by calculating the three terms G=G 1 +G 2 +G 3 G=G_1+G_2+G_3 in Eq.", "individually (crosses).", "Note that GG increases with PP for this geometrical family.", "(b) |G 1 |/G|G_1|/G (asterisks), |G 2 |/G|G_2|/G (diamonds), and |G 3 |/G|G_3|/G (crosses) as defined in Eq.", "for the same data in (a).In Fig.", "REF (a), we show the structure of geometrical families in the $\\phi $ -$\\gamma $ plane for jammed packings of $N=6$ bidisperse circulo-lines with aspect ratio ${\\cal R}=2.0$ .", "As for jammed disk packings, we find that packings of circulo-lines can form upward parabolic geometrical families, e.g.", "from point (b) to (c) in panel (a).", "The configurations that correspond to the beginning and end of this geometrical family are shown in Fig.", "REF (b) and (c).", "These two configurations share the same contact network, but the relative angles that the circulo-lines make with each other are different, e.g.", "circulo-lines 3 and 4 in panel (c) are tilted away from the horizontal axis compared to those in panel (b).", "Because pairs of circulo-lines can form three different types of contacts, the contact network can change when particles $i$ and $j$ form or break a contact, as well as when the type of contact between $i$ and $j$ changes (e.g.", "from an end-end to an end-middle contact).", "(See Sec. .)", "As a result, packings of circulo-lines possess more point changes and shorter geometrical families in the $\\phi $ -$\\gamma $ plane than disk packings.", "For example, the average terminus $P_{\\rm end}$ of the lowest-pressure geometrical family is a factor of 5 larger for $N=6$ disk packings compared to $N=6$ packings of circulo-lines with ${\\cal R}=2$ .", "In response to an infinitesimal increase in shear strain at point (c) in Fig.", "REF (a), the system undergoes a jump change to point (d).", "There are a total of eight changes in the contact network across the jump change, e.g.", "circulo-lines 1 and 2 are in contact in Fig.", "REF (c), but they are not in contact in (d) and the types of contacts between circulo-lines 3 and 4 change between Fig.", "REF (c) and (d) (from end-middle to middle-end or from middle-end to end-middle).", "At a jump change in packings of circulo-lines, the packing fraction, shear stress, and shear modulus also change discontinuously.", "(See Sec.", "REF below.)", "Another upward geometrical family occurs between points (d) and (e) in Fig.", "REF (a).", "At point (e), the system undergoes a point change.", "As shown in Fig.", "REF (e) and (f), the system loses a single contact across the point change.", "The packing fraction and shear stress are continuous, whereas we will show in Sec.", "REF that the shear modulus is discontinuous across a point change (for purely repulsive, linear spring interactions).", "In contrast to jammed disk packings, the geometrical families for packings of circulo-lines can form both upward and downward parabolic segments in the $\\phi $ -$\\gamma $ plane.", "One of the downward geometrical families is highlighted from $\\gamma \\sim 0.82$ to 1 in Fig.", "REF (a).", "In the next section, we show that the sign of the curvature of the parabolic geometrical families determines whether the shear modulus of the geometrical family increases or decreases with pressure.", "Figure: The shear modulus GG versus the pressure PP for N=16N=16 jammed packings of bidisperse circulo-lines with aspect ratio ℛ=2.0{\\cal R} =2.0.", "We show 50 different low-pressure geometrical families (black circles) and each geometrical family ends at a point or jump change in the interparticle contact network.", "The blue solid lines are best fits to G=G 0 +ηPG=G_0 + \\eta P." ], [ "Stress-dilatancy relation", "We now derive an exact expression for the shear modulus in terms of derivatives of the total potential energy $U$ , packing fraction $\\phi $ , and pressure $P$ with respect to the shear strain $\\gamma $ .", "This expression will enable us to understand the behavior of $G(P)$ within geometrical families.", "If we take infinitesmal steps $d\\phi $ and $d\\gamma $ along a geometrical family in the $\\phi $ -$\\gamma $ plane, the change in the total potential energy is $dU = -PdA - \\Sigma _{xy}Ad\\gamma ,$ where $A=L^2$ is the area of the system and $dA/A = -d\\phi /\\phi $ .", "After rearranging Eq.", "REF , we find the following expression that relates the shear stress to the dilatancy $-\\phi ^{-1} (d\\phi /d\\gamma )_{P}$ at finite pressure [28]: $\\Sigma = \\frac{1}{L^2} \\left(\\frac{dU}{d\\gamma }\\right)_P - \\frac{P}{\\phi } \\left(\\frac{d\\phi }{d\\gamma }\\right)_P.$ The shear modulus can be obtained by calculating the derivative $d \\Sigma /d\\gamma $ at constant packing fraction: $\\begin{aligned}G = \\left( \\frac{d \\Sigma }{d\\gamma } \\right)_{\\phi } =\\frac{1}{L^2}\\left(\\frac{d\\left(\\frac{dU}{d\\gamma }\\right)_{P}}{d\\gamma } \\right)_\\phi \\\\ - \\frac{P}{\\phi }\\left(\\frac{\\left(\\frac{d\\phi }{d\\gamma }\\right)_P}{d\\gamma } \\right)_\\phi - \\frac{1}{\\phi }\\left(\\frac{dP}{d\\gamma }\\right)_\\phi \\left(\\frac{d\\phi }{d\\gamma } \\right)_P.\\end{aligned}$ The shear modulus is a sum of three terms, $G=G_1+G_2+G_3$ , where $G_1=L^{-2} (d (dU/d\\gamma )_P/d\\gamma )_{\\phi }$ includes mixed derivatives of $U$ with respect to $\\gamma $ at fixed $\\phi $ and at fixed $P$ , $G_2=-P\\phi ^{-1} (d (d\\phi /d\\gamma )_{P}/d\\gamma )_{\\phi }$ is proportional to the derivative of the dilatancy with respect to $\\gamma $ at fixed $\\phi $ , and $G_3=-\\phi ^{-1} (dP/d\\gamma )_{\\phi } (d\\phi /d\\gamma )_{P}$ includes shear strain derivatives of $P$ at fixed $\\phi $ and of $\\phi $ at fixed $P$ .", "Eq.", "REF is verified numerically for a low-pressure geometrical family of circulo-line packings for which $G$ increases with $P$ in Fig.", "REF (a).", "For all disk and circulo-line packings that we have considered, $|G_1| \\ll |G_2|$ or $|G_1| \\ll |G_3|$ , and thus the first term $G_1$ in Eq.", "REF can be neglected.", "Further, we show in Fig.", "REF (b) for an $N=16$ packing of circulo-lines with ${\\cal R}=2$ that $|G_3| > |G_2|$ for $P < P^{\\prime }$ and $|G_2| > |G_3|$ for $P > P^{\\prime }$ .", "($P^{\\prime } \\sim 10^{-5}$ for this particular system size and aspect ratio.)", "We find that $G \\sim G_3 \\sim G_0>0$ in the zero-pressure limit, and thus the shear modulus for low-pressure geometrical families can be approximated as $G \\sim G_0 + \\eta P$ , where $\\eta $ is determined by the negative curvature of the geometrical families in the $\\phi $ -$\\gamma $ plane.", "In particular, $\\eta >0$ ($\\eta <0$ ) for downward (upward) geometrical families in the $\\phi $ -$\\gamma $ plane.", "In Fig.", "REF , we show the shear modulus versus pressure for 50 low-pressure geometrical families of $N=16$ bidisperse packings of circulo-lines with ${\\cal R}=2$ .", "We show that the shear modulus obeys $G \\sim G_0 + \\eta P$ within each geometrical family.", "For this system size and aspect ratio, approximately half of the geometrical families have $\\eta > 0$ and the other half have $\\eta <0$ .", "The fact that a finite fraction of geometrical families possesses upward geometrical families with $\\eta >0$ strongly influences the ensemble-averaged shear modulus $\\langle G\\rangle $ as discussed in Sec.", "REF ." ], [ "Affine and nonaffine contributions to the shear modulus for geometrical families", "The shear modulus can be decomposed into the affine and non-affine contributions: $G=G_a-G_{na}$ , where the affine contribution, $G_a=L^{-2} d^2U/d\\gamma ^2$ , is obtained by applying a global, affine simple shear strain and the non-affine contribution, $G_{na}$ , includes the relaxation process during potential energy minimization following the applied affine shear strain.", "Numerous prior studies have shown that the non-affine response dominates the shear modulus in disk packings near jamming onset  [31], [32], [33], [34], [35].", "Similar calculations of the non-affine contribution to the shear modulus of jammed packings of non-spherical particles have not been performed.", "In previous studies [24], we calculated $G_{na} = G - G_a$ for isostatic geometrical families of jammed disk packings.", "These results are also shown in Fig.", "REF (a) for $N=16$ disk packings.", "We find that $G_{na}$ increases with pressure, which causes jammed disk packings to soften (i.e.", "$G$ decreases) along compression geometrical families.", "When the geometrical families persist to high pressure, we find that $G_{na}$ increases rapidly near point and jump changes in the contact network, which causes deviations from the simple form $G_{na} \\sim G_0^{\\prime } + \\eta ^{\\prime } P$ .", "We now describe similar studies of $G_{na}$ for low-pressure geometrical families of circulo-lines, as shown in Fig.", "REF (b) for ${\\cal R}=2.0$ .", "(We show explicit expressions for the affine shear modulus $G_a$ for packings of circulo-lines in Appendix .)", "We identify several results concerning $G_{na}$ for packings of circulo-lines that are different from the results for disk packings.", "First, as discussed above, the average pressure range over which the first, low-pressure geometrical families persist is smaller for packings of circulo-lines, and $G_{na}(P)$ is well-fit by $G_{na}(P) \\sim G_0^{\\prime } +\\eta ^{\\prime } P$ .", "Second, $G_{na}$ can either increase or decrease with pressure (i.e.", "$\\eta ^{\\prime } >0$ or $\\eta ^{\\prime } <0$ ).", "In particular, packings of circulo-lines can harden (i.e.", "$G$ increases) along compression geometrical families, in contrast to disk packings.", "In addition, the ensemble-averaged $\\langle G_{na} \\rangle $ for the first geometrical family is typically larger for packings of circulo-lines compared to that for disk packings.", "For example, for packings of circulo-lines with ${\\cal R}=2.0$ , $\\langle G_{na}\\rangle $ is more than a factor of 2 larger than $\\langle G_{na}\\rangle $ for disk packings.", "Figure: (a) The non-affine contribution to the shear modulus G na G_{na} versus pressure PP for 30 low-pressure geometrical families of N=16N=16 jammed disk packings (black dots).", "(b) G na G_{na} versus PP for 57 low-pressure geometrical families of N=16N=16 bidisperse circulo-line packings with ℛ=2.0{\\cal R}=2.0 (black dots).", "Each geometrical family ends at a point or jump change in the interparticle contact network.", "The solid blue lines in both panels are fits to G na ∼G 0 ' +η ' PG_{na} \\sim G^{\\prime }_{0} + \\eta ^{\\prime } P." ], [ "Point and jump changes in the contact network", "In previous studies [22] of jammed packings of disks that interact via purely repulsive, linear spring potentials, we found that the shear modulus possesses discontinuous jumps when the packings undergo point and jump changes in the interparticle contact networks during applied deformations (such as shear or compression).", "Point changes are additions or removals of a single contact in the network.", "In contrast, jump changes are caused by mechanical instabilities and involve multiple contact changes and significant particle motion.", "In Fig.", "REF (a), we show a scatter plot of changes in $G$ and the potential energy $U$ between jammed $N=128$ disk packings that are separated by small compression steps over a range of pressure $10^{-3} < P < 10^{-2}$ .", "To generate Fig.", "REF (a), we identify the pressures at the beginning and end of all geometrical families in this pressure range ($10^{-3} < P < 10^{-2}$ ) and compare $G$ and $U$ between two packings before the contact network change and one before and one after the contact network change.", "We identify three regions of clustered points in Fig.", "REF (a): regions with 1) large $|\\Delta G|$ and large $|\\Delta U|$ , 2) large $|\\Delta G|$ and small $|\\Delta U|$ , or 3) small $|\\Delta G|$ and small $|\\Delta U|$ .", "The interparticle contact networks change for the systems in regions 1 and 2, and data in these regions correspond to jump and point changes, respectively.", "For the data in region 3, the contact network does not change, and $|\\Delta G|$ and $|\\Delta U|$ will decrease with improved force balance.", "In Fig.", "REF (b), we show similar data for $|\\Delta G|$ and $|\\Delta U|$ for $N=128$ jammed packings of circulo-lines with ${\\cal R} =2.0$ over the same pressure range as studied for jammed disk packings.", "As found previously for disk packings, jammed packings of circulo-lines undergo point and jump changes in the contact network, which lead to discontinuous jumps in $G$ .", "As found previously, point and jump changes can be differentiated because point changes have vanishing $|\\Delta U|$ , whereas jump changes have non-zero values of $|\\Delta U|$ .", "For the jammed disk packings considered in Fig.", "REF (a), jump changes accounted for $\\sim 0.13$ of the changes in the contact network.", "However, jump changes accounted for only $\\sim 0.03$ of the changes in the contact network for jammed circulo-line packings over the same range of pressure.", "Since point changes can involve changes in the type of contact between the same pair of particles, point changes are much more frequent in packings of circulo-lines compared to disk packings.", "Figure: (a) Scatter plot of the change in the shear modulus |ΔG||\\Delta G| versus the change in the potential energy |ΔU||\\Delta U| measured between packings separated by small compression steps for N=128N=128 disk packings.", "(b) Same data as in (a), but for N=128N=128 circulo-line packings with ℛ=2.0{\\cal R} =2.0.", "The red dots in region 3 indicate comparisons between two packings with the same interparticle contact networks.", "The blue points in regions 1 and 2 correspond to jump and point changes, respectively." ], [ "Ensemble-averaged shear modulus", "Numerous prior studies have shown that the ensemble-averaged shear modulus $\\langle G \\rangle $ for packings of frictionless, spherical particles increases as a power-law in pressure when $P > P^{**}$  [17], [12], where $P^{**} \\sim N^{-2}$ decreases with increasing system size.", "For finite-sized systems, we have used the following scaling form for the ensemble-averaged shear modulus [21]: $\\langle G \\rangle = \\langle G_0 \\rangle + \\frac{b P^{\\chi }}{1 + cP^{\\chi -\\beta }},$ where $\\langle G_0 \\rangle \\sim N^{-1}$ , $b$ and $c$ are constants, $\\chi $ and $\\beta $ are power-law exponents, and the large-pressure scaling exponent $\\beta \\sim 0.5$ for packings of spherical particles with repulsive linear spring interactions.", "(See Fig.", "REF (a).)", "In Fig.", "REF (b), we show $\\langle G\\rangle $ versus $P$ for packings of circulo-lines with ${\\cal R} = 1.5$ for several system sizes.", "As for disk packings, $\\langle G \\rangle = \\langle G_0\\rangle \\sim N^{-1}$ in the zero-pressure limit and $\\langle G(P) \\rangle $ increases as a power-law at large pressure.", "For packings of circulo-lines with ${\\cal R} \\gtrsim 1.2$ , we find that a scaling function with a single power-law (i.e.", "$c=0$ in Eq.", "REF ) provides a better description of $\\langle G \\rangle $ versus $P$ over the full range of pressure.", "We find that the power-law exponent $\\chi \\sim 0.9$ for ${\\cal R} =1.5$ , suggesting that the pressure-dependent mechanical properties of jammed packings of circulo-lines differ from those of jammed disk packings.", "In Fig.", "REF , we show $\\langle G \\rangle $ (normalized by the zero-pressure value $\\langle G_0\\rangle $ ) versus $P$ over a range of aspect ratios (for a single system size $N=128$ ).", "In panel (a), we compare $\\langle G\\rangle /\\langle G_0\\rangle $ for packings of circulo-lines with large aspect ratios (${\\cal R} =1.5$ and 2) and for disk packings.", "$\\langle G \\rangle /\\langle G_0\\rangle $ for the packings with large aspect ratios has a robust large-pressure scaling exponent $\\chi $ that is larger than that for disk packings.", "The inset to Fig.", "REF (b) shows that $0.8 \\lesssim \\chi \\lesssim 0.9$ for ${\\cal R} > 1.2$ .", "In panel (b), we show $\\langle G\\rangle /\\langle G_0\\rangle $ versus $P$ for small aspect ratios ${\\cal R} \\le 1.2$ .", "$\\langle G\\rangle /\\langle G_0\\rangle $ no longer has a single large-pressure scaling exponent.", "The curves have a steep region in the intermediate pressure regime from $10^{-5}$ to $10^{-3.5}$ , with scaling exponents that are comparable to those for higher aspect ratios.", "However, at larger pressures above a characteristic pressure, $P>P^*$ , the curves bend over and have scaling exponents that are much less than those in the inset to Fig.", "REF (b).", "The data also suggests that $P^*$ decreases as the aspect ratio decreases, indicating that the range of pressure over which the elevated scaling exponent occurs decreases as ${\\cal R} \\rightarrow 1$ .", "To explain the increase in the power-law scaling exponent for packings of circulo-lines, we decompose the ensemble-averaged shear modulus $\\langle G\\rangle $ into contributions from geometrical families, $\\langle G_f \\rangle $ , and from discontinuous jumps caused by changes in the interparticle contact network, $\\langle G_r\\rangle $ : $\\langle G \\rangle = \\langle G_f \\rangle + \\langle G_r \\rangle $  [21], [24].", "In Fig.", "REF (a), we compare $|\\langle G_f\\rangle |/\\langle G_0\\rangle $ and $\\langle G_r \\rangle /\\langle G_0 \\rangle $ for packings of circulo-lines with ${\\cal R} = 1.5$ and for disk packings both with $N=128$ .", "For jammed disk packings, $\\langle G_f \\rangle /\\langle G_0\\rangle $ decreases monotonically with increasing pressure, and thus there is a characteristic pressure $\\langle P_0\\rangle $ at which $\\langle G_f \\rangle = 0$ and above which $\\langle G_f \\rangle <0$ .", "For $N=128$ jammed disk packings, $\\langle P_0 \\rangle \\sim 10^{-5}$ .", "Thus, for $P > \\langle P_0 \\rangle $ , the difference between the rearrangement and geometrical family contributions, $\\langle G\\rangle = \\langle G_r\\rangle - |\\langle G_f\\rangle |$ , determines the power-law scaling behavior of the shear modulus with pressure for jammed disk packings.", "A key difference between jammed disk packings and packings of circulo-lines is that circulo-line packings possess a finite fraction of geometrical families whose shear modulus increases with pressure (see inset of Fig.", "REF (a)), which can cause $\\langle G_f \\rangle /\\langle G_0\\rangle $ to increase with pressure.", "For ${\\cal R} =1.5$ circulo-line packings with $N=128$ , we find that $\\langle G_f\\rangle /\\langle G_0\\rangle $ increases over a wide pressure range from $10^{-7} \\lesssim P \\lesssim 10^{-3.5}$ .", "For $P \\gtrsim 10^{-3.5}$ , the fraction of geometrical families that increases with pressure, $p_{\\rm up}$ , decreases dramatically, and $\\langle G_f\\rangle /\\langle G_0\\rangle $ begins decreasing with pressure.", "$\\langle G_f\\rangle /\\langle G_0\\rangle $ reaches zero near $\\langle P_0 \\rangle \\sim 10^{-2.5}$ and continues decreasing with further increases in pressure.", "In addition, the rearrangement contribution $\\langle G_r\\rangle /\\langle G_0\\rangle $ is larger for disk packings compared to packings of circulo-lines with ${\\cal R}=1.5$ , as shown in Fig.", "REF (a).", "Thus, even though the frequency of contact network changes is enhanced for packings of circulo-lines (see Sec.", "REF ), the discontinuous jumps in the shear modulus are sufficiently small for circulo-line packings that $\\langle G_r\\rangle /\\langle G_0\\rangle $ is larger for disk packings.", "Since $\\langle G_r \\rangle /\\langle G_0\\rangle $ is larger for jammed disk packings compared to circulo-line packings and $\\langle G_f\\rangle /\\langle G_0\\rangle $ increases with pressure over a wide range of pressure, it is clear that the existence of geometrical families with shear moduli that increase with pressure is responsible for the elevanted power-law scaling exponent for packings of circulo-lines with ${\\cal R} \\gtrsim 1.2$ .", "For jammed packings of circulo-lines with ${\\cal R} \\lesssim 1.2$ , we do not find a single power-law scaling exponent for $\\langle G\\rangle $ versus $P$ .", "Instead, $\\langle G(P)\\rangle $ has a power-law exponent $\\beta \\sim 0.8$ -$0.9$ for intermediate pressures and then the exponent decreases for $P \\gtrsim 10^{-2.5}$ .", "(See Fig.", "REF (b).)", "In Fig.", "REF (b), we show that $\\langle G_r \\rangle $ is much larger for circulo-line packings with ${\\cal R} =1.1$ than that with $1.5$ .", "In particular, $\\langle G_r \\rangle $ for ${\\cal R}=1.1$ is comparable to that for jammed disk packings for $P \\gtrsim 10^{-4}$ and $\\langle G_f\\rangle > 0$ over a much narrower range of pressure.", "The two results cause the lack of single power-law scaling for packings of circulo-lines with ${\\cal R} \\lesssim 1.2$ ." ], [ "Conclusions and future directions", "In this article, we studied the structural and mechanical properties of jammed packings of circulo-lines with frictionless, purely repulsive, linear spring interactions.", "We found several important results for jammed packings of circulo-lines that are different from those for jammed packings of spherical particles.", "First, we showed that packings of circulo-lines posses geometrical families that can be both concave upward or concave downward in the packing fraction-shear strain ($\\phi $ -$\\gamma $ ) plane.", "In contrast, the geometrical families are nearly always concave upward in the $\\phi $ -$\\gamma $ plane, especially at low pressure, for jammed packings of spherical particles.", "We then derived a stress-dilatancy relation for packings at finite pressure, which allowed us to show that the shear modulus for low-pressure geometrical families obeys $G_f = G_0 +\\eta P$ to linear order in pressure, where the sign of $\\eta $ is determined by the negative curvature of geometrical families in the $\\phi $ -$\\gamma $ plane.", "Thus, the shear modulus of low-pressure geometrical families increases with pressure when $d^2\\phi /d\\gamma ^2 < 0$ and decreases with pressure when $d^2\\phi /d\\gamma ^2 > 0$ .", "The fact that the shear modulus of geometrical families can increase with pressure has a profound effect on the pressure-dependent, ensemble-averaged shear modulus $\\langle G(P)\\rangle $ .", "In particular, we found that $\\langle G(P)\\rangle $ for jammed packings of circulo-lines with aspect ratios ${\\cal R} \\gtrsim 1.2$ displays robust power-law scaling over a wide range of pressure, but the scaling exponent ($\\beta \\sim 0.8$ -$0.9$ ) is nearly a factor of two larger than that for jammed disk packings ($\\beta \\sim 0.5$ ).", "For smaller aspect ratios, ${\\cal R} \\lesssim 1.2$ , $\\langle G(P)\\rangle $ does not possess a single power-law scaling exponent over the same range of pressure.", "To understand the origin of this behavior, we decomposed $\\langle G\\rangle $ into separate contributions from geometrical families, $\\langle G_f \\rangle $ , and from changes in the contact network, $\\langle G_r\\rangle $ : $\\langle G\\rangle /\\langle G_0\\rangle = \\langle G_f \\rangle /\\langle G_0 \\rangle + \\langle G_r\\rangle /\\langle G_0 \\rangle $ , where $\\langle G_0 \\rangle $ is the value of $\\langle G\\rangle $ in the zero-pressure limit.", "In general, we found that $\\langle G_r\\rangle /\\langle G_0 \\rangle $ is larger for disk packings compared to that for packings of circulo-lines, even though the frequency of changes in the contact network is larger for packings of circulo-lines.", "In contrast, we found that $\\langle G_f\\rangle /\\langle G_0 \\rangle $ is much larger for packings of circulo-lines.", "In fact, $\\langle G_f\\rangle /\\langle G_0 \\rangle <0$ for disk packings, whereas it can be positive for packings of circulo-lines in the pressure regime where the power-law scaling exponent is larger than that for disk packings.", "Thus, the presence of geometrical families with shear moduli that increase with pressure gives rise to important changes in the pressure-dependent mechanical properties for jammed packings of circulo-lines.", "These results suggest several promising areas of future research.", "First, in the present studies, we did not examine in detail how properties of jammed packings of circulo-lines in the ${\\cal R} \\rightarrow 1$ limit compare to those for jammed disk packings.", "Two issues arise in the ${\\cal R} \\rightarrow 1$ limit.", "1) Non-circular particles always possess 3 degrees of freedom per particle, whereas smooth disks possess only 2 nontrivial degrees of freedom per particle.", "Thus, in future studies, we will compare the properties of jammed packings of circulo-lines to those for packings of weakly frictional or bumpy particles, which both possess three degrees of freedom per particle [36].", "2) The current force model for circulo-lines, which considers forces between the end and middle sections of pairs of circulo-lines, does not converge to the force model obtained from Eq.", "REF in the ${\\cal R}\\rightarrow 1$ limit for packings with finite pressure.", "A force law that is a function of the square-root of the area of overlap between pairs of circulo-lines is a more promising model.", "In addition, we know that jammed packings of circulo-lines possess concave upward and concave downward geometrical families in the $\\phi $ -$\\gamma $ plane.", "Do jammed packings of other non-spherical particle shapes also possess concave upward and concave downward geometrical families?", "Can we find paticle shapes for which jammed packings only possess concave downward geometrical families in the $\\phi $ -$\\gamma $ plane?", "We have shown in previous studies that the power-law scaling exponent for $\\langle G(P)\\rangle $ for jammed packings of ellipse-shaped particles is also elevated relative to that for jammed disk packings [25], [26].", "Thus, it is likely that jammed packings of ellipse-shaped particles also possess concave downward geometrical families in the $\\phi $ -$\\gamma $ plane.", "Since the power-law scaling exponent $\\beta $ for $\\langle G(P)\\rangle $ depends on properties of the geometrical families, the frequency of contact network changes, and the size of the discontinuous jumps in $G$ caused by the contact network changes, it seems likely that the power-law scaling exponent $\\beta $ will depend sensitively on particle shape.", "Thus, it will be important to study $\\langle G(P)\\rangle $ and other mechanical properties for packings of many different particle shapes in both two- and three-dimensions.", "We acknowledge support from the Army Research Laboratory under Grant No.", "W911NF-17-1-0164 (P.W., N.O., and C.O.", "), NSF Grants No.", "DBI-1755494 (P.T.", "), No.", "CBET-2002782 (CO.), and No.", "CBET-2002797 (M.S.", "), and China Scholarship Council Grant No.", "201906340202 (S.Z.).", "This work was also supported by the High Performance Computing facilities operated by Yale’s Center for Research Computing and computing resources provided by the Army Research Laboratory Defense University Research Instrumentation Pro- gram Grant No.", "W911NF-18-1-0252." ], [ "Affine shear modulus", "In Sec.", "REF in the main text, we calculated the non-affine shear modulus $G_{na}=G-G_a$ for low-pressure geometrical families for jammed packings of circulo-lines.", "In this Appendix, we derive an expression for the affine contribution to the shear modulus, $G_a$ , for jammed packings of circulo-lines.", "For a globally affine simple shear strain, the particle positions and orientations of each circulo-line change according to Eqs.", "REF and REF , and the affine contribution to the shear stress is $\\Sigma _a=L^{-2} {\\frac{dU(r_{ij})}{d\\gamma }},$ where the contact distance vector satisfies $\\vec{r}_{ij}=\\vec{c}_{ij}+\\lambda _j \\hat{u}_j-\\lambda _i \\hat{u}_i,$ $\\vec{c}_{ij}$ =($x_{ij}$ ,$y_{ij}$ ) is the the separation vector between the centers of mass of particles $i$ and $j$ , and $\\hat{u}_i=(\\cos \\theta _i,\\sin \\theta _i)$ .", "We set $\\lambda _i=l_i$ and $\\lambda _j=l_j$ for end-end contacts and set $\\lambda _i=(\\vec{r}_{ij}+\\lambda _j \\hat{u}_j) \\cdot \\hat{u}_i$ for end-middle contacts, where $\\lambda _j=l_j$ and particle $j$ is the particle whose end is in contact with the middle section of particle $i$ .", "The affine shear modulus is $G_a=\\frac{d\\Sigma _a}{d\\gamma }=L^{-2} \\frac{d^2U(r_{ij})}{d\\gamma ^2}.$ Using Eq.", "REF , we obtain the following for the second derivative of the total potential energy with respect to shear strain $\\gamma $ : $\\frac{d^2U(r_{ij})}{d\\gamma ^2}=\\frac{1}{\\sigma _{ij}^2}\\left( \\left(\\frac{dr_{ij}}{d\\gamma }\\right)^2-(\\sigma _{ij}-r_{ij})\\frac{d^2r_{ij}}{d\\gamma ^2}\\right).$ For both end-end and end-middle contacts, the first- and second-derivatives of the contact distance with strain are given by $\\frac{dr_{ij}}{d\\gamma }=\\frac{f_1 x_{ij}+f_2 y_{ij}}{r_{ij}}$ and $\\frac{d^2r_{ij}}{d\\gamma ^2}=-\\frac{(f_1 x_{ij}+f_2 y_{ij})^2}{r^3_{ij}} + \\frac{f_1^2+f_2^2+f_3 x_{ij}+f_4 y_{ij}}{r_{ij}},$ respectively, where $f_1$ , $f_2$ , $f_3$ , and $f_4$ are functions of $\\lambda _i$ , $\\lambda _j$ , $\\theta _i$ , and $\\theta _j$ .", "For end-end contacts, $f_1=y_{ij}-\\lambda _i \\sin ^3 \\theta _i+\\lambda _j \\sin ^3 \\theta _j,$ $f_2=\\lambda _i \\sin ^2 \\theta _i \\cos \\theta _i-\\lambda _j \\sin ^2 \\theta _j \\cos \\theta _j,$ $f_3=3\\lambda _i \\sin ^4 \\theta _i \\cos \\theta _i -3\\lambda _j \\sin ^4 \\theta _j \\cos \\theta _j,$ and $f_4= \\left( 3\\sin ^2 \\theta _i-2 \\right) \\cos \\theta _i - \\left(3\\sin ^2 \\theta _j -2\\right) \\cos ^2 \\theta _j.$ For end-middle contacts, $f_1$ , $f_2$ , $f_3$ , and $f_4$ obey different expressions.", "We find $f_1=y_{ij}-\\lambda _i \\sin ^3 \\theta _i +\\lambda _j \\sin ^3 \\theta _j +f_5 \\cos \\theta _j,$ $f_2=\\lambda _i \\sin ^2 \\theta _i \\cos \\theta _i-\\lambda _j \\sin ^2 \\theta _j \\cos \\theta _j + f_5 \\sin \\theta _j,$ $f_3 & = &3\\lambda _i \\sin ^4 \\theta _i \\cos \\theta _i-3\\lambda _j \\sin ^4 \\theta _j \\cos \\theta _j \\nonumber \\\\& & + 2f_5 \\sin ^3 \\theta _j + f_6 \\cos \\theta _j,$ and $f_4 & = & \\left(3\\sin ^2 \\theta _i -2\\right) \\cos \\theta _i-\\left(3\\sin ^2 \\theta _j-2 \\right) \\cos ^2 \\theta _j \\nonumber \\\\& & - 2f_5 \\sin ^2 \\theta _j \\cos \\theta _j + f_6 \\sin \\theta _j,$ where $f_5 & = & \\left( y_{ij}+\\lambda _j \\sin ^3 \\theta _j \\right) \\cos \\theta _i \\nonumber \\\\& & + \\left(x_{ij}+\\lambda _j \\cos \\theta _j \\right) \\sin ^3 \\theta _i \\nonumber \\\\& & -\\lambda _j \\sin ^2 \\theta _j \\cos \\theta _j/\\sin \\theta _i \\nonumber \\\\& & -\\left(y_{ij}+\\lambda _j \\sin \\theta _j \\right)\\sin ^2 \\theta _i \\cos \\theta _i$ and $f_6 & = &-3\\lambda _j \\cos \\theta _i \\sin ^4 \\theta _j \\cos \\theta _j \\nonumber \\\\& & +2\\left( y_{ij}+\\lambda _j \\sin ^3 \\theta _j \\right) \\sin ^3 \\theta _i \\nonumber \\\\& & -3\\left( x_{ij}+\\lambda _j \\cos \\theta _j \\right) \\sin ^4 \\theta _i \\cos \\theta _i \\nonumber \\\\& & -\\lambda _j \\sin ^3 \\theta _j \\left( 3\\sin ^2 \\theta _j-2 \\right) \\sin \\theta _i \\nonumber \\\\& & +2\\lambda _j \\sin ^2 \\theta _j \\cos \\theta _j \\sin ^2 \\theta _i \\cos \\theta _i \\nonumber \\\\& & -\\left(y_{ij}+\\lambda _j \\cos \\theta _j \\right)\\sin ^3 \\theta _i \\left(3\\sin ^2 \\theta _i-2\\right).$" ] ]
2105.11648
[ [ "Classical and uniform exponents of multiplicative $p$-adic approximation" ], [ "Abstract Let $p$ be a prime number and $\\xi$ an irrational $p$-adic number.", "Its irrationality exponent $\\mu (\\xi)$ is the supremum of the real numbers $\\mu$ for which the system of inequalities $$ 0 < \\max\\{|x|, |y|\\} \\le X, \\quad |y \\xi - x|_{p} \\leq X^{-\\hmu} $$ has a solution in integers $x, y$ for arbitrarily large real number $X$.", "Its multiplicative irrationality exponent $\\tmu (\\xi)$ (resp., uniform multiplicative irrationality exponent $\\htmu (\\xi)$) is the supremum of the real numbers $\\hmu$ for which the system of inequalities $$ 0 < |x y|^{1/2} \\le X, \\quad |y \\xi - x|_{p} \\leq X^{-\\hmu} $$ has a solution in integers $x, y$ for arbitrarily large (resp., for every sufficiently large) real number $X$.", "It is not difficult to show that $\\mu (\\xi) \\le \\tmu(\\xi) \\le 2 \\mu (\\xi)$ and $\\htmu (\\xi) \\le 4$.", "We establish that the ratio between the multiplicative irrationality exponent $\\tmu$ and the irrationality exponent $\\mu$ can take any given value in $[1, 2]$.", "Furthermore, we prove that $\\htmu (\\xi) \\le (5 + \\sqrt{5})/2$ for every $p$-adic number $\\xi$." ], [ "Introduction", "Let $\\alpha $ be an irrational real number.", "Its irrationality exponent $\\mu (\\alpha )$ is the supremum of the real numbers $\\mu $ for which $ 0 < |y \\alpha - x| \\le \\max \\lbrace |x|, |y| \\rbrace ^{-\\mu + 1}$ or, equivalently, $0 < |\\alpha - x/y| \\le \\max \\lbrace |x|, |y| \\rbrace ^{-\\mu }$ has infinitely many solutions in nonzero integers $x, y$ .", "Since, for all nonzero integers $x, y$ with $|y \\alpha - x| \\le 1$ , we have $\\min \\lbrace |x|, |y|\\rbrace \\ge \\min \\lbrace |\\alpha |, |\\alpha |^{-1}\\rbrace \\cdot \\max \\lbrace |x|, |y|\\rbrace - 1,$ the integers $|x|$ and $|y|$ in (REF ) have the same order of magnitude and we can replace $\\max \\lbrace |x|, |y| \\rbrace $ in (REF ) by $|x y|^{1/2}$ .", "The same observation does not hold for rational approximation in $p$ -adic fields, where similar definitions give rise to two different irrationality exponents.", "Throughout this paper, we let $p$ denote a prime number and $\\mathbb {Q}_{p}$ the field of $p$ -adic numbers.", "Let $\\xi $ be an irrational $p$ -adic number.", "The irrationality exponent $\\mu (\\xi )$ of $\\xi $ is the supremum of the real numbers $\\mu $ for which $ 0 < |y \\xi - x|_{p} \\le \\max \\lbrace |x|, |y| \\rbrace ^{-\\mu }$ has infinitely many solutions in nonzero integers $x, y$ .", "Unlike in the real case, the integers $|x|$ and $|y|$ in (REF ) do not necessarily have the same order of magnitude, and one of them can be much larger than the other one.", "This has recently been pointed out by de Mathan [13], who studied whether $p$ -adic numbers $\\xi $ such that $\\inf _{x, y \\ne 0} \\, |xy| \\cdot |y \\xi - x|_p > 0$ do exist; see also [7], [1].", "Consequently, it is meaningful to also define the multiplicative irrationality exponent ${\\mu ^{\\times }}(\\xi )$ of $\\xi $ as the supremum of the real numbers ${\\mu ^{\\times }}$ for which $ 0 < |y \\xi - x|_{p} \\le \\bigl (| x y |^{1/2} \\bigr )^{-{\\mu ^{\\times }}}$ has infinitely many solutions in nonzero integers $x, y$ .", "It follows from the Minkowski Theorem (see [10] or [11]) and the obvious inequalites $\\max \\lbrace |x|, |y|\\rbrace \\le |xy| \\le (\\max \\lbrace |x|, |y|\\rbrace )^2$ valid for all nonzero integers $x, y$ that we have $ 2 \\le \\mu (\\xi ) \\le {\\mu ^{\\times }}(\\xi ) \\le 2 \\mu (\\xi ).$ An easy covering argument shows that ${\\mu ^{\\times }}(\\xi ) = 2$ for almost all $p$ -adic number $\\xi $ .", "Furthermore, the right hand side inequality of (REF ) can be an equality: for any sufficiently large integer $c$ , the $p$ -adic number $\\xi _c = 1+ \\sum _{j \\ge 1} p^{c^j}$ is well approximated by integers constructed by truncating its Hensel expansion and it satisfies ${\\mu ^{\\times }}(\\xi _c) = 2 \\mu (\\xi _c)$ ; see Theorem REF .", "Our first results, gathered in Section 2, are concerned with the study of the spectra of the exponents of approximation $\\mu $ and ${\\mu ^{\\times }}$ , that is, the set of values taken by these exponents.", "We also investigate the spectrum of their quotient ${\\mu ^{\\times }}/ \\mu $ and show that it is equal to the whole interval $[1, 2]$ .", "Beside the exponents of approximation $\\mu $ and ${\\mu ^{\\times }}$ , we consider the uniform exponents ${\\widehat{\\mu }}$ and ${\\widehat{\\mu }^{\\times }}$ defined as follows.", "Definition 1.1 Let $\\xi $ be an irrational $p$ -adic number.", "The uniform irrationality exponent ${\\widehat{\\mu }}(\\xi )$ of $\\xi $ is the supremum of the real numbers ${\\widehat{\\mu }}$ for which the system $ 0 < \\max \\lbrace |x|, |y| \\rbrace \\le X, \\quad |y \\xi - x|_{p} \\le X^{-{\\widehat{\\mu }}}$ has a solution for every sufficiently large real number $X$ .", "The uniform multiplicative irrationality exponent ${\\widehat{\\mu }^{\\times }}(\\xi )$ of $\\xi $ is the supremum of the real numbers ${\\widehat{\\mu }^{\\times }}$ for which the system $ 0 < |x y|^{1/2} \\le X, \\quad |y \\xi - x|_{p} \\le X^{-{\\widehat{\\mu }^{\\times }}}$ has a solution for every sufficiently large real number $X$ .", "Let us note that, beside the classical exponent (where the points $(x, y)$ belong to a square of area $4 X^2$ centered at the origin) and the mulitplicative exponent (where the points $(x, y)$ belong to a set of area $16 X^2 (\\log X)$ bounded by four branches of hyperbola), we can as well consider weighted exponents (where the points $(x, y)$ belong to a rectangle of area $4 X^2$ centered at the origin).", "Although most of our results can be extended to the weighted setting, we restrict for simplicity our attention to the somehow most natural exponents ${\\mu ^{\\times }}$ and ${\\widehat{\\mu }^{\\times }}$ defined above.", "We point out that $x$ and $y$ are not assumed to be coprime in (REF ), (REF ), (REF ), nor in (REF ).", "Adding this assumption would not change the values of $\\mu (\\xi )$ and ${\\mu ^{\\times }}(\\xi )$ , but would change the values of the uniform exponents at some $p$ -adic numbers $\\xi $ .", "As in the real case, it is not difficult to show that ${\\widehat{\\mu }}(\\xi ) = 2$ for every irrational $p$ -adic number $\\xi $ (this follows from [12], see also Lemma REF below).", "This implies that every irrational $p$ -adic number $\\xi $ satisfies $ 2 = {\\widehat{\\mu }}(\\xi ) \\le {\\widehat{\\mu }^{\\times }}(\\xi ) \\le 2 {\\widehat{\\mu }}(\\xi ) = 4.$ Furthermore, the example of the $p$ -adic numbers $\\xi _c$ defined above shows that the exponent ${\\widehat{\\mu }^{\\times }}$ takes values exceeding 2; see Theorem REF .", "Thus, unlike ${\\widehat{\\mu }}$ , this exponent is not trivial and deserves to be studied more closely.", "Among several results, stated in Section 3, we prove that ${\\widehat{\\mu }^{\\times }}$ is bounded from above by $(5 + \\sqrt{5})/2$ , thereby improving the upper bound 4 given by (REF ).", "Thus, unlike (REF ), the inequalities (REF ) are not best possible.", "Throughout this paper, the numerical constants implied by $\\ll $ are always positive and depend at most on the prime number $p$ .", "Furthermore, the symbol $\\asymp $ means that both inequalities $\\ll $ and $\\gg $ hold." ], [ "On the spectra of $\\mu ^{{\\times }}, {\\widehat{\\mu }}^{{\\times }}$ , and {{formula:7122b62c-94e3-4117-9ac4-5bc7cbfacf0d}}", "We begin with explicit examples of lacunary Hensel expansions, which include the $p$ -adic numbers $\\xi _c$ defined in Section 1.", "Theorem 2.1 Let $(a_k)_{k \\ge 0}$ be an increasing sequence of non-negative integers with $a_0 = 0$ and $a_{k+1} \\ge 2 a_k$ for every sufficiently large integer $k$ .", "Define $\\xi =\\sum _{k=0}^{\\infty } p^{a_{k}} = 1 + p^{a_1} + p^{a_2} + \\cdots .$ Set $c=\\liminf _{k \\rightarrow \\infty } \\frac{a_{k+1}}{a_{k}}, \\quad d = \\limsup _{k \\rightarrow \\infty }\\frac{a_{k+1}}{a_{k}},$ where $c, d$ are in $[2, + \\infty ]$ .", "Then, we have $ \\mu (\\xi )= d, \\quad \\mu ^{{\\times }}(\\xi )= 2 d,$ and $ 3-\\frac{1}{c}\\; \\le \\; {\\widehat{\\mu }}^{{\\times }}(\\xi ) \\; \\le \\; 3+\\frac{1}{d-1}.$ The left hand equality of (REF ) has been established in [4], the best rational approximations being given by the integers $\\sum _{j=0}^J p^{a_{j}}$ , with $J \\ge 1$ , obtained by truncation of the Hensel expansion of $\\xi $ .", "In view of the definition of ${\\mu ^{\\times }}$ and of (REF ), this implies the right hand equality of (REF ).", "The left hand inequality of (REF ) is proved in Section 5, while the right hand inequality is derived from (REF ) below.", "For small values of $d$ , Theorem REF below slightly sharpens the right hand inequality of (REF ).", "We believe that the left hand inequality in (REF ) is actually an equality.", "Recall that a $p$ -adic Liouville number is, by definition, an irrational $p$ -adic number whose irrationality exponent is infinite.", "The case where $c$ and $d$ are infinite yields the following statement.", "Corollary 2.2 The $p$ -adic Liouville number $\\xi _{\\infty }:= \\sum _{j=1}^{\\infty } p^{j!", "}$ satisfies ${\\widehat{\\mu }}^{{\\times }}(\\xi _{\\infty })=3$ .", "Consequently, the spectrum of ${\\widehat{\\mu }}^{{\\times }}$ contains 3.", "Inequalities (REF ) motivate the study of the joint spectrum of the exponents $\\mu $ and ${\\mu ^{\\times }}$ and of the spectrum of their quotient ${\\mu ^{\\times }}/ \\mu $ , which, by (REF ), is included in the interval $[1, 2]$ .", "Theorem 2.3 For any pair of real numbers $(\\mu ,{\\mu ^{\\times }})$ satisfying $ {\\mu ^{\\times }}> 5+\\sqrt{17}, \\quad \\frac{{\\mu ^{\\times }}}{2} \\le \\mu \\le {\\mu ^{\\times }},$ there exists a $p$ -adic number $\\xi $ such that $\\mu ^{{\\times }} (\\xi )={\\mu ^{\\times }}$ and $\\mu (\\xi )=\\mu $ .", "Consequently, the spectrum of the quotient $\\mu ^{{\\times }} / \\mu $ is equal to the whole interval $[1,2]$ .", "The restriction ${\\mu ^{\\times }}> 5+\\sqrt{17}$ in Theorem REF comes from the proof and has no reason to be best possible.", "We believe that (REF ) can be replaced by the inequalities $ \\max \\lbrace 2, {{\\mu ^{\\times }}/ 2} \\rbrace \\le \\mu \\le {\\mu ^{\\times }}$ .", "Let $\\dim $ denote the Hausdorff dimension.", "The $p$ -adic analogue of the theorem of Jarník and Besicovitch [8], [9] asserts that, for every real number $\\mu \\ge 2$ , we have $\\dim (\\lbrace \\xi \\in \\mathbb {Q}_{p}: \\mu (\\xi )\\ge \\mu \\rbrace ) =\\dim (\\lbrace \\xi \\in \\mathbb {Q}_{p}: \\mu (\\xi ) = \\mu \\rbrace ) =\\frac{2}{\\mu };$ see [2] for a more general $p$ -adic result.", "Combining this result with (REF ) and an easy covering argument, we deduce that $\\dim (\\lbrace \\xi \\in \\mathbb {Q}_{p}: \\mu ^{{\\times }}(\\xi )\\ge {\\mu ^{\\times }}\\rbrace ) =\\dim (\\lbrace \\xi \\in \\mathbb {Q}_{p}: \\mu ^{{\\times }}(\\xi ) = {\\mu ^{\\times }}\\rbrace ) =\\frac{2}{{\\mu ^{\\times }}}$ holds for every real number ${\\mu ^{\\times }}\\ge 2$ .", "Consequently, the spectrum of $\\mu ^{{\\times }}$ is equal to the whole interval $[2, +\\infty ]$ .", "It would be interesting to construct explicitly, for any real number ${\\mu ^{\\times }}\\ge 2$ , a $p$ -adic number $\\xi _{\\mu ^{\\times }}$ satisfying ${\\mu ^{\\times }}(\\xi _{\\mu ^{\\times }}) = {\\mu ^{\\times }}$ .", "For ${\\mu ^{\\times }}\\ge 4$ , such examples are given in Theorem REF .", "Problem 2.4 For ${\\mu ^{\\times }}$ any real number with $2 \\le {\\mu ^{\\times }}< 4$ , construct explicitly a $p$ -adic number $\\xi _{\\mu ^{\\times }}$ such that ${\\mu ^{\\times }}(\\xi _{\\mu ^{\\times }}) = {\\mu ^{\\times }}$ .", "The algorithm presented in Section  below may be helpful for answering Problem REF .", "However, if we impose the additional natural condition $\\mu ^{\\times }(\\xi _{\\mu ^{\\times }})> \\mu (\\xi _{\\mu ^{\\times }})$ , new difficulties occur.", "In a subsequent work, we will study more closely the classical and uniform multiplicative exponents of $p$ -adic numbers whose Hensel expansion is given by a classical combinatorial sequence, like the Thue–Morse sequence or a Sturmian sequence.", "Let us just note that the $p$ -adic Thue–Morse number $\\xi _{TM} = 1 + p^3 + p^5 + p^6 + p^9 + p^{10} + \\ldots $ satisfies $\\mu (\\xi _{TM}) = 2$ (see [6]) and ${\\mu ^{\\times }}(\\xi _{TM}) \\ge 3$ , where presumably this inequality is in fact an equality." ], [ "Upper bounds for the uniform exponent ${\\widehat{\\mu }}^{{\\times }}$", "In the main result of this section, we improve the trivial upper bound 4 given in (REF ) for the exponent of uniform approximation ${\\widehat{\\mu }^{\\times }}$ .", "Theorem 3.1 Any irrational $p$ -adic number $\\xi $ satisfies $ {\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le 3 +\\frac{2}{\\mu ^{{\\times }}(\\xi )- 2},$ $ \\mu ^{{\\times }}(\\xi ) \\ge {\\widehat{\\mu }}^{{\\times }}(\\xi )^2 -3 {\\widehat{\\mu }}^{{\\times }}(\\xi ) + 3,$ and $ {\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le \\frac{5+\\sqrt{5}}{2} = 3.6180\\ldots $ The first assertion of Theorem REF is stronger than the third one only when $\\mu ^{{\\times }}(\\xi )$ exceeds $3 +\\sqrt{5} = 5.23\\ldots $ .", "The combination of (REF ) and (REF ) gives ${\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le 3 +\\frac{2}{{\\widehat{\\mu }}^{{\\times }}(\\xi )^2 - 3 {\\widehat{\\mu }}^{{\\times }}(\\xi ) + 1},$ thus $({\\widehat{\\mu }}^{{\\times }}(\\xi ) - 1) \\bigl ( {\\widehat{\\mu }}^{{\\times }}(\\xi )^2 - 5 {\\widehat{\\mu }}^{{\\times }}(\\xi ) + 5 \\bigr ) \\le 0,$ and we obtain (REF ).", "Therefore, to establish Theorem REF , it is sufficient to prove (REF ) and (REF ).", "Note that (REF ) is of interest only for putative $\\xi $ with ${\\widehat{\\mu }}^{{\\times }}(\\xi )> 3$ .", "Combined with (REF ) it implies that $ \\mu (\\xi )\\ge \\frac{{\\widehat{\\mu }}^{{\\times }}(\\xi )^2 - 3 {\\widehat{\\mu }}^{{\\times }}(\\xi ) + 3}{2}.$ In particular, if ${\\widehat{\\mu }}^{{\\times }}(\\xi )>(3+\\sqrt{13})/2=3.3027\\ldots $ then $\\mu (\\xi )>2$ , thus, $\\xi $ is very well approximable.", "In other words, if $\\mu (\\xi )=2$ then ${\\widehat{\\mu }}^{{\\times }}(\\xi )\\le (3+\\sqrt{13})/2$ .", "We display an immediate consequence of (REF ).", "Corollary 3.2 Any $p$ -adic Liouville number $\\xi $ satisfies ${\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le 3.$ In view of Corollary REF , the upper bound 3 for Liouville numbers obtained in Corollary REF is best possible.", "We cannot exclude that ${\\widehat{\\mu }^{\\times }}$ is always bounded by 3.", "For the proof of Theorem REF , we introduce the sequence $(x_{k}^{{\\times }}, y_{k}^{{\\times }})_{k \\ge 1}$ of multiplicative best approximations to $\\xi $ , defined in Section .", "We are able to get the stronger conclusion ${\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le 3$ under certain conditions.", "Theorem 3.3 Assume that at least one of the following two claims holds (i) There exist $c > 0$ and are arbitrarily large $k$ such that $|x_{k}^{{\\times }}|\\ge c |y_{k}^{{\\times }}|$ and $|x_{k+1}^{{\\times }}|\\ge c |y_{k+1}^{{\\times }}|$ ; (ii) There exist $c > 0$ and are arbitrarily large $k$ such that $|x_{k}^{{\\times }}|\\le c |y_{k}^{{\\times }}|$ and $|x_{k+1}^{{\\times }}|\\le c |y_{k+1}^{{\\times }}|$ .", "Then we have ${\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le 3$ .", "The upper bound $\\frac{5+\\sqrt{5}}{2}$ in Theorem REF is obtained when, simultaneously, $|x_{2k}^{{\\times }}|$ is very small compared to $|y_{2k}^{{\\times }}|$ and $|x_{2k + 1}^{{\\times }}|$ is very large compared to $|y_{2k + 1}^{{\\times }}|$ , or vice versa, for every sufficiently large integer $k$ .", "We cannot exclude the existence of a $p$ -adic number whose sequence of multiplicative best approximations has this property.", "The main difference with the classical setting occurs when we estimate the $p$ -adic value of the difference between distinct rational numbers.", "Let $x, y, x^{\\prime }, y^{\\prime }$ be nonzero integers, not divisible by $p$ and such that $ x y^{\\prime } \\ne x^{\\prime } y$ .", "Then, $|x/y - x^{\\prime }/y^{\\prime }|_p^{-1} = | x y^{\\prime } - x^{\\prime } y|_p^{-1}$ is at most equal to $|x y^{\\prime }| + |x^{\\prime } y|$ , which can be much larger than the product $| x y|^{1/2}$ times $|x^{\\prime } y^{\\prime }|^{1/2}$ , in particular when simultaneously $|x|$ is much larger than $|y|$ and $|y^{\\prime }|$ is much larger than $|x^{\\prime }|$ .", "Thus, we cannot avoid to use the trivial estimate $| x y^{\\prime } - x^{\\prime } y|_p^{-1} \\le 2 \\max \\lbrace |x|, |y|\\rbrace \\max \\lbrace |x^{\\prime }|, |y^{\\prime }|\\rbrace $ , which involves the sup norm.", "It follows from (REF ) that $\\dim (\\lbrace \\xi \\in \\mathbb {Q}_{p}: {\\widehat{\\mu }}^{{\\times }}(\\xi ) \\ge {\\mu ^{\\times }}\\rbrace )\\le \\frac{2}{({\\mu ^{\\times }})^2 - 3 {\\mu ^{\\times }}+ 3 }, \\quad {\\mu ^{\\times }}\\in \\biggl [3, \\frac{5+\\sqrt{5}}{2} \\biggr ].$ Our results motivate the following question.", "Problem 3.4 Determine the Hausdorff dimension of the sets $\\lbrace \\xi \\in \\mathbb {Q}_{p}: {\\widehat{\\mu }}^{{\\times }}(\\xi ) \\ge {\\mu ^{\\times }}\\rbrace , \\quad \\lbrace \\xi \\in \\mathbb {Q}_{p}: {\\widehat{\\mu }}^{{\\times }}(\\xi ) = {\\mu ^{\\times }}\\rbrace , \\quad {\\mu ^{\\times }}\\in \\biggl [2, \\frac{5+\\sqrt{5}}{2} \\biggr ].$ We end this section with a remark.", "It follows from Theorem REF that any $p$ -adic number $\\xi $ with ${\\widehat{\\mu }^{\\times }}(\\xi ) = \\frac{5+\\sqrt{5}}{2}$ also satisfies $\\mu ^{{\\times }}(\\xi )= 3 +\\sqrt{5}.", "$ A similar situation occurs with the extremal numbers defined by Roy [15].", "These are transcendental real numbers $\\alpha $ whose uniform exponent of quadratic approximation takes the maximal possible value, that is, for which we have ${\\widehat{w}}_2 (\\alpha ) = (3 + \\sqrt{5})/2$ .", "Roy [15] proved that they satisfy $ 1 + w_2^* (\\alpha ) = 3 +\\sqrt{5}, \\quad 1 + {\\widehat{w}}_2 (\\alpha ) = \\frac{5 +\\sqrt{5}}{2},$ where $w_2^*$ and ${\\widehat{w}}_2$ denote classical and uniform exponents of quadratic approximation.", "Subsequently, Moshchevitin [14] established that every irrational, non-quadratic real number $\\alpha $ satisfies $w_2^* (\\alpha ) \\ge {\\widehat{w}}_2 (\\alpha ) ({\\widehat{w}}_2 (\\alpha ) - 1),$ $ 1 + w_2^* (\\alpha ) \\ge (1 + {\\widehat{w}}_2 (\\alpha ) )^2 - 3 (1 + {\\widehat{w}}_2 (\\alpha ) ) + 3,$ with equality when $\\alpha $ satisfies (REF ).", "Furthermore, by [5], we also have $ 1 + {\\widehat{w}}_2 (\\alpha ) \\le 3 + \\frac{2}{(1 + w_2^* (\\alpha ) ) - 2},$ with equality when $\\alpha $ satisfies (REF ).", "Since (REF ) and (REF ) are analogous to (REF ) and (REF ), respectively, this may suggest that the bounds of Theorem REF are best possible." ], [ "Preparatory results", "First, we observe that in the definitions of the exponents of approximation $\\mu $ and $\\mu ^{{\\times }}$ , we can assume that the integers $x$ and $y$ are coprime.", "This is not the case for the uniform exponents.", "The next two statements are $p$ -adic analogues of classical results in the real case, which can be easily proved using the theory of continued fractions.", "They can most likely be found in the literature, but we choose to supply short proofs for the convenience of the reader.", "Lemma 4.1 Let $\\xi $ be in $\\mathbb {Q}_{p}$ .", "There do not exist two linearly independent integer pairs $(x_{1},y_{1})$ and $(x_{2},y_{2})$ , which, setting $X_{i}=\\max \\lbrace |x_{i}|,|y_{i}|\\rbrace $ for $i=1,2$ , satisfy $|y_{i}\\xi -x_{i}|_{p} < \\frac{1}{2}X_{1}^{-1} X_{2}^{-1}, \\quad i=1,2.$ In particular, for any real number $X>1$ , the system $\\max \\lbrace |x|,|y|\\rbrace \\le X, \\quad |y \\xi - x|_{p} < \\frac{1}{2}X^{-2},$ does not have two linearly independent integer solutions.", "Lemma REF easily implies that every irrational $p$ -adic number $\\xi $ satisfies ${\\widehat{\\mu }}(\\xi ) \\le 2$ , thus ${\\widehat{\\mu }}(\\xi ) = 2$ , a fact already stated in Section .", "Assume, on the contrary, that there are two linearly independent pairs of solutions $(x_1, y_1)$ and $(x_2, y_2)$ as in the statement of the lemma.", "Write $X=\\max \\lbrace X_{1},X_{2}\\rbrace $ .", "It follows from the identity $x_{1}y_{2}-x_{2}y_{1}= y_{1}(y_{2} \\xi - x_{2}) - y_{2}(y_1 \\xi - x_{1})$ that $|x_{1}y_{2}-x_{2}y_{1}|_{p} \\le \\max \\lbrace |y_{2} \\xi - x_{2}|_{p} \\; , \\; |y_1 \\xi - x_{1}|_{p} \\rbrace < \\frac{1}{2X_{1}X_{2}}.$ Since $x_{1}y_{2} \\ne x_{2}y_{1}$ , we get $|x_{1}y_{2}-x_{2}y_{1}|_{p}\\ge \\frac{1}{|x_{1}y_{2}-x_{2}y_{1}|}\\ge \\frac{1}{|x_{1}y_{2}|+|x_{2}y_{1}|}\\ge \\frac{1}{2X_{1}X_{2}},$ a contradiction.", "The next claim follows directly since $X_{i}\\le X$ for $i=1, 2$ implies that $X^{-2}\\le X_{1}^{-1}X_{2}^{-1}$ .", "For an irrational $p$ -adic number $\\xi $ and a real number $T\\ge 1$ , let $(x(T),y(T))$ denote the pair of coprime integers which minimizes $|y(T)\\xi - x(T)|_{p}$ among all the pairs $(x,y)$ of coprime integers with $\\max \\lbrace |x|,|y|\\rbrace \\le T$ .", "As $T$ increases, this gives rise to a sequence of best approximations $(x_{j},y_{j})$ , $j \\ge 1$ , to $\\xi $ with the properties $\\max \\lbrace |x_{1}|,|y_{1}|\\rbrace <\\max \\lbrace |x_{2}|,|y_{2}|\\rbrace <\\cdots , \\quad |y_{1}\\xi -x_{1}|_{p}>|y_{2}\\xi -x_{2}|_{p}>\\cdots .$ and $|y_{j}\\xi -x_{j}|_{p}$ minimizes $|y \\xi -x|_{p}$ among all the pairs $(x,y)$ of coprime integers with $\\max \\lbrace |x|,|y|\\rbrace \\le \\max \\lbrace |x_{j}|,|y_{j}|\\rbrace $ .", "Corollary 4.2 If $(x_{k},y_{k})$ is a best approximation to $\\xi $ in $\\mathbb {Q}_{p}$ and $\\tau _{k}$ is defined by $|y_{k}\\xi - x_{k}|_{p} = \\max \\lbrace |x_{k}|,|y_{k}|\\rbrace ^{-\\tau _{k}},$ then the next best approximation $(x_{k+1},y_{k+1})$ to $\\xi $ satisfies $\\frac{1}{2}\\max \\lbrace |x_{k}|,|y_{k}|\\rbrace ^{\\tau _{k}-1} \\le \\max \\lbrace |x_{k+1}|,|y_{k+1}|\\rbrace \\le (p+1) \\max \\lbrace |x_{k}|,|y_{k}|\\rbrace ^{\\tau _{k}-1}.$ Thus, $|y_{k}\\xi - x_{k}|_{p} \\asymp \\max \\lbrace |x_{k}|,|y_{k}|\\rbrace ^{-1}\\max \\lbrace |x_{k+1}|,|y_{k+1}|\\rbrace ^{-1}.$ Set $Q_{j}=\\max \\lbrace |x_{j}|,|y_{j}|\\rbrace $ for $j \\ge 1$ .", "Assume that $Q_{k+1}< C Q_{k}^{\\tau _{k}-1}$ for some $C<\\frac{1}{2}$ .", "Since any two best approximations are linearly independent, the inequalities $|y_{k+1} \\xi - x_{k+1}|_{p}< |y_{k}\\xi - x_{k}|_{p}=Q_{k}^{-\\tau _{k}}<CQ_{k}^{-1}Q_{k+1}^{-1}<\\frac{1}{2}Q_{k}^{-1}Q_{k+1}^{-1},$ contradict Lemma REF .", "This proves the left hand estimate.", "For the right hand estimate notice that, by [10], for every positive integer $h$ , there are integers $x, y$ , not both zero, such that $|y \\xi - x|_p \\le p^{-h}$ and $\\max \\lbrace |x|, |y|\\rbrace \\le p^{h/2}$ .", "Consequently, for every $Q$ there is a pair $(x,y)$ of integers with $\\max \\lbrace |x|,|y|\\rbrace \\le Q$ and $|y \\xi -x|_{p}\\le p Q^{-2}$ .", "Set $Q=(p+1) \\max \\lbrace |x_{k}|,|y_{k}|\\rbrace ^{\\tau _{k}-1}$ .", "For any positive integer $M$ satisfying $M\\cdot \\max \\lbrace |x_{k}|,|y_{k}|\\rbrace \\le Q$ we have $|My_{k}\\xi -M x_{k}|_{p}\\ge M^{-1}|y_{k}\\xi -x_{k}|_{p}\\ge \\frac{\\max \\lbrace |x_{k}|,|y_{k}|\\rbrace }{Q}\\cdot \\max \\lbrace |x_{k}|,|y_{k}|\\rbrace ^{-\\tau _{k}}> p Q^{-2}.$ This implies that $\\max \\lbrace |x_{k+1}|,|y_{k+1}|\\rbrace \\le Q$ and completes the proof." ], [ "Proof of Theorem ", "Let $\\xi $ be as in the theorem and define the rational integers $Q_{k}= \\sum _{j=0}^{k} p^{a_{j}}, \\quad k \\ge 1.$ Then with $c_{k}= a_{k+1}/ a_{k}$ for $k \\ge 1$ we get $|\\xi -Q_{k}|_{p} = \\Bigl |\\sum _{j=k+1}^{\\infty } p^{a_{j}} \\Bigr |_{p}\\asymp p^{-a_{k+1}}\\asymp Q_{k+1}^{-1} \\asymp Q_{k}^{-c_{k}}, \\quad k \\ge 1.$ This in particular shows that $\\mu (\\xi ) \\ge d$ and ${\\mu ^{\\times }}(\\xi ) \\ge 2 d$ , since there are arbitrarily large $k$ such that $c_k$ is arbitrarily close to $d$ .", "The equality $\\mu (\\xi ) = d$ has been established in [4].", "By (REF ), this gives ${\\mu ^{\\times }}(\\xi ) \\le 2 d$ and proves (REF ).", "Set $Q_{k}^{{\\times }} = \\sqrt{Q_{k}}$ for $k \\ge 1$ .", "For a given integer $X$ , let $k$ be the index defined by $Q_{k}^{{\\times }}\\le X< Q_{k+1}^{{\\times }}$ .", "We then have $Q_{k}^{{\\times }} = \\sqrt{Q_{k}\\cdot 1} \\le X, \\quad |\\xi -Q_{k}|_{p} \\asymp (Q_{k}^{{\\times }})^{ -2c_{k} }.$ Let $M$ be the largest integral power of $p$ smaller than $X/Q_{k}^{{\\times }}$ .", "Then $\\sqrt{(M\\cdot 1) \\cdot (M\\cdot Q_{k})} \\le X$ and $|M\\xi -MQ_{k}|_{p} \\ll M^{-1}(Q_{k}^{{\\times }})^{ -2c_{k} }\\ll \\frac{Q_{k}^{{\\times }}}{X}(Q_{k}^{{\\times }})^{ -2c_{k} }\\ll X^{\\frac{1-2c_{k} }{c_{k} } -1}= X^{-3+\\frac{1}{c_{k}} } .$ By definition of $c$ , we get the lower bound ${\\widehat{\\mu }}^{{\\times }}(\\xi )\\ge 3-1/c$ in (REF ).", "The upper bound follows from (REF )." ], [ "An auxiliary result", "The proof of Theorem REF is semi-constructive and uses Theorem 6.1 For any $\\tilde{\\mu }>2$ and any $\\epsilon >0$ , there exists $\\xi $ in $\\mathbb {Z}_{p}$ with the following properties.", "There exists a sequence $((x_{j,0}, x_{j,1}))_{j \\ge 1}$ of pairs of coprime integers not divisible by $p$ , whose moduli tend to infinity, and satisfy the following properties: We have $|x_{j,0}|\\asymp |x_{j,1}|, \\quad |x_{j,1}\\xi - x_{j,0}|_{p}\\asymp |x_{j,0}|^{- \\tilde{\\mu } }\\asymp |x_{j,1}|^{- \\tilde{\\mu } }, \\quad j\\ge 1,$ We have $\\lim _{j\\rightarrow \\infty } \\frac{\\log |x_{j+1,0}|}{\\log |x_{j,0}|}= \\infty $ For every integer pair $(z_{0},z_{1})$ linearly independent of any pair $(x_{j,0},x_{j,1})$ with $j \\ge 1$ , we have $ |z_{1}\\xi - z_{0}|_{p} \\gg \\max \\lbrace |z_{0}|, |z_{1}|\\rbrace ^{-2-\\epsilon }.$ The first property implies $\\mu (\\xi ) \\ge \\tilde{\\mu }$ .", "The second property states that there are large gaps between consecutive very good approximations.", "The third states that at most finitely of the other approximations are very good, thus (when $\\epsilon <\\mu -2$ ) we have $\\mu (\\xi ) = \\tilde{\\mu }$ .", "It follows that $((x_{j,0}, x_{j,1}))_{j \\ge 1}$ is a subsequence of the sequence of best approximations defined in Section , but may not contain all of them.", "We remark that (REF ) may be sharpened, indeed, using refined estimates the proof actually yields the lower bound $\\gg _{\\varepsilon } Z^{-2} (\\log Z)^{-1-\\varepsilon }$ , where $Z=\\max \\lbrace |z_{0}|, |z_{1}|\\rbrace $ and $\\varepsilon >0$ can be taken arbitrarily small.", "Fix $\\epsilon >0$ .", "We first construct a $p$ -adic number $\\xi $ in such a way that we control the quality of its best rational approximations, apart possibly some good approximations $(x, y)$ , for which $|y \\xi - x|\\gg \\max \\lbrace |x |, |y |\\rbrace ^{-2-\\epsilon }$ .", "More precisely, for a given sequence $(\\mu _{n})_{n\\ge 1}$ with $\\mu _{n}\\ge 2+\\epsilon $ for $n \\ge 1$ , we find $\\xi $ as the $p$ -adic limit of a sequence of rationals $p_{n}/q_{n}$ with $p_n\\asymp q_n$ and upon writing $L_{i}= |p_{i}q_{i+1}-p_{i+1}q_{i}|_{p}, \\quad H_i = \\max \\lbrace |p_i|, |q_i|\\rbrace , \\quad i \\ge 0,$ we have $ H_n^{-\\mu _n} \\le L_n \\le p H_n^{-\\mu _n}, \\quad n\\ge 1,$ and (REF ) holds for any $(z_0,z_1)$ linearly independent of all $(p_{n},q_{n})$ .", "We construct a sequence $p_{n}/q_{n}$ that will converge at some given rate with respect to the $p$ -metric to some $p$ -adic number $\\xi $ .", "We use Schneider's continued fraction algorithm; see e.g.", "[3].", "Start with $p_{-1}=1,\\; q_{-1}=0, \\quad p_{0}=0, \\; q_{0}=1.$ Then $|p_{-1}q_{0}-q_{-1}p_{0}|=1$ .", "Then recursively let $ p_{n+1}= p_{n}+b_{n+1}p_{n-1}, \\quad q_{n+1}=q_{n}+b_{n+1}q_{n-1}$ where each $b_{n}=p^{g_{n}}$ is an integer power of $p$ to be chosen at any step accordingly.", "For all $n$ , since $|b_{n}|_{p}=b_{n}^{-1}$ we calculate $|p_{n}q_{n+1}-p_{n+1}q_{n}|_{p} &= |p_{n}(q_{n}+b_{n+1}q_{n-1})-q_{n}(p_{n}+b_{n+1}p_{n-1})|_{p}\\\\&=|b_{n+1}(p_{n}q_{n-1}-p_{n-1}q_{n})|_{p}=\\frac{1}{b_{n+1}}\\cdot |p_{n-1}q_{n}-p_{n}q_{n-1}|_{p}.", "$ Setting $L_{i}= |p_{i}q_{i+1}-p_{i+1}q_{i}|_{p}$ and $H_i = \\max \\lbrace |p_i|, |q_i|\\rbrace $ for $i\\ge 0$ as above, we see that $ L_{n}= \\frac{1}{b_{n+1}}\\cdot L_{n-1}, \\quad n \\ge 1.$ It is easy to see that all $p_{i},q_{i}$ are not divisible by $p$ , hence $\\Bigl |\\frac{p_{n+1}}{q_{n+1}}-\\frac{p_{n}}{q_{n}} \\Bigr |_{p}=|p_{n}q_{n+1}-p_{n+1}q_{n}|_{p}= \\frac{1}{b_{n+1}}\\cdot |p_{n-1}q_{n}-p_{n}q_{n-1}|_{p}.$ Since $b_{n}\\ge p>1$ , the rational numbers $p_{n}/q_{n}$ form a Cauchy sequence and thus converge with respect to the $p$ -adic metric to some $p$ -adic number $\\xi $ .", "Observe that $|q_{n}\\xi -p_{n}|_{p}= \\Bigl |\\xi -\\frac{p_{n}}{q_{n}} \\Bigr |_{p}= \\Bigl |\\frac{p_{n+1}}{q_{n+1}}-\\frac{p_{n}}{q_{n}}\\Bigr |_{p}=|p_{n+1}q_{n}-p_{n}q_{n+1}|_{p},\\quad n\\ge 1,$ where the second identity holds because obviously $\\Bigl |\\xi -\\frac{p_{n+1}}{q_{n+1}} \\Bigr |_{p} < \\Bigl |\\xi -\\frac{p_{n}}{q_{n}} \\Bigr |_{p}.$ We get from (REF ) for all $n$ that $ H_{n+1} \\le H_n + b_{n+1}H_{n-1} \\le 2 \\max \\lbrace H_{n}, b_{n+1}H_{n-1}\\rbrace .$ Assume now that for some fixed integer $N$ we have constructed $p_{1}/q_{1}, \\ldots ,p_{N}/q_{N}$ with the desired approximation properties.", "We describe how to choose $b_{N+1}$ (or $g_{N+1}$ ) to get the next $p_{N+1}/q_{N+1}$ .", "Set $\\gamma _N= L_{N-1}H_{N}H_{N-1}$ and observe that the inequality $ L_{N-1}\\le \\gamma _N \\cdot H_{N}^{-1}H_{N-1}^{-1}.$ holds.", "Now, define recursively $g_{n+1} = \\left\\lfloor \\frac{ \\log H_n^{\\mu _n} L_{n-1} }{\\log p} \\right\\rfloor ,\\quad n\\ge N,$ which is the largest integer with $b_{n+1}=p^{g_{n+1}}\\le H_n^{\\mu _n} L_{n-1}$ .", "We readily conclude from (REF ) that then indeed (REF ) holds for all $n\\ge 1$ .", "By an easy induction, it follows from (REF ) that $p_k\\asymp q_k$ for $k\\ge 1$ .", "Moreover it is clear from the recursion (REF ) that $p_n$ and $q_n$ are coprime for all $n$ .", "It remains to be shown that there is no good approximation apart from the $p_n/q_n$ , i.e., that (REF ) holds.", "For this we first estimate the growth of the height sequence $(H_{n})_{n\\ge 1}$ .", "For the initial value $n=N$ , in case the maximum in (REF ) is $b_{n+1}H_{n-1}=b_{N+1} H_{N-1}$ , by (REF ) and (REF ) we have $ H_{N+1} \\le 2b_{N+1}H_{N-1}= 2\\frac{L_{N-1}}{L_N}H_{N-1} &\\le 2L_{N-1} H_N^{\\mu _N} H_{N-1}\\nonumber \\\\& = 2H_N^{\\mu _N}(L_{N-1}H_{N-1})\\le 2\\gamma _N H_{N}^{\\mu _{N}-1},$ where we used our induction assumption (REF ).", "Then by (REF ) in view of (REF ) for $n$ (which we verified above) we have $L_{N}\\le p H_{N}^{-\\mu _{N}}\\le 2p\\gamma _{N}\\cdot H_{N+1}^{-1}H_{N}^{-1}.$ In other words, in the next step similar to (REF ) we have $L_{N}\\le \\gamma _{N+1} H_{N+1}^{-1}H_{N}^{-1}, \\quad \\gamma _{N+1}=2p \\gamma _{N}.$ Thus similarly as in (REF ) above we infer $H_{N+2} \\le 2\\gamma _{N+1} H_{N+1}^{\\mu _{N+1}-1}.$ Iterating this process we see that for all $n\\ge N$ and some fixed $c>1$ we get $L_{n}\\le \\gamma _N(2p)^n\\cdot H_{n+1}^{-1}H_{n}^{-1}\\ll c^{n}\\cdot H_{n+1}^{-1}H_{n}^{-1}$ and $ H_{n} \\le \\gamma _N(4p)^n\\cdot H_{n-1}^{\\mu _{n-1}-1}\\ll c^{n}\\cdot H_{n-1}^{\\mu _{n-1}-1}.$ Otherwise, if the maximum in (REF ) is $H_{N+1}$ , then we directly get $H_{N+1}\\le 2H_{N}\\le 2H_{N}^{\\mu _{N}-1},$ since $\\mu _{n}\\ge 2+\\epsilon >2$ , which is even stronger than the estimates derived in the other case and we infer the same result.", "Notice that by Corollary REF we have $H_{n+1}\\ge H_n^{ \\min \\lbrace \\mu _n , 2+\\epsilon \\rbrace -1 }/2 = H_n^{1+\\epsilon }/2$ for all large $n$ .", "Thus the sequence $(\\log H_{n})_{n \\ge 2}$ grows exponentially fast, so in particular $ c^{k}= H_{k}^{o(1)}, \\quad k\\rightarrow \\infty .$ It then follows from (REF ) that $ L_{k}\\ll _{\\varepsilon } H_{k+1}^{-1+\\varepsilon }H_{k}^{-1+\\varepsilon },\\quad H_{k}\\ll _{\\varepsilon } H_{k-1}^{\\mu _{k-1}-1+\\varepsilon }, \\quad k \\ge 2,$ for every $\\varepsilon >0$ and some implicit positive constants depending only on $\\varepsilon $ .", "Now assume that an integer pair $(z_0, z_1)$ is not among the $(p_n,q_n)$ and satisfies $ | z_1 \\xi - z_0| \\le Z^{-2-\\epsilon /2},$ where $Z=\\max \\lbrace |z_0|, |z_1| \\rbrace $ .", "We may assume that $(z_0, z_1)$ is linearly independent to all $(p_n,q_n)$ and that $z_0$ and $z_1$ are coprime.", "Let $k$ be the index with $H_{k-1}\\le Z< H_{k}$ .", "By coprimality and since $|q_n\\xi -p_n|\\le \\max \\lbrace p_n, q_n\\rbrace ^{-2-\\epsilon } < p_{n}^{-1}q_{n}^{-1}/2,$ Lemma REF implies that the pairs $(p_n,q_n)$ are best approximations.", "Similarly, by ${\\rm gcd} (z_0,z_1)=1$ and (REF ) similarly the pair $(z_0,z_1)$ is a best approximation as well.", "Thus, on the one hand, by Corollary REF we have $Z\\ge \\frac{H_{k-1}^{\\mu _{k-1}-1}}{2}, \\quad k\\ge k_0,$ combined with (REF ) and $Z<H_k$ we get $\\frac{\\log H_{k}}{\\log Z} \\le 1+\\eta ,$ for arbitrarily small $\\eta >0$ and large enough $k$ .", "On the other hand, again by (REF ) and Corollary REF we must have $H_{k}\\ge \\frac{Z^{1+\\epsilon /2}}{2}.$ By choosing $\\eta =\\epsilon /3$ , we end up with a contradiction for large $k$ .", "Thus, (REF ) cannot hold if $Z$ is large enough.", "We choose for $(\\mu _{n})_{n \\ge 1}$ the sequence $2+\\epsilon ,\\tilde{\\mu },2+\\epsilon ,2+\\epsilon ,\\ldots ,2+\\epsilon ,\\tilde{\\mu },2+\\epsilon ,2+\\epsilon ,\\ldots ,$ with very long blocks of $2+\\epsilon $ separating two occurrences of $\\tilde{\\mu }$ .", "We identify $x_{j,1}=q_{\\sigma (j)}$ and $x_{j,0}=p_{\\sigma (j)}$ for all $j\\ge 1$ where the injective map $\\sigma : \\mathbb {N} \\rightarrow \\mathbb {N}$ is defined so that $\\sigma (j)$ is the $j$ -th index where $\\mu _n=\\tilde{\\mu }$ .", "The property ${\\rm gcd} (p, x_{j,0} x_{j,1}) = 1 $ holds since we noticed that ${\\rm gcd} (p, p_{i} q_i) = 1$ for all $i$ .", "Moreover, the large gaps guarantee the second claim of the theorem.", "It then follows from the observations above that ${\\rm gcd}(x_{j,0}, x_{j,1})=1$ and $x_{j,0}\\asymp x_{j,1}$ for all $j$ , and the estimate $|x_{j,1} \\xi - x_{j,0}|_p \\asymp x_{j,0}^{-\\tilde{\\mu }}$ is immediate from (REF ).", "For the remaining $p_n/q_n$ with $n$ not in the image of $\\sigma $ , we have $\\mu _n=2+\\epsilon $ , and the estimate (REF ) is implied by (REF ) again.", "For all other pairs $(z_0,z_1)$ we have already shown that (REF ) does not hold if $\\max \\lbrace |z_0|, |z_1|\\rbrace $ is large enough.", "This completes the proof." ], [ "Proof of Theorem ", "We prove our Theorem REF using a similar strategy as in [16].", "The idea is to start with a $p$ -adic number $\\zeta $ given by Theorem REF and to change its Hensel expansion by replacing its digits by 0 in certain large intervals $J_{i}$ in order to obtain a $p$ -adic number $\\xi $ with the requested properties.", "This will induce good integer approximations to $\\xi $ and thereby imply that $\\mu ^{{\\times }}(\\xi )$ is rather large.", "We will see that the good approximations $x_{j,0}/x_{j,1}$ to $\\zeta $ give rise to equally good rational approximations $y_{j,0}/y_{j,1}$ to $\\xi $ , thereby showing $\\mu (\\xi )\\ge \\mu $ as well.", "The most technical part is to show that there are no better rational approximations, that is, to verify the upper bounds $\\mu (\\xi )\\le \\mu , \\mu ^{{\\times }}(\\xi )\\le \\mu ^{{\\times }}$ .", "Here we essentially use the method invented in [16] to show that putative good approximations to $\\xi $ would induce good approximations to $\\zeta $ which are not among the $x_{j,0}/x_{j,1}$ , in contradiction with Theorem REF .", "In the proof below, all $\\varepsilon _j$ are positive and can be taken arbitrarily close to 0.", "Fix $t$ in $[1, 2]$ and $\\mu >2$ .", "Let $\\zeta $ be in $\\mathbb {Z}_{p}$ which satisfies the hypotheses of Theorem REF for small enough $\\epsilon >0$ depending on $\\mu $ (this will be made more precise later) and with $\\mu (\\zeta ) = \\tilde{\\mu } = t \\mu .$ Let $(x_{j,0}, x_{j,1})_{j \\ge 1}$ denote the sequence of integer pairs given by Theorem REF .", "Without loss of generality, we assume $x_{j,0}>0$ for $j \\ge 1$ and that $x_{1,0}$ and $|x_{1,1}|$ are large.", "Let the Hensel expansion of $\\zeta $ be $\\zeta = \\sum _{i=0}^{\\infty } a_{i}p^{i},\\quad a_{i}\\in \\lbrace 0,1,\\ldots ,p-1\\rbrace .$ For $j \\ge 1$ , set $\\sigma _{j}= \\lfloor \\log x_{j,0}/\\log p\\rfloor $ so that $x_{j,0}\\asymp |x_{j,1}| \\asymp p^{\\sigma _{j}}.$ Then the second claim of Theorem REF implies that $\\sigma _{j+1}/\\sigma _{j}$ tends to infinity with $j$ .", "Partition the integers greater than or equal to $\\sigma _1$ into the intervals $I_{j}:=[\\sigma _{j}, \\sigma _{j+1})\\cap \\mathbb {Z}$ .", "We construct $\\xi $ with the desired properties by manipulating the Hensel expansion of $\\zeta $ .", "First, we derive from the sequence $(\\sigma _{j})_{j\\ge 1}$ two other positive integer sequences $(\\tau _{j})_{j\\ge 1},(\\nu _{j})_{j\\ge 1}$ defined by $ \\nu _{j}=\\lfloor t \\mu \\sigma _{j}\\rfloor +C, \\quad \\tau _{j}=\\lfloor \\mu \\nu _{j}\\rfloor ,$ for some large positive integer constant $C$ .", "For $x_{1,0}$ and $|x_{1,1}|$ sufficiently large, we have $\\sigma _{1}<\\nu _{1}<\\tau _{1}<\\sigma _{2}<\\nu _{2}<\\tau _{2}<\\cdots ,$ and since the quotient $\\sigma _{j+1} / \\sigma _j$ tends to infinity with $j$ we also have $ \\lim _{j\\rightarrow \\infty } \\frac{\\sigma _{j+1}}{\\tau _{j}}= \\infty .$ Let $J_{j}=\\lbrace \\nu _{j},\\nu _{j}+1,\\ldots ,\\tau _{j}\\rbrace =[\\nu _{j}, \\mu \\nu _{j}]\\cap \\mathbb {Z},\\quad j\\ge 1,$ so that $J_{j}\\subseteq I_{j}$ , for $j$ sufficiently large.", "Consider the $p$ -adic number $\\xi = \\sum _{i=0}^{\\infty } b_{i}p^{i},\\quad b_{i}\\in \\lbrace 0,1,\\ldots ,p-1\\rbrace ,$ derived from $\\zeta $ by setting $b_i= {\\left\\lbrace \\begin{array}{ll} 0, \\quad i\\in \\cup _{j} (J_{j}\\setminus \\lbrace \\nu _j , \\tau _{j}\\rbrace ), \\\\1, \\quad i\\in \\cup _{j} \\lbrace \\nu _j , \\tau _{j}\\rbrace , \\\\a_{i}, \\quad i \\notin \\cup _{j} J_{j}.", "\\end{array}\\right.", "}$ In other words the Hensel expansions of $\\xi $ and $\\zeta $ coincide outside the intervals $J_{j}$ , whereas the digits of $\\xi $ are all zero inside $J_{j}$ , except at the first and last position of any $J_{j}$ , where for technical reasons we put the digit 1.", "We will show that $ \\mu ^{{\\times }}(\\xi )= 2\\mu , \\quad \\mu (\\xi )=\\tilde{\\mu }= t \\mu ,$ if $\\mu $ is sufficiently large.", "This will prove the theorem as $t$ is arbitrary in $[1, 2]$ .", "We start with the easiest of the four inequalities, namely $ \\mu ^{{\\times }}(\\xi )\\ge 2\\mu .$ Define the integers $ N_{j}= \\sum _{i=0}^{\\nu _{j}}b_{i}p^{i}, \\quad j\\ge 1.$ Clearly $N_j \\ll p^{\\nu _j}$ .", "Morever, as $\\xi $ has zero digits at places ranging from $\\nu _j+1$ to $\\tau _j-1\\approx \\mu \\nu _j$ , the integers $N_j$ approximate $\\xi $ at the order roughly $\\mu $ , hence $ |\\xi -N_{j}|_{p}\\ll p^{-\\tau _{j}} \\ll p^{-\\nu _{j} \\mu }\\ll N_{j}^{-\\mu }=(\\sqrt{1\\cdot N_{j}})^{-2\\mu }.$ We directly deduce (REF ) from (REF ).", "We remark that $|\\xi -N_{j}|_{p}\\ge p^{-\\tau _{j}}$ since $b_{\\tau _{j}}=1$ , and furthermore, since $b_{\\nu _j}=1$ , we get in fact $N_j \\asymp p^{\\nu _j}$ .", "So we can refine (REF ) as $ |\\xi -N_{j}|_{p} \\asymp p^{-\\nu _j \\mu }\\asymp N_j^{-\\mu }.$ Next we show that $ \\mu (\\xi )\\ge t \\mu .$ By (REF ) the pairs $(x_{0},x_{1})=(N_{j}, 1)$ only induce approximations of quality $\\mu $ .", "By manipulating the pairs $(x_{j,0}, x_{j,1})$ associated to $\\zeta $ , we construct better approximating sequences $(y_{j,0})_{j\\ge 1}, (y_{j,1})_{j\\ge 1}$ such that, for any given $\\varepsilon _1>0$ and sufficiently large $j\\ge j_{0}(\\varepsilon _1)$ , we have $ |y_{j,1}\\xi -y_{j,0}|_{p} \\ll \\max \\lbrace |y_{j,0}|, |y_{j,1}| \\rbrace ^{-t \\mu +\\varepsilon _1}.$ This obviously implies (REF ).", "To construct suitable $y_{j,0},y_{j,1}$ , recall that $|x_{j,0}|\\asymp |x_{j,1}|\\asymp p^{\\sigma _{j}}$ and $\\sigma _{j}<\\nu _{j}<\\tau _{j}<\\sigma _{j+1}$ for $j \\ge 1$ , with $\\lim _{j \\rightarrow \\infty } \\frac{\\nu _{j}}{\\sigma _{j}} = t \\mu , \\quad \\lim _{j \\rightarrow \\infty } \\frac{\\sigma _{j+1}}{\\tau _{j}} = + \\infty , \\quad \\lim _{j \\rightarrow \\infty } \\frac{\\tau _{j}}{\\nu _{j}} = \\mu .$ For $i\\ge 1$ define $ u_{i}= \\sum _{j\\in J_{i} } a_{j}p^{j} -p^{\\nu _{i}} -p^{\\tau _{i}}, \\qquad u^{(i)}=u_{1}+u_{2}+\\cdots +u_{i}.$ Notice that by construction $\\zeta -\\xi $ is the infinite sum $u_1 + u_2 + \\ldots $ .", "Moreover, assuming that $\\epsilon < (\\mu - 2)/2$ , we note that $ p^{\\tau _i(\\frac{1}{2} - \\epsilon )}\\ll |u^{(i)}|\\ll p^{\\tau _i},\\qquad \\qquad i\\ge 1.$ The right estimate is obvious.", "If the left one is not satisfied, then $a_j=0$ for $\\lfloor \\eta _i\\rfloor \\le j\\le \\tau _i-1$ , where $\\eta _i:= (\\frac{1}{2}-\\epsilon )\\tau _i$ and $a_{\\tau _{i}}=1$ .", "But then the integer $M_i= \\sum _{j\\le \\lfloor \\eta _i\\rfloor } a_jp^{j}$ satisfies $| \\zeta - M_i|_p \\ll p^{-\\tau _i}\\ll M_i^{- \\tau _i / \\eta _i }\\ll M_i^{- 1 / (\\frac{1}{2}-\\epsilon ) }= M_{i}^{-2-\\epsilon - \\frac{3\\epsilon +2\\epsilon ^2}{1-2\\epsilon } },$ a contradiction with Theorem REF for large $i$ .", "We claim that if we set $ y_{j,0}= x_{j,0}- u^{(j-1)}x_{j,1}, \\quad y_{j,1}= x_{j,1},$ then indeed (REF ) holds.", "We rearrange $ |y_{j,1}\\xi -y_{j,0}|_{p}&=|x_{j,1}(\\xi +u^{(j-1)})-x_{j,0}|_{p}\\\\&=|x_{j,1}(\\xi +u^{(j-1)}-\\zeta )+(x_{j,1}\\zeta -x_{j,0})|_{p}.", "\\nonumber \\\\&\\le \\max \\lbrace |x_{j,1}(\\xi +u^{(j-1)}-\\zeta )|_p, |x_{j,1}\\zeta -x_{j,0}|_{p} \\rbrace .", "\\nonumber $ By assumption the latter term satisfies $ |x_{j,1}\\zeta -x_{j,0}|_{p} \\ll x_{j,0}^{-t \\mu }.$ To estimate the former expression, note that by construction the Hensel expansions of $\\xi +u^{(j-1)}$ and $\\zeta $ coincide up to digit $\\nu _{j}-1$ (last digit before the interval $J_{j}$ starts).", "Thus, we have $ |x_{j,1}(\\xi +u^{(j-1)}-\\zeta )|_{p}\\le |\\xi +u^{(j-1)}-\\zeta |_{p} \\ll p^{-\\nu _{j}}\\ll x_{j,0}^{-\\nu _{j}/\\sigma _{j}}\\ll x_{j,0}^{-t \\mu },$ where the last estimate follows from (REF ).", "By combining (REF ), (REF ), and (REF ), we derive $ |y_{j,1}\\xi -y_{j,0}|_{p} \\ll x_{j,0}^{-t \\mu }\\ll |x_{j,1}|^{-t \\mu }= |y_{j,1}|^{-t \\mu }.$ Now for given $\\varepsilon _2>0$ and large $j\\ge j_{0}(\\varepsilon _2)$ , we get from (REF ) the estimate $|u^{(j-1)}|\\ll p^{\\tau _{j-1}}< p^{\\varepsilon _2 \\sigma _{j}}\\ll x_{j,0}^{\\varepsilon _2}.$ Combined with (REF ) and recalling that $x_{j,0}\\asymp |x_{j,1}|$ we infer $ |y_{j,0}|=|x_{j,0}-u^{(j-1)}x_{j,1}| \\ll x_{j,0}+|u^{(j-1)}|\\cdot |x_{j,1}|\\ll |x_{j,1}|^{1+\\varepsilon _2}=|y_{j,1}|^{1+\\varepsilon _2},$ hence we derive (REF ) from (REF ), and consequently (REF ) follows.", "At this point we notice that the reverse inequality $|y_{j,0}|\\gg |y_{j,1}|$ follows similarly via $ |y_{j,0}|=|x_{j,0}-u^{(j-1)}x_{j,1}|\\gg |x_{j,1}|\\cdot |u^{(j-1)}|\\ge |x_{j,1}|=|y_{j,1}|,$ where we use that $x_{j,0}\\asymp |x_{j,1}|$ and the fact that $|u^{(j)}|$ tends to infinity with $j$ by (REF ).", "So we keep in mind for the sequel that all the integers $x_{j,0}, |x_{j,1}|, |y_{j,0}|, |y_{j,1}|$ are of comparable size, in the sense that, for every $\\delta > 0$ and for every sufficiently large $j$ , we have $\\max \\lbrace x_{j,0}, |x_{j,1}|, |y_{j,0}|, |y_{j,1}| \\rbrace \\le (\\min \\lbrace x_{j,0}, |x_{j,1}|, |y_{j,0}|, |y_{j,1}| \\rbrace )^{1 + \\delta }.$ Next we show the reverse estimate $ \\mu (\\xi )\\le t \\mu .$ Assume otherwise that there are integers $x,y$ with $\\max \\lbrace |x|,|y|\\rbrace $ arbitrarily large and $\\theta > t \\mu $ such that $ |y \\xi -x|_{p} \\le \\max \\lbrace |x|,|y|\\rbrace ^{-\\theta }.$ We may assume that $x$ and $y$ are coprime, by the comments in Section .", "We distinguish two cases.", "Case 1: The pair $(x,y)$ is among the pairs $(y_{j,0}, y_{j,1})$ defined in (REF ) above.", "We show the reverse estimate to (REF ), that is, $ |y_{j,1}\\xi -y_{j,0}|_{p} \\gg \\max \\lbrace |y_{j,0}|, |y_{j,1}| \\rbrace ^{-t \\mu }, \\quad j\\ge 1.$ This clearly contradicts (REF ) for these pairs.", "By assumption the reverse estimate to (REF ) holds as well, i.e.", "$|x_{j,1}\\zeta -x_{j,0}|_{p} \\gg x_{j,0}^{-t \\mu }.$ Recall that for $a,b$ in $\\mathbb {Q}_{p}$ if $|a|_{p}\\ne |b|_{p}$ then $|a+b|_{p}=\\max \\lbrace |a|_{p},|b|_{p}\\rbrace $ .", "Now by (REF ) the upper bound in (REF ) with the parameter $C$ taken large enough will be strictly smaller than this value $x_{j,0}^{-t \\mu }$ , hence applying the above argument to $a= x_{j,1}\\zeta -x_{j,0} , \\quad b= x_{j,1}(\\xi +u^{(j-1)}-\\zeta )$ allows us, in view of (REF ), to derive $|y_{j,1}\\xi -y_{j,0}|_{p} \\gg x_{j,0}^{-t \\mu } \\gg |x_{j,1}|^{-t \\mu }=|y_{j,1}|^{-t \\mu }\\ge \\max \\lbrace |y_{j,0}|, |y_{j,1}| \\rbrace ^{-t \\mu },$ our desired lower bound (REF ).", "Case 2: The pair $(x,y)$ is not among the $(y_{j,0}, y_{j,1})$ .", "Write $H= \\max \\lbrace |x|, |y|\\rbrace .$ In fact we show that then $ |y \\xi - x|_{p} \\gg H^{-\\mu -\\varepsilon _3}.$ Since $\\mu \\le t \\mu $ this clearly implies (REF ).", "Note that the bound is optimal as by (REF ) it is attained with $\\varepsilon _3=0$ by $(x,y)=(N_j, 1)$ .", "However, by the same argument, we can exclude these pairs in our investigation.", "For other pairs, we verify (REF ) indirectly by showing that any pair $(x,y)$ that violates the inequality induces a reasonably good rational approximation to $\\zeta $ which is not among the $x_{j,0}/x_{j,1}$ , contradicting the third claim of Theorem REF .", "So, assume that for some $(x,y)$ as above we have $ |y \\xi - x|_{p} \\le H^{-\\mu -\\varepsilon _3}.$ Below (REF ) we noticed that $|y_{j,1}|=|x_{j,1}|$ and $|y_{j,0}|$ are of comparable size, all being roughly equal to $x_{j,0}\\asymp |x_{j,1}|$ .", "In particular, the sequences $(|y_{j,0}|)_{j \\ge 1}$ and $(|y_{j,1}|)_{j \\ge 1}$ are increasing.", "For a pair $(x,y)$ satisfying (REF ), let $h$ be the index with $\\max \\lbrace |y_{h,0}|, |y_{h,1}| \\rbrace < H \\le \\max \\lbrace |y_{h+1,0}|, |y_{h+1,1}| \\rbrace .$ In view of (REF ) and (REF ), we derive from Corollary  REF $ \\max \\lbrace |y_{h+1,0}|, |y_{h+1,1}|\\rbrace ^{\\frac{1}{\\mu - 1}+\\varepsilon _4}\\gg H \\gg \\max \\lbrace |y_{h,0}|, |y_{h,1}|\\rbrace ^{t \\mu - 1 -\\varepsilon _4}.$ However, the right hand estimate is not sufficient.", "We show the stronger lower bound $ H \\gg \\max \\lbrace |y_{h,0}|, |y_{h,1}|\\rbrace ^{t \\mu (\\mu - 1)},$ again by application of Lemma REF .", "For simplicity write $s= \\frac{\\log H}{\\log x_{h,0}}.$ Recalling that all $x_{j,0}, x_{j,1}, y_{j,0}, y_{j,1}$ are in absolute value roughly of the same size, we have to show $s\\ge t \\mu (\\mu - 1) -\\varepsilon _{5}$ for arbitrarily small $\\varepsilon _{5}>0$ .", "According to (REF ), upon increasing $\\varepsilon _{4}$ to take into account the implied constants if necessary, we can assume $s\\ge t \\mu - 1 -\\varepsilon _{6}$ for arbitrarily small $\\varepsilon _{6}>0$ .", "On the one hand, with $N_{j}$ as in (REF ) in view of (REF ) we have $\\max \\lbrace |y \\xi - x|_{p} , |\\xi -N_{h}|_{p} \\rbrace \\ll \\max \\lbrace H^{-\\mu -\\varepsilon _3} , p^{-\\nu _h \\mu } \\rbrace $ and, since $H= x_{h,0}^{s},\\quad p^{\\nu _{h}} \\asymp x_{h,0}^{\\nu _{h}/\\sigma _{h}}\\asymp x_{h,0}^{t \\mu },$ we get $\\max \\lbrace |y \\xi - x|_{p} , |\\xi -N_{h}|_{p} \\rbrace \\ll x_{h,0}^{- \\mu \\min \\lbrace t \\mu ,s \\rbrace }.$ On the other hand, as ${\\rm gcd}(x,y)=1$ and ${\\rm gcd}(N_j,1)=1$ and we have assumed $(x,y)\\ne (N_{j},1)$ , these pairs are linearly independent.", "Hence, from Lemma REF and $N_h\\ll p^{\\nu _{h}}\\ll x_{h,0}^{t \\mu }$ , we get $\\max \\lbrace |y \\xi - x|_{p} , |\\xi -N_{h}|_{p} \\rbrace \\gg H^{-1}N_{h}^{-1}\\gg x_{h,0}^{-s- t \\mu }.$ This gives the lower bound $ s+ t \\mu +\\varepsilon _{7} \\ge \\mu \\min \\lbrace s, t \\mu \\rbrace .$ If $s \\le t \\mu $ , then we get $s\\le t \\mu / (\\mu - 1) + \\varepsilon _{8}$ .", "However, in view of $s\\ge t \\mu - 1 -\\varepsilon _{6}$ noticed above as $\\varepsilon _{6}, \\varepsilon _{8}$ can be arbitrarily small this gives a contradiction as soon as $\\mu > 1 + \\frac{t \\mu }{t \\mu - 1} = 2+ \\frac{1}{t\\mu -1}.$ Since $t \\mu \\ge \\mu >2$ a sufficient criterion is $ \\mu > 3.$ If $s > t \\mu $ , then we derive from (REF ) that $s+ t \\mu +\\varepsilon _{7} \\ge t \\mu ^2$ or equivalently $s\\ge t \\mu ( \\mu - 1)-\\varepsilon _{7}$ .", "Thus, as $\\varepsilon _{7}$ can be arbitrarily small, we have shown (REF ).", "Next observe that the triangle inequality gives $|x + yu^{(h)}-y\\zeta |_{p} = |(x-y\\xi )+y(\\xi +u^{(h)}-\\zeta )|_{p}\\le \\max \\lbrace |x-y\\xi |_{p} , |y(\\xi +u^{(h)}-\\zeta )|_{p}\\rbrace ,$ so combined with (REF ) applied for $j=h+1$ and with (REF ) we conclude $ |x+y u^{(h)}- y \\zeta |_{p} \\ll \\max \\lbrace \\; x_{h+1,0}^{- t \\mu }\\;,\\; H^{-\\mu -\\varepsilon _{3} }\\; \\rbrace .$ From (REF ) and as $|y_{h,0}|\\ll |y_{h,1}|^{1+\\varepsilon _2}=|x_{h,1}|^{1+\\varepsilon _2}$ by (REF ) and $\\mu >2$ and $t\\ge 1$ , we check that $H \\ll x_{h+1,0}^{ \\frac{1}{\\mu - 1} +\\varepsilon _{9} } \\ll x_{h+1,0}^{t},$ so the right hand expression in (REF ) is larger.", "With (REF ) and (REF ), and since $|y_{j,1}|$ , $|y_{j,0}|$ and $x_{j,0}$ are of comparable size and $\\tau _{h}/\\sigma _{h}$ tends to $(t\\mu )\\mu = t \\mu ^2$ by (REF ), we estimate $\\max \\lbrace |y|, |x+yu^{(h)}| \\rbrace &\\le |x|+|y|\\cdot |u^{(h)}|\\le H+H \\cdot p^{\\tau _{h}+1} \\\\ &\\ll H\\cdot x_{h,0}^{\\tau _{h}/\\sigma _{h}}\\ll H\\cdot (H^{\\frac{1}{(\\mu -1)t\\mu }})^{\\tau _{h}/\\sigma _{h}}\\ll H^{2+ \\frac{1}{\\mu - 1} }.$ To sum up, we have shown $ \\max \\lbrace |y|, |x+yu^{(h)}| \\rbrace &\\ll H^{2+ \\frac{1}{\\mu - 1} }, \\\\|x+yu^{(h)}-y \\zeta |_{p} &\\le H^{-\\mu -\\varepsilon _{3} }.", "\\nonumber $ From (REF ), we get $-\\frac{\\log |x+ yu^{(h)}-y \\zeta |_{p}}{\\log \\max \\lbrace |y|, |x+yu^{(h)}| \\rbrace }& \\ge (\\mu +\\varepsilon _{3})\\cdot \\frac{\\log H}{\\log \\max \\lbrace |y|, |x+yu^{(h)}|\\rbrace }\\\\& \\ge (\\mu +\\varepsilon _{3})\\cdot \\left(2+\\frac{1}{\\mu - 1}\\right)^{-1} - \\varepsilon _{10}> \\frac{\\mu ^{2} - \\mu }{2\\mu - 1} - \\varepsilon _{10}.$ Thus we have found integers $z_{0},z_{1}$ with $-\\frac{\\log |z_{1}\\zeta -z_{0}|_{p}}{\\log \\max \\lbrace |z_{0}|,|z_{1}|\\rbrace }\\ge \\frac{\\mu ^{2} - \\mu }{2\\mu -1} - \\varepsilon _{10}.$ Thereby, as $\\varepsilon _{10}$ can be arbitrarily small, if $ \\mu >\\frac{5+\\sqrt{17}}{2},$ then $\\frac{\\mu ^{2} - \\mu }{2\\mu -1} > 2$ and we have constructed an approximation of order greater than 2 to $\\zeta $ .", "If $\\epsilon >0$ from Theorem REF for our $\\zeta $ has been chosen small enough (depending on the given $\\mu $ ), concretely for $\\epsilon = \\frac{1}{2} \\biggl ( \\frac{\\mu ^{2} - \\mu }{2\\mu -1} - 2 \\biggr ),$ by the assumptions of the theorem and since ${\\rm gcd}(x,y)=1$ and ${\\rm gcd}(x_{j,0},x_{j,1}) = 1$ , this implies $(x+yu^{(h)},y)= (x_{j,0},x_{j,1})$ for some $j$ .", "We assume this is the case and will derive a contradiction.", "We first show that $j$ cannot exceed $h$ .", "Note that the case $j=h+1$ has already been treated in Case 1.", "Since $x_{j,0}\\asymp |x_{j,1}|$ and $|u^{(j)}|$ tends to infinity with $j$ by (REF ), we must have $|x|\\asymp |y|\\cdot |u^{(h)}|\\asymp x_{j,0}\\cdot p^{\\tau _h}\\asymp x_{j,0}\\cdot x_{h,0}^{\\tau _h /\\sigma _h }\\asymp x_{j,0}\\cdot x_{h,0}^{t\\mu ^2 }.$ If $j\\ge h+2$ then this clearly contradicts $|x|\\le H\\ll x_{h+1,0}$ from (REF ).", "Now, we assume that $j \\le h$ .", "It follows from (REF ) that $|\\xi y-x|_{p} &= | \\xi x_{j,1} - x_{j,0} + x_{j,1}u^{(h)}|_{p} \\\\&= | (u^{(h)}+\\xi -\\zeta ) x_{j,1} + (\\zeta x_{j,1}- x_{j,0})|_{p} \\\\&\\ge |\\zeta x_{j,1}- x_{j,0}|_{p} - |(u^{(h)}+\\xi -\\zeta ) x_{j,1}|_{p}\\\\&\\gg x_{j,0}^{-t\\mu } - x_{h+1,0}^{-t\\mu }.$ Now the crude estimate $x_{h+1,0}\\gg x_{h,0}^{ \\sigma _{h+1}/\\sigma _{h} }\\gg x_{h,0}^{ \\tau _{h}/\\sigma _{h} }\\gg x_{h,0}^{t\\mu ^2}\\gg x_{h,0}^{4}\\gg x_{j,0}^{4}$ suffices to derive $|\\xi y-x|_{p} \\gg x_{j,0}^{-t\\mu }.$ But on the other hand by assumption (REF ) and (REF ) we get $|\\xi y-x|_{p} \\ll H^{-\\mu }\\ll x_{j,0}^{-t\\mu ^2(\\mu -1)}.$ The combination of the latter inequalities gives the desired contradiction.", "We see that $\\mu >(5+\\sqrt{17})/2$ is the most restrictive one among the conditions (REF ), (REF ) we have collected on the way, which imposes $\\mu ^{\\times }=2\\mu > 5+\\sqrt{17}$ , that is, the restriction made in the theorem.", "Finally we show $ \\mu ^{{\\times }}(\\xi ) \\le 2\\mu .$ We again distinguish between rationals $y_{j,0}/y_{j,1}$ and other rationals.", "Concerning the first type, again because $|y_{j,0}|$ and $|y_{j,1}|$ are of comparable size and from (REF ) we indeed derive that $|y_{j,1}\\xi -y_{j,0}|_{p} \\gg \\left(\\sqrt{|y_{j,0} \\cdot y_{j,1}|}\\right)^{- t \\mu -\\varepsilon _{11} }.$ Thus the exponent restricted to this case satisfies $\\mu ^{{\\times }}(\\xi )\\le t \\mu +\\varepsilon _{11}\\le 2\\mu +\\varepsilon _{11}$ .", "Since $\\varepsilon _{11}$ can be taken arbitrarily small the claim follows.", "Finally in the latter case where $(x,y)\\ne (y_{j,0},y_{j,1})$ we conclude with (REF ) via $|y \\xi -x|_{p} \\gg \\max \\lbrace |x|,|y| \\rbrace ^{- \\mu -\\varepsilon _3}\\ge \\left(\\sqrt{|xy|} \\right)^{-2\\mu -2\\varepsilon _3},$ again giving $\\mu ^{{\\times }}(\\xi )\\le 2\\mu +2\\varepsilon _3$ and the claim (REF ) follows as $\\varepsilon _3$ can be taken arbitrarily small.", "The proof of (REF ) and thus of the theorem is complete." ], [ "Proofs of Theorem ", "Assume that for some $\\mu \\ge 2$ , there exist non-zero coprime integers $x,y$ with $|x y|$ arbitrarily large and such that $ |y \\xi - x|_{p} = (Q^{{\\times }})^{ -\\mu }, \\quad Q^{{\\times }} =\\sqrt{|xy|}.$ Set $Q=\\max \\lbrace |x|,|y|\\rbrace $ .", "Define $A$ in $[1,2]$ and $\\tau $ by $Q=(Q^{{\\times }})^{ A}, \\quad \\tau = \\frac{\\mu -A}{2}.$ In the sequel, $\\varepsilon _1, \\varepsilon _2, \\ldots $ denote positive real numbers that can be taken arbitrarily small as $Q^{\\times }$ tends to infinity.", "Set $X= (Q^{{\\times }})^{ \\tau -\\varepsilon _1}.$ By (REF ), for any positive integer $M$ with $\\sqrt{| Mx \\cdot My | }=MQ^{{\\times }}\\le X$ , we have $|My \\xi -Mx|_{p} \\ge M^{-1}(Q^{{\\times }})^{ -\\mu } & \\ge \\frac{Q^{{\\times }}}{X }(Q^{{\\times }})^{ -\\mu }\\nonumber \\\\&\\ge X^{-\\frac{\\mu - 1}{\\tau }-1 - \\varepsilon _{2} }= X^{-\\frac{2(\\mu - 1)}{\\mu - A}-1 - \\varepsilon _{2} }.$ Consider an integer pair $(\\tilde{x},\\tilde{y})$ linearly independent to $(x,y)$ and with $\\sqrt{|\\tilde{x}\\tilde{y}|}\\le X$ .", "Set $\\tilde{X}:=\\max \\lbrace |\\tilde{x}|,|\\tilde{y}| \\rbrace $ and observe that $\\tilde{X} \\le X^{2}$ .", "By construction, we have $|y \\xi - x|_{p} =(Q^{{\\times }})^{ -\\mu } = Q^{-\\frac{\\mu }{A} }\\le Q^{-1 }X^{-2} \\le Q^{-1 }\\tilde{X}^{-1},$ where we have used that $X^{2} = (Q^{{\\times }})^{\\mu -A-2 \\varepsilon _1} =Q^{\\frac{\\mu -A}{A} -\\varepsilon _3 }.$ It then follows from Lemma REF that $|\\tilde{y}\\xi - \\tilde{x}|_{p} \\gg Q^{-1} \\tilde{X}^{-1}\\gg Q^{-1}X^{-2}& = X^{-2- \\frac{A}{\\tau -\\varepsilon _1} } \\nonumber \\\\& = X^{-2- \\frac{A}{\\tau } -\\varepsilon _{4}}=X^{- \\frac{2 \\mu }{\\mu -A} -\\varepsilon _{4}}.", "$ Since $\\mu $ can be chosen arbitrarily close to $\\mu ^{{\\times }}(\\xi )$ , we deduce from (REF ) and (REF ) that $ {\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le \\sup _{A\\in [1,2] }\\max \\left\\lbrace \\frac{3\\mu ^{{\\times }}(\\xi ) - 2 - A}{\\mu ^{{\\times }}(\\xi )-A}\\; , \\; \\frac{2 \\mu ^{{\\times }}(\\xi ) }{\\mu ^{{\\times }}(\\xi ) -A} \\right\\rbrace .$ For $\\mu ^{{\\times }}(\\xi )\\ge 4$ it is readily checked that, for any $A$ in $[1,2]$ , the quantity on the left is greater than or equal to the quantity on the right.", "Hence, for $\\mu ^{{\\times }}(\\xi )\\ge 4$ , we have proved that $ {\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le \\sup _{A\\in [1,2] } \\frac{3\\mu ^{{\\times }}(\\xi ) - 2 - A}{\\mu ^{{\\times }}(\\xi )-A}= 3+\\frac{2}{\\mu ^{{\\times }}(\\xi )-2}.$ Since (REF ) clearly holds if $\\mu ^{{\\times }}(\\xi ) < 4$ (recall that ${\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le 4$ is always true), this proves the first claim (REF ) of the theorem.", "Actually, in the preceding proof, we have shown a slightly stronger result than (REF ), which we state in the following corollary for later use.", "Corollary 8.1 Let $\\xi $ be an irrational $p$ -adic number.", "Assume that there exist $A$ in $[1, 2]$ and an infinite sequence ${\\mathcal {S}}$ of pairs of nonzero integers $(x, y)$ such that $\\limsup _{(x,y) \\in {\\mathcal {S}}, \\max \\lbrace |x|, |y|\\rbrace \\rightarrow \\infty } \\, \\frac{- \\log |y \\xi - x|_{p}}{ \\log \\sqrt{|xy|}} = \\mu ^{{\\times }}(\\xi )$ and $\\lim _{(x,y) \\in {\\mathcal {S}}, \\max \\lbrace |x|, |y|\\rbrace \\rightarrow \\infty } \\, \\frac{ \\log \\max \\lbrace |x|, |y|\\rbrace }{ \\log \\sqrt{|xy|}} = A.$ Then, we have $ {\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le \\max \\left\\lbrace \\frac{3\\mu ^{{\\times }}(\\xi ) - 2 - A}{\\mu ^{{\\times }}(\\xi )-A}\\; , \\; \\frac{2 \\mu ^{{\\times }}(\\xi ) }{\\mu ^{{\\times }}(\\xi ) -A} \\right\\rbrace .$ In particular, if we have $A=1$ , that is, if $|x|$ and $|y|$ are of comparable size for every pair $(x, y)$ in ${\\mathcal {S}}$ , then we obtain ${\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le 3$ .", "The estimate (REF ) comes directly from the proof of Theorem REF above.", "For the last assertion, observe that when $A=1$ the left hand side of (REF ) is equal to 3, while the right hand side is at most equal to 3 when $\\mu ^{{\\times }}(\\xi )\\ge 3$ .", "For a given $p$ -adic number $\\xi $ , we define the sequence of multiplicative best approximation pairs $((x_{k}^{{\\times }},y_{k}^{{\\times }}))_{k \\ge 1}$ in a similar way as for the usual approximation problem, by looking at the pair of coprime integers $(x,y)$ minimizing $|y \\xi - x|_{p}$ among all the integer pairs with $0<\\sqrt{|xy|}\\le X$ , and letting the positive real number $X$ grow to infinity.", "Write $Q_{k}^{{\\times }}= \\sqrt{|x_{k}^{{\\times }} y_{k}^{{\\times }}|}$ for $k \\ge 1$ .", "By construction, we have $Q_{1}^{{\\times }}<Q_{2}^{{\\times }}<\\cdots , \\quad |y_{1}^{{\\times }}\\xi - x_{1}^{{\\times }}|_{p}> |y_{2}^{{\\times }}\\xi - x_{2}^{{\\times }}|_{p} > \\cdots .$ Furthermore, $|y_{k+1}^{{\\times }}\\xi - x_{k+1}^{{\\times }}|_{p}$ is smaller than $|My_{k}^{{\\times }}\\xi - Mx_{k}^{{\\times }}|_{p}$ as soon as the positive integer $M$ satisfies $M<Q_{k+1}^{{\\times }}/Q_{k}^{{\\times }}$ .", "Observe that, by the remark on the coprimality of $x$ and $y$ following Definition REF , we have $\\mu ^{{\\times }}(\\xi ) = \\limsup _{k \\rightarrow \\infty }\\, \\frac{- \\log |y_{k}^{{\\times }}\\xi - y_{k}^{{\\times }}|_{p}}{\\log Q_k^{{\\times }}}$ and $ \\widehat{\\mu }^{{\\times }}(\\xi )= 1 + \\liminf _{k \\rightarrow \\infty } \\, \\frac{- \\log |y_{k}^{{\\times }}\\xi - y_{k}^{{\\times }}|_{p} -\\log Q_{k}^{{\\times }} }{\\log Q_{k+1}^{{\\times }}}.$ We begin with an auxiliary result on the sequence of best approximations.", "Lemma 8.2 With the above notation, we have $\\limsup _{k \\rightarrow \\infty } \\, \\frac{\\log Q_{k+1}^{{\\times }}}{\\log Q_{k}^{{\\times }} }\\le \\frac{\\mu ^{{\\times }}(\\xi ) - 1}{{\\widehat{\\mu }}^{{\\times }}(\\xi ) - 1}.$ By the definitions of the limsup and of the liminf, we get that, for every $\\varepsilon > 0$ and every large $k$ , we have $(\\mu ^{{\\times }}(\\xi ) + \\varepsilon ) \\log Q_k^{{\\times }} \\ge - \\log |y_{k}^{{\\times }}\\xi - x_{k}^{{\\times }}|_{p}$ and $(\\widehat{\\mu }^{{\\times }}(\\xi ) - 1 - \\varepsilon ) \\log Q_{k+1}^{{\\times }}\\le - \\log |y_{k}^{{\\times }}\\xi - x_{k}^{{\\times }}|_{p} - \\log Q_{k}^{{\\times }}.$ This gives $(\\widehat{\\mu }^{{\\times }}(\\xi ) - 1 - \\varepsilon ) \\log Q_{k+1}^{{\\times }} \\le (\\mu ^{{\\times }}(\\xi ) - 1 + \\varepsilon ) \\log Q_k^{{\\times }},$ and the lemma follows.", "We establish (i) and observe that (ii) can be proved analogously.", "Assume that we have $|x_{k}^{{\\times }} | \\gg |y_{k}^{{\\times }}|$ and $|x_{k+1}^{{\\times }}| \\gg |y_{k+1}^{{\\times }}|$ .", "Recall that $Q_{k}^{{\\times }}=\\sqrt{ |x_{k}^{{\\times }} y_{k}^{{\\times }} |}$ .", "Define $\\alpha _k$ and $\\beta _k$ by $(Q_{k}^{{\\times }})^{ \\alpha _{k}}= |x_{k}^{{\\times }}|,\\quad (Q_{k}^{{\\times }})^{ \\beta _{k}}= |y_{k}^{{\\times }}|,$ and note that $\\alpha _{k}+\\beta _{k}=2$ .", "Define $\\mu _k$ (it would be more appropriate to write $\\mu _k^{\\times }$ , but for the sake of readability we choose to drop the ${}^{\\times }$ ) by $|y_{k+1}^{{\\times }} \\xi - x_{k+1}^{{\\times }}|_{p} < |y_{k}^{{\\times }} \\xi - x_{k}^{{\\times }}|_{p}= (Q_{k}^{{\\times }})^{-\\mu _k}.$ Now, as in Lemma REF , we get $|x_{k+1}^{{\\times }}y_{k}^{{\\times }} - x_{k}^{{\\times }}y_{k+1}^{{\\times }}|_{p}= |y_{k+1}^{{\\times }}(y_{k}^{{\\times }} \\xi - x_{k}^{{\\times }})- y_{k}^{{\\times }}(y_{k+1}^{{\\times }} \\xi - x_{k+1}^{{\\times }})|_{p}\\le (Q_{k}^{{\\times }})^{-\\mu _k}.$ Thus, $|x_{k+1}^{{\\times }}y_{k}^{{\\times }} - x_{k}^{{\\times }}y_{k+1}^{{\\times }}| \\ge (Q_{k}^{{\\times }})^{ \\mu _k},$ which implies that $\\max \\lbrace |x_{k+1}^{{\\times }}y_{k}^{{\\times }}| , |x_{k}^{{\\times }}y_{k+1}^{{\\times }}| \\rbrace \\gg (Q_{k}^{{\\times }})^{ \\mu _k}.$ Set $\\delta _{k}= \\min \\lbrace \\alpha _{k}, \\beta _{k} \\rbrace $ .", "Since $|x_k^{{\\times }}| \\gg |y_k^{{\\times }}|$ , we may assume that $\\delta _{k}= \\beta _{k}$ (if necessary, we absorb the numerical constant in $\\ll $ ).", "We have either $|x_{k+1}^{{\\times }}| \\gg \\frac{(Q_{k}^{{\\times }})^{ \\mu _k} }{|y_{k}^{{\\times }}|}=(Q_{k}^{{\\times }})^{ \\mu _k-\\beta _{k}},$ which gives $Q_{k+1}^{{\\times }} = \\sqrt{ |x_{k+1}^{{\\times }} y_{k+1}^{{\\times }} | } \\ge \\sqrt{|x_{k+1}^{{\\times }}|} \\gg (Q_{k}^{{\\times }})^{ \\frac{\\mu _k -\\delta _{k} }{2} },$ or $|x_{k+1}^{{\\times }}| \\gg |y_{k+1}^{{\\times }}| \\gg \\frac{(Q_{k}^{{\\times }})^{ \\mu _k} }{|x_{k}^{{\\times }}|} = (Q_{k}^{{\\times }})^{ \\mu _k -\\alpha _{k} } \\gg (Q_{k}^{{\\times }})^{ \\mu _k- 2+\\delta _{k}},$ which gives $Q_{k+1}^{{\\times }} = \\sqrt{ |x_{k+1}^{{\\times }} y_{k+1}^{{\\times }} | } \\ge (Q_{k}^{{\\times }})^{ \\mu _k-2+\\delta _{k}}.$ To sum up, we have proved that $\\frac{ \\log Q_{k+1}^{{\\times }} }{ \\log Q_{k}^{{\\times }} } \\ge \\min \\Bigl \\lbrace \\frac{\\mu _k-\\delta _{k} }{2} , \\mu _k - 2 +\\delta _{k} \\Bigr \\rbrace .$ Let ${\\varepsilon }> 0$ be a given real number.", "For $k$ large enough, it then follows from the proof of Lemma REF that $ \\frac{ \\log Q_{k+1}^{{\\times }} }{ \\log Q_{k}^{{\\times }} } \\le \\frac{\\mu _k - 1}{ {\\widehat{\\mu }}^{{\\times }}(\\xi ) - 1 } + {\\varepsilon }.$ We deduce that ${\\widehat{\\mu }}^{{\\times }}(\\xi ) & \\le 1 + \\max \\Bigl \\lbrace \\frac{2\\mu _k - 2}{\\mu _k -\\delta _{k}},\\frac{\\mu _k - 1}{\\mu _k-2+\\delta _{k}} \\Bigr \\rbrace + {\\varepsilon }\\\\& \\le 1 + \\max \\Bigl \\lbrace \\frac{2\\mu _k - 2}{\\mu _k -1},\\frac{\\mu _k - 1}{\\mu _k-2} \\Bigr \\rbrace + {\\varepsilon }= 1 + \\max \\Bigl \\lbrace 2,\\frac{\\mu _k - 1}{\\mu _k-2} \\Bigr \\rbrace + {\\varepsilon }.$ If $\\mu _k \\le 3$ for arbitrarily large $k$ as above, then the upper bound ${\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le 3$ follows from (REF ).", "Otherwise, we have $\\mu _k > 3$ for every sufficiently large $k$ and, since ${\\varepsilon }$ can be taken arbitrarily small, we conclude that ${\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le 3$ , as asserted.", "First, note that (REF ) clearly holds when ${\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le 3$ , since we then have ${\\widehat{\\mu }}^{{\\times }}(\\xi )^2 - 3 {\\widehat{\\mu }}^{{\\times }}(\\xi ) + 3 \\le {\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le \\mu ^{{\\times }}(\\xi ).$ Consequently, we assume throughout this proof that ${\\widehat{\\mu }}^{{\\times }}(\\xi ) > 3$ .", "By (REF ), we then have $3 < \\mu ^{{\\times }}(\\xi ) < + \\infty .$ Observe also that (REF ) can be rewritten as $ {\\widehat{\\mu }^{\\times }}(\\xi )\\le \\frac{3+\\sqrt{4\\mu ^{{\\times }}(\\xi )-3}}{2}.$ Define $\\mu _k$ by $|x_{k}^{{\\times }}\\xi -y_{k}^{{\\times }}|_{p} =\\Bigl ( \\sqrt{|x_{k}^{{\\times }}y_{k}^{{\\times }}| } \\Bigr )^{-\\mu _k}=(Q_{k}^{{\\times }})^{-\\mu _k},$ and $\\alpha _{k}, \\beta _{k}, \\delta _{k}$ by $(Q_{k}^{{\\times }})^{ \\alpha _{k}}= |x_{k}^{{\\times }}|,\\quad (Q_{k}^{{\\times }})^{ \\beta _{k}}= |y_{k}^{{\\times }}|, \\quad \\delta _{k}=\\min \\lbrace \\alpha _{k},\\beta _{k} \\rbrace .$ We can assume that $ \\mu ^{{\\times }}(\\xi )= \\limsup _{\\ell \\rightarrow \\infty } \\, \\mu _{2 \\ell },$ and, in view of Theorem REF , that for all large even integers $k$ we have $ |x_{k}^{{\\times }}| > |y_{k}^{{\\times }}|, \\quad |x_{k+1}^{{\\times }}| < |y_{k+1}^{{\\times }}|.$ Then $\\delta _k = \\beta _k$ .", "Below, $k$ denotes a sufficiently large even integer.", "Let ${\\varepsilon }> 0$ be a given real number.", "Proceeding as in the preceding proof, but with the pairs $(x_{k}^{{\\times }}, y_{k}^{{\\times }})$ and $(x_{k+2}^{{\\times }}, y_{k+2}^{{\\times }})$ which satisfy the inequalities $|x_{k+2}^{{\\times }} | > |y_{k+2}^{{\\times }}|$ and $|x_{k}^{{\\times }}| > |y_{k}^{{\\times }}|$ , we get $\\frac{ \\log Q_{k+2}^{{\\times }} }{ \\log Q_{k}^{{\\times }} } \\ge \\min \\left\\lbrace \\frac{\\mu _k -\\delta _{k} }{2} ,\\; \\mu _k-2+\\delta _{k} \\right\\rbrace + {\\varepsilon },$ for $k$ large.", "On the other hand, from the proof of Lemma REF we get $\\frac{ \\log Q_{k+2}^{{\\times }} }{ \\log Q_{k}^{{\\times }} } =\\frac{ \\log Q_{k+2}^{{\\times }} }{ \\log Q_{k+1}^{{\\times }} } \\cdot \\frac{ \\log Q_{k+1}^{{\\times }} }{ \\log Q_{k}^{{\\times }} } \\le \\left(\\frac{\\mu _k - 1}{ {\\widehat{\\mu }}^{{\\times }}(\\xi ) - 1}\\right)^{2}+ {\\varepsilon }.$ The combination of the latter inequalities gives ${\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le 1 +\\sqrt{ \\max \\Bigl \\lbrace \\frac{2(\\mu _k-1)^{2} }{\\mu _k -\\delta _{k} },\\frac{ (\\mu _k-1)^{2} }{\\mu _k-2+\\delta _{k} } \\Bigr \\rbrace } + \\tilde{{\\varepsilon }},$ where $\\tilde{{\\varepsilon }}$ tends to 0 with ${\\varepsilon }$ .", "By considering a sequence $(k_j)_{j \\ge 1}$ of even integers such that $\\mu ^{{\\times }}(\\xi )= \\limsup _{j \\rightarrow \\infty } \\, \\mu _{k_j}$ and putting $\\overline{\\delta }= \\limsup _{j \\rightarrow \\infty } \\delta _{k_j},$ we deduce that ${\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le 1 +\\max \\biggl \\lbrace \\sqrt{ \\frac{2}{\\mu ^{{\\times }}(\\xi ) - \\overline{\\delta } } } (\\mu ^{{\\times }}(\\xi )-1),\\frac{ \\mu ^{{\\times }}(\\xi ) -1}{\\sqrt{\\mu ^{{\\times }}(\\xi ) -2+\\overline{\\delta } }} \\biggr \\rbrace .$ Observe that in the maximum the right hand term is larger than the left hand term if and only if $\\mu ^{{\\times }}(\\xi ) \\le 4 - 3 \\overline{\\delta }$ .", "Consequently, if $\\mu ^{{\\times }}(\\xi ) \\le 4 - 3 \\overline{\\delta } \\le 4$ , then we get ${\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le 1 + \\frac{\\mu ^{{\\times }}(\\xi ) - 1}{ \\sqrt{\\mu ^{{\\times }}(\\xi )-2}}.$ Taking into account that $\\mu ^{{\\times }}(\\xi ) > 3$ , a rapid calculation shows that this inequality implies (REF ), as wanted.", "So we may assume $\\mu ^{{\\times }}(\\xi ) > 4 - 3 \\overline{\\delta }$ and thus ${\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le 1 + \\sqrt{ \\frac{2}{\\mu ^{{\\times }}(\\xi ) - \\overline{\\delta } } } (\\mu ^{{\\times }}(\\xi )-1).$ Observe that Corollary REF applied with $A =2-\\overline{\\delta }$ gives ${\\widehat{\\mu }}^{{\\times }}(\\xi ) \\le \\max \\left\\lbrace \\frac{3\\mu ^{{\\times }}(\\xi ) - 2 - A}{\\mu ^{{\\times }}(\\xi )-A}, \\;\\frac{2 \\mu ^{{\\times }}(\\xi )}{\\mu ^{{\\times }}(\\xi ) -A} \\right\\rbrace ,$ where the maximum is given by the left hand term if and only if we have $A\\le \\mu ^{{\\times }}(\\xi )-2$ .", "We distinguish two cases.", "Case 1: Assume that $A\\le \\mu ^{{\\times }}(\\xi )-2$ , that is, $\\mu ^{{\\times }}(\\xi )\\ge 4-\\overline{\\delta }$ .", "Then $ {\\widehat{\\mu }^{\\times }}(\\xi ) \\le \\min \\left\\lbrace \\frac{3\\mu ^{{\\times }}(\\xi ) -4+ \\overline{\\delta } }{ \\mu ^{{\\times }}(\\xi ) -2+\\overline{\\delta } } \\;,\\; \\sqrt{\\frac{2}{\\mu ^{{\\times }}(\\xi )- \\overline{\\delta } }} (\\mu ^{{\\times }}(\\xi )-1)+1 \\right\\rbrace $ holds.", "In view of (REF ), we can assume that $\\frac{3\\mu ^{{\\times }}(\\xi ) -4 + \\overline{\\delta } }{\\mu ^{{\\times }}(\\xi ) - 2 + \\overline{\\delta } }> \\frac{3+\\sqrt{4\\mu ^{{\\times }}(\\xi )-3}}{2},$ that is, $\\overline{\\delta }< \\frac{3 \\mu ^{{\\times }}(\\xi ) - 2 - (\\mu ^{{\\times }}(\\xi ) - 2)\\sqrt{4\\mu ^{{\\times }}(\\xi ) - 3}}{1 + \\sqrt{4\\mu ^{{\\times }}(\\xi ) - 3}}.$ Using this bound for $\\overline{\\delta }$ , we derive from (REF ) that ${\\widehat{\\mu }^{\\times }}(\\xi ) < 1 + \\sqrt{\\frac{(\\mu ^{{\\times }}(\\xi ) - 1)( 1 + \\sqrt{4\\mu ^{{\\times }}(\\xi ) - 3})}{\\sqrt{4\\mu ^{{\\times }}(\\xi ) - 3} - 1} }= 1 + \\frac{1+\\sqrt{4\\mu ^{{\\times }}(\\xi )-3}}{2},$ which gives the bound (REF ).", "Case 2: We assume that $A>\\mu ^{{\\times }}(\\xi )-2$ , that is, $\\mu ^{{\\times }}(\\xi )< 4-\\overline{\\delta }\\le 4$ .", "Then (REF ) gives ${\\widehat{\\mu }^{\\times }}(\\xi ) \\le 2+\\frac{2A }{\\mu ^{{\\times }}(\\xi ) -A } = \\frac{2\\mu ^{{\\times }}(\\xi ) }{ \\mu ^{{\\times }}(\\xi ) -2+\\overline{\\delta } }$ and we get $ {\\widehat{\\mu }^{\\times }}(\\xi ) \\le \\min \\left\\lbrace \\frac{2\\mu ^{{\\times }}(\\xi ) }{ \\mu ^{{\\times }}(\\xi ) -2+\\overline{\\delta } } \\;,\\; \\sqrt{\\frac{2}{\\mu ^{{\\times }}(\\xi )- \\overline{\\delta } }} (\\mu ^{{\\times }}(\\xi )-1)+1 \\right\\rbrace .$ In view of (REF ), we can assume that $\\frac{2\\mu ^{{\\times }}(\\xi ) }{ \\mu ^{{\\times }}(\\xi ) -2+\\overline{\\delta } }> \\frac{3+\\sqrt{4\\mu ^{{\\times }}(\\xi )-3}}{2},$ that is, $\\overline{\\delta }< \\frac{\\mu ^{{\\times }}(\\xi ) + 6 - (\\mu ^{{\\times }}(\\xi ) - 2) \\sqrt{4\\mu ^{{\\times }}(\\xi ) - 3}}{ 3 + \\sqrt{4\\mu ^{{\\times }}(\\xi ) - 3}}.$ Using this bound for $\\overline{\\delta }$ , we derive from (REF ) that ${\\widehat{\\mu }^{\\times }}(\\xi ) < 1 + \\sqrt{\\frac{ (\\mu ^{{\\times }}(\\xi ) - 1)^2( 3 + \\sqrt{4\\mu ^{{\\times }}(\\xi ) - 3})}{(\\mu ^{{\\times }}(\\xi ) - 3) + (\\mu ^{{\\times }}(\\xi ) - 1)\\sqrt{4\\mu ^{{\\times }}(\\xi ) - 3} } }.$ A careful computation shows that, since $\\mu ^{{\\times }}(\\xi ) \\ge 3$ , we get (REF ).", "As noticed above Corollary REF , the upper bound (REF ) follows from (REF ) and (REF ).", "The proof of Theorem REF is complete." ] ]
2105.11779
[ [ "On the weak norm of $\\mathscr{U}_p$-residuals of all subgroups of a\n finite group" ], [ "Abstract Let $\\mathscr{F}$ be a formation and $G$ a finite group.", "The weak norm of a subgroup $H$ in $G$ with respect to $\\mathscr{F}$ is defined by $N_{\\mathscr{F}}(G,H)=\\underset{T\\leq H}{\\bigcap}N_G(T^{\\mathscr{F}})$.", "In particular, $N_{\\mathscr{F}}(G)=N_{\\mathscr{F}}(G,G)$.", "Let $N^i_{\\mathscr{F}}(G)$,$i\\geq 1$, be a upper series of $G$ by setting $N^0_{\\mathscr{F}}(G)=1$, $N^{i+1}_{\\mathscr{F}}(G)/N^i_{\\mathscr{F}}(G)=N_{\\mathscr{F}}(G/N^i_{\\mathscr{F}}(G))$ and denoted by $N^{\\infty}_{\\mathscr{F}}(G)$ the terminal term of the series.", "In this paper, for the case $\\mathscr{F}\\in\\{\\mathscr{U}_p,\\mathscr{U}\\}$, where $\\mathscr{U}_p$($\\mathscr{U}$,respectively) is the class of all finite $p$-supersolvable groups(supersolvable groups,respectively), we characterize the structure of some given finite groups by the properties of weak norm of some subgroups in $G$ with respect to $\\mathscr{F}$.", "Some of our main results may regard as a continuation of many nice previous work." ], [ "Introduction", "Throughout this paper, all groups are finite.", "We use the standard terminology and notations as in [1].", "For the specific, we denote $G$ : a finite group.", "$|G|$ : the order of $G$ .", "$\\pi (G)$ : the set of prime divisors of $|G|$ .", "$p$ : a prime.", "$G_p$ : the Sylow $p$ -subgroup of $G$ .", "$F_p(G)$ : the $p$ -Fitting subgroup of $G$ .", "$\\mathfrak {G}$ : the class of all finite groups.", "${F}$ : a formation of groups, that is a class of finite groups satisfying the following: (1) if $G\\in {F}$ and $N$ is a normal subgroup of $G$ , then $G/N\\in {F}$ , and (2) if $N_1$ ,$N_2$ are normal subgroups of $G$ such that $G/N_i\\in {F}(i=1,2)$ , then $G/(N_1\\cap N_2)\\in {F}$ .", "$G^{{F}}$ : the ${F}$ -residual of $G$ , that is the intersection of all those normal subgroups $N$ of $G$ such that $G/N\\in {F}$ .", "${F}_1{F}_2$ : the formation product or Gaschëtz product of ${F}_1$ and ${F}_2$ , that is the class $\\lbrace G\\in \\mathfrak {G} \\mid G^{{F}_2}\\in {F}_1 \\rbrace $ .", "In particular, denote ${F}^2={F}{F}$ .", "${A}$ : the class of all Abelian groups.", "${N}$ : the class of all nilpotent groups.", "${N}_p$ : the class of all $p$ -nilpotent groups.", "${U}$ : the class of all supersolvable groups.", "${U}_p$ : the class of all $p$ -supersolvable groups.", "$l_p(G)$ : the $p$ -length of a $p$ -solvable group $G$ , that is the number of the $p$ -factor groups in the upper $p$ -series of $G$ : $1=P_0(G)\\unlhd M_0(G)\\unlhd P_1(G)\\unlhd M_1(G)\\unlhd \\cdots \\unlhd P_n(G)\\unlhd M_n(G)=G$ such that $M_i(G)/P_i(G)=O_{p^{\\prime }}(G/P_i(G))$ and $P_i(G)/M_{i-1}(G)=O_p(G/M_{i-1}(G))$ .", "$h_p(G)$ : the $p$ -Fitting length of a $p$ -solvable group $G$ , that is the smallest positive integer $n$ such that $1=F^0_p(G)\\le F^1_p(G)\\le \\cdots \\le F^{n-1}_p(G)\\le F^n_p(G)=G,$ where $F^1_p(G)=F_p(G)$ and $F^{i+1}_p(G)/F^i_p(G)=F_p(G/F^i_p(G))$ for $i=1,2,\\cdots ,n-1$ .", "$h(G)$ : the Fitting length of a solvable group $G$ , that is the positive integer $n$ such that $1=F^0(G)\\le F^1(G)\\le \\cdots \\le F^{n-1}(G)\\le F^n(G)=G,$ where $F^1(G)=F(G)$ and $F^{i+1}(G)/F^i(G)=F(G/F^i(G))$ for $i=1,2,\\cdots ,n-1$ .", "$N(G)$ : the intersection of the normalizers of all subgroups of $G$ .", "$S(G)$ (or $N_{{N}}(G)$ ): the intersection of the normalizers of the nilpotent residuals of all subgroups of $G$ .", "$N^{{N}_p}(G)$ : the intersection of the normalizers of the $p$ -nilpotent residuals of all subgroups of $G$ .", "It is interesting to characterize the structure of a given group by using of some special subgroups.", "For example, Gashëtz and N. Itô [1] proved that $G$ is solvable with Fitting length at most 3 if all minimal subgroups of $G$ are normal.", "Also, it is well know that $G$ is nilpotent if $G^{\\prime }$ normalizers each subgroup of $G$ (see Baer's theorem in [2]).", "Further more, R. Baer in [3] defined the subgroup $N(G)$ , the norm of a group $G$ .", "Obviously, a group $G$ is a Dedeking group if and only if $G=N(G)$ .", "The norm of a group has many other good properties and has studied further by many scholars.", "In recent years, some weaker versions of the concepts of norm of groups have been introduced.", "Let ${F}$ be a non-empty formation.", "Recently, Su and Wang in [4], [5] introduced the subgroup $N_{{F}}(G)$ , the norm of ${F}$ -residual of a group $G$ as follows $N_{{F}}(G)= \\underset{H\\le G}{\\bigcap }{N_G(H^{{F}})}.", "\\qquad \\mathrm {(\\star )}$ As in [4], we set $N^0_{{F}}(G)=1$ and if $N^i_{{F}}(G)$ is defined, set $N^{i+1}_{{F}}(G)/N^i_{{F}}(G)=N_{{F}}(G/N^i_{{F}}(G))$ .", "The subgroup $N^{\\infty }_{{F}}(G)$ is the terminal term of the ascending series.", "In fact, $N^{\\infty }_{{F}}(G)=N^k_{{F}}(G)$ for some integer $k$ such that $N^k_{{F}}(G)=N^{k+1}_{{F}}(G)=N^{k+2}_{{F}}(G)=\\cdots $ .", "Many scholars also call $N_{{F}}(G)$ (${F}={A},{N}, {N}_p$ , respectively) the generalized norm of group $G$ .", "Obviously, $N_{{F}}(G)$ is a characteristic subgroup and every element of $N_{{F}}(G)$ normalize the ${F}$ -residual of each subgroup of $G$ .", "The so called norm of ${F}$ -residual has many other nice properties and also closely related to the global properties of a given group.", "For some given formation ${F}$ , there are many papers devoted to study the $p$ -length, Fitting length, solvability, ($p$ -)nilpotency and so on.", "For example, for the case ${F}={A}$ , Li and Shen in [6] denoted $N_{{{A}}}(G)$ by $D(G)$ .", "They fingered out that $G$ is solvable with Fitting length at most 3 if all elements of $G$ of prime order are in $D(G)$ (see [6]).", "It is a dual problem of Gashëtz and N. Itô[1].", "Li and Shen also defined the $D$ -group, i.e., $G=D(G)$ , they characterized the relationship between $D(G)$ and $G$ .", "Shen, Shi and Qian in [7] considered the case ${F}={N}$ , they denoted $N_{{N}}(G)$ by $S(G)$ .", "Shen et al., deeply studied the dual problem of Gashëtz and N. Itô and characterized the ${F}_{nn}$ -groups by means of the subgroup $S^{\\infty }(G)$ , where $S^{\\infty }(G)=N^{\\infty }_{{N}}(G)$ and ${F}_{nn}$ -groups are class of groups belong to ${N}{N}$ .", "They also introduced the $S$ -group, i.e., $G=S(G)$ and given some sufficient and necessary conditions involved $S$ -groups.", "Meanwhile, Gong and Guo in [8] also consider the case ${F}={N}$ and given some meaningful conclusions.", "In the case ${F}={N}_p$ , Guo and Li in [9] introduced the norm of ${N}_p$ -residual of a group $G$ , they denoted $N_{{N}_p}(G)$ by $N^{{N}_p}(G)$ .", "As a local version of Gong's results, Li and Guo characterized the relationship between $C_G(G^{{N}_p})$ and $N^{{N}_p}(G)$ .", "In particular, Li and Guo in [9] also investigated the relationship between $N_{{N}}(G)$ and $N^{{N}_p}(G)$ .", "For more detail and other relevant conclusions about the norm of ${F}$ -residual, please see [7], [4], [6], [5], [11], [9], [10], [12], [8], [13].", "We wonder whether the above conclusions hold for general formations.", "A natural idea is to replace ${F}({F}\\in \\lbrace {N}_p,{N},{A}\\rbrace )$ by ${U}_p$ or ${U}$ in $(\\star )$ .", "Remark 1.1 (1) In general, for a group $G$ and some $p\\in \\pi (G)$ , $N_{{U}_p}(G)\\ne N_{{F}}(G)$ is possible, where ${F}\\in \\lbrace {N}_p,{N},{A}\\rbrace $ .", "For example, let $G=S_4$ , the symmetric group of degree 4.", "Obviously, $G$ is a 3-supersolvable non-3-nilpotent group, so $G=N_{{U}_3}(G)$ .", "Pick a subgroup $H\\cong S_3$ , the symmetric group of degree 3, but $N_G(H^{{N}_3})=N_G(S^{{N}_3}_3)=N_G(C_3)=S_3<S_4$ , so $N_{{N}_3}(G)<G$ .", "(2) The condition in definition $(\\star )$ that the intersection of the ${F}$ -residuals of all subgroups of $G$ may be too strong and some of the subgroups may be redundant whenever ${F}={U}_p$ (see Example REF below).", "(3) The case that $N_{{U}}(G)=N_{{U}_p}(G)=1$ is possible for a solvable group $G$ and $p\\in \\pi (G)$ .", "Let $H=\\langle a,b\\mid a^3=b^3=1,[a,b]=1\\rangle \\cong C_3\\times C_3$ and $Q_8=\\langle c,d\\mid c^4=1,c^2=d^2=e,c^d=c^{-1}\\rangle $ .", "Considering the irreducible action of $Q_8$ on $H$ by $a^c=a^{-1}b,b^c=ab,a^d=b^{-1},b^d=a$ , denote $T=H\\rtimes Q_8\\cong (C_3\\times C_3)\\rtimes Q_8$ .", "Let $C=\\langle f\\rangle $ be a cyclic group of order 3 and let $C$ act on $T$ by $a^f=b^{-1},b^f=ab^{-1},c^f=d^3,d^f=cd$ .", "Then $G=\\langle a,b,c,d,e,f\\rangle \\cong ((C_3\\times C_3)\\rtimes Q_8)\\rtimes C_3$ is solvable(IdGroup=[216,153]).", "It is easy to see that $G^{{U}_2}=\\langle a,b,c,d,e\\rangle \\cong (C_3\\times C_3)\\rtimes Q_8$ and $C_G(G^{{U}_2})=1$ .", "By Theorem REF , $N_{{U}_2}(G)=1$ .", "Further more, by Theorem REF , $N_{{U}}(G)=1$ .", "So we have $N_{{U}}(G)=N_{{U}_2}(G)=1$ .", "Recall that the weak centralizer of $H$ in $G$ , $C^{\\ast }_G(H)$ , introduced in [14], is defined by $C^{\\ast }_G(H)=\\bigcap \\lbrace N_G(K):K\\le H\\rbrace .$ In the above investigation, we introduce the following more general and interesting definition.", "Definition 1.2 Let $G$ be a group and ${F}$ a formation.", "We define $N_{{F}}(G,H)$ , the weak norm of $H$ in $G$ with respect to ${F}$ as follows: $N_{{F}}(G,H)=\\underset{T\\le H}{\\bigcap }{N_G(T^{{F}})}.$ In particular, $N_{{U}}(G,H)=\\underset{T\\le H}{\\bigcap }{N_G(T^{{U}})}$ and $N_{{U}_p}(G,H)=\\underset{T\\le H}{\\bigcap }{N_G(T^{{U}_p})}$ .", "Example 1.3 Let $G=\\langle a,b,c,d\\rangle \\rtimes \\langle e,f\\rangle \\cong C^4_2\\rtimes C_6$ , where $e^2=f^3=1$ and $a^e=ac,b^e=bd,c^e=c,d^e=d,a^f=b,b^f=ab,c^f=d,d^f=cd$ (IdGroup=[96,70] in GAP [15]).", "Let $H=\\langle a,b,f\\rangle $ , then $N_{{U}_2}(G)=N_{{U}_2}(G,H)=\\langle a,b,c,d,f\\rangle \\cong C^4_2\\rtimes C_3<G$ .", "Obviously, $N_{{F}}(G)=N_{{F}}(G,G)\\le N_{{F}}(G,H)\\le G$ .", "Without causing confusion, we call $N_{{F}}(G)=N_{{F}}(G,G)$ the norm of $G$ with respect to ${F}$ .", "Moreover, for a subgroup $H$ of $G$ , we call $G$ is an ${N}_1$ -group with respect to $H$ and ${F}$ if $G=N_{{F}}(G,H)$ .", "In this paper, we mainly investigate properties of $N_{{F}}(G,H)$ and the influence of $N_{{F}}(G,H)$ on the structure of group $G$ .", "Actually, we mainly consider the case ${F}\\in \\lbrace {U},{U}_p\\rbrace $ .", "Our main work may be regard as the continuation of some conclusions in [6], [7], [9], [8]." ], [ "Preliminaries", "In this section, we always assume that ${F}$ is a non-empty formation and $G$ is a group.", "We first give some important lemmas.", "Lemma 2.1 Let $H,K,N$ be subgroups of $G$ and $N\\unlhd G$ .", "Then (1) If $H\\le K$ , then $N_{{F}}(G,K)\\le N_{{F}}(G,H)$ ; (2) $K\\cap N_{{F}}(G,H)\\le N_{{F}}(K,K\\cap H)$ , in particular, if $H\\le K$ , then $K\\cap N_{{F}}(G,H)\\le N_{{F}}(K,H)$ .", "(3) If $N\\le H$ , then $N_{{F}}(G,H)N/N\\le N_{{F}}(G/N,H/N)$ .", "(1) By definition, $N_{{F}}(G,K)=\\underset{T\\le K}{\\bigcap }{N_G(T^{{F}})}\\le \\underset{T\\le H}{\\bigcap }{N_G(T^{{F}})}=N_{{F}}(G,H)$ .", "(2) Obviously, $K\\cap N_{{F}}(G,H)=K\\cap (\\underset{T\\le H}{\\bigcap }{N_G(T^{{F}})})\\le \\underset{T\\le H\\cap K}{\\bigcap }{N_K(T^{{F}})}=N_{{F}}(K,H\\cap K)$ .", "In particular, if $H\\le K$ , then $K\\cap N_{{F}}(G,H)\\le N_{{F}}(K,H)$ .", "(3) Let $x\\in N_{{F}}(G,H)$ , then $x$ normalize $T^{{F}}$ for every $T\\le H$ , so $xN$ normalize $T^{{F}}N/N=(TN/N)^{{F}}$ .", "Thus every element of $N_{{F}}(G,H)N/N$ normalize $(T/N)^{{F}}$ for all subgroups $T/N$ of $H/N$ , so $N_{{F}}(G,H)N/N\\le N_{{F}}(G/N,H/N)$ .", "$\\Box $ As a corollary, we have Lemma 2.2 Let $G$ be a group, let $K$ be a subgroup of $G$ and $N$ a normal subgroup of $G$ , then (1) $K\\cap N_{{F}}(G)\\le N_{{F}}(K)$ .", "(2) $N_{{F}}(G)N/N\\le N_{{F}}(G/N)$ .", "Lemma 2.3 [4] Let $G$ be a group, let $K$ be a subgroup of $G$ and $N$ a normal subgroup of $G$ , then (1) $K\\cap N^{\\infty }_{{F}}(G)\\le N^{\\infty }_{{F}}(K)$ .", "(2) $N^{\\infty }_{{F}}(G)N/N\\le N^{\\infty }_{{F}}(G/N)$ .", "(3) If $N\\le N^{\\infty }_{{F}}(G)$ , then $N^{\\infty }_{{F}}(G)/N=N^{\\infty }_{{F}}(G/N)$ .", "Lemma 2.4 Let $G$ be a $p$ -solvable group.", "(1) If $N\\unlhd G$ , then $l_p(G/N)\\le l_p(G)$ .", "(2) If $U\\le G$ , then $l_p(U)\\le l_p(G)$ .", "(3) Let $N_1$ and $N_2$ be two normal subgroups of $G$ , then $l_p(G/(N_1\\cap N_2))\\le max\\lbrace l_p(G/N_1),l_p(G/N_2)\\rbrace $ .", "(4) $l_p(G/\\Phi (G))=l_p(G)$ .", "(5) If $N$ is a normal $p^{\\prime }$ -group of $G$ , then $l_p(G/N)=l_p(G)$ .", "The proof of (1)-(4) follows from [1] and (5) is obviously.", "$\\Box $ Lemma 2.5 Let $G$ be a group and $N$ a normal subgroup of $G$ .", "Then (1) if $N/N\\cap \\Phi (G)$ is $p$ -nilpotent, then $N$ is $p$ -nilpotent; (2) Let $H$ be a subgroup of $G$ and $N$ be a $p^{\\prime }$ -group, if $HN/N$ is $p$ -nilpotent, then $H$ is $p$ -nilpotent.", "The statement (1) is directly form [16], and (2) is easy.", "$\\Box $ Lemma 2.6 Let $G$ be a $p$ -solvable group and $H$ a subgroup of $G$ , let $H,N,A,B$ be subgroups of $G$ and $N,A,B$ are normal in $G$ , then (1) $h_p(H)\\le h_p(G)$ .", "(2) $h_p(G/N)\\le h_p(G)$ .", "(3) If $G=A\\times B$ , then $h_p(G)=\\max \\lbrace h_p(A),h_p(B)\\rbrace $ .", "(4) If $h_p(G/A)\\le k$ and $h_p(G/N)\\le k$ , then $h_p(G/(A\\cap B))\\le k$ .", "(5) $h_p(G/\\Phi (G)))=h_p(G)$ .", "(6) If $N$ is a $p^{\\prime }$ -group, then $h_p(G/N)=h_p(G)$ .", "Let $1=N_0\\le N_1\\le N_2\\le \\cdots \\le N_r=G$ be the shortest normal chain of $G$ with $p$ -nilpotent factors $N_i/N_{i-1}$ for all $i=1,2,\\cdots ,r$ .", "(1) If $H\\le G$ , then $1=N_0\\cap H\\le N_1\\cap H\\le N_2\\cap H\\le \\cdots \\le N_r\\cap H=G\\cap H=H$ is a normal chain of $H$ with $N_i\\cap H/N_{i-1}\\cap H\\cong (N_i\\cap H)N_{i-1}/N_{i-1}$ a $p$ -nilpotent factors for all $i=1,2,\\cdots ,r$ , so $h_p(H)\\le h_p(G)$ .", "(2) If $N\\unlhd G$ , obviously, $\\bar{1}=N_0N/N\\le N_1N/N\\le N_2N/N\\le \\cdots \\le N_r/N=G/N$ is a normal chain of $G/N$ .", "Note that $N_iN/N/N_{i-1}N/N=N_iN/N_{i-1}N\\cong N_i/N_{i-1}(N_i\\cap N)\\le N_i/N_{i-1}$ is $p$ -nilpotent, so $h_p(G/N)\\le h_p(G)$ .", "(3) Let $1=A_0\\le A_1\\le A_2\\le \\cdots \\le A_r=A$ and $1=B_0\\le B_1\\le B_2\\le \\cdots \\le B_t=B$ be the shortest normal chain of $A$ and $B$ with $p$ -nilpotent factors respectively.", "Without loss of generality, assume $r\\le t$ , then $1=A_0B_0\\le A_1B_1\\le A_2B_2\\le \\cdots \\le A_rB_r\\le A_rB_{r+1}\\le A_rB_{r+2}\\le A_rB_t=AB$ is a normal chain.", "Since $A_iB_i/A_{i-1}B_{i-1}\\cong A_i/A_{i-1}(A_i\\cap B_{i-1})\\cdot B_i/B_{i-1}(B_i\\cap A_{i-1})$ whenever $i\\le r$ and $A_rB_j/A_rB_{j-1}\\cong B_j/B_{j-1}(B_j\\cap A_i)$ whenever $r<j\\le t$ are $p$ -nilpotent.", "So $h_p(G)=\\max \\lbrace h_p(A),h_p(B)\\rbrace $ .", "(4) Since $G/(A\\cap B)\\cong G/A\\times G/B$ , the result follows from (1) and (3).", "(5) Assume that $\\bar{1}=\\Phi (G)/\\Phi (G)=T_0/\\Phi (G)\\le T_1/\\Phi (G)\\le T_2/\\Phi (G)\\le \\cdots \\le T_s/\\Phi (G)=G/\\Phi (G)$ is the normal chain of $G$ with $p$ -nilpotent factors, so $s\\le r$ and $T_i/T_{i-1}\\cong T_i/\\Phi (G)/T_{i-1}/\\Phi (G)$ is $p$ -nilpotent.", "Since $T_1/\\Phi (G)$ is $p$ -nilpotent, then $T_1$ is $p$ -nilpotent by Lemma REF , so $1\\le T_1\\le T_2\\le \\cdots \\le T_s=G$ is a normal chain with $p$ -nilpotent factors.", "Now we have $r\\le s$ , hence $s=r$ .", "(6) Let $\\bar{1}=N_0/N\\le N_1/N\\le N_2/N\\le \\cdots \\le N_r/N=G/N$ be a normal chain of $G/N$ with $p$ -nilpotent factors $N_i/N/N_{i-1}/N\\cong N_i/N_{i-1}$ for all $i=1,2,\\cdots ,r$ .", "Now by Lemma REF , $N_1$ is $p$ -nilpotent since $N$ is a $p^{\\prime }$ -group, so $1\\le N_1\\le N_2\\le \\cdots \\le N_r=G$ is a normal chain of $G$ with $p$ -nilpotent factors $N_i/N_{i-1}$ for all $i=1,2,\\cdots ,r$ , which implies that $h_p(G)\\le h_p(G/N)$ , now by (1), we have $h_p(G/N)=h_p(G)$ .", "$\\Box $ As a local vision of [17], we have Lemma 2.7 Let $G$ be a solvable group, then $l_p(G)\\le 1$ if $h(G)\\le 2$ for every $p\\in \\pi (G)$ .", "Lemma 2.8 [18] Let ${F}$ be a saturated formation.", "(1) Assume that $G$ is a group such that $G$ does not belong to ${F}$ , but all its proper subgroups belong to ${F}$ .", "Then $F^{\\prime }(G)/\\Phi (G)$ is the unique minimal normal subgroup of $G/\\Phi (G)$ , where $F^{\\prime }(G)=Soc(G \\mod {\\Phi }(G))$ , and $F^{\\prime }(G)=G^{{F}}\\Phi (G)$ .", "In addition, if the derived subgroup of $G^{{F}}$ is a proper subgroup of $G^{{F}}$ , then $G^{{F}}$ is a soluble group.", "Furthermore, if $G^{{F}}$ is soluble, then $F^{\\prime }(G)=F(G)$ , the Fitting subgroup of $G$ .", "Moreover $(G^{{F}})^{\\prime }=T\\cap G^{{F}}$ for every maximal subgroup $T$ of $G$ such that $G/T_G\\notin {F}$ and $F^{\\prime }(G)T=G$ .", "(2) Assume that $G$ is a group such that $G$ does not belong to ${F}$ and there exists a maximal subgroup $M$ of $G$ such that $M\\in {F}$ and $G=MF(G)$ .", "Then $G^{{F}}/(G^{{F}})^{\\prime }$ is a chief factor of $G$ , $G^{{F}}$ is a $p$ -group for some prime $p$ , $G^{{F}}$ has exponent $p$ if $p>2$ and exponent at most 4 if $p=2$ .", "Moreover, either $G^{{F}}$ is elementary abelian or $(G^{{F}})^{\\prime } = Z(G^{{F}})=\\Phi (G^{{F}})$ is an elementary abelian group.", "Lemma 2.9 Let $G$ be a $p$ -solvable minimal non-$p$ -supersolvable group, then $G^{{U}_p}$ is a $p$ -group.", "By Lemma REF (1), $G/\\Phi (G)$ has the unique minimal normal subgroup $F^{\\prime }(G)/\\Phi (G)$ , where $F^{\\prime }(G)=G^{{U}_p}\\Phi (G)$ , so $F^{\\prime }(G)/\\Phi (G)$ is an elementary abelian $p$ -group since $G$ is a $p$ -solvable group.", "Otherwise, $F^{\\prime }(G)/\\Phi (G)$ is a $p^{\\prime }$ -group and the $p$ -supersolvability of $G/\\Phi (G)/F^{\\prime }(G)/\\Phi (G)$ implies that $G/\\Phi (G)$ is $p$ -supersolvable, thus $G$ is $p$ -supersolvable, a contradiction.", "Furthermore, $\\Phi (G)=1$ .", "Now we have $G^{{U}_p}$ is solvable and $F^{\\prime }(G)=F(G)$ , $G=F(G)T$ for some maximal subgroup $T$ of $G$ .", "Now by Lemma REF (2), $G^{{U}_p}$ is a $p$ -group.", "$\\Box $ Lemma 2.10 Let $G$ be a group and $\\pi (G)=\\lbrace p_1,p_2,\\cdots ,p_n\\rbrace $ , then $G^{{U}}=G^{{U}_{p_1}}G^{{U}_{p_2}}\\cdots G^{{U}_{p_n}}$ .", "It follows from the supersolvability of $G/G^{{U}}$ that $G^{{U}_{p_i}}\\le G^{{U}}$ for all $p_i\\in \\pi (G)$ , so $G^{{U}_{p_1}}G^{{U}_{p_2}}\\cdots G^{{U}_{p_n}}\\le G^{{U}}$ .", "Conversely, obviously, $G/(G^{{U}_{p_1}}G^{{U}_{p_2}}\\cdots G^{{U}_{p_n}})\\in {U}_{p_i}$ for any $p_i\\in \\pi (G)$ , so $G^{{U}}\\le G^{{U}_{p_1}}G^{{U}_{p_2}}\\cdots G^{{U}_{p_n}}$ , as desired.", "$\\Box $" ], [ "Basic properties", "Now we give some elementary properties.", "Firstly, we have Proposition 3.1 Let $G=A\\times B$ with $(|A|,|B|)=1$ and $H\\le G$ , then $N_{{U}_p}(G,H)=N_{{U}_p}(A,A\\cap H)\\times B$ whenever $p\\in \\pi (A)$ or $A\\times N_{{U}_p}(B,B\\cap H)$ whenever $p\\in \\pi (B)$ .", "By hypothesis, we may assume that $p\\in \\pi (A)\\setminus \\pi (B)$ .", "Note that $H=(H\\cap A)\\times (H\\cap B)$ and $T=(T\\cap A)\\times (T\\cap B)$ , so $T^{{U}_p}=(T\\cap A)^{{U}_p}\\times (T\\cap B)^{{U}_p}=(T\\cap A)^{{U}_p},$ where $T\\le H$ .", "Now $N_G(T^{{U}_p})=N_{A\\times B}(T^{{U}_p})=N_A((T\\cap A)^{{U}_p})\\times B.$ We have $N_{{U}_p}(G,H) &=\\underset{T\\le H}{\\bigcap }{N_G(T^{{U}_p})} \\\\{}&=(\\underset{T\\le H}{\\bigcap }{N_A((T\\cap A)^{{U}_p})})\\times B \\\\{}&=(\\underset{T\\le H\\cap A}{\\bigcap }{N_A(T^{{U}_p})})\\times B \\\\{}&=N_{{U}_p}(A,A\\cap H)\\times B.$ Similarly, if $p\\in \\pi (B)\\backslash \\pi (A)$ , we have $N_{{U}_p}(G,H)=A\\times N_{{U}_p}(B,B\\cap H)$ .", "$\\Box $ Proposition 3.2 Let $G$ be a $p$ -solvable group, then the ${U}_p$ -residual of $N_{{U}_p}(G)$ is a $p$ -group.", "Denote $X=N_{{U}_p}(G)$ .", "We first prove this result in the case that $X=G$ .", "Assume $G\\in {U}_p$ , obviously $G^{{U}_p}=1$ is a $p$ -group.", "Assume now that $G\\notin {U}_p$ , if $O_p(G)>1$ , then by Lemma REF (2), $G/O_p(G)=N_{{U}_p}(G/O_p(G))$ .", "By induction on $|G|$ , $(G/O_p(G))^{{U}_p}=N^{{U}_p}_{{U}_p}(G/O_p(G))$ is a $p$ -group.", "Obviously $G^{{U}_p}$ is a $p$ -group.", "If $O_p(G)=1$ , then for any $K<G$ , by Lemma REF (1), $K=N_{{U}_p}(K)$ , so by induction, $K^{{U}_p}$ is a $p$ -group.", "Note that $K^{{U}_p}\\unlhd G$ , so $K^{{U}_p}\\le O_p(G)=1$ , this implies that $G$ is a $p$ -solvable minimal non-$p$ -supersolvable group, now by Lemma REF , $G^{{U}_p}$ is a $p$ -group.", "Assume now that $X<G$ , by Lemma REF (1), $X=X\\cap N_{{U}_p}(G)\\le N_{{U}_p}(X)\\le X$ , so $X=N_{{U}_p}(X)$ .", "As a similar argument above, we also have the conclusion.", "$\\Box $ Denote $Z_{\\infty }(G)$ be the terminal term of the ascending central series of $G$ .", "As we know, $Z_{\\infty }(G)=\\bigcap \\lbrace N\\unlhd G \\mid Z(G/N)=1\\rbrace $ , we have the following similar result.", "Theorem 3.3 Let $G$ be a group, then $N^{\\infty }_{{U}_p}(G)=\\bigcap \\lbrace N\\unlhd G\\mid N_{{U}_p}(G/N)=1\\rbrace $ .", "Let $N\\unlhd G$ such that $N_{{U}_p}(G/N)=1$ , then $N^{\\infty }_{{U}_p}(G/N)=1$ .", "By Lemma REF (2), $N^{\\infty }_{{U}_p}(G)N/N\\le N^{\\infty }_{{U}_p}(G/N)=1$ , so $N^{\\infty }_{{U}_p}(G)\\le N$ , thus $N^{\\infty }_{{U}_p}(G)\\le \\bigcap \\lbrace N\\unlhd G\\mid N_{{U}_p}(G/N)=1\\rbrace $ .", "Conversely, choice the least positive integer $n$ such that $N^{\\infty }_{{U}_p}(G)=N^n_{{U}_p}(G)$ .", "By definition, $N_{{U}_p}(G/N^n_{{U}_p}(G))=N^{n+1}_{{U}_p}(G)/N^n_{{U}_p}(G)=N^{n}_{{U}_p}(G)/N^n_{{U}_p}(G)=1$ .", "Obviously, $\\bigcap \\lbrace N\\unlhd G\\mid N_{{U}_p}(G/N)=1\\rbrace \\le N^{\\infty }_{{U}_p}(G)$ .", "$\\Box $ Now we give some characterizations between $N_{{U}_p}(G,H)$ and $C_G(H^{{U}_p})$ .", "Theorem 3.4 Let $G$ be a $p$ -solvable group and $H$ a normal subgroup of $G$ , then $N_{{U}_p}(G,H)=1$ if and only if $C_G(H^{{U}_p})=1$ .", "As we know, $C_G(H^{{U}_p})\\le N_{{U}_p}(G,H)$ , the necessary is obviously.", "Assume now $C_G(H^{{U}_p})=1$ , we prove that $N_{{U}_p}(G,H)=1$ .", "If not, let $N$ be a minimal normal subgroup of $G$ contained in $N_{{U}_p}(G,H)$ , if $N\\nleq H$ , then $N$ normalize $H^{{U}_p}$ by definition, we have $[N,H^{{U}_p}]\\le N\\cap H^{{U}_p}=1$ by the minimal normality of $N$ , so $N\\le C_G(H^{{U}_p})=1$ , a contradiction.", "If $H<G$ , then $(H,H)$ satisfies our hypothesis.", "By induction on $|G||H|$ and Lemma REF (2), $N\\le H\\cap N_{{U}_p}(G,H)\\le N_{{U}_p}(H,H)=1$ , a contradiction, thus $H=G$ .", "Now by the $p$ -solvability of $G$ , $N$ is a $p^{\\prime }$ -group or an elementary abelian $p$ -group.", "If $N$ is a $p^{\\prime }$ -group, then $G/C_G(N)\\in {U}_p$ , otherwise, there is a minimal non-$p$ -supersolvable subgroup $T/C_G(N)$ of $G/C_G(N)$ .", "Let $T=C_G(N)L$ such that $C_G(N)\\cap L\\le \\Phi (L)$ .", "Since $L/\\Phi (L)\\le T/C_G(N)\\cong L/L\\cap C_G(N)$ is a minimal non-$p$ -supersolvable group, we have $(L/\\Phi (L))^{{U}_p}=L^{{U}_p}\\Phi (L)/\\Phi (L)$ is a $p$ -group by Lemma REF .", "Now let $P\\in Syl_p(L^{{U}_p})$ , then $L^{{U}_p}=P(L^{{U}_p}\\cap \\Phi (L))$ .", "By Frattini argument, $L=N_L(P)L^{{U}_p}=N_L(P)P(L^{{U}_p}\\cap \\Phi (L))=N_L(P)$ , we have $P\\unlhd L^{{U}_p}$ and $N$ normalize $P$ , so $P\\le C_G(N)\\cap L\\le \\Phi (L)$ , thus $(L/\\Phi (L))^{{U}_p}=1$ , $L\\in {U}_p$ , contrary to our choice of $T/C_G(N)$ .", "Consequently, $G^{{U}_p}\\le C_G(N)$ , hence $N\\le C_G(G^{{U}_p})=1$ , a contradiction.", "So $N$ is an elementary abelian $p$ -group and $N\\le C_G(N)$ .", "Note that $\\Phi (G)\\le F(G)\\le C_G(N)$ , we consider the following two cases: Case 1.", "$\\Phi (G)=C_G(N)$ .", "In this case, $F(G)=C_G(N)=\\Phi (G)\\le F_p(G)$ , so $N\\le \\Phi (G)$ .", "By the $p$ -solvability of $G$ , $F_p(G/\\Phi (G))=F_p(G)/\\Phi (G)\\ne 1$ , so $\\Phi (G)=F(G)<F_p(G)$ .", "Now follows from $F_p(G)/O_{p^{\\prime }}(G)=O_{p^{\\prime }p}(G)/O_{p^{\\prime }}(G)=O_p(G/O_{p^{\\prime }}(G))$ that $O_{p^{\\prime }}(G)>1$ , so $O_{p^{\\prime }}(G)\\le C_G(N)=\\Phi (G)$ .", "On the other hand, Let $P\\in Syl_p(F_p(G))$ , then $F_p(G)=[O_{p^{\\prime }}(G)]P$ .", "By the Frattini argument, we have $G=F_p(G)N_G(P)=O_{p^{\\prime }}(G)N_G(P)=N_G(P)$ , so $F_p(G)=P\\times O_{p^{\\prime }}(G)=F(G)$ , a contradiction.", "Case 2.", "$\\Phi (G)<C_G(N)$ .", "In this case, if $N\\nleq \\Phi (G)$ , then there exists some maximal subgroup $M$ of $G$ such that $G=NM$ and $N\\cap M=1$ .", "Obviously, $(G/N)^{{U}_p}=G^{{U}_p}N/N\\cong G^{{U}_p}/G^{{U}_p}\\cap N\\cong M^{{U}_p}$ and $N$ normalize $M^{{U}_p}$ .", "By the minimal normality of $N$ , we have $G^{{U}_p}\\cap N=1$ or $G^{{U}_p}\\cap N=N$ .", "If the former holds, then $N\\le C_G(G^{{U}_p})=1$ , a contradiction.", "If the later holds, then $G^{{U}_p}/N\\cong M^{{U}_p}$ .", "On the other hand, $[N,M^{{U}_p}]=1$ , so $N\\le C_G(M^{{U}_p})\\cap C_G(N)=C_G(G^{{U}_p})=1$ since $G^{{U}_p}=NM^{{U}_p}=N\\times M^{{U}_p}$ , a contradiction.", "Now if $G=C_G(N)$ , then $N\\le C_G(G^{{U}_p})=1$ , a contradiction.", "So $C_G(N)<G$ and we may chose a non-$p$ -supersolvable subgroup $L$ of $G$ such that $G=C_G(N)L$ and $C_G(N)\\cap L\\le \\Phi (L)$ .", "Obviously, $N$ normalize $L^{{U}_p}$ and $N\\cap L^{{U}_p}\\unlhd G$ .", "Since $N$ is a minimal normal subgroup of $G$ , we have $N\\cap L^{{U}_p}=1$ or $N\\cap L^{{U}_p}=N$ .", "If $N\\cap L^{{U}_p}=1$ , then $L^{{U}_p}\\le C_G(N)\\cap L$ , so $G/C_G(N)\\cong L/L\\cap C_G(N)\\in {U}_p$ , thus $G^{{U}_p}\\le C_G(N)$ and hence $N\\le C_G(G^{{U}_p})=1$ , a contradiction.", "Which implies that $N\\le L^{{U}_p}$ .", "Now by the fact $G=C_G(N)L$ that $N$ is a minimal normal subgroup of $L$ .", "So $F(L)= C_L(N)=C_G(N)\\cap L=\\Phi (L)$ and $F(L)<F_p(L)$ , as a similar argument of Case 1, we also have a contradiction that $F_p(L)=F(L)$ .", "$\\Box $ As corollaries of Theorem REF , we have Corollary 3.5 Let $G$ be a solvable group and $H$ a normal subgroup of $G$ , then $N_{{U}}(G,H)=1$ if and only if $C_G(H^{{U}})=1$ .", "Corollary 3.6 Let $G$ be a $p$ -solvable group.", "Then $N_{{U}_p}(G)=1$ if and only if $C_G(G^{{U}_p})=1$ .", "Corollary 3.7 Let $G$ be a $p$ -solvable group, if $Z(G^{{U}_p})=1$ , then $C_G(G^{{U}_p})=N_{{U}_p}(G)$ .", "By Corollary REF , we may assume that $C_G(G^{{U}_p})>1$ .", "Moreover, obviously, $C_G(G^{{U}_p})\\le N_{{U}_p}(G)$ .", "Now consider $G/C_G(G^{{U}_p})$ , let $gC_G(G^{{U}_p})\\in C_{G/C_G(G^{{U}_p})}((G/C_G(G^{{U}_p})^{{U}_p})=C_{G/C_G(G^{{U}_p})}(G^{{U}_p}C_G(G^{{U}_p})/C_G(G^{{U}_p})),$ then $[g,G^{{U}_p}]\\le C_G(G^{{U}_p})\\cap G^{{U}_p}=Z(G^{{U}_p})=1$ , thus $g\\in C_G(G^{{U}_p})$ and $C_{G/C_G(G^{{U}_p})}(G/C_G(G^{{U}_p})^{{U}_p})=\\bar{1}$ .", "By Corollary REF , we have $N_{{U}_p}(G/C_G(G^{{U}_p}))=\\bar{1}$ , so by Lemma REF (2), $N_{{U}_p}(G)\\le C_G(G^{{U}_p})$ , as desired.", "$\\Box $ Corollary 3.8 Let $G$ be a $p$ -solvable group, then $Z(G^{{U}_p})=1$ if and only if $G^{{U}_p}\\cap N_{{U}_p}(G)=1$ .", "Since $Z(G^{{U}_p})\\le G^{{U}_p}\\cap N_{{U}_p}(G)$ , so the “only if\" part is obviously.", "Now assume that $Z(G^{{U}_p})=1$ , it follows form Corollary REF that $C_G(G^{{U}_p})=N_{{U}_p}(G)$ , so $G^{{U}_p}\\cap N_{{U}_p}(G)=G^{{U}_p}\\cap C_G(G^{{U}_p})=Z(G^{{U}_p})=1$ , as desired.", "$\\Box $ Theorem 3.9 Let $G$ be a group, then $Z_{\\infty }(G^{{U}_p})\\le N^{\\infty }_{{U}_p}(G)$ .", "If $Z(G^{{U}_p})=1$ , obviously the result holds.", "So we assume $Z(G^{{U}_p})>1$ .", "Now, $Z(G^{{U}_p})\\le N_{{U}_p}(G)$ by definition.", "By induction on $|G|$ , $Z_{\\infty }(G^{{U}_p}/Z(G^{{U}_p}))=Z_{\\infty }((G/Z(G^{{U}_p}))^{{U}_p})\\le N^{\\infty }_{{U}_p}(G/Z(G^{{U}_p}))$ , so $Z_{\\infty }(G^{{U}_p})\\le N^{\\infty }_{{U}_p}(G)$ by Lemma REF (3).", "$\\Box $ As a similar argument above, we have Corollary 3.10 Let $G$ be a group, then $Z_{\\infty }(G^{{U}})\\le N^{\\infty }_{{U}_p}(G)$ for every $p\\in \\pi (G)$ .", "At the end of this section, we investigate the relationship between $N_{{U}_p}(G,H)$ and $N_{{U}}(G,H)$ .", "Lemma 3.11 Let $G$ be a group and $H$ a subgroup of $G$ , then $\\underset{p\\in \\pi (H)}{\\bigcap }N_{{U}_{p}}(G,H)\\le N_{{U}}(G,H)$ .", "Denote $T=\\underset{p\\in \\pi (G)}{\\bigcap }N_{{U}_{p}}(G,H)$ , then for any subgroup $K$ of $H$ , by definition, $T$ normalize $K^{{U}_p}$ for every $p\\in \\pi (K)$ .", "Now by Lemma REF , $T$ normalize $K^{{U}}$ for any $K\\le H$ , so $T\\le N_{{U}}(G,H)$ , as desired.", "$\\Box $ Lemma 3.12 Let $G$ be a group and $H$ a subgroup of $G$ .", "If $H^{{U}}$ is a $p$ -group for some $p\\in \\pi (H)$ , where $\\pi (H)=\\lbrace p_1,p_2,\\cdots ,p_n\\rbrace $ , then $\\underset{p_i\\in \\pi (H)}{\\bigcap }N_{{U}_{p_i}}(G,H)= N_{{U}}(G,H)$ .", "By Lemma REF , we only need to prove $N_{{U}}(G,H)\\le \\underset{p_i\\in \\pi (H)}{\\bigcap }N_{{U}_{p_i}}(G,H)$ .", "Without loss of generality , we assume $p=p_1$ .", "By Lemma REF , $H^{{U}_{p_i}}$ is a $p$ -group for every $p_i\\in \\pi (H)$ .", "Now if $j\\ne 1$ , the $p_j$ -supersolvability of $H/H^{{U}_{p_j}}$ implies that $H\\in {U}_{p_j}$ .", "Thus by Lemma REF , $K^{{U}}=K^{{U}_p}$ for every subgroup $K$ of $H$ and $G=N_{{U}_{p_j}}(G,H)$ for every $p\\ne p_j\\in \\pi (H)$ , so $\\underset{p_i\\in \\pi (H)}{\\bigcap }N_{{U}_{p_i}}(G,H)=N_{{U}_{p}}(G,H)=N_{{U}}(G,H)$ , as desired.", "$\\Box $ Theorem 3.13 Let $G$ be a solvable group and $H$ a normal subgroup of $G$ , then $\\underset{p\\in \\pi (H)}{\\bigcap }N_{{U}_{p}}(G,H)=G$ if and only if $N_{{U}}(G,H)=G$ .", "Denote $\\pi (H)=\\lbrace p_1,p_2,\\cdots ,p_n\\rbrace $ , the “$\\Rightarrow $ \" part of this result follows from Lemma REF .", "Now we prove the “$\\Leftarrow $ \" part, this is equivalent to prove that $N_{{U}_{p_i}}(G,H)=G$ for arbitrary $p_i\\in \\pi (H)$ .", "Assume false, then there is at least one prime, say $p$ in $\\pi (H)$ such that $N_{{U}_p}(G,H)<G$ .", "If $p\\notin \\pi (H^{{U}_p})$ , then the $p$ -supersolvability of $H/H^{{U}_p}$ implies that $H\\in {U}_p$ , then $H^{{U}_p}=1$ , so $G=N_{{U}_p}(G,H)$ , a contradiction.", "By Lemma REF (1), $H=N_{{U}}(H)$ .", "So $H^{{U}}$ is nilpotent by [5].", "Note that $p\\in \\pi (H^{{U}})$ , so there is a minimal normal subgroup $N$ of $G$ contained in $O_p(H^{{U}})$ .", "By Lemma REF , $G/N=N_{{U}}(G/N,H/N)$ .", "By induction on $|G||H|$ , we have $G/N=N_{{U}_{p_i}}(G/N,H/N)$ for any $p_i\\in \\pi (H)$ .", "Now by Proposition REF , $(H/N)^{{U}_{p_i}}$ is a $p_i$ -group for any $p_i\\in \\pi (H)$ .", "In particular, $H^{{U}_p}$ is a $p$ -group.", "If there exists some $p\\ne p_i\\in \\pi (H)$ such that $H^{{U}_{p_i}}\\ne 1$ , then as a similar proof above, there is a minimal normal subgroup $T$ of $G$ contained in $O_{p_i}(H^{{U}_{p_i}})$ such that $G/T=N_{{U}_p}(G/T,H/T)$ .", "For any subgroup $K$ of $H$ , we have $(KT/T)^{{U}_{p}}\\unlhd G/T$ , so $K^{{U}_{p}}T\\unlhd G$ .", "Note that $[K^{{U}_{p}},T]\\le [H^{{U}_p},T]\\le H^{{U}_{p}}\\cap T=1$ since $p\\ne p_i$ , hence $K^{{U}_{p}}$ char $K^{{U}_{p}}T\\unlhd G$ , which implies that $K^{{U}_{p}}\\unlhd G$ .", "This is also contrary to that $G=N_{{U}_{p}}(G,H)$ .", "So $H^{{U}_{p_i}}=1$ for any $p\\ne p_i\\in \\pi (H)$ , thus by Lemma REF , $H^{{U}}=H^{{U}_{p_1}}H^{{U}_{p_2}}\\cdots H^{{U}_{p_n}}=H^{{U}_p}$ is a $p$ -group, now by Lemma REF , we have $\\underset{p_i\\in \\pi (H)}{\\bigcap }N_{{U}_{p_i}}(G,H)= N_{{U}}(G,H)$ , a contradiction.", "$\\Box $ Theorem 3.14 Let $G$ be a solvable group and $H$ a normal subgroup of $G$ , then $\\underset{p\\in \\pi (H)}{\\bigcap }N_{{U}_p}(G,H)=1$ if and only if $N_{{U}}(G,H)=1$ .", "Assume that $N_{{U}}(G,H)=1$ , then by Lemma REF , $\\underset{p\\in \\pi (H)}{\\bigcap }N_{{U}_{p}}(G,H)=1$ .", "Now assume that $\\underset{p\\in \\pi (H)}{\\bigcap }N_{{U}_{p}}(G,H)=1$ , since $C_G(H^{{U}})\\le C_G(H^{{U}_p})\\le N_{{U}_p}(G,H)$ , so $1=C_G(H^{{U}})\\le \\underset{p\\in \\pi (H)}{\\bigcap }N_{{U}_p}(G,H)$ .", "By Corollary REF , $N_{{U}}(G,H)=1$ .", "$\\Box $" ], [ "${N}_1$ -group", "Let $G$ be a group and $H$ a subgroup of $G$ .", "We now consider the case that $G=N_{{U}_p}(G,H)$ , i.e., $G$ is an ${N}_1$ -group with respect to $H$ and ${U}_p$ .", "Obviously, $G$ is an ${N}_1$ -group with respect to $H$ and ${U}_p$ if and only if $K^{{U}_p}\\unlhd G$ for every $K\\le H$ .", "There are many groups which are ${N}_1$ -groups, here we list some of them.", "Proposition 4.1 The following groups are ${N}_1$ -groups with respect to some subgroup $H$ and ${U}_p$ : (1) All ($p$ -)supersolvable groups.", "(2) Groups with $H$ a ($p$ -)supersolvable subgroup.", "(3) Groups all of whose non-($p$ -)supersolvable subgroups are normal.", "(4) Groups with $G^{{U}_p}$ a cyclic subgroup.", "(5) Groups with a normal minimal non-$p$ -supersolvable subgroup $H$ .", "Proposition 4.2 Let $G$ be an ${N}_1$ -group with respect to $H$ and ${U}_p$ , then (1) For any $K\\le G$ , $K$ is an ${N}_1$ -group with respect to $H\\cap K$ and ${U}_p$ ; (2) Let $N\\unlhd G$ and $N\\le H$ , then $G/N$ is an ${N}_1$ -group with respect to $H/N$ and ${U}_p$ ; The proof of (1) and (2) follows from Lemmas REF (2)(3) respectively.", "$\\Box $ In general, if both of a normal subgroup $N$ and corresponding quotient group $G/N$ are ${N}_1$ -groups with respect to themselves and ${U}_p$ , then $G$ may be not an ${N}_1$ -group.", "Example 4.3 Let $G=\\langle a,b,c,d\\rangle \\rtimes \\langle e,f\\rangle \\cong C^4_2\\rtimes C_6$ , where $e^2=f^3=1$ and $a^e=ac,b^e=bd,c^e=c,d^e=d,a^f=b,b^f=ab,c^f=d,d^f=cd$ (IdGroup=[96,70] in GAP [15]).", "Let $N=\\langle a,b,c,d,e\\rangle $ , then $N=N_{{U}_2}(N)$ and $G/N=N_{{U}_2}(G/N)$ .", "Now let $H=\\langle a,b,f\\rangle \\cong A_4$ , the alternating group of degree 4, then $H^{{U}_2}=\\langle a,b\\rangle $ is not normal in $G$ , so $G$ is not an ${N}_1$ -group with respect to itself and ${U}_2$ .", "In the case ${F}\\in \\lbrace {A},{N}\\rbrace $ , the solvability of $N_{{F}}(G)$ has been investigated by many scholars.", "For example, see [6], [7].", "For the case ${F}={U}$ , we have Theorem 4.4 Let $G$ be an ${N}_1$ -group with respect to itself and ${U}$ .", "Then $G$ is solvable.", "In particular, $N_{{U}}(G)$ and $N^{\\infty }_{{U}}$ are solvable.", "Obviously, every subgroup and every homomorphic image of $G$ satisfies our hypothesis.", "If $G$ is not solvable, then $G$ is a non-abelian simple group.", "Let $H$ be a proper subgroup of $G$ , then $H^{{U}}\\unlhd G$ , so $H^{{U}}=1$ and $H$ is supersolvable.", "This implies that $G$ is a minimal non-supersolvable group.", "Now by Doerk's result [19], $G$ is solvable, a contradiction.", "$\\Box $ Let $H$ be a subgroup of $G$ , for the case of ${F}={U}_p$ , the solvability of $N_{{U}_p}(G,H)$ does not always holds.", "Here, we have Theorem 4.5 Let $G$ be a group and $H$ a subgroup of $G$ , if $G$ is an ${N}_1$ -group with respect to $H$ and ${U}_p$ , where $p$ is the smallest prime in $\\pi (H)$ , then $H$ is $p$ -solvable.", "Assume that the result is false and let the pair $(G,H)$ be a counterexample with $|G||H|$ minimal.", "By Feit-Thompson's theorem, we may assume that $p=2$ .", "If $H<G$ , then by Lemma REF (2), $H=N_{{U}_2}(H,H)=N_{{U}_2}(H)$ , so $H$ is 2-solvable by the choice of $(G,H)$ , a contradiction.", "Now we assume that $H=G$ , by hypothesis, $G=N_{{U}_2}(G)$ .", "If $G$ is a non-abelian simple group, we prove the contradiction that $G$ is a solvable simple group.", "Let $K$ be any proper subgroup of $G$ , then $K^{{U}_2}\\unlhd G$ , so $K^{{U}_2}=1$ and $K$ is 2-supersolvable.", "Which implies that $G$ is a non-2-supersolvable group all of whose proper subgroups are 2-supersolvable.", "Now by [20], $G$ is solvable, a contradiction.", "Let $N$ be a minimal normal subgroup of $G$ , then Lemma REF implies that $G/N=N_{{U}_2}(G/N)$ and $N=N_{{U}_2}(N)$ , thus $G/N$ and $N$ are 2-solvable by our choice of $(G,H)$ , hence $G$ is 2-solvable, the finial contradiction complement our proof.", "$\\Box $ Corollary 4.6 Let $G$ be a group and $p$ is the smallest prime in $\\pi (G)$ .", "If $G$ is an ${N}_1$ -group with respect to itself and ${U}_p$ , then $G$ is $p$ -solvable.", "In Theorem REF , the restriction that $p$ is smallest prime in $\\pi (H)$ is necessary.", "Furthermore, in general, we can't obtain the $p$ -solvability of $G$ even though $p=2$ .", "Example 4.7 Let $G=SL(2,5)$ (IdGroup=[120,5], see [15]), then $\\pi (G)=\\lbrace 2,3,5\\rbrace $ and $G$ is not 2-solvable and not 3-solvable.", "But $G=N_{{U}_3}(G)$ .", "Furthermore, $G$ has a normal subgroup, say $H\\cong C_2$ , obviously, $G=N_{{U}_2}(G,H)$ .", "But we have the following obvious conclusions: Theorem 4.8 Let $G$ be a group and $H$ a subgroup of $G$ .", "Assume $G$ is an ${N}_1$ -group with respect to $H$ and ${U}_p$ , where $p$ is the smallest prime divisor of $|H|$ .", "Then $G$ is $p$ -solvable if one of the statement holds: (1) $H\\unlhd G$ and $G/H$ is $p$ -solvable; (2) $G^{{U}_p}\\le H$ ; Theorem 4.9 Let $G$ be a group and $P$ a Sylow $p$ -subgroup of $G$ .", "Assume that $P\\unlhd G$ and $G$ is an ${N}_1$ -group with itself and ${U}_p$ , then $G$ is solvable.", "Theorem 4.10 Let $G$ be a $p$ -solvable ${N}_1$ -group with respect to itself and ${U}_p$ , then (1) $G^{{U}_p}$ is a $p$ -group.", "(2) $N_{{U}_p}(G/N)>1$ for any proper normal subgroup $N$ of $G$ .", "(3) $h_p(G)\\le 3$ ; (1) It follows from Proposition REF .", "(2) Let $N$ be a proper normal subgroup of $G$ , by Lemma REF (2), $G/N=N_{{U}_p}(G/N)$ , so $N_{{U}_p}(G/N)>1$ by our choice of $N$ .", "(3) By (1), $G^{{U}_p}$ is a $p$ -group, then $G^{{U}_p}\\le F_p(G)$ .", "Consider $\\bar{G}=G/F_p(G)$ , then $\\bar{G}$ is $p$ -supersolvable, thus $\\bar{G}^{\\prime }=G^{\\prime }F_p(G)/F_p(G)$ is $p$ -nilpotent.", "So $h_p(G)\\le h_p(\\bar{G})+1\\le h_p(\\bar{G}/\\bar{G}^{\\prime })+h_p(\\bar{G}^{\\prime })+1=3$ .", "$\\Box $ As a corollary of Theorem REF , we have Corollary 4.11 Let $G$ be a ${N}_1$ -group with respect to itself and ${U}_p$ for every $p\\in \\pi (G)$ , then (1) $G$ is solvable.", "(2) $G=N_{{U}}(G)$ .", "(3) $G^{{U}}$ is nilpotent.", "(4) $G/N\\in {N}{U}$ for every normal subgroup $N$ of $G$ .", "(5) $h(G)\\le 3$ .", "(6) $l_r(G)\\le 2$ for every $r\\in \\pi (G)$ .", "(1) Let $p\\in \\pi (G)$ be the smallest prime, if $p>2$ , obviously $G$ is solvable by the well know Feit-Thompson Theorem.", "If $p=2$ , then by Theorem REF , $G$ is 2-solvable, so $G$ is solvable, as desired.", "(2) By hypothesis, $G=N_{{U}_p}(G)$ for every $p\\in \\pi (G)$ .", "Now by Theorem REF , $G=N_{{U}}(G)$ .", "(3) Let $p\\in \\pi (G)$ , if $G\\in {U}_p$ , then $G^{{U}_p}=1$ , if $G\\notin {U}_p$ , by Theorem REF (1), $G^{{U}_p}$ is a $p$ -group.", "Now assume $\\pi (G)=\\lbrace p_1,p_2,\\cdots ,p_n\\rbrace $ , then by Lemma REF , $G^{{U}}=G^{{U}_{p_1}}G^{{U}_{p_2}}\\cdots G^{{U}_{p_n}}$ , so $G^{{U}}$ is nilpotent.", "(4) By (2), $G^{{U}}$ is nilpotent.", "Let $N$ be a proper normal subgroup of $G$ , then $(G/N)^{{U}}=G^{{U}}N/N\\cong G^{{U}}/G^{{U}}\\cap N$ is nilpotent, as desired.", "(5) It follows from Theorem REF (3).", "(6) If $O_{r^{\\prime }}(G)>1$ , then $N_{{U}_p}(G/O_{r^{\\prime }}(G))=G/O_{r^{\\prime }}(G)$ for every $p\\in \\pi (G)$ by Lemma REF (2), so $G/O_{r^{\\prime }}(G)$ satisfies our hypothesis.", "By induction, $l_r(G/O_{r^{\\prime }}(G))\\le 2$ , which implies that $l_r(G)\\le 2$ .", "As a similarly argument above and by Lemma REF , we have $l_r(G)=l_r(G/\\Phi (G))\\le 2$ , hence we may assume that $O_{r^{\\prime }}(G)=\\Phi (G)=1$ , so $F(G)=F_r(G)=O_r(G)$ .", "Now consider $G/F(G)$ , by (5), $h(G/F(G))\\le 2$ .", "Now by Lemma REF , $l_r(G/F(G))=l_r(G/O_r(G))\\le 1$ , so $l_r(G)\\le l_r(G/O_r(G))+1\\le 2$ , as desired.", "$\\Box $ Recall that the well-know results of P.Hall and D.J.S.", "Robinson as follows: Theorem 4.12 Let $G$ be a group and $N$ a nilpotent normal subgroup of $G$ .", "Then (1) (P. Hall) If $G/N^{\\prime }$ is nilpotent, then $G$ is nilpotent.", "(2) (D.J.S.", "Robinson).", "If $G/N^{\\prime }$ is supersolvable, then $G$ is supersolvable.", "Here we have Theorem 4.13 Let $G$ be a $p$ -solvable ${N}_1$ -group with respect to itself and ${U}_p$ .", "Then $G\\in {N}$ if $G^{{N}}\\le G^{{U}^2_p}$ .", "Theorem 4.14 Let $G$ be an ${N}_1$ -group with respect to itself and ${U}$ .", "Then $G\\in {U}$ if $G^{{U}}\\le G^{{U}^2}$ ." ], [ "Applications", "Theorem 5.1 Let $G$ be a $p$ -solvable group, then the following statements are equivalent: (1) $G\\in {N}_p{U}_p$ .", "(2) $G/N_{{U}_p}(G)\\in {N}_p{U}_p$ .", "(1)$\\Rightarrow $ (2).", "It is obviously.", "(2) $\\Rightarrow $ (1).", "Assume the result is false and let $G$ be a counterexample of minimal order.", "If $N_{{U}_p}(G)=1$ , there is nothing to prove.", "Now we assume that $N_{{U}_p}(G)>1$ .", "Let $N$ be a minimal normal subgroup of $G$ contained in $N_{{U}_p}(G)$ .", "Then either $N$ is an elementary abelian $p$ -group or a $p^{\\prime }$ -group.", "Note that $G/N/N_{{U}_p}(G/N)\\le G/N/N_{{U}_p}(G)/N\\cong G/N_{{U}_p}(G)$ by Lemma REF (2), then $G/N\\in {N}_p{U}_p$ by the choice of $G$ .", "So we may assume that $N$ is an elementary abelian $p$ -group, otherwise, $G^{{U}_p}N/N\\cong G^{{U}_p}/(G^{{U}_p}\\cap N) \\in {N}_p$ , so $G\\in {N}_p{U}_p$ by Lemma REF (2), a contradiction.", "Furthermore, if $N\\le \\Phi (G)$ , then $G/\\Phi (G)\\in {N}_p{U}_p$ , so $G^{{U}_p}\\Phi (G)/\\Phi (G)\\in {N}_p$ , hence $G\\in {N}_p{U}_p$ by Lemma REF (1), a contradiction again.", "Now there exist some maximal subgroup $M$ of $G$ such that $G=MN$ and $M\\cap N=1$ .", "Also we have $M^{{U}_p}=(G/N)^{{U}_p}\\in {N}_p$ .", "Note that $N\\le N_{{U}_p}(G)$ , then $N$ normalize $M^{{U}_p}$ , thus $[N,M^{{U}_p}]=1$ and $NM^{{U}_p}=N\\times M^{{U}_p}\\unlhd G$ .", "Since $G/NM^{{U}_p}=MN/NM^{{U}_p}=MNM^{{U}_p}/NM^{{U}_p}\\cong M/M\\cap NM^{{U}_p}=M/M^{{U}_p}\\in {U}_p$ , then $G^{{U}_p}\\le NM^{{U}_p}\\in {N}_p$ , a contradiction.", "$\\Box $ Corollary 5.2 Let $G$ be a $p$ -solvable group, if $G=N^{\\infty }_{{U}_p}(G)$ , then (1) $G\\in {N}_p{U}_p$ .", "(2) $N^{\\infty }_{{U}_p}(G/N)>1$ for any proper normal subgroup $N$ of $G$ .", "(3) $h_p(G)\\le 3$ ; (1) Set $G=N^n_{{U}_p}(G)$ for some positive integer $n$ .", "If $n=1$ , the result follows from Theorem REF (1).", "If $n\\ge 2$ , then $N_{{U}_p}(G)<G$ .", "By Lemma REF , $G/N_{{U}_p}(G)=N^{\\infty }_{{U}_p}(G/N_{{U}_p}(G))$ , so $G/N_{{U}_p}(G)$ satisfies our hypothesis, thus $G/N_{{U}_p}(G)\\in {N}_p{U}_p$ by induction on $|G|$ .", "Now by Theorem REF , $G\\in {N}_p{U}_p$ , as desired.", "(2) If there exists some proper normal subgroup $N$ of $G$ such that $N^{\\infty }_{{U}_p}(G/N)=1$ , then $G=N^{\\infty }_{{U}_p}(G)\\le N$ by Lemma REF (3), that is impossible.", "The proof of (3) is similar to Theorem REF (3).", "$\\Box $ We have proved in Theorem REF that $G$ is solvable if $G=N^{\\infty }_{{U}}(G)$ .", "Now as similar argument above, we have Corollary 5.3 Let $G$ be a group, if $G=N^{\\infty }_{{U}}(G)$ , then (1) $G\\in {N}{U}$ .", "(2) $h(G)\\le 3$ .", "(3) $l_p(G)\\le 2$ for every $p\\in \\pi (G)$ .", "Theorem 5.4 Let $G$ be a group and $p\\in \\pi (G)$ .", "(1) Assume $G$ is $p$ -solvable, then $h_p(G)\\le k$ if and only if $h_p(G/N^{\\infty }_{{U}_p}(G))\\le k$ , where $k\\ge 3$ .", "(2) Assume $G$ is solvable, then $l_p(G)\\le k$ if and only if $l_p(G/N^{\\infty }_{{U}}(G))\\le k$ , where $k\\ge 2$ .", "We only prove (1) since the proof of (2) is similar and based on Corollary REF (4).", "The “$\\Rightarrow $ \" part of this theorem is obviously, so we prove the“$\\Leftarrow $ \" part.", "Setting $N^{\\infty }_{{U}_p}(G)=N^n_{{U}_p}(G)$ for some positive integer $n$ .", "If $G=N^n_{{U}_p}(G)$ , then the result follows form Corollary REF (3).", "So we assume that $N^n_{{U}_p}(G)<G$ .", "Assume that the result is false and let the pair $(G,n)$ be a counterexample with $n|G|$ minimal.", "For the case $n=1$ , we have $h_p(G/N_{{U}_p}(G))\\le k$ by hypothesis.", "Let $N$ be a minimal normal subgroup of $G$ , then by Lemma REF (2), $G/N/N_{{U}_p}(G/N)\\le G/N/N_{{U}_p}(G)N/N\\cong G/N_{{U}_p}(G)N\\cong G/N_{{U}_p}(G)/N_{{U}_p}(G)N/N_{{U}_p}(G)$ , so $h_p(G/N/N_{{U}_p}(G/N))\\le k$ by Lemma REF (2), thus $h_p(G/N)\\le k$ by the choice of $(G,1)$ .", "Suppose that $N$ is not unique, that is $G$ has at least two minimal normal subgroups, say $N_1$ , $N_2$ and $N_1\\ne N_2$ , then as above, $h_p(G/N_1),h_p(G/N_2)\\le k$ , so $h_p(G)\\le k$ by Lemma REF (4), a contradiction.", "With similar argument and by using of Lemma REF (5), we have $\\Phi (G)=1$ .", "Furthermore, if $O_{p^{\\prime }}(G)>1$ , then it follows from Lemma REF (6) that $h_p(G/O_{p^{\\prime }}(G))=h_p(G)\\le k$ , a contradiction.", "So $F_p(G)=F(G)=C_G(N)=N\\le N_{{U}_p}(G)$ .", "Therefore, $G=MN$ , $M\\cap N=1$ for some maximal subgroup $M$ of $G$ .", "If $M^{{U}_p}=1$ , that is $G/N$ is $p$ -supersolvable, so $G/N=N_{{U}_p}(G/N)=N^{\\infty }_{{U}_p}(G/N)=N^{\\infty }_{{U}_p}(G)/N$ , thus $G=N^{\\infty }_{{U}_p}(G)$ , a contradiction.", "If $M^{{U}_p}>1$ , then $N$ normalize $M^{{U}_p}$ , so $[N,M^{{U}_p}]=1$ and hence $M^{{U}_p}\\le C_G(N)=N$ , $M^{{U}_p}\\le N\\cap M=1$ , a contradiction again.", "Now assume that $n>1$ , since $G/N^n_{{U}_p}(G)\\cong G/N^{n-1}_{{U}_p}(G)/N^n_{{U}_p}(G)/N^{n-1}_{{U}_p}(G)=G/N^{n-1}_{{U}_p}(G)/N_{{U}_p}(G/N^{n-1}_{{U}_p}(G))$ , so by induction and combining the proof of the case $n=1$ , we have that $h_p(G/N^{n-1}_{{U}_p}(G))\\le k$ , hence $h_p(G)\\le k$ by our choice of $(G,n)$ , a contradiction.", "$\\Box $ Theorem 5.5 Let $G$ be a $p$ -solvable group.", "If all elements of order $p$ and order 4(if $p=2$ ) of $G$ are in $N^{\\infty }_{{U}_p}(G)$ , then $h_p(G)\\le 3$ .", "Assume that the result is false and let $G$ be a counterexample with minimal order, we prove this result by several steps: Step 1.", "Let $H<G$ , then $h_p(H)\\le 3$ .", "Let $H$ be a proper subgroup of $G$ , then all elements of $H$ of order $p$ and order 4(if $p=2$ ) are contained in $H\\cap N^{\\infty }_{{U}_p}(G)\\le N^{\\infty }_{{U}_p}(H)$ by Lemma REF (1), so $H$ satisfies our hypothesis, $h_p(H)\\le 3$ by the choice of $G$ .", "Step 2.", "$O_{p^{\\prime }}(G)=1$ , $C_G(O_p(G))\\le O_p(G)=F(G)$ .", "Let $T=O_{p^{\\prime }}(G)$ , if $T\\ne 1$ , for any $xT\\in N^{\\infty }_{{U}_p}(G)T/T$ with $|x|=p$ or $|x|=4$ (if $p=2$ ), $xT\\le N^{\\infty }_{{U}_p}(G/T)$ by Lemma REF (3), so $G/T$ satisfies our hypothesis, thus $h_p(G/T)\\le 3$ by the choice of $G$ .", "It is easy to see that $h_p(G)\\le 3$ , a contradiction.", "Further more, we have $C_G(O_p(G))\\le O_p(G)=F(G)$ since $G$ is $p$ -solvable and $O_{p^{\\prime }}(G)=1$ .", "Step 3.", "$\\Phi (G)<O_p(G)$ .", "By (2), $\\Phi (G)\\le O_p(G)$ , if $O_p(G)=\\Phi (G)$ , then $O_{pp^{\\prime }}(G)/\\Phi (G)=O_{p^{\\prime }}(G/\\Phi (G))>1$ since $G$ is $p$ -solvable and $O_p(G/\\Phi (G))=1$ .", "Let $O_{pp^{\\prime }}(G)=\\Phi (G)T$ , where $T>1$ is a $p^{\\prime }$ -group, then $G=N_G(T)O_{pp^{\\prime }}(G)=N_G(T)$ by Frattini argument, it follows that $T\\unlhd G$ and then $T\\le O_{p^{\\prime }}(G)=1$ by Step 2, a contradiction.", "Step 4.", "Finial contradiction.", "Denote by ${F}_p$ the class of all $p$ -solvable groups whose $p$ -Fitting length are less than 3, then by Lemma REF , ${F}_p$ is a saturated formation.", "Now by Step 1, $G$ is an ${F}_p$ -critical group, i.e, $G$ is not belongs to ${F}_p$ , but all proper subgroups of $G$ belong to ${F}_p$ .", "Now by Steps 1,3, there exists some maximal subgroup $M$ of $G$ such that $G=O_p(G)M=F(G)M$ and $M\\in {F}_p$ , so by Lemma REF (2), $G^{{F}_p}$ is a $p$ -group and $G^{{F}_p}$ has exponent $p$ if $p>2$ and exponent at most 4 if $p=2$ , hence $G^{{F}_p}\\le O_p(G)$ and $G^{{F}_p}\\le N^{\\infty }_{{U}_p}(G)$ , it follows that $h_p(G/N^{\\infty }_{{U}_p}(G))\\le 3$ .", "Now by Theorem REF (2), $h_p(G)\\le 3$ , a contradiction.", "$\\Box $ Corollary 5.6 Let $G$ be a $p$ -solvable group, if every cyclic subgroup of order prime $p$ and order 4(if $p=2$ ) of $G$ is contained in $N_{{U}_p}(G)$ , then $h_p(G)\\le 3$ .", "Combining Theorem REF , Lemma REF and Lemma REF , we have the following Theorem REF .", "It's proof is similar to Theorem REF .", "Theorem 5.7 Let $G$ be a solvable group, if every cyclic subgroup of order prime $p$ and order 4(if $p=2$ ) of $G$ is contained in $N^{\\infty }_{{U}}(G)$ , then $l_p(G)\\le 2$ .", "As a local vision of Theorem REF , we have the following question.", "Question 5.8 Let $G$ be a $p$ -solvable group, if every cyclic subgroup of order prime $p$ and order 4(if $p=2$ ) of $G$ is contained in $N^{\\infty }_{{U}_p}(G)$ , does $l_p(G)\\le 2$ ?", "Remark 5.9 (1) In Theorem REF , Corollary REF and Question REF , the Theorem REF implies the hypothesises that $G$ is $p$ -solvable are necessary.", "For example, consider $G=A_5$ , the alternating group of degree 5, then $G=N_{{U}_3}(G)=N_{{U}_5}(G)$ , so $G$ satisfies our condition, but $G$ is simple.", "(2) In Theorem REF , the solvability of $G$ is necessary.", "For example, choice the non-solvable group $G=C_7\\times A_5$ (IdGroup=[420,13]) and let $p=7$ .", "Then $N_{{U}}(G)=N^{\\infty }_{{U}}(G)=C_7$ and every cyclic subgroup of order 7 is in $C_7$ .", "(3) In Question REF , the integer 2 is the minimum upper bound.", "For example, let $G=C_5\\times S_4$ (IdGroup=[120,37] in GAP).", "Obviously, $G$ is 2-solvable and the proper non-2-supersolvable groups of $G$ isomorphism to one of $A_4,S_4,C_5\\times A_4$ .", "As $A_4,S_4,C_5\\times A_4$ are normal in $G$ , so $G=N_{{U}_2}(G)$ .", "Note that $1<C_5<C_{10}\\times C_2<C_5\\times A_4<C_5\\times S_4=G$ is the upper 2-series of $G$ , so $l_2(G)=2$ ." ] ]
2105.11637
[ [ "Silica Raman scattering probe for absolute calibration of Thomson\n scattering spectrometers" ], [ "Abstract We have developed a solid state probe for an absolute irradiance calibration of the Thomson scattering system on the Large Plasma Device (LAPD), based on Raman scattering off silica.", "Measurements performed with a triple-grating spectrometer have investigated the intensities of a pulsed laser beam Raman scattered off crystalline and amorphous silica over a range of temperatures of relevance to the LAPD (299 - 498 K).", "The data were compared with Rayleigh and Raman scattering intensities in gaseous nitrogen.", "The measurements show that Raman scattering off quartz allows rapid and accurate alignment and calibration of Thomson scattering systems in plasma physics experiments that cannot be calibrated using conventional methods." ], [ "Introduction", "Thomson scattering (TS) is a reliable model-independent, non-intrusive diagnostic tool in the studies of plasma scattering of electromagnetic radiation.", "Photons, provided by a pulsed laser beam, scatter off free electrons, which allows characterization of plasma density and temperature with high spatial and temporal resolution [1], [2].", "Absolute calibration of the detection system and measurement conditions in situ is essential to avoiding systematic errors.", "This is typically accomplished via Rayleigh [3] or Raman scattering [4] off gas.", "Rayleigh scattering is the elastic scattering off electrons bound to heavy particles that produces a signal at the laser wavelength [5].", "Therefore, stray light reduction measures must be taken to distinguish the Rayleigh signal from stray light [6].", "Raman scattering is characterized by specific frequency changes of incident photons as a result of a transitions in rotational and vibrational states of the molecule [7].", "The Raman signals appear on both sides of the photon wavelength, separate from the stray light, but within the TS spectral range [8].", "Previous studies examined calibration of TS with Raman scattering off O$_2$ [9], H$_2$ [8], and N$_2$ [7].", "A potential drawback of this approach is that the Raman scattering intensity is less than that of Rayleigh by several orders of magnitude and requires much longer exposure times [10].", "As such, using Raman scattering in conjunction with Rayleigh scattering can calibrate the photon collection efficiency with greater precision, while validating that the intensity response of the detector is linear.", "A TS diagnostic is currently being developed for the Large Plasma Device (LAPD) [11] at the University of California Los Angeles, which can facilitate research on collisionless shocks [12], magnetic reconnection [13], and mini-magnetospheres [14].", "This diagnostic will be described in the future but is based on the same spectrometer used for this work.", "Given that the LAPD is a user facility, it cannot be filled with gas regularly for alignment and calibration.", "We propose an absolute calibration via Raman scattering off a quartz crystal probe.", "To our knowledge, this is the first time that Raman scattering off quartz and fused silica are being used for TS calibration.", "This method offers significant advantages over other techniques that require the placement of a calibrated light source into the scattering volume [15].", "We have cross calibrated crystalline (quartz) and amorphous silica (fused silica) using both Rayleigh and Raman scattering off nitrogen over a range of pressures.", "Quartz has the advantage of producing distinct and bright Raman lines that simplify calibration, while amorphous silica is orientation independent and produces a photon count comparable to TS.", "The Raman spectrum of quartz and line intensities depend on the crystal temperature.", "During LAPD plasma operation the crystal can heat up to temperatures as high as 200oC, significantly affecting the calibration factor.", "However, the crystal temperature can be directly deduced from the measured Raman Stokes to anti-Stokes line intensity ratio.", "We present Raman line intensity measurements for crystal temperatures between 299 and 498 K." ], [ "Experimental setup", "A schematic drawing of the scattering setup is shown in figure REF .", "The laser source employed for our measurements is a diode pumped optical parametric oscillator (OPO), tuned to the second harmonic of Nd:YAG ($\\lambda _i=532.1$ nm, E$_{\\rm {pulse}}$ =10-14 mJ, pulse length $\\tau _L$ = 4 ns).", "The repetition rate was set to 1 Hz to match the parameters that will be used in the LAPD.", "Raman spectra in static nitrogen were also recorded at 50 Hz.", "The laser linewidth is 5 cm$^{-1}$ which is slightly less than the spectrometer resolution.", "The laser emits pulses with shot-to-shot energy fluctuations of 10$\\%$ .", "Pulse energy was recorded on every shot by two photodiodes read out via a 16 bit digitizer that monitors a fraction of the laser beam energy leaking through a turning mirror.", "Intensities of all spectra were post-processed to account for laser energy fluctuations.", "The two energy pickups agree within better than 2$\\%$ .", "Figure: Overview of the experimental setup showing (a) schematic view of the experimental setup, (b) internal arrangement of the spectrometer, and (c) magnified 3D view of the scattering geometry a. Electromagnetic waves are generated by the optical parametric oscillator (OPO).", "Mirrors are used to guide the beam to the vacuum chamber.", "The viewing dump (VD) and baffles inside and outside the chamber reduce stray light contribution.", "Photodiodes (PD) along the path of the laser are utilized for energy measurement.", "A focusing lens (FL) is placed in front of the vacuum chamber in order to focus the beam into the scattering volume (SV).", "Beam is trapped by a beam dump (BD).", "Scattered light is collected with a collection lens (CL) and focused onto a single fiber, which feeds directly into the spectrometer.", "b.", "Light that enters the spectrometer through the input slit is reflected by mirrors (M) and dispersed by three diffraction gratings (DG) as it is passes through the notch filter and is focused onto the intermediate slit (IS) on its way to the detector (ICCD camera).", "c. Crystal is sandwiched between two heaters (white).", "Beam is directed towards the crystal, scattered, and collected at Θ=90 o \\Theta =90^o by the collection lens.The laser beam was focused into the 38 $\\times $ 19 $\\times $ 19 mm$^3$ cuboid crystal mounted in the center of a spherical 75 cm diameter vacuum chamber using a slow f/100 spherical lens with a focal length of f=50 cm.", "This configuration produced a \"pencil\" beam with a diameter of 0.5 mm full-width at half maximum (FWHM) over a Rayleigh length of several centimeters, at a fluence around 10 J/cm$^2$ .", "The beam entered the vacuum chamber through a flat anti-reflex coated quartz window and was incident on the crystal normal to one of the elongated surfaces as shown in figure REF c. Scattered light was collected in an f/20 cone perpendicular to the probe beam ($\\Theta =90^o$ ) to match what will be used in the LAPD and to maximize scattering probability and spatial resolution [7].", "The laser beam was linearly polarized perpendicular to the direction of the collected light ($\\varphi = 90^o$ ).", "A collection lens with a focal length of 10 cm was placed such as to focus the scattered beam at f/5 into a single 200 $\\mu m$ optical fiber with a numerical aperture of NA=0.22, coupled directly to the spectrometer input slit.", "The scattering volume is defined by the intersection of the focused laser beam and the projection of the fiber core cross section onto the beam.", "In order to reduce alignment sensitivity, the collection system was designed such that the 4$\\times $ magnified fiber projection exceeds the beam size by a factor of two.", "Several measures were taken to achieve high stray light rejection.", "A viewing dump and baffles were placed inside the scattering chamber around the laser beam and the collection branch.", "Irises were placed along the path of the beam to filter non-collimated beam components.", "The scattering volume was produced in the center of the chamber, away from its walls and windows.", "The beam terminates in a beam trap 100 cm from the scattering volume to minimize stray light via temporal filtering.", "For Rayleigh and Raman scattering measurements in nitrogen the crystal was removed but the scattering volume stayed the same.", "The chamber was backfilled with 99.998$\\%$ pure nitrogen gas at room temperature at pressures ranging from 0.03 to 1 standard atmospheres.", "The gas was allowed to settle after every pressure change and the pressure was monitored by a capacitance manometer.", "Since the Raman scattering signal is several orders of magnitude dimmer than the unshifted Rayleigh signal and stray light, a triple-grating spectrometer (TGS) was used that enhances the performance of the diagnostic by increased stray light rejection.", "Specifically, a triple grating f/4 Czerney-Turner type imaging spectrometer with an 0.5 m focal length and three 1200 grooves/mm holographic gratings was used.", "As illustrated in figure REF b, scattered light from the fiber entering the spectrometer through the input slit is collimated and guided through a notch filter and intermediate slit by a number of internal mirrors before it reaches the detector.", "The first two gratings are used in subtractive dispersion to reject more stray light.", "The last grating disperses the light onto the detector.", "Toroidal mirrors were used to image the input slit onto the intermediate slit and detector.", "The TGS was equipped with a notch mask between the double-subtractive spectrometers to block wavelengths around 532 nm.", "We used 0.75 mm, or 1 mm wide and 50 $\\mu $ m thick stainless steel notch masks as needed, corresponding to blocking widths of 1.5 nm and 2 nm, respectively.", "The spectral resolution was measured to be 0.21 nm for a 100 $\\mu $ m input slit and a 1.0 mm intermediate slit, with a full wavelength coverage of 19.4 nm.", "The total transmission through the spectrometer was measured to be 25$\\%$ and is mostly determined by the reflectivity of the three aluminum-coated holographic gratings blazed at 500 nm.", "Spectra were recorded on an image intensified charge-coupled device (ICCD) camera, directly mounted to the output of the third spectrometer stage.", "The camera was equipped with a Generation III photocathode with 50$\\%$ quantum efficiency at 532 nm.", "All spectra were recorded at maximum micro-channel plate gain of 233, and with an exposure time of 10 ns, with the 4 ns beam centered in the 10 ns window to account for jitter.", "In order to increase the pixel count a 2$\\times $ 2 pixel hardware binning was applied to all images.", "The spectra were further software binned over the entire fiber and slit vertically into 512 total horizontal bins with 0.038 nm/bin.", "An equal number of data and background shots were recorded back to back.", "The 20 e$^-$ readout noise of the ICCD cooled to -20oC is negligible when binning the spectrum and averaging multiple shots.", "In the absence of plasma fluorescence and since ambient light was eliminated by the short exposure time, the noise was determined solely by the shot noise of the laser.", "The signal to noise (SNR) therefore scaled as SNR $\\sim \\sqrt{N}$ , where $N$ is the number of laser shots.", "We typically averaged 100 laser shots and 100 background shots at 1 Hz for each Rayleigh spectrum, while the Raman spectra in nitrogen were averaged over 25,000 laser shots and an equal number of background shots to increase SNR.", "We investigated Raman scattering off synthetic crystal quartz and fused silica cuboids of identical dimensions.", "Optical surfaces for laser entry and exit and scattered light collection were polished at $\\lambda $ /10 and 20-10 scratch-dig, while the other three surfaces were frosted.", "The crystals' surfaces were aligned to be normal to the beam within $\\pm 1 ^o$ using the laser back-reflection.", "Refraction at the crystal interface is negligible as long as alignment is kept within reasonable limits.", "For example, yawing the crystal by as large an angle as $\\pm $ 5o only displaces the beam in the center of the cuboid by $\\pm $ 0.2 mm, which is less than the diameter of the scattering source and does not affect the collection efficiency.", "This has been confirmed in the measurements below.", "The optical axis of the crystal was aligned parallel to the collection axis.", "Quartz and fused silica produce Raman emission within the spectral range of a typical Thomson scattering spectrometer and have a high optical damage threshold [16] in excess of 100 J/cm$^2$ so that they can be placed in the focus of the pulsed probe beam.", "In this experiment the crystals were mounted on a kinematic stage.", "In the LAPD they will be inserted on a standard probe shaft using a motorized stage [17] via a vacuum interlock for rapid TS alignment and calibration.", "In order to investigate the Raman spectra as a function of temperature the quartz crystal was sandwiched between two 25 W ceramic heaters and the temperature was stabilized with a thermocouple and proportional-integral-derivative (PID) controller.", "The maximum temperature was limited to 225oC by heat conduction in the ambient air and the ceramic optical post and the wire leads." ], [ "Notch filter", "Figure REF a shows the instrument function for a 100 $\\mu $ m slit and without the notch filter, measured by scattering the probe-laser off a frosted glass diffuser.", "The peak amplitude was scaled to the Rayleigh scattering signal amplitude in nitrogen at 750 torr, to produce the equivalent Rayleigh scattering profile without the Raman spectrum.", "The instrument profile is the convolution of the Gaussian contribution due to the aberrated slit width, and the Lorentzian profile caused by diffraction off the three gratings.", "A Raman scattered spectrum recorded at identical conditions with the notch filter is shown for comparison (orange line) and is three orders of magnitude smaller in amplitude than the Rayleigh signal.", "Without a notch filter, the broad Lorentzian wings of the instrument broadened Rayleigh signal overpower the much fainter Raman signal.", "The 1.5 nm width notch reduces the Rayleigh power at 532 nm by five orders of magnitude (blue curve) and the wings by three, and makes the detection of the faint Raman signal possible.", "The notch reduces the intensity of the wings outside its 1.5 nm width, since it removes 532 nm light before it diffracts off the final two gratings.", "The total extinction of the 532 nm instrument broadened line was 2$\\cdot 10^4$ , equivalent to an optical density of 4.3.", "The 2 mm wide notch in combination with a 1 mm intermediate slit had an optical density of 5.0.", "Figure: (a) Comparison of the Rayleigh (black) and Raman scattering spectra (orange) of nitrogen at 750 torr for a 100 μ\\mu m slit.", "Since the instrument broadened Rayleigh signal cannot be separated from the Raman spectrum we actually plot the stray light scaled to the Rayleigh scattering amplitude, which also represents the instrument function.Without the notch, the broad Lorentzian wings of the Rayleigh line overpower the much fainter Raman spectrum.", "The 1.5 nm notch reduces the central 532 nm wavelength by five orders of magnitude and the wings by three (blue line), and reveal the Raman spectrum.", "(b) The measured Rayleigh scattering intensity in N 2 _2 increases linearly with pressure as expected.", "The absence of a signal at zero pressure shows that stray light was effectively eliminated in this experiment." ], [ "Absolute spectrometer calibration", "For light scattering off microscopic particles, and when the scattering cross section depends on the scattering direction, as is the case for Thomson, Rayleigh, or Raman scattering, the scattered power $P_s$ is $P_s = P_i \\cdot n\\cdot L_{det} \\cdot {\\frac{d\\sigma }{d\\Omega }}\\cdot \\Delta \\Omega \\ ,$ where $P_i$ is the incident power, $n$ is the density of scattering particles, $L_{det}$ is the length of the scattering volume, $d\\sigma /d\\Omega $ is the differential cross section, and $\\Delta \\Omega $ is the solid angle of detection.", "The total number of photons emitted per pulse is the total energy divided by the energy of one photon or $E_i/(h\\nu _i)$ , where $\\nu _i$ is the frequency of the incident laser light.", "However, only a very small fraction of these photons encounter and scatter off the particles.", "Laser parameters can be expressed in terms of the power and intensity of photons in the scattering volume.", "Since $P_i = E_i/\\tau _L$ , where $\\tau _L$ is the pulse length, and intensity $I_L = P_i/A$ is power per cross sectional area, the number of photons supplied by the light source per pulse can be quantified as ${\\tau _L \\cdot I_L}/{(h\\nu _i)}$ .", "The scattering volume $\\Delta V =A\\cdot L_{det}$ , and solid angle of detection, $\\Delta \\Omega $ are determined by the collection lens.", "Thus, the total number of photons scattered into the solid angle of the collection lens is [18]: $N = \\frac{\\tau _L \\cdot I_L}{h\\nu _i} \\Delta V \\Delta \\Omega \\cdot n\\cdot \\frac{d \\sigma }{d\\Omega }\\ .$ Transmission efficiency through optical components should also be taken into consideration since only a fraction the photons scattered into the solid angle $\\Delta \\Omega $ make it to the detector.", "The optical transmission through the vacuum window, lens, optical fiber, and spectrometer is only around $\\mu =0.1$ , mostly due to the low reflectivity of the three diffraction gratings in the triple spectrometer.", "Photons are converted to photo-electrons in the iCCD camera with quantum efficiency $\\eta =0.5$ , which are multiplied by the MCP of gain $G=233$ before being recorded on the iCCD.", "The total number of photo-electrons (counts) on the detector area is then $N_{pe} = \\underbrace{ \\overbrace{ \\frac{\\tau _L \\cdot I_L}{h\\nu _i}}^{\\rm {laser}} \\overbrace{ \\Delta V \\Delta \\Omega }^{\\rm {collection}} \\cdot \\overbrace{ \\mu \\ \\eta \\ G}^{\\rm {optics}}}_{k} \\cdot \\ n \\cdot \\frac{d\\sigma }{d\\Omega }\\ .$ The experimental throughput parameter $k$ depends on the probe laser parameters, the collection optics and scattering volume, transmission through optical components and the spectrometer, and the camera sensitivity and gain.", "For Thomson scattering, the total number of counts $N_T$ in a measured spectrum is proportional to the electron density $n_e$ $N_T = k\\cdot n_e \\frac{d\\sigma _T}{d\\Omega }\\ ,$ where the differential cross section is $\\frac{d\\sigma _T}{d\\Omega }=r_e^2$ for scattering perpendicular to the probe beam, and $r_e = 2.818\\cdot 10^{-15}$ m is the classical electron radius.", "If $k$ is known, $n_e$ can be deduced from the measured TS counts.", "Since $k$ is difficult to calculate accurately, it is typically calibrated in situ.", "This can be accomplished, for example, via Rayleigh or Raman scattering.", "The total number of counts in the Rayleigh scattering spectrum is given by $N_R = k\\cdot n_{\\rm {gas}} \\frac{d\\sigma _R}{d\\Omega }\\ ,$ where the differential cross section for nitrogen at 532 nm is $d\\sigma _R / d\\Omega = 6.07\\cdot 10^{-32}\\ m^2$ for scattering perpendicular to the probe beam [19].", "Using a 100 $\\mu $ m input slit, we measured (4.7 $\\pm $ 0.2) $\\cdot 10^4$ total counts at 10 mJ and 750 torr, corresponding to $k= (3.13\\pm 0.14) \\cdot 10^{12}\\ cm^{-1}$ .", "Figure REF b shows the measured Rayleigh scattering intensity in nitrogen as a function of pressure.", "The absence of a signal at zero pressure indicates that stray light was effectively eliminated in this experiment.", "The slope of the linear fit is proportional to $k$ .", "Figure REF shows a raw Raman scattering ICCD image averaged over 25,000 laser shots (a) and the corresponding spectrum of the red-shifted Stokes lines (b).", "The Raman spectrum is composed of a large number of narrow peaks on both sides of the laser line, blocked by the notch and shown in the center of figure REF a.", "Figure: (a) Raw ICCD image showing the Raman spectrum in nitrogen with the laser line blocked by the notch in the center.", "(b) Measured Stokes Raman spectrum (black) and the best fit of the theoretical spectrum (orange).", "The total area under the fit is used to determine the measured Raman scattering signal count.", "(c) Calculated rotational state densities at room temperature and their spectral distribution.", "The statistical weight factor g J g_J causes the alternating amplitudes of adjacent transition lines.", "The differential cross section decreases with wavelength (dashed line).Each line corresponds to a different rovibrational transition from one rotational state $J$ to another $J^{\\prime }$ , induced by the inelastic scattering process.", "Only transitions $J\\rightarrow J + 2$ (Stokes) and $J\\rightarrow J - 2$ (anti-Stokes) are allowed, where the integer $J$ is the rotational quantum number.", "In the following we will review the analytical expressions needed to fit the rotational Raman spectrum as described in more detail elsewhere [20], [21], [19].", "The wavelength of the red-shifted Stokes Raman lines may be approximated by $\\lambda _{J\\rightarrow J+2} \\approx \\lambda _i + \\frac{\\lambda _i^2}{hc}\\cdot B(4J + 6) \\ ,$ where the rotational constant is $B=2.48\\cdot 10^{-4} eV$ for nitrogen and $hc=1240\\ eV\\cdot nm$ .", "The total counts observed for a single Raman line $J$ is given by $N_{J\\rightarrow J^{\\prime }} = k\\cdot n_J \\frac{d\\sigma _{J\\rightarrow J^{\\prime }}}{d\\Omega }\\ $ where the density $n_J$ of a rotational state at temperature $T$ is determined by the Boltzmann distribution $n_J = n_{\\rm {gas}}\\frac{g_J(2J+1)}{Q} exp\\left(-\\frac{E_J}{k_BT}\\right).$ Here $g_J$ is a statistical weight factor, which is $g_J = 6$ or 3 for even or odd $J$ , respectively.", "$Q$ is the partition sum [22] $Q = \\sum _J g_J(2J+1) \\cdot exp\\left(- \\frac{E_J}{k_BT}\\right) \\approx \\frac{9 k_B T}{B}\\ ,$ and the energy of a rotational state $J$ is given by $E_J = B\\cdot J(J+1).$ Figure REF c shows the densities of the Stokes rotational states visible at room temperature and their spectral distribution.", "The statistical weight factor causes the alternating amplitudes of adjacent transition lines.", "The differential cross section of an individual transition $d\\sigma _{J\\rightarrow J^{\\prime }}/d\\Omega $ depends on the Placzek-Teller coefficients, and the polarizability anisotropy, which stems from measurements [20] and was interpolated at 532 nm [19].", "The cross section varies with wavelength as shown in figure REF c around a weighted average of 3.8$\\cdot 10^{-34} m^2$ .", "The total number of counts in the brightest Stokes Raman line ($J = 6 \\rightarrow 8$ ) is given by $N_{\\rm {6\\rightarrow 8}} = \\ k\\cdot n_{\\rm {gas}} \\cdot 4.20\\cdot 10^{-35} m^2\\ .$ At room temperature (T=295 K) only the lowest 20-25 lines are visible on either side of the spectrum and the total number of counts in all visible Stokes lines is given by $N_S = \\sum _{J=0}^{25} N_{J\\rightarrow J+2} = k\\cdot n_{\\rm {gas}} \\cdot 3.82\\cdot 10^{-34} m^2\\ .$ Similarly, the total number of counts in the anti-Stokes spectrum is given by $N_{\\rm {AS}} = \\sum _{J=2}^{27} N_{J\\rightarrow J-2} = k\\cdot n_{\\rm {gas}} \\cdot 2.68\\cdot 10^{-34} m^2\\ ,$ and the total counts in the entire rotational Raman spectrum by $N_{\\rm {Raman}} = N_{\\rm {AS}} + N_S \\ = \\ k\\cdot n_{\\rm {gas}} \\cdot 6.49\\cdot 10^{-34} m^2\\ .$ Given that $k$ remains constant, dividing equation REF by equation REF shows that the total number of counts in the Rayleigh spectrum is 94 times that of the total counts in the rotational Raman spectrum $N_R = 93.7 \\cdot N_{Raman}\\ ,$ and the ratio between the intensities of Rayleigh and the brightest Raman line ($J=6\\rightarrow 8$ ) is $N_R = 1.47\\cdot 10^3 \\cdot N_{6\\rightarrow 8}\\ .$ The Rayleigh to Raman amplitude ratio is larger than the total integrated intensity ratio, since the Raman spectrum is much wider.", "Similarly, the ratio between the measured counts in the Raman spectrum and in the Thomson spectrum could then be used to deduce the electron density $n_e$ [19] $\\frac{N_{\\rm {Raman}}}{N_T} = 8.15\\cdot 10^5 \\ \\cdot \\frac{n_{\\rm {gas}}}{n_e}\\ .$ The spectral resolution in this experiment did only allow the separation of the even-J lines while the weaker odd-J lines were buried in the wings.", "Furthermore, the lines closest to $\\lambda _i$ were partially suppressed by the notch.", "Instead of directly integrating the measured Raman spectrum, the total count was therefore determined from the area under the synthetic fit to the measured spectrum.", "The counts in the Raman fine-structure spectrum as a function of wavelength is $I_{\\rm {fs}}(\\lambda ) = k \\sum _{J} n_J \\frac{d\\sigma _{J\\rightarrow J^{\\prime }}}{d\\Omega } \\delta (\\lambda - \\lambda _{J\\rightarrow J^{\\prime }})\\ ,$ where $\\delta (\\lambda - \\lambda _{J\\rightarrow J^{\\prime }})$ is the Dirac delta function.", "Natural line broadening and pressure broadening are negligible, and the width of the measured Raman peaks is determined solely by the instrument function.", "The synthetic Raman spectrum $I_{\\rm {fit}} = I_{\\rm {fs}} \\circledast I_{\\rm {instr}}$ can then be constructed via convolution of the calculated fine structure spectrum $I_{\\rm {fs}}$ and the experimentally measured instrument function $I_{\\rm {instr}}$ .", "Given that the width of a bin is the same for the experimental and the synthetic spectrum (0.038 nm/bin) the sum of the synthetic spectrum is equivalent to equation REF .", "$N_{\\rm {Raman}} = \\sum _{\\lambda } I_{\\rm {fit}}\\ \\ .$ Using a 100 $\\mu $ m slit we measured a total Raman scattering signal of N$_{\\rm {Raman}}$ =560$\\pm $ 80 counts at 10 mJ and 750 torr (figure REF b).", "This measured ratio of total Rayleigh to Raman scattered photons of $N_{\\rm R}/N_{\\rm {Raman}}$ = 104$\\pm 15$ agrees well with equation REF , which indicates that the camera response is linear over the dynamic range used.", "The measured ratio of Rayleigh peak amplitude to J$_{6\\rightarrow 8}$ Raman peak amplitude is slightly larger than predicted by equation REF , since adjacent lines overlap." ], [ "Raman scattering off silica", "Figure REF compares the Raman spectra of crystalline (a) and amorphous (b) silica.", "The data seamlessly combine multiple spectra obtained with different spectrometer indications to extend the spectral range.", "The spectra show both the red-shifted Stokes and blue-shifted anti-Stokes lines on either side of the 532 nm laser line.", "Rayleigh scattering peaks have amplitudes several orders of magnitude higher than the Raman spectra, and were blocked by a 2 nm wide notch.", "Quartz spectra were integrated over 200 laser shots and 200 background shots, while the fainter spectra of amorphous silica were averaged over 5000 laser shots.", "Figure: Raman spectra of crystalline (a) and amorpheous silica (b) obtained with a 2 nm wide notch mask.", "The area under the 127 cm -1 ^{-1} line in quartz is used for calibration (shaded blue), after subtracting the broad Lorentzian wings of the 207 cm -1 ^{-1} line (orange dashed line).", "The spectrum shows both the Stokes and anti-Stokes lines on either side of the notch filter (shaded orange).The quartz Raman spectrum shows distinct peaks at 127 cm$^{-1}$ , 207 cm$^{-1}$ , and 463 cm$^{-1}$ , that are slightly broader than the instrument function.", "The red-shifted Stokes lines are typically used for sample analysis purposes since they are brighter and less temperature dependent than their blue-shifted anti-Stokes counterparts.", "We used the 127 cm$^{-1}$ peak as the calibration reference since it is closest to the laser line but outside the notch.", "It overlaps with the wide Lorentzian wings of the adjacent 207 cm$^{-1}$ line [23], which must be subtracted.", "The strongest peak at 463 cm$^{-1}$ , which corresponds to symmetric stretching-bending modes of Si-O-Si [24], [25] is outside the spectral range of a typical Thomson scattering diagnostic.", "By contrast, the Raman spectrum of amorphous silica shows a broad band emission with a maximum at around 450 cm$^{-1}$ , which is due to the Si-O-Si bond rocking and bending in the SiO$_4$ tetrahedra [26].", "It is interesting to note, that the total integrated intensity in the amorpheous and in the crystalline silica Raman spectra are almost identical.", "However, in the quartz spectrum intensity is concentrated in a few bright lines, and the amplitudes are an order of magnitude larger.", "In fused silica the Rayleigh signal is more than two orders of magnitude higher in amplitude than the Raman signal.", "In quartz the Rayleigh signal amplitude is one order of magnitude higher than the Raman amplitude.", "The Rayleigh signal of fused silica is also an order of magnitude brighter than the Rayleigh signal from quartz.", "Figure REF compiles the measured intensity of the 127 cm$^{-1}$ quartz Raman line versus the Rayleigh and total Raman scattering signal from nitrogen for various experimental configurations.", "The Raman intensity axis (top) was scaled by a factor of 94 relative to the Rayleigh scattering axis (bottom), consistent with equation REF .", "This data were collected over the course of several weeks and the collection optics and crystal were realigned for each measurement pair.", "While the total photon count decreases with slit width, the ratio between the Rayleigh and quartz intensities, and the ratio between the Raman and quartz intensities stays constant.", "This confirms that the quartz Raman line can be used as a robust and reproducible light source for an absolute spectrometer calibration.", "Figure: Measured intensity of the 127 cm -1 ^{-1} Raman line from quartz versus the Rayleigh and Raman intensities from nitrogen for different experimental configurations.", "The Raman axis (top) was scaled down by a factor of 94 with respect to the Rayleigh axis (bottom).", "While the intensities all decrease with slit width, the ratio stays constant.", "The star-shaped markers shows data obtained with a 150 μ\\mu m slit while intentionally misaligning the collection branch relative to the probe laser beam.The linear fit between the quartz and Rayleigh intensities is given by $N_{\\rm {quartz}} = (129.7 \\pm 0.1)\\ torr\\cdot \\frac{N_R}{p}$ where $p$ is the gas pressure in torr used when measuring the Rayleigh scattering signal $N_R$ .", "At $p=760$ torr the relation can be expressed as $N_{\\rm {quartz}} = (0.1707 \\pm 0.0002)\\cdot N_R$ Since the relation between the Thomson scattering count and the Rayleigh count is $n_e = \\frac{N_T}{N_R} \\cdot \\frac{1}{131} \\cdot n_{gas}$ where n$_{gas} = 2.5\\cdot 10^{19}\\ cm^{-3}$ , $n_e = \\frac{N_T}{N_{\\rm {quartz}}} \\cdot (3.258 \\pm 0.003) \\cdot {10^{16}} cm^{-3}\\ .$ The good agreement between the Rayleigh and the Raman calibration despite more than three orders of magnitude difference in pixel counts confirms that the camera response is linear over the dynamic range used.", "When using the broad fused silica Raman spectrum for calibration, signal amplitude rather than integrated counts can be used.", "Comparing fused silica Raman amplitude to Rayleigh integrated intensity is appropriate since the fused silica spectrum is not instrument broadened.", "At 540 nm the Raman signal of fused silica was measured to be (3.3 $\\pm $ 0.3) $\\cdot 10^4$ counts/nm.", "At identical experimental conditions the measured integrated Rayleigh scattering signal in nitrogen at one standard atmosphere was (2.7$\\pm $ 0.1)$\\cdot 10^{4}$ counts.", "In combination with equation REF the fused silica cross calibration factor is $n_e = \\frac{N_T}{N_{\\rm {SiO_2}}} \\cdot (2.3 \\pm 0.2) \\cdot {10^{16}}\\ \\frac{cm^{-3}}{nm}\\ ,$ where $N_{\\rm {SiO_2}}$ is measured in counts/nm.", "Table REF summarizes all cross-calibration factors between silica and nitrogen.", "Table: Overview of the cross calibration factors at room temperature.It was important to investigate refraction at the crystal interface as significant crystal misalignment could displace the scattering volume.", "For that purpose, signal intensities were investigated as the crystal was intentionally misaligned by $\\pm $  2.5o about each axis.", "The intensity of Raman scattered light did not change as a result of the misalignments.", "Similarly, the star-shaped markers in figure REF represent data obtained with a 150 $\\mu $ m slit while intentionally misaligning the collection branch relative to the probe laser beam.", "While a mismatch decreases the measured intensity, the ratio between the quartz and the Rayleigh lines remains constant.", "These results are indicative of the scattering volume not being affected by refraction as long as the crystal alignment is kept within reasonable limits." ], [ "Temperature dependence of the quartz spectrum", "When using a solid state calibration probe in the LAPD during plasma operation the plasma source can heat the crystal to temperatures close to 200oC.", "The intensity of the 127 $cm^{-1}$ Raman line was therefore calibrated as a function of temperature.", "Temperature was controlled and monitored using ceramic heaters and a thermocouple, respectively.", "Simultaneously, the variation of the Raman spectrum with temperature was measured and compared with theoretical predictions in order to develop a Raman scattering based temperature diagnostic for the crystal.", "Figure REF a compares the Raman spectra for room temperature (black curve) and for 225oC (orange curve).", "Line intensities and widths generally increase with temperature, while the Raman shifts decrease.", "The 207 cm$^{-1}$ line shifts and broadens the most.", "Its Lorentzian line width increases from 0.84 nm (FWHM) at room temperature to 1.58 nm at 225oC.", "Both the line intensities of the Stokes and anti-Stokes lines increase with temperature.", "The anti-Stokes intensities increase faster than the Stokes intensities.", "The Stokes to anti-Stokes intensity ratio therefore decreases with temperature [27], [28], [29] $\\frac{I_{S}}{I_{AS}} = \\left(\\frac{\\nu _i - \\tilde{\\nu }}{\\nu _i + \\tilde{\\nu }}\\right)^n exp\\left(\\frac{h\\tilde{\\nu }}{k_B T}\\right)\\ ,$ where $\\nu _i = c/\\lambda _i = 5.64\\cdot 10^{14}$ Hz is the frequency of the laser beam, $\\tilde{\\nu }=$ 127 cm$^{-1}\\cdot c =3.83\\cdot 10^{12}$ Hz is the transition frequency, and $c$ is the speed of light.", "The term in front of the exponential vanishes for Raman lines close to the laser wavelength.", "Previous experiments have found good agreement with various $n$ .", "While Landsberg and Mandelstam [30] measured Stokes to anti-Stokes ratios in quartz that agreed best with the ordinary Boltzmann distribution function ($n=0$ ), spectra based on energy detection of the signals generally agree well with $n=4$ , while $n=3$ is appropriate when spectra are recorded using photon counting in ICCDs [31].", "Intensity ratios measured in this experiment shown in Fig.", "REF b are consistent with $n=3$ .", "Line positions shift linearly with temperature (Fig.", "REF b) consistent with other observations [32].", "The spectral shift of the 207 cm$^{-1}$ line is a particularly sensitive temperature diagnostic.", "For example, based on the fit in figure REF c the crystal temperature can be determined from the measured line shift $\\Delta \\tilde{\\nu }$ using $T = (297 - 16.3\\ cm\\cdot \\Delta \\tilde{\\nu })\\ K$ with an accuracy of 7 K. Figure: (a) The Raman spectra of quartz at room temperature (black) and at 225oC (orange) show that line intensities and line width increase with temperature, while the Raman shift decreases.", "(b) The measured Stokes versus anti-Stokes line intensity ratio agrees well with equation for n=3n=3.", "(c) The 127 cm -1 ^{-1} and 207 cm -1 ^{-1} Raman shifts decrease linearly with temperature and can be used as a more accurate temperature diagnostic.", "(d) The measured increase of the 127 cm -1 ^{-1} line intensity with temperature agrees well with equation .The intensity of the Stokes lines increases with temperature [33] as $I_{S}(T) \\sim \\left(\\nu _i - \\tilde{\\nu }(T)\\right)^4 \\frac{1}{1- exp\\left(\\frac{-h\\tilde{\\nu }(T)}{k_B T}\\right)\\ }\\ ,$ where the transition frequency $\\tilde{\\nu }$ of the Raman line also varies with temperature.", "This is consistent with the observations.", "Figure REF c shows the measured ratio of the 127 cm$^{-1}$ Stokes line intensity at temperature $T$ and its intensity at T=299 K. The ratio increases by 50$\\%$ for a temperature change of 200 K. A change in temperature of the order of the accuracy of the temperature measurement the intensity only changes by 1.5$\\%$ .", "Equation REF fits the observed intensity change very well.", "Over the temperature range of interest the ratio increases approximately linearly with temperature as $\\frac{I_T}{I_{299\\ K}} = 2.28\\cdot 10^{-3} K^{-1} \\cdot T + 0.320\\ .$ The temperature dependent TS cross calibration relation for quartz is therefore $n_e = \\frac{N_T}{N_{\\rm {quartz}}} \\left( 7.4\\cdot 10^{-3} \\frac{T}{K} + 1.0\\right)\\cdot 10^{16}\\ cm^{-3}\\ .$" ], [ "Summary", "We have used Rayleigh and Raman scattering off nitrogen over a broad pressure range to cross calibrate the Raman spectra of crystalline (quartz) and amorphous fused silica.", "The Raman spectrum of quartz shows three bright peaks at 127 cm$^{-1}$ , 207 cm$^{-1}$ , and 463 cm$^{-1}$ .", "Since it is nearest to the laser line but beyond the notch, we used the 127 cm$^{-1}$ line as the calibration reference.", "Raman scattering off fused silica results in a broad band emission around 450 cm$^{-1}$ .", "Since the ratios between the measured Raman line intensities in quartz and the Raman and Rayleigh line intensities in nitrogen do not vary with the experimental parameters, the quartz Raman line can be used as a stable and reproducible signal for an absolute spectrometer calibration.", "Measurements confirmed that the calibration obtained with Raman scattering off nitrogen fell in line with that of Rayleigh scattering to show the camera response is linear over the range used.", "Measured line intensities and line width increase with temperature, while Raman shifts at 127 cm$^{-1}$ and 207 cm$^{-1}$ decrease linearly with temperature.", "The crystal temperature can be directly determined from the measured line intensity ratios and line shifts.", "These measurements show that Raman scattering off quartz allows an accurate calibration of TS systems in plasma physics experiments, such as those to be performed in the LAPD, where a calibration via scattering off gases is not possible.", "This work was supported by the Department of Energy under contract numbers DE-SC0019011 and DE-SC0021133." ] ]
2105.11643
[ [ "Unbiased Asymmetric Reinforcement Learning under Partial Observability" ], [ "Abstract In partially observable reinforcement learning, offline training gives access to latent information which is not available during online training and/or execution, such as the system state.", "Asymmetric actor-critic methods exploit such information by training a history-based policy via a state-based critic.", "However, many asymmetric methods lack theoretical foundation, and are only evaluated on limited domains.", "We examine the theory of asymmetric actor-critic methods which use state-based critics, and expose fundamental issues which undermine the validity of a common variant, and limit its ability to address partial observability.", "We propose an unbiased asymmetric actor-critic variant which is able to exploit state information while remaining theoretically sound, maintaining the validity of the policy gradient theorem, and introducing no bias and relatively low variance into the training process.", "An empirical evaluation performed on domains which exhibit significant partial observability confirms our analysis, demonstrating that unbiased asymmetric actor-critic converges to better policies and/or faster than symmetric and biased asymmetric baselines." ], [ "Introduction", "Partial observability is a key characteristic of many real-world reinforcement learning (RL) problems where the agent lacks access to the system state, and is restricted to operate based on the observable past, a.k.a.", "the history.", "Such control problems are commonly encoded as partially observable Markov decision processes (POMDPs) [11], which are the focus of a significant amount of research effort.", "Offline learning and online execution is an RL framework where an agent is trained in a simulated offline environment before operating in the real online environment, which offers the possibility of using latent information not generally available in online learning, such as the simulated system state or even the state belief from the agent's perspective [21], [12], [10], [20], [29], [2].", "Offline learning methods are in principle able to exploit this additional information to achieve better online performance, so long as the resulting agent does not use the latent information during online execution.", "Specifically, actor-critic methods [26], [13] are able to adopt this approach via critic asymmetry, where the policy and critic models receive different types of information (e.g., the history and a latent state) [21], [6], [16], [14], [27], [31]; this is possible because the critic is purely a training construct, and is not required for the agent to operate online.", "By the very nature of actor-critic methods, critic models which are unable or slow to learn accurate values act as a performance bottleneck on the policy.", "Consequently, critic asymmetry is a powerful tool which, if carried out with rigor, may provide significant benefits and bootstrap the learning of the overall agent.", "Unfortunately, existing asymmetric methods use asymmetric information heuristically, and rely extensively on empirical experimentation on selected environments to show their validity [21], [6], [16], [14], [27], [31], [22], [17], [23], [20]; however, the lack of a sound theoretical foundation leaves many doubts on whether these methods are able to generalize to other environments.", "Our main contributions are: [label=()] we analyze a standard variant of asymmetric actor-critic and expose analytical issues associated with the use of a state critic, namely that the state value function is, for most environments, either ill-defined, or is well-defined but causes learning bias; we develop the asymmetric policy gradient theorem for partially observable control, an extension of the policy gradient theorem which explicitly features latent state information; we propose a novel unbiased asymmetric actor-critic method, which lacks the analytical issues of biased asymmetric actor-critic and is, to the best of our knowledge, the first of its kind to be theoretically motivated and sound; we validate our theoretical findings through empirical evaluations on environments which feature significant amounts of partial observability, and demonstrate the performance gains of our unbiased variant.", "This work opens the door for other principled asymmetric policy gradient methods that can learn with partial observability.", "In particular, although we focus on advantage actor-critic (A2C), our method can easily be extended to other critic-based learning methods such as off-policy actor-critic [5], [28], (deep) deterministic policy gradient [24], [15], and asynchronous actor-critic [19].", "Similarly, offline training is the dominant paradigm in multi-agent RL with many asymmetric actor-critic methods that could be similarly improved [6], [16], [14], [27], [31], [22], [17], [23].", "Related Work The use of latent information during offline training has been successfully adopted in a variety of policy-based methods [21], [6], [16], [31], [14], [27], [4], [29] and value-based methods [22], [17], [23], [4].", "Among the single-agent methods, asymmetric actor-critic for robot learning [21] uses a reactive variant of DDPG with a state-based critic to help address partial observability; belief-grounded networks [20] use an belief-reconstruction auxiliary task to train history representations; and [29] and [2] use a fully observable agent trained offline on latent state information to train a partially observable agent via imitation.", "Asymmetric learning has also been very popular in the multi-agent setting: COMA [6] uses reactive control and a shared asymmetric critic which can receive either the joint observations of all agents or the system state to solve cooperative tasks; MADDPG [16] and M3DDPG [14] use the same form of asymmetry with individual asymmetric critics to solve cooperative-competitive tasks; R-MADDPG [27] uses recurrent models to represent non-reactive control, and the centralized critic uses the entire histories of all agents; and CM3 [31] uses a state critic for reactive control.", "Asymmetry is also used in multi-agent value-based methods; QMIX [22], MAVEN [17], and WQMIX [23] all train individual Q-models using a centralized but factored Q-model, itself trained using state, joint histories, and joint actions.", "Background In this section, we review the topics relevant to our work, i.e., POMDPs, the reinforcement learning graphical model, standard actor-critic, and asymmetric actor-critic.", "Notation We denote sets with calligraphy $\\mathcal {X}$ , set elements with lowercase $x\\in \\mathcal {X}$ , random variables (RVs) with uppercase $X$ , and the set of distributions over a set as $\\Delta \\mathcal {X}$ .", "Occasionally, we will need absolute and/or relative time indices; We use subscript $x_t$ to indicate absolute time, and superscript $x{k}$ to indicate the relative time of variables, e.g., $x{0}$ marks the beginning of a sequence happening at an undetermined absolute time, and $x{k}$ represents the variable $k$ steps later.", "We also use the bar notation to represent a sequence of superscripted variables $\\bar{x}\\equiv (x{0}, x{1}, x{2}, \\ldots )$ .", "POMDPs A POMDP [11] is a discrete-time partially observable control problem described by a tuple $\\langle , , , , ,, \\gamma \\rangle $ consisting of: state, action and observation spaces $$ , $$ , and $$ ; transition function $\\colon \\times \\rightarrow \\Delta $ ; observation function $\\colon \\times \\times \\rightarrow \\Delta $ ; reward function $\\colon \\times \\rightarrow $ ; and discount factor $\\gamma \\in \\left[0,1\\right]$ .", "The goal is that of maximizing the expected discounted sum of rewards $\\left[ \\sum _t \\gamma ^t R(S_t, A_t \\right]$ , a.k.a.", "the expected return.", "In the partial observable setting, the agent lacks access to the underlying system state, and action selection is based on the observable history $h$ , i.e., the sequences of past actions and observations.", "We denote the space of all histories as $\\doteq \\left(\\times \\right)^*$ , and the space of histories of length $l$ as $_l \\doteq \\left(\\times \\right)^l$ .", "Generally, an agent operating under partial observability might have to consider the entire history to achieve optimal behavior [25], i.e., its policy should represent a mapping $\\pi \\colon \\rightarrow \\Delta $ .", "The belief-state $b\\doteq \\rightarrow \\Delta $ is the conditional distribution over states given the observable history, i.e., $b(h) = \\Pr (S\\mid h)$ , and a sufficient statistic of the history for optimal control [11].", "We define the history reward function as $(h, a) \\doteq _{s\\mid h}\\left[ (s, a)\\right]$ ; from the agent's perspective, this is the reward function of the decision process.", "We denote the last observation in a history $h$ as $o_h$ , and say that an agent is reactive if its policy $\\pi \\colon \\rightarrow \\Delta $ uses the last observation rather than the entire history.", "A policy's history value function ${V^\\pi }\\colon \\rightarrow $ represents the expected future discounted returns when the agent finds itself in history $h$ , ${V^\\pi }(h{0}) = _{\\bar{s}, \\bar{a}\\mid h{0}}\\left[\\sum _{k=0}^\\infty \\gamma ^k R(s{k}, a{k}) \\right] \\,, $ which supports an indirect recursive Bellman form, V(h) = a (a; h) Q(h, a)  , Q(h, a) = (h, a) + oh, a[ V(hao) ]  .", "The RL Graphical Model Figure: The graphical model induced by the environment-agent pair.", "RVs areshown as solid nodes, observed RVs are shown in gray, and latent RVs inwhite.", "The history RVs, shown as dashed nodes, are aggregates of otherRVs, i.e., the previous actions and observations.Some of the theory and results developed in this document concerns whether certain (RVs) of interest are well-defined; therefore, we review the RVs defined in the partially observable control case.", "Together, the environment and the agent induce a graphical model (see fig:graphmodel) over timed RVs $S_t$ , $A_t$ , and $O_t$ .", "Note that only timed RVs are defined directly, and there are no intrinsically time-less RVs.", "Any other RV must be defined in terms of the available ones, e.g.", "we can define a joint RV representing the timed history RVs $H_t \\doteq (A_0, O_0, \\ldots , A_{t-1}, O_{t-1})$ .", "Sometimes it is possible to define a limiting (stationary) state RV $S\\doteq \\lim _{t\\rightarrow \\infty } S_t$ ; However, it is never possible to define a limiting (stationary) history RV $H$ , since the sample space of each timed RV $H_t$ is different and $\\lim _{t\\rightarrow \\infty } H_t$ does not exist.", "A probability is a numeric value associated with the assignment of a value $x$ from a sample space $\\mathcal {X}$ to an RV $X$ , e.g., $\\Pr (X=x)$ .", "Although it is common to use simplified notation and informally omit the RV assignment (e.g., $\\Pr (x)$ ), it must always be implicitly clear which RV is involved in the assignment.", "In the reinforcement learning graphical model, a probability is well-defined if and only if [label=()] it is grounded (implicitly or explicitly) to timed RVs (or functions thereof); or it is time-invariant (i.e., it can be grounded to any time index).", "For example, $\\Pr (s^{\\prime }\\mid s, a)$ is implicitly grounded to the RVs of a state transition $\\Pr (S_{t+1}=s^{\\prime }\\mid S_t=s, A_t=a)$ , and although the time-index $t$ is not clear from context, the probability is time-invariant and thus well defined.", "As another example, $\\Pr (s\\mid h)$ is implicitly grounded to the RVs of a belief $\\Pr (S_t=s\\mid H_t=h)$ ; in this case, the time-index $t$ can be contextually grounded to the history length $t=|h|$ , so the probability is well defined.", "(Symmetric) Actor-Critic for POMDPs Policy gradient methods [26] for fully observable control problems can be adapted to partial observable control problems by replacing occurrences of the system state $s$ with the history $h$ , which is the Markov-state of a history-MDP equivalent to the POMDP.", "In advantage actor-critic methods (A2C) [13], a policy model $\\pi \\colon \\rightarrow \\Delta $ parameterized by $\\theta $ is trained using gradients estimated from sample data, while a critic model ${\\hat{V}}\\colon \\rightarrow $ parameterized by $\\vartheta $ is trained to predict history values ${V^\\pi }(h)$ .", "Note that we annotate parametric critic models with a hat ${\\hat{V}}$ , to distinguish them from their analytical counterparts ${V^\\pi }$ .", "In A2C, the critic is used to bootstrap return estimates and as a baseline, both of which are techniques for the reduction of estimation variance [7].", "The overall A2C objective is $(\\theta , \\vartheta ) \\doteq _\\text{policy}(\\theta ) + _\\text{critic}(\\vartheta ) +\\lambda _\\text{neg-entropy}(\\theta ) \\,.", "$ Policy Loss The policy loss $_\\text{policy}$ is the agent's performance, i.e., the episodic return $_\\text{policy}(\\theta ) \\doteq -\\left[ \\sum _{t=0}^\\infty \\gamma ^t (s_t, a_t)\\right] \\,.$ The policy gradient theorem [26], [13] provides an analytical expression for the policy loss gradient w.r.t.", "the policy parameters, $\\nabla _\\theta _\\text{policy}(\\theta ) = -\\left[ \\sum _{t=0}^\\infty \\gamma ^t{Q^\\pi }(h_t, a_t) \\nabla _\\theta \\log \\pi (a_t; h_t) \\right] \\,.$ In A2C, the value function ${Q^\\pi }(h_t, a_t)$ is replaced with the temporal difference (TD) error $\\delta _t$ , policy() = -[ t=0t t (at; ht) ]  , t (st, at) + V(ht+1) - V(ht)  , to reduce variance at the cost of introducing modeling bias.", "Critic Loss The critic is trained to minimize the TD error of the history-states; to make the policy and critic losses scale similarly to environments with different episode lengths, we adopt an unconventional time-discounted variant of the critic loss, $_\\text{critic}(\\vartheta ) \\doteq \\left[ \\sum _{t=0}^\\infty \\gamma ^t \\delta _t^2\\right] \\,,$ the gradient of which should propagate through ${\\hat{V}}(h_t)$ , but not through the bootstrapping component ${\\hat{V}}(h_{t+1})$ .", "Negative-Entropy Loss Finally, the negative-entropy loss is commonly used, in combination with a decaying weight $\\lambda $ , to avoid premature convergence of the policy model and to promote exploration [30].", "As with the critic loss, we employ a time-discounted variant of the negative-entropy loss, $_\\text{neg-entropy}(\\theta ) \\doteq - \\left[ \\sum _t \\gamma ^t \\left[ \\pi (A_t; h_t)\\right] \\right] \\,.$ Asymmetric Actor-Critic for POMDPs While asymmetric actor-critic can be understood to be an entire family of methods which use critic asymmetry, for the remainder of this document we will be specifically referring to a non-reactive and non-deterministic variant of the work by [21], which uses critic asymmetry to address image-based robot learning.", "Their work uses a reactive variant of deep deterministic policy gradient (DDPG) [15] trained in simulation, and replaces the reactive observation critic ${\\hat{V}}(o)$ with a state critic ${\\hat{V}}(s)$ ; the variant we will be analyzing applies the same critic substitution to A2C.", "In practice, this state-based asymmetry is implemented by replacing the TD error of eq:tderror:h (used in both the policy and critic losses) with $\\delta _t = (s_t, a_t) + \\gamma {\\hat{V}}(s_{t+1}) - {\\hat{V}}(s_t) \\,,$ while every other aspect of A2C remains unchanged.", "Although [21] claim that their work addresses partial observability, their evaluation is based on reactive environments which are virtually fully observable: while only an image is available to the agent, each image gives a virtually complete and collusion-free view of the entire workspace.", "In practice, the images are low-level representations of the full state.", "Theory of Asymmetric Actor-Critic In this section, we analyze the theoretical implications of using a state critic as described in sec:bg:aa2c, and expose critical issues.", "The primary result will be that the state value function ${V^\\pi }(s)$ of a non-reactive agent under partial observability is generally ill-defined.", "Then, we show that the state value function ${V^\\pi }(s)$ of a reactive agent under partial observability is well-defined, but introduces a bias into the training process which may undermine learning.", "Finally, we show that the state value function ${V^\\pi }(s)$ of a reactive agent under an observation function which is equivalent to full observability is both well-defined and unbiased.", "Note that replacing the history critic is intrinsically questionable: the policy gradient theorem for POMDPs (eq:policy-gradient) specifically requires history values, and substituting them for another value which has a different expectation will result in the gradient estimates losing their theoretical guarantees of correctness.", "Therefore, we analyze state values ${V^\\pi }(s)$ as estimators of history values ${V^\\pi }(h)$ , and consider the corresponding estimation bias, i.e., the difference between $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ and ${V^\\pi }(h)$ for a given history $h$ .", "Informally, the fundamental issue with ${V^\\pi }(s)$ is that the state does not contain sufficient information to determine the agent's future behavior (which depends on the history) and is thus unable to meaningfully represent expected future returns.", "Ironically, state values suffer from a problem we call history aliasing, i.e., being unable to infer the agent's history from the system's state.", "Even when ${V^\\pi }(s)$ is numerically well-defined, it usually introduces a bias caused by the imperfect correlation between histories and states; in essence, the average value of histories inferred from the current state is not an accurate estimate of the true current history's value.", "We begin with the definition of the state value function, ${V^\\pi }(s{0}) = _{\\bar{s}, \\bar{a} \\mid s{0}}\\left[\\sum _{k=0}^\\infty \\gamma ^k R(s{k}, a{k}) \\right] \\,,$ which, if well-defined, supports an indirect recursive Bellman form, V(s) = a (as) Q(s, a)  , Q(s, a) = R(s, a) + s's, a[ V(s') ]  .", "From eq:vs, we note the term $\\Pr (a\\mid s)$ , which encodes the likelihood of an action being taken from a given state.", "Because the agent policy acts on histories, this term is not directly available, but must be derived indirectly by integrating over possible histories; and because there is no contextual information available to limit the integration to histories of a specific length, we can only integrate over the space of all possible histories, $\\Pr (a\\mid s) = \\sum _{h\\in } \\Pr (h\\mid s) \\pi (a; h) \\,.", "$ eq:pras reveals the probability term $\\Pr (h\\mid s)$ , which encodes the likelihood of an history having taken place given a state.", "While $\\Pr (h\\mid s)$ may look harmless, it is the underlying cause of serious analytical issues.", "As discussed in sec:graphmodel, a probability term is only well-defined if associated with well-defined RVs, and the fundamental issue with $\\Pr (h\\mid s)$ is that such RVs do not exist.", "On one hand, we cannot used timed RVs $\\Pr (H_t=h\\mid S_t=s)$ , because eq:pras integrates over the sample space of all histories, and not just those of a given length $t$ .", "On the other hand, we cannot use time-less RVs $\\Pr (H=h\\mid S=s)$ , because time-less RVs do not exist in the RL graphical model.", "Ultimately, $\\Pr (h\\mid s)$ is ill-defined, which causes $\\Pr (a\\mid s)$ and ${V^\\pi }(s)$ itself to be ill-defined.", "Theorem 4.1 For an arbitrary POMDP and policy, a time-invariant state value function ${V^\\pi }(s)$ is ill-defined.", "(proof in sec:proofs).", "In relation to the training procedure of a state critic model, the practical implications of an ill-defined value function are not obvious; even though the analytical value function is ill-defined mathematically, the critic's training process performs valid calculations on sample data which results in valid updates of the critic parameters.", "However, convergence is unlikely, since empirical convergence to a meaningful value virtually requires the existence of a theoretical convergence value.", "Rather, it is likely that the critic's training target will shift indefinitely, depending on the recent training data, inhibiting the convergence of the critic model even under ideal training circumstances, which itself will cause instabilities and divergence in the policy model; this is verified empirically in sec:evaluation.", "A natural solution to the underlying issue of state value functions ${V^\\pi }(s)$ is to define and employ timed value functions $V_t^\\pi (s)$ ; in sec:special:timed, we show that timed value functions indeed address the primary issue, although learning a critic to model them is likely to pose a significantly harder learning challenge, due to the need to generalize well and accurately over different time-steps.", "Rather, in the next subsections, we show special cases of the general control problem which make non-timed ${V^\\pi }(s)$ well-defined; in some cases, this will lead to other theoretical issues involving the introduction of estimation bias.", "Reactive Policy under Partial Observability We show that ${V^\\pi }(s)$ is well-defined if we make two assumptions about the agent and environment: [label=()] that the policy is reactive; and that the POMDP observation function depends only on the current state, $\\colon \\rightarrow \\Delta $ , rather than the entire state transition.", "Under these assumptions, we can expand $\\Pr (a\\mid s)$ by integrating over the space of all observations (rather than all histories), $\\Pr (a\\mid s) = \\sum _{o\\in } \\Pr (o\\mid s) \\pi (a; o) \\,.$ In this case, the term $\\Pr (o\\mid s)$ can be grounded to timed RVs $\\Pr (O_t=o \\mid S_t=s)$ ; because that probability is time-invariant, it is well-defined, meaning that ${V^\\pi }(s)$ is well-defined in this case.", "However, we show that ${V^\\pi }(s)$ is biased compared to ${V^\\pi }(h)$ .", "Theorem 4.2 If the POMDP observation function depends only on the current state, $\\colon \\rightarrow \\Delta $ , and the policy is reactive, then ${V^\\pi }(s)$ is not necessarily an unbiased estimate of ${V^\\pi }(h)$ , i.e., it is not guaranteed that ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s)\\right]$ .", "(proof in sec:proofs).", "Although we were able to show that the value function is well-defined in the case of reactive control, there are still two significant issues: [label=()] reactive policies are inadequate to solve many POMDPs; and the bias of the value function ${V^\\pi }(s)$ may influence the agent learning capabilities catastrophically.", "Broadly speaking, this bias is caused by the fact that hidden in ${V^\\pi }(s)$ is an expectation over observations $o$ which, while conditioned on the state $s$ , are not necessarily consistent with the true history $h$ .", "We discuss the cause of this bias more formally in the corresponding proof.", "Reactive Policy under Full Observability We show that the state value function is not only well-defined but also unbiased if we make two assumptions about the agent and environment: [label=()] that the policy is reactive; and that the there is a bijective abstraction $\\phi \\colon \\rightarrow $ between observations and states.", "The abstraction $\\phi $ encodes the fact that the environment is not truly partially observable, but rather that states and observations essentially contain the same information, albeit at different levels of abstraction, akin to the problems used by [21].", "For example, an image displaying a workspace without occlusions could be a low-level abstraction (observation), while a concise vector representation of the object poses in the workspace could be a high-level abstraction (state).", "In this case, the action probability term $\\Pr (a\\mid s)$ does not need to be obtained indirectly by integrating other variables; rather, the state-observation bijection can be used to directly relate it to the policy model, (as) = (a; I(s))  .", "Contrary to the previous cases, the overall state value function ${V^\\pi }(s)$ is not only well-defined, but also unbiased.", "Theorem 4.3 If the POMDP states and observations are related by a bijection $\\phi \\colon \\rightarrow $ , and the policy is reactive, then ${V^\\pi }(s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ .", "(proof in sec:proofs).", "The benefit of using a state critic under this scenario is that the critic model can avoid learning a representation of the observations before learning the values [21].", "Naturally, the main disadvantage of this scenario is that most POMDPs do not satisfy the bijective abstraction assumption, which is virtually equivalent to full observability.", "Nonetheless, if a control problem only deviates mildly from full observability, it is very possible that a state critic might benefit the learning agent.", "Unbiased Asymmetric Actor-Critic In this section, we introduce unbiased asymmetric actor-critic, an actor-critic variant which is able to exploit asymmetric state information during offline training while avoiding the issues of state value functions exposed in sec:aa2c.", "Consider a history-state value function ${V^\\pi }(h,s)$  [1], which represents the expected future discounted returns obtained when the history is $h$ and the state is $s$ , ${V^\\pi }(h{0}, s{0}) = _{\\bar{s}, \\bar{a} \\mid h{0},s{0}}\\left[ \\sum _{k=0}^\\infty \\gamma ^k R(s{k}, a{k})\\right] \\,, $ which supports an indirect recursive Bellman form, V(h, s) = a (a; h) Q(h, s, a)  , Q(h, s, a) = (s, a) + s',os,a[ V(hao, s') ]  .", "Providing the history information makes the history-state value function ${V^\\pi }(h, s)$ not only well-defined even for non-reactive policies, but also an unbiased estimate of ${V^\\pi }(h)$ .", "Theorem 5.1 For an arbitrary POMDP and policy, ${V^\\pi }(h, s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(h, s)\\right]$ .", "(proof in sec:proofs).", "As we have done for state values ${V^\\pi }(s)$ , we are interested in the properties of history-state values ${V^\\pi }(h, s)$ in relation to history values ${V^\\pi }(h)$ .", "thm:vhs shows that history and history-state values are related by ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(h, s) \\right]$ , i.e., history-state values are interpretable as Monte Carlo (MC) estimate of the respective history values.", "In expectation, history-state values provides the same information as the history values, therefore an asymmetric variant of the policy gradient theorem also holds.", "Theorem 5.2 (Asymmetric Policy Gradient) The policy gradient can be expressed using history-state values, $\\nabla _\\theta _\\text{policy}(\\theta ) = -\\left[ \\sum _t \\gamma ^t {Q^\\pi }(h_t,s_t, a_t) \\nabla _\\theta \\log \\pi (a_t; h_t) \\right] \\,.$ (proof in sec:proofs).", "As estimators, history-state values ${V^\\pi }(h, s)$ can be described in terms of their bias and variance w.r.t.", "history values ${V^\\pi }(h)$ .", "Beyond providing the inspiration for the MC interpretation, thm:vhs already proves that ${V^\\pi }(h, s)$ is unbiased, while its variance is dynamic and depends on the history $h$ via the belief-state $\\Pr (S\\mid h)$ ; in particular, low-uncertainty belief-states result in relatively low variance, and deterministic belief-states result in no variance.", "Given that operating optimally in a partially observable environment generally involves information-gathering strategies associated with low-uncertainty belief-states, the practical variance of the history-state value is likely to be relatively low once the agent has learned to solve the task to some degree of success.", "Inspired by thm:asymmetric-policy-gradient, we propose unbiased asymmetric A2C, which uses a history-state critic ${\\hat{V}}\\colon \\times \\rightarrow $ trained to model history-state values ${V^\\pi }(h, s)$ , policy() = -[ t t t (at; ht) ]  , t = R(st, at) + V(ht+1, st+1) - V(ht, st)  .", "Because ${\\hat{V}}(h, s)$ receives the history $h$ as input, it can still predict reasonable estimates of the agent's expected future discounted returns; and because it receives the state $s$ as input, it is still able to exploit state information while introducing no bias into the learning process, e.g., for the purposes of bootstrapping the learning of critic values and/or aiding the learning of history representations.", "Interpretations of State Although the history-state value is analytically well-defined, It is worthwhile to question why the inclusion of the state information should help the actor-critic agent at all.", "We attempt to address this open question, and consider two competing interpretations, which we call state-as-information and state-as-a-feature.", "State as Information Under this interpretation, state information is valuable because it is latent information unavailable in the history, which results in more informative values.", "However, this interpretation is flawed for two reasons: [label=()] The policy gradient theorem specifically requires ${V^\\pi }(h)$ , which contains precisely the correct information required to accurately estimate policy gradients.", "In this context, there is no such thing as “more informative values” than history values.", "In theory, the history-state value in thm:asymmetric-policy-gradient could use any other state sampled according to $\\tilde{s}\\sim b(h)$ , rather than the true system state, which would also result in the same analytical bias and variance properties.", "In practice, we use the true system state primarily due to it being directly available during simulation; however, we believe that its identity as the true system state is analytically irrelevant, which leads to the next interpretation of state.", "State as a Feature We conjecture an alternative interpretation according to which the state can be seen as a stochastic high-level feature of the history.", "Consider a history critic ${\\hat{V}}(h)$ to appropriately model the value function ${V^\\pi }(h)$ , the model must first learn an adequate history representation, which is in and of itself a significant learning challenge.", "The critic model would likely benefit from receiving auxiliary high-level black-box features of the history $\\phi (h)$ .", "The resulting critic ${\\hat{V}}(h,\\phi (h))$ remains fundamentally a history critic, the supplementary features being exclusively a modeling construct.", "Next, we consider what kind of high-level features $\\phi (h)$ would be useful for control.", "While the specifics of what makes a good history representation depend strongly on the task, there is a natural choice which is arguably useful in many cases: the belief-state $b(h)$ .", "Because the belief-state is a sufficient statistic of the history for control, providing it to the critic model ${\\hat{V}}(h, b(h))$ is likely to greatly improve its ability to generalize across histories.", "Finally, we conjecture that any state sampled according to the belief-state $s\\sim b(h)$ —including the true system state—can be considered a stochastic realization of the belief-state feature, resulting in the history-state critic ${\\hat{V}}(h, s)$ .", "According to this interpretation, the importance of the state in the history-state critic is not in its identity as the true system state, but as a stochastic realization of hypothetical belief-state features, and presumably any other state sampled from the belief-state $\\tilde{s}\\sim b(h)$ could be equivalently used.", "Evaluation [tb] Each method follows the same algorithm structure, but uses different types of critics to compute the TD errors $\\delta _t$ (see eq:tderror:h,eq:tderror:s,eq:tderror:hs).", "Values $N$ , $B$ , and $E$ vary by environment.", "Input: epochs $N$ , episode batch $B$ , evaluation period $E$ epoch in 1 ...$N$ training_episodes $\\leftarrow $ sample_episodes($\\pi $ , $B$ ) update $\\theta ,\\vartheta $ via $\\nabla ($ training_episodes$)$ (see eq:a2c) epoch $\\bmod $ evaluation_period = 0 evaluation_episodes $\\leftarrow $ sample_episodes($\\pi $ , $E$ ) report empirical_returns(evaluation_episodes) In this section, we empirically evaluate three actor-critic variants: A2C(h), standard actor-critic with a history critic ${\\hat{V}}(h)$ as described in sec:a2c; A2C(s), asymmetric actor-critic with a state critic ${\\hat{V}}(s)$ as described in sec:aa2c; and A2C(h,s), unbiased asymmetric actor-critic with a history-state critic ${\\hat{V}}(h,s)$ as described in sec:uaa2c.", "Each method is trained and evaluated according to alg:code.", "Broadly speaking, each method uses a gated recurrent unit (GRU) [3] to compute history features, and two feed-forward networks which represent the policy action probabilities and critic values; sec:architectures contains a more detailed description of the used architectures.", "We perform evaluations on seven gridworld environments which exhibit significant partial observability: Shopping-5, Shopping-6, Heavenhell-3, Heavenhell-4, and Rocksample-5-6 are flat POMDPshttps://github.com/abaisero/gym-pomdps, where states and observations are represented by categorical indices, while Keydoor and Ninerooms are gridverse POMDPshttps://github.com/abaisero/gym-gridverse, where states and observations are represented by tensors of categorical indices which encode spatial relationships and other cell information.", "See sec:environments for a detailed descriptions and graphical representations of all environments.", "Results and Discussion Each method is evaluated in one of two ways: [label=()] we show empirical learning curve statistics for all environments, and we show how critic values change for important history-state pairs over the course of training in Heavenhell-4.", "Learning Curves Figure: Learning curve statistics over 20 independent training runs, whereeach run is periodically evaluated 20 times.", "Shaded areas are centeredaround the empirical mean performance, and show 2 standard errors (of themean).fig:learningperformance depicts the performance results in all the environments.", "First, we note that the symmetric baseline A2C(h) does not always learn to solve the task, but succeeds fully in fig:shoppingv,fig:heavenhelliii,fig:gvkeydoor, partially in fig:heavenhelliv,fig:rocksamplevvi,fig:gvninerooms, and fails in fig:shoppingvi.", "The asymmetric baseline A2C(s) also performs inconsistently across environments, with a mixture of successful and failing cases.", "We particularly note the strange learning curves of A2C(s) in fig:shoppingv,fig:shoppingvi, where performance improves quickly during the early training, but fails to improve further or even becomes unstable and collapses later on.", "While the exact dynamics of this collapse are not completely clear, we believe it is likely that this is a consequence of modeling the critic after an analytically unstable state value function, making convergence to stable values impossible.", "Using our proposed history-state critic, A2C(h,s) consistently exhibits either faster convergence and/or higher final performance in virtually all the flat environments, and competitive performance in the gridverse environments, taking a mild lead towards the end of Ninerooms.", "These results strongly demonstrate the importance of exploiting asymmetric information in ways which are theoretically justified and sound, as done in our work.", "Critic Values Figure: Critic value statistics for 4 history-state pairs, evaluatedthroughout the training procedure via 20 independent runs.In the top row the agent did not visit the priest, while in the bottom rowthe agent did visit the priest.To further inspect the behavior of each critic, we show the evolution of critic values for important history-state pairs over the course of training.", "We perform this evaluation on Heavenhell-4, and use 4 deliberately chosen history-state pairs.", "In each case the agent is located at the fork, and the 4 cases differ according to heaven's location (left or right) and whether the agent has previously visited the priest.", "fig:criticvalues shows the resulting critic values, with one figure for each of the chosen history-state pairs.", "In each scenario, we note that all critic values exhibit convergence properties which resemble those of the agent's performance in fig:heavenhelliv.", "This empirically confirms that agent performance and critic quality are strongly correlated factors; although it is not possible to make conclusions about the causality in this situation, we strongly believe that it is the critics which act as a learning bottleneck on policies.", "Notably, the critics which focus on a single aspect of the joint history-state show the exact same values for different history-state; namely, A2C(s) is identical in the top and bottom plots, while A2C(h) is identical in the top-left and top-right plots.", "Although in practice left and right plots are similar for all critics, A2C(h,s) is the only critic capable of representing different values in each of the 4 scenarios, as none of its curves are identical.", "We also note that the state critic ${\\hat{V}}(s)$ is muct less stable than the others, and shows no signs of convergence, which is consistent with our analysis in sec:aa2c.", "In contract, the history-state critic exhibits good convergence properties despite itself also using state information, which is consistent with our analysis in sec:uaa2c.", "Finally, we note again that the history-state critic ${\\hat{V}}(h, s)$ converges significantly faster than the history critic ${\\hat{V}}(h)$ , confirming that state information is useful for training.", "Conclusions Asymmetric methods trained offline in simulated environments can use information which is normally unavailable in partially observable RL, such as the true system state.", "While the idea of exploiting such information has potential, current state-of-the-art methods are powered by empirical results rather than theoretical analysis.", "In this work, we exposed profound theoretical issues with a standard variant of asymmetric actor-critic, and proposed an unbiased asymmetric actor-critic variant which is analytically sound and theoretically justified.", "Empirical results confirm our analysis, the weaknesses of state critics and the strengths of history-state critics.", "Although we applied the history-state value function only to A2C, the same concepts are easily extensible to other critic-based RL methods [24], [15], [5], [19].", "In future work, we aim to extend the theory of history-state policy value functions ${Q^\\pi }(h, s, a)$ to optimal value functions $Q^*(h, s, a)$ , and develop theoretically sound asymmetric variants of other deep RL methods such as DQN [18], soft Q-learning [8], and soft actor-critic [9].", "We also plan to extend the theory and approach to multi-agent methods, potentially bringing theoretical rigor and improved performance [6], [16], [14], [27], [31], [22], [17], [23].", "Timed Value Functions sec:aa2c shows that for a general POMDP and policy $\\pi \\colon \\rightarrow \\Delta $ , the state value function ${V^\\pi }(s)$ is not generally well-defined due to issues caused by the lack of time information.", "In this section, we consider addressing the primary issue by providing explicit time-index information via timed value functions, $V_t^\\pi (s)$ and $Q_t^\\pi (s, a)$ , which represent the expected returns obtained when the agent finds itself in a state $s$ at time $t$ , Vt(s) = a (At=aSt=s) Qt(s, a)  , Qt(s, a) = R(s, a) + s's, a[ Vt+1(s') ]  .", "Once again, we analyze the state-dependent action distribution term to verify correctness, and expand it by integrating over histories; this time, we can use the explicit time-index information to integrate over histories of a given length only, $\\Pr (A_t=a\\mid S_t=s) = \\sum _{h\\in _t} \\Pr (H_t=h\\mid S_t=s) \\pi (a; h)\\,.", "$ Because eq:sumrule:timed is now restricted to histories of a given length $t$ , the probability term $\\Pr (H_t=h\\mid S_t=s)$ is well-defined, which means $\\Pr (A_t=a\\mid S_t=s)$ is well-defined, and $V_t^\\pi (s)$ is well-defined.", "Introducing the time-index makes the value function $V_t^\\pi (s)$ well-defined.", "However, its utility for the purpose of asymmetric reinforcement learning remains unclear because [label=()] it is still not formally proven whether the timed value function is unbiased, i.e., whether $V_t^\\pi (h) = _{s\\mid h}\\left[V_t^\\pi (s) \\right]$ , and it is harder for timed value critics to generalize appropriately across the additional discrete input $t$ .", "Proofs This section contains the proofs omitted from the main body of the document.", "For the sake of clarity, we repeat the main statements before showing the respective proof.", "thm:vsTheorem REF For an arbitrary POMDP and policy, a time-invariant state value function ${V^\\pi }(s)$ is ill-defined.", "The state value function, defined as ${V^\\pi }(s) = \\sum _{a\\in } \\Pr (a\\mid s) {Q^\\pi }(s, a) \\,,$ requires the state-conditioned action probability term $\\Pr (a\\mid s)$ .", "Because a partially observable policy depends on the history and not the state, the state-conditioned action probability term must be expanded by integrating over the space of all histories, $\\Pr (a\\mid s) = \\sum _{h\\in } \\Pr (h\\mid s) \\pi (a; h) \\,, $ which requires the state-conditioned history probability term $\\Pr (h\\mid s)$ .", "However, it is impossible to associate that term to well-defined RVs: Clearly, the reinforcement learning graphical model (see sec:graphmodel) does not define a time-less history RV $H$ , so the term cannot be explicitly written as $\\Pr (H=h\\mid S=s)$ .", "Further, the probability associated with timed RVs $\\Pr (H_t=h\\mid S_t=s)$ is clearly not time-invariant, hence identifying the time-index is crucial.", "Finally, it is not possible to restrict the integration in eq:pras:app only to histories of a given time-index.", "Overall, the probability term $\\Pr (h\\mid s)$ is necessarily time-variant, which means that $\\Pr (a\\mid s)$ is itself time-variant, and therefore the state value function ${V^\\pi }(s)$ is time-variant, and a time-invariant state value function ${V^\\pi }(s)$ is ill-defined.", "thm:vs.reactive.poTheorem REF If the POMDP observation function depends only on the current state, $\\colon \\rightarrow \\Delta $ , and the policy is reactive, then ${V^\\pi }(s)$ is not necessarily an unbiased estimate of ${V^\\pi }(h)$ , i.e., it is not guaranteed that ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s)\\right]$ .", "As one of the salient theorems of our work, we will cover it from different angles and provide three proofs: first one which is simple but does not go into the detail about the causes of the bias, then a more detailed and analytical one, and finally one by example.", "[First proof] Consider two histories $h, h^{\\prime }\\in $ which are different, $h\\ne h^{\\prime }$ , but are associated with the same belief $b(h) = b(h^{\\prime })$ ; a fairly common occurrence in many POMDPs.", "On one hand, because the two histories are different, the agent's next action may differ, which may then lead to different immediate rewards and future trajectories, i.e., the respective values would also differ, ${V^\\pi }(h) \\ne {V^\\pi }(h^{\\prime }) \\,.$ On the other hand, because the two beliefs are the same, then the expected state values must be the same, $_{s\\mid h}\\left[ {V^\\pi }(s) \\right] = _{s\\mid h^{\\prime }}\\left[ {V^\\pi }(s)\\right] \\,.$ Therefore, it is not guaranteed that ${V^\\pi }(h) = _{s\\mid h}\\left[{V^\\pi }(s) \\right]$ .", "[Second proof, by contradiction] First, we assume that ${V^\\pi }(s)$ is unbiased and show that ${Q^\\pi }(s,a)$ (as defined by eq:qsa) is unbiased, sh[ Q(s, a) ] = sh[ (s, a) + s's, a[ V(s') ] ] = sh[ (s, a) ] + sh[ s's, a[ V(s') ] ] = sh[ (s, a) ] + s'h, a[ V(s') ] = sh[ (s, a) ] + oh, a s'hao[ V(s') ] = (h, a) + oh, a[ V(hao) ] = Q(h, a)  .", "Next, we show that even if ${Q^\\pi }(s, a)$ is unbiased, ${V^\\pi }(s)$ (as defined by eq:vs) is biased, which contradicts the original assumption.", "To do that, we expand the expected state value function $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ and the history value function ${V^\\pi }(h)$ , and show that there is a concrete difference between them: sh[ V(s) ] = sh[ a (as) Q(s, a) ] = sh[ a os[ (a; o) ] Q(s, a) ]  , V(h) = a (a; oh) Q(h, a) = a (a; oh) sh[ Q(s, a) ] = sh[ a os[ (a; oh) ] Q(s, a) ]  .", "eq:proof:vs,eq:proof:vh differ in terms of which observation is used by the policy; in eq:proof:vs, an observation $o$ inferred from a state $s$ inferred from the history $h$ is used, while in eq:proof:vh the final observation $o_h$ of the history $h$ is used.", "These two observations $o$ and $o_h$ are not generally the same, and the respective expectations are similarly not generally the same.", "The nested expectation in eq:proof:vs can be interpreted as a lossy round-trip inference from history to state and from state back to observation $h\\rightarrow s\\rightarrow o$ .", "Although histories and states tend to be somewhat correlated, both state aliasing and history aliasing make the roundtip conversion imperfect, causing a mismatch between the expected state value function $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ and the history value function ${V^\\pi }(h)$ in the general control case of a general POMDP.", "[Third proof, by example] This is a proof by example (with a proof by contradiction element).", "We will define the good/bad POMDP and, for a specific policy and history, first calculate $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ exactly, and then ${V^\\pi }(h)$ using bootstrapping (while also assuming ${V^\\pi }(hao) =_{s^{\\prime }\\mid hao}\\left[ {V^\\pi }(s^{\\prime }) \\right]$ ).", "We show that the two values are numerically different.", "In the good/bad POMDP, $= \\lbrace \\texttt {GOOD}, \\texttt {BAD}\\rbrace , = \\lbrace \\texttt {GOOD},\\texttt {BAD}\\rbrace $ , $= \\lbrace \\texttt {GOOD}, \\texttt {BAD}\\rbrace $ ; At times, we will use the shorthands $\\texttt {G}$ and $\\texttt {B}$ .", "The initial state distribution is uniform, and each state deterministically transitions into itself.", "The GOOD state always emits the GOOD observation, while the BAD state emits a random observation.", "Consider the reward function such that $R(s, a) = \\left[ a = \\texttt {GOOD}\\right]$ , i.e., the agent receives a reward whenever it choses the GOOD action.", "We will denote a history as the concatenation of alternating observations and actions, starting with an observation.", "To keep the notation compact, we will occasionally use symbols G and B to represent GOOD and BAD states, observations and actions.", "Consider a deterministic policy $\\pi (a; h) = \\left[ a = o_h \\right]$ which returns the action corresponding to the last observation.", "Note that this POMDP and this policy satisfy the requirements to guarantee that ${V^\\pi }(s)$ is well defined.", "Next, we calculate the state values ${V^\\pi }(s)$ .", "The $\\texttt {GOOD}$ state always emits the $\\texttt {GOOD}$ observation, so the agent will always choose the $\\texttt {GOOD}$ action and receive a reward of 1, then the state will always transition into itself, V(s=GOOD) = 1 + V(s=GOOD) = 11 -  .", "On the other hand, the $\\texttt {BAD}$ state will only emit the $\\texttt {GOOD}$ observation half of the times, so the agent will only choose the $\\texttt {GOOD}$ action and receive a reward of 1 half of the times, then the state will always transition into itself, V(s=BAD) = 12 + V(s=BAD) = 12(1 - )  .", "Next, we consider the history $h=\\texttt {G}$ after a single initial GOOD observation, and calculate the history value ${V^\\pi }(h)$ .", "Before proceeding, we need to calculate a few intermediate quantities, such as the belief-distribution: (s=Gh=G) (h=Gs=G) (s=G) = 1 12 = 12  , (s=Bh=G) (h=Gs=B) (s=B) = 12 12 = 14  , therefore (s=Gh=G) = 23  , (s=Bh=G) = 13  .", "We also calculate the belief-state distribution after two other histories.", "First $h=\\texttt {G}\\texttt {G}\\texttt {G}$ , (s=Gh=GGG) (h=GGGs=G) (s=G) = 1 12 = 12  , (s=Bh=GGG) (h=GGGs=B) (s=B) = 14 12 = 18  , therefore (s=Gh=GGG) = 45  , (s=Bh=GGG) = 15  .", "Then $h=\\texttt {G}\\texttt {G}\\texttt {B}$ , (s=Gh=GGB) (h=GGBs=G) (s=G) = 0 12 = 0  , (s=Bh=GGB) (h=GGBs=B) (s=B) = 14 12 = 18  , therefore (s=Gh=GGB) = 0  , (s=Bh=GGB) = 1  .", "We also need to calculate the observation emission probabilities, (o=Gh=G, a=G) = (s=Gh=G) (o=Gs=G) = + (s=Bh=G) (o=Gs=B) = 23 1 + 13 12 = 56  , (o=Bh=G, a=G) = (s=Gh=G) (o=Bs=G) = + (s=Bh=G) (o=Bs=B) = 23 0 + 13 12 = 16  .", "Next, we calculate ${V^\\pi }(h=\\texttt {G})$ under the assumption that the equality holds, V(h=G) = sh=G[ V(s) ] = (s=Gh=G) V(s=G) + (s=Bh=G) V(s=B) = 23 11- + 13 12(1-) = 56(1-)  .", "We can also apply the equality to other histories, V(h=GGG) = sh=GGG[ V(s) ] = (s=Gh=GGG) V(s=G) + (s=Bh=GGG) V(s=B) = 45 11- + 15 12(1-) = 910(1-)  , V(h=GGB) = sh=GGB[ V(s) ] = (s=Gh=GGB) V(s=G) + (s=Bh=GGB) V(s=B) = 0 11- + 1 12(1-) = 12(1-)  .", "Next, we calculate ${V^\\pi }(h=\\texttt {G})$ , this time by bootstrapping first, and then using the equality.", "Note that with the given history $h=\\texttt {G}$ , the agent will choose action $a=\\texttt {GOOD}$ .", "Then, V(h=G) = R(h=G, a=G) + oh=G, a=G[ V(hao=GGo) ] = 1 + ( $\\Pr (o=\\texttt {G}\\mid h=\\texttt {G}, a=\\texttt {G}) {V^\\pi }(hao=\\texttt {G}\\texttt {G}\\texttt {G}) \\\\+ \\Pr (o=\\texttt {B}\\mid h=\\texttt {G}, a=\\texttt {G}) {V^\\pi }(hao=\\texttt {G}\\texttt {G}\\texttt {B})$ ) = 1 + 56 910(1-) + 16 12(1-) = 60-6060(1-) + 4560(1-) + 560(1-) = 60-1060(1-) = 6-6(1-)  .", "The values from eq:proof:value:1,eq:proof:value:2 contradict each other, therefore, for this POMDP, policy, and history, ${V^\\pi }(h) \\ne _{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ .", "thm:vs.reactive.foTheorem REF If the POMDP states and observations are related by a bijection $\\phi \\colon \\rightarrow $ , and the policy is reactive, then ${V^\\pi }(s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ .", "Although there is no intrinsic state uncertainty, we continue to use probabilistic notation for notational consistency and simplicity.", "In the following derivation, we use the fact that states and observations contain the same system information to determine the first action and reward.", "This process can be repeated iteratively for all future actions and rewards (omitted, but represented by the ellipsis), sh[ V(s) ] = sh[ a (as) Q(s, a) ] = sh[ a (a; I(s)) Q(s, a) ] = sh[ a (a; oh) Q(s, a) ] = a (a; oh) sh[ Q(s, a) ] = a (a; oh) sh[ R(s, a) + s's, a[ V(s') ] ] = a (a; oh) ( R(h, a) + s'h, a[ V(s') ] ) = a (a; oh) ( R(h, a) + oh, a[ s'hao[ V(s') ] ] ) = $\\vdots $ = a (a; oh) ( R(h, a) + oh, a[ V(hao) ] ) = a (a; oh) Q(h, a) = V(h)  .", "In this case, the bijection between observations and states removes any error from the round-trip inference $h\\rightarrow s\\rightarrow o$ is perfect, which was the cause of the bias in thm:vs.reactive.po.", "thm:vhsTheorem REF For an arbitrary POMDP and policy, ${V^\\pi }(h, s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(h, s) \\right]$ Follows from eq:vh-raw,eq:vhs-raw, V(h0) = s,ah0[ k k R(sk, ak) ] = s0h0 s,ah0,s0[ k k R(sk, ak) ] = s0h0[ V(h0, s0) ]  .", "thm:asymmetric-policy-gradientTheorem REF (Asymmetric Policy Gradient) The policy gradient can be expressed using history-state values, $\\nabla _\\theta _\\text{policy}(\\theta ) = -\\left[ \\sum _t \\gamma ^t {Q^\\pi }(h_t,s_t, a_t) \\nabla _\\theta \\log \\pi (a_t; h_t) \\right] \\,.$ Following thm:vhs, we have Q(h, a) = R(h, a) + oh, a[ V(hao) ] = R(h, a) + oh, a[ s'h, a, o[ V(hao, s') ] ] = R(h, a) + s', oh, a[ V(hao, s') ] = sh[ R(s, a) + s', os, a[ V(hao, s') ] ] = sh[ Q(h, s, a) ]  .", "Therefore, policy() = -[ t t Q(ht, at) (at; ht) ] = -t t ht,at[ Q(ht, at) (at; ht) ] = -t t ht,at[ stht[ Q(ht, st, at) ] (at; ht) ] = -t t ht,st,at[ Q(ht, st, at) (at; ht) ] = -[ t t Q(ht, st, at) (at; ht) ]  .", "Environments Table: Environment properties.The environments used in the evaluation can be split into two groups, Shopping-5, Shopping-6, Heavenhell-3, Heavenhell-4 and Rocksample-5-6 are flat POMDPs, while Keydoor and Ninerooms are gridverse POMDPs.", "Flat POMDPs Figure: Layout of the Shopping environments.Figure: Layout of the Heavenhell environments.Figure: Layout of the Rocksample environment.In the flat POMDPshttps://github.com/abaisero/gym-pomdps, states, actions and observations are encoded by categorical indices with no inherent metric, which are intrinsically equally (dis)similar to each other.", "While it is not possible to generalize across states and observations via feature extraction, the primary challenge in these POMDPs is that of generalizing across different histories.", "Because the flat POMDPs are finite, their state, action and observation spaces have well-defined sizes, shown in tab:environments; note, however, that the size of the state space is not a significant measure of the complexity of partially observable tasks, while the time required to solve the task (i.e., history length) is a more relevant measure.", "Shopping This environment simulates an agent going to a shop to buy an item it forgot.", "The agent navigates a $5\\times 5$ or $6\\times 6$ gridworld trying to locate and select a randomly positioned item.", "The agent's position is fully observable, while the item's position is only observed when queried.", "fig:map:shopping depicts the gridworlds encoded by Shopping-5 and Shopping-6.", "States and Observations States encode the position of the agent and the position of the item in a single integer.", "Observations encode the position of the agent or the position of the item in a single integer.", "Actions Each time-step, the agent must choose an action from the set { LEFT, RIGHT, UP, DOWN, QUERY, BUY }.", "If the agent chooses the QUERY action, it observes the item's position, otherwise it observes its own position.", "To solve the task optimally, the agents needs to query the item's position and remember it, navigate to it, and then buy it.", "Rewards The agent receives the following reward signal: a reward of $-1.0$ for moving; a reward of $-2.0$ for performing a QUERY action; a reward of $-5.0$ for performing a BUY action in the wrong cell; and a reward of $10.0$ for performing a BUY action in the correct cell.", "Heavenhell The agent navigates a corridor-like gridworld composed of a fork and 3 dead-ends.", "Two dead-ends are exits which lead to heaven or hell, although the agent does not know which is which, while the third dead-end leads to a priest who can help the agent identify the heaven exit.", "fig:map:heavenhell depicts the gridworlds encoded by Heavenhell-3 and Heavenhell-4.", "States and Observations States encode the position of the agent and the position of the exit to heaven.", "Observations encode the position of the agent or the position of the exit to heaven.", "Actions Each time-step, the agent must choose an action from the set { NORTH, SOUTH, EAST, WEST }.", "If the agent is at the priest, it observes heaven's location, otherwise it observes its own position.", "To solve the task, the agent needs to navigate to the priest, then back to the fork, and on to heaven.", "Rewards The agent receives a sparse reward signal composed of: a reward of $1.0$ for exiting to heaven; and a reward of $-1.0$ for exiting to hell.", "Rocksample This environment simulates an agent navigating a landscape to collect research material.", "The agent navigates a $5\\times 5$ gridworld which contains 6 rocks, each having either good or bad research value.", "The agent's position is fully observable, while each rock's goodness is randomly sampled and unobserved unless the agent checks it.", "Each rock's goodness can be queried via a stochastic check action, whose reliability decays with the agent-rock distance.", "fig:map:rocksample depicts the gridworld encoded by Rocksample-5-6.", "States and Observations States encode the position of the agent and whether each rock is good or bad; the positions of the rocks are fixed so they do not need to be tracked by the state.", "Observations encode the position of the agent or whether the checked rock is good or bad.", "Actions Each time-step, the agent must choose an action from the set { NORTH, SOUTH, EAST, WEST, CHECK_0, CHECK_1, CHECK_2, CHECK_3, CHECK_4, CHECK_5, SAMPLE, }.", "If the agent chose a CHECK_* action, it observes whether the corresponding rock is good or bad, otherwise it observes its own position.", "To solve the task, the agent needs to sample all the good rocks (and none of the bad rocks), dynamically adapting its path depending on the stochastic information gained by checking each rock.", "Rewards The agent receives a sparse reward signal composed of: a reward of $10.0$ for sampling a good rock; and a reward of $-10.0$ for sampling a bad rock.", "Gridverse POMDPs Figure: KeydoorFigure: NineroomsIn the gridverse POMDPshttps://github.com/abaisero/gym-gridverse, actions are still encoded by categorical indices, while states and observations are encoded as structures which do have an inherent similarity metric; they are split into two components: A grid component: a $6\\times H\\times W$ volume of categorical indices which encode cell types, colors, statuses, agent presence, and the spatial relationships between them.", "The observation grid component is a slice of the corresponding state grid component made to match the agent's perspective: it is rotated to be a first-person view, and cells hidden behind walls are occluded, as shown in fig:keydoor:observation,fig:ninerooms:observation.", "An agent component: a 6-dimensional array of categorical indices representing the agent position and orientation, and the item held by the agent if any.", "The position and orientations in the state agent component are absolute, while those in the observation component are relative to the agent's perspective—they are essentially constant, and not necessary for control.", "Keydoor The agent navigates a $5\\times 5$ gridworld split into two rooms split by a wall and a locked door; on one side is the agent and a key which can be used to open the door, while on the other side is the goal.", "The positions of the agent, the key, the door, and the goal are randomly sampled such that two instances of the same problem are unlikely to be the same.", "fig:keydoor shows state and observation frames taken from an instance of the Keydoor environment.", "States and Observations The state grid component is a $6\\times 5\\times 5$ volume.", "The observation grid component is a $6\\times 7\\times 7$ volume.", "Actions Each time-step, the agent must choose an action from the set { MOVE_FORWARD, MOVE_BACKWARD, MOVE_LEFT, MOVE_RIGHT, TURN_LEFT, TURN_RIGHT, PICK_N_DROP, ACTUATE }.", "The MOVE_* actions result in a movement depending on the agent's orientation, while the TURN_* allows the agent to change its orientation.", "With the PICK_N_DROP action, the agent can pick and/or drop the key from/to the cell in front, while with the ACTUATE action, the agent can open and/or close the door.", "Rewards The agent receives a dense reward signal composed as the sum of the following terms: a living reward of $-0.05$ for every time-step; a reward of $0.2$ for stepping closer to the goal, and $-0.2$ for stepping further away (ignoring walls); a reward of $1.0$ for picking up the key, and $-1.0$ for dropping it; a reward of $1.0$ for opening the door, and $-1.0$ for closing it; and a reward of $5.0$ for reaching the goal.", "Ninerooms The agent navigates a $13\\times 13$ maze-like gridworld trying to locate and reach the goal.", "fig:ninerooms shows state and observation frames taken from an instance of the Ninerooms environment.", "States and Observations The state grid component is a $6\\times 13\\times 13$ volume.", "The observation grid component is a $6\\times 7\\times 7$ volume.", "Actions Each time-step, the agent must choose an action from the set { MOVE_FORWARD, MOVE_BACKWARD, MOVE_LEFT, MOVE_RIGHT, TURN_LEFT, TURN_RIGHT }.", "The MOVE_* actions result in a movement depending on the agent's orientation, while the TURN_* allows the agent to change its orientation.", "Rewards The agent receives a dense reward signal composed as the sum of the following terms: a living reward of $-0.05$ for every time-step; a reward of $0.2$ for stepping closer to the goal, and $-0.2$ for stepping further away (ignoring walls); and a reward of $5.0$ for reaching the goal.", "Architectures and Hyper-Parameters Figure: A2C architecture.", "The state, action, and observationrepresentations φ(s)\\phi (s), φ(a)\\phi (a), and φ(o)\\phi (o) are those shown infig:architecture:representations:flat orfig:architecture:representations:gridverse, depending on whetherthe type of enviroment.", "Dotted lines are present or omitted depending onwhether a history critic V ^(h){\\hat{V}}(h), state critic V ^(s){\\hat{V}}(s), orhistory-state critic V ^(h,s){\\hat{V}}(h, s) is being modeled.In this section, we describe the architectures used by the policy and critic models; a general overview is shown in fig:architecture.", "The general architecture will be similar for all domains; however there will some differences to accommodate the fact that the flat POMDPs provide states and observations as categorical indices, while the gridverse POMDPs provide states and observations as volumes and arrays of categorical indices, whose structures include spatial relationships.", "Features Extraction for Flat POMDPs These components are shown in fig:architecture:representations:flat.", "Because flat POMDPs provide states, actions and observations as categorical indices, and we use 128-dimensional embedding models to represent each of them.", "Features Extraction for Gridverse POMDPs These components are shown in fig:architecture:representations:gridverse.", "Because gridverse POMDPs provide actions as categorical indices, and we use 128-dimensional embedding models to represent them.", "On the other hand, states and observations are provided in the format described in sec:environments:gridverse, each composed of a grid and an agent component: grid The grid component is a $6\\times H\\times W$ volume of categorical data, which we pass through 4-dimensional embeddings, resulting in a $24\\times H\\times W$ volume of numeric data.", "We further pass that data to two layers of CNN followed by ReLU non-linearities, resulting in a $16\\times H\\times W$ volume of data, finally flattened into a $16 HW$ -dimensional array.", "agent The agent component is a 6-dimensional array of categorical data, which we pass through 4-dimensional embeddings, resulting in a 24-dimensional array of numeric data.", "The grid and agent representations are then concatenated to form the state or observation representation $\\phi (s)$ or $\\phi (o)$ .", "Remainder of the Architecture These components are shown in fig:architecture:a2c.", "Concatenated action and observation embeddings form the to a 128-dimensional single-layer gated recurrent unit (GRU) [3], which acts as a history representation $\\phi (h)$ .", "The history representation is then fed into separate policy and critic models, each a 2-layer 128-dimensional feedforward model with ReLU non-linearities.", "A practical issue we found with A2C(h,s) is that the history and state representations $\\phi (h)$ and $\\phi (s)$ contain features with very different orders of magniture; To address this, we use layer-normalization.", "To guarantee a fair comparison with the other methods, we do the same for other critics as well.", "Next, we describe architectural details specific to each method: A2C(h): The history representation passes through layer-normalization first, and then is fed to the critic model; A2C(s): The state embedding passes through layer-normalization first, and then is fed to the critic model; A2C(h,s): The history representation and state embedding individually pass through layer-normalization first, and then are concatenated to form a single feature vector, which is then fed to the critic model." ], [ "Related Work", "The use of latent information during offline training has been successfully adopted in a variety of policy-based methods [21], [6], [16], [31], [14], [27], [4], [29] and value-based methods [22], [17], [23], [4].", "Among the single-agent methods, asymmetric actor-critic for robot learning [21] uses a reactive variant of DDPG with a state-based critic to help address partial observability; belief-grounded networks [20] use an belief-reconstruction auxiliary task to train history representations; and [29] and [2] use a fully observable agent trained offline on latent state information to train a partially observable agent via imitation.", "Asymmetric learning has also been very popular in the multi-agent setting: COMA [6] uses reactive control and a shared asymmetric critic which can receive either the joint observations of all agents or the system state to solve cooperative tasks; MADDPG [16] and M3DDPG [14] use the same form of asymmetry with individual asymmetric critics to solve cooperative-competitive tasks; R-MADDPG [27] uses recurrent models to represent non-reactive control, and the centralized critic uses the entire histories of all agents; and CM3 [31] uses a state critic for reactive control.", "Asymmetry is also used in multi-agent value-based methods; QMIX [22], MAVEN [17], and WQMIX [23] all train individual Q-models using a centralized but factored Q-model, itself trained using state, joint histories, and joint actions." ], [ "Background", "In this section, we review the topics relevant to our work, i.e., POMDPs, the reinforcement learning graphical model, standard actor-critic, and asymmetric actor-critic." ], [ "Notation", "We denote sets with calligraphy $\\mathcal {X}$ , set elements with lowercase $x\\in \\mathcal {X}$ , random variables (RVs) with uppercase $X$ , and the set of distributions over a set as $\\Delta \\mathcal {X}$ .", "Occasionally, we will need absolute and/or relative time indices; We use subscript $x_t$ to indicate absolute time, and superscript $x{k}$ to indicate the relative time of variables, e.g., $x{0}$ marks the beginning of a sequence happening at an undetermined absolute time, and $x{k}$ represents the variable $k$ steps later.", "We also use the bar notation to represent a sequence of superscripted variables $\\bar{x}\\equiv (x{0}, x{1}, x{2}, \\ldots )$ .", "A POMDP [11] is a discrete-time partially observable control problem described by a tuple $\\langle , , , , ,, \\gamma \\rangle $ consisting of: state, action and observation spaces $$ , $$ , and $$ ; transition function $\\colon \\times \\rightarrow \\Delta $ ; observation function $\\colon \\times \\times \\rightarrow \\Delta $ ; reward function $\\colon \\times \\rightarrow $ ; and discount factor $\\gamma \\in \\left[0,1\\right]$ .", "The goal is that of maximizing the expected discounted sum of rewards $\\left[ \\sum _t \\gamma ^t R(S_t, A_t \\right]$ , a.k.a.", "the expected return.", "In the partial observable setting, the agent lacks access to the underlying system state, and action selection is based on the observable history $h$ , i.e., the sequences of past actions and observations.", "We denote the space of all histories as $\\doteq \\left(\\times \\right)^*$ , and the space of histories of length $l$ as $_l \\doteq \\left(\\times \\right)^l$ .", "Generally, an agent operating under partial observability might have to consider the entire history to achieve optimal behavior [25], i.e., its policy should represent a mapping $\\pi \\colon \\rightarrow \\Delta $ .", "The belief-state $b\\doteq \\rightarrow \\Delta $ is the conditional distribution over states given the observable history, i.e., $b(h) = \\Pr (S\\mid h)$ , and a sufficient statistic of the history for optimal control [11].", "We define the history reward function as $(h, a) \\doteq _{s\\mid h}\\left[ (s, a)\\right]$ ; from the agent's perspective, this is the reward function of the decision process.", "We denote the last observation in a history $h$ as $o_h$ , and say that an agent is reactive if its policy $\\pi \\colon \\rightarrow \\Delta $ uses the last observation rather than the entire history.", "A policy's history value function ${V^\\pi }\\colon \\rightarrow $ represents the expected future discounted returns when the agent finds itself in history $h$ , ${V^\\pi }(h{0}) = _{\\bar{s}, \\bar{a}\\mid h{0}}\\left[\\sum _{k=0}^\\infty \\gamma ^k R(s{k}, a{k}) \\right] \\,, $ which supports an indirect recursive Bellman form, V(h) = a (a; h) Q(h, a)  , Q(h, a) = (h, a) + oh, a[ V(hao) ]  ." ], [ "The RL Graphical Model", "Some of the theory and results developed in this document concerns whether certain (RVs) of interest are well-defined; therefore, we review the RVs defined in the partially observable control case.", "Together, the environment and the agent induce a graphical model (see fig:graphmodel) over timed RVs $S_t$ , $A_t$ , and $O_t$ .", "Note that only timed RVs are defined directly, and there are no intrinsically time-less RVs.", "Any other RV must be defined in terms of the available ones, e.g.", "we can define a joint RV representing the timed history RVs $H_t \\doteq (A_0, O_0, \\ldots , A_{t-1}, O_{t-1})$ .", "Sometimes it is possible to define a limiting (stationary) state RV $S\\doteq \\lim _{t\\rightarrow \\infty } S_t$ ; However, it is never possible to define a limiting (stationary) history RV $H$ , since the sample space of each timed RV $H_t$ is different and $\\lim _{t\\rightarrow \\infty } H_t$ does not exist.", "A probability is a numeric value associated with the assignment of a value $x$ from a sample space $\\mathcal {X}$ to an RV $X$ , e.g., $\\Pr (X=x)$ .", "Although it is common to use simplified notation and informally omit the RV assignment (e.g., $\\Pr (x)$ ), it must always be implicitly clear which RV is involved in the assignment.", "In the reinforcement learning graphical model, a probability is well-defined if and only if [label=()] it is grounded (implicitly or explicitly) to timed RVs (or functions thereof); or it is time-invariant (i.e., it can be grounded to any time index).", "For example, $\\Pr (s^{\\prime }\\mid s, a)$ is implicitly grounded to the RVs of a state transition $\\Pr (S_{t+1}=s^{\\prime }\\mid S_t=s, A_t=a)$ , and although the time-index $t$ is not clear from context, the probability is time-invariant and thus well defined.", "As another example, $\\Pr (s\\mid h)$ is implicitly grounded to the RVs of a belief $\\Pr (S_t=s\\mid H_t=h)$ ; in this case, the time-index $t$ can be contextually grounded to the history length $t=|h|$ , so the probability is well defined.", "(Symmetric) Actor-Critic for POMDPs Policy gradient methods [26] for fully observable control problems can be adapted to partial observable control problems by replacing occurrences of the system state $s$ with the history $h$ , which is the Markov-state of a history-MDP equivalent to the POMDP.", "In advantage actor-critic methods (A2C) [13], a policy model $\\pi \\colon \\rightarrow \\Delta $ parameterized by $\\theta $ is trained using gradients estimated from sample data, while a critic model ${\\hat{V}}\\colon \\rightarrow $ parameterized by $\\vartheta $ is trained to predict history values ${V^\\pi }(h)$ .", "Note that we annotate parametric critic models with a hat ${\\hat{V}}$ , to distinguish them from their analytical counterparts ${V^\\pi }$ .", "In A2C, the critic is used to bootstrap return estimates and as a baseline, both of which are techniques for the reduction of estimation variance [7].", "The overall A2C objective is $(\\theta , \\vartheta ) \\doteq _\\text{policy}(\\theta ) + _\\text{critic}(\\vartheta ) +\\lambda _\\text{neg-entropy}(\\theta ) \\,.", "$ Policy Loss The policy loss $_\\text{policy}$ is the agent's performance, i.e., the episodic return $_\\text{policy}(\\theta ) \\doteq -\\left[ \\sum _{t=0}^\\infty \\gamma ^t (s_t, a_t)\\right] \\,.$ The policy gradient theorem [26], [13] provides an analytical expression for the policy loss gradient w.r.t.", "the policy parameters, $\\nabla _\\theta _\\text{policy}(\\theta ) = -\\left[ \\sum _{t=0}^\\infty \\gamma ^t{Q^\\pi }(h_t, a_t) \\nabla _\\theta \\log \\pi (a_t; h_t) \\right] \\,.$ In A2C, the value function ${Q^\\pi }(h_t, a_t)$ is replaced with the temporal difference (TD) error $\\delta _t$ , policy() = -[ t=0t t (at; ht) ]  , t (st, at) + V(ht+1) - V(ht)  , to reduce variance at the cost of introducing modeling bias.", "Critic Loss The critic is trained to minimize the TD error of the history-states; to make the policy and critic losses scale similarly to environments with different episode lengths, we adopt an unconventional time-discounted variant of the critic loss, $_\\text{critic}(\\vartheta ) \\doteq \\left[ \\sum _{t=0}^\\infty \\gamma ^t \\delta _t^2\\right] \\,,$ the gradient of which should propagate through ${\\hat{V}}(h_t)$ , but not through the bootstrapping component ${\\hat{V}}(h_{t+1})$ .", "Negative-Entropy Loss Finally, the negative-entropy loss is commonly used, in combination with a decaying weight $\\lambda $ , to avoid premature convergence of the policy model and to promote exploration [30].", "As with the critic loss, we employ a time-discounted variant of the negative-entropy loss, $_\\text{neg-entropy}(\\theta ) \\doteq - \\left[ \\sum _t \\gamma ^t \\left[ \\pi (A_t; h_t)\\right] \\right] \\,.$ Asymmetric Actor-Critic for POMDPs While asymmetric actor-critic can be understood to be an entire family of methods which use critic asymmetry, for the remainder of this document we will be specifically referring to a non-reactive and non-deterministic variant of the work by [21], which uses critic asymmetry to address image-based robot learning.", "Their work uses a reactive variant of deep deterministic policy gradient (DDPG) [15] trained in simulation, and replaces the reactive observation critic ${\\hat{V}}(o)$ with a state critic ${\\hat{V}}(s)$ ; the variant we will be analyzing applies the same critic substitution to A2C.", "In practice, this state-based asymmetry is implemented by replacing the TD error of eq:tderror:h (used in both the policy and critic losses) with $\\delta _t = (s_t, a_t) + \\gamma {\\hat{V}}(s_{t+1}) - {\\hat{V}}(s_t) \\,,$ while every other aspect of A2C remains unchanged.", "Although [21] claim that their work addresses partial observability, their evaluation is based on reactive environments which are virtually fully observable: while only an image is available to the agent, each image gives a virtually complete and collusion-free view of the entire workspace.", "In practice, the images are low-level representations of the full state.", "Theory of Asymmetric Actor-Critic In this section, we analyze the theoretical implications of using a state critic as described in sec:bg:aa2c, and expose critical issues.", "The primary result will be that the state value function ${V^\\pi }(s)$ of a non-reactive agent under partial observability is generally ill-defined.", "Then, we show that the state value function ${V^\\pi }(s)$ of a reactive agent under partial observability is well-defined, but introduces a bias into the training process which may undermine learning.", "Finally, we show that the state value function ${V^\\pi }(s)$ of a reactive agent under an observation function which is equivalent to full observability is both well-defined and unbiased.", "Note that replacing the history critic is intrinsically questionable: the policy gradient theorem for POMDPs (eq:policy-gradient) specifically requires history values, and substituting them for another value which has a different expectation will result in the gradient estimates losing their theoretical guarantees of correctness.", "Therefore, we analyze state values ${V^\\pi }(s)$ as estimators of history values ${V^\\pi }(h)$ , and consider the corresponding estimation bias, i.e., the difference between $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ and ${V^\\pi }(h)$ for a given history $h$ .", "Informally, the fundamental issue with ${V^\\pi }(s)$ is that the state does not contain sufficient information to determine the agent's future behavior (which depends on the history) and is thus unable to meaningfully represent expected future returns.", "Ironically, state values suffer from a problem we call history aliasing, i.e., being unable to infer the agent's history from the system's state.", "Even when ${V^\\pi }(s)$ is numerically well-defined, it usually introduces a bias caused by the imperfect correlation between histories and states; in essence, the average value of histories inferred from the current state is not an accurate estimate of the true current history's value.", "We begin with the definition of the state value function, ${V^\\pi }(s{0}) = _{\\bar{s}, \\bar{a} \\mid s{0}}\\left[\\sum _{k=0}^\\infty \\gamma ^k R(s{k}, a{k}) \\right] \\,,$ which, if well-defined, supports an indirect recursive Bellman form, V(s) = a (as) Q(s, a)  , Q(s, a) = R(s, a) + s's, a[ V(s') ]  .", "From eq:vs, we note the term $\\Pr (a\\mid s)$ , which encodes the likelihood of an action being taken from a given state.", "Because the agent policy acts on histories, this term is not directly available, but must be derived indirectly by integrating over possible histories; and because there is no contextual information available to limit the integration to histories of a specific length, we can only integrate over the space of all possible histories, $\\Pr (a\\mid s) = \\sum _{h\\in } \\Pr (h\\mid s) \\pi (a; h) \\,.", "$ eq:pras reveals the probability term $\\Pr (h\\mid s)$ , which encodes the likelihood of an history having taken place given a state.", "While $\\Pr (h\\mid s)$ may look harmless, it is the underlying cause of serious analytical issues.", "As discussed in sec:graphmodel, a probability term is only well-defined if associated with well-defined RVs, and the fundamental issue with $\\Pr (h\\mid s)$ is that such RVs do not exist.", "On one hand, we cannot used timed RVs $\\Pr (H_t=h\\mid S_t=s)$ , because eq:pras integrates over the sample space of all histories, and not just those of a given length $t$ .", "On the other hand, we cannot use time-less RVs $\\Pr (H=h\\mid S=s)$ , because time-less RVs do not exist in the RL graphical model.", "Ultimately, $\\Pr (h\\mid s)$ is ill-defined, which causes $\\Pr (a\\mid s)$ and ${V^\\pi }(s)$ itself to be ill-defined.", "Theorem 4.1 For an arbitrary POMDP and policy, a time-invariant state value function ${V^\\pi }(s)$ is ill-defined.", "(proof in sec:proofs).", "In relation to the training procedure of a state critic model, the practical implications of an ill-defined value function are not obvious; even though the analytical value function is ill-defined mathematically, the critic's training process performs valid calculations on sample data which results in valid updates of the critic parameters.", "However, convergence is unlikely, since empirical convergence to a meaningful value virtually requires the existence of a theoretical convergence value.", "Rather, it is likely that the critic's training target will shift indefinitely, depending on the recent training data, inhibiting the convergence of the critic model even under ideal training circumstances, which itself will cause instabilities and divergence in the policy model; this is verified empirically in sec:evaluation.", "A natural solution to the underlying issue of state value functions ${V^\\pi }(s)$ is to define and employ timed value functions $V_t^\\pi (s)$ ; in sec:special:timed, we show that timed value functions indeed address the primary issue, although learning a critic to model them is likely to pose a significantly harder learning challenge, due to the need to generalize well and accurately over different time-steps.", "Rather, in the next subsections, we show special cases of the general control problem which make non-timed ${V^\\pi }(s)$ well-defined; in some cases, this will lead to other theoretical issues involving the introduction of estimation bias.", "Reactive Policy under Partial Observability We show that ${V^\\pi }(s)$ is well-defined if we make two assumptions about the agent and environment: [label=()] that the policy is reactive; and that the POMDP observation function depends only on the current state, $\\colon \\rightarrow \\Delta $ , rather than the entire state transition.", "Under these assumptions, we can expand $\\Pr (a\\mid s)$ by integrating over the space of all observations (rather than all histories), $\\Pr (a\\mid s) = \\sum _{o\\in } \\Pr (o\\mid s) \\pi (a; o) \\,.$ In this case, the term $\\Pr (o\\mid s)$ can be grounded to timed RVs $\\Pr (O_t=o \\mid S_t=s)$ ; because that probability is time-invariant, it is well-defined, meaning that ${V^\\pi }(s)$ is well-defined in this case.", "However, we show that ${V^\\pi }(s)$ is biased compared to ${V^\\pi }(h)$ .", "Theorem 4.2 If the POMDP observation function depends only on the current state, $\\colon \\rightarrow \\Delta $ , and the policy is reactive, then ${V^\\pi }(s)$ is not necessarily an unbiased estimate of ${V^\\pi }(h)$ , i.e., it is not guaranteed that ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s)\\right]$ .", "(proof in sec:proofs).", "Although we were able to show that the value function is well-defined in the case of reactive control, there are still two significant issues: [label=()] reactive policies are inadequate to solve many POMDPs; and the bias of the value function ${V^\\pi }(s)$ may influence the agent learning capabilities catastrophically.", "Broadly speaking, this bias is caused by the fact that hidden in ${V^\\pi }(s)$ is an expectation over observations $o$ which, while conditioned on the state $s$ , are not necessarily consistent with the true history $h$ .", "We discuss the cause of this bias more formally in the corresponding proof.", "Reactive Policy under Full Observability We show that the state value function is not only well-defined but also unbiased if we make two assumptions about the agent and environment: [label=()] that the policy is reactive; and that the there is a bijective abstraction $\\phi \\colon \\rightarrow $ between observations and states.", "The abstraction $\\phi $ encodes the fact that the environment is not truly partially observable, but rather that states and observations essentially contain the same information, albeit at different levels of abstraction, akin to the problems used by [21].", "For example, an image displaying a workspace without occlusions could be a low-level abstraction (observation), while a concise vector representation of the object poses in the workspace could be a high-level abstraction (state).", "In this case, the action probability term $\\Pr (a\\mid s)$ does not need to be obtained indirectly by integrating other variables; rather, the state-observation bijection can be used to directly relate it to the policy model, (as) = (a; I(s))  .", "Contrary to the previous cases, the overall state value function ${V^\\pi }(s)$ is not only well-defined, but also unbiased.", "Theorem 4.3 If the POMDP states and observations are related by a bijection $\\phi \\colon \\rightarrow $ , and the policy is reactive, then ${V^\\pi }(s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ .", "(proof in sec:proofs).", "The benefit of using a state critic under this scenario is that the critic model can avoid learning a representation of the observations before learning the values [21].", "Naturally, the main disadvantage of this scenario is that most POMDPs do not satisfy the bijective abstraction assumption, which is virtually equivalent to full observability.", "Nonetheless, if a control problem only deviates mildly from full observability, it is very possible that a state critic might benefit the learning agent.", "Unbiased Asymmetric Actor-Critic In this section, we introduce unbiased asymmetric actor-critic, an actor-critic variant which is able to exploit asymmetric state information during offline training while avoiding the issues of state value functions exposed in sec:aa2c.", "Consider a history-state value function ${V^\\pi }(h,s)$  [1], which represents the expected future discounted returns obtained when the history is $h$ and the state is $s$ , ${V^\\pi }(h{0}, s{0}) = _{\\bar{s}, \\bar{a} \\mid h{0},s{0}}\\left[ \\sum _{k=0}^\\infty \\gamma ^k R(s{k}, a{k})\\right] \\,, $ which supports an indirect recursive Bellman form, V(h, s) = a (a; h) Q(h, s, a)  , Q(h, s, a) = (s, a) + s',os,a[ V(hao, s') ]  .", "Providing the history information makes the history-state value function ${V^\\pi }(h, s)$ not only well-defined even for non-reactive policies, but also an unbiased estimate of ${V^\\pi }(h)$ .", "Theorem 5.1 For an arbitrary POMDP and policy, ${V^\\pi }(h, s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(h, s)\\right]$ .", "(proof in sec:proofs).", "As we have done for state values ${V^\\pi }(s)$ , we are interested in the properties of history-state values ${V^\\pi }(h, s)$ in relation to history values ${V^\\pi }(h)$ .", "thm:vhs shows that history and history-state values are related by ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(h, s) \\right]$ , i.e., history-state values are interpretable as Monte Carlo (MC) estimate of the respective history values.", "In expectation, history-state values provides the same information as the history values, therefore an asymmetric variant of the policy gradient theorem also holds.", "Theorem 5.2 (Asymmetric Policy Gradient) The policy gradient can be expressed using history-state values, $\\nabla _\\theta _\\text{policy}(\\theta ) = -\\left[ \\sum _t \\gamma ^t {Q^\\pi }(h_t,s_t, a_t) \\nabla _\\theta \\log \\pi (a_t; h_t) \\right] \\,.$ (proof in sec:proofs).", "As estimators, history-state values ${V^\\pi }(h, s)$ can be described in terms of their bias and variance w.r.t.", "history values ${V^\\pi }(h)$ .", "Beyond providing the inspiration for the MC interpretation, thm:vhs already proves that ${V^\\pi }(h, s)$ is unbiased, while its variance is dynamic and depends on the history $h$ via the belief-state $\\Pr (S\\mid h)$ ; in particular, low-uncertainty belief-states result in relatively low variance, and deterministic belief-states result in no variance.", "Given that operating optimally in a partially observable environment generally involves information-gathering strategies associated with low-uncertainty belief-states, the practical variance of the history-state value is likely to be relatively low once the agent has learned to solve the task to some degree of success.", "Inspired by thm:asymmetric-policy-gradient, we propose unbiased asymmetric A2C, which uses a history-state critic ${\\hat{V}}\\colon \\times \\rightarrow $ trained to model history-state values ${V^\\pi }(h, s)$ , policy() = -[ t t t (at; ht) ]  , t = R(st, at) + V(ht+1, st+1) - V(ht, st)  .", "Because ${\\hat{V}}(h, s)$ receives the history $h$ as input, it can still predict reasonable estimates of the agent's expected future discounted returns; and because it receives the state $s$ as input, it is still able to exploit state information while introducing no bias into the learning process, e.g., for the purposes of bootstrapping the learning of critic values and/or aiding the learning of history representations.", "Interpretations of State Although the history-state value is analytically well-defined, It is worthwhile to question why the inclusion of the state information should help the actor-critic agent at all.", "We attempt to address this open question, and consider two competing interpretations, which we call state-as-information and state-as-a-feature.", "State as Information Under this interpretation, state information is valuable because it is latent information unavailable in the history, which results in more informative values.", "However, this interpretation is flawed for two reasons: [label=()] The policy gradient theorem specifically requires ${V^\\pi }(h)$ , which contains precisely the correct information required to accurately estimate policy gradients.", "In this context, there is no such thing as “more informative values” than history values.", "In theory, the history-state value in thm:asymmetric-policy-gradient could use any other state sampled according to $\\tilde{s}\\sim b(h)$ , rather than the true system state, which would also result in the same analytical bias and variance properties.", "In practice, we use the true system state primarily due to it being directly available during simulation; however, we believe that its identity as the true system state is analytically irrelevant, which leads to the next interpretation of state.", "State as a Feature We conjecture an alternative interpretation according to which the state can be seen as a stochastic high-level feature of the history.", "Consider a history critic ${\\hat{V}}(h)$ to appropriately model the value function ${V^\\pi }(h)$ , the model must first learn an adequate history representation, which is in and of itself a significant learning challenge.", "The critic model would likely benefit from receiving auxiliary high-level black-box features of the history $\\phi (h)$ .", "The resulting critic ${\\hat{V}}(h,\\phi (h))$ remains fundamentally a history critic, the supplementary features being exclusively a modeling construct.", "Next, we consider what kind of high-level features $\\phi (h)$ would be useful for control.", "While the specifics of what makes a good history representation depend strongly on the task, there is a natural choice which is arguably useful in many cases: the belief-state $b(h)$ .", "Because the belief-state is a sufficient statistic of the history for control, providing it to the critic model ${\\hat{V}}(h, b(h))$ is likely to greatly improve its ability to generalize across histories.", "Finally, we conjecture that any state sampled according to the belief-state $s\\sim b(h)$ —including the true system state—can be considered a stochastic realization of the belief-state feature, resulting in the history-state critic ${\\hat{V}}(h, s)$ .", "According to this interpretation, the importance of the state in the history-state critic is not in its identity as the true system state, but as a stochastic realization of hypothetical belief-state features, and presumably any other state sampled from the belief-state $\\tilde{s}\\sim b(h)$ could be equivalently used.", "Evaluation [tb] Each method follows the same algorithm structure, but uses different types of critics to compute the TD errors $\\delta _t$ (see eq:tderror:h,eq:tderror:s,eq:tderror:hs).", "Values $N$ , $B$ , and $E$ vary by environment.", "Input: epochs $N$ , episode batch $B$ , evaluation period $E$ epoch in 1 ...$N$ training_episodes $\\leftarrow $ sample_episodes($\\pi $ , $B$ ) update $\\theta ,\\vartheta $ via $\\nabla ($ training_episodes$)$ (see eq:a2c) epoch $\\bmod $ evaluation_period = 0 evaluation_episodes $\\leftarrow $ sample_episodes($\\pi $ , $E$ ) report empirical_returns(evaluation_episodes) In this section, we empirically evaluate three actor-critic variants: A2C(h), standard actor-critic with a history critic ${\\hat{V}}(h)$ as described in sec:a2c; A2C(s), asymmetric actor-critic with a state critic ${\\hat{V}}(s)$ as described in sec:aa2c; and A2C(h,s), unbiased asymmetric actor-critic with a history-state critic ${\\hat{V}}(h,s)$ as described in sec:uaa2c.", "Each method is trained and evaluated according to alg:code.", "Broadly speaking, each method uses a gated recurrent unit (GRU) [3] to compute history features, and two feed-forward networks which represent the policy action probabilities and critic values; sec:architectures contains a more detailed description of the used architectures.", "We perform evaluations on seven gridworld environments which exhibit significant partial observability: Shopping-5, Shopping-6, Heavenhell-3, Heavenhell-4, and Rocksample-5-6 are flat POMDPshttps://github.com/abaisero/gym-pomdps, where states and observations are represented by categorical indices, while Keydoor and Ninerooms are gridverse POMDPshttps://github.com/abaisero/gym-gridverse, where states and observations are represented by tensors of categorical indices which encode spatial relationships and other cell information.", "See sec:environments for a detailed descriptions and graphical representations of all environments.", "Results and Discussion Each method is evaluated in one of two ways: [label=()] we show empirical learning curve statistics for all environments, and we show how critic values change for important history-state pairs over the course of training in Heavenhell-4.", "Learning Curves Figure: Learning curve statistics over 20 independent training runs, whereeach run is periodically evaluated 20 times.", "Shaded areas are centeredaround the empirical mean performance, and show 2 standard errors (of themean).fig:learningperformance depicts the performance results in all the environments.", "First, we note that the symmetric baseline A2C(h) does not always learn to solve the task, but succeeds fully in fig:shoppingv,fig:heavenhelliii,fig:gvkeydoor, partially in fig:heavenhelliv,fig:rocksamplevvi,fig:gvninerooms, and fails in fig:shoppingvi.", "The asymmetric baseline A2C(s) also performs inconsistently across environments, with a mixture of successful and failing cases.", "We particularly note the strange learning curves of A2C(s) in fig:shoppingv,fig:shoppingvi, where performance improves quickly during the early training, but fails to improve further or even becomes unstable and collapses later on.", "While the exact dynamics of this collapse are not completely clear, we believe it is likely that this is a consequence of modeling the critic after an analytically unstable state value function, making convergence to stable values impossible.", "Using our proposed history-state critic, A2C(h,s) consistently exhibits either faster convergence and/or higher final performance in virtually all the flat environments, and competitive performance in the gridverse environments, taking a mild lead towards the end of Ninerooms.", "These results strongly demonstrate the importance of exploiting asymmetric information in ways which are theoretically justified and sound, as done in our work.", "Critic Values Figure: Critic value statistics for 4 history-state pairs, evaluatedthroughout the training procedure via 20 independent runs.In the top row the agent did not visit the priest, while in the bottom rowthe agent did visit the priest.To further inspect the behavior of each critic, we show the evolution of critic values for important history-state pairs over the course of training.", "We perform this evaluation on Heavenhell-4, and use 4 deliberately chosen history-state pairs.", "In each case the agent is located at the fork, and the 4 cases differ according to heaven's location (left or right) and whether the agent has previously visited the priest.", "fig:criticvalues shows the resulting critic values, with one figure for each of the chosen history-state pairs.", "In each scenario, we note that all critic values exhibit convergence properties which resemble those of the agent's performance in fig:heavenhelliv.", "This empirically confirms that agent performance and critic quality are strongly correlated factors; although it is not possible to make conclusions about the causality in this situation, we strongly believe that it is the critics which act as a learning bottleneck on policies.", "Notably, the critics which focus on a single aspect of the joint history-state show the exact same values for different history-state; namely, A2C(s) is identical in the top and bottom plots, while A2C(h) is identical in the top-left and top-right plots.", "Although in practice left and right plots are similar for all critics, A2C(h,s) is the only critic capable of representing different values in each of the 4 scenarios, as none of its curves are identical.", "We also note that the state critic ${\\hat{V}}(s)$ is muct less stable than the others, and shows no signs of convergence, which is consistent with our analysis in sec:aa2c.", "In contract, the history-state critic exhibits good convergence properties despite itself also using state information, which is consistent with our analysis in sec:uaa2c.", "Finally, we note again that the history-state critic ${\\hat{V}}(h, s)$ converges significantly faster than the history critic ${\\hat{V}}(h)$ , confirming that state information is useful for training.", "Conclusions Asymmetric methods trained offline in simulated environments can use information which is normally unavailable in partially observable RL, such as the true system state.", "While the idea of exploiting such information has potential, current state-of-the-art methods are powered by empirical results rather than theoretical analysis.", "In this work, we exposed profound theoretical issues with a standard variant of asymmetric actor-critic, and proposed an unbiased asymmetric actor-critic variant which is analytically sound and theoretically justified.", "Empirical results confirm our analysis, the weaknesses of state critics and the strengths of history-state critics.", "Although we applied the history-state value function only to A2C, the same concepts are easily extensible to other critic-based RL methods [24], [15], [5], [19].", "In future work, we aim to extend the theory of history-state policy value functions ${Q^\\pi }(h, s, a)$ to optimal value functions $Q^*(h, s, a)$ , and develop theoretically sound asymmetric variants of other deep RL methods such as DQN [18], soft Q-learning [8], and soft actor-critic [9].", "We also plan to extend the theory and approach to multi-agent methods, potentially bringing theoretical rigor and improved performance [6], [16], [14], [27], [31], [22], [17], [23].", "Timed Value Functions sec:aa2c shows that for a general POMDP and policy $\\pi \\colon \\rightarrow \\Delta $ , the state value function ${V^\\pi }(s)$ is not generally well-defined due to issues caused by the lack of time information.", "In this section, we consider addressing the primary issue by providing explicit time-index information via timed value functions, $V_t^\\pi (s)$ and $Q_t^\\pi (s, a)$ , which represent the expected returns obtained when the agent finds itself in a state $s$ at time $t$ , Vt(s) = a (At=aSt=s) Qt(s, a)  , Qt(s, a) = R(s, a) + s's, a[ Vt+1(s') ]  .", "Once again, we analyze the state-dependent action distribution term to verify correctness, and expand it by integrating over histories; this time, we can use the explicit time-index information to integrate over histories of a given length only, $\\Pr (A_t=a\\mid S_t=s) = \\sum _{h\\in _t} \\Pr (H_t=h\\mid S_t=s) \\pi (a; h)\\,.", "$ Because eq:sumrule:timed is now restricted to histories of a given length $t$ , the probability term $\\Pr (H_t=h\\mid S_t=s)$ is well-defined, which means $\\Pr (A_t=a\\mid S_t=s)$ is well-defined, and $V_t^\\pi (s)$ is well-defined.", "Introducing the time-index makes the value function $V_t^\\pi (s)$ well-defined.", "However, its utility for the purpose of asymmetric reinforcement learning remains unclear because [label=()] it is still not formally proven whether the timed value function is unbiased, i.e., whether $V_t^\\pi (h) = _{s\\mid h}\\left[V_t^\\pi (s) \\right]$ , and it is harder for timed value critics to generalize appropriately across the additional discrete input $t$ .", "Proofs This section contains the proofs omitted from the main body of the document.", "For the sake of clarity, we repeat the main statements before showing the respective proof.", "thm:vsTheorem REF For an arbitrary POMDP and policy, a time-invariant state value function ${V^\\pi }(s)$ is ill-defined.", "The state value function, defined as ${V^\\pi }(s) = \\sum _{a\\in } \\Pr (a\\mid s) {Q^\\pi }(s, a) \\,,$ requires the state-conditioned action probability term $\\Pr (a\\mid s)$ .", "Because a partially observable policy depends on the history and not the state, the state-conditioned action probability term must be expanded by integrating over the space of all histories, $\\Pr (a\\mid s) = \\sum _{h\\in } \\Pr (h\\mid s) \\pi (a; h) \\,, $ which requires the state-conditioned history probability term $\\Pr (h\\mid s)$ .", "However, it is impossible to associate that term to well-defined RVs: Clearly, the reinforcement learning graphical model (see sec:graphmodel) does not define a time-less history RV $H$ , so the term cannot be explicitly written as $\\Pr (H=h\\mid S=s)$ .", "Further, the probability associated with timed RVs $\\Pr (H_t=h\\mid S_t=s)$ is clearly not time-invariant, hence identifying the time-index is crucial.", "Finally, it is not possible to restrict the integration in eq:pras:app only to histories of a given time-index.", "Overall, the probability term $\\Pr (h\\mid s)$ is necessarily time-variant, which means that $\\Pr (a\\mid s)$ is itself time-variant, and therefore the state value function ${V^\\pi }(s)$ is time-variant, and a time-invariant state value function ${V^\\pi }(s)$ is ill-defined.", "thm:vs.reactive.poTheorem REF If the POMDP observation function depends only on the current state, $\\colon \\rightarrow \\Delta $ , and the policy is reactive, then ${V^\\pi }(s)$ is not necessarily an unbiased estimate of ${V^\\pi }(h)$ , i.e., it is not guaranteed that ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s)\\right]$ .", "As one of the salient theorems of our work, we will cover it from different angles and provide three proofs: first one which is simple but does not go into the detail about the causes of the bias, then a more detailed and analytical one, and finally one by example.", "[First proof] Consider two histories $h, h^{\\prime }\\in $ which are different, $h\\ne h^{\\prime }$ , but are associated with the same belief $b(h) = b(h^{\\prime })$ ; a fairly common occurrence in many POMDPs.", "On one hand, because the two histories are different, the agent's next action may differ, which may then lead to different immediate rewards and future trajectories, i.e., the respective values would also differ, ${V^\\pi }(h) \\ne {V^\\pi }(h^{\\prime }) \\,.$ On the other hand, because the two beliefs are the same, then the expected state values must be the same, $_{s\\mid h}\\left[ {V^\\pi }(s) \\right] = _{s\\mid h^{\\prime }}\\left[ {V^\\pi }(s)\\right] \\,.$ Therefore, it is not guaranteed that ${V^\\pi }(h) = _{s\\mid h}\\left[{V^\\pi }(s) \\right]$ .", "[Second proof, by contradiction] First, we assume that ${V^\\pi }(s)$ is unbiased and show that ${Q^\\pi }(s,a)$ (as defined by eq:qsa) is unbiased, sh[ Q(s, a) ] = sh[ (s, a) + s's, a[ V(s') ] ] = sh[ (s, a) ] + sh[ s's, a[ V(s') ] ] = sh[ (s, a) ] + s'h, a[ V(s') ] = sh[ (s, a) ] + oh, a s'hao[ V(s') ] = (h, a) + oh, a[ V(hao) ] = Q(h, a)  .", "Next, we show that even if ${Q^\\pi }(s, a)$ is unbiased, ${V^\\pi }(s)$ (as defined by eq:vs) is biased, which contradicts the original assumption.", "To do that, we expand the expected state value function $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ and the history value function ${V^\\pi }(h)$ , and show that there is a concrete difference between them: sh[ V(s) ] = sh[ a (as) Q(s, a) ] = sh[ a os[ (a; o) ] Q(s, a) ]  , V(h) = a (a; oh) Q(h, a) = a (a; oh) sh[ Q(s, a) ] = sh[ a os[ (a; oh) ] Q(s, a) ]  .", "eq:proof:vs,eq:proof:vh differ in terms of which observation is used by the policy; in eq:proof:vs, an observation $o$ inferred from a state $s$ inferred from the history $h$ is used, while in eq:proof:vh the final observation $o_h$ of the history $h$ is used.", "These two observations $o$ and $o_h$ are not generally the same, and the respective expectations are similarly not generally the same.", "The nested expectation in eq:proof:vs can be interpreted as a lossy round-trip inference from history to state and from state back to observation $h\\rightarrow s\\rightarrow o$ .", "Although histories and states tend to be somewhat correlated, both state aliasing and history aliasing make the roundtip conversion imperfect, causing a mismatch between the expected state value function $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ and the history value function ${V^\\pi }(h)$ in the general control case of a general POMDP.", "[Third proof, by example] This is a proof by example (with a proof by contradiction element).", "We will define the good/bad POMDP and, for a specific policy and history, first calculate $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ exactly, and then ${V^\\pi }(h)$ using bootstrapping (while also assuming ${V^\\pi }(hao) =_{s^{\\prime }\\mid hao}\\left[ {V^\\pi }(s^{\\prime }) \\right]$ ).", "We show that the two values are numerically different.", "In the good/bad POMDP, $= \\lbrace \\texttt {GOOD}, \\texttt {BAD}\\rbrace , = \\lbrace \\texttt {GOOD},\\texttt {BAD}\\rbrace $ , $= \\lbrace \\texttt {GOOD}, \\texttt {BAD}\\rbrace $ ; At times, we will use the shorthands $\\texttt {G}$ and $\\texttt {B}$ .", "The initial state distribution is uniform, and each state deterministically transitions into itself.", "The GOOD state always emits the GOOD observation, while the BAD state emits a random observation.", "Consider the reward function such that $R(s, a) = \\left[ a = \\texttt {GOOD}\\right]$ , i.e., the agent receives a reward whenever it choses the GOOD action.", "We will denote a history as the concatenation of alternating observations and actions, starting with an observation.", "To keep the notation compact, we will occasionally use symbols G and B to represent GOOD and BAD states, observations and actions.", "Consider a deterministic policy $\\pi (a; h) = \\left[ a = o_h \\right]$ which returns the action corresponding to the last observation.", "Note that this POMDP and this policy satisfy the requirements to guarantee that ${V^\\pi }(s)$ is well defined.", "Next, we calculate the state values ${V^\\pi }(s)$ .", "The $\\texttt {GOOD}$ state always emits the $\\texttt {GOOD}$ observation, so the agent will always choose the $\\texttt {GOOD}$ action and receive a reward of 1, then the state will always transition into itself, V(s=GOOD) = 1 + V(s=GOOD) = 11 -  .", "On the other hand, the $\\texttt {BAD}$ state will only emit the $\\texttt {GOOD}$ observation half of the times, so the agent will only choose the $\\texttt {GOOD}$ action and receive a reward of 1 half of the times, then the state will always transition into itself, V(s=BAD) = 12 + V(s=BAD) = 12(1 - )  .", "Next, we consider the history $h=\\texttt {G}$ after a single initial GOOD observation, and calculate the history value ${V^\\pi }(h)$ .", "Before proceeding, we need to calculate a few intermediate quantities, such as the belief-distribution: (s=Gh=G) (h=Gs=G) (s=G) = 1 12 = 12  , (s=Bh=G) (h=Gs=B) (s=B) = 12 12 = 14  , therefore (s=Gh=G) = 23  , (s=Bh=G) = 13  .", "We also calculate the belief-state distribution after two other histories.", "First $h=\\texttt {G}\\texttt {G}\\texttt {G}$ , (s=Gh=GGG) (h=GGGs=G) (s=G) = 1 12 = 12  , (s=Bh=GGG) (h=GGGs=B) (s=B) = 14 12 = 18  , therefore (s=Gh=GGG) = 45  , (s=Bh=GGG) = 15  .", "Then $h=\\texttt {G}\\texttt {G}\\texttt {B}$ , (s=Gh=GGB) (h=GGBs=G) (s=G) = 0 12 = 0  , (s=Bh=GGB) (h=GGBs=B) (s=B) = 14 12 = 18  , therefore (s=Gh=GGB) = 0  , (s=Bh=GGB) = 1  .", "We also need to calculate the observation emission probabilities, (o=Gh=G, a=G) = (s=Gh=G) (o=Gs=G) = + (s=Bh=G) (o=Gs=B) = 23 1 + 13 12 = 56  , (o=Bh=G, a=G) = (s=Gh=G) (o=Bs=G) = + (s=Bh=G) (o=Bs=B) = 23 0 + 13 12 = 16  .", "Next, we calculate ${V^\\pi }(h=\\texttt {G})$ under the assumption that the equality holds, V(h=G) = sh=G[ V(s) ] = (s=Gh=G) V(s=G) + (s=Bh=G) V(s=B) = 23 11- + 13 12(1-) = 56(1-)  .", "We can also apply the equality to other histories, V(h=GGG) = sh=GGG[ V(s) ] = (s=Gh=GGG) V(s=G) + (s=Bh=GGG) V(s=B) = 45 11- + 15 12(1-) = 910(1-)  , V(h=GGB) = sh=GGB[ V(s) ] = (s=Gh=GGB) V(s=G) + (s=Bh=GGB) V(s=B) = 0 11- + 1 12(1-) = 12(1-)  .", "Next, we calculate ${V^\\pi }(h=\\texttt {G})$ , this time by bootstrapping first, and then using the equality.", "Note that with the given history $h=\\texttt {G}$ , the agent will choose action $a=\\texttt {GOOD}$ .", "Then, V(h=G) = R(h=G, a=G) + oh=G, a=G[ V(hao=GGo) ] = 1 + ( $\\Pr (o=\\texttt {G}\\mid h=\\texttt {G}, a=\\texttt {G}) {V^\\pi }(hao=\\texttt {G}\\texttt {G}\\texttt {G}) \\\\+ \\Pr (o=\\texttt {B}\\mid h=\\texttt {G}, a=\\texttt {G}) {V^\\pi }(hao=\\texttt {G}\\texttt {G}\\texttt {B})$ ) = 1 + 56 910(1-) + 16 12(1-) = 60-6060(1-) + 4560(1-) + 560(1-) = 60-1060(1-) = 6-6(1-)  .", "The values from eq:proof:value:1,eq:proof:value:2 contradict each other, therefore, for this POMDP, policy, and history, ${V^\\pi }(h) \\ne _{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ .", "thm:vs.reactive.foTheorem REF If the POMDP states and observations are related by a bijection $\\phi \\colon \\rightarrow $ , and the policy is reactive, then ${V^\\pi }(s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ .", "Although there is no intrinsic state uncertainty, we continue to use probabilistic notation for notational consistency and simplicity.", "In the following derivation, we use the fact that states and observations contain the same system information to determine the first action and reward.", "This process can be repeated iteratively for all future actions and rewards (omitted, but represented by the ellipsis), sh[ V(s) ] = sh[ a (as) Q(s, a) ] = sh[ a (a; I(s)) Q(s, a) ] = sh[ a (a; oh) Q(s, a) ] = a (a; oh) sh[ Q(s, a) ] = a (a; oh) sh[ R(s, a) + s's, a[ V(s') ] ] = a (a; oh) ( R(h, a) + s'h, a[ V(s') ] ) = a (a; oh) ( R(h, a) + oh, a[ s'hao[ V(s') ] ] ) = $\\vdots $ = a (a; oh) ( R(h, a) + oh, a[ V(hao) ] ) = a (a; oh) Q(h, a) = V(h)  .", "In this case, the bijection between observations and states removes any error from the round-trip inference $h\\rightarrow s\\rightarrow o$ is perfect, which was the cause of the bias in thm:vs.reactive.po.", "thm:vhsTheorem REF For an arbitrary POMDP and policy, ${V^\\pi }(h, s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(h, s) \\right]$ Follows from eq:vh-raw,eq:vhs-raw, V(h0) = s,ah0[ k k R(sk, ak) ] = s0h0 s,ah0,s0[ k k R(sk, ak) ] = s0h0[ V(h0, s0) ]  .", "thm:asymmetric-policy-gradientTheorem REF (Asymmetric Policy Gradient) The policy gradient can be expressed using history-state values, $\\nabla _\\theta _\\text{policy}(\\theta ) = -\\left[ \\sum _t \\gamma ^t {Q^\\pi }(h_t,s_t, a_t) \\nabla _\\theta \\log \\pi (a_t; h_t) \\right] \\,.$ Following thm:vhs, we have Q(h, a) = R(h, a) + oh, a[ V(hao) ] = R(h, a) + oh, a[ s'h, a, o[ V(hao, s') ] ] = R(h, a) + s', oh, a[ V(hao, s') ] = sh[ R(s, a) + s', os, a[ V(hao, s') ] ] = sh[ Q(h, s, a) ]  .", "Therefore, policy() = -[ t t Q(ht, at) (at; ht) ] = -t t ht,at[ Q(ht, at) (at; ht) ] = -t t ht,at[ stht[ Q(ht, st, at) ] (at; ht) ] = -t t ht,st,at[ Q(ht, st, at) (at; ht) ] = -[ t t Q(ht, st, at) (at; ht) ]  .", "Environments Table: Environment properties.The environments used in the evaluation can be split into two groups, Shopping-5, Shopping-6, Heavenhell-3, Heavenhell-4 and Rocksample-5-6 are flat POMDPs, while Keydoor and Ninerooms are gridverse POMDPs.", "Flat POMDPs Figure: Layout of the Shopping environments.Figure: Layout of the Heavenhell environments.Figure: Layout of the Rocksample environment.In the flat POMDPshttps://github.com/abaisero/gym-pomdps, states, actions and observations are encoded by categorical indices with no inherent metric, which are intrinsically equally (dis)similar to each other.", "While it is not possible to generalize across states and observations via feature extraction, the primary challenge in these POMDPs is that of generalizing across different histories.", "Because the flat POMDPs are finite, their state, action and observation spaces have well-defined sizes, shown in tab:environments; note, however, that the size of the state space is not a significant measure of the complexity of partially observable tasks, while the time required to solve the task (i.e., history length) is a more relevant measure.", "Shopping This environment simulates an agent going to a shop to buy an item it forgot.", "The agent navigates a $5\\times 5$ or $6\\times 6$ gridworld trying to locate and select a randomly positioned item.", "The agent's position is fully observable, while the item's position is only observed when queried.", "fig:map:shopping depicts the gridworlds encoded by Shopping-5 and Shopping-6.", "States and Observations States encode the position of the agent and the position of the item in a single integer.", "Observations encode the position of the agent or the position of the item in a single integer.", "Actions Each time-step, the agent must choose an action from the set { LEFT, RIGHT, UP, DOWN, QUERY, BUY }.", "If the agent chooses the QUERY action, it observes the item's position, otherwise it observes its own position.", "To solve the task optimally, the agents needs to query the item's position and remember it, navigate to it, and then buy it.", "Rewards The agent receives the following reward signal: a reward of $-1.0$ for moving; a reward of $-2.0$ for performing a QUERY action; a reward of $-5.0$ for performing a BUY action in the wrong cell; and a reward of $10.0$ for performing a BUY action in the correct cell.", "Heavenhell The agent navigates a corridor-like gridworld composed of a fork and 3 dead-ends.", "Two dead-ends are exits which lead to heaven or hell, although the agent does not know which is which, while the third dead-end leads to a priest who can help the agent identify the heaven exit.", "fig:map:heavenhell depicts the gridworlds encoded by Heavenhell-3 and Heavenhell-4.", "States and Observations States encode the position of the agent and the position of the exit to heaven.", "Observations encode the position of the agent or the position of the exit to heaven.", "Actions Each time-step, the agent must choose an action from the set { NORTH, SOUTH, EAST, WEST }.", "If the agent is at the priest, it observes heaven's location, otherwise it observes its own position.", "To solve the task, the agent needs to navigate to the priest, then back to the fork, and on to heaven.", "Rewards The agent receives a sparse reward signal composed of: a reward of $1.0$ for exiting to heaven; and a reward of $-1.0$ for exiting to hell.", "Rocksample This environment simulates an agent navigating a landscape to collect research material.", "The agent navigates a $5\\times 5$ gridworld which contains 6 rocks, each having either good or bad research value.", "The agent's position is fully observable, while each rock's goodness is randomly sampled and unobserved unless the agent checks it.", "Each rock's goodness can be queried via a stochastic check action, whose reliability decays with the agent-rock distance.", "fig:map:rocksample depicts the gridworld encoded by Rocksample-5-6.", "States and Observations States encode the position of the agent and whether each rock is good or bad; the positions of the rocks are fixed so they do not need to be tracked by the state.", "Observations encode the position of the agent or whether the checked rock is good or bad.", "Actions Each time-step, the agent must choose an action from the set { NORTH, SOUTH, EAST, WEST, CHECK_0, CHECK_1, CHECK_2, CHECK_3, CHECK_4, CHECK_5, SAMPLE, }.", "If the agent chose a CHECK_* action, it observes whether the corresponding rock is good or bad, otherwise it observes its own position.", "To solve the task, the agent needs to sample all the good rocks (and none of the bad rocks), dynamically adapting its path depending on the stochastic information gained by checking each rock.", "Rewards The agent receives a sparse reward signal composed of: a reward of $10.0$ for sampling a good rock; and a reward of $-10.0$ for sampling a bad rock.", "Gridverse POMDPs Figure: KeydoorFigure: NineroomsIn the gridverse POMDPshttps://github.com/abaisero/gym-gridverse, actions are still encoded by categorical indices, while states and observations are encoded as structures which do have an inherent similarity metric; they are split into two components: A grid component: a $6\\times H\\times W$ volume of categorical indices which encode cell types, colors, statuses, agent presence, and the spatial relationships between them.", "The observation grid component is a slice of the corresponding state grid component made to match the agent's perspective: it is rotated to be a first-person view, and cells hidden behind walls are occluded, as shown in fig:keydoor:observation,fig:ninerooms:observation.", "An agent component: a 6-dimensional array of categorical indices representing the agent position and orientation, and the item held by the agent if any.", "The position and orientations in the state agent component are absolute, while those in the observation component are relative to the agent's perspective—they are essentially constant, and not necessary for control.", "Keydoor The agent navigates a $5\\times 5$ gridworld split into two rooms split by a wall and a locked door; on one side is the agent and a key which can be used to open the door, while on the other side is the goal.", "The positions of the agent, the key, the door, and the goal are randomly sampled such that two instances of the same problem are unlikely to be the same.", "fig:keydoor shows state and observation frames taken from an instance of the Keydoor environment.", "States and Observations The state grid component is a $6\\times 5\\times 5$ volume.", "The observation grid component is a $6\\times 7\\times 7$ volume.", "Actions Each time-step, the agent must choose an action from the set { MOVE_FORWARD, MOVE_BACKWARD, MOVE_LEFT, MOVE_RIGHT, TURN_LEFT, TURN_RIGHT, PICK_N_DROP, ACTUATE }.", "The MOVE_* actions result in a movement depending on the agent's orientation, while the TURN_* allows the agent to change its orientation.", "With the PICK_N_DROP action, the agent can pick and/or drop the key from/to the cell in front, while with the ACTUATE action, the agent can open and/or close the door.", "Rewards The agent receives a dense reward signal composed as the sum of the following terms: a living reward of $-0.05$ for every time-step; a reward of $0.2$ for stepping closer to the goal, and $-0.2$ for stepping further away (ignoring walls); a reward of $1.0$ for picking up the key, and $-1.0$ for dropping it; a reward of $1.0$ for opening the door, and $-1.0$ for closing it; and a reward of $5.0$ for reaching the goal.", "Ninerooms The agent navigates a $13\\times 13$ maze-like gridworld trying to locate and reach the goal.", "fig:ninerooms shows state and observation frames taken from an instance of the Ninerooms environment.", "States and Observations The state grid component is a $6\\times 13\\times 13$ volume.", "The observation grid component is a $6\\times 7\\times 7$ volume.", "Actions Each time-step, the agent must choose an action from the set { MOVE_FORWARD, MOVE_BACKWARD, MOVE_LEFT, MOVE_RIGHT, TURN_LEFT, TURN_RIGHT }.", "The MOVE_* actions result in a movement depending on the agent's orientation, while the TURN_* allows the agent to change its orientation.", "Rewards The agent receives a dense reward signal composed as the sum of the following terms: a living reward of $-0.05$ for every time-step; a reward of $0.2$ for stepping closer to the goal, and $-0.2$ for stepping further away (ignoring walls); and a reward of $5.0$ for reaching the goal.", "Architectures and Hyper-Parameters Figure: A2C architecture.", "The state, action, and observationrepresentations φ(s)\\phi (s), φ(a)\\phi (a), and φ(o)\\phi (o) are those shown infig:architecture:representations:flat orfig:architecture:representations:gridverse, depending on whetherthe type of enviroment.", "Dotted lines are present or omitted depending onwhether a history critic V ^(h){\\hat{V}}(h), state critic V ^(s){\\hat{V}}(s), orhistory-state critic V ^(h,s){\\hat{V}}(h, s) is being modeled.In this section, we describe the architectures used by the policy and critic models; a general overview is shown in fig:architecture.", "The general architecture will be similar for all domains; however there will some differences to accommodate the fact that the flat POMDPs provide states and observations as categorical indices, while the gridverse POMDPs provide states and observations as volumes and arrays of categorical indices, whose structures include spatial relationships.", "Features Extraction for Flat POMDPs These components are shown in fig:architecture:representations:flat.", "Because flat POMDPs provide states, actions and observations as categorical indices, and we use 128-dimensional embedding models to represent each of them.", "Features Extraction for Gridverse POMDPs These components are shown in fig:architecture:representations:gridverse.", "Because gridverse POMDPs provide actions as categorical indices, and we use 128-dimensional embedding models to represent them.", "On the other hand, states and observations are provided in the format described in sec:environments:gridverse, each composed of a grid and an agent component: grid The grid component is a $6\\times H\\times W$ volume of categorical data, which we pass through 4-dimensional embeddings, resulting in a $24\\times H\\times W$ volume of numeric data.", "We further pass that data to two layers of CNN followed by ReLU non-linearities, resulting in a $16\\times H\\times W$ volume of data, finally flattened into a $16 HW$ -dimensional array.", "agent The agent component is a 6-dimensional array of categorical data, which we pass through 4-dimensional embeddings, resulting in a 24-dimensional array of numeric data.", "The grid and agent representations are then concatenated to form the state or observation representation $\\phi (s)$ or $\\phi (o)$ .", "Remainder of the Architecture These components are shown in fig:architecture:a2c.", "Concatenated action and observation embeddings form the to a 128-dimensional single-layer gated recurrent unit (GRU) [3], which acts as a history representation $\\phi (h)$ .", "The history representation is then fed into separate policy and critic models, each a 2-layer 128-dimensional feedforward model with ReLU non-linearities.", "A practical issue we found with A2C(h,s) is that the history and state representations $\\phi (h)$ and $\\phi (s)$ contain features with very different orders of magniture; To address this, we use layer-normalization.", "To guarantee a fair comparison with the other methods, we do the same for other critics as well.", "Next, we describe architectural details specific to each method: A2C(h): The history representation passes through layer-normalization first, and then is fed to the critic model; A2C(s): The state embedding passes through layer-normalization first, and then is fed to the critic model; A2C(h,s): The history representation and state embedding individually pass through layer-normalization first, and then are concatenated to form a single feature vector, which is then fed to the critic model." ], [ "Theory of Asymmetric Actor-Critic", "In this section, we analyze the theoretical implications of using a state critic as described in sec:bg:aa2c, and expose critical issues.", "The primary result will be that the state value function ${V^\\pi }(s)$ of a non-reactive agent under partial observability is generally ill-defined.", "Then, we show that the state value function ${V^\\pi }(s)$ of a reactive agent under partial observability is well-defined, but introduces a bias into the training process which may undermine learning.", "Finally, we show that the state value function ${V^\\pi }(s)$ of a reactive agent under an observation function which is equivalent to full observability is both well-defined and unbiased.", "Note that replacing the history critic is intrinsically questionable: the policy gradient theorem for POMDPs (eq:policy-gradient) specifically requires history values, and substituting them for another value which has a different expectation will result in the gradient estimates losing their theoretical guarantees of correctness.", "Therefore, we analyze state values ${V^\\pi }(s)$ as estimators of history values ${V^\\pi }(h)$ , and consider the corresponding estimation bias, i.e., the difference between $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ and ${V^\\pi }(h)$ for a given history $h$ .", "Informally, the fundamental issue with ${V^\\pi }(s)$ is that the state does not contain sufficient information to determine the agent's future behavior (which depends on the history) and is thus unable to meaningfully represent expected future returns.", "Ironically, state values suffer from a problem we call history aliasing, i.e., being unable to infer the agent's history from the system's state.", "Even when ${V^\\pi }(s)$ is numerically well-defined, it usually introduces a bias caused by the imperfect correlation between histories and states; in essence, the average value of histories inferred from the current state is not an accurate estimate of the true current history's value.", "We begin with the definition of the state value function, ${V^\\pi }(s{0}) = _{\\bar{s}, \\bar{a} \\mid s{0}}\\left[\\sum _{k=0}^\\infty \\gamma ^k R(s{k}, a{k}) \\right] \\,,$ which, if well-defined, supports an indirect recursive Bellman form, V(s) = a (as) Q(s, a)  , Q(s, a) = R(s, a) + s's, a[ V(s') ]  .", "From eq:vs, we note the term $\\Pr (a\\mid s)$ , which encodes the likelihood of an action being taken from a given state.", "Because the agent policy acts on histories, this term is not directly available, but must be derived indirectly by integrating over possible histories; and because there is no contextual information available to limit the integration to histories of a specific length, we can only integrate over the space of all possible histories, $\\Pr (a\\mid s) = \\sum _{h\\in } \\Pr (h\\mid s) \\pi (a; h) \\,.", "$ eq:pras reveals the probability term $\\Pr (h\\mid s)$ , which encodes the likelihood of an history having taken place given a state.", "While $\\Pr (h\\mid s)$ may look harmless, it is the underlying cause of serious analytical issues.", "As discussed in sec:graphmodel, a probability term is only well-defined if associated with well-defined RVs, and the fundamental issue with $\\Pr (h\\mid s)$ is that such RVs do not exist.", "On one hand, we cannot used timed RVs $\\Pr (H_t=h\\mid S_t=s)$ , because eq:pras integrates over the sample space of all histories, and not just those of a given length $t$ .", "On the other hand, we cannot use time-less RVs $\\Pr (H=h\\mid S=s)$ , because time-less RVs do not exist in the RL graphical model.", "Ultimately, $\\Pr (h\\mid s)$ is ill-defined, which causes $\\Pr (a\\mid s)$ and ${V^\\pi }(s)$ itself to be ill-defined.", "Theorem 4.1 For an arbitrary POMDP and policy, a time-invariant state value function ${V^\\pi }(s)$ is ill-defined.", "(proof in sec:proofs).", "In relation to the training procedure of a state critic model, the practical implications of an ill-defined value function are not obvious; even though the analytical value function is ill-defined mathematically, the critic's training process performs valid calculations on sample data which results in valid updates of the critic parameters.", "However, convergence is unlikely, since empirical convergence to a meaningful value virtually requires the existence of a theoretical convergence value.", "Rather, it is likely that the critic's training target will shift indefinitely, depending on the recent training data, inhibiting the convergence of the critic model even under ideal training circumstances, which itself will cause instabilities and divergence in the policy model; this is verified empirically in sec:evaluation.", "A natural solution to the underlying issue of state value functions ${V^\\pi }(s)$ is to define and employ timed value functions $V_t^\\pi (s)$ ; in sec:special:timed, we show that timed value functions indeed address the primary issue, although learning a critic to model them is likely to pose a significantly harder learning challenge, due to the need to generalize well and accurately over different time-steps.", "Rather, in the next subsections, we show special cases of the general control problem which make non-timed ${V^\\pi }(s)$ well-defined; in some cases, this will lead to other theoretical issues involving the introduction of estimation bias." ], [ "Reactive Policy under Partial\nObservability", "We show that ${V^\\pi }(s)$ is well-defined if we make two assumptions about the agent and environment: [label=()] that the policy is reactive; and that the POMDP observation function depends only on the current state, $\\colon \\rightarrow \\Delta $ , rather than the entire state transition.", "Under these assumptions, we can expand $\\Pr (a\\mid s)$ by integrating over the space of all observations (rather than all histories), $\\Pr (a\\mid s) = \\sum _{o\\in } \\Pr (o\\mid s) \\pi (a; o) \\,.$ In this case, the term $\\Pr (o\\mid s)$ can be grounded to timed RVs $\\Pr (O_t=o \\mid S_t=s)$ ; because that probability is time-invariant, it is well-defined, meaning that ${V^\\pi }(s)$ is well-defined in this case.", "However, we show that ${V^\\pi }(s)$ is biased compared to ${V^\\pi }(h)$ .", "Theorem 4.2 If the POMDP observation function depends only on the current state, $\\colon \\rightarrow \\Delta $ , and the policy is reactive, then ${V^\\pi }(s)$ is not necessarily an unbiased estimate of ${V^\\pi }(h)$ , i.e., it is not guaranteed that ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s)\\right]$ .", "(proof in sec:proofs).", "Although we were able to show that the value function is well-defined in the case of reactive control, there are still two significant issues: [label=()] reactive policies are inadequate to solve many POMDPs; and the bias of the value function ${V^\\pi }(s)$ may influence the agent learning capabilities catastrophically.", "Broadly speaking, this bias is caused by the fact that hidden in ${V^\\pi }(s)$ is an expectation over observations $o$ which, while conditioned on the state $s$ , are not necessarily consistent with the true history $h$ .", "We discuss the cause of this bias more formally in the corresponding proof.", "Reactive Policy under Full Observability We show that the state value function is not only well-defined but also unbiased if we make two assumptions about the agent and environment: [label=()] that the policy is reactive; and that the there is a bijective abstraction $\\phi \\colon \\rightarrow $ between observations and states.", "The abstraction $\\phi $ encodes the fact that the environment is not truly partially observable, but rather that states and observations essentially contain the same information, albeit at different levels of abstraction, akin to the problems used by [21].", "For example, an image displaying a workspace without occlusions could be a low-level abstraction (observation), while a concise vector representation of the object poses in the workspace could be a high-level abstraction (state).", "In this case, the action probability term $\\Pr (a\\mid s)$ does not need to be obtained indirectly by integrating other variables; rather, the state-observation bijection can be used to directly relate it to the policy model, (as) = (a; I(s))  .", "Contrary to the previous cases, the overall state value function ${V^\\pi }(s)$ is not only well-defined, but also unbiased.", "Theorem 4.3 If the POMDP states and observations are related by a bijection $\\phi \\colon \\rightarrow $ , and the policy is reactive, then ${V^\\pi }(s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ .", "(proof in sec:proofs).", "The benefit of using a state critic under this scenario is that the critic model can avoid learning a representation of the observations before learning the values [21].", "Naturally, the main disadvantage of this scenario is that most POMDPs do not satisfy the bijective abstraction assumption, which is virtually equivalent to full observability.", "Nonetheless, if a control problem only deviates mildly from full observability, it is very possible that a state critic might benefit the learning agent.", "Unbiased Asymmetric Actor-Critic In this section, we introduce unbiased asymmetric actor-critic, an actor-critic variant which is able to exploit asymmetric state information during offline training while avoiding the issues of state value functions exposed in sec:aa2c.", "Consider a history-state value function ${V^\\pi }(h,s)$  [1], which represents the expected future discounted returns obtained when the history is $h$ and the state is $s$ , ${V^\\pi }(h{0}, s{0}) = _{\\bar{s}, \\bar{a} \\mid h{0},s{0}}\\left[ \\sum _{k=0}^\\infty \\gamma ^k R(s{k}, a{k})\\right] \\,, $ which supports an indirect recursive Bellman form, V(h, s) = a (a; h) Q(h, s, a)  , Q(h, s, a) = (s, a) + s',os,a[ V(hao, s') ]  .", "Providing the history information makes the history-state value function ${V^\\pi }(h, s)$ not only well-defined even for non-reactive policies, but also an unbiased estimate of ${V^\\pi }(h)$ .", "Theorem 5.1 For an arbitrary POMDP and policy, ${V^\\pi }(h, s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(h, s)\\right]$ .", "(proof in sec:proofs).", "As we have done for state values ${V^\\pi }(s)$ , we are interested in the properties of history-state values ${V^\\pi }(h, s)$ in relation to history values ${V^\\pi }(h)$ .", "thm:vhs shows that history and history-state values are related by ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(h, s) \\right]$ , i.e., history-state values are interpretable as Monte Carlo (MC) estimate of the respective history values.", "In expectation, history-state values provides the same information as the history values, therefore an asymmetric variant of the policy gradient theorem also holds.", "Theorem 5.2 (Asymmetric Policy Gradient) The policy gradient can be expressed using history-state values, $\\nabla _\\theta _\\text{policy}(\\theta ) = -\\left[ \\sum _t \\gamma ^t {Q^\\pi }(h_t,s_t, a_t) \\nabla _\\theta \\log \\pi (a_t; h_t) \\right] \\,.$ (proof in sec:proofs).", "As estimators, history-state values ${V^\\pi }(h, s)$ can be described in terms of their bias and variance w.r.t.", "history values ${V^\\pi }(h)$ .", "Beyond providing the inspiration for the MC interpretation, thm:vhs already proves that ${V^\\pi }(h, s)$ is unbiased, while its variance is dynamic and depends on the history $h$ via the belief-state $\\Pr (S\\mid h)$ ; in particular, low-uncertainty belief-states result in relatively low variance, and deterministic belief-states result in no variance.", "Given that operating optimally in a partially observable environment generally involves information-gathering strategies associated with low-uncertainty belief-states, the practical variance of the history-state value is likely to be relatively low once the agent has learned to solve the task to some degree of success.", "Inspired by thm:asymmetric-policy-gradient, we propose unbiased asymmetric A2C, which uses a history-state critic ${\\hat{V}}\\colon \\times \\rightarrow $ trained to model history-state values ${V^\\pi }(h, s)$ , policy() = -[ t t t (at; ht) ]  , t = R(st, at) + V(ht+1, st+1) - V(ht, st)  .", "Because ${\\hat{V}}(h, s)$ receives the history $h$ as input, it can still predict reasonable estimates of the agent's expected future discounted returns; and because it receives the state $s$ as input, it is still able to exploit state information while introducing no bias into the learning process, e.g., for the purposes of bootstrapping the learning of critic values and/or aiding the learning of history representations.", "Interpretations of State Although the history-state value is analytically well-defined, It is worthwhile to question why the inclusion of the state information should help the actor-critic agent at all.", "We attempt to address this open question, and consider two competing interpretations, which we call state-as-information and state-as-a-feature.", "State as Information Under this interpretation, state information is valuable because it is latent information unavailable in the history, which results in more informative values.", "However, this interpretation is flawed for two reasons: [label=()] The policy gradient theorem specifically requires ${V^\\pi }(h)$ , which contains precisely the correct information required to accurately estimate policy gradients.", "In this context, there is no such thing as “more informative values” than history values.", "In theory, the history-state value in thm:asymmetric-policy-gradient could use any other state sampled according to $\\tilde{s}\\sim b(h)$ , rather than the true system state, which would also result in the same analytical bias and variance properties.", "In practice, we use the true system state primarily due to it being directly available during simulation; however, we believe that its identity as the true system state is analytically irrelevant, which leads to the next interpretation of state.", "State as a Feature We conjecture an alternative interpretation according to which the state can be seen as a stochastic high-level feature of the history.", "Consider a history critic ${\\hat{V}}(h)$ to appropriately model the value function ${V^\\pi }(h)$ , the model must first learn an adequate history representation, which is in and of itself a significant learning challenge.", "The critic model would likely benefit from receiving auxiliary high-level black-box features of the history $\\phi (h)$ .", "The resulting critic ${\\hat{V}}(h,\\phi (h))$ remains fundamentally a history critic, the supplementary features being exclusively a modeling construct.", "Next, we consider what kind of high-level features $\\phi (h)$ would be useful for control.", "While the specifics of what makes a good history representation depend strongly on the task, there is a natural choice which is arguably useful in many cases: the belief-state $b(h)$ .", "Because the belief-state is a sufficient statistic of the history for control, providing it to the critic model ${\\hat{V}}(h, b(h))$ is likely to greatly improve its ability to generalize across histories.", "Finally, we conjecture that any state sampled according to the belief-state $s\\sim b(h)$ —including the true system state—can be considered a stochastic realization of the belief-state feature, resulting in the history-state critic ${\\hat{V}}(h, s)$ .", "According to this interpretation, the importance of the state in the history-state critic is not in its identity as the true system state, but as a stochastic realization of hypothetical belief-state features, and presumably any other state sampled from the belief-state $\\tilde{s}\\sim b(h)$ could be equivalently used.", "Evaluation [tb] Each method follows the same algorithm structure, but uses different types of critics to compute the TD errors $\\delta _t$ (see eq:tderror:h,eq:tderror:s,eq:tderror:hs).", "Values $N$ , $B$ , and $E$ vary by environment.", "Input: epochs $N$ , episode batch $B$ , evaluation period $E$ epoch in 1 ...$N$ training_episodes $\\leftarrow $ sample_episodes($\\pi $ , $B$ ) update $\\theta ,\\vartheta $ via $\\nabla ($ training_episodes$)$ (see eq:a2c) epoch $\\bmod $ evaluation_period = 0 evaluation_episodes $\\leftarrow $ sample_episodes($\\pi $ , $E$ ) report empirical_returns(evaluation_episodes) In this section, we empirically evaluate three actor-critic variants: A2C(h), standard actor-critic with a history critic ${\\hat{V}}(h)$ as described in sec:a2c; A2C(s), asymmetric actor-critic with a state critic ${\\hat{V}}(s)$ as described in sec:aa2c; and A2C(h,s), unbiased asymmetric actor-critic with a history-state critic ${\\hat{V}}(h,s)$ as described in sec:uaa2c.", "Each method is trained and evaluated according to alg:code.", "Broadly speaking, each method uses a gated recurrent unit (GRU) [3] to compute history features, and two feed-forward networks which represent the policy action probabilities and critic values; sec:architectures contains a more detailed description of the used architectures.", "We perform evaluations on seven gridworld environments which exhibit significant partial observability: Shopping-5, Shopping-6, Heavenhell-3, Heavenhell-4, and Rocksample-5-6 are flat POMDPshttps://github.com/abaisero/gym-pomdps, where states and observations are represented by categorical indices, while Keydoor and Ninerooms are gridverse POMDPshttps://github.com/abaisero/gym-gridverse, where states and observations are represented by tensors of categorical indices which encode spatial relationships and other cell information.", "See sec:environments for a detailed descriptions and graphical representations of all environments.", "Results and Discussion Each method is evaluated in one of two ways: [label=()] we show empirical learning curve statistics for all environments, and we show how critic values change for important history-state pairs over the course of training in Heavenhell-4.", "Learning Curves Figure: Learning curve statistics over 20 independent training runs, whereeach run is periodically evaluated 20 times.", "Shaded areas are centeredaround the empirical mean performance, and show 2 standard errors (of themean).fig:learningperformance depicts the performance results in all the environments.", "First, we note that the symmetric baseline A2C(h) does not always learn to solve the task, but succeeds fully in fig:shoppingv,fig:heavenhelliii,fig:gvkeydoor, partially in fig:heavenhelliv,fig:rocksamplevvi,fig:gvninerooms, and fails in fig:shoppingvi.", "The asymmetric baseline A2C(s) also performs inconsistently across environments, with a mixture of successful and failing cases.", "We particularly note the strange learning curves of A2C(s) in fig:shoppingv,fig:shoppingvi, where performance improves quickly during the early training, but fails to improve further or even becomes unstable and collapses later on.", "While the exact dynamics of this collapse are not completely clear, we believe it is likely that this is a consequence of modeling the critic after an analytically unstable state value function, making convergence to stable values impossible.", "Using our proposed history-state critic, A2C(h,s) consistently exhibits either faster convergence and/or higher final performance in virtually all the flat environments, and competitive performance in the gridverse environments, taking a mild lead towards the end of Ninerooms.", "These results strongly demonstrate the importance of exploiting asymmetric information in ways which are theoretically justified and sound, as done in our work.", "Critic Values Figure: Critic value statistics for 4 history-state pairs, evaluatedthroughout the training procedure via 20 independent runs.In the top row the agent did not visit the priest, while in the bottom rowthe agent did visit the priest.To further inspect the behavior of each critic, we show the evolution of critic values for important history-state pairs over the course of training.", "We perform this evaluation on Heavenhell-4, and use 4 deliberately chosen history-state pairs.", "In each case the agent is located at the fork, and the 4 cases differ according to heaven's location (left or right) and whether the agent has previously visited the priest.", "fig:criticvalues shows the resulting critic values, with one figure for each of the chosen history-state pairs.", "In each scenario, we note that all critic values exhibit convergence properties which resemble those of the agent's performance in fig:heavenhelliv.", "This empirically confirms that agent performance and critic quality are strongly correlated factors; although it is not possible to make conclusions about the causality in this situation, we strongly believe that it is the critics which act as a learning bottleneck on policies.", "Notably, the critics which focus on a single aspect of the joint history-state show the exact same values for different history-state; namely, A2C(s) is identical in the top and bottom plots, while A2C(h) is identical in the top-left and top-right plots.", "Although in practice left and right plots are similar for all critics, A2C(h,s) is the only critic capable of representing different values in each of the 4 scenarios, as none of its curves are identical.", "We also note that the state critic ${\\hat{V}}(s)$ is muct less stable than the others, and shows no signs of convergence, which is consistent with our analysis in sec:aa2c.", "In contract, the history-state critic exhibits good convergence properties despite itself also using state information, which is consistent with our analysis in sec:uaa2c.", "Finally, we note again that the history-state critic ${\\hat{V}}(h, s)$ converges significantly faster than the history critic ${\\hat{V}}(h)$ , confirming that state information is useful for training.", "Conclusions Asymmetric methods trained offline in simulated environments can use information which is normally unavailable in partially observable RL, such as the true system state.", "While the idea of exploiting such information has potential, current state-of-the-art methods are powered by empirical results rather than theoretical analysis.", "In this work, we exposed profound theoretical issues with a standard variant of asymmetric actor-critic, and proposed an unbiased asymmetric actor-critic variant which is analytically sound and theoretically justified.", "Empirical results confirm our analysis, the weaknesses of state critics and the strengths of history-state critics.", "Although we applied the history-state value function only to A2C, the same concepts are easily extensible to other critic-based RL methods [24], [15], [5], [19].", "In future work, we aim to extend the theory of history-state policy value functions ${Q^\\pi }(h, s, a)$ to optimal value functions $Q^*(h, s, a)$ , and develop theoretically sound asymmetric variants of other deep RL methods such as DQN [18], soft Q-learning [8], and soft actor-critic [9].", "We also plan to extend the theory and approach to multi-agent methods, potentially bringing theoretical rigor and improved performance [6], [16], [14], [27], [31], [22], [17], [23].", "Timed Value Functions sec:aa2c shows that for a general POMDP and policy $\\pi \\colon \\rightarrow \\Delta $ , the state value function ${V^\\pi }(s)$ is not generally well-defined due to issues caused by the lack of time information.", "In this section, we consider addressing the primary issue by providing explicit time-index information via timed value functions, $V_t^\\pi (s)$ and $Q_t^\\pi (s, a)$ , which represent the expected returns obtained when the agent finds itself in a state $s$ at time $t$ , Vt(s) = a (At=aSt=s) Qt(s, a)  , Qt(s, a) = R(s, a) + s's, a[ Vt+1(s') ]  .", "Once again, we analyze the state-dependent action distribution term to verify correctness, and expand it by integrating over histories; this time, we can use the explicit time-index information to integrate over histories of a given length only, $\\Pr (A_t=a\\mid S_t=s) = \\sum _{h\\in _t} \\Pr (H_t=h\\mid S_t=s) \\pi (a; h)\\,.", "$ Because eq:sumrule:timed is now restricted to histories of a given length $t$ , the probability term $\\Pr (H_t=h\\mid S_t=s)$ is well-defined, which means $\\Pr (A_t=a\\mid S_t=s)$ is well-defined, and $V_t^\\pi (s)$ is well-defined.", "Introducing the time-index makes the value function $V_t^\\pi (s)$ well-defined.", "However, its utility for the purpose of asymmetric reinforcement learning remains unclear because [label=()] it is still not formally proven whether the timed value function is unbiased, i.e., whether $V_t^\\pi (h) = _{s\\mid h}\\left[V_t^\\pi (s) \\right]$ , and it is harder for timed value critics to generalize appropriately across the additional discrete input $t$ .", "Proofs This section contains the proofs omitted from the main body of the document.", "For the sake of clarity, we repeat the main statements before showing the respective proof.", "thm:vsTheorem REF For an arbitrary POMDP and policy, a time-invariant state value function ${V^\\pi }(s)$ is ill-defined.", "The state value function, defined as ${V^\\pi }(s) = \\sum _{a\\in } \\Pr (a\\mid s) {Q^\\pi }(s, a) \\,,$ requires the state-conditioned action probability term $\\Pr (a\\mid s)$ .", "Because a partially observable policy depends on the history and not the state, the state-conditioned action probability term must be expanded by integrating over the space of all histories, $\\Pr (a\\mid s) = \\sum _{h\\in } \\Pr (h\\mid s) \\pi (a; h) \\,, $ which requires the state-conditioned history probability term $\\Pr (h\\mid s)$ .", "However, it is impossible to associate that term to well-defined RVs: Clearly, the reinforcement learning graphical model (see sec:graphmodel) does not define a time-less history RV $H$ , so the term cannot be explicitly written as $\\Pr (H=h\\mid S=s)$ .", "Further, the probability associated with timed RVs $\\Pr (H_t=h\\mid S_t=s)$ is clearly not time-invariant, hence identifying the time-index is crucial.", "Finally, it is not possible to restrict the integration in eq:pras:app only to histories of a given time-index.", "Overall, the probability term $\\Pr (h\\mid s)$ is necessarily time-variant, which means that $\\Pr (a\\mid s)$ is itself time-variant, and therefore the state value function ${V^\\pi }(s)$ is time-variant, and a time-invariant state value function ${V^\\pi }(s)$ is ill-defined.", "thm:vs.reactive.poTheorem REF If the POMDP observation function depends only on the current state, $\\colon \\rightarrow \\Delta $ , and the policy is reactive, then ${V^\\pi }(s)$ is not necessarily an unbiased estimate of ${V^\\pi }(h)$ , i.e., it is not guaranteed that ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s)\\right]$ .", "As one of the salient theorems of our work, we will cover it from different angles and provide three proofs: first one which is simple but does not go into the detail about the causes of the bias, then a more detailed and analytical one, and finally one by example.", "[First proof] Consider two histories $h, h^{\\prime }\\in $ which are different, $h\\ne h^{\\prime }$ , but are associated with the same belief $b(h) = b(h^{\\prime })$ ; a fairly common occurrence in many POMDPs.", "On one hand, because the two histories are different, the agent's next action may differ, which may then lead to different immediate rewards and future trajectories, i.e., the respective values would also differ, ${V^\\pi }(h) \\ne {V^\\pi }(h^{\\prime }) \\,.$ On the other hand, because the two beliefs are the same, then the expected state values must be the same, $_{s\\mid h}\\left[ {V^\\pi }(s) \\right] = _{s\\mid h^{\\prime }}\\left[ {V^\\pi }(s)\\right] \\,.$ Therefore, it is not guaranteed that ${V^\\pi }(h) = _{s\\mid h}\\left[{V^\\pi }(s) \\right]$ .", "[Second proof, by contradiction] First, we assume that ${V^\\pi }(s)$ is unbiased and show that ${Q^\\pi }(s,a)$ (as defined by eq:qsa) is unbiased, sh[ Q(s, a) ] = sh[ (s, a) + s's, a[ V(s') ] ] = sh[ (s, a) ] + sh[ s's, a[ V(s') ] ] = sh[ (s, a) ] + s'h, a[ V(s') ] = sh[ (s, a) ] + oh, a s'hao[ V(s') ] = (h, a) + oh, a[ V(hao) ] = Q(h, a)  .", "Next, we show that even if ${Q^\\pi }(s, a)$ is unbiased, ${V^\\pi }(s)$ (as defined by eq:vs) is biased, which contradicts the original assumption.", "To do that, we expand the expected state value function $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ and the history value function ${V^\\pi }(h)$ , and show that there is a concrete difference between them: sh[ V(s) ] = sh[ a (as) Q(s, a) ] = sh[ a os[ (a; o) ] Q(s, a) ]  , V(h) = a (a; oh) Q(h, a) = a (a; oh) sh[ Q(s, a) ] = sh[ a os[ (a; oh) ] Q(s, a) ]  .", "eq:proof:vs,eq:proof:vh differ in terms of which observation is used by the policy; in eq:proof:vs, an observation $o$ inferred from a state $s$ inferred from the history $h$ is used, while in eq:proof:vh the final observation $o_h$ of the history $h$ is used.", "These two observations $o$ and $o_h$ are not generally the same, and the respective expectations are similarly not generally the same.", "The nested expectation in eq:proof:vs can be interpreted as a lossy round-trip inference from history to state and from state back to observation $h\\rightarrow s\\rightarrow o$ .", "Although histories and states tend to be somewhat correlated, both state aliasing and history aliasing make the roundtip conversion imperfect, causing a mismatch between the expected state value function $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ and the history value function ${V^\\pi }(h)$ in the general control case of a general POMDP.", "[Third proof, by example] This is a proof by example (with a proof by contradiction element).", "We will define the good/bad POMDP and, for a specific policy and history, first calculate $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ exactly, and then ${V^\\pi }(h)$ using bootstrapping (while also assuming ${V^\\pi }(hao) =_{s^{\\prime }\\mid hao}\\left[ {V^\\pi }(s^{\\prime }) \\right]$ ).", "We show that the two values are numerically different.", "In the good/bad POMDP, $= \\lbrace \\texttt {GOOD}, \\texttt {BAD}\\rbrace , = \\lbrace \\texttt {GOOD},\\texttt {BAD}\\rbrace $ , $= \\lbrace \\texttt {GOOD}, \\texttt {BAD}\\rbrace $ ; At times, we will use the shorthands $\\texttt {G}$ and $\\texttt {B}$ .", "The initial state distribution is uniform, and each state deterministically transitions into itself.", "The GOOD state always emits the GOOD observation, while the BAD state emits a random observation.", "Consider the reward function such that $R(s, a) = \\left[ a = \\texttt {GOOD}\\right]$ , i.e., the agent receives a reward whenever it choses the GOOD action.", "We will denote a history as the concatenation of alternating observations and actions, starting with an observation.", "To keep the notation compact, we will occasionally use symbols G and B to represent GOOD and BAD states, observations and actions.", "Consider a deterministic policy $\\pi (a; h) = \\left[ a = o_h \\right]$ which returns the action corresponding to the last observation.", "Note that this POMDP and this policy satisfy the requirements to guarantee that ${V^\\pi }(s)$ is well defined.", "Next, we calculate the state values ${V^\\pi }(s)$ .", "The $\\texttt {GOOD}$ state always emits the $\\texttt {GOOD}$ observation, so the agent will always choose the $\\texttt {GOOD}$ action and receive a reward of 1, then the state will always transition into itself, V(s=GOOD) = 1 + V(s=GOOD) = 11 -  .", "On the other hand, the $\\texttt {BAD}$ state will only emit the $\\texttt {GOOD}$ observation half of the times, so the agent will only choose the $\\texttt {GOOD}$ action and receive a reward of 1 half of the times, then the state will always transition into itself, V(s=BAD) = 12 + V(s=BAD) = 12(1 - )  .", "Next, we consider the history $h=\\texttt {G}$ after a single initial GOOD observation, and calculate the history value ${V^\\pi }(h)$ .", "Before proceeding, we need to calculate a few intermediate quantities, such as the belief-distribution: (s=Gh=G) (h=Gs=G) (s=G) = 1 12 = 12  , (s=Bh=G) (h=Gs=B) (s=B) = 12 12 = 14  , therefore (s=Gh=G) = 23  , (s=Bh=G) = 13  .", "We also calculate the belief-state distribution after two other histories.", "First $h=\\texttt {G}\\texttt {G}\\texttt {G}$ , (s=Gh=GGG) (h=GGGs=G) (s=G) = 1 12 = 12  , (s=Bh=GGG) (h=GGGs=B) (s=B) = 14 12 = 18  , therefore (s=Gh=GGG) = 45  , (s=Bh=GGG) = 15  .", "Then $h=\\texttt {G}\\texttt {G}\\texttt {B}$ , (s=Gh=GGB) (h=GGBs=G) (s=G) = 0 12 = 0  , (s=Bh=GGB) (h=GGBs=B) (s=B) = 14 12 = 18  , therefore (s=Gh=GGB) = 0  , (s=Bh=GGB) = 1  .", "We also need to calculate the observation emission probabilities, (o=Gh=G, a=G) = (s=Gh=G) (o=Gs=G) = + (s=Bh=G) (o=Gs=B) = 23 1 + 13 12 = 56  , (o=Bh=G, a=G) = (s=Gh=G) (o=Bs=G) = + (s=Bh=G) (o=Bs=B) = 23 0 + 13 12 = 16  .", "Next, we calculate ${V^\\pi }(h=\\texttt {G})$ under the assumption that the equality holds, V(h=G) = sh=G[ V(s) ] = (s=Gh=G) V(s=G) + (s=Bh=G) V(s=B) = 23 11- + 13 12(1-) = 56(1-)  .", "We can also apply the equality to other histories, V(h=GGG) = sh=GGG[ V(s) ] = (s=Gh=GGG) V(s=G) + (s=Bh=GGG) V(s=B) = 45 11- + 15 12(1-) = 910(1-)  , V(h=GGB) = sh=GGB[ V(s) ] = (s=Gh=GGB) V(s=G) + (s=Bh=GGB) V(s=B) = 0 11- + 1 12(1-) = 12(1-)  .", "Next, we calculate ${V^\\pi }(h=\\texttt {G})$ , this time by bootstrapping first, and then using the equality.", "Note that with the given history $h=\\texttt {G}$ , the agent will choose action $a=\\texttt {GOOD}$ .", "Then, V(h=G) = R(h=G, a=G) + oh=G, a=G[ V(hao=GGo) ] = 1 + ( $\\Pr (o=\\texttt {G}\\mid h=\\texttt {G}, a=\\texttt {G}) {V^\\pi }(hao=\\texttt {G}\\texttt {G}\\texttt {G}) \\\\+ \\Pr (o=\\texttt {B}\\mid h=\\texttt {G}, a=\\texttt {G}) {V^\\pi }(hao=\\texttt {G}\\texttt {G}\\texttt {B})$ ) = 1 + 56 910(1-) + 16 12(1-) = 60-6060(1-) + 4560(1-) + 560(1-) = 60-1060(1-) = 6-6(1-)  .", "The values from eq:proof:value:1,eq:proof:value:2 contradict each other, therefore, for this POMDP, policy, and history, ${V^\\pi }(h) \\ne _{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ .", "thm:vs.reactive.foTheorem REF If the POMDP states and observations are related by a bijection $\\phi \\colon \\rightarrow $ , and the policy is reactive, then ${V^\\pi }(s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ .", "Although there is no intrinsic state uncertainty, we continue to use probabilistic notation for notational consistency and simplicity.", "In the following derivation, we use the fact that states and observations contain the same system information to determine the first action and reward.", "This process can be repeated iteratively for all future actions and rewards (omitted, but represented by the ellipsis), sh[ V(s) ] = sh[ a (as) Q(s, a) ] = sh[ a (a; I(s)) Q(s, a) ] = sh[ a (a; oh) Q(s, a) ] = a (a; oh) sh[ Q(s, a) ] = a (a; oh) sh[ R(s, a) + s's, a[ V(s') ] ] = a (a; oh) ( R(h, a) + s'h, a[ V(s') ] ) = a (a; oh) ( R(h, a) + oh, a[ s'hao[ V(s') ] ] ) = $\\vdots $ = a (a; oh) ( R(h, a) + oh, a[ V(hao) ] ) = a (a; oh) Q(h, a) = V(h)  .", "In this case, the bijection between observations and states removes any error from the round-trip inference $h\\rightarrow s\\rightarrow o$ is perfect, which was the cause of the bias in thm:vs.reactive.po.", "thm:vhsTheorem REF For an arbitrary POMDP and policy, ${V^\\pi }(h, s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(h, s) \\right]$ Follows from eq:vh-raw,eq:vhs-raw, V(h0) = s,ah0[ k k R(sk, ak) ] = s0h0 s,ah0,s0[ k k R(sk, ak) ] = s0h0[ V(h0, s0) ]  .", "thm:asymmetric-policy-gradientTheorem REF (Asymmetric Policy Gradient) The policy gradient can be expressed using history-state values, $\\nabla _\\theta _\\text{policy}(\\theta ) = -\\left[ \\sum _t \\gamma ^t {Q^\\pi }(h_t,s_t, a_t) \\nabla _\\theta \\log \\pi (a_t; h_t) \\right] \\,.$ Following thm:vhs, we have Q(h, a) = R(h, a) + oh, a[ V(hao) ] = R(h, a) + oh, a[ s'h, a, o[ V(hao, s') ] ] = R(h, a) + s', oh, a[ V(hao, s') ] = sh[ R(s, a) + s', os, a[ V(hao, s') ] ] = sh[ Q(h, s, a) ]  .", "Therefore, policy() = -[ t t Q(ht, at) (at; ht) ] = -t t ht,at[ Q(ht, at) (at; ht) ] = -t t ht,at[ stht[ Q(ht, st, at) ] (at; ht) ] = -t t ht,st,at[ Q(ht, st, at) (at; ht) ] = -[ t t Q(ht, st, at) (at; ht) ]  .", "Environments Table: Environment properties.The environments used in the evaluation can be split into two groups, Shopping-5, Shopping-6, Heavenhell-3, Heavenhell-4 and Rocksample-5-6 are flat POMDPs, while Keydoor and Ninerooms are gridverse POMDPs.", "Flat POMDPs Figure: Layout of the Shopping environments.Figure: Layout of the Heavenhell environments.Figure: Layout of the Rocksample environment.In the flat POMDPshttps://github.com/abaisero/gym-pomdps, states, actions and observations are encoded by categorical indices with no inherent metric, which are intrinsically equally (dis)similar to each other.", "While it is not possible to generalize across states and observations via feature extraction, the primary challenge in these POMDPs is that of generalizing across different histories.", "Because the flat POMDPs are finite, their state, action and observation spaces have well-defined sizes, shown in tab:environments; note, however, that the size of the state space is not a significant measure of the complexity of partially observable tasks, while the time required to solve the task (i.e., history length) is a more relevant measure.", "Shopping This environment simulates an agent going to a shop to buy an item it forgot.", "The agent navigates a $5\\times 5$ or $6\\times 6$ gridworld trying to locate and select a randomly positioned item.", "The agent's position is fully observable, while the item's position is only observed when queried.", "fig:map:shopping depicts the gridworlds encoded by Shopping-5 and Shopping-6.", "States and Observations States encode the position of the agent and the position of the item in a single integer.", "Observations encode the position of the agent or the position of the item in a single integer.", "Actions Each time-step, the agent must choose an action from the set { LEFT, RIGHT, UP, DOWN, QUERY, BUY }.", "If the agent chooses the QUERY action, it observes the item's position, otherwise it observes its own position.", "To solve the task optimally, the agents needs to query the item's position and remember it, navigate to it, and then buy it.", "Rewards The agent receives the following reward signal: a reward of $-1.0$ for moving; a reward of $-2.0$ for performing a QUERY action; a reward of $-5.0$ for performing a BUY action in the wrong cell; and a reward of $10.0$ for performing a BUY action in the correct cell.", "Heavenhell The agent navigates a corridor-like gridworld composed of a fork and 3 dead-ends.", "Two dead-ends are exits which lead to heaven or hell, although the agent does not know which is which, while the third dead-end leads to a priest who can help the agent identify the heaven exit.", "fig:map:heavenhell depicts the gridworlds encoded by Heavenhell-3 and Heavenhell-4.", "States and Observations States encode the position of the agent and the position of the exit to heaven.", "Observations encode the position of the agent or the position of the exit to heaven.", "Actions Each time-step, the agent must choose an action from the set { NORTH, SOUTH, EAST, WEST }.", "If the agent is at the priest, it observes heaven's location, otherwise it observes its own position.", "To solve the task, the agent needs to navigate to the priest, then back to the fork, and on to heaven.", "Rewards The agent receives a sparse reward signal composed of: a reward of $1.0$ for exiting to heaven; and a reward of $-1.0$ for exiting to hell.", "Rocksample This environment simulates an agent navigating a landscape to collect research material.", "The agent navigates a $5\\times 5$ gridworld which contains 6 rocks, each having either good or bad research value.", "The agent's position is fully observable, while each rock's goodness is randomly sampled and unobserved unless the agent checks it.", "Each rock's goodness can be queried via a stochastic check action, whose reliability decays with the agent-rock distance.", "fig:map:rocksample depicts the gridworld encoded by Rocksample-5-6.", "States and Observations States encode the position of the agent and whether each rock is good or bad; the positions of the rocks are fixed so they do not need to be tracked by the state.", "Observations encode the position of the agent or whether the checked rock is good or bad.", "Actions Each time-step, the agent must choose an action from the set { NORTH, SOUTH, EAST, WEST, CHECK_0, CHECK_1, CHECK_2, CHECK_3, CHECK_4, CHECK_5, SAMPLE, }.", "If the agent chose a CHECK_* action, it observes whether the corresponding rock is good or bad, otherwise it observes its own position.", "To solve the task, the agent needs to sample all the good rocks (and none of the bad rocks), dynamically adapting its path depending on the stochastic information gained by checking each rock.", "Rewards The agent receives a sparse reward signal composed of: a reward of $10.0$ for sampling a good rock; and a reward of $-10.0$ for sampling a bad rock.", "Gridverse POMDPs Figure: KeydoorFigure: NineroomsIn the gridverse POMDPshttps://github.com/abaisero/gym-gridverse, actions are still encoded by categorical indices, while states and observations are encoded as structures which do have an inherent similarity metric; they are split into two components: A grid component: a $6\\times H\\times W$ volume of categorical indices which encode cell types, colors, statuses, agent presence, and the spatial relationships between them.", "The observation grid component is a slice of the corresponding state grid component made to match the agent's perspective: it is rotated to be a first-person view, and cells hidden behind walls are occluded, as shown in fig:keydoor:observation,fig:ninerooms:observation.", "An agent component: a 6-dimensional array of categorical indices representing the agent position and orientation, and the item held by the agent if any.", "The position and orientations in the state agent component are absolute, while those in the observation component are relative to the agent's perspective—they are essentially constant, and not necessary for control.", "Keydoor The agent navigates a $5\\times 5$ gridworld split into two rooms split by a wall and a locked door; on one side is the agent and a key which can be used to open the door, while on the other side is the goal.", "The positions of the agent, the key, the door, and the goal are randomly sampled such that two instances of the same problem are unlikely to be the same.", "fig:keydoor shows state and observation frames taken from an instance of the Keydoor environment.", "States and Observations The state grid component is a $6\\times 5\\times 5$ volume.", "The observation grid component is a $6\\times 7\\times 7$ volume.", "Actions Each time-step, the agent must choose an action from the set { MOVE_FORWARD, MOVE_BACKWARD, MOVE_LEFT, MOVE_RIGHT, TURN_LEFT, TURN_RIGHT, PICK_N_DROP, ACTUATE }.", "The MOVE_* actions result in a movement depending on the agent's orientation, while the TURN_* allows the agent to change its orientation.", "With the PICK_N_DROP action, the agent can pick and/or drop the key from/to the cell in front, while with the ACTUATE action, the agent can open and/or close the door.", "Rewards The agent receives a dense reward signal composed as the sum of the following terms: a living reward of $-0.05$ for every time-step; a reward of $0.2$ for stepping closer to the goal, and $-0.2$ for stepping further away (ignoring walls); a reward of $1.0$ for picking up the key, and $-1.0$ for dropping it; a reward of $1.0$ for opening the door, and $-1.0$ for closing it; and a reward of $5.0$ for reaching the goal.", "Ninerooms The agent navigates a $13\\times 13$ maze-like gridworld trying to locate and reach the goal.", "fig:ninerooms shows state and observation frames taken from an instance of the Ninerooms environment.", "States and Observations The state grid component is a $6\\times 13\\times 13$ volume.", "The observation grid component is a $6\\times 7\\times 7$ volume.", "Actions Each time-step, the agent must choose an action from the set { MOVE_FORWARD, MOVE_BACKWARD, MOVE_LEFT, MOVE_RIGHT, TURN_LEFT, TURN_RIGHT }.", "The MOVE_* actions result in a movement depending on the agent's orientation, while the TURN_* allows the agent to change its orientation.", "Rewards The agent receives a dense reward signal composed as the sum of the following terms: a living reward of $-0.05$ for every time-step; a reward of $0.2$ for stepping closer to the goal, and $-0.2$ for stepping further away (ignoring walls); and a reward of $5.0$ for reaching the goal.", "Architectures and Hyper-Parameters Figure: A2C architecture.", "The state, action, and observationrepresentations φ(s)\\phi (s), φ(a)\\phi (a), and φ(o)\\phi (o) are those shown infig:architecture:representations:flat orfig:architecture:representations:gridverse, depending on whetherthe type of enviroment.", "Dotted lines are present or omitted depending onwhether a history critic V ^(h){\\hat{V}}(h), state critic V ^(s){\\hat{V}}(s), orhistory-state critic V ^(h,s){\\hat{V}}(h, s) is being modeled.In this section, we describe the architectures used by the policy and critic models; a general overview is shown in fig:architecture.", "The general architecture will be similar for all domains; however there will some differences to accommodate the fact that the flat POMDPs provide states and observations as categorical indices, while the gridverse POMDPs provide states and observations as volumes and arrays of categorical indices, whose structures include spatial relationships.", "Features Extraction for Flat POMDPs These components are shown in fig:architecture:representations:flat.", "Because flat POMDPs provide states, actions and observations as categorical indices, and we use 128-dimensional embedding models to represent each of them.", "Features Extraction for Gridverse POMDPs These components are shown in fig:architecture:representations:gridverse.", "Because gridverse POMDPs provide actions as categorical indices, and we use 128-dimensional embedding models to represent them.", "On the other hand, states and observations are provided in the format described in sec:environments:gridverse, each composed of a grid and an agent component: grid The grid component is a $6\\times H\\times W$ volume of categorical data, which we pass through 4-dimensional embeddings, resulting in a $24\\times H\\times W$ volume of numeric data.", "We further pass that data to two layers of CNN followed by ReLU non-linearities, resulting in a $16\\times H\\times W$ volume of data, finally flattened into a $16 HW$ -dimensional array.", "agent The agent component is a 6-dimensional array of categorical data, which we pass through 4-dimensional embeddings, resulting in a 24-dimensional array of numeric data.", "The grid and agent representations are then concatenated to form the state or observation representation $\\phi (s)$ or $\\phi (o)$ .", "Remainder of the Architecture These components are shown in fig:architecture:a2c.", "Concatenated action and observation embeddings form the to a 128-dimensional single-layer gated recurrent unit (GRU) [3], which acts as a history representation $\\phi (h)$ .", "The history representation is then fed into separate policy and critic models, each a 2-layer 128-dimensional feedforward model with ReLU non-linearities.", "A practical issue we found with A2C(h,s) is that the history and state representations $\\phi (h)$ and $\\phi (s)$ contain features with very different orders of magniture; To address this, we use layer-normalization.", "To guarantee a fair comparison with the other methods, we do the same for other critics as well.", "Next, we describe architectural details specific to each method: A2C(h): The history representation passes through layer-normalization first, and then is fed to the critic model; A2C(s): The state embedding passes through layer-normalization first, and then is fed to the critic model; A2C(h,s): The history representation and state embedding individually pass through layer-normalization first, and then are concatenated to form a single feature vector, which is then fed to the critic model." ], [ "Unbiased Asymmetric Actor-Critic", "In this section, we introduce unbiased asymmetric actor-critic, an actor-critic variant which is able to exploit asymmetric state information during offline training while avoiding the issues of state value functions exposed in sec:aa2c.", "Consider a history-state value function ${V^\\pi }(h,s)$  [1], which represents the expected future discounted returns obtained when the history is $h$ and the state is $s$ , ${V^\\pi }(h{0}, s{0}) = _{\\bar{s}, \\bar{a} \\mid h{0},s{0}}\\left[ \\sum _{k=0}^\\infty \\gamma ^k R(s{k}, a{k})\\right] \\,, $ which supports an indirect recursive Bellman form, V(h, s) = a (a; h) Q(h, s, a)  , Q(h, s, a) = (s, a) + s',os,a[ V(hao, s') ]  .", "Providing the history information makes the history-state value function ${V^\\pi }(h, s)$ not only well-defined even for non-reactive policies, but also an unbiased estimate of ${V^\\pi }(h)$ .", "Theorem 5.1 For an arbitrary POMDP and policy, ${V^\\pi }(h, s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(h, s)\\right]$ .", "(proof in sec:proofs).", "As we have done for state values ${V^\\pi }(s)$ , we are interested in the properties of history-state values ${V^\\pi }(h, s)$ in relation to history values ${V^\\pi }(h)$ .", "thm:vhs shows that history and history-state values are related by ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(h, s) \\right]$ , i.e., history-state values are interpretable as Monte Carlo (MC) estimate of the respective history values.", "In expectation, history-state values provides the same information as the history values, therefore an asymmetric variant of the policy gradient theorem also holds.", "Theorem 5.2 (Asymmetric Policy Gradient) The policy gradient can be expressed using history-state values, $\\nabla _\\theta _\\text{policy}(\\theta ) = -\\left[ \\sum _t \\gamma ^t {Q^\\pi }(h_t,s_t, a_t) \\nabla _\\theta \\log \\pi (a_t; h_t) \\right] \\,.$ (proof in sec:proofs).", "As estimators, history-state values ${V^\\pi }(h, s)$ can be described in terms of their bias and variance w.r.t.", "history values ${V^\\pi }(h)$ .", "Beyond providing the inspiration for the MC interpretation, thm:vhs already proves that ${V^\\pi }(h, s)$ is unbiased, while its variance is dynamic and depends on the history $h$ via the belief-state $\\Pr (S\\mid h)$ ; in particular, low-uncertainty belief-states result in relatively low variance, and deterministic belief-states result in no variance.", "Given that operating optimally in a partially observable environment generally involves information-gathering strategies associated with low-uncertainty belief-states, the practical variance of the history-state value is likely to be relatively low once the agent has learned to solve the task to some degree of success.", "Inspired by thm:asymmetric-policy-gradient, we propose unbiased asymmetric A2C, which uses a history-state critic ${\\hat{V}}\\colon \\times \\rightarrow $ trained to model history-state values ${V^\\pi }(h, s)$ , policy() = -[ t t t (at; ht) ]  , t = R(st, at) + V(ht+1, st+1) - V(ht, st)  .", "Because ${\\hat{V}}(h, s)$ receives the history $h$ as input, it can still predict reasonable estimates of the agent's expected future discounted returns; and because it receives the state $s$ as input, it is still able to exploit state information while introducing no bias into the learning process, e.g., for the purposes of bootstrapping the learning of critic values and/or aiding the learning of history representations." ], [ "Interpretations of State", "Although the history-state value is analytically well-defined, It is worthwhile to question why the inclusion of the state information should help the actor-critic agent at all.", "We attempt to address this open question, and consider two competing interpretations, which we call state-as-information and state-as-a-feature." ], [ "State as Information", "Under this interpretation, state information is valuable because it is latent information unavailable in the history, which results in more informative values.", "However, this interpretation is flawed for two reasons: [label=()] The policy gradient theorem specifically requires ${V^\\pi }(h)$ , which contains precisely the correct information required to accurately estimate policy gradients.", "In this context, there is no such thing as “more informative values” than history values.", "In theory, the history-state value in thm:asymmetric-policy-gradient could use any other state sampled according to $\\tilde{s}\\sim b(h)$ , rather than the true system state, which would also result in the same analytical bias and variance properties.", "In practice, we use the true system state primarily due to it being directly available during simulation; however, we believe that its identity as the true system state is analytically irrelevant, which leads to the next interpretation of state.", "State as a Feature We conjecture an alternative interpretation according to which the state can be seen as a stochastic high-level feature of the history.", "Consider a history critic ${\\hat{V}}(h)$ to appropriately model the value function ${V^\\pi }(h)$ , the model must first learn an adequate history representation, which is in and of itself a significant learning challenge.", "The critic model would likely benefit from receiving auxiliary high-level black-box features of the history $\\phi (h)$ .", "The resulting critic ${\\hat{V}}(h,\\phi (h))$ remains fundamentally a history critic, the supplementary features being exclusively a modeling construct.", "Next, we consider what kind of high-level features $\\phi (h)$ would be useful for control.", "While the specifics of what makes a good history representation depend strongly on the task, there is a natural choice which is arguably useful in many cases: the belief-state $b(h)$ .", "Because the belief-state is a sufficient statistic of the history for control, providing it to the critic model ${\\hat{V}}(h, b(h))$ is likely to greatly improve its ability to generalize across histories.", "Finally, we conjecture that any state sampled according to the belief-state $s\\sim b(h)$ —including the true system state—can be considered a stochastic realization of the belief-state feature, resulting in the history-state critic ${\\hat{V}}(h, s)$ .", "According to this interpretation, the importance of the state in the history-state critic is not in its identity as the true system state, but as a stochastic realization of hypothetical belief-state features, and presumably any other state sampled from the belief-state $\\tilde{s}\\sim b(h)$ could be equivalently used.", "Evaluation [tb] Each method follows the same algorithm structure, but uses different types of critics to compute the TD errors $\\delta _t$ (see eq:tderror:h,eq:tderror:s,eq:tderror:hs).", "Values $N$ , $B$ , and $E$ vary by environment.", "Input: epochs $N$ , episode batch $B$ , evaluation period $E$ epoch in 1 ...$N$ training_episodes $\\leftarrow $ sample_episodes($\\pi $ , $B$ ) update $\\theta ,\\vartheta $ via $\\nabla ($ training_episodes$)$ (see eq:a2c) epoch $\\bmod $ evaluation_period = 0 evaluation_episodes $\\leftarrow $ sample_episodes($\\pi $ , $E$ ) report empirical_returns(evaluation_episodes) In this section, we empirically evaluate three actor-critic variants: A2C(h), standard actor-critic with a history critic ${\\hat{V}}(h)$ as described in sec:a2c; A2C(s), asymmetric actor-critic with a state critic ${\\hat{V}}(s)$ as described in sec:aa2c; and A2C(h,s), unbiased asymmetric actor-critic with a history-state critic ${\\hat{V}}(h,s)$ as described in sec:uaa2c.", "Each method is trained and evaluated according to alg:code.", "Broadly speaking, each method uses a gated recurrent unit (GRU) [3] to compute history features, and two feed-forward networks which represent the policy action probabilities and critic values; sec:architectures contains a more detailed description of the used architectures.", "We perform evaluations on seven gridworld environments which exhibit significant partial observability: Shopping-5, Shopping-6, Heavenhell-3, Heavenhell-4, and Rocksample-5-6 are flat POMDPshttps://github.com/abaisero/gym-pomdps, where states and observations are represented by categorical indices, while Keydoor and Ninerooms are gridverse POMDPshttps://github.com/abaisero/gym-gridverse, where states and observations are represented by tensors of categorical indices which encode spatial relationships and other cell information.", "See sec:environments for a detailed descriptions and graphical representations of all environments.", "Results and Discussion Each method is evaluated in one of two ways: [label=()] we show empirical learning curve statistics for all environments, and we show how critic values change for important history-state pairs over the course of training in Heavenhell-4.", "Learning Curves Figure: Learning curve statistics over 20 independent training runs, whereeach run is periodically evaluated 20 times.", "Shaded areas are centeredaround the empirical mean performance, and show 2 standard errors (of themean).fig:learningperformance depicts the performance results in all the environments.", "First, we note that the symmetric baseline A2C(h) does not always learn to solve the task, but succeeds fully in fig:shoppingv,fig:heavenhelliii,fig:gvkeydoor, partially in fig:heavenhelliv,fig:rocksamplevvi,fig:gvninerooms, and fails in fig:shoppingvi.", "The asymmetric baseline A2C(s) also performs inconsistently across environments, with a mixture of successful and failing cases.", "We particularly note the strange learning curves of A2C(s) in fig:shoppingv,fig:shoppingvi, where performance improves quickly during the early training, but fails to improve further or even becomes unstable and collapses later on.", "While the exact dynamics of this collapse are not completely clear, we believe it is likely that this is a consequence of modeling the critic after an analytically unstable state value function, making convergence to stable values impossible.", "Using our proposed history-state critic, A2C(h,s) consistently exhibits either faster convergence and/or higher final performance in virtually all the flat environments, and competitive performance in the gridverse environments, taking a mild lead towards the end of Ninerooms.", "These results strongly demonstrate the importance of exploiting asymmetric information in ways which are theoretically justified and sound, as done in our work.", "Critic Values Figure: Critic value statistics for 4 history-state pairs, evaluatedthroughout the training procedure via 20 independent runs.In the top row the agent did not visit the priest, while in the bottom rowthe agent did visit the priest.To further inspect the behavior of each critic, we show the evolution of critic values for important history-state pairs over the course of training.", "We perform this evaluation on Heavenhell-4, and use 4 deliberately chosen history-state pairs.", "In each case the agent is located at the fork, and the 4 cases differ according to heaven's location (left or right) and whether the agent has previously visited the priest.", "fig:criticvalues shows the resulting critic values, with one figure for each of the chosen history-state pairs.", "In each scenario, we note that all critic values exhibit convergence properties which resemble those of the agent's performance in fig:heavenhelliv.", "This empirically confirms that agent performance and critic quality are strongly correlated factors; although it is not possible to make conclusions about the causality in this situation, we strongly believe that it is the critics which act as a learning bottleneck on policies.", "Notably, the critics which focus on a single aspect of the joint history-state show the exact same values for different history-state; namely, A2C(s) is identical in the top and bottom plots, while A2C(h) is identical in the top-left and top-right plots.", "Although in practice left and right plots are similar for all critics, A2C(h,s) is the only critic capable of representing different values in each of the 4 scenarios, as none of its curves are identical.", "We also note that the state critic ${\\hat{V}}(s)$ is muct less stable than the others, and shows no signs of convergence, which is consistent with our analysis in sec:aa2c.", "In contract, the history-state critic exhibits good convergence properties despite itself also using state information, which is consistent with our analysis in sec:uaa2c.", "Finally, we note again that the history-state critic ${\\hat{V}}(h, s)$ converges significantly faster than the history critic ${\\hat{V}}(h)$ , confirming that state information is useful for training.", "Conclusions Asymmetric methods trained offline in simulated environments can use information which is normally unavailable in partially observable RL, such as the true system state.", "While the idea of exploiting such information has potential, current state-of-the-art methods are powered by empirical results rather than theoretical analysis.", "In this work, we exposed profound theoretical issues with a standard variant of asymmetric actor-critic, and proposed an unbiased asymmetric actor-critic variant which is analytically sound and theoretically justified.", "Empirical results confirm our analysis, the weaknesses of state critics and the strengths of history-state critics.", "Although we applied the history-state value function only to A2C, the same concepts are easily extensible to other critic-based RL methods [24], [15], [5], [19].", "In future work, we aim to extend the theory of history-state policy value functions ${Q^\\pi }(h, s, a)$ to optimal value functions $Q^*(h, s, a)$ , and develop theoretically sound asymmetric variants of other deep RL methods such as DQN [18], soft Q-learning [8], and soft actor-critic [9].", "We also plan to extend the theory and approach to multi-agent methods, potentially bringing theoretical rigor and improved performance [6], [16], [14], [27], [31], [22], [17], [23].", "Timed Value Functions sec:aa2c shows that for a general POMDP and policy $\\pi \\colon \\rightarrow \\Delta $ , the state value function ${V^\\pi }(s)$ is not generally well-defined due to issues caused by the lack of time information.", "In this section, we consider addressing the primary issue by providing explicit time-index information via timed value functions, $V_t^\\pi (s)$ and $Q_t^\\pi (s, a)$ , which represent the expected returns obtained when the agent finds itself in a state $s$ at time $t$ , Vt(s) = a (At=aSt=s) Qt(s, a)  , Qt(s, a) = R(s, a) + s's, a[ Vt+1(s') ]  .", "Once again, we analyze the state-dependent action distribution term to verify correctness, and expand it by integrating over histories; this time, we can use the explicit time-index information to integrate over histories of a given length only, $\\Pr (A_t=a\\mid S_t=s) = \\sum _{h\\in _t} \\Pr (H_t=h\\mid S_t=s) \\pi (a; h)\\,.", "$ Because eq:sumrule:timed is now restricted to histories of a given length $t$ , the probability term $\\Pr (H_t=h\\mid S_t=s)$ is well-defined, which means $\\Pr (A_t=a\\mid S_t=s)$ is well-defined, and $V_t^\\pi (s)$ is well-defined.", "Introducing the time-index makes the value function $V_t^\\pi (s)$ well-defined.", "However, its utility for the purpose of asymmetric reinforcement learning remains unclear because [label=()] it is still not formally proven whether the timed value function is unbiased, i.e., whether $V_t^\\pi (h) = _{s\\mid h}\\left[V_t^\\pi (s) \\right]$ , and it is harder for timed value critics to generalize appropriately across the additional discrete input $t$ .", "Proofs This section contains the proofs omitted from the main body of the document.", "For the sake of clarity, we repeat the main statements before showing the respective proof.", "thm:vsTheorem REF For an arbitrary POMDP and policy, a time-invariant state value function ${V^\\pi }(s)$ is ill-defined.", "The state value function, defined as ${V^\\pi }(s) = \\sum _{a\\in } \\Pr (a\\mid s) {Q^\\pi }(s, a) \\,,$ requires the state-conditioned action probability term $\\Pr (a\\mid s)$ .", "Because a partially observable policy depends on the history and not the state, the state-conditioned action probability term must be expanded by integrating over the space of all histories, $\\Pr (a\\mid s) = \\sum _{h\\in } \\Pr (h\\mid s) \\pi (a; h) \\,, $ which requires the state-conditioned history probability term $\\Pr (h\\mid s)$ .", "However, it is impossible to associate that term to well-defined RVs: Clearly, the reinforcement learning graphical model (see sec:graphmodel) does not define a time-less history RV $H$ , so the term cannot be explicitly written as $\\Pr (H=h\\mid S=s)$ .", "Further, the probability associated with timed RVs $\\Pr (H_t=h\\mid S_t=s)$ is clearly not time-invariant, hence identifying the time-index is crucial.", "Finally, it is not possible to restrict the integration in eq:pras:app only to histories of a given time-index.", "Overall, the probability term $\\Pr (h\\mid s)$ is necessarily time-variant, which means that $\\Pr (a\\mid s)$ is itself time-variant, and therefore the state value function ${V^\\pi }(s)$ is time-variant, and a time-invariant state value function ${V^\\pi }(s)$ is ill-defined.", "thm:vs.reactive.poTheorem REF If the POMDP observation function depends only on the current state, $\\colon \\rightarrow \\Delta $ , and the policy is reactive, then ${V^\\pi }(s)$ is not necessarily an unbiased estimate of ${V^\\pi }(h)$ , i.e., it is not guaranteed that ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s)\\right]$ .", "As one of the salient theorems of our work, we will cover it from different angles and provide three proofs: first one which is simple but does not go into the detail about the causes of the bias, then a more detailed and analytical one, and finally one by example.", "[First proof] Consider two histories $h, h^{\\prime }\\in $ which are different, $h\\ne h^{\\prime }$ , but are associated with the same belief $b(h) = b(h^{\\prime })$ ; a fairly common occurrence in many POMDPs.", "On one hand, because the two histories are different, the agent's next action may differ, which may then lead to different immediate rewards and future trajectories, i.e., the respective values would also differ, ${V^\\pi }(h) \\ne {V^\\pi }(h^{\\prime }) \\,.$ On the other hand, because the two beliefs are the same, then the expected state values must be the same, $_{s\\mid h}\\left[ {V^\\pi }(s) \\right] = _{s\\mid h^{\\prime }}\\left[ {V^\\pi }(s)\\right] \\,.$ Therefore, it is not guaranteed that ${V^\\pi }(h) = _{s\\mid h}\\left[{V^\\pi }(s) \\right]$ .", "[Second proof, by contradiction] First, we assume that ${V^\\pi }(s)$ is unbiased and show that ${Q^\\pi }(s,a)$ (as defined by eq:qsa) is unbiased, sh[ Q(s, a) ] = sh[ (s, a) + s's, a[ V(s') ] ] = sh[ (s, a) ] + sh[ s's, a[ V(s') ] ] = sh[ (s, a) ] + s'h, a[ V(s') ] = sh[ (s, a) ] + oh, a s'hao[ V(s') ] = (h, a) + oh, a[ V(hao) ] = Q(h, a)  .", "Next, we show that even if ${Q^\\pi }(s, a)$ is unbiased, ${V^\\pi }(s)$ (as defined by eq:vs) is biased, which contradicts the original assumption.", "To do that, we expand the expected state value function $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ and the history value function ${V^\\pi }(h)$ , and show that there is a concrete difference between them: sh[ V(s) ] = sh[ a (as) Q(s, a) ] = sh[ a os[ (a; o) ] Q(s, a) ]  , V(h) = a (a; oh) Q(h, a) = a (a; oh) sh[ Q(s, a) ] = sh[ a os[ (a; oh) ] Q(s, a) ]  .", "eq:proof:vs,eq:proof:vh differ in terms of which observation is used by the policy; in eq:proof:vs, an observation $o$ inferred from a state $s$ inferred from the history $h$ is used, while in eq:proof:vh the final observation $o_h$ of the history $h$ is used.", "These two observations $o$ and $o_h$ are not generally the same, and the respective expectations are similarly not generally the same.", "The nested expectation in eq:proof:vs can be interpreted as a lossy round-trip inference from history to state and from state back to observation $h\\rightarrow s\\rightarrow o$ .", "Although histories and states tend to be somewhat correlated, both state aliasing and history aliasing make the roundtip conversion imperfect, causing a mismatch between the expected state value function $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ and the history value function ${V^\\pi }(h)$ in the general control case of a general POMDP.", "[Third proof, by example] This is a proof by example (with a proof by contradiction element).", "We will define the good/bad POMDP and, for a specific policy and history, first calculate $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ exactly, and then ${V^\\pi }(h)$ using bootstrapping (while also assuming ${V^\\pi }(hao) =_{s^{\\prime }\\mid hao}\\left[ {V^\\pi }(s^{\\prime }) \\right]$ ).", "We show that the two values are numerically different.", "In the good/bad POMDP, $= \\lbrace \\texttt {GOOD}, \\texttt {BAD}\\rbrace , = \\lbrace \\texttt {GOOD},\\texttt {BAD}\\rbrace $ , $= \\lbrace \\texttt {GOOD}, \\texttt {BAD}\\rbrace $ ; At times, we will use the shorthands $\\texttt {G}$ and $\\texttt {B}$ .", "The initial state distribution is uniform, and each state deterministically transitions into itself.", "The GOOD state always emits the GOOD observation, while the BAD state emits a random observation.", "Consider the reward function such that $R(s, a) = \\left[ a = \\texttt {GOOD}\\right]$ , i.e., the agent receives a reward whenever it choses the GOOD action.", "We will denote a history as the concatenation of alternating observations and actions, starting with an observation.", "To keep the notation compact, we will occasionally use symbols G and B to represent GOOD and BAD states, observations and actions.", "Consider a deterministic policy $\\pi (a; h) = \\left[ a = o_h \\right]$ which returns the action corresponding to the last observation.", "Note that this POMDP and this policy satisfy the requirements to guarantee that ${V^\\pi }(s)$ is well defined.", "Next, we calculate the state values ${V^\\pi }(s)$ .", "The $\\texttt {GOOD}$ state always emits the $\\texttt {GOOD}$ observation, so the agent will always choose the $\\texttt {GOOD}$ action and receive a reward of 1, then the state will always transition into itself, V(s=GOOD) = 1 + V(s=GOOD) = 11 -  .", "On the other hand, the $\\texttt {BAD}$ state will only emit the $\\texttt {GOOD}$ observation half of the times, so the agent will only choose the $\\texttt {GOOD}$ action and receive a reward of 1 half of the times, then the state will always transition into itself, V(s=BAD) = 12 + V(s=BAD) = 12(1 - )  .", "Next, we consider the history $h=\\texttt {G}$ after a single initial GOOD observation, and calculate the history value ${V^\\pi }(h)$ .", "Before proceeding, we need to calculate a few intermediate quantities, such as the belief-distribution: (s=Gh=G) (h=Gs=G) (s=G) = 1 12 = 12  , (s=Bh=G) (h=Gs=B) (s=B) = 12 12 = 14  , therefore (s=Gh=G) = 23  , (s=Bh=G) = 13  .", "We also calculate the belief-state distribution after two other histories.", "First $h=\\texttt {G}\\texttt {G}\\texttt {G}$ , (s=Gh=GGG) (h=GGGs=G) (s=G) = 1 12 = 12  , (s=Bh=GGG) (h=GGGs=B) (s=B) = 14 12 = 18  , therefore (s=Gh=GGG) = 45  , (s=Bh=GGG) = 15  .", "Then $h=\\texttt {G}\\texttt {G}\\texttt {B}$ , (s=Gh=GGB) (h=GGBs=G) (s=G) = 0 12 = 0  , (s=Bh=GGB) (h=GGBs=B) (s=B) = 14 12 = 18  , therefore (s=Gh=GGB) = 0  , (s=Bh=GGB) = 1  .", "We also need to calculate the observation emission probabilities, (o=Gh=G, a=G) = (s=Gh=G) (o=Gs=G) = + (s=Bh=G) (o=Gs=B) = 23 1 + 13 12 = 56  , (o=Bh=G, a=G) = (s=Gh=G) (o=Bs=G) = + (s=Bh=G) (o=Bs=B) = 23 0 + 13 12 = 16  .", "Next, we calculate ${V^\\pi }(h=\\texttt {G})$ under the assumption that the equality holds, V(h=G) = sh=G[ V(s) ] = (s=Gh=G) V(s=G) + (s=Bh=G) V(s=B) = 23 11- + 13 12(1-) = 56(1-)  .", "We can also apply the equality to other histories, V(h=GGG) = sh=GGG[ V(s) ] = (s=Gh=GGG) V(s=G) + (s=Bh=GGG) V(s=B) = 45 11- + 15 12(1-) = 910(1-)  , V(h=GGB) = sh=GGB[ V(s) ] = (s=Gh=GGB) V(s=G) + (s=Bh=GGB) V(s=B) = 0 11- + 1 12(1-) = 12(1-)  .", "Next, we calculate ${V^\\pi }(h=\\texttt {G})$ , this time by bootstrapping first, and then using the equality.", "Note that with the given history $h=\\texttt {G}$ , the agent will choose action $a=\\texttt {GOOD}$ .", "Then, V(h=G) = R(h=G, a=G) + oh=G, a=G[ V(hao=GGo) ] = 1 + ( $\\Pr (o=\\texttt {G}\\mid h=\\texttt {G}, a=\\texttt {G}) {V^\\pi }(hao=\\texttt {G}\\texttt {G}\\texttt {G}) \\\\+ \\Pr (o=\\texttt {B}\\mid h=\\texttt {G}, a=\\texttt {G}) {V^\\pi }(hao=\\texttt {G}\\texttt {G}\\texttt {B})$ ) = 1 + 56 910(1-) + 16 12(1-) = 60-6060(1-) + 4560(1-) + 560(1-) = 60-1060(1-) = 6-6(1-)  .", "The values from eq:proof:value:1,eq:proof:value:2 contradict each other, therefore, for this POMDP, policy, and history, ${V^\\pi }(h) \\ne _{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ .", "thm:vs.reactive.foTheorem REF If the POMDP states and observations are related by a bijection $\\phi \\colon \\rightarrow $ , and the policy is reactive, then ${V^\\pi }(s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ .", "Although there is no intrinsic state uncertainty, we continue to use probabilistic notation for notational consistency and simplicity.", "In the following derivation, we use the fact that states and observations contain the same system information to determine the first action and reward.", "This process can be repeated iteratively for all future actions and rewards (omitted, but represented by the ellipsis), sh[ V(s) ] = sh[ a (as) Q(s, a) ] = sh[ a (a; I(s)) Q(s, a) ] = sh[ a (a; oh) Q(s, a) ] = a (a; oh) sh[ Q(s, a) ] = a (a; oh) sh[ R(s, a) + s's, a[ V(s') ] ] = a (a; oh) ( R(h, a) + s'h, a[ V(s') ] ) = a (a; oh) ( R(h, a) + oh, a[ s'hao[ V(s') ] ] ) = $\\vdots $ = a (a; oh) ( R(h, a) + oh, a[ V(hao) ] ) = a (a; oh) Q(h, a) = V(h)  .", "In this case, the bijection between observations and states removes any error from the round-trip inference $h\\rightarrow s\\rightarrow o$ is perfect, which was the cause of the bias in thm:vs.reactive.po.", "thm:vhsTheorem REF For an arbitrary POMDP and policy, ${V^\\pi }(h, s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(h, s) \\right]$ Follows from eq:vh-raw,eq:vhs-raw, V(h0) = s,ah0[ k k R(sk, ak) ] = s0h0 s,ah0,s0[ k k R(sk, ak) ] = s0h0[ V(h0, s0) ]  .", "thm:asymmetric-policy-gradientTheorem REF (Asymmetric Policy Gradient) The policy gradient can be expressed using history-state values, $\\nabla _\\theta _\\text{policy}(\\theta ) = -\\left[ \\sum _t \\gamma ^t {Q^\\pi }(h_t,s_t, a_t) \\nabla _\\theta \\log \\pi (a_t; h_t) \\right] \\,.$ Following thm:vhs, we have Q(h, a) = R(h, a) + oh, a[ V(hao) ] = R(h, a) + oh, a[ s'h, a, o[ V(hao, s') ] ] = R(h, a) + s', oh, a[ V(hao, s') ] = sh[ R(s, a) + s', os, a[ V(hao, s') ] ] = sh[ Q(h, s, a) ]  .", "Therefore, policy() = -[ t t Q(ht, at) (at; ht) ] = -t t ht,at[ Q(ht, at) (at; ht) ] = -t t ht,at[ stht[ Q(ht, st, at) ] (at; ht) ] = -t t ht,st,at[ Q(ht, st, at) (at; ht) ] = -[ t t Q(ht, st, at) (at; ht) ]  .", "Environments Table: Environment properties.The environments used in the evaluation can be split into two groups, Shopping-5, Shopping-6, Heavenhell-3, Heavenhell-4 and Rocksample-5-6 are flat POMDPs, while Keydoor and Ninerooms are gridverse POMDPs.", "Flat POMDPs Figure: Layout of the Shopping environments.Figure: Layout of the Heavenhell environments.Figure: Layout of the Rocksample environment.In the flat POMDPshttps://github.com/abaisero/gym-pomdps, states, actions and observations are encoded by categorical indices with no inherent metric, which are intrinsically equally (dis)similar to each other.", "While it is not possible to generalize across states and observations via feature extraction, the primary challenge in these POMDPs is that of generalizing across different histories.", "Because the flat POMDPs are finite, their state, action and observation spaces have well-defined sizes, shown in tab:environments; note, however, that the size of the state space is not a significant measure of the complexity of partially observable tasks, while the time required to solve the task (i.e., history length) is a more relevant measure.", "Shopping This environment simulates an agent going to a shop to buy an item it forgot.", "The agent navigates a $5\\times 5$ or $6\\times 6$ gridworld trying to locate and select a randomly positioned item.", "The agent's position is fully observable, while the item's position is only observed when queried.", "fig:map:shopping depicts the gridworlds encoded by Shopping-5 and Shopping-6.", "States and Observations States encode the position of the agent and the position of the item in a single integer.", "Observations encode the position of the agent or the position of the item in a single integer.", "Actions Each time-step, the agent must choose an action from the set { LEFT, RIGHT, UP, DOWN, QUERY, BUY }.", "If the agent chooses the QUERY action, it observes the item's position, otherwise it observes its own position.", "To solve the task optimally, the agents needs to query the item's position and remember it, navigate to it, and then buy it.", "Rewards The agent receives the following reward signal: a reward of $-1.0$ for moving; a reward of $-2.0$ for performing a QUERY action; a reward of $-5.0$ for performing a BUY action in the wrong cell; and a reward of $10.0$ for performing a BUY action in the correct cell.", "Heavenhell The agent navigates a corridor-like gridworld composed of a fork and 3 dead-ends.", "Two dead-ends are exits which lead to heaven or hell, although the agent does not know which is which, while the third dead-end leads to a priest who can help the agent identify the heaven exit.", "fig:map:heavenhell depicts the gridworlds encoded by Heavenhell-3 and Heavenhell-4.", "States and Observations States encode the position of the agent and the position of the exit to heaven.", "Observations encode the position of the agent or the position of the exit to heaven.", "Actions Each time-step, the agent must choose an action from the set { NORTH, SOUTH, EAST, WEST }.", "If the agent is at the priest, it observes heaven's location, otherwise it observes its own position.", "To solve the task, the agent needs to navigate to the priest, then back to the fork, and on to heaven.", "Rewards The agent receives a sparse reward signal composed of: a reward of $1.0$ for exiting to heaven; and a reward of $-1.0$ for exiting to hell.", "Rocksample This environment simulates an agent navigating a landscape to collect research material.", "The agent navigates a $5\\times 5$ gridworld which contains 6 rocks, each having either good or bad research value.", "The agent's position is fully observable, while each rock's goodness is randomly sampled and unobserved unless the agent checks it.", "Each rock's goodness can be queried via a stochastic check action, whose reliability decays with the agent-rock distance.", "fig:map:rocksample depicts the gridworld encoded by Rocksample-5-6.", "States and Observations States encode the position of the agent and whether each rock is good or bad; the positions of the rocks are fixed so they do not need to be tracked by the state.", "Observations encode the position of the agent or whether the checked rock is good or bad.", "Actions Each time-step, the agent must choose an action from the set { NORTH, SOUTH, EAST, WEST, CHECK_0, CHECK_1, CHECK_2, CHECK_3, CHECK_4, CHECK_5, SAMPLE, }.", "If the agent chose a CHECK_* action, it observes whether the corresponding rock is good or bad, otherwise it observes its own position.", "To solve the task, the agent needs to sample all the good rocks (and none of the bad rocks), dynamically adapting its path depending on the stochastic information gained by checking each rock.", "Rewards The agent receives a sparse reward signal composed of: a reward of $10.0$ for sampling a good rock; and a reward of $-10.0$ for sampling a bad rock.", "Gridverse POMDPs Figure: KeydoorFigure: NineroomsIn the gridverse POMDPshttps://github.com/abaisero/gym-gridverse, actions are still encoded by categorical indices, while states and observations are encoded as structures which do have an inherent similarity metric; they are split into two components: A grid component: a $6\\times H\\times W$ volume of categorical indices which encode cell types, colors, statuses, agent presence, and the spatial relationships between them.", "The observation grid component is a slice of the corresponding state grid component made to match the agent's perspective: it is rotated to be a first-person view, and cells hidden behind walls are occluded, as shown in fig:keydoor:observation,fig:ninerooms:observation.", "An agent component: a 6-dimensional array of categorical indices representing the agent position and orientation, and the item held by the agent if any.", "The position and orientations in the state agent component are absolute, while those in the observation component are relative to the agent's perspective—they are essentially constant, and not necessary for control.", "Keydoor The agent navigates a $5\\times 5$ gridworld split into two rooms split by a wall and a locked door; on one side is the agent and a key which can be used to open the door, while on the other side is the goal.", "The positions of the agent, the key, the door, and the goal are randomly sampled such that two instances of the same problem are unlikely to be the same.", "fig:keydoor shows state and observation frames taken from an instance of the Keydoor environment.", "States and Observations The state grid component is a $6\\times 5\\times 5$ volume.", "The observation grid component is a $6\\times 7\\times 7$ volume.", "Actions Each time-step, the agent must choose an action from the set { MOVE_FORWARD, MOVE_BACKWARD, MOVE_LEFT, MOVE_RIGHT, TURN_LEFT, TURN_RIGHT, PICK_N_DROP, ACTUATE }.", "The MOVE_* actions result in a movement depending on the agent's orientation, while the TURN_* allows the agent to change its orientation.", "With the PICK_N_DROP action, the agent can pick and/or drop the key from/to the cell in front, while with the ACTUATE action, the agent can open and/or close the door.", "Rewards The agent receives a dense reward signal composed as the sum of the following terms: a living reward of $-0.05$ for every time-step; a reward of $0.2$ for stepping closer to the goal, and $-0.2$ for stepping further away (ignoring walls); a reward of $1.0$ for picking up the key, and $-1.0$ for dropping it; a reward of $1.0$ for opening the door, and $-1.0$ for closing it; and a reward of $5.0$ for reaching the goal.", "Ninerooms The agent navigates a $13\\times 13$ maze-like gridworld trying to locate and reach the goal.", "fig:ninerooms shows state and observation frames taken from an instance of the Ninerooms environment.", "States and Observations The state grid component is a $6\\times 13\\times 13$ volume.", "The observation grid component is a $6\\times 7\\times 7$ volume.", "Actions Each time-step, the agent must choose an action from the set { MOVE_FORWARD, MOVE_BACKWARD, MOVE_LEFT, MOVE_RIGHT, TURN_LEFT, TURN_RIGHT }.", "The MOVE_* actions result in a movement depending on the agent's orientation, while the TURN_* allows the agent to change its orientation.", "Rewards The agent receives a dense reward signal composed as the sum of the following terms: a living reward of $-0.05$ for every time-step; a reward of $0.2$ for stepping closer to the goal, and $-0.2$ for stepping further away (ignoring walls); and a reward of $5.0$ for reaching the goal.", "Architectures and Hyper-Parameters Figure: A2C architecture.", "The state, action, and observationrepresentations φ(s)\\phi (s), φ(a)\\phi (a), and φ(o)\\phi (o) are those shown infig:architecture:representations:flat orfig:architecture:representations:gridverse, depending on whetherthe type of enviroment.", "Dotted lines are present or omitted depending onwhether a history critic V ^(h){\\hat{V}}(h), state critic V ^(s){\\hat{V}}(s), orhistory-state critic V ^(h,s){\\hat{V}}(h, s) is being modeled.In this section, we describe the architectures used by the policy and critic models; a general overview is shown in fig:architecture.", "The general architecture will be similar for all domains; however there will some differences to accommodate the fact that the flat POMDPs provide states and observations as categorical indices, while the gridverse POMDPs provide states and observations as volumes and arrays of categorical indices, whose structures include spatial relationships.", "Features Extraction for Flat POMDPs These components are shown in fig:architecture:representations:flat.", "Because flat POMDPs provide states, actions and observations as categorical indices, and we use 128-dimensional embedding models to represent each of them.", "Features Extraction for Gridverse POMDPs These components are shown in fig:architecture:representations:gridverse.", "Because gridverse POMDPs provide actions as categorical indices, and we use 128-dimensional embedding models to represent them.", "On the other hand, states and observations are provided in the format described in sec:environments:gridverse, each composed of a grid and an agent component: grid The grid component is a $6\\times H\\times W$ volume of categorical data, which we pass through 4-dimensional embeddings, resulting in a $24\\times H\\times W$ volume of numeric data.", "We further pass that data to two layers of CNN followed by ReLU non-linearities, resulting in a $16\\times H\\times W$ volume of data, finally flattened into a $16 HW$ -dimensional array.", "agent The agent component is a 6-dimensional array of categorical data, which we pass through 4-dimensional embeddings, resulting in a 24-dimensional array of numeric data.", "The grid and agent representations are then concatenated to form the state or observation representation $\\phi (s)$ or $\\phi (o)$ .", "Remainder of the Architecture These components are shown in fig:architecture:a2c.", "Concatenated action and observation embeddings form the to a 128-dimensional single-layer gated recurrent unit (GRU) [3], which acts as a history representation $\\phi (h)$ .", "The history representation is then fed into separate policy and critic models, each a 2-layer 128-dimensional feedforward model with ReLU non-linearities.", "A practical issue we found with A2C(h,s) is that the history and state representations $\\phi (h)$ and $\\phi (s)$ contain features with very different orders of magniture; To address this, we use layer-normalization.", "To guarantee a fair comparison with the other methods, we do the same for other critics as well.", "Next, we describe architectural details specific to each method: A2C(h): The history representation passes through layer-normalization first, and then is fed to the critic model; A2C(s): The state embedding passes through layer-normalization first, and then is fed to the critic model; A2C(h,s): The history representation and state embedding individually pass through layer-normalization first, and then are concatenated to form a single feature vector, which is then fed to the critic model." ], [ "Evaluation", "[tb] Each method follows the same algorithm structure, but uses different types of critics to compute the TD errors $\\delta _t$ (see eq:tderror:h,eq:tderror:s,eq:tderror:hs).", "Values $N$ , $B$ , and $E$ vary by environment.", "Input: epochs $N$ , episode batch $B$ , evaluation period $E$ epoch in 1 ...$N$ training_episodes $\\leftarrow $ sample_episodes($\\pi $ , $B$ ) update $\\theta ,\\vartheta $ via $\\nabla ($ training_episodes$)$ (see eq:a2c) epoch $\\bmod $ evaluation_period = 0 evaluation_episodes $\\leftarrow $ sample_episodes($\\pi $ , $E$ ) report empirical_returns(evaluation_episodes) In this section, we empirically evaluate three actor-critic variants: A2C(h), standard actor-critic with a history critic ${\\hat{V}}(h)$ as described in sec:a2c; A2C(s), asymmetric actor-critic with a state critic ${\\hat{V}}(s)$ as described in sec:aa2c; and A2C(h,s), unbiased asymmetric actor-critic with a history-state critic ${\\hat{V}}(h,s)$ as described in sec:uaa2c.", "Each method is trained and evaluated according to alg:code.", "Broadly speaking, each method uses a gated recurrent unit (GRU) [3] to compute history features, and two feed-forward networks which represent the policy action probabilities and critic values; sec:architectures contains a more detailed description of the used architectures.", "We perform evaluations on seven gridworld environments which exhibit significant partial observability: Shopping-5, Shopping-6, Heavenhell-3, Heavenhell-4, and Rocksample-5-6 are flat POMDPshttps://github.com/abaisero/gym-pomdps, where states and observations are represented by categorical indices, while Keydoor and Ninerooms are gridverse POMDPshttps://github.com/abaisero/gym-gridverse, where states and observations are represented by tensors of categorical indices which encode spatial relationships and other cell information.", "See sec:environments for a detailed descriptions and graphical representations of all environments." ], [ "Results and Discussion", "Each method is evaluated in one of two ways: [label=()] we show empirical learning curve statistics for all environments, and we show how critic values change for important history-state pairs over the course of training in Heavenhell-4.", "Learning Curves Figure: Learning curve statistics over 20 independent training runs, whereeach run is periodically evaluated 20 times.", "Shaded areas are centeredaround the empirical mean performance, and show 2 standard errors (of themean).fig:learningperformance depicts the performance results in all the environments.", "First, we note that the symmetric baseline A2C(h) does not always learn to solve the task, but succeeds fully in fig:shoppingv,fig:heavenhelliii,fig:gvkeydoor, partially in fig:heavenhelliv,fig:rocksamplevvi,fig:gvninerooms, and fails in fig:shoppingvi.", "The asymmetric baseline A2C(s) also performs inconsistently across environments, with a mixture of successful and failing cases.", "We particularly note the strange learning curves of A2C(s) in fig:shoppingv,fig:shoppingvi, where performance improves quickly during the early training, but fails to improve further or even becomes unstable and collapses later on.", "While the exact dynamics of this collapse are not completely clear, we believe it is likely that this is a consequence of modeling the critic after an analytically unstable state value function, making convergence to stable values impossible.", "Using our proposed history-state critic, A2C(h,s) consistently exhibits either faster convergence and/or higher final performance in virtually all the flat environments, and competitive performance in the gridverse environments, taking a mild lead towards the end of Ninerooms.", "These results strongly demonstrate the importance of exploiting asymmetric information in ways which are theoretically justified and sound, as done in our work.", "Critic Values Figure: Critic value statistics for 4 history-state pairs, evaluatedthroughout the training procedure via 20 independent runs.In the top row the agent did not visit the priest, while in the bottom rowthe agent did visit the priest.To further inspect the behavior of each critic, we show the evolution of critic values for important history-state pairs over the course of training.", "We perform this evaluation on Heavenhell-4, and use 4 deliberately chosen history-state pairs.", "In each case the agent is located at the fork, and the 4 cases differ according to heaven's location (left or right) and whether the agent has previously visited the priest.", "fig:criticvalues shows the resulting critic values, with one figure for each of the chosen history-state pairs.", "In each scenario, we note that all critic values exhibit convergence properties which resemble those of the agent's performance in fig:heavenhelliv.", "This empirically confirms that agent performance and critic quality are strongly correlated factors; although it is not possible to make conclusions about the causality in this situation, we strongly believe that it is the critics which act as a learning bottleneck on policies.", "Notably, the critics which focus on a single aspect of the joint history-state show the exact same values for different history-state; namely, A2C(s) is identical in the top and bottom plots, while A2C(h) is identical in the top-left and top-right plots.", "Although in practice left and right plots are similar for all critics, A2C(h,s) is the only critic capable of representing different values in each of the 4 scenarios, as none of its curves are identical.", "We also note that the state critic ${\\hat{V}}(s)$ is muct less stable than the others, and shows no signs of convergence, which is consistent with our analysis in sec:aa2c.", "In contract, the history-state critic exhibits good convergence properties despite itself also using state information, which is consistent with our analysis in sec:uaa2c.", "Finally, we note again that the history-state critic ${\\hat{V}}(h, s)$ converges significantly faster than the history critic ${\\hat{V}}(h)$ , confirming that state information is useful for training.", "Conclusions Asymmetric methods trained offline in simulated environments can use information which is normally unavailable in partially observable RL, such as the true system state.", "While the idea of exploiting such information has potential, current state-of-the-art methods are powered by empirical results rather than theoretical analysis.", "In this work, we exposed profound theoretical issues with a standard variant of asymmetric actor-critic, and proposed an unbiased asymmetric actor-critic variant which is analytically sound and theoretically justified.", "Empirical results confirm our analysis, the weaknesses of state critics and the strengths of history-state critics.", "Although we applied the history-state value function only to A2C, the same concepts are easily extensible to other critic-based RL methods [24], [15], [5], [19].", "In future work, we aim to extend the theory of history-state policy value functions ${Q^\\pi }(h, s, a)$ to optimal value functions $Q^*(h, s, a)$ , and develop theoretically sound asymmetric variants of other deep RL methods such as DQN [18], soft Q-learning [8], and soft actor-critic [9].", "We also plan to extend the theory and approach to multi-agent methods, potentially bringing theoretical rigor and improved performance [6], [16], [14], [27], [31], [22], [17], [23].", "Timed Value Functions sec:aa2c shows that for a general POMDP and policy $\\pi \\colon \\rightarrow \\Delta $ , the state value function ${V^\\pi }(s)$ is not generally well-defined due to issues caused by the lack of time information.", "In this section, we consider addressing the primary issue by providing explicit time-index information via timed value functions, $V_t^\\pi (s)$ and $Q_t^\\pi (s, a)$ , which represent the expected returns obtained when the agent finds itself in a state $s$ at time $t$ , Vt(s) = a (At=aSt=s) Qt(s, a)  , Qt(s, a) = R(s, a) + s's, a[ Vt+1(s') ]  .", "Once again, we analyze the state-dependent action distribution term to verify correctness, and expand it by integrating over histories; this time, we can use the explicit time-index information to integrate over histories of a given length only, $\\Pr (A_t=a\\mid S_t=s) = \\sum _{h\\in _t} \\Pr (H_t=h\\mid S_t=s) \\pi (a; h)\\,.", "$ Because eq:sumrule:timed is now restricted to histories of a given length $t$ , the probability term $\\Pr (H_t=h\\mid S_t=s)$ is well-defined, which means $\\Pr (A_t=a\\mid S_t=s)$ is well-defined, and $V_t^\\pi (s)$ is well-defined.", "Introducing the time-index makes the value function $V_t^\\pi (s)$ well-defined.", "However, its utility for the purpose of asymmetric reinforcement learning remains unclear because [label=()] it is still not formally proven whether the timed value function is unbiased, i.e., whether $V_t^\\pi (h) = _{s\\mid h}\\left[V_t^\\pi (s) \\right]$ , and it is harder for timed value critics to generalize appropriately across the additional discrete input $t$ .", "Proofs This section contains the proofs omitted from the main body of the document.", "For the sake of clarity, we repeat the main statements before showing the respective proof.", "thm:vsTheorem REF For an arbitrary POMDP and policy, a time-invariant state value function ${V^\\pi }(s)$ is ill-defined.", "The state value function, defined as ${V^\\pi }(s) = \\sum _{a\\in } \\Pr (a\\mid s) {Q^\\pi }(s, a) \\,,$ requires the state-conditioned action probability term $\\Pr (a\\mid s)$ .", "Because a partially observable policy depends on the history and not the state, the state-conditioned action probability term must be expanded by integrating over the space of all histories, $\\Pr (a\\mid s) = \\sum _{h\\in } \\Pr (h\\mid s) \\pi (a; h) \\,, $ which requires the state-conditioned history probability term $\\Pr (h\\mid s)$ .", "However, it is impossible to associate that term to well-defined RVs: Clearly, the reinforcement learning graphical model (see sec:graphmodel) does not define a time-less history RV $H$ , so the term cannot be explicitly written as $\\Pr (H=h\\mid S=s)$ .", "Further, the probability associated with timed RVs $\\Pr (H_t=h\\mid S_t=s)$ is clearly not time-invariant, hence identifying the time-index is crucial.", "Finally, it is not possible to restrict the integration in eq:pras:app only to histories of a given time-index.", "Overall, the probability term $\\Pr (h\\mid s)$ is necessarily time-variant, which means that $\\Pr (a\\mid s)$ is itself time-variant, and therefore the state value function ${V^\\pi }(s)$ is time-variant, and a time-invariant state value function ${V^\\pi }(s)$ is ill-defined.", "thm:vs.reactive.poTheorem REF If the POMDP observation function depends only on the current state, $\\colon \\rightarrow \\Delta $ , and the policy is reactive, then ${V^\\pi }(s)$ is not necessarily an unbiased estimate of ${V^\\pi }(h)$ , i.e., it is not guaranteed that ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s)\\right]$ .", "As one of the salient theorems of our work, we will cover it from different angles and provide three proofs: first one which is simple but does not go into the detail about the causes of the bias, then a more detailed and analytical one, and finally one by example.", "[First proof] Consider two histories $h, h^{\\prime }\\in $ which are different, $h\\ne h^{\\prime }$ , but are associated with the same belief $b(h) = b(h^{\\prime })$ ; a fairly common occurrence in many POMDPs.", "On one hand, because the two histories are different, the agent's next action may differ, which may then lead to different immediate rewards and future trajectories, i.e., the respective values would also differ, ${V^\\pi }(h) \\ne {V^\\pi }(h^{\\prime }) \\,.$ On the other hand, because the two beliefs are the same, then the expected state values must be the same, $_{s\\mid h}\\left[ {V^\\pi }(s) \\right] = _{s\\mid h^{\\prime }}\\left[ {V^\\pi }(s)\\right] \\,.$ Therefore, it is not guaranteed that ${V^\\pi }(h) = _{s\\mid h}\\left[{V^\\pi }(s) \\right]$ .", "[Second proof, by contradiction] First, we assume that ${V^\\pi }(s)$ is unbiased and show that ${Q^\\pi }(s,a)$ (as defined by eq:qsa) is unbiased, sh[ Q(s, a) ] = sh[ (s, a) + s's, a[ V(s') ] ] = sh[ (s, a) ] + sh[ s's, a[ V(s') ] ] = sh[ (s, a) ] + s'h, a[ V(s') ] = sh[ (s, a) ] + oh, a s'hao[ V(s') ] = (h, a) + oh, a[ V(hao) ] = Q(h, a)  .", "Next, we show that even if ${Q^\\pi }(s, a)$ is unbiased, ${V^\\pi }(s)$ (as defined by eq:vs) is biased, which contradicts the original assumption.", "To do that, we expand the expected state value function $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ and the history value function ${V^\\pi }(h)$ , and show that there is a concrete difference between them: sh[ V(s) ] = sh[ a (as) Q(s, a) ] = sh[ a os[ (a; o) ] Q(s, a) ]  , V(h) = a (a; oh) Q(h, a) = a (a; oh) sh[ Q(s, a) ] = sh[ a os[ (a; oh) ] Q(s, a) ]  .", "eq:proof:vs,eq:proof:vh differ in terms of which observation is used by the policy; in eq:proof:vs, an observation $o$ inferred from a state $s$ inferred from the history $h$ is used, while in eq:proof:vh the final observation $o_h$ of the history $h$ is used.", "These two observations $o$ and $o_h$ are not generally the same, and the respective expectations are similarly not generally the same.", "The nested expectation in eq:proof:vs can be interpreted as a lossy round-trip inference from history to state and from state back to observation $h\\rightarrow s\\rightarrow o$ .", "Although histories and states tend to be somewhat correlated, both state aliasing and history aliasing make the roundtip conversion imperfect, causing a mismatch between the expected state value function $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ and the history value function ${V^\\pi }(h)$ in the general control case of a general POMDP.", "[Third proof, by example] This is a proof by example (with a proof by contradiction element).", "We will define the good/bad POMDP and, for a specific policy and history, first calculate $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ exactly, and then ${V^\\pi }(h)$ using bootstrapping (while also assuming ${V^\\pi }(hao) =_{s^{\\prime }\\mid hao}\\left[ {V^\\pi }(s^{\\prime }) \\right]$ ).", "We show that the two values are numerically different.", "In the good/bad POMDP, $= \\lbrace \\texttt {GOOD}, \\texttt {BAD}\\rbrace , = \\lbrace \\texttt {GOOD},\\texttt {BAD}\\rbrace $ , $= \\lbrace \\texttt {GOOD}, \\texttt {BAD}\\rbrace $ ; At times, we will use the shorthands $\\texttt {G}$ and $\\texttt {B}$ .", "The initial state distribution is uniform, and each state deterministically transitions into itself.", "The GOOD state always emits the GOOD observation, while the BAD state emits a random observation.", "Consider the reward function such that $R(s, a) = \\left[ a = \\texttt {GOOD}\\right]$ , i.e., the agent receives a reward whenever it choses the GOOD action.", "We will denote a history as the concatenation of alternating observations and actions, starting with an observation.", "To keep the notation compact, we will occasionally use symbols G and B to represent GOOD and BAD states, observations and actions.", "Consider a deterministic policy $\\pi (a; h) = \\left[ a = o_h \\right]$ which returns the action corresponding to the last observation.", "Note that this POMDP and this policy satisfy the requirements to guarantee that ${V^\\pi }(s)$ is well defined.", "Next, we calculate the state values ${V^\\pi }(s)$ .", "The $\\texttt {GOOD}$ state always emits the $\\texttt {GOOD}$ observation, so the agent will always choose the $\\texttt {GOOD}$ action and receive a reward of 1, then the state will always transition into itself, V(s=GOOD) = 1 + V(s=GOOD) = 11 -  .", "On the other hand, the $\\texttt {BAD}$ state will only emit the $\\texttt {GOOD}$ observation half of the times, so the agent will only choose the $\\texttt {GOOD}$ action and receive a reward of 1 half of the times, then the state will always transition into itself, V(s=BAD) = 12 + V(s=BAD) = 12(1 - )  .", "Next, we consider the history $h=\\texttt {G}$ after a single initial GOOD observation, and calculate the history value ${V^\\pi }(h)$ .", "Before proceeding, we need to calculate a few intermediate quantities, such as the belief-distribution: (s=Gh=G) (h=Gs=G) (s=G) = 1 12 = 12  , (s=Bh=G) (h=Gs=B) (s=B) = 12 12 = 14  , therefore (s=Gh=G) = 23  , (s=Bh=G) = 13  .", "We also calculate the belief-state distribution after two other histories.", "First $h=\\texttt {G}\\texttt {G}\\texttt {G}$ , (s=Gh=GGG) (h=GGGs=G) (s=G) = 1 12 = 12  , (s=Bh=GGG) (h=GGGs=B) (s=B) = 14 12 = 18  , therefore (s=Gh=GGG) = 45  , (s=Bh=GGG) = 15  .", "Then $h=\\texttt {G}\\texttt {G}\\texttt {B}$ , (s=Gh=GGB) (h=GGBs=G) (s=G) = 0 12 = 0  , (s=Bh=GGB) (h=GGBs=B) (s=B) = 14 12 = 18  , therefore (s=Gh=GGB) = 0  , (s=Bh=GGB) = 1  .", "We also need to calculate the observation emission probabilities, (o=Gh=G, a=G) = (s=Gh=G) (o=Gs=G) = + (s=Bh=G) (o=Gs=B) = 23 1 + 13 12 = 56  , (o=Bh=G, a=G) = (s=Gh=G) (o=Bs=G) = + (s=Bh=G) (o=Bs=B) = 23 0 + 13 12 = 16  .", "Next, we calculate ${V^\\pi }(h=\\texttt {G})$ under the assumption that the equality holds, V(h=G) = sh=G[ V(s) ] = (s=Gh=G) V(s=G) + (s=Bh=G) V(s=B) = 23 11- + 13 12(1-) = 56(1-)  .", "We can also apply the equality to other histories, V(h=GGG) = sh=GGG[ V(s) ] = (s=Gh=GGG) V(s=G) + (s=Bh=GGG) V(s=B) = 45 11- + 15 12(1-) = 910(1-)  , V(h=GGB) = sh=GGB[ V(s) ] = (s=Gh=GGB) V(s=G) + (s=Bh=GGB) V(s=B) = 0 11- + 1 12(1-) = 12(1-)  .", "Next, we calculate ${V^\\pi }(h=\\texttt {G})$ , this time by bootstrapping first, and then using the equality.", "Note that with the given history $h=\\texttt {G}$ , the agent will choose action $a=\\texttt {GOOD}$ .", "Then, V(h=G) = R(h=G, a=G) + oh=G, a=G[ V(hao=GGo) ] = 1 + ( $\\Pr (o=\\texttt {G}\\mid h=\\texttt {G}, a=\\texttt {G}) {V^\\pi }(hao=\\texttt {G}\\texttt {G}\\texttt {G}) \\\\+ \\Pr (o=\\texttt {B}\\mid h=\\texttt {G}, a=\\texttt {G}) {V^\\pi }(hao=\\texttt {G}\\texttt {G}\\texttt {B})$ ) = 1 + 56 910(1-) + 16 12(1-) = 60-6060(1-) + 4560(1-) + 560(1-) = 60-1060(1-) = 6-6(1-)  .", "The values from eq:proof:value:1,eq:proof:value:2 contradict each other, therefore, for this POMDP, policy, and history, ${V^\\pi }(h) \\ne _{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ .", "thm:vs.reactive.foTheorem REF If the POMDP states and observations are related by a bijection $\\phi \\colon \\rightarrow $ , and the policy is reactive, then ${V^\\pi }(s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ .", "Although there is no intrinsic state uncertainty, we continue to use probabilistic notation for notational consistency and simplicity.", "In the following derivation, we use the fact that states and observations contain the same system information to determine the first action and reward.", "This process can be repeated iteratively for all future actions and rewards (omitted, but represented by the ellipsis), sh[ V(s) ] = sh[ a (as) Q(s, a) ] = sh[ a (a; I(s)) Q(s, a) ] = sh[ a (a; oh) Q(s, a) ] = a (a; oh) sh[ Q(s, a) ] = a (a; oh) sh[ R(s, a) + s's, a[ V(s') ] ] = a (a; oh) ( R(h, a) + s'h, a[ V(s') ] ) = a (a; oh) ( R(h, a) + oh, a[ s'hao[ V(s') ] ] ) = $\\vdots $ = a (a; oh) ( R(h, a) + oh, a[ V(hao) ] ) = a (a; oh) Q(h, a) = V(h)  .", "In this case, the bijection between observations and states removes any error from the round-trip inference $h\\rightarrow s\\rightarrow o$ is perfect, which was the cause of the bias in thm:vs.reactive.po.", "thm:vhsTheorem REF For an arbitrary POMDP and policy, ${V^\\pi }(h, s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(h, s) \\right]$ Follows from eq:vh-raw,eq:vhs-raw, V(h0) = s,ah0[ k k R(sk, ak) ] = s0h0 s,ah0,s0[ k k R(sk, ak) ] = s0h0[ V(h0, s0) ]  .", "thm:asymmetric-policy-gradientTheorem REF (Asymmetric Policy Gradient) The policy gradient can be expressed using history-state values, $\\nabla _\\theta _\\text{policy}(\\theta ) = -\\left[ \\sum _t \\gamma ^t {Q^\\pi }(h_t,s_t, a_t) \\nabla _\\theta \\log \\pi (a_t; h_t) \\right] \\,.$ Following thm:vhs, we have Q(h, a) = R(h, a) + oh, a[ V(hao) ] = R(h, a) + oh, a[ s'h, a, o[ V(hao, s') ] ] = R(h, a) + s', oh, a[ V(hao, s') ] = sh[ R(s, a) + s', os, a[ V(hao, s') ] ] = sh[ Q(h, s, a) ]  .", "Therefore, policy() = -[ t t Q(ht, at) (at; ht) ] = -t t ht,at[ Q(ht, at) (at; ht) ] = -t t ht,at[ stht[ Q(ht, st, at) ] (at; ht) ] = -t t ht,st,at[ Q(ht, st, at) (at; ht) ] = -[ t t Q(ht, st, at) (at; ht) ]  .", "Environments Table: Environment properties.The environments used in the evaluation can be split into two groups, Shopping-5, Shopping-6, Heavenhell-3, Heavenhell-4 and Rocksample-5-6 are flat POMDPs, while Keydoor and Ninerooms are gridverse POMDPs.", "Flat POMDPs Figure: Layout of the Shopping environments.Figure: Layout of the Heavenhell environments.Figure: Layout of the Rocksample environment.In the flat POMDPshttps://github.com/abaisero/gym-pomdps, states, actions and observations are encoded by categorical indices with no inherent metric, which are intrinsically equally (dis)similar to each other.", "While it is not possible to generalize across states and observations via feature extraction, the primary challenge in these POMDPs is that of generalizing across different histories.", "Because the flat POMDPs are finite, their state, action and observation spaces have well-defined sizes, shown in tab:environments; note, however, that the size of the state space is not a significant measure of the complexity of partially observable tasks, while the time required to solve the task (i.e., history length) is a more relevant measure.", "Shopping This environment simulates an agent going to a shop to buy an item it forgot.", "The agent navigates a $5\\times 5$ or $6\\times 6$ gridworld trying to locate and select a randomly positioned item.", "The agent's position is fully observable, while the item's position is only observed when queried.", "fig:map:shopping depicts the gridworlds encoded by Shopping-5 and Shopping-6.", "States and Observations States encode the position of the agent and the position of the item in a single integer.", "Observations encode the position of the agent or the position of the item in a single integer.", "Actions Each time-step, the agent must choose an action from the set { LEFT, RIGHT, UP, DOWN, QUERY, BUY }.", "If the agent chooses the QUERY action, it observes the item's position, otherwise it observes its own position.", "To solve the task optimally, the agents needs to query the item's position and remember it, navigate to it, and then buy it.", "Rewards The agent receives the following reward signal: a reward of $-1.0$ for moving; a reward of $-2.0$ for performing a QUERY action; a reward of $-5.0$ for performing a BUY action in the wrong cell; and a reward of $10.0$ for performing a BUY action in the correct cell.", "Heavenhell The agent navigates a corridor-like gridworld composed of a fork and 3 dead-ends.", "Two dead-ends are exits which lead to heaven or hell, although the agent does not know which is which, while the third dead-end leads to a priest who can help the agent identify the heaven exit.", "fig:map:heavenhell depicts the gridworlds encoded by Heavenhell-3 and Heavenhell-4.", "States and Observations States encode the position of the agent and the position of the exit to heaven.", "Observations encode the position of the agent or the position of the exit to heaven.", "Actions Each time-step, the agent must choose an action from the set { NORTH, SOUTH, EAST, WEST }.", "If the agent is at the priest, it observes heaven's location, otherwise it observes its own position.", "To solve the task, the agent needs to navigate to the priest, then back to the fork, and on to heaven.", "Rewards The agent receives a sparse reward signal composed of: a reward of $1.0$ for exiting to heaven; and a reward of $-1.0$ for exiting to hell.", "Rocksample This environment simulates an agent navigating a landscape to collect research material.", "The agent navigates a $5\\times 5$ gridworld which contains 6 rocks, each having either good or bad research value.", "The agent's position is fully observable, while each rock's goodness is randomly sampled and unobserved unless the agent checks it.", "Each rock's goodness can be queried via a stochastic check action, whose reliability decays with the agent-rock distance.", "fig:map:rocksample depicts the gridworld encoded by Rocksample-5-6.", "States and Observations States encode the position of the agent and whether each rock is good or bad; the positions of the rocks are fixed so they do not need to be tracked by the state.", "Observations encode the position of the agent or whether the checked rock is good or bad.", "Actions Each time-step, the agent must choose an action from the set { NORTH, SOUTH, EAST, WEST, CHECK_0, CHECK_1, CHECK_2, CHECK_3, CHECK_4, CHECK_5, SAMPLE, }.", "If the agent chose a CHECK_* action, it observes whether the corresponding rock is good or bad, otherwise it observes its own position.", "To solve the task, the agent needs to sample all the good rocks (and none of the bad rocks), dynamically adapting its path depending on the stochastic information gained by checking each rock.", "Rewards The agent receives a sparse reward signal composed of: a reward of $10.0$ for sampling a good rock; and a reward of $-10.0$ for sampling a bad rock.", "Gridverse POMDPs Figure: KeydoorFigure: NineroomsIn the gridverse POMDPshttps://github.com/abaisero/gym-gridverse, actions are still encoded by categorical indices, while states and observations are encoded as structures which do have an inherent similarity metric; they are split into two components: A grid component: a $6\\times H\\times W$ volume of categorical indices which encode cell types, colors, statuses, agent presence, and the spatial relationships between them.", "The observation grid component is a slice of the corresponding state grid component made to match the agent's perspective: it is rotated to be a first-person view, and cells hidden behind walls are occluded, as shown in fig:keydoor:observation,fig:ninerooms:observation.", "An agent component: a 6-dimensional array of categorical indices representing the agent position and orientation, and the item held by the agent if any.", "The position and orientations in the state agent component are absolute, while those in the observation component are relative to the agent's perspective—they are essentially constant, and not necessary for control.", "Keydoor The agent navigates a $5\\times 5$ gridworld split into two rooms split by a wall and a locked door; on one side is the agent and a key which can be used to open the door, while on the other side is the goal.", "The positions of the agent, the key, the door, and the goal are randomly sampled such that two instances of the same problem are unlikely to be the same.", "fig:keydoor shows state and observation frames taken from an instance of the Keydoor environment.", "States and Observations The state grid component is a $6\\times 5\\times 5$ volume.", "The observation grid component is a $6\\times 7\\times 7$ volume.", "Actions Each time-step, the agent must choose an action from the set { MOVE_FORWARD, MOVE_BACKWARD, MOVE_LEFT, MOVE_RIGHT, TURN_LEFT, TURN_RIGHT, PICK_N_DROP, ACTUATE }.", "The MOVE_* actions result in a movement depending on the agent's orientation, while the TURN_* allows the agent to change its orientation.", "With the PICK_N_DROP action, the agent can pick and/or drop the key from/to the cell in front, while with the ACTUATE action, the agent can open and/or close the door.", "Rewards The agent receives a dense reward signal composed as the sum of the following terms: a living reward of $-0.05$ for every time-step; a reward of $0.2$ for stepping closer to the goal, and $-0.2$ for stepping further away (ignoring walls); a reward of $1.0$ for picking up the key, and $-1.0$ for dropping it; a reward of $1.0$ for opening the door, and $-1.0$ for closing it; and a reward of $5.0$ for reaching the goal.", "Ninerooms The agent navigates a $13\\times 13$ maze-like gridworld trying to locate and reach the goal.", "fig:ninerooms shows state and observation frames taken from an instance of the Ninerooms environment.", "States and Observations The state grid component is a $6\\times 13\\times 13$ volume.", "The observation grid component is a $6\\times 7\\times 7$ volume.", "Actions Each time-step, the agent must choose an action from the set { MOVE_FORWARD, MOVE_BACKWARD, MOVE_LEFT, MOVE_RIGHT, TURN_LEFT, TURN_RIGHT }.", "The MOVE_* actions result in a movement depending on the agent's orientation, while the TURN_* allows the agent to change its orientation.", "Rewards The agent receives a dense reward signal composed as the sum of the following terms: a living reward of $-0.05$ for every time-step; a reward of $0.2$ for stepping closer to the goal, and $-0.2$ for stepping further away (ignoring walls); and a reward of $5.0$ for reaching the goal.", "Architectures and Hyper-Parameters Figure: A2C architecture.", "The state, action, and observationrepresentations φ(s)\\phi (s), φ(a)\\phi (a), and φ(o)\\phi (o) are those shown infig:architecture:representations:flat orfig:architecture:representations:gridverse, depending on whetherthe type of enviroment.", "Dotted lines are present or omitted depending onwhether a history critic V ^(h){\\hat{V}}(h), state critic V ^(s){\\hat{V}}(s), orhistory-state critic V ^(h,s){\\hat{V}}(h, s) is being modeled.In this section, we describe the architectures used by the policy and critic models; a general overview is shown in fig:architecture.", "The general architecture will be similar for all domains; however there will some differences to accommodate the fact that the flat POMDPs provide states and observations as categorical indices, while the gridverse POMDPs provide states and observations as volumes and arrays of categorical indices, whose structures include spatial relationships.", "Features Extraction for Flat POMDPs These components are shown in fig:architecture:representations:flat.", "Because flat POMDPs provide states, actions and observations as categorical indices, and we use 128-dimensional embedding models to represent each of them.", "Features Extraction for Gridverse POMDPs These components are shown in fig:architecture:representations:gridverse.", "Because gridverse POMDPs provide actions as categorical indices, and we use 128-dimensional embedding models to represent them.", "On the other hand, states and observations are provided in the format described in sec:environments:gridverse, each composed of a grid and an agent component: grid The grid component is a $6\\times H\\times W$ volume of categorical data, which we pass through 4-dimensional embeddings, resulting in a $24\\times H\\times W$ volume of numeric data.", "We further pass that data to two layers of CNN followed by ReLU non-linearities, resulting in a $16\\times H\\times W$ volume of data, finally flattened into a $16 HW$ -dimensional array.", "agent The agent component is a 6-dimensional array of categorical data, which we pass through 4-dimensional embeddings, resulting in a 24-dimensional array of numeric data.", "The grid and agent representations are then concatenated to form the state or observation representation $\\phi (s)$ or $\\phi (o)$ .", "Remainder of the Architecture These components are shown in fig:architecture:a2c.", "Concatenated action and observation embeddings form the to a 128-dimensional single-layer gated recurrent unit (GRU) [3], which acts as a history representation $\\phi (h)$ .", "The history representation is then fed into separate policy and critic models, each a 2-layer 128-dimensional feedforward model with ReLU non-linearities.", "A practical issue we found with A2C(h,s) is that the history and state representations $\\phi (h)$ and $\\phi (s)$ contain features with very different orders of magniture; To address this, we use layer-normalization.", "To guarantee a fair comparison with the other methods, we do the same for other critics as well.", "Next, we describe architectural details specific to each method: A2C(h): The history representation passes through layer-normalization first, and then is fed to the critic model; A2C(s): The state embedding passes through layer-normalization first, and then is fed to the critic model; A2C(h,s): The history representation and state embedding individually pass through layer-normalization first, and then are concatenated to form a single feature vector, which is then fed to the critic model." ], [ "Conclusions", "Asymmetric methods trained offline in simulated environments can use information which is normally unavailable in partially observable RL, such as the true system state.", "While the idea of exploiting such information has potential, current state-of-the-art methods are powered by empirical results rather than theoretical analysis.", "In this work, we exposed profound theoretical issues with a standard variant of asymmetric actor-critic, and proposed an unbiased asymmetric actor-critic variant which is analytically sound and theoretically justified.", "Empirical results confirm our analysis, the weaknesses of state critics and the strengths of history-state critics.", "Although we applied the history-state value function only to A2C, the same concepts are easily extensible to other critic-based RL methods [24], [15], [5], [19].", "In future work, we aim to extend the theory of history-state policy value functions ${Q^\\pi }(h, s, a)$ to optimal value functions $Q^*(h, s, a)$ , and develop theoretically sound asymmetric variants of other deep RL methods such as DQN [18], soft Q-learning [8], and soft actor-critic [9].", "We also plan to extend the theory and approach to multi-agent methods, potentially bringing theoretical rigor and improved performance [6], [16], [14], [27], [31], [22], [17], [23]." ], [ "Timed Value Functions", "sec:aa2c shows that for a general POMDP and policy $\\pi \\colon \\rightarrow \\Delta $ , the state value function ${V^\\pi }(s)$ is not generally well-defined due to issues caused by the lack of time information.", "In this section, we consider addressing the primary issue by providing explicit time-index information via timed value functions, $V_t^\\pi (s)$ and $Q_t^\\pi (s, a)$ , which represent the expected returns obtained when the agent finds itself in a state $s$ at time $t$ , Vt(s) = a (At=aSt=s) Qt(s, a)  , Qt(s, a) = R(s, a) + s's, a[ Vt+1(s') ]  .", "Once again, we analyze the state-dependent action distribution term to verify correctness, and expand it by integrating over histories; this time, we can use the explicit time-index information to integrate over histories of a given length only, $\\Pr (A_t=a\\mid S_t=s) = \\sum _{h\\in _t} \\Pr (H_t=h\\mid S_t=s) \\pi (a; h)\\,.", "$ Because eq:sumrule:timed is now restricted to histories of a given length $t$ , the probability term $\\Pr (H_t=h\\mid S_t=s)$ is well-defined, which means $\\Pr (A_t=a\\mid S_t=s)$ is well-defined, and $V_t^\\pi (s)$ is well-defined.", "Introducing the time-index makes the value function $V_t^\\pi (s)$ well-defined.", "However, its utility for the purpose of asymmetric reinforcement learning remains unclear because [label=()] it is still not formally proven whether the timed value function is unbiased, i.e., whether $V_t^\\pi (h) = _{s\\mid h}\\left[V_t^\\pi (s) \\right]$ , and it is harder for timed value critics to generalize appropriately across the additional discrete input $t$ .", "Proofs This section contains the proofs omitted from the main body of the document.", "For the sake of clarity, we repeat the main statements before showing the respective proof.", "thm:vsTheorem REF For an arbitrary POMDP and policy, a time-invariant state value function ${V^\\pi }(s)$ is ill-defined.", "The state value function, defined as ${V^\\pi }(s) = \\sum _{a\\in } \\Pr (a\\mid s) {Q^\\pi }(s, a) \\,,$ requires the state-conditioned action probability term $\\Pr (a\\mid s)$ .", "Because a partially observable policy depends on the history and not the state, the state-conditioned action probability term must be expanded by integrating over the space of all histories, $\\Pr (a\\mid s) = \\sum _{h\\in } \\Pr (h\\mid s) \\pi (a; h) \\,, $ which requires the state-conditioned history probability term $\\Pr (h\\mid s)$ .", "However, it is impossible to associate that term to well-defined RVs: Clearly, the reinforcement learning graphical model (see sec:graphmodel) does not define a time-less history RV $H$ , so the term cannot be explicitly written as $\\Pr (H=h\\mid S=s)$ .", "Further, the probability associated with timed RVs $\\Pr (H_t=h\\mid S_t=s)$ is clearly not time-invariant, hence identifying the time-index is crucial.", "Finally, it is not possible to restrict the integration in eq:pras:app only to histories of a given time-index.", "Overall, the probability term $\\Pr (h\\mid s)$ is necessarily time-variant, which means that $\\Pr (a\\mid s)$ is itself time-variant, and therefore the state value function ${V^\\pi }(s)$ is time-variant, and a time-invariant state value function ${V^\\pi }(s)$ is ill-defined.", "thm:vs.reactive.poTheorem REF If the POMDP observation function depends only on the current state, $\\colon \\rightarrow \\Delta $ , and the policy is reactive, then ${V^\\pi }(s)$ is not necessarily an unbiased estimate of ${V^\\pi }(h)$ , i.e., it is not guaranteed that ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s)\\right]$ .", "As one of the salient theorems of our work, we will cover it from different angles and provide three proofs: first one which is simple but does not go into the detail about the causes of the bias, then a more detailed and analytical one, and finally one by example.", "[First proof] Consider two histories $h, h^{\\prime }\\in $ which are different, $h\\ne h^{\\prime }$ , but are associated with the same belief $b(h) = b(h^{\\prime })$ ; a fairly common occurrence in many POMDPs.", "On one hand, because the two histories are different, the agent's next action may differ, which may then lead to different immediate rewards and future trajectories, i.e., the respective values would also differ, ${V^\\pi }(h) \\ne {V^\\pi }(h^{\\prime }) \\,.$ On the other hand, because the two beliefs are the same, then the expected state values must be the same, $_{s\\mid h}\\left[ {V^\\pi }(s) \\right] = _{s\\mid h^{\\prime }}\\left[ {V^\\pi }(s)\\right] \\,.$ Therefore, it is not guaranteed that ${V^\\pi }(h) = _{s\\mid h}\\left[{V^\\pi }(s) \\right]$ .", "[Second proof, by contradiction] First, we assume that ${V^\\pi }(s)$ is unbiased and show that ${Q^\\pi }(s,a)$ (as defined by eq:qsa) is unbiased, sh[ Q(s, a) ] = sh[ (s, a) + s's, a[ V(s') ] ] = sh[ (s, a) ] + sh[ s's, a[ V(s') ] ] = sh[ (s, a) ] + s'h, a[ V(s') ] = sh[ (s, a) ] + oh, a s'hao[ V(s') ] = (h, a) + oh, a[ V(hao) ] = Q(h, a)  .", "Next, we show that even if ${Q^\\pi }(s, a)$ is unbiased, ${V^\\pi }(s)$ (as defined by eq:vs) is biased, which contradicts the original assumption.", "To do that, we expand the expected state value function $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ and the history value function ${V^\\pi }(h)$ , and show that there is a concrete difference between them: sh[ V(s) ] = sh[ a (as) Q(s, a) ] = sh[ a os[ (a; o) ] Q(s, a) ]  , V(h) = a (a; oh) Q(h, a) = a (a; oh) sh[ Q(s, a) ] = sh[ a os[ (a; oh) ] Q(s, a) ]  .", "eq:proof:vs,eq:proof:vh differ in terms of which observation is used by the policy; in eq:proof:vs, an observation $o$ inferred from a state $s$ inferred from the history $h$ is used, while in eq:proof:vh the final observation $o_h$ of the history $h$ is used.", "These two observations $o$ and $o_h$ are not generally the same, and the respective expectations are similarly not generally the same.", "The nested expectation in eq:proof:vs can be interpreted as a lossy round-trip inference from history to state and from state back to observation $h\\rightarrow s\\rightarrow o$ .", "Although histories and states tend to be somewhat correlated, both state aliasing and history aliasing make the roundtip conversion imperfect, causing a mismatch between the expected state value function $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ and the history value function ${V^\\pi }(h)$ in the general control case of a general POMDP.", "[Third proof, by example] This is a proof by example (with a proof by contradiction element).", "We will define the good/bad POMDP and, for a specific policy and history, first calculate $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ exactly, and then ${V^\\pi }(h)$ using bootstrapping (while also assuming ${V^\\pi }(hao) =_{s^{\\prime }\\mid hao}\\left[ {V^\\pi }(s^{\\prime }) \\right]$ ).", "We show that the two values are numerically different.", "In the good/bad POMDP, $= \\lbrace \\texttt {GOOD}, \\texttt {BAD}\\rbrace , = \\lbrace \\texttt {GOOD},\\texttt {BAD}\\rbrace $ , $= \\lbrace \\texttt {GOOD}, \\texttt {BAD}\\rbrace $ ; At times, we will use the shorthands $\\texttt {G}$ and $\\texttt {B}$ .", "The initial state distribution is uniform, and each state deterministically transitions into itself.", "The GOOD state always emits the GOOD observation, while the BAD state emits a random observation.", "Consider the reward function such that $R(s, a) = \\left[ a = \\texttt {GOOD}\\right]$ , i.e., the agent receives a reward whenever it choses the GOOD action.", "We will denote a history as the concatenation of alternating observations and actions, starting with an observation.", "To keep the notation compact, we will occasionally use symbols G and B to represent GOOD and BAD states, observations and actions.", "Consider a deterministic policy $\\pi (a; h) = \\left[ a = o_h \\right]$ which returns the action corresponding to the last observation.", "Note that this POMDP and this policy satisfy the requirements to guarantee that ${V^\\pi }(s)$ is well defined.", "Next, we calculate the state values ${V^\\pi }(s)$ .", "The $\\texttt {GOOD}$ state always emits the $\\texttt {GOOD}$ observation, so the agent will always choose the $\\texttt {GOOD}$ action and receive a reward of 1, then the state will always transition into itself, V(s=GOOD) = 1 + V(s=GOOD) = 11 -  .", "On the other hand, the $\\texttt {BAD}$ state will only emit the $\\texttt {GOOD}$ observation half of the times, so the agent will only choose the $\\texttt {GOOD}$ action and receive a reward of 1 half of the times, then the state will always transition into itself, V(s=BAD) = 12 + V(s=BAD) = 12(1 - )  .", "Next, we consider the history $h=\\texttt {G}$ after a single initial GOOD observation, and calculate the history value ${V^\\pi }(h)$ .", "Before proceeding, we need to calculate a few intermediate quantities, such as the belief-distribution: (s=Gh=G) (h=Gs=G) (s=G) = 1 12 = 12  , (s=Bh=G) (h=Gs=B) (s=B) = 12 12 = 14  , therefore (s=Gh=G) = 23  , (s=Bh=G) = 13  .", "We also calculate the belief-state distribution after two other histories.", "First $h=\\texttt {G}\\texttt {G}\\texttt {G}$ , (s=Gh=GGG) (h=GGGs=G) (s=G) = 1 12 = 12  , (s=Bh=GGG) (h=GGGs=B) (s=B) = 14 12 = 18  , therefore (s=Gh=GGG) = 45  , (s=Bh=GGG) = 15  .", "Then $h=\\texttt {G}\\texttt {G}\\texttt {B}$ , (s=Gh=GGB) (h=GGBs=G) (s=G) = 0 12 = 0  , (s=Bh=GGB) (h=GGBs=B) (s=B) = 14 12 = 18  , therefore (s=Gh=GGB) = 0  , (s=Bh=GGB) = 1  .", "We also need to calculate the observation emission probabilities, (o=Gh=G, a=G) = (s=Gh=G) (o=Gs=G) = + (s=Bh=G) (o=Gs=B) = 23 1 + 13 12 = 56  , (o=Bh=G, a=G) = (s=Gh=G) (o=Bs=G) = + (s=Bh=G) (o=Bs=B) = 23 0 + 13 12 = 16  .", "Next, we calculate ${V^\\pi }(h=\\texttt {G})$ under the assumption that the equality holds, V(h=G) = sh=G[ V(s) ] = (s=Gh=G) V(s=G) + (s=Bh=G) V(s=B) = 23 11- + 13 12(1-) = 56(1-)  .", "We can also apply the equality to other histories, V(h=GGG) = sh=GGG[ V(s) ] = (s=Gh=GGG) V(s=G) + (s=Bh=GGG) V(s=B) = 45 11- + 15 12(1-) = 910(1-)  , V(h=GGB) = sh=GGB[ V(s) ] = (s=Gh=GGB) V(s=G) + (s=Bh=GGB) V(s=B) = 0 11- + 1 12(1-) = 12(1-)  .", "Next, we calculate ${V^\\pi }(h=\\texttt {G})$ , this time by bootstrapping first, and then using the equality.", "Note that with the given history $h=\\texttt {G}$ , the agent will choose action $a=\\texttt {GOOD}$ .", "Then, V(h=G) = R(h=G, a=G) + oh=G, a=G[ V(hao=GGo) ] = 1 + ( $\\Pr (o=\\texttt {G}\\mid h=\\texttt {G}, a=\\texttt {G}) {V^\\pi }(hao=\\texttt {G}\\texttt {G}\\texttt {G}) \\\\+ \\Pr (o=\\texttt {B}\\mid h=\\texttt {G}, a=\\texttt {G}) {V^\\pi }(hao=\\texttt {G}\\texttt {G}\\texttt {B})$ ) = 1 + 56 910(1-) + 16 12(1-) = 60-6060(1-) + 4560(1-) + 560(1-) = 60-1060(1-) = 6-6(1-)  .", "The values from eq:proof:value:1,eq:proof:value:2 contradict each other, therefore, for this POMDP, policy, and history, ${V^\\pi }(h) \\ne _{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ .", "thm:vs.reactive.foTheorem REF If the POMDP states and observations are related by a bijection $\\phi \\colon \\rightarrow $ , and the policy is reactive, then ${V^\\pi }(s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ .", "Although there is no intrinsic state uncertainty, we continue to use probabilistic notation for notational consistency and simplicity.", "In the following derivation, we use the fact that states and observations contain the same system information to determine the first action and reward.", "This process can be repeated iteratively for all future actions and rewards (omitted, but represented by the ellipsis), sh[ V(s) ] = sh[ a (as) Q(s, a) ] = sh[ a (a; I(s)) Q(s, a) ] = sh[ a (a; oh) Q(s, a) ] = a (a; oh) sh[ Q(s, a) ] = a (a; oh) sh[ R(s, a) + s's, a[ V(s') ] ] = a (a; oh) ( R(h, a) + s'h, a[ V(s') ] ) = a (a; oh) ( R(h, a) + oh, a[ s'hao[ V(s') ] ] ) = $\\vdots $ = a (a; oh) ( R(h, a) + oh, a[ V(hao) ] ) = a (a; oh) Q(h, a) = V(h)  .", "In this case, the bijection between observations and states removes any error from the round-trip inference $h\\rightarrow s\\rightarrow o$ is perfect, which was the cause of the bias in thm:vs.reactive.po.", "thm:vhsTheorem REF For an arbitrary POMDP and policy, ${V^\\pi }(h, s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(h, s) \\right]$ Follows from eq:vh-raw,eq:vhs-raw, V(h0) = s,ah0[ k k R(sk, ak) ] = s0h0 s,ah0,s0[ k k R(sk, ak) ] = s0h0[ V(h0, s0) ]  .", "thm:asymmetric-policy-gradientTheorem REF (Asymmetric Policy Gradient) The policy gradient can be expressed using history-state values, $\\nabla _\\theta _\\text{policy}(\\theta ) = -\\left[ \\sum _t \\gamma ^t {Q^\\pi }(h_t,s_t, a_t) \\nabla _\\theta \\log \\pi (a_t; h_t) \\right] \\,.$ Following thm:vhs, we have Q(h, a) = R(h, a) + oh, a[ V(hao) ] = R(h, a) + oh, a[ s'h, a, o[ V(hao, s') ] ] = R(h, a) + s', oh, a[ V(hao, s') ] = sh[ R(s, a) + s', os, a[ V(hao, s') ] ] = sh[ Q(h, s, a) ]  .", "Therefore, policy() = -[ t t Q(ht, at) (at; ht) ] = -t t ht,at[ Q(ht, at) (at; ht) ] = -t t ht,at[ stht[ Q(ht, st, at) ] (at; ht) ] = -t t ht,st,at[ Q(ht, st, at) (at; ht) ] = -[ t t Q(ht, st, at) (at; ht) ]  .", "Environments Table: Environment properties.The environments used in the evaluation can be split into two groups, Shopping-5, Shopping-6, Heavenhell-3, Heavenhell-4 and Rocksample-5-6 are flat POMDPs, while Keydoor and Ninerooms are gridverse POMDPs.", "Flat POMDPs Figure: Layout of the Shopping environments.Figure: Layout of the Heavenhell environments.Figure: Layout of the Rocksample environment.In the flat POMDPshttps://github.com/abaisero/gym-pomdps, states, actions and observations are encoded by categorical indices with no inherent metric, which are intrinsically equally (dis)similar to each other.", "While it is not possible to generalize across states and observations via feature extraction, the primary challenge in these POMDPs is that of generalizing across different histories.", "Because the flat POMDPs are finite, their state, action and observation spaces have well-defined sizes, shown in tab:environments; note, however, that the size of the state space is not a significant measure of the complexity of partially observable tasks, while the time required to solve the task (i.e., history length) is a more relevant measure.", "Shopping This environment simulates an agent going to a shop to buy an item it forgot.", "The agent navigates a $5\\times 5$ or $6\\times 6$ gridworld trying to locate and select a randomly positioned item.", "The agent's position is fully observable, while the item's position is only observed when queried.", "fig:map:shopping depicts the gridworlds encoded by Shopping-5 and Shopping-6.", "States and Observations States encode the position of the agent and the position of the item in a single integer.", "Observations encode the position of the agent or the position of the item in a single integer.", "Actions Each time-step, the agent must choose an action from the set { LEFT, RIGHT, UP, DOWN, QUERY, BUY }.", "If the agent chooses the QUERY action, it observes the item's position, otherwise it observes its own position.", "To solve the task optimally, the agents needs to query the item's position and remember it, navigate to it, and then buy it.", "Rewards The agent receives the following reward signal: a reward of $-1.0$ for moving; a reward of $-2.0$ for performing a QUERY action; a reward of $-5.0$ for performing a BUY action in the wrong cell; and a reward of $10.0$ for performing a BUY action in the correct cell.", "Heavenhell The agent navigates a corridor-like gridworld composed of a fork and 3 dead-ends.", "Two dead-ends are exits which lead to heaven or hell, although the agent does not know which is which, while the third dead-end leads to a priest who can help the agent identify the heaven exit.", "fig:map:heavenhell depicts the gridworlds encoded by Heavenhell-3 and Heavenhell-4.", "States and Observations States encode the position of the agent and the position of the exit to heaven.", "Observations encode the position of the agent or the position of the exit to heaven.", "Actions Each time-step, the agent must choose an action from the set { NORTH, SOUTH, EAST, WEST }.", "If the agent is at the priest, it observes heaven's location, otherwise it observes its own position.", "To solve the task, the agent needs to navigate to the priest, then back to the fork, and on to heaven.", "Rewards The agent receives a sparse reward signal composed of: a reward of $1.0$ for exiting to heaven; and a reward of $-1.0$ for exiting to hell.", "Rocksample This environment simulates an agent navigating a landscape to collect research material.", "The agent navigates a $5\\times 5$ gridworld which contains 6 rocks, each having either good or bad research value.", "The agent's position is fully observable, while each rock's goodness is randomly sampled and unobserved unless the agent checks it.", "Each rock's goodness can be queried via a stochastic check action, whose reliability decays with the agent-rock distance.", "fig:map:rocksample depicts the gridworld encoded by Rocksample-5-6.", "States and Observations States encode the position of the agent and whether each rock is good or bad; the positions of the rocks are fixed so they do not need to be tracked by the state.", "Observations encode the position of the agent or whether the checked rock is good or bad.", "Actions Each time-step, the agent must choose an action from the set { NORTH, SOUTH, EAST, WEST, CHECK_0, CHECK_1, CHECK_2, CHECK_3, CHECK_4, CHECK_5, SAMPLE, }.", "If the agent chose a CHECK_* action, it observes whether the corresponding rock is good or bad, otherwise it observes its own position.", "To solve the task, the agent needs to sample all the good rocks (and none of the bad rocks), dynamically adapting its path depending on the stochastic information gained by checking each rock.", "Rewards The agent receives a sparse reward signal composed of: a reward of $10.0$ for sampling a good rock; and a reward of $-10.0$ for sampling a bad rock.", "Gridverse POMDPs Figure: KeydoorFigure: NineroomsIn the gridverse POMDPshttps://github.com/abaisero/gym-gridverse, actions are still encoded by categorical indices, while states and observations are encoded as structures which do have an inherent similarity metric; they are split into two components: A grid component: a $6\\times H\\times W$ volume of categorical indices which encode cell types, colors, statuses, agent presence, and the spatial relationships between them.", "The observation grid component is a slice of the corresponding state grid component made to match the agent's perspective: it is rotated to be a first-person view, and cells hidden behind walls are occluded, as shown in fig:keydoor:observation,fig:ninerooms:observation.", "An agent component: a 6-dimensional array of categorical indices representing the agent position and orientation, and the item held by the agent if any.", "The position and orientations in the state agent component are absolute, while those in the observation component are relative to the agent's perspective—they are essentially constant, and not necessary for control.", "Keydoor The agent navigates a $5\\times 5$ gridworld split into two rooms split by a wall and a locked door; on one side is the agent and a key which can be used to open the door, while on the other side is the goal.", "The positions of the agent, the key, the door, and the goal are randomly sampled such that two instances of the same problem are unlikely to be the same.", "fig:keydoor shows state and observation frames taken from an instance of the Keydoor environment.", "States and Observations The state grid component is a $6\\times 5\\times 5$ volume.", "The observation grid component is a $6\\times 7\\times 7$ volume.", "Actions Each time-step, the agent must choose an action from the set { MOVE_FORWARD, MOVE_BACKWARD, MOVE_LEFT, MOVE_RIGHT, TURN_LEFT, TURN_RIGHT, PICK_N_DROP, ACTUATE }.", "The MOVE_* actions result in a movement depending on the agent's orientation, while the TURN_* allows the agent to change its orientation.", "With the PICK_N_DROP action, the agent can pick and/or drop the key from/to the cell in front, while with the ACTUATE action, the agent can open and/or close the door.", "Rewards The agent receives a dense reward signal composed as the sum of the following terms: a living reward of $-0.05$ for every time-step; a reward of $0.2$ for stepping closer to the goal, and $-0.2$ for stepping further away (ignoring walls); a reward of $1.0$ for picking up the key, and $-1.0$ for dropping it; a reward of $1.0$ for opening the door, and $-1.0$ for closing it; and a reward of $5.0$ for reaching the goal.", "Ninerooms The agent navigates a $13\\times 13$ maze-like gridworld trying to locate and reach the goal.", "fig:ninerooms shows state and observation frames taken from an instance of the Ninerooms environment.", "States and Observations The state grid component is a $6\\times 13\\times 13$ volume.", "The observation grid component is a $6\\times 7\\times 7$ volume.", "Actions Each time-step, the agent must choose an action from the set { MOVE_FORWARD, MOVE_BACKWARD, MOVE_LEFT, MOVE_RIGHT, TURN_LEFT, TURN_RIGHT }.", "The MOVE_* actions result in a movement depending on the agent's orientation, while the TURN_* allows the agent to change its orientation.", "Rewards The agent receives a dense reward signal composed as the sum of the following terms: a living reward of $-0.05$ for every time-step; a reward of $0.2$ for stepping closer to the goal, and $-0.2$ for stepping further away (ignoring walls); and a reward of $5.0$ for reaching the goal.", "Architectures and Hyper-Parameters Figure: A2C architecture.", "The state, action, and observationrepresentations φ(s)\\phi (s), φ(a)\\phi (a), and φ(o)\\phi (o) are those shown infig:architecture:representations:flat orfig:architecture:representations:gridverse, depending on whetherthe type of enviroment.", "Dotted lines are present or omitted depending onwhether a history critic V ^(h){\\hat{V}}(h), state critic V ^(s){\\hat{V}}(s), orhistory-state critic V ^(h,s){\\hat{V}}(h, s) is being modeled.In this section, we describe the architectures used by the policy and critic models; a general overview is shown in fig:architecture.", "The general architecture will be similar for all domains; however there will some differences to accommodate the fact that the flat POMDPs provide states and observations as categorical indices, while the gridverse POMDPs provide states and observations as volumes and arrays of categorical indices, whose structures include spatial relationships.", "Features Extraction for Flat POMDPs These components are shown in fig:architecture:representations:flat.", "Because flat POMDPs provide states, actions and observations as categorical indices, and we use 128-dimensional embedding models to represent each of them.", "Features Extraction for Gridverse POMDPs These components are shown in fig:architecture:representations:gridverse.", "Because gridverse POMDPs provide actions as categorical indices, and we use 128-dimensional embedding models to represent them.", "On the other hand, states and observations are provided in the format described in sec:environments:gridverse, each composed of a grid and an agent component: grid The grid component is a $6\\times H\\times W$ volume of categorical data, which we pass through 4-dimensional embeddings, resulting in a $24\\times H\\times W$ volume of numeric data.", "We further pass that data to two layers of CNN followed by ReLU non-linearities, resulting in a $16\\times H\\times W$ volume of data, finally flattened into a $16 HW$ -dimensional array.", "agent The agent component is a 6-dimensional array of categorical data, which we pass through 4-dimensional embeddings, resulting in a 24-dimensional array of numeric data.", "The grid and agent representations are then concatenated to form the state or observation representation $\\phi (s)$ or $\\phi (o)$ .", "Remainder of the Architecture These components are shown in fig:architecture:a2c.", "Concatenated action and observation embeddings form the to a 128-dimensional single-layer gated recurrent unit (GRU) [3], which acts as a history representation $\\phi (h)$ .", "The history representation is then fed into separate policy and critic models, each a 2-layer 128-dimensional feedforward model with ReLU non-linearities.", "A practical issue we found with A2C(h,s) is that the history and state representations $\\phi (h)$ and $\\phi (s)$ contain features with very different orders of magniture; To address this, we use layer-normalization.", "To guarantee a fair comparison with the other methods, we do the same for other critics as well.", "Next, we describe architectural details specific to each method: A2C(h): The history representation passes through layer-normalization first, and then is fed to the critic model; A2C(s): The state embedding passes through layer-normalization first, and then is fed to the critic model; A2C(h,s): The history representation and state embedding individually pass through layer-normalization first, and then are concatenated to form a single feature vector, which is then fed to the critic model." ], [ "Proofs", "This section contains the proofs omitted from the main body of the document.", "For the sake of clarity, we repeat the main statements before showing the respective proof.", "thm:vsTheorem REF For an arbitrary POMDP and policy, a time-invariant state value function ${V^\\pi }(s)$ is ill-defined.", "The state value function, defined as ${V^\\pi }(s) = \\sum _{a\\in } \\Pr (a\\mid s) {Q^\\pi }(s, a) \\,,$ requires the state-conditioned action probability term $\\Pr (a\\mid s)$ .", "Because a partially observable policy depends on the history and not the state, the state-conditioned action probability term must be expanded by integrating over the space of all histories, $\\Pr (a\\mid s) = \\sum _{h\\in } \\Pr (h\\mid s) \\pi (a; h) \\,, $ which requires the state-conditioned history probability term $\\Pr (h\\mid s)$ .", "However, it is impossible to associate that term to well-defined RVs: Clearly, the reinforcement learning graphical model (see sec:graphmodel) does not define a time-less history RV $H$ , so the term cannot be explicitly written as $\\Pr (H=h\\mid S=s)$ .", "Further, the probability associated with timed RVs $\\Pr (H_t=h\\mid S_t=s)$ is clearly not time-invariant, hence identifying the time-index is crucial.", "Finally, it is not possible to restrict the integration in eq:pras:app only to histories of a given time-index.", "Overall, the probability term $\\Pr (h\\mid s)$ is necessarily time-variant, which means that $\\Pr (a\\mid s)$ is itself time-variant, and therefore the state value function ${V^\\pi }(s)$ is time-variant, and a time-invariant state value function ${V^\\pi }(s)$ is ill-defined.", "thm:vs.reactive.poTheorem REF If the POMDP observation function depends only on the current state, $\\colon \\rightarrow \\Delta $ , and the policy is reactive, then ${V^\\pi }(s)$ is not necessarily an unbiased estimate of ${V^\\pi }(h)$ , i.e., it is not guaranteed that ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s)\\right]$ .", "As one of the salient theorems of our work, we will cover it from different angles and provide three proofs: first one which is simple but does not go into the detail about the causes of the bias, then a more detailed and analytical one, and finally one by example.", "[First proof] Consider two histories $h, h^{\\prime }\\in $ which are different, $h\\ne h^{\\prime }$ , but are associated with the same belief $b(h) = b(h^{\\prime })$ ; a fairly common occurrence in many POMDPs.", "On one hand, because the two histories are different, the agent's next action may differ, which may then lead to different immediate rewards and future trajectories, i.e., the respective values would also differ, ${V^\\pi }(h) \\ne {V^\\pi }(h^{\\prime }) \\,.$ On the other hand, because the two beliefs are the same, then the expected state values must be the same, $_{s\\mid h}\\left[ {V^\\pi }(s) \\right] = _{s\\mid h^{\\prime }}\\left[ {V^\\pi }(s)\\right] \\,.$ Therefore, it is not guaranteed that ${V^\\pi }(h) = _{s\\mid h}\\left[{V^\\pi }(s) \\right]$ .", "[Second proof, by contradiction] First, we assume that ${V^\\pi }(s)$ is unbiased and show that ${Q^\\pi }(s,a)$ (as defined by eq:qsa) is unbiased, sh[ Q(s, a) ] = sh[ (s, a) + s's, a[ V(s') ] ] = sh[ (s, a) ] + sh[ s's, a[ V(s') ] ] = sh[ (s, a) ] + s'h, a[ V(s') ] = sh[ (s, a) ] + oh, a s'hao[ V(s') ] = (h, a) + oh, a[ V(hao) ] = Q(h, a)  .", "Next, we show that even if ${Q^\\pi }(s, a)$ is unbiased, ${V^\\pi }(s)$ (as defined by eq:vs) is biased, which contradicts the original assumption.", "To do that, we expand the expected state value function $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ and the history value function ${V^\\pi }(h)$ , and show that there is a concrete difference between them: sh[ V(s) ] = sh[ a (as) Q(s, a) ] = sh[ a os[ (a; o) ] Q(s, a) ]  , V(h) = a (a; oh) Q(h, a) = a (a; oh) sh[ Q(s, a) ] = sh[ a os[ (a; oh) ] Q(s, a) ]  .", "eq:proof:vs,eq:proof:vh differ in terms of which observation is used by the policy; in eq:proof:vs, an observation $o$ inferred from a state $s$ inferred from the history $h$ is used, while in eq:proof:vh the final observation $o_h$ of the history $h$ is used.", "These two observations $o$ and $o_h$ are not generally the same, and the respective expectations are similarly not generally the same.", "The nested expectation in eq:proof:vs can be interpreted as a lossy round-trip inference from history to state and from state back to observation $h\\rightarrow s\\rightarrow o$ .", "Although histories and states tend to be somewhat correlated, both state aliasing and history aliasing make the roundtip conversion imperfect, causing a mismatch between the expected state value function $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ and the history value function ${V^\\pi }(h)$ in the general control case of a general POMDP.", "[Third proof, by example] This is a proof by example (with a proof by contradiction element).", "We will define the good/bad POMDP and, for a specific policy and history, first calculate $_{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ exactly, and then ${V^\\pi }(h)$ using bootstrapping (while also assuming ${V^\\pi }(hao) =_{s^{\\prime }\\mid hao}\\left[ {V^\\pi }(s^{\\prime }) \\right]$ ).", "We show that the two values are numerically different.", "In the good/bad POMDP, $= \\lbrace \\texttt {GOOD}, \\texttt {BAD}\\rbrace , = \\lbrace \\texttt {GOOD},\\texttt {BAD}\\rbrace $ , $= \\lbrace \\texttt {GOOD}, \\texttt {BAD}\\rbrace $ ; At times, we will use the shorthands $\\texttt {G}$ and $\\texttt {B}$ .", "The initial state distribution is uniform, and each state deterministically transitions into itself.", "The GOOD state always emits the GOOD observation, while the BAD state emits a random observation.", "Consider the reward function such that $R(s, a) = \\left[ a = \\texttt {GOOD}\\right]$ , i.e., the agent receives a reward whenever it choses the GOOD action.", "We will denote a history as the concatenation of alternating observations and actions, starting with an observation.", "To keep the notation compact, we will occasionally use symbols G and B to represent GOOD and BAD states, observations and actions.", "Consider a deterministic policy $\\pi (a; h) = \\left[ a = o_h \\right]$ which returns the action corresponding to the last observation.", "Note that this POMDP and this policy satisfy the requirements to guarantee that ${V^\\pi }(s)$ is well defined.", "Next, we calculate the state values ${V^\\pi }(s)$ .", "The $\\texttt {GOOD}$ state always emits the $\\texttt {GOOD}$ observation, so the agent will always choose the $\\texttt {GOOD}$ action and receive a reward of 1, then the state will always transition into itself, V(s=GOOD) = 1 + V(s=GOOD) = 11 -  .", "On the other hand, the $\\texttt {BAD}$ state will only emit the $\\texttt {GOOD}$ observation half of the times, so the agent will only choose the $\\texttt {GOOD}$ action and receive a reward of 1 half of the times, then the state will always transition into itself, V(s=BAD) = 12 + V(s=BAD) = 12(1 - )  .", "Next, we consider the history $h=\\texttt {G}$ after a single initial GOOD observation, and calculate the history value ${V^\\pi }(h)$ .", "Before proceeding, we need to calculate a few intermediate quantities, such as the belief-distribution: (s=Gh=G) (h=Gs=G) (s=G) = 1 12 = 12  , (s=Bh=G) (h=Gs=B) (s=B) = 12 12 = 14  , therefore (s=Gh=G) = 23  , (s=Bh=G) = 13  .", "We also calculate the belief-state distribution after two other histories.", "First $h=\\texttt {G}\\texttt {G}\\texttt {G}$ , (s=Gh=GGG) (h=GGGs=G) (s=G) = 1 12 = 12  , (s=Bh=GGG) (h=GGGs=B) (s=B) = 14 12 = 18  , therefore (s=Gh=GGG) = 45  , (s=Bh=GGG) = 15  .", "Then $h=\\texttt {G}\\texttt {G}\\texttt {B}$ , (s=Gh=GGB) (h=GGBs=G) (s=G) = 0 12 = 0  , (s=Bh=GGB) (h=GGBs=B) (s=B) = 14 12 = 18  , therefore (s=Gh=GGB) = 0  , (s=Bh=GGB) = 1  .", "We also need to calculate the observation emission probabilities, (o=Gh=G, a=G) = (s=Gh=G) (o=Gs=G) = + (s=Bh=G) (o=Gs=B) = 23 1 + 13 12 = 56  , (o=Bh=G, a=G) = (s=Gh=G) (o=Bs=G) = + (s=Bh=G) (o=Bs=B) = 23 0 + 13 12 = 16  .", "Next, we calculate ${V^\\pi }(h=\\texttt {G})$ under the assumption that the equality holds, V(h=G) = sh=G[ V(s) ] = (s=Gh=G) V(s=G) + (s=Bh=G) V(s=B) = 23 11- + 13 12(1-) = 56(1-)  .", "We can also apply the equality to other histories, V(h=GGG) = sh=GGG[ V(s) ] = (s=Gh=GGG) V(s=G) + (s=Bh=GGG) V(s=B) = 45 11- + 15 12(1-) = 910(1-)  , V(h=GGB) = sh=GGB[ V(s) ] = (s=Gh=GGB) V(s=G) + (s=Bh=GGB) V(s=B) = 0 11- + 1 12(1-) = 12(1-)  .", "Next, we calculate ${V^\\pi }(h=\\texttt {G})$ , this time by bootstrapping first, and then using the equality.", "Note that with the given history $h=\\texttt {G}$ , the agent will choose action $a=\\texttt {GOOD}$ .", "Then, V(h=G) = R(h=G, a=G) + oh=G, a=G[ V(hao=GGo) ] = 1 + ( $\\Pr (o=\\texttt {G}\\mid h=\\texttt {G}, a=\\texttt {G}) {V^\\pi }(hao=\\texttt {G}\\texttt {G}\\texttt {G}) \\\\+ \\Pr (o=\\texttt {B}\\mid h=\\texttt {G}, a=\\texttt {G}) {V^\\pi }(hao=\\texttt {G}\\texttt {G}\\texttt {B})$ ) = 1 + 56 910(1-) + 16 12(1-) = 60-6060(1-) + 4560(1-) + 560(1-) = 60-1060(1-) = 6-6(1-)  .", "The values from eq:proof:value:1,eq:proof:value:2 contradict each other, therefore, for this POMDP, policy, and history, ${V^\\pi }(h) \\ne _{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ .", "thm:vs.reactive.foTheorem REF If the POMDP states and observations are related by a bijection $\\phi \\colon \\rightarrow $ , and the policy is reactive, then ${V^\\pi }(s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(s) \\right]$ .", "Although there is no intrinsic state uncertainty, we continue to use probabilistic notation for notational consistency and simplicity.", "In the following derivation, we use the fact that states and observations contain the same system information to determine the first action and reward.", "This process can be repeated iteratively for all future actions and rewards (omitted, but represented by the ellipsis), sh[ V(s) ] = sh[ a (as) Q(s, a) ] = sh[ a (a; I(s)) Q(s, a) ] = sh[ a (a; oh) Q(s, a) ] = a (a; oh) sh[ Q(s, a) ] = a (a; oh) sh[ R(s, a) + s's, a[ V(s') ] ] = a (a; oh) ( R(h, a) + s'h, a[ V(s') ] ) = a (a; oh) ( R(h, a) + oh, a[ s'hao[ V(s') ] ] ) = $\\vdots $ = a (a; oh) ( R(h, a) + oh, a[ V(hao) ] ) = a (a; oh) Q(h, a) = V(h)  .", "In this case, the bijection between observations and states removes any error from the round-trip inference $h\\rightarrow s\\rightarrow o$ is perfect, which was the cause of the bias in thm:vs.reactive.po.", "thm:vhsTheorem REF For an arbitrary POMDP and policy, ${V^\\pi }(h, s)$ is an unbiased estimate of ${V^\\pi }(h)$ , i.e., ${V^\\pi }(h) = _{s\\mid h}\\left[ {V^\\pi }(h, s) \\right]$ Follows from eq:vh-raw,eq:vhs-raw, V(h0) = s,ah0[ k k R(sk, ak) ] = s0h0 s,ah0,s0[ k k R(sk, ak) ] = s0h0[ V(h0, s0) ]  .", "thm:asymmetric-policy-gradientTheorem REF (Asymmetric Policy Gradient) The policy gradient can be expressed using history-state values, $\\nabla _\\theta _\\text{policy}(\\theta ) = -\\left[ \\sum _t \\gamma ^t {Q^\\pi }(h_t,s_t, a_t) \\nabla _\\theta \\log \\pi (a_t; h_t) \\right] \\,.$ Following thm:vhs, we have Q(h, a) = R(h, a) + oh, a[ V(hao) ] = R(h, a) + oh, a[ s'h, a, o[ V(hao, s') ] ] = R(h, a) + s', oh, a[ V(hao, s') ] = sh[ R(s, a) + s', os, a[ V(hao, s') ] ] = sh[ Q(h, s, a) ]  .", "Therefore, policy() = -[ t t Q(ht, at) (at; ht) ] = -t t ht,at[ Q(ht, at) (at; ht) ] = -t t ht,at[ stht[ Q(ht, st, at) ] (at; ht) ] = -t t ht,st,at[ Q(ht, st, at) (at; ht) ] = -[ t t Q(ht, st, at) (at; ht) ]  ." ], [ "Environments", "The environments used in the evaluation can be split into two groups, Shopping-5, Shopping-6, Heavenhell-3, Heavenhell-4 and Rocksample-5-6 are flat POMDPs, while Keydoor and Ninerooms are gridverse POMDPs." ], [ "Flat POMDPs", "In the flat POMDPshttps://github.com/abaisero/gym-pomdps, states, actions and observations are encoded by categorical indices with no inherent metric, which are intrinsically equally (dis)similar to each other.", "While it is not possible to generalize across states and observations via feature extraction, the primary challenge in these POMDPs is that of generalizing across different histories.", "Because the flat POMDPs are finite, their state, action and observation spaces have well-defined sizes, shown in tab:environments; note, however, that the size of the state space is not a significant measure of the complexity of partially observable tasks, while the time required to solve the task (i.e., history length) is a more relevant measure.", "This environment simulates an agent going to a shop to buy an item it forgot.", "The agent navigates a $5\\times 5$ or $6\\times 6$ gridworld trying to locate and select a randomly positioned item.", "The agent's position is fully observable, while the item's position is only observed when queried.", "fig:map:shopping depicts the gridworlds encoded by Shopping-5 and Shopping-6.", "States encode the position of the agent and the position of the item in a single integer.", "Observations encode the position of the agent or the position of the item in a single integer." ], [ "Actions", "Each time-step, the agent must choose an action from the set { LEFT, RIGHT, UP, DOWN, QUERY, BUY }.", "If the agent chooses the QUERY action, it observes the item's position, otherwise it observes its own position.", "To solve the task optimally, the agents needs to query the item's position and remember it, navigate to it, and then buy it." ], [ "Rewards", "The agent receives the following reward signal: a reward of $-1.0$ for moving; a reward of $-2.0$ for performing a QUERY action; a reward of $-5.0$ for performing a BUY action in the wrong cell; and a reward of $10.0$ for performing a BUY action in the correct cell." ], [ "Heavenhell", "The agent navigates a corridor-like gridworld composed of a fork and 3 dead-ends.", "Two dead-ends are exits which lead to heaven or hell, although the agent does not know which is which, while the third dead-end leads to a priest who can help the agent identify the heaven exit.", "fig:map:heavenhell depicts the gridworlds encoded by Heavenhell-3 and Heavenhell-4." ], [ "States and Observations", "States encode the position of the agent and the position of the exit to heaven.", "Observations encode the position of the agent or the position of the exit to heaven." ], [ "Actions", "Each time-step, the agent must choose an action from the set { NORTH, SOUTH, EAST, WEST }.", "If the agent is at the priest, it observes heaven's location, otherwise it observes its own position.", "To solve the task, the agent needs to navigate to the priest, then back to the fork, and on to heaven." ], [ "Rewards", "The agent receives a sparse reward signal composed of: a reward of $1.0$ for exiting to heaven; and a reward of $-1.0$ for exiting to hell.", "This environment simulates an agent navigating a landscape to collect research material.", "The agent navigates a $5\\times 5$ gridworld which contains 6 rocks, each having either good or bad research value.", "The agent's position is fully observable, while each rock's goodness is randomly sampled and unobserved unless the agent checks it.", "Each rock's goodness can be queried via a stochastic check action, whose reliability decays with the agent-rock distance.", "fig:map:rocksample depicts the gridworld encoded by Rocksample-5-6." ], [ "States and Observations", "States encode the position of the agent and whether each rock is good or bad; the positions of the rocks are fixed so they do not need to be tracked by the state.", "Observations encode the position of the agent or whether the checked rock is good or bad." ], [ "Actions", "Each time-step, the agent must choose an action from the set { NORTH, SOUTH, EAST, WEST, CHECK_0, CHECK_1, CHECK_2, CHECK_3, CHECK_4, CHECK_5, SAMPLE, }.", "If the agent chose a CHECK_* action, it observes whether the corresponding rock is good or bad, otherwise it observes its own position.", "To solve the task, the agent needs to sample all the good rocks (and none of the bad rocks), dynamically adapting its path depending on the stochastic information gained by checking each rock." ], [ "Rewards", "The agent receives a sparse reward signal composed of: a reward of $10.0$ for sampling a good rock; and a reward of $-10.0$ for sampling a bad rock." ], [ "Gridverse POMDPs", "In the gridverse POMDPshttps://github.com/abaisero/gym-gridverse, actions are still encoded by categorical indices, while states and observations are encoded as structures which do have an inherent similarity metric; they are split into two components: A grid component: a $6\\times H\\times W$ volume of categorical indices which encode cell types, colors, statuses, agent presence, and the spatial relationships between them.", "The observation grid component is a slice of the corresponding state grid component made to match the agent's perspective: it is rotated to be a first-person view, and cells hidden behind walls are occluded, as shown in fig:keydoor:observation,fig:ninerooms:observation.", "An agent component: a 6-dimensional array of categorical indices representing the agent position and orientation, and the item held by the agent if any.", "The position and orientations in the state agent component are absolute, while those in the observation component are relative to the agent's perspective—they are essentially constant, and not necessary for control." ], [ "Keydoor", "The agent navigates a $5\\times 5$ gridworld split into two rooms split by a wall and a locked door; on one side is the agent and a key which can be used to open the door, while on the other side is the goal.", "The positions of the agent, the key, the door, and the goal are randomly sampled such that two instances of the same problem are unlikely to be the same.", "fig:keydoor shows state and observation frames taken from an instance of the Keydoor environment.", "The state grid component is a $6\\times 5\\times 5$ volume.", "The observation grid component is a $6\\times 7\\times 7$ volume." ], [ "Actions", "Each time-step, the agent must choose an action from the set { MOVE_FORWARD, MOVE_BACKWARD, MOVE_LEFT, MOVE_RIGHT, TURN_LEFT, TURN_RIGHT, PICK_N_DROP, ACTUATE }.", "The MOVE_* actions result in a movement depending on the agent's orientation, while the TURN_* allows the agent to change its orientation.", "With the PICK_N_DROP action, the agent can pick and/or drop the key from/to the cell in front, while with the ACTUATE action, the agent can open and/or close the door." ], [ "Rewards", "The agent receives a dense reward signal composed as the sum of the following terms: a living reward of $-0.05$ for every time-step; a reward of $0.2$ for stepping closer to the goal, and $-0.2$ for stepping further away (ignoring walls); a reward of $1.0$ for picking up the key, and $-1.0$ for dropping it; a reward of $1.0$ for opening the door, and $-1.0$ for closing it; and a reward of $5.0$ for reaching the goal." ], [ "Ninerooms", "The agent navigates a $13\\times 13$ maze-like gridworld trying to locate and reach the goal.", "fig:ninerooms shows state and observation frames taken from an instance of the Ninerooms environment." ], [ "States and Observations", "The state grid component is a $6\\times 13\\times 13$ volume.", "The observation grid component is a $6\\times 7\\times 7$ volume." ], [ "Actions", "Each time-step, the agent must choose an action from the set { MOVE_FORWARD, MOVE_BACKWARD, MOVE_LEFT, MOVE_RIGHT, TURN_LEFT, TURN_RIGHT }.", "The MOVE_* actions result in a movement depending on the agent's orientation, while the TURN_* allows the agent to change its orientation." ], [ "Rewards", "The agent receives a dense reward signal composed as the sum of the following terms: a living reward of $-0.05$ for every time-step; a reward of $0.2$ for stepping closer to the goal, and $-0.2$ for stepping further away (ignoring walls); and a reward of $5.0$ for reaching the goal." ], [ "Architectures and Hyper-Parameters", "In this section, we describe the architectures used by the policy and critic models; a general overview is shown in fig:architecture.", "The general architecture will be similar for all domains; however there will some differences to accommodate the fact that the flat POMDPs provide states and observations as categorical indices, while the gridverse POMDPs provide states and observations as volumes and arrays of categorical indices, whose structures include spatial relationships." ], [ "Features Extraction for Flat POMDPs", "These components are shown in fig:architecture:representations:flat.", "Because flat POMDPs provide states, actions and observations as categorical indices, and we use 128-dimensional embedding models to represent each of them." ], [ "Features Extraction for Gridverse POMDPs", "These components are shown in fig:architecture:representations:gridverse.", "Because gridverse POMDPs provide actions as categorical indices, and we use 128-dimensional embedding models to represent them.", "On the other hand, states and observations are provided in the format described in sec:environments:gridverse, each composed of a grid and an agent component: grid The grid component is a $6\\times H\\times W$ volume of categorical data, which we pass through 4-dimensional embeddings, resulting in a $24\\times H\\times W$ volume of numeric data.", "We further pass that data to two layers of CNN followed by ReLU non-linearities, resulting in a $16\\times H\\times W$ volume of data, finally flattened into a $16 HW$ -dimensional array.", "agent The agent component is a 6-dimensional array of categorical data, which we pass through 4-dimensional embeddings, resulting in a 24-dimensional array of numeric data.", "The grid and agent representations are then concatenated to form the state or observation representation $\\phi (s)$ or $\\phi (o)$ ." ], [ "Remainder of the Architecture", "These components are shown in fig:architecture:a2c.", "Concatenated action and observation embeddings form the to a 128-dimensional single-layer gated recurrent unit (GRU) [3], which acts as a history representation $\\phi (h)$ .", "The history representation is then fed into separate policy and critic models, each a 2-layer 128-dimensional feedforward model with ReLU non-linearities.", "A practical issue we found with A2C(h,s) is that the history and state representations $\\phi (h)$ and $\\phi (s)$ contain features with very different orders of magniture; To address this, we use layer-normalization.", "To guarantee a fair comparison with the other methods, we do the same for other critics as well.", "Next, we describe architectural details specific to each method: A2C(h): The history representation passes through layer-normalization first, and then is fed to the critic model; A2C(s): The state embedding passes through layer-normalization first, and then is fed to the critic model; A2C(h,s): The history representation and state embedding individually pass through layer-normalization first, and then are concatenated to form a single feature vector, which is then fed to the critic model." ] ]
2105.11674
[ [ "Argument Undermining: Counter-Argument Generation by Attacking Weak\n Premises" ], [ "Abstract Text generation has received a lot of attention in computational argumentation research as of recent.", "A particularly challenging task is the generation of counter-arguments.", "So far, approaches primarily focus on rebutting a given conclusion, yet other ways to counter an argument exist.", "In this work, we go beyond previous research by exploring argument undermining, that is, countering an argument by attacking one of its premises.", "We hypothesize that identifying the argument's weak premises is key to effective countering.", "Accordingly, we propose a pipeline approach that first assesses the premises' strength and then generates a counter-argument targeting the weak ones.", "On the one hand, both manual and automatic evaluation proves the importance of identifying weak premises in counter-argument generation.", "On the other hand, when considering correctness and content richness, human annotators favored our approach over state-of-the-art counter-argument generation." ], [ "Introduction", "Following walton:2009, a counter-argument can be defined as an attack on a specific argument by arguing against either its claim (called rebuttal), the validity of reasoning of its premises toward its claim (undercut), or the validity of one of its premises (undermining).", "Not only the mining and retrieval of counter-arguments have been studied [18], [24], recent works also tackled the generation of counter-arguments.", "Among these, bilu:2015 and hidey:2019 studied the task of contrastive claim generation, the former in a partly rule-based manner, the latter data-driven.", "Moreover, hua:2019 proposed a neural approach.", "So far, however, research focused only on rebutting a given argument, ignoring the other aforementioned types.", "We expand this research by studying to what extent argument undermining can be utilized in counter-argument generation.", "Table: An example argument (claim + premises) and a counter-argument in response to it, taken from Reddit changemyview.", "The italicized premise part was quoted by the user who stated the counter-argument.In argument undermining, the validity of some premises is questioned.", "Such a phenomenon can be observed often in online discussions on social media.", "For example, in the discussion excerpt in Table REF , taken from the Reddit forum changemyview,https://en.wikipedia.org/wiki/R/changemyview a user contests the whole stated argument (claim and premises) refers to the specific premise highlighted in the table (on Reddit, it is the quoted part of the text).", "This implies two steps: first, to identify a potentially weak and thus attackable premise in the argument, and second, to counter it.", "In this work, we propose to tackle the task of counter-argument generation by attacking one of the weak premises of an argument.", "We hypothesize that identifying a weak premise is key to effective counter-argument generation—especially when the argument is of high complexity, comprising multiple interlinked claims and premises, making it hard to comprehend the argument as a single unit.", "Figure REF illustrates our two-step pipeline approach: it detects premises that may be attackable and then generates a counter-argument addressing one or more of these premises.", "To identify weak premises, we build on the work of jo:2020, who classify attackable sentences using BERT.", "In contrast, we rank premises based on their attackability concerning the argument's main claim, utilizing the learning-to-rank approach of han:2020.", "For the second step, similar to wolf:2019, we fine-tune a pre-trained transformer-based language model [19], in a multi-task learning setting: next-token classification and counter-argument classification.", "Figure: Argument undermining: Instead of countering a given argument directly, our approach first ranks the argument's premises by weakness.", "Then, an attack focused on the weakest premises is generated.In our experiments, we make use of the changemyview (CMV) dataset of jo:2020, where each instance is a post consisting of a title (say, an argument's claim) and a text (the argument's premises).", "Some of the sentences in the text are quoted by comments to the post.", "These sentences are considered to be weak/attackable premisesOur assumption is that each sentence represents a premise supporting the main claim mentioned in the title of the post.", "We further extend the dataset by collecting texts from comments, defining counter-arguments.", "To analyze our approach, we evaluate both of its steps individually as well as in combination.", "In particular, we first compare our ranking model for detecting attackable premises to jo:2020, observing significant improvements in the effectiveness.", "Second, given the ground-truth attackable premise (the quoted sentences), we evaluate our counter-argument generation model against several baselines.", "Our automatic evaluation provides evidence that training the model with the weak premise annotated significantly boosts the scores across all metrics.", "We additionally confirm these results by a manual evaluation, indicating that our approach is better than the baseline in 56% of the cases.", "Finally, we apply our generation model based on the automatically detected weak premises and compare it to the approach of hua:2019, which generates counter-arguments with opposing stance to the argument (i.e., rebuttals).", "While the automatic evaluation here was not in favor of our approach, the manual evaluation gave evidence of the favorability of our approach on all three tested quality dimensions.", "To summarize, our contributions areCode and resources can be found https://github.com/webis-de/ACL-21: A model for detecting premise attackability, achieving state-of-the-art effectiveness.", "A new approach to counter-argument generation that identifies and attacks weak premises.", "Empirical evidence of the importance of considering a specific attackable premise in the argument when generating a counter-argument." ], [ "Related Work", "Recently, text generation has gained much interest in computational argumentation, both for single claims and complete arguments.", "bilu:2015 composed opposing claims combining rules with classifiers, whereas hidey:2019 tackled an analog task with neural methods.", "alshomary:2020 reconstructed implicit claims from argument premises using triplet neural networks, and gretz:2020 explored ability of GPT-2 to generate claims on topics.", "Recently, alshomary:2021 studied how to encode specific beliefs into generated claims.", "sato:2015 generated full arguments in a largely rule-based way.", "el-baff:2019 modeled argument synthesis as a language modeling task, and [21] studied the neural generation of arguments on a topic with controlled aspects and stance.", "Unlike all these, we deal with counter-arguments.", "Research exists for mining attack relations [5], [4], [16], mining counter-considerations from text [18], and retrieving counter-arguments [24], [15].", "However, only the consecutive works of hua:2018,hua:2019 addressed the generation task.", "Their latest neural approach takes an argument or claim as input and generates a counter-argument rebutting it.", "Differently, we consider countering an argument by attacking one of its premises, known as undermining [25].", "Part of our approach is to identify attackable premises, which can be studied from an argument quality perspective.", "That is, a premise is attackable when it lacks specific quality criteria.", "A significant body of research has studied argument quality assessment, with a comprehensive survey of quality criteria presented in wachsmuth:2017a.", "Implicitly, we target criteria such as a premise's acceptability or relevance.", "Still, we follow jo:2020 in deriving attackability from the sentences of posts that users in the Reddit forum CMV attack.", "These sentences represent premises supporting the claim encoded in the post's title.", "The authors experimented with different features that potentially reflect weaknesses in the premises.", "Their best model for identifying attackable premises is a BERT-based classifier.", "We use their data to learn weak premise identification, but we address it as a learning-to-rank task.", "As for text generation, significant advances have been made through fine-tuning large pre-trained language models [22] on target tasks.", "We also benefit from this by utilizing a pre-trained transformer-based language model [6], and we fine-tune it in a multi-task fashion similar to wolf:2019." ], [ "Approach", "As sketched in Figure  REF , our pipeline approach counters an argument by attacking the validity of one of its potentially weak premises.", "This section presents the two main steps of our approach: first, the ranking of weak and thus attackable premises, and second, the generation of the weak premises' attack." ], [ "Weak-Premise Ranking", "Given an argument in the form of a claim and a set of premises, the task is to identify the argument's attackable premises.", "Unlike previous work [14], we model the task as a ranking task instead of a classification task, in which, for each argument, we learn to rank its premises by their weakness relevant to the claim.", "Our hypothesis here is that the attackability of a premise can be better learned when considering both the claim and other premises of the argument.", "We operationalize the weak-premise ranking similar to the ranking approach of han:2020.", "In particular, given a set of premises and the claim, we first represent each premise by concatenating its tokens with the claim's tokens, separated by special tokens [cls] and [sep]: [cls] claim_tokens [sep] premise_tokens [sep] Next, the resulting sequences are passed through a BERT model to obtain a vector representation for every premise.", "Each vector is then projected through a dense layer to get a score $\\hat{y}$ that reflects the weakness of the premise.", "Finally, a list-wise objective function (we use a Softmax loss) is optimized jointly on all premises of an argument as follows: $l(y, \\hat{y}) \\;=\\; - \\sum _{i=1}^{n} y_i \\cdot \\log \\Big (\\frac{exp(\\hat{y_i})}{\\sum _{j=1}^{n} exp(\\hat{y_{j}})}\\Big ),$ where $y$ is a binary ground-truth label reflecting whether the given premise is attackable ($y=1$ ) or not ($y=0$ ).", "Given training data, we can thus learn to rank premises by weakness." ], [ "Premise Attack Generation", "Given the ranking step's output, we identify the $k$ highest-ranked premises in an argument to be attackable (in our experiments, we test $k = 1$ and $k =3$ ).", "Then we generate a counter-argument putting the identified attackable premises into the focus.", "To this end, we follow wolf:2019 in using transfer learning and fine-tune a pre-trained transformer-based generation model on our task.", "Figure: Architecture of our approach: Given an argument, a weak premise, and a counter, three embedding representations are generated and fed to the transformer to obtain hidden states from which the language model and classification heads learn the Next-token prediction and Counter-argument classification tasks respectivelyIn our fine-tuning process, the input is a sequence of tokens created from two segments, the argument and the counter-argument: [bos] arg_tokens [counter] counter_tokens [eos] The final token embedding is then a result of concatenating three embeddings: word and positional embeddings learned in the pre-training process, as well as a token-type embedding learned in the fine-tuning process.", "Here, the token type reflects whether the token belongs to the argument in general, to a weak premise, or the counter-argument.", "Now, we train our model jointly on two tasks: Next-token prediction.", "Given a sequence of tokens, predict the next one.", "Counter-argument classification.", "Given two concatenated segments, is the second a counter-argument to the first.", "The first task is similar to the next-sentence prediction task introduced in [6], which was shown to be beneficial for multiple representation-learning tasks.", "Figure REF shows the architecture of our generation model.", "For training, we augment a given set of training sequences $D$ by adding distracting sequences, which we use, for each argument and its weak premise, a non-relevant text instead of the counter-argument.", "Given a sequence of tokens $d = (t_1, t_2, \\cdots , t_n) \\in D$ , we then optimize the following two loss functions jointly with equal weighting: $L_1(\\Theta ) & = & \\sum _{d \\in D} \\sum _{t_i \\in d}^{} \\log P(t_i \\,|\\, t_{i-k}, \\cdots ,t_{i-1}; \\Theta ), \\\\L_2(\\Theta ) & = & \\sum _{d_j \\in D}^{} \\log P(y_j \\,|\\, t_1, \\cdots , t_n; \\Theta ),$ where $\\Theta $ denotes the weights of the model, $k$ is the number of previous tokens, and $y_j$ is the ground-truth label of the sequence, indicating if the second segment of the sequence is a counter or not." ], [ "Data", "As proposed, the presented approach models the task of counter-argument generation as an attack on a potentially attackable premise.", "Such behavior is widely observed on the Reddit forum changemyview (CMV).", "In particular, a user writes a new post that presents reasons supporting the pro or con stance towards a given topic (captured in the title of the post), asking the CMV community to challenge the presented view.", "In turn, other users quote specific segments of the post (usually a few sentences) and seek to counter them in their comment.", "An example has already been given in Figure REF .", "The structure induced by CMV defines a suitable data source for our study.", "Specifically, we create the following distantly supervised mapping: The title of the post denotes the claim of the user's argument; the text of the post denotes the concatenated set of the argument's premises; the quoted sentence(s) denote the attackable (weak) premises; and the quoting sentences from the comment denote the counter-argument.", "In our work, we build on the CMV dataset of jo:2020, where each instance contains a post, a title, and a set of attackable sentences (those quoted in the comments).", "We use the same split as the authors, consisting of 25.8k posts for training, 8.7k for validation, and 8.5k for testing.", "We extend their dataset by further collecting the quoting sentences from the comments (i.e., the counter-arguments).", "The final dataset compiles 111.9k triples of argument (claim and premises), weak premise (one sentence or more), and a counter-argument (a set of sentences), split into 67.6k training, 23k validation, and 22.3k for testing instances." ], [ "Evaluation", "In the following, we present the experiments we carried out to evaluate both steps of our approach individually as well as in a pipelined approach.", "On the one hand, we aim to assess the applicability of identifying weak premises in an argument and the impact of targeting them in the process of counter-argument generation.", "On the other hand, our goal is to assess how well counter-argument generation via undermining is compared to other known counter-argument generation approaches." ], [ "Weak-Premise Ranking", "As presented, we approached the task of finding attackable premises by learning to rank premises by their weakness with respect to the main claim." ], [ "Approach", "Based on the code of han:2020 available in the Tensorflow learn-to-rank framework [17], we used a list-wise optimization technique that considers the order of all premises in the same argument.We also experimented with point-wise and pairwise techniques, but list-wise was best.", "We trained our ranking approach on the CMV dataset's training split and referred to it as bert-ltr below.Training details can be found in the appendix." ], [ "Baselines", "We compare our approach to the Bert-based classifier introduced by jo:2020, trained on the same training split using the authors' code.", "We use their trained model to score each premise and then rank all premises in an argument accordingly.", "We call this bert-classifier.", "As jo:2020, we also consider a random baseline as well as a baseline that ranks premises based on sentence length." ], [ "Measures", "To assess the effectiveness, we followed jo:2020 in computing the precision of putting a weak premise in the first rank ($P@1$ ), as well as the accuracy of having at least a weak premise ranked in the top three ($A@3$ )." ], [ "Results", "Table REF shows the weak-premise ranking results.", "We managed to almost exactly reproduce the values of jo:2020 for all three baselines.", "Our approach, bert-ltr, achieves the best scores according to both measures.", "In terms of a one-tailed dependent student-$t$ test, the differences between bert-ltr and bert-classifier results are significant with at least 95% confidence.", "These results support our hypothesis of the importance of tackling the task as a ranking task with respect to the main claim.", "Below, we will use our weak-premise ranking model in the overall approach, i.e., to automatically select attackable premises in an argument.", "Table: Weak-premise ranking: Precision of ranking a weak premise highest (P@1) and accuracy for the top three (A@3) of all evaluated approaches.", "Results with * are significantly better than bert-classifier at p<.05p < .05." ], [ "Premise Attack Generation", "Next, we evaluate our hypothesis of the importance of identifying weak premises in the process of counter-argument generation.", "To focus on this step, we use the ground-truth weak premises in our data.", "These are the quoted sentences in the post, considered potentially attackable premises." ], [ "Approach", "We used OpenAI GPT as a pre-trained language model.", "We trained two versions of our generation model: our-model-w/ with an extra special token ([weak]), surrounding the attackable sentences to give an extra signal to our model, and once our-model-w/o without it.", "We fine-tuned both versions with the same settings using the transformers library [26] for six epochsWe stopped at six epochs because we observed no gain in terms of validation loss.", "We left all other hyperparameters to their default values.", "As mentioned, the model's input is a sequence of tokens constructed from the argument (with weak premises highlighted) and either the correct counter or a distracting sequence.", "We randomly select one sentence from the original post to be the distracting sequence for each input instance.", "Table: Premise attack generation: METEOR and BLEU scores of the output of each evaluated approach compared to the ground-truth counter sentences and the full comment (argument).", "Values marked with * are significantly better than counter-baseline at p<.05p < .05." ], [ "Baseline", "We compare our model to a GPT-based model fine-tuned on a sequence of tokens representing a pair of an argument (title and post) and a counter-argument.", "We consider this as a general counter-argument generation model, trained without any consideration of weak premises.", "We train the baseline using the same setting as our model.", "We refer to it as counter-baseline." ], [ "Automatic Evaluation", "To assess the importance of selecting attackable sentences, we evaluated the effectiveness of our model in different inference settings in terms of what is being attacked: (1) the claim of the argument, (2) a random premise, or (3) a weak premise given in the ground-truth data.", "In the random setting, we randomly selected three premises from the argument, and we generated one counter for each.", "The final result is the average of the results for each.", "Figure: Premise attack generation: Mean token overlap between the ground-truth weak premises and the counters generated by each evaluated approach.We computed METEOR and BLEU scores, comparing the generated premises to (a) the exact counter sentences of the quoted weak premise and (b) the full argument.", "We performed this automatic evaluation on 1k posts from the test split." ], [ "Results", "As shown in Table REF , the best results are achieved by our-model-w/o in all cases when identifying the weak premises in the input.", "Encoding the knowledge about weak premises as token types is sufficient, and adding an extra special token doesn't help.", "Although the differences between our best model and the baseline are not big, they are significant according to the one-tailed dependent student-$t$ test with a confidence level of 95%.", "For both versions of our model, best scores are achieved when considering the weak premises as the target (except for the first METEOR column).", "However, not all these differences are significant.", "This gives evidence that exploiting information about weak premises in the training of counter-argument generation approaches can improve their effectiveness.", "To further assess the relationship between the generated counters and the attacked premises, we computed the proportion of covered content tokens in the weak premise for the two versions of our model and the baseline.", "Figure shows a histogram of the percentages.", "Clearly, both versions of our model have higher coverage of the annotated weak premise than the baseline.", "Table: Premise attack generation: Percentage of cases where each given approach was seen as more relevant and more appropriate, respectively, according to majority vote and the full agreement in the manual evaluation on 50 examples.", "The bottom line shows the mean pairwise inter-annotator agreement." ], [ "Manual Evaluation", "To analyze the generated counter-arguments more thoroughly, we carried a manual evaluation study on a sample of 50 random examples.", "Two authors of the paper inspected the sample comparing the two versions of our model.", "The results were in favor of our-model-w/o.", "Therefore, we compared only our-model-w/o against the counter-baseline.", "In particular, we assessed the relevance and appropriateness of the output of the two for each example.", "Given an argument, the highlighted premise to be attacked, and the two counters, we asked three annotators who hold an academic degree and are fluent in English (no author of this paper) to answer two questions: Which text is more relevant to the highlighted premise?", "Which text is more appropriate for being used as a counter-argument?" ], [ "Results", "As shown in Table REF , considering the majority vote, annotators favored our model in 56% of the cases in both tasks.", "These results give further evidence supporting our hypothesis of the importance of identifying weak premises.", "Considering the given task as a ranking task, we used Kendall's $\\tau $ to compute the annotator's agreement.", "The mean pairwise agreement was 0.41 for the relevance assessment and 0.23 for appropriateness.", "Clearly, assessing the text's appropriateness of being a counter-argument is more subjective and more challenging to judge than the relevance task." ], [ "Overall Approach", "Finally, we assess the overall effectiveness of our counter-generation approach, that is, when we identify weak premises automatically using Bert-ltr and then generating a counter-argument using our-model-w/o, focusing on the selected weak premises." ], [ "Approach", "Due to the limited P@1 value of our ranking model (see Table REF ), we evaluate two variations of our overall approach that differ in terms of what premises to attack.", "The first variant attacks the weakest premise.", "In the second, we first generate three counters considering each of the top three weak premises.", "Then, we select the counter that has the most content-token overlap with the corresponding weak premise.", "Table: Overall approach: METEOR and BLUE scores of the two variants with different attacked targets, the counter-baseline, and hua:2019." ], [ "Baselines", "On the one hand, we compare our approach to the counter-baseline from the previous section.", "On the other hand, we consider the state-of-the-art counter-argument generation approach of hua:2019, an LSTM-based Seq2seq model with two decoders, one for selecting talking points (phrases) and the other for generating the counter given the selection." ], [ "Automatic Evaluation", "While the approach of hua:2019 learns from a dataset collected from the same source (CMV), it requires retrieving relevant argumentative texts with a stance opposite to the input argument.", "Due to the complexity of the data preparation, we decided instead to evaluate all approaches on the test split of hua:2019.We verified that all posts in their test split do not appear in our training split.", "As a result, hua:2019's approach is trained on their training split, while our approach is trained on our training split, and then both are evaluated on the same test split of hua:2019.", "This can be considered a somewhat unfair setting for our approach due to certain domain differences since the dataset of hua:2019 comprises political topics only.", "Similar to Section REF , we generated counters for 1k examples and computed METEOR and BLEU scores of the generated counters with respect to the ground-truth counters, which are here full arguments (CMV comments)." ], [ "Results", "Table REF shows that our approach outperforms the counter-baseline in both settings, even with weak premises selected automatically.", "Considering the top-3 weak premises instead of the top-1 improves the results.", "The best scores are achieved by Hua and Wang, though.", "A reason for this may be the slight domain difference between our model's training data and the test data used for evaluation.", "Another observation is that the scores of both our approach and the baseline increase compared to Table REF .", "This is likely to be caused by the higher number of ground-truth references for each instance in data of hua:2019 compared to the test split of our data, making it more likely to have token overlaps." ], [ "Manual Evaluation", "Given the known limited reliability of automatic generation evaluation, we conducted another user study to evaluate the quality of the generated counters by our model and Hua and Wang.", "We evaluate the same quality dimensions used previously by hua:2019: Content Richness.", "The diversity of aspects covered by a counter-argument.", "Correctness.", "The relevance of a counter-argument to the given argument and their degree of disagreement.", "Grammaticality.", "The grammatical correctness and fluency of a counter-argument.", "We used the Upwork crowd working platform to recruit three annotators with English proficiency and experience in editorial work.Upwork, http://upwork.com We asked each of them to evaluate a sample of 100 examples.", "Each contained an argument (claim and premises) and two counters (one of each approach).", "We asked the annotators to compare the counters and assess each with a score from 1 (worst) to 5 (best) for each quality dimension.", "Table: Overall approach: Average scores of the three annotators for the three evaluated quality dimensions of the counter-arguments generated by our approach and the one of hua:2019.", "1 is worst, 5 is best.", "The bottom line shows the inter-annotator agreeement." ], [ "Results", "The results are presented in Table REF .", "Unlike in the automatic evaluation, the annotators gave, on average, higher scores on all quality dimensions to our generated counters than to those of Hua and Wang.We note that the scores of Hua and Wang in Table REF are notably lower than those reported by hua:2019.", "We believe this to be due to the comparison with our approach that affected the annotator's scores.", "Bringing knowledge from pre-trained language models (GPT) generally seems to contribute to the grammaticality and the richness of the generated counters.", "In terms of generating a correct counter, focusing the generation model on a specific weak premise in an argument seems to help (2.65 vs. 1.81), even though the results are far from perfect.", "Manual inspection revealed that far from all generated arguments are counters to exactly what is in the argument, indicating more room to work on this topic.", "Figure: Example counter-arguments generated by our approach and by the approach of hua:2019.", "The italicized premise segment was identified as the weak premise by our approach.Krippendorff's $\\alpha $ values show that the annotators had a fair agreement on grammaticallity and correctness tasks (given the subjectiveness of the tasks), but only slight agreement on content richness.", "We, therefore, think that the results for the latter should not be overinterpreted.", "In Figure REF , we show an example argument in favor of income inequality.", "Our approach considers the premise “being poor does not entitle someone to the cash of the rich people”.", "It then generates a counter-argument on the topic of inequality, focusing on the fact that “being poor limits the ability to contribute to society\".", "In contrast, the counter-argument generated by Hua and Wang diverges to address “low-income housing” which is less relevant to the topic.", "More examples of generated counters can be found in Figure 5 in the appendix." ], [ "Conclusion", "In this work, we have proposed a new approach to counter-argument generation.", "The approach focuses on argument undermining rather than rebuttal, aiming to expand the research in this area.", "The underlying hypothesis is that identifying weak premises in an argument is essential for effective countering.", "To account for this hypothesis, our approach first ranks the argument's premises by weakness and then generates a counter-argument to attack the weakest ones.", "In our experiments, we have first evaluated each step individually.", "We have observed state-of-the-art results in the weak-premise identification task.", "Our results also support the need for identifying weak premises to generate better attacks.", "We have also evaluated the overall approach against the state-of-the-art approach of hua:2019.", "While we did not beat that approach in automatic evaluation scores, independent annotators favored the counter-arguments generated by our approach across all evaluated quality dimensions.", "We conclude that our approach improves the state of the art in counter-argument generation in different respects, providing support for our hypothesis.", "Still, the limited manual evaluation scores imply notable room for improvement.", "Most importantly, controlling the stance of the generated counters is yet to be fully solved." ], [ "Ethical Statement", "We acknowledge that ethical issues might arise from our work.", "First, we would like to ensure that we did not violate user privacy when using data from public platforms.", "By reusing pre-trained models, our approach might have inherited some forms of bias.", "Mitigating such bias is still ongoing research.", "It is worth noting that our experiments show that our approach is far from being ready to be used as an end technology.", "Our goal is to advance the research on this task." ], [ "Acknowledgments", "This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Center “On-The-Fly Computing” (SFB 901/3) under the project number 160364472." ], [ "Models and Training Specifications", "All our models were trained on one 24 GeForce GTX 1080 GPU." ], [ "Weak-Premise Ranking", "We use the code of [9] to train our ranking model Bert-ltr, with BERT-Base, Uncased for embedding (12-layer, 768-hidden, 12-heads, 110M parameters).", "The model was trained for 100k steps, which took almost 9 hours of training." ], [ "Premise Attack Generation", "We use the transformers library to train our generation models, with GPT as a pre-training language model (12-layer, 768-hidden, 12-heads, 110M parameters.", "OpenAI GPT English model).", "The model was fine-tuned for six epochs, which took almost one day of training.", "In the generation time, we use top-$k$ sampling technique [13], with the following parameters: top-$k$ =50, top-$p$ =0.95, and temperature=1.0.", "We generate counters of a minimum of 100 tokens and a maximum of 150 tokens." ], [ "Evaluation Measures", "The BLEU and METEOR scores are computed using the NLTK library (https://www.nltk.org/).", "Significance tests were performed using the Scipy library (https://www.scipy.org/)." ], [ "Example Counters", "In this section, we present example counters generated by our approach and hua:2019 Figure: A list of examples of counter-arguments generated by our approach and by the approach of Hua and Wang (2019).", "The italicized premise segment was identified as the weak premise by our approach." ] ]
2105.11752
[ [ "Kernel Knockoffs Selection for Nonparametric Additive Models" ], [ "Abstract Thanks to its fine balance between model flexibility and interpretability, the nonparametric additive model has been widely used, and variable selection for this type of model has been frequently studied.", "However, none of the existing solutions can control the false discovery rate (FDR) unless the sample size tends to infinity.", "The knockoff framework is a recent proposal that can address this issue, but few knockoff solutions are directly applicable to nonparametric models.", "In this article, we propose a novel kernel knockoffs selection procedure for the nonparametric additive model.", "We integrate three key components: the knockoffs, the subsampling for stability, and the random feature mapping for nonparametric function approximation.", "We show that the proposed method is guaranteed to control the FDR for any sample size, and achieves a power that approaches one as the sample size tends to infinity.", "We demonstrate the efficacy of our method through intensive simulations and comparisons with the alternative solutions.", "Our proposal thus makes useful contributions to the methodology of nonparametric variable selection, FDR-based inference, as well as knockoffs." ], [ "Introduction", "In the past decades, the nonparametric additive model has been widely used in statistics and machine learning, thanks to its fine balance between model flexibility and model interpretability [42], [23], [50].", "For a univariate response variable $Y \\in \\mathbb {R}$ and $p$ predictor variables $=(X_1,\\ldots ,X_p)^{\\mbox{\\tiny {\\sf T}}}\\in ^p \\subseteq \\mathbb {R}^p$ , the model postulates that, $ Y = \\mu +\\sum _{j=1}^pf_j(X_j)+\\epsilon ,$ where $\\mu $ is the intercept, $f_j : \\mapsto \\mathbb {R}$ , with $\\mathbb {E}_{X_j}[f_j(X_j)]=0$ , $j=1,\\ldots ,p$ , are the component functions that are modeled nonparametrically, and $\\epsilon \\sim (0,\\sigma ^2)$ is the random error, with unknown $\\sigma $ .", "Furthermore, we assume throughout this article that the component functions $f_j$ 's reside in a reproducing kernel Hilbert space [3], [48].", "Variable selection for the nonparametric additive model dates back to [30], and has seen substantial developments ever since [31], [37], [25], [27], [50].", "In particular, [30] proposed a component selection and smoothing operator (COSSO) penalty that extends the Lasso penalty to the nonparametric additive model, and penalized the sum of the reproducing kernel Hilbert space norms of the component functions.", "Meanwhile, [31] and [37] both employed basis expansion and penalized the sparsity and smoothness seminorms.", "[25] employed the group Lasso penalty to obtain an initial estimator and to reduce the dimension of the problem, then employed the adaptive group Lasso to select nonzero components.", "This family of methods guarantee the asymptotic optimality of the function estimation and the selection consistency as the sample size tends to infinity.", "However, none has studied the FDR control under the finite sample setting.", "There has been another family of solutions that target simultaneous testing of multiple hypotheses and concentrate on controlling some forms of false discovery [9], [19], [43], [44], [45]; see also [11] for a review.", "Nevertheless, none of the existing solutions in this family directly addresses the problem of variable selection for the nonparametric additive model while controlling the false discovery at the same time.", "More recently, [6] proposed a powerful framework called knockoffs that effectively controls the false discovery rate (FDR) for variable selection in the linear model with a finite sample size.", "The key idea is to construct a set of so-called “knockoff variables\" that are not associated with the response conditioning on the original variables, while the structure of the knockoff variables mimics that of the original ones.", "It then computes an importance score for each variable, and selects those that have considerably higher scores than their knockoff counterparts.", "This framework is fairly general, but cannot be directly applied beyond the linear model.", "There have then been numerous generalizations of the knockoffs framework; see [7] and many references therein.", "Related to our target of nonparametric additive model, [14] developed an “expansion first\" strategy that performs feature expansion first, then constructs the knockoffs based on the expanded features, and they proposed to employ a group Lasso penalty for subsequent variable selection.", "They actually still studied the linear regression setting; however, their proposal can, in principle, be extended to the nonparametric additive model.", "[12] proposed a model-X knockoffs extension that works for random designs of predictors and allows the conditional distribution of the response given the predictors to be arbitrary and unknown.", "Nevertheless, they mostly focused on the step of how to generate the knockoff variables.", "[20] further built on the model-X knockoffs framework, and developed a knockoffs-based variable selection procedure that is applicable to the nonparametric additive model.", "It employs data splitting, uses half of the data to estimate the predictor precision matrix and screen the predictors, and uses the other half to perform knockoffs based on some empirical norm of the estimated component functions.", "These pioneering works have opened the door for knockoffs-based selection for the nonparametric additive model.", "However, they may be computationally intensive, suffer a limited power, and lack theoretical power analysis, in the context of nonparametric selection.", "In this article, we propose a novel kernel knockoffs selection procedure for the nonparametric additive model (REF ).", "We build on and integrate three key components: the knockoffs, the subsampling for stability, and the random feature mapping for nonparametric function approximation in RKHS.", "Specifically, we employ the random feature mapping [35] to approximate the component function $f_j$ , and construct a projection operator between the RKHS and the original predictor space.", "Such a projection allows us to define an analog of the effect size of the individual predictor in the setting of nonparametric additive model.", "We then construct the importance score based on the projected component function $f_j$ , instead of the original predictor $X_j$ or its knockoff.", "Moreover, we note that the random features may introduce additional stochastic errors, which can disturb the order of the variables entering the model and lead to both false positives and false negatives.", "We thus further employ the subsampling strategy to improve the selection stability [32].", "That is, we subsample the data and apply the random feature mapping multiple times, and compute the importance score as the difference of selection frequencies over subsampling replications between each predictor and its knockoff counterpart.", "We show that the proposed method is guaranteed to control the FDR below the nominal level under any finite sample size, and achieves a power that approaches one as the sample size tends to infinity.", "Our proposal makes useful contributions to the methodology of nonparametric variable selection, FDR-based inference, as well as knockoffs.", "First of all, whereas the methods such as [30], [37] obtain the variable selection consistency asymptotically, there has been no existing method that controls the FDR in the setting of nonparametric additive model with a finite sample size.", "Actually, the finite-sample FDR control for nonparametric variable selection has been a long-standing and open question, and our proposal is among the first solution for this kind of question.", "Second, our proposed FDR control is different from the existing subsampling based methods for FDR control.", "More specifically, [4] proposed a selection method based on bootstrap replications, which may suffer from a limited power under a finite sample size.", "[32] proposed a stability-based procedure to select the variables with the selection frequencies exceeding a threshold level over the entire solution path, which guarantees the control of the expected number of false positives, but may fail to control the FDR.", "[29] proposed to use a mixture model for the distribution of selection frequencies, whereas [1], [24] suggested to estimate this distribution via permutations.", "However, such a distribution estimation requires either strict parametric assumptions, or requires expensive computations.", "By contrast, our method does not require estimation of the distribution of selection frequencies, but still manages to control the FDR.", "Finally, our method also expands the scope of the currently fast growing area of knockoffs.", "Specifically, we generalize the knockoffs from the linear model to the nonparametric additive model.", "In addition, compared to the “expansion first\" strategy of [14], our method adopts a “knockoffs first\" strategy, which leads to an easier construction of the knockoff variables, ensures a good statistical power, and is much more economical computationally.", "Compared to the model-X based knockoffs method of [20], our method integrates subsampling and develops a new importance score that is built on the selection probability of the variables and their knockoffs.", "As a result, we show that our solution is more powerful and also more robust to the data distribution.", "We later numerically compare with several key alternative solutions.", "We further remark on some theoretical challenges and our contributions in theory.", "First, the classical kernel methods usually involve non-separable variables and their knockoffs, which renders the FDR control infeasible.", "By contrast, we employ the random feature mapping, which ensures the exchangeability of the null variables and their knockoffs, and in turn achieves the finite-sample FDR control.", "On the other hand, the random feature mapping introduces an extra layer of randomness.", "We employ subsampling and an importance score based on averaging over multiple subsampling replications, and show this technique can handle the extra randomness.", "Second, there is generally a lack of formal power analysis for the knockoffs-based methods in the literature, except for [20] for only the linear model.", "Nevertheless, the power analysis for the nonparametric additive model involves nonlinear dependences between the predictors and response, which adds another level of difficulty.", "We employ some functional data analysis techniques and concentration inequalities for functional empirical processes to study the spectral properties.", "Finally, our theoretical tools are readily applicable to the FDR control for more general nonparametric models, e.g., the functional analysis of variance type models that involve higher-order interactions [49], [30].", "The rest of the article is organized as follows.", "Section begins with the problem setup.", "Section develops the kernel knockoffs procedure.", "Section establishes the theoretical guarantees on the FDR and power.", "Section presents the simulations, and also an analysis of brain imaging data.", "The supplementary appendix collects all technical proofs." ], [ "Kernel learning", "Throughout this article, we consider regression functions that reside in an infinite-dimensional reproducing kernel Hilbert space.", "We begin with a Mercer kernel $K:\\mathcal {X}\\times \\mathcal {X}\\rightarrow \\mathbb {R}$ , $K(X,X^{\\prime })=\\sum _{\\nu =1}^\\infty \\widetilde{\\lambda }_\\nu \\widetilde{\\psi }_\\nu (X)\\widetilde{\\psi }_\\nu (X^{\\prime }),$ where $\\lbrace \\widetilde{\\psi }_\\nu \\rbrace _{\\nu =1}^\\infty $ are eigenfunctions, $\\lbrace \\widetilde{\\lambda }_\\nu \\rbrace _{\\nu =1}^\\infty $ are eigenvalues of the integral operator defined by the kernel function, and $\\widetilde{\\lambda }_\\nu \\widetilde{\\psi }_\\nu (X) = \\int _K(X,X^{\\prime })\\widetilde{\\psi }_\\nu (X^{\\prime })dX^{\\prime }$ [33].", "We consider the RHKS $_1$ generated by this kernel, which is defined as the closure of linear combinations of the basis functions $\\lbrace \\widetilde{\\psi }_\\nu \\rbrace _{\\nu =1}^\\infty $ , along with an RKHS norm $\\Vert \\cdot \\Vert _K$ , $_1 = \\overline{\\left\\lbrace f:\\rightarrow \\mathbb {R}| f(X) = \\widetilde{}(X)^{\\mbox{\\tiny {\\sf T}}}\\widetilde{},\\text{ and }\\Vert f\\Vert _K<\\infty \\text{ with }\\Vert f\\Vert _K^2=\\sum _{\\nu =1}^\\infty \\frac{\\widetilde{c}_\\nu ^2}{\\widetilde{\\lambda }_\\nu }\\right\\rbrace },$ where $\\widetilde{}(X)$ is an infinite-dimensional vector with the $\\nu $ th element equal to $\\sqrt{\\widetilde{\\lambda }_\\nu }\\widetilde{\\psi }_\\nu (X)$ , and $\\widetilde{}$ is an infinite-dimensional coefficient vector with the $\\nu $ th element $\\widetilde{c}_\\nu $ , $\\nu = 1, 2, \\ldots $ .", "Next, define the kernel $K_p:^p\\times ^p\\rightarrow \\mathbb {R}$ , $K_p\\Big ( (X_1,\\ldots ,X_p)^{\\mbox{\\tiny {\\sf T}}},(X^{\\prime }_1,\\ldots ,X_p^{\\prime })^{\\mbox{\\tiny {\\sf T}}}\\Big ) = K(X_1,X_1^{\\prime }) + \\ldots + K(X_p,X_p^{\\prime }).$ The RKHS $_p$ generated by $K_p$ is of the form [3], $_p = _1 \\oplus \\ldots \\oplus _1 = \\Big \\lbrace f : \\mathcal {X}^p \\rightarrow \\mathbb {R} \\; | \\; f() = f(X_1,\\ldots ,X_p) = f_1(X_1) + \\ldots + f_p(X_p),\\\\f_j\\in \\mathcal {H}_1, \\text{ and } \\mathbb {E}[f_j(X_j)]=0, \\; j=1,\\ldots ,p\\Big \\rbrace .$ Suppose the observed training data $\\lbrace (_i, y_i) \\rbrace _{i=1}^{n}$ consist of $n$ i.i.d.", "copies of $(, Y)$ following the nonparametric additive model (REF ), with $_i \\in \\mathbb {R}^p, y_i \\in \\mathbb {R}$ .", "The representer theorem [48] shows that the solution to the kernel learning problem when restricting $f\\in \\mathcal {H}_p$ , $\\underset{f\\in \\mathcal {H}_p}{\\min }\\left[n^{-1} \\sum _{i=1}^n\\mathcal {L}(f(_i),y_i)+\\lambda \\Vert f\\Vert _{K_p}^2\\right],$ for some loss function $\\mathcal {L}$ , the kernel $K_p$ , and the penalty parameter $\\lambda $ , is of the form, $ \\widetilde{f}() = \\sum _{i=1}^n\\alpha _iK_p(,_i),$ where $= (\\alpha _1, \\ldots , \\alpha _n)^{\\mbox{\\tiny {\\sf T}}}\\in \\mathbb {R}^n$ are the corresponding coefficients.", "This in effect turns an infinity-dimensional optimization problem to an optimization problem over $n$ parameters.", "This minimizer can be further written as, for any $\\in ^p$ , $ \\widetilde{f}() = \\widetilde{}_p()^{\\mbox{\\tiny {\\sf T}}}\\widetilde{}_p,$ where $\\widetilde{}_p() = \\left[ \\widetilde{}(X_1)^{\\mbox{\\tiny {\\sf T}}},\\ldots ,\\widetilde{}(X_p)^{\\mbox{\\tiny {\\sf T}}}\\right]^{\\mbox{\\tiny {\\sf T}}}$ assembles $\\widetilde{}(X_j)$ 's and is an infinite-dimensional vector, and $\\widetilde{}_p = \\Big [ \\widetilde{}_p(_1),\\ldots ,\\widetilde{}_p(_{n}) \\Big ] $ is the infinite-dimensional coefficient vector." ], [ "Variable selection for nonparametric additive models", "Next, we formally frame variable selection in the context of nonparametric additive models.", "We say a variable $X_j$ is null if and only if $Y$ is independent of $X_j$ conditional on all other variables $_{-j} = \\lbrace X_1,\\ldots ,X_p\\rbrace \\backslash \\lbrace X_j\\rbrace $ , i.e., $Y \\protect \\mathchoice{\\protect \\mathrel {\\unknown.", "{\\displaystyle \\perp }\\hspace{1.111pt}{\\displaystyle \\perp }}}{\\protect \\mathrel {\\unknown.", "{\\textstyle \\perp }\\hspace{1.111pt}{\\textstyle \\perp }}}{\\protect \\mathrel {\\unknown.", "{\\scriptstyle \\perp }\\hspace{1.111pt}{\\scriptstyle \\perp }}}{\\protect \\mathrel {\\unknown.", "{\\scriptscriptstyle \\perp }\\hspace{1.111pt}{\\scriptscriptstyle \\perp }}}X_j | _{-j}$ , and say $X_j$ is non-null otherwise [28].", "Let $\\subseteq \\lbrace 1,\\ldots ,p\\rbrace $ denote the indices of all the non-null variables, and $^\\perp \\subseteq \\lbrace 1,\\ldots ,p\\rbrace $ the indices of all the null variables, or equivalently, the complement set of $$ .", "Let $|\\cdot |$ denote the cardinality, and $\\widehat{\\mathcal {S}}$ the indices of variables selected by some selection procedure.", "Our goal is to discover as many non-null variables as possible while controlling the FDR, which is defined as, $\\text{FDR} = \\mathbb {E}\\left[\\frac{|\\widehat{\\mathcal {S}}\\cap \\mathcal {S}^\\perp |}{|\\widehat{\\mathcal {S}}|\\vee 1}\\right].$ We next establish the identifiability of the problem under the following condition.", "Assumption 1 (Irrepresentable Condition in RKHS) For any $j\\in \\lbrace 1,\\ldots ,p\\rbrace $ , and any functions $g_k\\in \\mathcal {H}_1, k \\ne j$ , $f_j(X_j) \\ne \\sum _{k=1; k \\ne j}^p g_k(X_k)$ .", "This condition simply says that the component function $f_j(X_j)$ in model (REF ) can not be strictly written as a linear combination of some functions of other variables $X_k, k \\ne j$ .", "This is a fairly mild condition, and its parametric counterpart that $X_j \\ne \\sum _{k=1;k\\ne j}^p\\beta _kX_k$ for any $\\beta _k\\in \\mathbb {R}$ has been commonly imposed in the linear model scenario [12].", "Under this condition, we establish the equivalence between variable selection and selection of the component functions $f_j$ in model (REF ).", "In other words, testing the hypothesis that $X_j$ is null is the same as testing whether $f_j = 0$ .", "Proposition 1 Suppose the nonparametric additive model (REF ) and Assumption REF hold.", "Then $X_j \\in \\mathcal {S}^\\perp $ if and only if $f_j = 0$ , for $j=1,\\ldots ,p$ .", "We next propose a selection procedure for model (REF ) that aims to control the FDR below a nominal level $q \\in (0,1)$ with a finite sample size, while achieving a good power." ], [ "Algorithm", "Our kernel knockoffs selection procedure consists of six main steps.", "Step 1 is to generate the knockoff variables.", "Step 2 is to subsample without replacement half of the sample observations.", "Step 3 is to construct the random features for both the original and knockoff variables.", "Step 4 is to solve the coefficient vector through a group Lasso penalized regression based on the subsamples, which in effect leads to the selection of a set of important variables.", "In addition, Steps 2 to 4 are carried out repeatedly over a number of subsampling replications.", "Step 5 is to compute the importance score for each original variable, which is defined as the empirical selection frequency based on multiple subsampling replications.", "Finally, Step 6 is to apply a knockoffs filter to the importance scores to produce the final set of selected variables under the given FDR level, as well as the final estimate of the component functions.", "We summarize our procedure in Algorithm REF first, then discuss each step in detail.", "[t!]", "Kernel knockoffs selection procedure for nonparametric additive models [1] Input: Training data $\\lbrace (x_i,y_i)\\rbrace _{i=1}^n$ , the number of random features $r$ , the number of subsampling replications $L$ , and the nominal FDR level $q\\in [0,1]$ .", "Step 1: Construct the knockoff variables $\\lbrace \\widetilde{}_i\\rbrace _{i=1}^n$ to augment the original variables $\\lbrace _i\\rbrace _{i=1}^n$ using the second-order knockoffs or the deep knockoffs machine.", "$\\ell =1$ to $L$ Step 2: Draw without replacement to obtain a subsample $I_\\ell \\subset \\lbrace 1,\\ldots ,n\\rbrace $ of size $\\lfloor n/2\\rfloor $ .", "Step 3: Sample $2p$ of i.i.d.", "$r$ -dimensional random features $\\lbrace w_\\nu ,b_\\nu \\rbrace _{\\nu =1}^r$ by (REF ), and construct the augmented random feature vector $_{2p}()$ by (REF ).", "Step 4: Solve the coefficient vector $\\widehat{}_{2p}(I_\\ell )$ by (REF ), and record the selected variables.", "Step 5: Compute the importance score by (REF ), i.e., the empirical selection frequency, $\\lbrace \\widehat{\\Pi }_{j}\\rbrace _{j\\in [2p]}$ based on the $L$ estimates of $\\lbrace \\widehat{}_{2p}(I_\\ell )\\rbrace _{\\ell \\in [L]}$ .", "Step 6: Apply the knockoffs filter by (REF ) at the nominal FDR level $q$ .", "Output: the set of selected variables $\\widehat{\\mathcal {S}}$ , and the function estimate $\\widehat{f}^{\\text{RF}}()$ ." ], [ "Knockoff variable construction", "A random vector $\\widetilde{}\\in \\mathbb {R}^p$ is said to be a knockoff copy of $\\in \\mathbb {R}^p$ [12] if $ (,\\widetilde{}) \\overset{d}{=}(,\\widetilde{})_{\\text{swap}(\\mathcal {A})}, \\; \\text{ for any } \\; \\mathcal {A}\\subseteq \\lbrace 1,\\ldots ,p\\rbrace , \\quad \\textrm { and } \\quad Y\\protect \\mathchoice{\\protect \\mathrel {\\unknown.", "{\\displaystyle \\perp }\\hspace{1.111pt}{\\displaystyle \\perp }}}{\\protect \\mathrel {\\unknown.", "{\\textstyle \\perp }\\hspace{1.111pt}{\\textstyle \\perp }}}{\\protect \\mathrel {\\unknown.", "{\\scriptstyle \\perp }\\hspace{1.111pt}{\\scriptstyle \\perp }}}{\\protect \\mathrel {\\unknown.", "{\\scriptscriptstyle \\perp }\\hspace{1.111pt}{\\scriptscriptstyle \\perp }}}\\widetilde{} \\ | \\ ,$ where the symbol $\\overset{d}{=}$ denotes the equality in distribution, and $\\text{swap}(j)$ is the operator swapping $X_j$ with $\\widetilde{X}_j$ .", "For instance, if $p=3$ and $\\mathcal {A}=\\lbrace 1,3\\rbrace $ , then $(X_1,X_2,X_3,\\widetilde{X}_1,\\widetilde{X}_2,\\widetilde{X}_3)_{\\text{swap}(\\mathcal {A})}$ becomes $(\\widetilde{X}_1,X_2,\\widetilde{X}_3,X_1,\\widetilde{X}_2,X_3)$ .", "There have been numerous ways proposed to construct the knockoff variables.", "We adopt two particular constructions, depending on the data.", "The first is the second-order knockoffs construction [12], which generates the knockoffs by matching only the first two moments of the two distributions.", "In this case, $\\widetilde{}$ is a second-order knockoff copy of $$ if $\\mathbb {E}[] = \\mathbb {E}[\\widetilde{}], \\quad \\text{ and } \\quad \\text{cov}[(,\\widetilde{})] =\\begin{bmatrix}& -\\text{diag}() \\\\-\\text{diag}() & \\end{bmatrix},$ where $$ is the covariance matrix of $$ , and $$ is a $p$ -dimensional vector such that $\\text{cov}[(,\\widetilde{})]$ is positive semi-definite.", "To ensure a good statistical power, $$ should be chosen as large as possible, so that the original and knockoff variables are differentiable [12].", "This strategy is implemented in practice by approximating the distribution of $$ as the multivariate normal, and is employed in numerous knockoffs-based applications [7].", "The second is the deep knockoffs machine [40], which generates the knockoff variables using deep generative models.", "The key idea is to iteratively refine a knockoff sampling mechanism until a criterion measuring the validity of the produced knockoffs is optimized.", "This strategy is shown to be able to match higher-order moments, and also achieve a better approximation of exchangeability.", "In our construction of knockoff variables, we employ the second-order knockoffs when there is clear evidence that the predictor variables approximately follow a multivariate normal distribution, and employ the deep knockoffs machine otherwise.", "Given the training samples $\\left\\lbrace (_i, y_i) \\right\\rbrace _{i=1}^{n}$ , we first augment with the knockoff samples $\\left\\lbrace \\widetilde{}_i \\right\\rbrace _{i=1}^{n}$ , and form the data $\\left\\lbrace (_i, \\widetilde{}_i, y_i) \\right\\rbrace _{i=1}^{n}$ , where $_i = (x_{i,1},\\ldots ,x_{i,p})^{\\mbox{\\tiny {\\sf T}}}\\in ^p$ , and $\\widetilde{}_i = (\\widetilde{x}_{i,1},\\ldots ,\\widetilde{x}_{i,p})^{\\mbox{\\tiny {\\sf T}}}\\in \\mathbb {R}^p$ ." ], [ "Random feature mapping", "We next construct the random features for both original and knockoff variables.", "The key idea is to employ the random feature mapping [35], [8] to approximate the kernel function, which enables us to construct a projection operator between the RKHS and the original predictor space.", "Specifically, if the kernel functions that generate $\\mathcal {H}_1$ are shift-invariant, i.e., $K(X,X^{\\prime }) = K(X-X^{\\prime })$ , and integrate to one, i.e., $\\int _K(X-X^{\\prime }) d(X-X^{\\prime }) = 1$ , then the Bochners theorem [10] states that such kernel functions satisfy the Fourier expansion: $\\begin{aligned}K(X-X^{\\prime }) &=\\int _\\mathbb {R}p(w)\\exp \\left\\lbrace \\sqrt{-1}w(X-X^{\\prime })\\right\\rbrace dw,\\end{aligned}$ where $p(w)$ is a probability density defined by $ p(w) = \\int _K(X)e^{-2\\pi \\sqrt{-1} wX}dX.$ We note that many kernel functions are shift-invariant and integrate to one.", "For example, for the Laplacian kernel $K(X-X^{\\prime })=c_1e^{-|X-X^{\\prime }|}$ , $p(w) = c_2 / [\\pi (1+w^2)]$ , where $c_1,c_2$ are two normalization constants.", "It is then shown that [35], [8] the minimizer in (REF ) can be approximated by, $ \\widehat{f}^{\\text{RF}}() = _p()^{\\mbox{\\tiny {\\sf T}}}_p,$ where $_p() = \\left[ (X_1)^{\\mbox{\\tiny {\\sf T}}},\\ldots ,(X_p)^{\\mbox{\\tiny {\\sf T}}}\\right]^{\\mbox{\\tiny {\\sf T}}}\\in \\mathbb {R}^{pr}$ , and $(X_j) = \\left[ \\psi _1(X_j),\\ldots ,\\psi _r(X_j) \\right]^{\\mbox{\\tiny {\\sf T}}}\\in \\mathbb {R}^r$ is a vector of $r$ Fourier bases with the frequencies drawn from the density $p(w) $ , i.e., $\\begin{aligned}& \\omega _{j,\\nu } \\overset{\\text{i.i.d.", "}}{\\sim } p(\\omega ), \\quad \\quad b_{j,\\nu } \\overset{\\text{i.i.d.", "}}{\\sim } \\textrm {Uniform}[0,2\\pi ], \\\\&\\psi _\\nu (X_j) = \\sqrt{\\frac{2}{r}}\\cos (X_j \\omega _{j,\\nu } + b_{j,\\nu }), \\quad \\quad j=1,\\ldots ,p, \\; \\nu =1,\\ldots ,r.\\end{aligned}$ The use of random feature mapping achieves potentially substantially dimension reduction.", "More specifically, the estimator in (REF ) only requires to learn the $pr$ -dimensional coefficient $_p$ , compared to the estimator in (REF ) that involves an infinite-dimensional vector $\\widetilde{}_p$ .", "[41] showed that the random feature mapping obtains an optimal bias-variance tradeoff if $r$ scales at a certain rate and $r/n\\rightarrow 0$ when $n$ grows.", "They further proved that the estimator in (REF ) can achieve the minimax optimal estimation error.", "Beyond the estimation optimality, we note that the random feature mapping also efficiently reduces the computational complexity.", "That is, the computation complexity of the estimator in (REF ) is only $O(nr^2)$ , compared to the computation complexity of the kernel estimator in (REF ) that is $O(n^3)$ .", "The saving of the computation is substantial if $r/n\\rightarrow 0$ as $n$ grows.", "In our setting of kernel knockoffs selection, we construct the random features for both original and knockoff variables and obtain the augmented random feature vector as, $ _{2p}() = \\left((X_1)^{\\mbox{\\tiny {\\sf T}}},\\ldots , (X_p)^{\\mbox{\\tiny {\\sf T}}},(\\widetilde{X}_1)^{\\mbox{\\tiny {\\sf T}}},\\ldots ,(\\widetilde{X}_p)^{\\mbox{\\tiny {\\sf T}}}\\right)^{\\mbox{\\tiny {\\sf T}}}\\in \\mathbb {R}^{2pr},$ where $(X_j) = \\left[ \\psi _1(X_j),\\ldots ,\\psi _r(X_j) \\right]^{\\mbox{\\tiny {\\sf T}}}\\in \\mathbb {R}^r$ , and $(\\widetilde{X}_j) = \\left[ \\psi _1(\\widetilde{X}_j),\\ldots ,\\psi _r(\\widetilde{X}_j) \\right]^{\\mbox{\\tiny {\\sf T}}}\\in \\mathbb {R}^r$ , $j=1,\\ldots ,p$ , are two sets of $r$ -dimensional random features that are independently sampled from (REF ).", "Then the minimizer in (REF ) can be approximated by, $ \\widehat{f}() = \\Psi _{2p}()^{\\mbox{\\tiny {\\sf T}}}_{2p}.$ Meanwhile, we note that the randomness of the features generated from (REF ) may alter the ranking of variable significances.", "As such, we couple the random feature mapping with knockoffs and subsampling to achieve the desired FDR control and power." ], [ "Resampling, importance score, and knockoffs filtering", "We adopt the subsampling scheme similarly as that in [32], [18].", "Specifically, we subsample half of the training samples without replacement, and let $I$ denote the set of subsample indices out of $\\lbrace 1,\\ldots ,n\\rbrace $ of size $\\lfloor n/2 \\rfloor $ , where $\\lfloor n/2 \\rfloor $ is the largest integer no greater than $n/2$ .", "We then estimate the coefficient vector $_{2p} = (c_1^{\\mbox{\\tiny {\\sf T}}}, \\ldots , c_{2p}^{\\mbox{\\tiny {\\sf T}}})^{\\mbox{\\tiny {\\sf T}}}\\in \\mathbb {R}^{2pr}$ in (REF ), in which each $c_j\\in \\mathbb {R}^r$ for $j=1,\\ldots ,2p$ , via a group Lasso penalized regression based on the subsample $I$ of the observations, $ \\begin{aligned}\\underset{\\underset{j=1,\\ldots ,2p}{c_{j} \\in \\mathbb {R}^{r}}}{\\min }~\\frac{1}{|I|}\\sum _{i\\in I} \\left[y_i - \\bar{y}(I) - \\sum _{j=1}^{p}(x_{i,j})^{\\mbox{\\tiny {\\sf T}}}c_{j} - \\sum _{j=p+1}^{2p}(\\widetilde{x}_{i,j-p})^{\\mbox{\\tiny {\\sf T}}}c_{j} \\right]^2 + \\tau \\sum _{j=1}^{2p} \\Vert c_{j}\\Vert _2,\\end{aligned}$ where $\\bar{y}(I)=\\sum _{i\\in I}y_i/|I|$ is the empirical mean, and $\\tau \\ge 0$ is the penalty parameter.", "Let $\\widehat{}_{2p}(I) = \\left( \\widehat{c}_1^{\\mbox{\\tiny {\\sf T}}}(I),\\ldots ,\\widehat{c}_{2p}^{\\mbox{\\tiny {\\sf T}}}(I) \\right)^{\\mbox{\\tiny {\\sf T}}}$ denote the minimizer of (REF ).", "We remark that the group Lasso penalty in (REF ) encourages the entire vector $c_j\\in \\mathbb {R}^{r}$ to be shrunk to zero, for $j = 1, \\ldots , 2p$ .", "Consequently, estimating $_{2p}$ via (REF ) in effect leads to the selection of important variables among all $2p$ candidate variables $(X_1, \\ldots , X_p, \\widetilde{X}_1, \\ldots , \\widetilde{X}_p)$ .", "We also remark that, our group Lasso penalty $\\sum _{j=1}^{2p}\\Vert c_j\\Vert _2$ in (REF ) is different from the COSSO penalty used in [30], which takes the form $\\sum _{j=1}^{p}\\Vert (x_{i,j})^{\\mbox{\\tiny {\\sf T}}}c_j\\Vert _{\\mathcal {H}_1}+\\sum _{j=p+1}^{2p}\\Vert (\\widetilde{x}_{i,j-p})^{\\mbox{\\tiny {\\sf T}}}c_j\\Vert _{\\mathcal {H}_1}$ .", "Since the random feature mapping generally cannot form orthogonal bases, there is no closed-form representation of the RKHS norms $\\Vert (x_{i,j})^{\\mbox{\\tiny {\\sf T}}}c_j\\Vert _{\\mathcal {H}_1}$ and $\\Vert (\\widetilde{x}_{i,j-p})^{\\mbox{\\tiny {\\sf T}}}c_j\\Vert _{\\mathcal {H}_1}$ in our setting.", "As a result, the COSSO penalty is difficult to implement, and instead we adopt the group Lasso penalty in (REF ) that also yields the desired theoretical properties.", "Given the penalized estimate $\\widehat{}_{2p}(I)$ , we obtain an estimate of the selected variable indices $\\widehat{\\mathcal {S}}(I) \\subseteq \\lbrace 1,\\ldots ,2p\\rbrace $ .", "That is, for each $j \\in \\lbrace 1,\\ldots ,p\\rbrace $ , $j\\in \\widehat{\\mathcal {S}}(I)$ if $\\widehat{c}_j(I)\\ne \\mathbf {0}$ and the original variable $X_j$ is selected, and $(j+p)\\in \\widehat{\\mathcal {S}}(I)$ if $\\widehat{c}_{j+p}(I)\\ne \\mathbf {0}$ and the knockoff variable $\\widetilde{X}_j$ is selected.", "Then the probability of being in the selected set $\\widehat{\\mathcal {S}}(I)$ is $ \\widehat{\\Pi }_j = \\mathbb {P}\\lbrace j\\in \\widehat{\\mathcal {S}}(I)\\rbrace , \\;\\; \\text{ for } \\; j=1,\\ldots ,2p,$ where $\\mathbb {P}$ is with respect to both subsampling $I$ and the random features.", "We note that $\\widehat{\\Pi }_j$ can be estimated accurately using the empirical selection frequencies [32].", "Specifically, we repeat the above subsampling and coefficient estimation procedure $L$ times, each time for a subsample $I_\\ell , \\ell = 1,\\ldots ,L$ .", "We then obtain the selected variable indices $\\widehat{}(I_\\ell )$ for $I_\\ell $ , and compute (REF ) using the empirical selection frequency as the percentage of times the $j$ th variable, $j=1,\\ldots ,2p$ , is included in $\\lbrace \\widehat{}(I_\\ell ) \\rbrace _{\\ell =1}^{L}$ .", "Next, we define the importance score for the original variable $X_j$ , $j=1,\\ldots ,p$ , as, $ \\Delta _j = \\widehat{\\Pi }_j- \\widehat{\\Pi }_{j+p}.$ We comment that $\\Delta _j$ in (REF ) is calculated for only one run of the knockoffs procedure, i.e., we generate the knockoffs only once.", "This is different from the derandomized knockoffs method recently proposed by [39], which aggregates the selection results across multiple runs of knockoffs to reduce the randomness of the knockoffs generation.", "Finally, given the target nominal FDR level $q$ , we apply a knockoffs filter [6] to the importance scores to produce the final set of selected variables, $ T = \\min \\left\\lbrace t\\in \\lbrace |\\Delta _j|:|\\Delta _j|>0\\rbrace : \\frac{\\# \\lbrace j: \\Delta _j \\le -t \\rbrace }{\\# \\lbrace j : \\Delta _j \\ge t\\rbrace } \\le q\\right\\rbrace \\quad (\\text{knockoffs}).$ Set $T=\\infty $ if the above set is empty.", "Another commonly used but slightly more conservative knockoffs filter [6], [12] is, $T_+ = \\min \\left\\lbrace t\\in \\lbrace |\\Delta _j|:|\\Delta _j|>0\\rbrace : \\frac{\\# \\lbrace j: \\Delta _j \\le -t \\rbrace +1}{\\# \\lbrace j : \\Delta _j \\ge t\\rbrace } \\le q\\right\\rbrace \\quad (\\text{knockoffs+}).$ In our simulations, we have experimented with both filers, which produce very similar results, so we only present the results based on $T$ .", "Given the threshold value $T$ , the final set of selected variables is, $ \\widehat{\\mathcal {S}} = \\Big \\lbrace j\\in \\lbrace 1,\\ldots ,p\\rbrace :\\Delta _j \\ge T \\Big \\rbrace .$ We then reestimate $_p$ in (REF ) using all the sample observations as, $\\widehat{}_p^{\\text{RF}} = \\left( (\\widehat{c}_1^{\\text{RF}})^{\\mbox{\\tiny {\\sf T}}}, \\ldots , (\\widehat{c}_{p}^{\\text{RF}})^{\\mbox{\\tiny {\\sf T}}}\\right)^{\\mbox{\\tiny {\\sf T}}}= \\underset{{c_{j} \\in \\mathbb {R}^{r},j\\in \\widehat{}}}{\\arg \\min }~\\frac{1}{n}\\sum _{i=1}^n \\left[ y_i- \\bar{y}(I) -\\sum _{j\\in \\widehat{}}(x_{i,j})^{\\mbox{\\tiny {\\sf T}}}c_j \\right]^2.$ We obtain the final knockoffs-based kernel regression estimator as, $ \\widehat{f}^{\\text{RF}}() =_p()^{\\mbox{\\tiny {\\sf T}}}\\widehat{}_p^{\\text{RF}}.$" ], [ "Parameter tuning", "Algorithm REF involves some hyperparameters, including the number of random features $r$ , the penalty parameter $\\tau $ , the number of subsampling replications $L$ , and the subsampling sample size.", "We next discuss their choices.", "For the number of random features $r$ , we start with an initial set $\\Xi $ of candidate values for $r$ .", "For each working rank $r \\in \\Xi $ , we calculate the selection frequencies $\\big \\lbrace \\widehat{\\Pi }_{j,r} \\big \\rbrace _{j\\in [2p],r\\in \\Xi }$ , and the standard deviation $ \\widehat{\\sigma }_r={\\rm sd} \\left( \\big \\lbrace \\widehat{\\Pi }_{j,r} \\big \\rbrace _{j\\in [2p]} \\right)$ , with a relatively small number for the subsampling replications.", "We then choose the value of $r \\in \\Xi $ that maximizes the following criterion that balances the selection standard deviation and model complexity, $ \\widehat{r} = _{r \\in \\Xi } 2 p \\widehat{\\sigma }_r - \\ln (r).$ We have observed that, through our numerical simulations, when we start from a small value of $r$ , the selection frequencies of both original variables and their knockoffs counterparts are close to zero.", "As $r$ increases, it starts to separate the truly important variables from the null variables and knockoffs, where the selection frequencies of those truly important variables grow positively, and correspondingly, the standard deviation $\\widehat{\\sigma }_r$ increases.", "Meanwhile, the log penalty term helps balance the model complexity.", "For the regularization parameter $\\tau $ in (REF ), it does not need to be fixed in advance and can be chosen in data-driven approach by minimizing the commonly used BIC criterion, $ \\widehat{\\tau } = _{\\tau \\ge 0} \\log [{\\rm RSS}(\\tau )] + r \\frac{\\log n }{n } | \\widehat{}(\\tau ) |,$ where $\\rm {RSS}(\\tau )$ is the cross-validation residual sum of squares, and $| \\widehat{}(\\tau ) |$ is the number of selected variables in $\\widehat{}(\\tau )$ .", "For the number of subsampling replications $L$ , our numerical experiments have found that $L=100$ leads to a competitive performance in both FDR control and power of selection.", "We also comment that, the computation can be easily parallelized, since it requires no information sharing different subsamples.", "For the subsampling sample size, again, our experiments have found that, when it is no smaller than $\\lfloor n/2 \\rfloor $ , the method achieves both a good FDR and power." ], [ "FDR Control", "We show that our proposed kernel knockoffs selection procedure controls the false discovery rate under any given nominal level for a finite sample size.", "We first show that the importance score $\\Delta _j$ in (REF ) has a symmetric distribution for a null variable $X_j \\in ^\\perp $ , and is equally likely to be positive or negative.", "The symmetric property of null variables is crucial for the knockoffs procedure, which then chooses a data-dependent threshold while having the FDR under control [6].", "Theorem 1 Suppose Assumption REF holds.", "Let $(s_1, \\ldots , s_p)$ be a set of independent random variables, such that $s_j = \\pm 1$ with probability $1/2$ if $j \\in \\mathcal {S}^\\perp $ , and $s_j=1$ if $j \\in \\mathcal {S}$ .", "Then, $(\\Delta _1,\\ldots , \\Delta _p) \\overset{d}{=} (\\Delta _1 \\cdot s_1, \\ldots , \\Delta _p \\cdot s_p).$ Next, we show that our selection procedure successfully controls the false discovery under any finite sample size.", "The result holds regardless of the distribution or the number of predictors, and does not require any knowledge of the noise level.", "The false discovery here is measured by both the FDR, and the modified FDR, which is defined as, $\\text{mFDR} = \\mathbb {E}\\left[\\frac{|\\widehat{\\mathcal {S}}\\cap \\mathcal {S}^\\perp |}{|\\widehat{\\mathcal {S}}|+1/q}\\right],$ Theorem 2 For any $q\\in [0,1]$ , the selected set $\\widehat{\\mathcal {S}}$ in (REF ) satisfies that $\\text{mFDR} \\le q$ , and $\\text{FDR} \\le q$ , for any finite sample size $n$ ." ], [ "Power Analysis", "Next, we show that our proposed kernel knockoffs selection procedure achieves a power that approaches one as the sample size tends to infinity.", "We first note that, the theoretical power analysis for the knockoffs methods is largely missing in the current literature, with the only exception of [20], who studied the power for linear regressions under the model-X knockoffs framework.", "We also remark that, as is common for all knockoffs selection methods, the power of our knockoffs-based method is usually no greater than that of the group Lasso-based selection method.", "This is because the proposed knockoffs procedure is built on top of the group Lasso selection in (REF ).", "In a sense, the knockoffs procedure further selects variables from the set of variables identified by group Lasso applied to the augmented predictor set to achieve the FDR control.", "Therefore, the key of our power analysis is to investigate how much power loss that the knockoffs procedure would induce.", "We introduce some regularity conditions.", "Assumption 2 The number of nonzero component functions, i.e., $||$ , is bounded.", "Assumption 3 Suppose there exists a constant $C_{\\min }>0$ such that the minimal eigenvalue of matrix $\\mathbb {E}[\\frac{1}{n}_{}^{\\mbox{\\tiny {\\sf T}}}_{}]$ satisfies that, $\\Lambda _{\\min }\\left(\\mathbb {E}\\left[\\frac{1}{n}_{}^{\\mbox{\\tiny {\\sf T}}}_{}\\right]\\right)\\ge \\frac{1}{2}C_{\\min },$ where the expectation is taken over the random features and $_\\in \\mathbb {R}^{n\\times 2r||}$ is the design matrix with the $i$ th row equal to $\\left[ (x_{i,j_1})^{\\mbox{\\tiny {\\sf T}}}, \\ldots , (x_{i,j_{||}})^{\\mbox{\\tiny {\\sf T}}}, (\\widetilde{x}_{i,j_1})^{\\mbox{\\tiny {\\sf T}}}, \\ldots , (\\widetilde{x}_{i,j_{||}})^{\\mbox{\\tiny {\\sf T}}}\\right]$ , $i=1,\\ldots ,n$ , $=\\lbrace j_1,\\ldots ,j_{||}\\rbrace $ .", "Assumption 4 Let $\\eta _R\\equiv c_\\eta \\left\\lbrace n^{-\\beta /(2\\beta +1)}+[(\\log p)/n]^{1/2} \\right\\rbrace $ for some constant $c_\\eta >0$ .", "Suppose $\\min _{j\\in }\\Vert f_{j}(X_j)\\Vert _{L_2(X_j)}\\ge \\kappa _n\\eta _R$ , for some slowly diverging sequence $\\kappa _n\\rightarrow \\infty $ , as $n\\rightarrow \\infty $ , where the RKHS $_1$ is embedded to a $\\beta $ th order Sobolev space with $\\beta >1$ .", "Assumption 5 Suppose there exists a constant $0 \\le \\xi _{} < 1$ such that, $\\max _{j\\notin }\\left\\Vert \\lbrace _{j}(I)\\rbrace ^{\\mbox{\\tiny {\\sf T}}}_{}(I)[_{}(I)^{\\mbox{\\tiny {\\sf T}}}_{}(I)]^{-1}\\right\\Vert _{2}\\le \\xi _{}, \\quad \\textrm { and } \\quad \\frac{\\xi _{}\\sqrt{||}+1}{\\tau }\\eta _{R}+\\xi _{}\\sqrt{||}<1.$ All these conditions are reasonable and are commonly imposed in the literature.", "Specifically, Assumption REF concerns the overall complexity of the component functions.", "Similar assumptions have been adopted in sparse additive models over RKHS [27], [36], [15].", "Assumption REF ensures the identifiability among the $||$ submatrices of $_{}$ .", "The same condition has been used in [51], [38].", "Assumption REF imposes some regularity on the minimum regulatory effect.", "Similar assumptions have been used in Lasso regressions [38], [36], [20].", "Assumption REF reflects the intuition that the large number of irrelevant variables cannot exert an overly strong effect on the relevant variables.", "Besides, the second inequality characterizes the relationship between $\\xi _{}$ , the sparse tuning parameter $\\tau $ , and the sparsity level $||$ .", "This condition is again standard for Lasso regressions [51], [38].", "Next, we characterize the statistical power of the proposed kernel knockoffs procedure.", "For the true set $$ and the selected set $\\widehat{}$ , the power is defined as $\\text{Power}(\\widehat{\\mathcal {S}}) = \\mathbb {E}\\left[\\frac{|\\widehat{\\mathcal {S}}\\cap \\mathcal {S}|}{|\\mathcal {S}|\\vee 1}\\right],$ Theorem 3 Suppose Assumptions REF –REF hold, and the number of random features $r \\ge c_rn^{2\\beta /(2\\beta +1)}$ for some $c_r>0$ .", "Then, the selected set $\\widehat{}$ in (REF ) satisfies that, $\\text{Power}(\\widehat{}) \\rightarrow 1$ , as $n \\rightarrow \\infty $ .", "Together, Theorems REF and REF show that our proposed selection method is able to achieve both the finite-sample FDR control and the asymptotic power that approaches one.", "We also remark that, our results hold for both the low-dimensional and high-dimensional settings, where the number of predictors $p$ can be either smaller or larger than the sample size $n$ ." ], [ "Numerical Studies", "We examine the finite-sample performance of our proposed method with the varying signal strength, the predictor distribution, the nonparametric component function, the sample size and the number of predictors.", "We compare with several alternative solutions.", "We also illustrate with an analysis of brain imaging data for Alzheimer’s disease." ], [ "Alternative methods for comparison", "We abbreviate our proposed kernel knockoffs selection method as KKO.", "We solve the group Lasso penalized problem in (REF ) using the R package grpreg.", "We employ the Laplacian kernel, and tune the hyperparameters following Section REF .", "We set the target FDR level at $q=0.2$ following [20].", "We also briefly comment that, in addition to the reproducing kernel approach, the spline basis expansion is another commonly used approach in the nonparametric additive modeling.", "But it involves a totally different set of methodological tools and theoretical analysis, and we leave it as future research.", "We compare our method with three main competitors.", "The first competitor is the nonparametric selection method for sparse additive models (SPAM) of [37], which combines B-spline basis expansion with grouped Lasso.", "We set the number of B-spline expansions at $\\lceil n^{1/5}\\rceil $ , i.e., the largest integer no greater than $n^{1/5}$ , and tune the sparsity penalty by generalized cross-validation.", "We implement the method using the R package SAM.", "We did not compare with the COSSO method of [30], due to that the code is not available, and SPAM usually achieves a similar and sometimes more competitive performance than COSSO.", "The second competitor is the linear knockoffs (LKO) selection method, and we implement it using the R package knockoffs.", "The third competitor is the graphical nonlinear knockoffs (RANK) selection method of [20].", "We follow the same parameter setup as in [20], and implement their method based on the R package gamsel.", "We also briefly compare our method with that of [14].", "More specifically, [14] adopted an “expansion first\" strategy, which first performs feature expansion $\\widetilde{}(X_j)$ , $j=1,\\ldots ,p$ , then constructs the knockoffs based on the expanded features.", "By contrast, we adopt the “knockoffs first\" approach, which constructs the knockoffs directly for the variables $\\lbrace X_j\\rbrace _{j=1}^p$ , then performs the random feature expansion on both original variables and their knockoffs.", "There are two advantages of doing the knockoffs first.", "First, it ensures a better knockoffs construction and eventually a better statistical power.", "That is, to construct a good knockoff variable using either the second-order knockoffs or the deep knockoffs, it requires a reasonably slow eigenvalue decay of the covariance $$ , so that the original variables and their knockoffs are differentiable [12].", "We consider a simulation example replicated 500 times, where the predictors follow a multivariate normal distribution, $n=500, p=5$ , and we employ the Laplacian kernel with $r=3$ .", "Figure REF (a) shows the eigenvalue decay of the sample covariance matrix (blue line), and the random kernel expansion (red line).", "It is seen that the former decays more slowly than the latter, and therefore it is better to construct the knockoffs based on the original variables.", "Second, the “expansion first\" leads to larger correlations between the original variables and their knockoffs, compared to the “knockoffs first\", as shown in Figure REF (b), which would in turn make the subsequent group Lasso selection harder.", "Finally, the “expansion first” is computationally more expensive.", "This is because the “expansion first” approach requires generating the knockoffs for $pr$ -dimensional variables, whereas our “knockoffs first\" approach only requires generating the knockoffs for $p$ -dimensional variables.", "Figure: Comparison between the “expansion first\" strategy and the “knockoffs first\" strategy.", "Left panel: the eigenvalue decay of the sample covariance matrix (blue line) and the random kernel expansion (red line).", "Right panel: the marginal sample correlations between the original variables and the knockoff variables (blue line), between the random kernel expansion and the “expansion first” knockoff variables (red line), and between the random kernel expansion and the “knockoffs first” knockoff variables (green line)." ], [ "Varying signal strength, predictor design, and component functions", "We first study the performance with the varying signal strength and the predictor distribution.", "We simulate the response, $Y = \\sum _{j \\in } \\theta _j f_j (X_j ) + \\epsilon $ , where $$ is the set of relevant predictors with $|| = 10$ , and $\\epsilon $ is a standard normal error.", "We sample $\\theta _j$ independently from a uniform distribution $(-\\theta , \\theta )$ for some positive constant $\\theta $ .", "The magnitude of $\\theta $ reflects the strength of the signal, and we vary $\\theta = \\lbrace 0.1,1,10,50,100\\rbrace $ .", "We simulate the predictors independently from three different distributions, a multivariate normal distribution with mean zero and covariance $\\Sigma _{ij}=0.3^{|i-j|}$ , a mixture normal distribution, with an equal probability from three multivariate normal distributions, all with mean zero, and different covariances where $\\Sigma _{1,ij}=0.1^{|i-j|}$ , $\\Sigma _{2,ij}=0.3^{|i-j|}$ , and $\\Sigma _{3,ij}=0.5^{|i-j|}$ , and a uniform $(-2, 2)$ distribution.", "We employ the second-order knockoffs when the predictor distribution is normal, and the deep knockoffs otherwise, to generate the knockoff variables.", "We first consider a trigonometric polynomial component function, and fix the number of predictors at $p=50$ , and the sample size at $n=800$ .", "We later consider other forms of component functions, and different $(p, n)$ .", "$ f_j (x)= u_{j,1} \\sin (c_{j,1} x) + u_{j,2}\\cos (c_{j,2} x) + u_{j,3} \\sin ^2 (c_{j,3} x) +u_{j,4} \\cos ^2 (c_{j,4} x),$ where $u_{j,k}$ follows a uniform $(1,2)$ distribution, and $c_{j,k}$ follows a uniform $(1,10)$ , for $k=1,2,3,4$ .", "Figure REF reports the average FDR and power over 500 data replications for the four methods with varying signal strength $\\theta $ and three different predictor distributions.", "It is seen that our method successfully controls the FDR blow the expected level, and at the same time achieves the best power.", "Besides, the performance is robust with respect to different signal strengths or the predictor distributions.", "By comparison, the alternative methods are much more sensitive in terms of the FDR control, and the powers are consistently lower.", "Moreover, the linear knockoffs method often fails to control the FDR.", "Figure: Empirical performance and comparison in terms of FDP and TPR with the varying signal strength and the predictor distribution.", "Four methods are compared: the nonparametric selection method for sparse additive models (SPAM) of , the linear knockoffs (LKO) of , the graphical nonlinear knockoffs (RANK) of , and our proposed kernel knockoffs (KKO).Figure: Empirical performance and comparison in terms of FDP and TPR with the varying signal strength and the component function.", "Four methods are compared: the nonparametric selection method for sparse additive models (SPAM) of , the linear knockoffs (LKO) of , the graphical nonlinear knockoffs (RANK) of , and our proposed kernel knockoffs (KKO).Next, we consider more forms of component functions.", "The first is a sin-ratio function, $ f_j (x) = \\frac{\\sin (c_{j,1} x)}{ 2- \\sin (c_{j,2} x) },$ where $c_{j,k}$ follows a uniform $(1,10)$ distribution for $k=1,2$ , and $|| = 10$ .", "We note that it is generally more difficult to estimate the sin-ratio function (REF ) compared to the trigonometric polynomial function (REF ).", "The second is a mixed additive model, where we sample the component function with an equal probability from (REF ) or (REF ).", "We fix the predictor distribution as the multivariate normal, $p=50$ and $n=800$ .", "We continue to vary the signal strength $\\theta = \\lbrace 0.1,1,10,50,100\\rbrace $ .", "Figure REF reports the average FDP and TPR based on 500 data replications.", "It is seen again that our method achieves the best power while controlling the FDR under the nominal level for the new component functions." ], [ "Varying sample size and dimension", "Next, we investigate the empirical performance with the varying sample size $n$ and the number of predictors $p$ .", "For the varying $n$ , we consider the trigonometric polynomial function (REF ) with $p = 50, || = 10, \\theta = 10$ and the multivariate normal predictor distribution.", "We vary the sample size $n = \\lbrace 400, 600, 800, 1200, 1500, 2000\\rbrace $ .", "Figure REF reports the average FDP and TPR based on 500 data replications.", "It is seen that our method successfully control the FDR at all sample sizes, while its power quickly increases and $n$ increases, and dominates the powers of all the competitive methods considerably.", "Besides, both LKO and RANK have inflated FDR especially when the sample size is small.", "Figure: Empirical performance and comparison in terms of FDP and TPR with the varying sample size nn.", "Four methods are compared: the nonparametric selection method for sparse additive models (SPAM) of , the linear knockoffs (LKO) of , the graphical nonlinear knockoffs (RANK) of , and our proposed kernel knockoffs (KKO).For the varying $p$ , we consider the trigonometric polynomial function (REF ) with $n = 800, || = 10, \\theta = 10$ and the multivariate normal predictor distribution.", "We vary the number of predictors $p = \\lbrace 30, 50, 70, 90, 120, 150, 200\\rbrace $ .", "Figure REF reports the average FDP and TPR based on 500 data replications.", "It is seen that our method again achieves the best performance in both FDR and power.", "We also remark that, our proposed method in principle can handle the high-dimensional regime.", "Actually, the theoretical guarantees in Section are all established for both $p < n$ and $p > n$ .", "However, the current bottleneck is how to construct the knockoff variables in the high-dimensional setting [21], [20].", "Once the knockoffs are generated, the rest of the procedure remains the same.", "Figure: Empirical performance and comparison in terms of FDP and TPR with the varying number of predictors pp.", "Four methods are compared: the nonparametric selection method for sparse additive models (SPAM) of , the linear knockoffs (LKO) of , the graphical nonlinear knockoffs (RANK) of , and our proposed kernel knockoffs (KKO)." ], [ "Brain imaging data analysis", "We illustrate the proposed method with a brain imaging data analysis to study the Alzheimer's disease (AD).", "AD is an irreversible neurodegenerative disorder, and is characterized by progressive impairment of cognitive and memory functions, loss of bodily functions, and ultimately death.", "It is the leading form of dementia in elderly subjects, and is the sixth leading cause of death in the United States.", "Over 5.5 million Americans were affected by AD in 2018, and without any effective treatment or prevention, this number is projected to triple by 2050 [2].", "Brain atrophy as reflected by brain grey matter cortical thickness is a well-known biomarker for AD.", "We study a dataset with $n = 697$ subjects, each of whom received an anatomical magnetic resonance imaging (MRI) scan that measures cortical thickness.", "The MRI image has been preprocessed by the standard pipeline, and is summarized in the form of a vector of cortical thickness measurements for a set of parcellated brain regions-of-interest.", "There are $p=68$ regions in total.", "Brain parcellation is particularly useful to facilitate the interpretation, and has been frequently employed in brain imaging analysis [22], [26].", "In addition to the MRI image, for each subject, the data also records a composite cognitive score, which combines numerous tests that assess episodic memory, timed executive function, and global cognition [16].", "Our goal is to study the association between the composite cognitive score and the vector of brain cortical thickness, and identify individual brain regions with strong associations.", "Intuitively, a linear model is inadequate to capture such an association, and we turn to the nonparametric additive model instead.", "We apply the proposed kernel knockoffs selection procedure.", "As the distribution of the predictors is not necessarily normal, we employ the deep knockoffs machine to generate the knockoff variables.", "We continue to set the target FDR level at $q=0.2$ .", "Table: Brain regions identified by the kernel knockoffs selection procedure.", "“l-\" stands for the left hemisphere, and “r-\" stands for the right hemisphere.Table REF reports the ten brain regions selected by our method.", "These findings agree with and support the current literature on AD research.", "Particularly, the middle temporal gyrus is located on the temporal lobe, and is associated with processes of recognition of known faces and accessing word meaning while reading.", "Middle temporal lobe atrophy is common in AD as well as its prodromal stage, mild cognitive impairment [47].", "The superior parietal lobe is involved with attention, visuospatial perception, and spatial orientation.", "Damage to the parietal lobe is common in AD, and leads to problems with performing gestures and skilled movements [34].", "The fusiform is linked with various neural pathways related to recognition.", "The inferior parietal lobe is involved in perception of emotions.", "The superior temporal gyrus is involved in auditory processing, and has also been implicated as a critical structure in social cognition.", "Numerous studies have found involvement of these brain regions in the development of AD [13], [17], [34].", "The precuneus is associated with episodic memory, visuospatial processing, reflections upon self, and aspects of consciousness, and is found to be an AD-signature region [5].", "Finally, the entorhinal cortex functions as a hub in a widespread network for memory, navigation and the perception of time.", "It is found implicated in the early stages of AD, and is one of the most heavily damaged cortices in AD [46]." ] ]
2105.11659
[ [ "A new look at old friends. I. Imaging classical radio galaxies with\n uGMRT and MeerKAT" ], [ "Abstract We have undertaken a systematic study of FRI and FRII radio galaxies with the upgraded Giant Metrewave Radio Telescope (uGMRT) and MeerKAT.", "The main goal is to explore whether the unprecedented few $\\mu$Jy sensitivity reached in the range 550-1712 MHz at the resolution of $\\sim4^{\\prime\\prime}-7^{\\prime\\prime}$ reveals new features in the radio emission which might need us to revise our current classification scheme for classical radio galaxies.", "In this paper we present the results for the first set of four radio galaxies, i.e.", "4C 12.02, 4C 12.03, CGCG 044-046 and CGCG 021-063.", "The sources have been selected from the 4C sample with well-defined criteria, and have been imaged with the uGMRT in the range 550-850 MHz (band 4) and with the MeerKAT in the range 856-1712 MHz (L-band).", "Full resolution images are presented for all sources in the sample, together with MeerKAT in-band spectral images.", "Additionally, the uGMRT-MeerKAT spectral image and MeerKAT L-band polarisation structure are provided for CGCG 044-046.", "Our images contain a wealth of morphological details, such as filamentary structure in the emission from the lobes, radio emission beyond the hot-spots in three sources, and misalignments.", "We briefly discuss the overall properties of CGCG 044-046 in the light of the local environment as well, and show possible restarted activity in 4C 12.03 which needs to be confirmed.", "We conclude that at least for the sources presented here, the classical FRI/FRII morphological classification still holds with the current improved imaging capabilities, but the richness in details also suggests caution in the systematic morphological classification carried out with automatic procedures in surveys with poorer sensitivity and angular resolution." ], [ "Introduction", "The morphology of extragalactic radio sources of high and low luminosity, set out by [18] has suggested the framework for the study of the physics, origin, and evolution of extragalactic radio sources.", "After more than 46 years, the classification of extended extragalactic radio sources in FR I and FR II types is still used to separate low power and high power radio galaxies respectively.", "However, our knowledge of extragalactic radio sources has improved considerably since then.", "Table: Source properties.It is known that FR I radio galaxies (radio powers typically below $10^{24}$ W Hz$^{-1}$ at 1.4 GHz) with their symmetric prominent jets and lobes, are associated with red passive galaxies, i.e.", "evolved galaxies which usually show little to no evidence of nuclear activity in other bands of the spectrum.", "More recently [4], [28], these have been associated with low excitation emission-line galaxies [34] at moderate luminosities.", "Deviations from the straight (180$^{\\circ }$ ) FR I morphology is common for optical hosts in galaxy clusters, where radio galaxies may form spectacular tails as the radio jets are exposed to a combination of effects, such as galaxy motion through the intracluster medium (ICM), a.k.a., tailed radio sources [74], [62], bulk motions of the ICM (i.e.", "“cluster weather” [8]), shocks in the ICM [60] and other phenomena related to the formation of clusters [6].", "On the other hand, FR II radio galaxies (radio powers above $10^{24}$ W Hz$^{-1}$ at 1.4 GHz) are associated both with quasars and galaxies and the optical host usually shows other indicators of nuclear activity, such as high excitation emission lines [34]; see also [28].", "The jets of these sources are often faint (sometimes barely visible) and asymmetric, and the radio spectral information indicates that the lobes are the result of back-flow from the hot spots.", "A lot of work has been done to understand the different behaviour of jets in FR Is and FR IIs [5], [47].", "Very-long baseline interferometric studies of samples of FR I and FR II radio galaxies show that jets are relativistic at their origin in both classes of sources [80], [22]; however jets in FR I decelerate closer to the core compared to FR II, most likely due to differences in some combination of jet power and propagating medium.", "It is commonly stated that FR I preferentially reside in dense environments, such as clusters and groups of galaxies, and FR II are found in less dense environments [32], but the most famous FR II radio galaxy, Cygnus-A, is at the centre of a galaxy group [3].", "FR IIs appear to populate the Universe to much larger redshifts.", "This is partly due to selection effects related to the sensitivity and resolution of the radio interferometers used so far in large surveys.", "Imaging of large regions of the sky with the current generation of radio interferometers, such as LOFAR (the LOw Frequency ARray), ASKAP (Australia SKA Pathfinder) and MeerKAT clearly shows a wealth of extended radio galaxies with a broad distribution of size, flux density and distance, which might change the paradigm of redshift distribution of FR Is and FR IIs [12], [59], [13].", "Table: Log of the observations using uGMRT and MeerKAT arrays.A small fraction of FR I and FR II radio galaxies [39], [2], shows Mpc-scale extent, posing the problem of the energy supply to the lobes, and a number of them show signs of restarted activity [14], [7], [56].", "Equally important is a new class of low power radio galaxies that has been recently characterised, the so-called FR 0 [1].", "The radio power of these sources is typical of FR Is.", "However, they are compact on the scale of a few arcseconds.", "High sensitivity observations at high angular resolution show that a fraction of them have double-sided jets on very small angular scales, but the majority remain compact [9].", "The nature of these sources and how they fit into the overall classification of radio galaxies is still uncertain.", "A wealth of amazing new images has been collected over the past few years with the current generation of radio interferometers, such as LOFAR, JVLA, MeerKAT and uGMRT, whose much improved sensitivity at arcsecond resolution over at least two orders of magnitude in frequency is revealing new features in the radio emission.", "A few remarkable examples are the amazing tails of NGC 326 detected with LOFAR [29], the low surface brightness emission surrounding several radio galaxies at the centre of groups [75], the filamentary structure within and outside the lobes of Fornax A [55], and the filaments of ESO 137-006 [71].", "The imaging capabilities available nowadays thus allow us to throw new light on our understanding of the radio galaxy phenomenon and address questions that have so far remained unanswered.", "In particular: (a) Why are the morphological classification in terms of FR I and FR II and the radio power so closely linked, and what do intermediate objects tell us?", "(b) Which mechanism produces radio loudness in the form of FR I or FR II in galaxies, what is the duty cycle of radio nuclear activity, and what triggers it?", "(c) Does the sharp classification in FR I and FR II types still hold when we improve our imaging capabilities to the level which can be currently achieved?", "(d) Do we need more morphological classes?", "To address at least the third and fourth of these questions, we started an imaging project of classical radio galaxies with uGMRT in 550–850 MHz (band-4) and MeerKAT in 856–1712 MHz (L-band), to explore their radio morphology with unprecedented sensitivity at arcsecond resolution, and study their spectral structure in the frequency range 0.5–1.7 GHz.", "Here we present our results of the pilot study for the first set of four radio galaxies.", "The paper is organised as follows.", "We present our target selection, observations and data analysis in Sec.", ", followed by the radio morphology in Sec. .", "In Sec.", ", we show the integrated radio spectra and spectral imaging for our sources.", "In Sec.", "we discuss our results, and concluding remarks are given in Sec. .", "We assume a $\\Lambda $ CDM cosmology with $\\Omega _{\\rm m}$ = 0.27, $\\Omega _{\\Lambda }$ = 0.73, and H$_0$ = 70 km s$^{-1}$ Mpc$^{-1}$ .", "We define spectral index, $\\alpha $ as, S$_\\nu $ $\\propto $ $\\nu ^\\alpha $ ; where S$_\\nu $ is the flux density at frequency $\\nu $ .", "Throughout, positions are given in J2000 coordinates." ], [ "Target selection and observations", "We selected a sample of FR I and FR II radio galaxies from the 4C catalogue [69] which meet the following selection criteria: (i) The source is hosted by a detected optical galaxy with spectroscopically determined redshift $z$ in the range between 0.05 and 0.20.", "This ensures (i) the detection of Mpc scale extended emission with uGMRT band-4, and (ii) a similar fraction of FR Is and FR IIs; (ii) The target source is in the declination range $-$ 10$^{\\circ }$ and $+$ 20$^{\\circ }$ to ensure proper visibility and comparable ($u,v$ )-coverage with both uGMRT and MeerKAT arrays at respective bands; and (iii) A clear double radio morphology is imaged at the 45$^{\\prime \\prime }$ angular resolution of the NRAO VLA Sky Survey [10].", "Figure: The images of our four sample sources, 4C 12.02 (top-left panel), 4C 12.03 (top-right panel), CGCG 044--046 (bottom-left panel) and CGCG 021--063 (bottom-right panel).", "The uGMRT and MeerKAT images have angular resolutions of ∼4 '' \\sim 4^{\\prime \\prime } and 7 '' ^{\\prime \\prime }, respectively (see also Table ).", "The radio contours of uGMRT and MeerKAT images are overlaid on the DSS-II (red band) optical images (in gray-scale).", "The red and black surface brightness contours, with magenta and blue being first negative surface brightness contours, correspond to uGMRT and MeerKAT images, respectively.", "The contour levels are rms ×\\times --1, 1, 2, 4, etc., and increases by a factor of two.", "The bar in the top-left corner of each panel image depicts the physical scale for our sample source.These well-defined selection criteria provided us with a total of 12 sources, 6 FR I and 6 FR II radio galaxies.", "We started our project observing the four sources reported in Table REF , which are also members of the GaLactic and Extragalactic All-sky MWA (GLEAM) 4-Jy sample [81], [82].", "We note that the morphological classification of CGCG 021–063 is challenging.", "The source looks like a FR II source in the new MeerKAT and uGMRT observations.", "This will be addressed in Sec.", "REF .", "In order to image the sample sources with matching sensitivity, ($u,v$ )-coverage and angular resolution with the two arrays, we observed each target for $\\sim $ 3 hours and $\\sim $ 3–6 hours respectively with uGMRT and MeerKAT.", "The observing logs for the uGMRT and MeerKAT observations are detailed in Table REF .", "The upgraded GMRT has a hybrid configuration [27], [79] with half of its 45 m diameter 30 antennas located in a central ($\\sim $ 1 km) compact array and with the remaining antennas distributed in a roughly `Y' shaped configuration giving $\\sim $ 25 km maximum baseline length.", "The antennas in the central square and in a `Y' shaped configuration provide baselines that are comparable to the VLA D-array and B-array configurations, respectively.", "Hence, a single observation with the uGMRT provides good angular resolution when mapping the detailed source structure with reasonably good sensitivity.", "MeerKAT is also a hybrid array of 64 13.5 m diameter dish antennas.", "Forty-eight of the 64 interlinked antennas are located in a core region of 1 km in diameter and the other 16 are located outside the core, giving a maximum baseline length of $\\sim $ 8 km [41].", "This configuration of MeerKAT is equivalent to a simultaneous hybrid of VLA B- C- and D-array configurations.", "The observations described in this paper were all carried out with the new dual polarisation (RR and LL) 550–850 MHz band (band-4) receivers of uGMRT, and with the dual linear polarisation (horizontal and vertical) 856–1712 MHz (L-band) receivers [54], [53] of MeerKAT.", "Figure: The image of one of our four sample source, 4C 12.02 is shown to give an idea of the capabilities of MeerKAT and uGMRT.", "The radio contours of the MeerKAT image (red contour), NVSS image (blue contour) and GLEAM image (black contour) are overlaid on the DSS-II (red band) optical image (in gray-scale).", "The red, blue, and black surface brightness contours correspond to the contour levels 0.045, 0.090, 0.18, 0.36, 0.72, 1.44, 2.88, 6.0, 12.0, 24.0, 48.0 and 100.0 mJy beam -1 ^{-1}, 30, 60, 120 and 240 mJy beam -1 ^{-1}, and 800, 1600 mJy beam -1 ^{-1}, respectively.", "We also show a compass at the bottom right location indicating the north and east directions.", "All through our uGMRT and MeerKAT images, we follow this convention." ], [ "Data reduction", "The calibration of the data presented in this paper was carried out following the standard procedures.", "However, as the MeerKAT data reduction requires a new approach, we summarize here some of the specific steps that have been performed for both the uGMRT and the MeerKAT in reducing these data.", "The uGMRT datasets were calibrated using a standard approach [49].", "Observations of a flux density calibrator at the beginning and at the end of each run were used to correct for flux density scale and bandpass shape; the phase-calibration source was used to correct for phases.", "The data analysis, in particular editing of bad data, gain and bandpass calibrations, were carried out using the NRAO Astronomical Image Processing System aips, following standard imaging procedures.", "We made an error in providing the observing set-up for the uGMRT observations of CGCG 044$-$ 046 source, which had 200 MHz bandwidth with the high spectral resolution, whereas the rest of the sources had 400 MHz bandwidth with a factor of two lower spectral resolution.", "Note that this error did not limit us in terms of the angular resolution or the sensitivity (see below).", "We used the [68] flux density scale using the coefficients in the aips task setjy.", "After the above preliminary calibration, the 300-MHzThe GMRT wideband correlator supports a bandwidth of 400 MHz, 200 MHz, ..., in multiples of 0.5.", "The usable bandwidth of band-4 is 300 MHz, from 550 MHz to 850 MHz.", "wide dataset was split into six 50 MHz sub-bands for 4C 12.02, 4C 12.03 and CGCG 023$-$ 061 and four 50 MHz sub-bands for CGCG 044$-$ 046.", "An area of $\\approx $  1.5$^\\circ $  $\\times $ 1.5$^\\circ $ was imaged, just bigger than the first null of the uGMRT primary beam in order to correct for antenna-based gains in the direction of each radio source.", "A standard self-calibration procedure was performed in aips on each sub-band [50].", "All the calibrated 50 MHz sub-bands data were further stitched together to form full-bandwidth calibrated visibility datasets.", "These visibilities were imaged using the tclean task in casa.", "A final amplitude-and-phase self-calibration was also carried out to the full-bandwidth calibrated dataset onto the same flux density scale using casa, and the final images were obtained using the task tclean.", "We used 3D imaging (gridder = `widefield') and Briggs weighting (robust = 0.5).", "The MeerKAT datasets were also calibrated using a standard approach, but employing some novel software.", "We used the CARACal pipelinehttps://github.com/caracal-pipeline/caracal [43], [42] for the initial data reduction.", "CARACal orchestrates standard reduction packages into a single workflow.", "In this instance, it combined the Tricolourhttps://github.com/ska-sa/tricolour flagger [36] for radio frequency interference flagging, and standard CASA tasks for reference calibration.", "For the epochs employing B1934$-$ 638 as the primary calibrator, we used the [68] scale to set its flux density scale.", "For the epoch employing the other standard MeerKAT calibration source, PKS 0408$-$ 65, we used a custom component-based field model provided in CARACal, converted into model visibilities via the MeqTrees package [61].", "After applying all the reference calibration, the 4096 spectral channels data was averaged down to 1024 spectral channels, and imaged using the WSCLEAN package [64].", "We used Briggs weighting (robust = 0), disabled multi-frequency weighting, employed the joined-channel deconvolution and (4th order) polynomial fitting options of WSCLEAN to make wideband multi-frequency synthesis images.", "We imaged an area of $\\approx 22\\times 22$ for the CGCG 021$-$ 063 and CGCG 044$-$ 046 fields.", "For the 4C 12.02 and 4C 12.03 fields, we imaged a larger area of $\\approx 31 \\times 22$ : these sources are sufficiently close that their sidelobes contribute to each other's fields at the sensitivity levels we reach, and thus needed to be deconvolved jointly.", "This was followed by a round of phase and delay self-calibration using the CubiCalhttps://github.com/ratt-ru/cubical package [44].", "The 4C 12.02 field exhibited direction-dependent (DD) effects due to a 0.64 Jy (apparent) off-axis source, which was successfully peeled using CubiCal.", "The other fields did not require DD calibration.", "The 4C 12.02 and 4C 12.03 maps were then slightly improved via a round of amplitude self-calibration using CubiCal.", "This resulted in noise-limited maps for three of our four sample sources.", "The resulting MeerKAT image of CGCG 021$-$ 063 retains some radial, north-south oriented artefacts related to the point spread function (psf) centered on the bright core of the source.", "We were only partially able to mitigate these via self-calibration.", "These are not due to DD effects, since the core is the dominant source and it is at the centre of the field, nor are they likely to be deconvolution artefacts, as the core is completely unresolved.", "We hypothesize that they are due to residual nonlinearities in the system response, exacerbated by the effective psf of this observation: since the source is only $15^\\prime $ off the equator, the psf exhibits pathologically high sidelobes in the north-south direction, at the 1–2% level (compared to $<0.1\\%$ level in the east-west direction).", "The dynamic range of $\\sim 50\\,000:1$ achieved here is comparable to that achieved with MeerKAT on other equatorial fields (Ian Heywood, private communication).", "Our final images resulted in an angular resolution of $\\sim $ 6$^{\\prime \\prime }$ for all our target sources.", "The residual amplitude errors in each image are of the order of 5% for uGMRT and 3% for MeerKAT.", "The final 550–850 MHz (band-4) uGMRT and MeerKAT images are shown in Fig.", "REF , REF and REF .", "The MeerKAT observations of CGCG 044$-$ 046 and CGCG 021$-$ 063 included a scan of a bright polarised calibrator, 3C 286, which allowed us to perform polarisation imaging.", "Polarisation calibration was done via CARACal, using the standard CASA approach.", "The calibrator B1934$-$ 638 was used to derive frequency-dependent leakages (D–Jones), while 3C 286 was used to calibrate cross-hand phase and delay (K/X–Jones).", "The joined-polarisation mode of WSClean was then used to deconvolve Stokes $I, Q, U$ and $V$ maps.", "The $Q$ and $U$ images of CGCG 021$-$ 063 suffered from the same artefacts as the total intensity image, so we did not use them in our further analysis.", "We include here results from our polarisation observations using MeerKAT for CGCG 044$-$ 046 (see Sec.", ")." ], [ "The images", "Our uGMRT and MeerKAT observations revealed several new sources in the fields of view in addition to deep images of our four targets sources.", "Fig.", "REF shows the uGMRT and MeerKAT contours overlaid on the DSS-II (red band) optical image.", "Fig.", "REF is an image of 4C 12.02 showing the MeerKAT radio contours together with radio contours of NVSS and GLEAM images overlaid on the DSS-II (red band) optical image, to faithfully present the capabilities of MeerKAT and uGMRT.", "The compass depicts north and east and all through we follow this convention.", "Fig.", "REF provides colour scale with contour levels overlaid for each dataset (left and right panels showing the uGMRT and MeerKAT image respectively).", "It clearly shows that in almost all cases, barring CGCG 044$-$ 046, the surface brightness declines sharply at the edges of the visible radio lobes and of the low surface brightness features.", "We thus conclude that we have not missed any part of the source that fades into the noise.", "Details on each source are provided below." ], [ "4C 12.02", "It is an FR II radio galaxy whose optical counterpart (a galaxy) is located at redshift $z$ = 0.143.", "[76] first noted it as a triple source, with a radio core, and two radio lobes located on opposite sides along the east-west direction.", "Recently, [86] reported it as a representative giant X-shaped radio source based on the GMRT TGSS_ADR1 [38].", "Our uGMRT and MeerKAT images are shown in Fig.", "REF (top-left panel) and Fig.", "REF (top-left and top-right panels respectively).", "The angular resolution of the uGMRT provides important insight into the morphology of the hot spots, while the MeerKAT image is better suited to highlight the details and extent of the lobes.", "The morphology of the back-flows from the lobes is similar to what has been found for PKS 2014$-$ 55 [11].", "Both hot spots are resolved with multiple peaks.", "There are at least two hot spots near the termination of the west jet and probably a string of them in the east one.", "Both the east and west lobes show substructure.", "The north-western lobe flares abruptly at the peak of the hot spot closer to the core, while the southern lobe expands perpendicular to the main axis of the radio galaxy.", "In the west lobe, the hot spot before the bow shock is clearly extended in an arc perpendicular to the source major axis, which may suggest some sort of reflected, internal shock.", "The hot spots in the east and west lobes are not exactly co-linear if the line goes through what appears to be the active galactic nucleus (AGN).", "It is not clear that a projection effect could do this.", "The total flux density of the source is S$_{\\rm 0.69~GHz}=3.50~\\pm $ 0.19 Jy and S$_{\\rm 1.28~GHz}=2.15~\\pm $ 0.07 Jy.", "The angular extent of the source is $\\sim 5^{\\prime }$ , corresponding to $\\sim $ 760 kpc, i.e.", "a giant radio galaxy [2]." ], [ "4C 12.03", "This X-shaped radio source is associated with an elliptical host galaxy [35], classified as a low emission line radio galaxy located at redshift $z$ = 0.156 [48].", "[50] reported that 240 MHz and 610 MHz GMRT images show a symmetrical structure whose extent is $\\sim 4^{\\prime }$ along both axes, corresponding to $\\sim $ 650 kpc.", "The overall X-shaped morphology has been explained by a million-year precession period [23], or as a rapid realignment of a central supermassive black hole accretion disk system due to a relatively recent merger of a supermassive binary black hole [50], [57] or as a result of disk instability [15].", "Our new uGMRT and MeerKAT images, shown in Fig.", "REF (bottom-left panel) and Fig.", "REF (middle-left and middle-right panels respectively), confirm the X-shaped radio morphology.", "The northern and southern jets leading to the north and south hot spots respectively, form the active axis, whereas the east-west axis forms the low-surface brightness wings.", "The overall north-south morphology of the radio galaxy seems to follow an arc, and shows a clear asymmetry: the northern component culminates in a clear hot spot, as in FR IIs.", "On the other hand, the southern component could be either FR I or FR II.", "Both the western and eastern low-surface brightness wings are suggestive of a hydrodynamic back-flow as seen in PKS 2014–55 [11].", "The angular resolution of both our images clearly shows the presence of two inner brightness peaks (labelled as \"inner lobes\" in Fig.", "REF , middle-right panel), forming an inner double structure, whose total extent is $\\sim $ 80 kpc.", "These two features are perfectly aligned with the large north-south axis [77].", "Thus, the structures of the source possibly place it in the category of restarted sources.", "The overall extent of this radio galaxy is comparable to the earlier images, but broader extended emission is detected in the east-west radio lobes, whose outer parts show a filamentary structure at both frequencies.", "The total flux density of the source is S$_{\\rm 0.69~GHz}=3.14~\\pm $ 0.18 Jy and S$_{\\rm 1.28~GHz}=2.06~\\pm $ 0.06 Jy." ], [ "CGCG 044$-$ 046", "The source is identified as a m = 14.2 cD galaxy and is associated with the Zwicky cluster 1313.7$+$ 0721 at $z$ = 0.050145 [87].", "Indeed the bent radio morphology is typical of radio galaxies at the centre of galaxy clusters and groups.", "Some notable examples are ESO 137–006 [71] and 3C 465 [17], but see also [63], and [19] for a recent analysis.", "The radio source was first studied in detail by [67] and [66], who noticed the asymmetry between the sharp bend of about 90$^{\\circ }$ in the eastern tail and the more gentle bend in the western one, with diffuse emission further out in both.", "This asymmetry is confirmed in both our images (see Fig.", "REF , upper-right panel, and Fig.", "REF , third row, left and right panels for uGMRT and MeerKAT respectively), whose sensitivity allows the detection of emission from the radio tails to much larger distances than previously detected at these frequencies (see also Sec.", "REF ).", "The morphology suggests that it possibly belongs to the wide-angle tail class of radio galaxies.", "The extension of the western tail was also detected at lower frequencies via GLEAM though at much lower angular resolution [81], [37].", "Table: The total intensity and spectral index for our sample sources.", "The integrated flux densities quoted are in Jy along with corresponding error-bars when available.The radio galaxy shows an inner pair of straight jets, similar to FR I jets, which culminate in two high surface brightness knots at $\\sim $ 50 kpc from the core.", "Beyond this region, the eastern jet flares to form a tail, which bends sharply by $\\sim $ 90$^{\\circ }$ , heading toward the south, consistent with [67].", "Our images clearly show another bend at the southern end of the tail, behind the tail itself, as is clear from the radio contours (see Fig.", "REF ) and from the spectral analysis (see Sec. ).", "The transition between the western jet and the tail is quite sharp.", "The tail shows very little transverse expansion and makes a few wiggles before fading.", "Both our images reveal filamentary structure in the tails, whose projected length is about 150 and 300 kpc respectively for the eastern and the western tails, as measured using the MeerKAT data.", "Such sharp bends cannot be explained by projection effects, and are suggestive of reflection or refraction at discontinuities.", "The total flux density of the source is S$_{\\rm 0.69~GHz}=3.24~\\pm $ 0.16 Jy and S$_{\\rm 1.28~GHz}=2.05~\\pm $ 0.07 Jy." ], [ "CGCG 021$-0$ 63", "Very little information is reported in the literature on the arcsecond-scale properties of this radio galaxy, characterised by a compact, i.e.", "$<$ 0.2 milli-arcsec in size [16], radio core.", "Using the FIRST survey image, [70] reported that this source has a resolved compact component, with triple extended structure and possibly a $S$ or $Z$ -shaped structure.", "Our uGMRT and MeerKAT images (Fig.", "REF , bottom-right panel, and bottom row in Fig.", "REF , left panel and right panel respectively) are dynamic range limited, most likely because of the strong compact core.", "However, they clearly show that the radio emission of CGCG 021–063 has two components: a radio galaxy, whose projected linear size is $\\sim 280$ kpc that is embedded in a low-surface brightness cocoon of radio emission.", "The inner radio galaxy and the lobes have an FR II morphology, with only one visible jet and two hot spots of moderate brightness.", "It is interesting to note that the west jet seems to propagate beyond the hot spot or working surface of the jet.", "There is filamentary structure in the east lobe which is well reproduced in both images and there are also clearly beads in the west jet.", "We interpret the cocoon of fainter emission embedding the radio galaxy as the result of the back-flow from both jets.", "This emission is fairly symmetrical, suggesting that there is little relative movement of the galaxy through the intra-galactic medium (IGM).", "No information on the local environment of the optical host is available in the literature.", "The total flux density of the source is S$_{\\rm 0.69~GHz}=3.77~\\pm $ 0.20 Jy and S$_{\\rm 1.28~GHz}=2.80~\\pm $ 0.09 Jy.", "The spectral index analysis is a useful tool to investigate the life cycle of radio sources and to better understand their radio morphologies.", "In the following we present the integrated radio spectra for all sources.", "We further show uGMRT-MeerKAT spectral index imaging for CGCG 044–046 and in-band MeerKAT spectral imaging for all sources." ], [ "Integrated radio spectra", "We complemented our flux density measurements with literature information.", "Table REF (Cols.", "2–9) lists the integrated flux density we collected for our sources over a broad range of frequencies along with error-bars.", "The corresponding integrated spectra are shown in Fig.", "REF .", "Figure: Integrated flux densities of our sample radio sources at 550–850 MHz band of the uGMRT and 856–1712 MHz band of MeerKAT;data at other frequencies are from the literature (see Table  for references).", "The error-bars, not plotted are less than two times the size of the symbols.", "The spectra (and the data-points) are shifted with respect to one another for clarity.Our data points are very well aligned with the literature data, which confirms the reliable calibration for both uGMRT and MeerKAT observations.", "Note that we have used an identical polygon for our uGMRT and MeerKAT images to determine integrated flux density, making sure that we do not include the obvious artefacts in the images.", "Inspection of Fig.", "REF clearly shows that the spectra of our sources have slightly different behaviours.", "Those of CGCG 044–046 and CGCG 021–063 are reasonably well-fitted by a single power law, with $\\alpha = -$ 0.52 $\\pm $ 0.09 and $-$ 0.44 $\\pm $ 0.11, respectively.", "Both these values are very flat, which is remarkable if we consider that the integrated flux density measurements include the diffuse emission tails and radio lobes, whose spectra are typically steeper, i.e.", "$\\alpha \\simeq -$ 0.8.", "This suggests that for both sources the active components, i.e.", "the core and inner jets, are the dominant source of emission over a broad range of frequencies.", "The spectra of 4C 12.02 and 4C 12.03 on the other hand, show a clear break at $\\sim $ 1.2 GHz.", "Further, for both sources the spectra have $\\alpha \\sim -$ 0.7 down to 1.284 MHz, which steepens to $\\alpha \\sim -$ 1.2 at higher frequencies, suggesting the dominant role of the radio lobes at low frequency, as qualitatively suggested by our total intensity images in both cases.", "Figure: MeerKAT in-band spectral index images of our four sample sources, 4C 12.02 (top-left panel), 4C 12.03 (top-right panel), CGCG 044--046 (bottom-left panel) and CGCG 021--063 (bottom-right panel).The total intensity radio contours from the MeerKAT data are overlaid on it, with the lowest radio contour plotted is three times the local rms noise and increasing by factors of 2.We have also marked the three distinct regions of emission beyond the compact flat spectrum core, the inner jets all the way to the hot spots, the central part of the tails, and the terminating part of the tails for the CGCG 044–046 source.It highlights three different regions of emission beyond the compact flat spectrum core: (1) the inner jets all the way to the hot spots; (2) the central part of the tails, labelled as `B' and `C', and (3) the terminating part of the tails, labelled as `A'.", "This feature, `A' is the end of the tail and is called as the eastern protrusion.Furthermore the sharp transition from the central to the terminating part of the tail in the western radio emission is marked as dashed magenta line (see also Sec.", ")." ], [ "Spectral index imaging", "We constructed the spectral index image via the standard direct method, i.e., $log\\left(\\frac{S_{\\nu _1}(x,y)}{S_{\\nu _2}(x,y)}\\right) \\div log\\left(\\frac{\\nu _1}{\\nu _2}\\right),$ where $S_{\\nu _1}(x,y)$ and $S_{\\nu _2}(x,y)$ are flux densities at pixel location $(x,y)$ for two frequencies, $\\nu 1$ and $\\nu 2$ .", "With this approach we performed MeerKAT in-band spectral index imaging for all sources in our sample.", "The MeerKAT datasets were imaged using the WSCLEAN package [64] via a joint deconvolution and 4th order polynomial fitting options of it to make wide-band multi-frequency synthesis images (see also Sec.", "REF ).", "This provides images at centre-frequencies and the corresponding in-band spectral index maps.", "The resulting images are shown in Fig.", "REF for all sources, with total intensity radio contours overlaid from the 856–1712 MHz (L-band) MeerKAT data.", "We further produced uGMRT-MeerKAT spectral index images.", "The uGMRT and MeerKAT full resolution images, which on average give $\\theta $ $\\approx $ 45 and 75 at full width half maximum, were restored to the same angular resolution of $\\theta $ = 8$^{\\prime \\prime }$ at both central frequencies for the spectral index imaging.", "We point out that the ($u,v$ ) coverage of the two arrays is nearly identical, hence we did not need to remove baselines in the ($u,v$ ) planes to match the range of accessible angular scales in the two datasets.", "We use these matched resolution images to construct the spectral index images.", "The errors in the flux densities are approximately 5% and 3% at band-4 and L-band of uGMRT and MeerKAT, respectively, including calibration errors.", "The final errors in the spectral indices have been estimated by propagating individual errors in quadrature.", "Here we show only the uGMRT-MeerKAT spectral index image which we obtained for CGCG 044–046 (Figure REF ).", "For this source the comparable sensitivity of the uGMRT and MeerKAT datasets provides the best result, which adds information to the MeerKAT in-band image.", "CGCG 021–063 is affected by residual artefacts due to the strong nuclear component, nevertheless the MeerKAT in-band spectral index image is reliable for the strongest features, i.e.", "the jets and the hot spots.", "The spectral index distribution in our sources (see Fig.", "REF ) shows a flat spectrum core in all cases.", "The spectral index image of 4C 12.02 is quite puzzling.", "The spectrum of the very bright hot spots is overall similar to that of the lobes, with $\\alpha \\simeq -0.8$ , suggesting that replenishment of fresh particles is reducing, or has stopped.", "Radio galaxies show steepening from the hot spots towards the core, which is consistent with what is found in 4C 12.03 [32], where the lobes are back-flow emission.", "Our image for 4C 12.03 shows a clear separation in the spectral index distribution between the north-south and east-west axis, the latter being considerably steeper ($\\alpha \\simeq -1.2$ to be compared to the values in the range $\\alpha \\simeq -0.8$ and $-0.9$ in the north-south lobes).", "Our results are consistent with earlier findings in [73] and [50].", "The source will be further discussed in Sec. .", "The spectral index image of CGCG 044–046 shows four clearly separated regions, with the spectral index steepening gradually from the flat to inverted spectrum core, to the ending parts of the tail, where it reaches values of $\\alpha \\simeq -1.5$ .", "This source will be further discussed in Sec. .", "Despite the overall poorer quality of the image, the spectral index distribution of CGCG 021–063 is quite interesting, with the north-western hot-spot considerably flatter than the surrounding emission from the lobe, i.e.", "$\\alpha \\simeq -0.8$ and $\\alpha \\simeq -1.2$ respectively.", "On the other hand, the eastern hot-spot and channels of emission feeding it (see bottom-left panel of Fig.", "REF ) have a similar spectral index, in the range $\\alpha \\simeq -0.9$ ." ], [ "Discussion", "The MeerKAT and uGMRT images presented in this paper cover a frequency range from 550 MHz to 1712 MHz at nearly identical angular resolutions and sensitivities, which is ideal for studying the radio morphology and spectral features in the jets and lobes of radio galaxies.", "The images obtained with the two arrays, for all the sources studied, are remarkably similar in detail, which gives us confidence in the image integrity, imaging processing, and calibration of the uGMRT and MeerKAT data.", "The remarkable coincidence of the radio contours with the galaxies in the optical overlay (Fig.", "REF ) gives us confidence in even the lowest contour in our images.", "Overall, our images confirm the morphological classification which is reported in Table REF , and which has been made on the basis of images at lower sensitivity and poorer angular resolution.", "The only exception is CGCG 021–063, which can be classified as FR II radio galaxy.", "At the same time, each source has revealed interesting features which open new questions and throw light on the complexity of the interplay of the radio plasma emitted by the host galaxy and the surrounding medium in which the jets propagate.", "Our total intensity and spectral index imaging suggest that we can broadly identify three different regimes in all our sources: a region close to the AGN (and within the optical host) where the FR I and FR II division is relevant and fairly clear, a region further from the AGN where the intergalactic medium (IGM) affects the hot spots, and a third region, which goes beyond the hot spots and bears information on the interaction between the tails and the external medium (as in the case of CGCG 044–046).", "Figure: Spectral index image using the uGMRT and MeerKAT for CGCG 044--046 radio galaxy.", "The radio contour plotted corresponds to a spectral index of --1.0.Furthermore, the high angular resolution and high sensitivity of our images clearly reveals that substructure in the hot spots is common, which we discuss below." ], [ "Multiple hot spots", "The western hot spot in 4C 12.02 has two peaks (labelled in Fig.", "REF , upper-right panel), and the one closer to the core is perpendicular to the direction of the lobe.", "The eastern hot spot has three peaks along the direction of the jet flow, and it is notable that the brightest peak is not the outermost one.", "Some faint emission is also detected beyond the northern hot spot of 4C 12.03, while the southern hot spot is broad and uniform in surface brightness, which is unusual.", "The bent morphology of CGCG 044–046 is typical of radio galaxies at the centre of groups and clusters of galaxies.", "At the same time, the structure of the hot spots in the inner part of the source poses the question of what is really bending the jets and how the flow propagates beyond their location.", "Finally, the north-western hot spot in CGCG 021–063 is not the terminating point of the jet, because the radio emission is detected beyond the hot spot.", "The hot spot itself has two peaks of comparable brightness and is elongated along the direction of propagation of the jet.", "The south-eastern hot spot is misaligned with the source axis and has complex morphology perpendicular to the jet axis.", "Hot spots are believed to be the result of some form of shock, such as the working surface of the jets when they hit the interstellar medium or the IGM, forming the termination shock, also called the standing or reflected shock front [32].", "The spectral index images shown in Fig.", "REF are overall consistent with this idea, but there are some interesting trends that should be noted.", "On average, the hot spots have flatter spectra than the jets and lobes.", "However, there are remarkable deviations.", "For example, 4C 12.02 is intriguing, since the spectral index of the hot spots is not different from the emission of the lobes, at least in the MeerKAT frequency range (856–1712 MHz).", "The integrated spectrum for this source also suggests that the compact features are not dominant in the emission at high frequency.", "Additionally, the spectral features of the south-eastern hot spot in CGCG 021–063 are not different from the spectrum of the emission of the adjacent regions of the lobe.", "Radio galaxies with multiple knots and hot spots have been known for quite some time, with the hot spots themselves detected even at wavelengths other than radio wavelengths [45], [31], [30].", "Our results suggest various possibilities, such as the propagation of the jets through contact discontinuities, or standing shocks.", "The sensitivity and resolution of our images allow detailed insight into the brightness distribution of the lobes.", "Furthermore, each of the sources presented here shows features that deserve attention, which are discussed in the next subsection.", "Figure: Flux densities as a function of frequency (spectra) for four distinct regions of 4C 12.03 radio galaxy at low radio frequencies (see Sec. ).", "The 240 MHz and 610 MHz measurements are from , the 690 MHz and 1284 MHz measurements are using uGMRT band-4 and MeerKAT L-band images presented here, and the 1.5 GHz measurements are from .", "The error-bars are smaller than the size of the symbols." ], [ "Jets and lobes", "The misalignment and asymmetry of the two lobes in 4C 12.02 is remarkable.", "Moreover, the transition between the hot spot and lobe in the western emission is very sharp.", "We should note that the size of this radio galaxy places it in the class of giant radio galaxies.", "The properties of the external medium can thus change considerably through almost 800 kpc and projection effects may play an important role [33], [46].", "Furthermore, despite the morphology with prominent hot spots in both directions of the emission, the uniform spectral index between the lobes and hot spots suggests that the latter are no longer replenished.", "The X-shaped morphology of 4C 12.03, as well as its spectral index distribution, led us to investigate whether the radiative age in the north-south and in the east-west axes differ, as would be expected if they traced two different cycles of radio activity.", "The trend of the integrated spectral index in 4C 12.03, shown in Fig.", "REF (green line) highlights the contribution of the east-west structure in the total flux density at high frequencies.", "Assuming an equipartition of energy between relativistic particles and magnetic field, we determine magnetic fields, $B_{\\rm eq}$ [58] for the regions encompassing hot spots along the north-south axis and `winged' diffuse radio lobes along the east-west axis.", "The north and the south hot spots have $B_{\\rm eq}$ = 3.8 $\\mu $ G (and $u_{\\rm eq}$ = 13.7 $\\times $ 10$^{-13}$ erg cm$^{-3}$ ) and $B_{\\rm eq}$ = 3.1 $\\mu $ G (and $u_{\\rm eq}$ = 9.5 $\\times $ 10$^{-13}$ erg cm$^{-3}$ ) respectively, whereas the east and the west diffuse lobes have $B_{\\rm eq}$ = 2.1 $\\mu $ G (and $u_{\\rm eq}$ = 4.5 $\\times $ 10$^{-13}$ erg cm$^{-3}$ ) and $B_{\\rm eq}$ = 1.4 $\\mu $ G (and $u_{\\rm eq}$ = 2.0 $\\times $ 10$^{-13}$ erg cm$^{-3}$ ), respectively.", "Here we have assumed the ratio of energy in the heavy particles to that in the electrons and the filling factor of the emitting regions to be unity.", "The sizes of the four regions are symmetrical rectangular regions, each being at least $\\sim $ 15–20 beams across.", "From Fig.", "REF we estimated a break frequency $\\nu _{\\rm br}$ =0.69 GHz for the east and west wings, and assumed $\\nu _{\\rm br}$ =1.2 GHz for the north and south lobes.", "This leads to radiative lifetimes [40] of 40.8 Myr and 41.4 Myr respectively for the north and the south lobes (this value should be considered an upper limit, as the assumed break frequency is a lower limit), and 52.1 Myr and 51.5 Myr for the east and west diffuse lobes, respectively.", "The estimates of both the equipartition magnetic field and the radiative age, though not conclusive, are consistent with the possibility that the east and west diffuse wings are formed due to hydrodynamic mechanism, i.e.", "back-flow from the north and south active lobes, as suggested also by the in-band spectral index map of 4C 12.03 (Fig.", "REF , top-right panel).", "4C 12.03 is also characterised by two peaks of emission along the north-south jet, symmetrically located with respect to the core and separated by $\\sim $ 80 kpc (Figs.", "REF and REF ).", "They are well visible also in the spectral index distribution (Fig.", "REF ).", "They could be either brightness peaks of the underlying jets feeding the hot spots, or an inner double source related to a new cycle of radio activity [77].", "Unfortunately, our data do not allow us to discriminate between the two alternatives.", "Finally, we point out that the active axis is not aligned with the axis of the inner double source.", "The northern hot spot is aligned at $\\sim $ 10 deg, and the southern one at $\\sim $ 40 deg.", "It is unclear what could be responsible for such misalignment over scales of several hundreds of kiloparsec.", "Figure: The radio polarisation image (upper-panel) and the rr-band SDSS (DR12) image (lower-panel) of CGCG 044--046 with the MeerKAT L-band total intensity radio contours overlaid on them.Also overlaid on the polarisation image are the EVPA orientation vectors.", "In both images, the lowest radio contour plotted is approximately three times the local rms noise (= 18.0 μ\\mu Jy beam -1 ^{-1} and 23.5 μ\\mu Jy beam -1 ^{-1}, in upper and lower panels, respectively) and increasing by factors of 2.The distribution of E vector position angles are shown, where the vectors have lengths proportional to the degree of polarisation pp.", "The length of the vector, 1 arcsec corresponds to 3.125% (the length of the vector corresponding to p = 4% is shown in the lower left corner).", "No vectors are shown where the polarised signal-to-noise is <<4:1.", "The vectors are shown slightly undersampled, at ∼\\sim 5 arcsec intervals, for clarity.", "Four bright galaxies that are also part of the Zwicky cluster 1313.7++0721, of which CGCG 044--046 is the dominant galaxy, are also marked (see also Sec.", ").Unfortunately, the artefacts affecting the images of CGCG 021–063 do not allow a detailed analysis of its features.", "However, the filamentary structure of the south-eastern lobe is clear and remarkable.", "The uGMRT total intensity image is suggestive of two parallel channels feeding the hot spots.", "Alternatively, they could be the outer walls of the propagating continuous stream of radio-emitting synchrotron plasma (labelled in Fig.", "REF , upper-left panel).", "Looking more closely at the uGMRT and MeerKAT images together, the two parallel channels in the south lobe are off the line joining the north hot spot through the radio core to the south hot spot, which contradicts the simple picture that the two parallel channels in the south lobe could be feeding the hot spots.", "It seems more likely that the two channels are leading away from the south hot spot, rather than leading towards it, which then suggests that these parallel channels are probably the reflected shocks or back-flow emission.", "The MeerKAT in-band spectral index in this region is very uniform, $\\simeq $ $-$ 0.9, with no apparent transition between the hot spot and the emission feeding it.", "The jets and hot spots are surrounded by a cocoon and the cocoon could be a result of back-flow emission.", "It is possible that projection effects play a role in this source, as suggested by the very compact core and the strong asymmetry between the north-east and south-west jets." ], [ "CGCG 044–046", "The overall observational properties of CGCG 044–046 source suggest that the local environment plays a major role.", "More specifically, the physical mechanism giving rise to bent tails is possibly due to the motion of the host galaxy through the ICM.", "The source extends $\\sim $ 414 kpc to the west and $\\sim $ 207 kpc to the south-east, i.e.", "well into the ICM.", "The source is remarkably similar to 3C 465, the prototype of wide-angle tail radio galaxy [17], [21].", "The total intensity and spectral index images highlight three different regions of emission beyond the compact flat spectrum core: (1) the inner jets all the way to the hot spots; (2) the central part of the tails and (3) the terminating part of the tails.", "The overlay of the MeerKAT contours and the $r$ -band SDSS image of the optical field shows that the hot spots are located outside the envelope of the optical galaxy, at least at the sensitivity level of the $r$ -band SDSS image.", "We point out that the m$_g$ = 16.7 galaxy just north of CGCG 044–046 and within its optical envelope (at least in projection) is VIII Zw 276 and is located at the cluster redshift, $z$ = 0.049567.", "The inner jets and the hot spots (region 1) have a fairly uniform spectral index, $\\simeq $ $-$ 0.6, which is quite flat.", "The transition between the inner jet and the central part of the tail is sharp in the western part of the radio galaxy, both in the total intensity image and in the spectral index, which steepens to $\\alpha $ $\\simeq $ $-$ 0.9.", "This part of the tail (region 2) shows prominent bending with edge brightening.", "A sharp spectral transition with similar values is seen beyond the eastern hot spot, too.", "In the eastern lobe, we note a flatter central ridge with $\\alpha \\approx -0.7$ surrounded by steeper emission with $\\alpha \\approx -0.9$ .", "In the western emission, the transition from the central to the terminating part of the tail (region 3) is sharp both in the total intensity image and spectral index, which steepens from values around $-$ 1.0 to values around $-$ 1.5.", "The eastern lobe, on the other hand, seems to bend behind the emission itself, and we believe that the eastern protrusion (labelled `A' in Fig.", "REF ) with a spectral index as steep as $-$ 1.6/$-$ 1.8 is the end of the tail.", "The origin of these sharp transitions in surface brightnesses and spectral properties is unclear.", "They could reflect intermittent activity in the radio emission of the AGN and/or significant interaction of the jets and lobes with discontinuities in the IGM.", "Further hints of substantial bending come from the polarisation properties.", "This source is the only one with polarisation calibration source included in our MeerKAT observation.", "[66] presented a polarisation study at higher frequency (VLA $L$ - and $C$ -band), and was able to constrain rotation measures across the source to within $\\pm 30\\,\\mathrm {rad}\\,\\mathrm {m}^{-2}$ .", "Although these values are under the detectable range with our frequency coverage, we used full-band $Q$ and $U$ maps and did not perform rotation measure analysis.", "The upper panel of Fig.", "REF shows the electric field vectors superimposed on the MeerKAT total intensity image.", "There is a remarkable asymmetry in polarisation between the western and eastern regions of the radio emission and the two tails differ significantly in polarisation fraction.", "In particular, the central and terminating part of the western tail is much more polarised than the south-eastern one.", "Our recovered polarisation vectors are consistent with [66], but provide more detail across the lobes.", "Considering that this radio galaxy is at the centre of a galaxy cluster, where dense ICM is usually found, we suggest that the source is not in the plane of the sky and interpret the polarisation asymmetry as due to Laing-Garrington effect [20], i.e., an external medium that is denser against the south-eastern lobe.", "As a final remark, we point out that the four bright galaxies with radio emission clearly visible in the lower panel of Fig.", "REF are all part of the same Zwicky cluster 1313.7$+$ 0721.", "In particular, using data gleaned from NED (labelled in Fig.", "REF , lower panel), their names from west to east are as follows: (i) WISEA J131604.49$+$ 070453.4 (m$_g$ =18.4) at $z$ = 0.049298; (ii) CGCG 044$-$ 047 (m = 15.6) at $z$ = 0.049551; (iii) WISEA J131631.79$+$ 070000.9 (no magnitude available) at $z$ = 0.049297; and (iv) WISEA J131636.85$+$ 070054.6 (m = 17.11) at $z$ = 0.047596." ], [ "Concluding remarks", "In this paper we have presented uGMRT and MeerKAT images of four radio galaxies belonging to a larger sample of 12 FR I and FR II radio galaxies selected from the 4C catalogue with the main goal of investigating whether the sharp differences between the FR I and FR II morphologies still hold with the improved imaging capabilities of the current generation of radio interferometers, or if we need more morphological classes.", "We have explored the radio morphology of our four targets - 4C 12.02, 4C 12.03, CGCG 044–046 and CGCG 021–063 - in the light of the superb image sensitivity reached by the uGMRT in the 550–850 MHz band, and by the MeerKAT in the 856–1712 MHz band.", "The collected information hence seamlessly spans more than 1 GHz in bandwidth.", "We have supplemented our total intensity datasets with MeerKAT in-band spectral imaging for all sources.", "Furthermore, for CGCG 044–046 we obtained the uGMRT-MeerKAT spectral structure and MeerKAT polarisation information.", "The MeerKAT and uGMRT images of all four radio galaxies presented here are remarkably similar in detail, which gives us confidence in image fidelity, imaging processes and calibration of the uGMRT and MeerKAT.", "Moreover, in all cases, the morphology in the uGMRT band-4 and in the MeerKAT L-band is the same, even in the finest details.", "While we conclude that the overall FR I–FR II classification scheme still holds, at least for our targets, the combination of $\\mu $ Jy beam$^{-1}$ sensitivity and high ($\\sim 5^{\\prime \\prime }$ to 7$^{\\prime \\prime }$ ) angular resolution over the full 550-1712 MHz range reveals very interesting features.", "For example, filamentary emission in the lobes and substructure in the hot spots is common.", "Moreover, in CGCG 044–046, CGCG 021–063 and in the north hot-spot of 4C 12.03 the radio emission extends beyond the hot spots themselves.", "In 4C 12.02 and 4C 12.03 the two hot-spots are not at 180$^{\\circ }$ with respect to the core, and it is unclear what may cause such slight misalignment over hundreds of kpc.", "The MeerKAT in-band spectral imaging has complemented and confirmed the morphological classification from the total intensity radio images and provided new insights into some of the radio sources.", "In particular, the uniform steep spectrum in the lobes and hot-spots of 4C 12.02 suggests that the hot-spots are no longer replenished by fresh electrons.", "The radio emission in the east-west axis of 4C 12.03 is steeper than the north-south axis, and one possibility is that the former is the result of hydrodynamic back-flow, as has been recently seen in PKS 2014–55 [11].", "This radio galaxy also shows two inner brightness peaks, which form an inner double $\\simeq $ 80 kpc in size, aligned with the north-south axis of emission.", "Further investigation will be required to determine if this is the signature of restarted activity.", "Finally, the MeerKAT in-band spectral structure and polarisation information for CGCG 044–046 shows that most likely the source is not in the plane of the sky and that the observed properties are strongly affected by the cluster environment.", "We interpret the asymmetries in the polarisation properties as due to the Laing–Garrington effect.", "Our work further shows that very good image sensitivity over a broad range of angular scales is necessary to perform a detailed study of radio galaxies, and warns against the accuracy of the morphological classification of radio galaxies made with source detection algorithms, especially for surveys made with radio telescopes which have much lower sensitivity to diffuse, low brightness emission.", "At least for the sources presented in this paper, we conclude that there is probably little value in establishing a more complicated morphological classification system.", "It appears to us that the FR I and FR II (and possibly FR 0) morphologies are common and basic and that the complexities are introduced by the local environment, buoyancy and motions through the IGM.", "All are probably at play, together with reflection, refraction or bifurcation of bow shocks at discontinuities in the IGM.", "Adding further descriptors (e.g.", "wide-angle tail or narrow-angle tail) may be useful to link the morphology to the environment in which the lobes are formed or propagate, whereas the FR I/II classification probably describes conditions closer to or more intrinsic to the AGN and host galaxy." ], [ "Acknowledgments", "We thank the anonymous referee for his/her comments that improved and broadened the scope of this paper.", "DVL acknowledges the support of the Department of Atomic Energy, Government of India, under project no.", "12-R&D-TFR-5.02-0700.", "TV acknowledges the support from the Ministero degli Affari Esteri e della Cooperazione Internazionale, Direzione Generale per la Promozione del Sistema Paese, Progetto di Grande Rilevanza ZA18GR02.", "OS's research is supported by the South African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation.", "DK acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no.", "679627).", "FL acknowledges financial support from the Italian Minister for Research and Education (MIUR), project FARE, project code R16PR59747, project name FORNAX-B.", "SVW acknowledges the financial assistance of the South African Radio Astronomy Observatory.", "We thank the staff of the GMRT who made these observations possible.", "The GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research.", "The MeerKAT telescope is operated by the South African Radio Astronomy Observatory, which is a facility of the National Research Foundation, an agency of the Department of Science and Innovation.", "We are grateful to the full MeerKAT team at SARAO for their work on building and commissioning MeerKAT.", "This research has made use of the NED, which is operated by the Jet Propulsion Laboratory, Caltech, under contract with the NASA, and NASA's Astrophysics Data System.", "The GMRT data underlying this article are available via the GMRT online archive facilityhttps://naps.ncra.tifr.res.in/goa/data/search [naps.ncra.tifr.res.in].", "The MeerKAT data will be publicly available via the SARAO archivehttps://archive.sarao.ac.za [archive.sarao.ac.za] (proposal ID SCI-20190418-BF-01) after the end of the proprietary period in mid-2021, but may be shared earlier upon reasonable request to the corresponding author.", "All data analyses packages used in this work are publicly available, and their URLs have been noted in the main text." ] ]
2105.11695
[ [ "GAN for Vision, KG for Relation: a Two-stage Deep Network for Zero-shot\n Action Recognition" ], [ "Abstract Zero-shot action recognition can recognize samples of unseen classes that are unavailable in training by exploring common latent semantic representation in samples.", "However, most methods neglected the connotative relation and extensional relation between the action classes, which leads to the poor generalization ability of the zero-shot learning.", "Furthermore, the learned classifier incline to predict the samples of seen class, which leads to poor classification performance.", "To solve the above problems, we propose a two-stage deep neural network for zero-shot action recognition, which consists of a feature generation sub-network serving as the sampling stage and a graph attention sub-network serving as the classification stage.", "In the sampling stage, we utilize a generative adversarial networks (GAN) trained by action features and word vectors of seen classes to synthesize the action features of unseen classes, which can balance the training sample data of seen classes and unseen classes.", "In the classification stage, we construct a knowledge graph (KG) based on the relationship between word vectors of action classes and related objects, and propose a graph convolution network (GCN) based on attention mechanism, which dynamically updates the relationship between action classes and objects, and enhances the generalization ability of zero-shot learning.", "In both stages, we all use word vectors as bridges for feature generation and classifier generalization from seen classes to unseen classes.", "We compare our method with state-of-the-art methods on UCF101 and HMDB51 datasets.", "Experimental results show that our proposed method improves the classification performance of the trained classifier and achieves higher accuracy." ], [ "Introduction", "Action recognition is an important research issue in the fields of machine learning and computer vision, and has wide applications in human-computer interaction, video monitoring, motion retrieval and sports video analysis [1], [2], [3], [4], [5], [6].", "With the rapid development of internet technology and emerging social media, monocular or multiview videos provide fruitful cues for action recognition[7], [8], [9].", "Hence, the increase of both the complexity of human actions and the number of video classes is also inevitable.", "On the other hand, annotating massive video data is pervasive and important for action recognition, but it is a tedious and inaccurate task.", "Since it is not only a time-consuming and expensive operation, but also easy to be influenced by subjective judgments of experts.", "At the same time, due to the limitation of extensibility of data classes, traditional action recognition methods are unsuitable for recognizing data from unseen classes and cannot support the realization of automatic annotation.", "Therefore, how to obtain potential information from labeled videos so as to effectively annotate unseen videos has become an urgent problem.", "Zero-shot action recognition [11], [12], [25] provides an effective means to settle this problem, which can recognize data of unseen classes (the classes without training samples), and has attracted extensive attention recently.", "The aim of zero-shot learning (ZSL) is to explore common latent semantic representation, and produce a trained model that can generalize to unseen classes.", "Existing zero-shot action recognition methods can be divided into two categories.", "Manually-defined attribute based methods utilize manually defined attributes for classification, which are easy to understand and implement, and only use the relationship between actions and attributes to distinguish new action classes.", "However, human subjectivity and the lack of domain knowledge make it difficult to identify a set of attributes that can describe all actions.", "In addition, although attributes can be viewed as data-driven learning, their semantic meaning may be unseen or inappropriate.", "Therefore, it is difficult to generalize attribute-based methods to large-scale scenarios.", "Word embedding based methods utilize semantic representation of action class names (e.g., word vectors) to model the relationship between actions and actions in semantic spaces.", "The word vectors used in these methods are acquired by natural language processing using massive amounts of text information.", "Therefore, these methods can overcome the limitation of attribute based methods.", "However, these methods can only express the relationship between actions implicitly in word vector spaces and hardly benefit from the other information of videos resulting in poor classification performance.", "In summary, the above research work on zero-shot action recognition neglect the connotative relation and extensional relation between action classes, which leads to a poor generalization ability of ZSL.", "In fact, humans can use empirically learned semantic knowledge to extend their ability to recognize large-scale concepts by virtue of the association between connotation and extension extension.", "Thus, using structured knowledge information to build relationships of concepts (e.g.", "actions and attributes) can transfer learned knowledge from seen classes to unseen classes.", "Recently, Graph Convolutional Networks (GCN) based methods [30], [23] applied knowledge graph (KG) to ZSL and achieved promising results.", "However, the adjacency matrix constructed by these methods remains unchanged after initial setting, which makes it unable to describe the changing relationships of nodes in the graph adaptively, resulting in incomplete knowledge transfer.", "In addition, most of ZSL methods cannot use the samples of unseen classes in training, which makes the training classifier more inclined to predict the samples of seen classes.", "Recently, Generative Adversarial Networks (GAN) based methods [37], [24] synthesized the features of unseen classes, which are used to train the classifier.", "The performance of these methods can effectively improved, which shows that providing the samples of unseen classes during training can make the learned classifier better adapt to the classification requirements of unseen classes.", "It can be seen from the above that GCN based methods and GAN based methods have different perspectives of analysis and thinking.", "GAN based zero-shot learning used the adversarial relationship between sample features to realize the generalization of the feature generation ability of seen class to unseen class.", "Knowledge graph based zero-shot learning used the extrinsic correlation between classes and attributes to realize the generalization of classification ability.", "In general, GAN based methods are analyzed from sample level, and GCN based methods are analyzed from the classifier level.", "For both of these methods, the word vector is a bridge from the seen classes to the unseen classes for feature generation or classifier generalization.", "On the whole, these two types of methods are complementary with each other and the performance will be better by exploring them simultaneously.", "Therefore, we consider comprehensively from two aspects, and propose a new zero-shot action recognition, i.e., joint Feature Generation network and Graph Attention network (FGGA) for ZSL.", "The main idea comes from [31], [32] which indicated that simultaneously exploiting information from different perspectives can complementary with each other for action recognition than single exploitation.", "On one hand, FGGA utlizes GAN to synthesize the features of unseen classes, which can reduce the imbalance between training samples of seen classes and unseen classes.", "On the other hand, FGGA constructs a KG based on the relationship between action class and related objects, and a graph convolution network based on attention mechanism.", "In fact, graph convolution network based on attention mechanism can effectively realize the dynamic expression of the relation between action classes and objects, which reflects the influence of knowledge update on model learning.", "Experimental results show that FGGA improves the classification performance of the trained classifier and can achieve higher accuracy.", "A flowchart of FGGA is illustrated in Figure REF .", "FGGA consists of two stages: sampling stage and classification stage.", "In the sampling stage, FGGA uses feature generation network to train generator, which is used to synthesize the features of unseen classes.", "In the second stage, FGGA uses graph attention network to train classifier, which is used to expand the generalization ability of the classifier.", "Figure: Illustration of FGGA.", "FGGA consists of two stages: sampling stage and classification stage.", "The red arrows indicate the sampling stage, which is the training process of feature generation network, The orange arrows indicate the classification stage, which is the training process of graph attention convolution network.", "Red cylinder and blue cylinder represent the object node and the class node, respectively.The main contributions of our work are summarized as follows.", "We propose a two-stage network for zero-shot action recognition: FGGA, which joints feature generation network and graph attention network, makes a comprehensive analysis from sample level and classifier level.", "FGGA adopts the conditional Wasserstein GAN with additional loss terms to train generator, which can transform the representation space of actions from the word vector space to the visual feature space, and the synthesized features of unseen class can be straightforwardly fed to typical classifiers.", "FGGA integrates an attention mechanism with GCN, and expresses the relationship between action class and related objects dynamically.", "The remainder of this paper is organized as follows.", "Section  introduces the related work.", "Section  elaborates FGGA.", "Section  conducts comparative experiments and evaluates the performance of FGGA.", "Section  gives the conclusion." ], [ "Related Work", "In this section, we briefly review the work related to our proposed method.", "We review ZSL methods, which can be roughly grouped into two categories: attribute based methods and word embedding based methods." ], [ "Attribute based methods", "Early research work on action recognition mainly considered a set of manually defined attributes to describe the spatio-temporal evolution of actions.", "Liu et al.", "[10] proposed a potential SVM model in which potential variables determine the importance of each attribute of each class.", "Lampert et al.", "[33] proposed a direct and indirect attribute prediction model, which used attribute-class and class-class correlation to predict unseen samples.", "Akata et al.", "[34] regarded the classification problem based on attributes as a label embedding problem, and each class was embedded in the space of attribute vectors.", "This method measured the matching degree of image and label embedding by constructing matching function.", "Gan et al.", "[27] studied how to accurately and robustly detect attributes in images or video, and the learned high-quality attribute detector can be generalized to different classes.", "The advantage of the attribute-based approach is that each attribute describes the shared features between classes, and it can transfer knowledge between classes well.", "In addition, attributes can also describe the details of the class, which can be used to predict the unseen class.", "However, the attribute-based method has several drawbacks: Firstly, each action is a complex combination of various human motions and human-object interactions.", "Human subjective factors and the lack of domain knowledge make it very difficult to determine a set of attributes used to describe all actions.", "Secondly, the relationship between classes and attributes needs to be defined manually, whenever a new class is created, the attribute of class and the corresponding relationship between attribute and class need to be changed again.", "Therefore, attribute-based methods are not applicable to ZSL problems of massive classes.", "Thirdly, attributes can be viewed as data-driven learning, but their semantic meaning may be unknown or inappropriate." ], [ "Word embedding based methods", "Since only action class names are needed to construct label embedding, word embedding based methods are gradually favored.", "Xu et al.", "[14] first applied word vector space as middle layer embedding to the zero shot learning for action recognition.", "Subsequently, Xu et al.", "[15] proposed manifold regularization regression and data enhancement strategies, which significantly enhanced semantic space mapping, and used post-processing strategies such as self-training to further improve accuracy.", "Alexiou et al.", "[16] explored broader semantic context information in the text domain (for example, synonyms) expressed as word vectors that enrich the action class.", "Qin et al.", "[17] adopted error-correcting output coding to solve the domain shift problem, which took the advantage of semantics of class layers and internal data structure.", "However, because of the semantic differences between visual and textual information, individual word vectors are not sufficient to distinguish between classes.", "Recently, inspired by the strong relationship between objects and action, some methods have achieved good performance by using objects as attributes.", "Jaine et al.", "[18] constructed a semantic embedding model by considering thousands of object classes.", "Mettes et al.", "[19] further designed spatial perception object embedding to classify actions.", "In addition, a few methods explored semantic relationships using inter-class relationships [20] and pair relationships [21].", "Gan et al.", "[22] used knowledge information to construct an analogy pool based on external ontology.", "However, these methods are not end-to-end.", "Gao et al.", "[23] proposed a ZSL framework based on two-stream graph convolutional networks and knowledge graphs, and established a relationship between objects and classes in an end-to-end manner.", "Oza et al.", "[48] used class conditioned auto-encoders and the training procedure was divided in two sub-tasks, where encoder learns the first task following the closed-set classification training pipeline, and decoder learns the second task by reconstructing conditioned on class identity.", "Mandal et al.", "[24] proposed an out-of-distribution (OD) detector for detecting features of unseen classes, which could be trained by using real features of seen classes and synthetic features of unseen classes.", "Compared with shared semantic embedding spaces consisting of manually defined class attributes, word embedding spaces are composed of the word vectors of all classes.", "In these spaces, each class is described by its word vector.", "These word vectors are learned by natural language processing techniques using massive amounts of text information.", "Therefore, ZSL methods based on word embedding spaces overcome the limitation of ZSL methods based on attribute spaces, as the latter require manual definitions of the corresponding relationship between attributes and classes.", "In summary, the state-of-the-art methods mainly use GAN or knowledge graph.", "However, these methods used GAN and knowledge graph for the problem of zero-shot learning independently.", "GAN based zero-shot learning [24] considered the problem from the perspective of missing unseen class samples, and synthesized visual representations of the unseen classes.", "Knowledge graph based zero-shot learning [23] considered the problem from the perspective of the unseen classes and its relationship to the seen classes, and built structured knowledge information of unseen classes and seen classes.", "Generally, GAN based method is analyzed from the perspective of sample, and knowledge graph based method is analyzed from the perspective of class.", "For both of these methods, the word vector is a bridge from the seen classes to the unseen classes for feature generation or classifier generalization.", "In this paper, We combine GAN and knowledge graph and propose joint feature generation network and graph attention network for zero-shot action recognition.", "We use word vectors as bridges, both the vision features and the classifiers of unseen actions are inferable from seen actions.", "Furthermore, the edges of knowledge graph of previous work are constant values, which means the relationships between nodes in the knowledge graph are constant.", "Recently, incorporation of attention [49], [50] into Deep learning model has attracted a lot of interests, showing great potential in performance improvement.", "In this paper, we use graph attention to dynamically expresses the relationship between nodes in the knowledge graph.", "Overall, our method enhances the generalization ability and discriminability of the model by dynamically updating the relationship between action classes and objects and synthesizing features of unseen classes." ], [ "Methodology", "We give the methodology of FGGA in this section.", "We denote scalars, vectors, matrices, and sets by nonbold letters, bold lowercase letters, bold uppercase letters, and calligraphic uppercase letters respectively.", "Denote $[\\mathbf {A}]_{ij}$ to be the $(i,j)$ th element of a matrix $\\mathbf {A}$ .", "Denote $\\mathbf {I}$ to be the identity matrix whose size is determined in context.", "Denote $|\\cdot |$ to be the cardinality operator." ], [ "Problem definition", "Let $\\mathcal {X}\\subseteq \\mathbb {R}^{{d_x}}$ be the set of all samples, and let $\\mathcal {Y}^s,\\mathcal {Y}^u$ be the set of labels of seen classes, labels of unseen classes, respectively, where the superscripts $s,u$ denote “seen\", “unseen\", respectively.", "Let $\\mathcal {E}\\subseteq {\\mathbb {R}^{{d_c}}}$ be the set of word vectors of all classes.", "Denote $\\mathcal {S} = \\lbrace {(\\mathbf {x},y,\\mathbf {c}(y))|\\mathbf {x} \\in \\mathcal {X},y \\in {\\mathcal {Y}^s},\\mathbf {c}(y) \\in \\mathcal {E}\\rbrace }$ to be the training set for seen classes, where $\\mathbf {x}$ represents feature of sample, $y$ represents the class-specific label of seen classes, $\\mathbf {c}(y)$ represents the class-specific embedding.", "Additionally, we denote ${\\cal U} = \\lbrace {(u,\\mathbf {c}(u))|u \\in {\\mathcal {Y}^u},\\mathbf {c}(u) \\in \\mathcal {E}\\rbrace }$ to be a set for unseen classes, which is available during training, where $u$ represents the class-specific label of unseen classes, $\\mathbf {c}(u) \\in {\\mathbb {R}^{{d_c}}}$ represents the class-specific embedding, and $\\mathcal {Y}^{u} \\cap \\mathcal {Y}^{s}=\\varnothing $ .", "Furthermore, we have an object set ${\\cal O} $ , which serves as attributes for describing different action classes.", "The goal of ZSL is to learn a classifier $f_{\\rm ZSL}: \\mathcal {X} \\rightarrow \\mathcal {Y}^{u},$ and the goal of generalized zero-shot learning (GZSL) is to learn a classifier $f_{\\rm GZSL}: \\mathcal {X} \\rightarrow \\mathcal {Y}^{s} \\cup \\mathcal {Y}^{u}, $ where the features of samples of unseen classes $\\mathcal {X}^{u}$ are completely unavailable during training." ], [ "Unseen class feature synthesis based on feature generation network", "Most of the ZSL methods train classifier only by data of seen classes during training, which is prone to be biased towards the samples of seen classes and not conducive to the samples of unseen classes.", "To solve this problem, sample imbalance between seen and unseen classes during training needs further exploration.", "One of the straightforward ideas is to add additional synthesize features of unseen classes without access to any samples of that class.", "As we all know, GAN can synthesize images of unseen objects through given semantic descriptions.", "In this paper, the objective of our study is human actions rather than images.", "Therefore, to apply GAN to our model, we want to synthesize features of samples using GAN.", "Given the training data of seen classes, we hope to use the word vectors of unseen classes to synthesize the features of unseen classes.", "Therefore, we use features and word vectors of seen classes to learn GAN.", "GAN consists of a generator G and a discriminator D, where the generator is used to generate “false\" samples and the discriminator is used to determine whether samples are real or synthetic.", "G and D constitute a dynamic “game process\".", "Under the background of synthesizing video features, the goal of G is to obtain random signals from prior distribution and synthesize video features close to real features to deceive the discriminating network D. The goal of D is to accurately distinguish the video features synthesized by G from real video features.", "Since we need to synthesize the features of unseen classes, we use the condition GAN [35], that is, the word vector $\\mathbf {c}(y)$ of seen classes is added in the modeling of G and D. The conditional generator takes random Gaussian noise and the word vector of seen classes as its input and video feature as its output.", "Then, the generator can synthesize the video features of unseen classes through the word vectors of unseen classes.", "The loss function of condition GAN is: ${\\mathcal {L}_{\\rm GAN}} = \\mathbb {E}[\\log D(\\mathbf {x},\\mathbf {c}(y))]+ \\mathbb {E}[\\log (1 - D(\\mathbf {\\tilde{x}},\\mathbf {c}(y)))]$ where $\\mathbf {\\tilde{x}} = G(\\mathbf {z},\\mathbf {c}(y))$ represents the synthetic feature, $\\mathbb {E}[\\cdot ]$ is the expectation, and $D:\\mathcal {X} \\times \\mathcal {E} \\rightarrow [0,1]$ is a multilayer perceptron with the sigmoid function as the last layer.", "Although condition GAN can obtain complex data distribution, it is difficult to train.", "Therefore, we use Wasserstein Generative Adversarial Network-Gradient Penalty (WGAN-GP, [36]) for training, since the algorithm is more stable when training.", "The loss function of WGAN-GP is: ${\\mathcal {L}_{\\rm WGAN}} = \\mathbb {E}[D(\\mathbf {x},\\mathbf {c}(y))] - \\mathbb {E}[D(\\mathbf {\\tilde{x}}, \\mathbf {c}(y))]- \\lambda \\mathbb {E}\\left[ {{{\\left( {{{\\left\\Vert {{\\nabla _{\\mathbf {\\widehat{x}}}}D(\\mathbf {\\widehat{x}}, \\mathbf {c}(y))} \\right\\Vert }_2} - 1} \\right)}^2}} \\right]$ where $\\mathbf {\\tilde{x}} = G(\\mathbf {z},\\mathbf {c}(y))$ , $\\mathbf {\\widehat{x}} = \\alpha \\mathbf {x} + (1 - \\alpha )\\mathbf {\\tilde{x}}$ , and $\\lambda $ is the gradient penalty weight.", "The first and second terms approximate the Wasserstein distance, and the third term is the gradient penalty term for D. Unlike the condition GAN, the discriminator here is $D:\\mathcal {X} \\times \\mathcal {E} \\rightarrow \\mathbb {R}$ , which removes the sigmoid layer and outputs a real value.", "$\\mathcal {L}_{\\rm WGAN}$ cannot guarantee that the synthetic features are well suited for training a discriminative classifier.", "We conjecture that this issue could be alleviated by encouraging the generator to construct features that have strong discriminating ability and can correctly reconstruct the word vector.", "To this end, following [37], we add a decoder to reconstruct the word vector of the synthetic feature.", "The cyc (cycle-consistency) loss function is used, which is given by ${\\mathcal {L}_{\\rm CYC}} = {\\left\\Vert {\\mathbf {\\hat{c}}(y) - \\mathbf {c}(y)} \\right\\Vert _2}$ where $\\mathbf {\\hat{c}}(y)$ represents the reconstructed word vector.", "The final objective for training the feature generation network is as follows: ${\\min _G}~{\\max _D}~ ({\\mathcal {L}_{\\rm WGAN}} + \\beta {\\mathcal {L}_{\\rm CYC}})$ where $\\beta $ is hyper-parameter." ], [ "Classifier training based on graph attention network", "To extract information from implicit representations (word vectors) and explicit representations (knowledge graphs), we use GCN to train classifier.", "Similar to [23], we construct a knowledge graph whose nodes are word vectors of the concepts of seen classes, unseen action classes and objects, and whose edges are defined by using the adjacency matrix later.", "Let $S=|\\mathcal {Y}^s|, U=|\\mathcal {Y}^u|, O = |\\mathcal {O}|$ .", "GCN takes the word vectors of $ S + U$ classes and $O$ objects information as input, and obtains all action class classifiers $\\lbrace {\\mathbf {w}_i}\\rbrace _{i = 1}^{S+U}$ and all object classifiers $\\lbrace {\\mathbf {w}_i}\\rbrace _{i = S+U + 1}^{S+U + O}$ through the transmission and calculation of information between each layer of GCN.", "Among them, $O$ object classifiers act as a bridge between seen classes and unseen classes.", "Each layer of GCN takes the feature matrix of the previous layer $\\mathbf {Z}^{(l - 1)}$ as input and outputs a new feature matrix $\\mathbf {Z}^{(l)}$ .", "The input of the first layer is a $k_{0} \\times (S+U + O)$ feature matrix $\\mathbf {Z}^{(0)}$ , where $k_{0}$ represents the dimension of the input feature.", "The convolution operation of each layer in the network can be expressed as $\\mathbf {Z}^{(l)} = \\mathbf {{D}}^{-\\frac{1}{2}} \\mathbf {\\widehat{A}} \\mathbf {{D}}^{-\\frac{1}{2}} \\left(\\mathbf {Z}^{(l - 1)}\\right)^{\\top } \\Phi ^{(l - 1)},~~l=1,\\ldots ,L$ where $L$ denotes the total layer number, $\\mathbf {Z}^{(l)} \\in \\mathbb {R}^{k_{l} \\times (S+U + O)}$ represents the feature matrix of the $l$ th layer, $1\\le l\\le L$ , $\\mathbf {\\widehat{A}} = \\mathbf {{A}}+ \\mathbf {I}$ , $\\mathbf {A}\\in \\mathbb {R}^{(S+U + O) \\times (S+U + O)}$ represents the adjacency matrix of the knowledge graph, $\\mathbf {{D}}\\in \\mathbb {R}^{(S+U + O) \\times (S+U + O)}$ is a diagonal matrix whose diagonal entries are given by ${[\\mathbf {{D}}]_{ii}} = \\sum _j [\\mathbf {\\widehat{A}}]_{ij}$ , and $\\Phi ^{(l - 1)}\\in \\mathbb {R}^{k_{l-1} \\times k_{l}}$ represents the parameter matrix of the $(l-1)$ th layer.", "Each layer is followed by a ReLU function.", "To further assign the trained action classifier with a stronger classification ability, we perform attention mechanism on the nodes, which update the relationship between action-object, object-object and action-action after each iteration.", "Specially, we define the attentional coefficient matrix $\\mathbf {B}$ as $[\\mathbf {B}]_{ij}={\\left\\lbrace \\begin{array}{ll}\\frac{\\mathbf {w}_{i}^{\\top }\\mathbf {w}_{j}}{\\Vert \\mathbf {w}_{i}\\Vert _{2}\\Vert \\mathbf {w}_{j}\\Vert _{2}},&\\text{if $\\mathbf {w}_{i}\\in \\mathcal {N}_k(\\mathbf {w}_{j})$~or~$\\mathbf {w}_{j}\\in \\mathcal {N}_k(\\mathbf {w}_{i})$ }\\\\0,&\\text{otherwise}\\end{array}\\right.", "}$ where ${{\\cal N}_k}({\\mathbf {w}_j})$ represents the set of the $k$ nearest neighbors of the $j$ th node.", "To make coefficients easily comparable across different nodes, we normalize them across all choices of node $j$ using the softmax function by: $[\\mathbf {A}]_{ij}=\\dfrac{\\exp ([\\mathbf {B}]_{ij})}{\\sum _{j^{\\prime } \\in {{\\cal N}(j)}}\\exp ([\\mathbf {B}]_{ij^{\\prime }})}$ where ${{\\cal N}(j)}$ represents the index set of all the neighbors of node $j$ .", "The cross entropy loss function of GCN in training is given by: $\\begin{split}\\mathcal {L}_{\\rm CE} &= - \\frac{1}{N}\\sum _{n = 1}^N {\\sum _{c = 1}^{S+U} {y_n^c} } \\log \\left( {p_n^c} \\right) \\\\p_n^c &= \\frac{\\exp \\left( {\\mathbf {w}_c^{\\top } \\mathbf {x}_n^c} \\right)}{\\sum _{c^{\\prime } = 1}^{S+U} \\exp \\left( \\mathbf {w}_{c^{\\prime }}^{\\top } \\mathbf {x}_n^{c^{\\prime }} \\right)}\\end{split}$ where $y_n^c\\in \\lbrace 0,1\\rbrace $ is a boolean variable indicating whether the $n$ th sample belongs to the $c$ th class, $N$ represents the sum of the number of training samples of the seen class and the number of synthetic samples of the unseen class, $\\mathbf {w}_c$ represents the classifier of $c$ th action class, $\\mathbf {x}_n^c$ represents the $n$ th sample which belongs to the $c$ th class, and $\\mathbf {w}_c^{\\top } \\mathbf {x}_n^c$ represents the predicted score of the $n$ th sample in the $c$ th class, and $p_n^c$ represents the predicted score with a softmax operation." ], [ "A two-stage deep network", "After training the feature generation network, we generate features of unseen classes by trained generator.", "Specifically, given the word vector $\\mathbf {c}(u)$ of an unseen class $u \\in {\\mathcal {Y}^u}$ and random Gaussian noise $z \\in {\\cal Z}$ , any feature $\\mathbf {\\tilde{x}}$ can be synthesized by $\\mathbf {\\tilde{x}} = G(z,\\mathbf {c}(u))$ , thus the synthesized training set $\\widetilde{\\cal U} = \\lbrace (\\mathbf {\\tilde{x}},u,\\mathbf {c}(u))\\rbrace $ can be obtained, which provides more training samples of unseen classes for the training classifier of graph convolution network based on attention mechanism, and enhances its classification performance and generalization ability.", "Therefore, we propose a two-stage network, which joints feature generation network and graph attention network as shown in Figure REF , which is analyzed from both sample level and classifier level.", "In this way, the training samples include the sample features of the seen class and unseen classes.", "During training, we first generate the features of unseen classes by sample level, and use them together with the features of seen classes as training samples.", "Then, both the classifiers of the seen classes and unseen classes are trained by classifier level.", "In the test stage, we use the classifier of unseen classes to classify test videos.", "This section verifies the effectiveness and superiority of FGGA on UCF101 [38] and HMDB51 [39].", "HMDB51 consists of 6,766 videos, including 51 action classes, while UCF101 consists of 13,320 videos, including 101 action classes (see Figures REF -REF ).", "We verify the effectiveness of FGGA using two different tasks: ZSL and GZSL.", "During the training phase, samples and attributes of seen classes are available for both ZSL and GZSL tasks.", "In the test stage, for ZSL, the trained model only evaluates data of unseen classes; For GZSL, the trained model evaluates the data of both seen and unseen classes.", "Figure: Some samples of HDMB51, which consists of 51 action classes." ], [ "Experimental settings", "$\\mathbf {Video~feature~extraction}$     We use Inflated 3D [40] as the initial feature of video.", "Appearance Inflated 3D features and stream Inflated 3D features are extracted from mixed 5c layer output of RGB Inflated 3D network and stream Inflated 3D network, respectively.", "Similar to [24], both networks are pre-trained on Kinetics dataset [40].", "These two features are 4096 dimensions respectively, and the feature of each video is the splicing of these two features, i.e., 8192 dimensions.", "$\\mathbf {Knowledge~graph~building}$     We use ConceptNet [41] to build knowledge graph.", "ConceptNet is a semantic network that contains a lot of information about the world that computers should know to help them do better searches and understand human intentions.", "It consists of nodes representing concepts that are expressed as natural language words or phrases and whose relationships are indicated.", "ConceptNet contains relational knowledge and is connected to a subset of DBPedia that includes knowledge extracted from information box Wikipedia.", "Wikipedia is the largest collaborative online encyclopedia with 3 million articles in English.", "Much of this knowledge comes from Wiktionary, a free multilingual dictionary that provides information about synonyms, antonyms, and concepts translated into hundreds of languages.", "More dictionary knowledge comes from open multilingual WordNet, which is an English vocabulary containing more than 100,000 word concepts, rich in semantic information.", "Something about intuitive associations of people for words comes from “games with a purpose\".", "Similar to [19], we use an English subgraph of about 1.5 million nodes and string matching to map concepts to nodes in ConceptNet.", "The most important thing in building a knowledge graph is to determine the relationships between these nodes.", "Specifically, if two nodes that connect to an edge can be found in ConceptNet, the initial correlation between the two nodes is expressed using the weight corresponding to the edge.", "Although the knowledge graph may have many types of edges, we follow the previous approach [19] to reduce it to a single matrix to effectively express semantic consistency and propagate information between nodes.", "$\\mathbf {Word~embedding}$     We use the images and videos from YFCC100M [19] to train skip Gram network.", "The training model generates a 500-dimensional word vector representation for each word.", "$\\mathbf {Network~structure}$     For our GAN, its generator is a three-layer fully connected network, whose output feature dimension is equal to video feature dimension, and the size of hidden layer is 4096.", "The decoder is also a three-layer fully connected network.", "The feature dimension of the output layer is equal to the word vector dimension of classes, and the size of the hidden layer is 4096.", "The discriminator is a two-layer fully connected network, of which the size of the dimension of the output/hidden layer feature is 1/4096, and the value of $\\beta $ is 0.01.", "For our GCN, it is composed of three graph convolution layers, and the output channel sizes are 8192, 4096 and 4096 respectively.", "We apply LeakyReLU as the activation function after each convolutional layer of the graph and performed $\\ell _2$ regularization on generated classifiers." ], [ "Results", "In this section, we introduce the comparison results between FGGA and other ZSL methods on ZSL and GZSL tasks, and take the average accuracy and standard deviation as the evaluation criteria for effectiveness.", "First we compared the results on the ZSL task, then we compared the results on the GZSL task.", "The comparative methods mainly include: SVE (Self-training with SVM and semantic Embedding [14]) performs self-training and data augmentation strategy ESZSL (Embarrassingly Simple Zero Shot Learning [42]) models the relationship between features and attributes SJE (Structured Joint Embedding [43]) learns a compatibility function, which relates image features and side information MTE (Multi-Task Embedding [13]) Performs multi-task visual-semantic mapping ZSECOC (Zero-Shot with Error-Correcting Output Codes [17]) performs category-level error-correcting output codes GA (Generative Model [44]) models each action class as a probability distribution UR (Universal Representation [45]) performs generalized multiple-instance learning and universal representation learning TS-GCN (Two-Stream Graph Convolutional Network [23]) models the three types of relationships by two stream GCN ConSE (Convex combination of semantic embedding [46]) performs image embedding system and semantic word embedding CLSWGAN (WGAN with class-level semantic information [47]) generates features instead of images and is trained with a novel classification loss CEWGAN (WGAN with class-embeddings [24]) performs WGAN with cosine embedding and cycle-consistency to synthesize CNN features of unseen class CEWGAN-OD (WGAN with class-embeddings and OD detector [24]) performs CEWGAN with out of distribution detector to synthesize CNN features of unseen class Figure: Some samples of UCF101, which which consists of 101 action classes.Table: ZSL performance comparison with state-of-the-art methods on HMDB51 and UCF101.Table: GZSL performance comparison with state-of-the-art methods on HMDB51 and UCF101.To make a fair comparison on the ZSL task, we use the partitioning strategy in [15], that is, 50% of the action class videos are used for training, and the other 50% of the action class videos are used for testing.", "For each dataset, we randomly select 10 partitions.", "Table REF shows experimental results on the ZSL task with respect to different types of feature extraction: Bag of words, Fisher Vector, Inflated 3D, with the same label embedding method: word2vec.", "As can be seen from the table, FGGA achieves the best accuracy on UCF101 and HMDB51.", "Compared with traditional methods (ZSECOC, UR, etc.", "), those methods based on deep learning (including FGGA) produce better performance.", "Compared with the recent methods UR, CLSWGAN and CEWGAN, FGGA improves 6.8%/10.8%, 2.1%/2.5% and 1%/1.4% respectively on HMDB51/UCF101.", "These results demonstrate the effectiveness of FGGA.", "In addition, the performance of FGGA has a smaller standard deviation, which indicates that FGGA has a relatively stable identification performance under different training and testing data partition.", "To make a fair comparison on the GZSL task, we use the strategy in [44], that is, 20% of samples of the seen classes were selected for testing and the rest samples were trained.", "Table REF shows the experimental results on the GZSL task.", "It can be seen from the table that FGGA is superior to other comparative methods on the two datasets, reaching 36.4% and 37.6% recognition accuracies respectively on HMDB51 and UCF101.", "Compared with CEWGAN which achieved best results currently, FGGA achieves better performance." ], [ "Effectiveness analysis", "In this section, we further compare with several baseline methods.", "We first compare the performance of FGGA and TS-GCN.", "To make a fair comparison with TS-GCN, we also use object scores feature [23] as the initial feature, and represent each sample by a graph structure.", "Therefore, the feature generation network is not applicable.", "Figure REF shows the comparison results of FGGA and TS-GCN under different epochs on UCF101, mainly to verify the effectiveness of the graph attention mechanism.", "As can be seen from the figure, the results of the two methods are equivalent when $epoch\\le 3$ .", "when $epoch>3$ the result of TS-GCN tends to be stable, FGGA still has an upward trend.", "In general, FGGA has better classification performance, which proves that the graph attention mechanism is effective.", "Table REF shows quantitative results of FGGA versus several GAN based ZSL methods, which includes the accuracy of the seen class, the accuracy of the unseen class, and the harmonic mean of them.", "Among these comparative methods, CLSWGAN applied GAN to ZSL earlier, and CEWGAN-OD is the best ZSL method based on GAN at present.", "As can be seen from Table REF , FGGA achieves the best accuracy.", "The results of FGGA and CEWGAN-OD were significantly better than those of CLSWGAN and CEWGAN.", "Compared with the two methods, FGGA improved the performance by 3.7%/5.2% and 2.8%/3.9% respectively on HMDB51/UCF101, which indicated that the design of a reasonable classifier could significantly improve the performance.", "Compared with CEWGAN-OD, FGGA is slightly less accurate in unseen classes, but it is significantly better in seen classes.", "The reason for better accuracy performance of CEWGAN-OD on unseen classes may be that CEWGAN-OD adds an OD detector which can detect whether each sample belongs to a seen class before classification, thus improving the classification performance.", "However, FGGA merely uses a common classifier for classification instead of OD detectors, which still achieves considerable performance.", "To analyze the sensitivity of FGGA to different word embedding methods, and make a fair comparison with CEWGAN, we test the performance on another Word2Vec embedding trained on GoogleNews as shown in Table REF .", "From the table we can see that although the performance of FGGA has some changes after changing the word vector, it can still achieve better performance.", "Overall, the above analysis demonstrates that FGGA outperforms the baseline methods.", "Table: Accuracy of GZSL versus state-of-the-art methods on HMDB51 and UCF101.Table: ZSL performance comparison with word vector trained on GoogleNews.Figure: Comparison results with different number of GCN layers on HMDB51 and UCF101.Figure: Comparison results of different feature dimensions on HMDB51." ], [ "Parameter analysis", "To verify the effect of GCN depth on FGGA performance, we compared FGGA performance at different levels.", "Figure REF shows the results of FGGA under different GCN layers, in which the number of output channels of two layers is 4096,4096 respectively.", "For the three layers, the number of output channels is 8192, 4096, 4096 respectively.", "The number of output channels for the three layers is 8192, 4096, 4096, 4096 respectively.", "As can be seen from the figure, the performance of the three-layer GCN is the same as that of the four-layer GCN, and the number of deeper layers does not get a higher recognition rate.", "The reason may be that the number of training samples is not sufficiently large, leading to an overfitting of the deeper network.", "To verify the influence of feature dimensions on FGGA, we compared the performance of FGGA under different feature dimensions, as shown in Figure REF .", "As can be seen from the figure, when the feature dimension is 512, the performance is obviously poor.", "When the feature dimension reaches 1024 dimensions, the effect is significantly improved, and the best recognition accuracy can be obtained when the feature dimension reaches 4096 dimensions." ], [ "Ablation study", "To further verify the effectiveness of feature generation network and graph attention network, we conducted ablation experiments by comparing FGGA with FGGA without using feature generation network (denoted by FGGA-NoFG), FGGA without using attentional mechanism (denoted by FGGA-NoAt) and model with only WGAN (denoted by only WGAN).", "Figure REF shows the comparative results, which indicates that feature generation network and graph attention convolutional network introduced in FGGA are both important and effective for ZSL.", "As for FGGA-NoFG, this method has no synthesis features of unseen classes in training, which leads to a poor classification performance of the learning classifier for unseen classes.", "For FGGA-NoAt, this method does not update the adjacent matrix dynamically in training, which prevents the relationship among action-action, action-object and object-object from being adjusted adaptively, so the classifier classification performance is poor.", "For only WGAN, this method has no cycle-consistency constraint term, and doesn't analyse the relationship of different classes.", "It results in the worst performance.", "Therefore, each component of FGGA contributes to FGGA in an indispensable fashion.", "Figure: Ablation results of FGGA on HMDB51 and UCF101.In this paper, we propose a new zero-shot action recognition framework: FGGA, which can effectively improve generalization and discrimination of model by analyzing relationship between the action classes and training sample imbalances of seen and unseen class.", "On the one hand, FGGA effectively circumvents the problem of unbalanced training samples of seen classes and unseen classes by establishing and generating the action features of the unseen class generated by the conditional Wasserstein GAN with additional loss terms.", "On the other hand, FGGA transfers the knowledge learned from the seen class to the unseen class by means of the prior knowledge graph, and proposes the graph convolution network based on the attention mechanism, which leads to promising classification performance of learned classifiers.", "Experimental results show that FGGA can achieve higher accuracy on two public datasets.", "Comprehensive performance studies have been conducted by comparing FGGA with state-of-the-art methods over two datasets.", "The effectiveness of FGGA is evidenced by its favorable performances compared with others." ] ]
2105.11789
[ [ "An Ultra-compact Object from Semi-classical Gravity" ], [ "Abstract In a recent report, Carballo-Rubio [1] utilizes the semi-classical theory of gravity to obtain a generalized Tolman-Oppenheimer-Volkoff (TOV) equation.", "This model has a new coupling constant $l_p$ implying two different modified TOV equation forms characterized by the sign of $p'$.", "The negative branch reduces to the ordinary GR-TOV in the limit of $l_p\\to0$, while the positive one does not.", "In the positive branch, Carballo-Rubio was able to find the exact solutions using the constant-$\\lambda$ trick.", "In this work, we investigate whether this theory's negative branch can also provide a different feature of the ultra-compact object compared to those obtained from the GR-TOV equation.", "We study ultra-compact objects with an isotropically ideal fluid matter where we use a simple but physically motivated equation of state $\\rho=p/w+\\rho_0$ with $w$=1 and $w$=1/3.", "In general, we obtain that the range of $l_p$ is very restricted and must not be equal to $r_c$.", "Here $r_c$ is the starting point of integration located at the center of the star.", "While $l_p$ should be set to be much larger than Planck length $L_\\text{Pl}$.", "Consequently, the mass-radius curves for the various value of $l_p$ for both $w$ cases are still indistinguishable from the standard GR-TOV results.", "Hence from the negative branch of $p'(r)$, the additional free parameter $l_p$ does not provide a significant effect compared to the standard GR-TOV equation results, even though $l_p$ is not in the limit of $l_p\\to0$ anymore.", "Therefore, similar to the conclusion in Ref.", "[3] with GR theory that the ultra-compact objects from negative branch of semi-classical gravity with a linear equation of state are unable to generate demanding gravitational echoes." ], [ "Introduction", "The recent excitement over gravitational wave astronomy by LIGO and Virgo [4] gave rise to many discussions over what kind of massive bodies produce the waves.", "From the astronomical point of view, the curiosity of synthesizing heavy metals such as silver and platinum from a collision of two neutron stars also triggered excitement.", "From the theoretical point of view, many attempts have been made to justify the existence of horizonless ultra-compact objects other than known compact objects such as neutron stars or white dwarfs when interacting with other massive bodies by probing its behavior through its waves.", "There are many proposals given, including gravastars [5], [6] where inside the surface of the star lies another de-Sitter space connected to the usual Schwarzschild space-time by a thin shell of ultrarelativistic matter.", "Interestingly recently, Carballo-Rubio [1] has claimed to predict exotic stars that have many similar features with gravastar using semiclassical gravity theory.", "The semiclassical theory of gravity is probably one of the earliest attempts by physicists to reconcile quantum field theory and gravity.", "It began with the remarkable result on black hole entropy and temperature and the Unruh effect (see the famous classic monograph [7] and for more recent one see [8]).", "In this theory, the concept of particle creation in general space-time is different from the usual one in Minkowski space-time.", "This ambiguity is mainly caused by the Bogolyubov transformation, which is similar to coordinate transformation.", "Although the problem seems unsolvable, it turns out that the energy-momentum tensor is indifferent to what “coordinate system” we chose.", "Even though there are infinities inside them, physicists had identified what kind of infinities leads to the physical quantum state, resulting in the so-called Hadamard state, i.e., a state whose two-point function has infinities whose form is the same as the one calculated in Minkowskian space-time.", "Another state exists, i.e., Boulware state, which had been extensively studied again since it describes a Schwarzschild black hole vacuum.", "Since the study of $n$ -point functions are related to regularization of infinities from expectation value of energy-momentum tensor, $\\langle \\hat{T}_{\\mu \\nu }\\rangle $ , hence the main interest in the semiclassical theory of gravity is mainly around renormalization of the energy-momentum tensor which is related to the Einstein tensor by $ G_{\\mu \\nu }=8\\pi G \\langle \\hat{T}_{\\mu \\nu } \\rangle .$ The resulting renormalization should satisfy Wald's renormalization axioms (see page 89 of [9]).", "It has been known that the methods of renormalization only can apply for some specific cases [10], and in the case of conformal field theory, these methods give rise to the so-called trace anomaly [7] which is a still unsolvable problem in semi-classical gravity theory.", "The value of the cosmological constant is restricted in a nontrivial way [11]; hence one cannot treat this constant as a free parameter in this theory.", "Here we provide a short discussion related to Boulware and Unruh vacuums.", "The definition of Boulware vacuum is a quantum state of the exterior of any massive body but singular at the horizon.", "In contrast, the Unruh vacuum is the quantum state of gravitational collapse, and it features Hawking radiation at large distances  [12], [13].", "In general, these two vacuums are inequivalent even though they could be interlinked classically by coordinate transformation.", "Please see Ref.", "[14] for a detailed discussion about how to relate both quantum vacuums.", "The authors of Ref.", "[15] investigated the dynamical evolution of a collapsed star within semiclassical gravity.", "They found that in some cases, the trapping horizon is prevented from forming in a semiclassical approach, and the new collapsed objects could exist.", "These objects make the confrontation with information paradox and run-away endpoint problem unnecessary.", "They also found that both quantum vacuums can describe the same exterior compact object locally as long as the horizon is unformed.", "If the horizon is formed, then the Boulware vacuum could no longer be used, and one should use Unruh vacuum to describe the object.", "Please see detail technical discussions of this matter in Ref. [15].", "A recent study by Ho and Matsuo [2] has shown that if the compact objects have no singularity nor horizons implies that arbitrary heavy objects can have a physical state in the Boulware vacuum as long as it has stationary state.", "Furthermore, Ho and Matsuo in other paper [16] have shown that it is physically sensible to consider the Boulware vacuum for any macroscopic radius of a compact object from nonperturbative analysis of the semiclassical Einstein equation.", "These results  [2], [16] are contrary to the common belief that Boulware vacuum becomes unphysical if the radius is smaller than the Schwarzchild radius since the energy-momentum tensor is divergent at the Schwarzchild radius.", "In this work, we study the compact object without horizon, and the radius is larger than the Schwarzchild radius.", "Therefore, using a Boulware vacuum is relatively safe.", "Meanwhile, on gravitational wave astronomy, there are some indications that all black holes cannot be observed except by their intense gravity effects.", "Therefore, there exists the possibility of the existence of the so-called black hole mimickers.", "These are compact horizonless astronomical objects, usually assumed to have spherical symmetry, whose mass is enormous, yet their size is relatively small.", "As an illustration, a neutron star's mass $M$ had been observed to be around $1.4-2$ solar mass, and its radius $R$ is roughly within the range of 10 to 20 km.", "These compact objects are mainly featured by their compactness $C=GM/R$ , a dimensionless quantity due to Newton's constant $G$ .", "Those objects can be categorized as follows [17]: compact objects, if $2C>1/3$ , ultra-compact objects, if $2C>2/3$ , objects violating Buchdhal limit, if $2C>8/9$ , clean-photon sphere objects, if $2C>1/(1+0.019)$ , and near-horizon quantum structures, if $2C>1/(1+10^{-40})$ .", "A black hole's compactness is $2C=1$ , so these exotic compact objects should have compactness less than the black hole's ($2C<1$ ).", "Suppose that any two massive objects had collided and formed another massive object in the so-called ring-down phase, this final object, in general, can be anything, including a black hole.", "Due to the gravitational field's intensity, these final objects will most likely be a black hole, but if it is not, it will be at least an ultra-compact object.", "By many analyses on a geodesic of light rays propagating near the object, perturbating its exterior gravitational field, people knew that the effective potential $V_{eff}$ from its gravity would affect its ring-down gravitational wave pulse [3].", "After two massive objects are combined into either a black hole or an ultra-compact object, its gravitational wave pulse will stop or produce echoes as the final object forms.", "This echo is the leaking of those gravitational waves that are trapped by the effective potential barrier located at radius $r=3GM$ ($V_{eff}^{\\prime }(3GM)=0$ and $V_{eff}^{\\prime \\prime }(3GM)<0$ ), which is the location of the so-called photon sphere.", "To have the photon sphere, then the object's radius should satisfy $R<3GM$ , which implies compactness $2C>2/3$ .", "It is worthy to note that the photon sphere's existence is due to the geodesic of light rays propagating near the massive object, but its existence does not imply gravitational wave echo, which came from the so-called TT-tensor part of metric perturbation.", "The potential wall location at $r=3GM$ is in coincidence with the photon sphere and is a byproduct of the spherical symmetry assumption, which is generally not true for rotating objects that obey axial symmetry.", "In the rest of this paper, we shall only assume spherical symmetry.", "The gravitational wave echo frequency $f_{echo}$ is in one-to-one correspondence on the object's compactness $C$ by [17] $\\tau _{echo}^{(approx)}\\sim 4M |\\log \\epsilon |,$ with $\\epsilon ={1/2C}-1$ and $f_{echo}=\\pi /\\tau _{echo}$ .", "This echo time delay can also be estimated by a calculation of its metric component $g_{tt}$ and $g_{rr}$ both inside and outside the star [18] $\\tau _{echo}^{(num)}=\\int _{0}^{3GM} \\sqrt{-{g_{rr}\\over g_{tt}}}~dr.$ The integration means that the trapped waves propagated from $r=3GM$ to $r=0$ and back to $r=3GM$ by assuming that the gravitational wave penetrates the ultra-compact object.", "(In fact, Eq.", "(REF ) came from Eq.", "(REF ) but integrated from $r=R$ to $r=3GM$ .)", "It had been calculated that the binary neutron stars merger GW170817 produced an approximation to a “tentative” echo frequency $f_{echo}\\simeq 72$ Hz from a 2.6-2.7 solar mass \"black hole\" remnant with dimensionless spin $0.84-0.87$  [19].", "This observation had been predicted to be compatible with a toy model of an incompressible star with mass $2-3$ solar mass [20].", "There are varieties of the proposed compact objects (see the compilation of the properties of those objects and the corresponding discussions in Ref. [17]).", "These proposed ultra-compact object studies are motivated by the hope that at least the echo from one of those objects will eventually be detected in the near future.", "Recently, Carballo-Rubio [1] proposed a modified TOV equation generated by semi-classical gravity theory.", "Solutions obeying boundary conditions are found.", "They arise from the pressure equation $p^{\\prime }$ with a positive sign and are identified with a nontrivial combination of the black stars and gravastars.", "This calculation is done by defining a suitably new constant $\\lambda >1$ that relates all pressure $p$ , mass $m$ , and energy density $\\rho $ , hence the EoS are unique.", "The form of pressure equation $p^{\\prime }$ with the negative sign has also been analyzed by Ho and Matsuo [2] to show, using constant energy density, that the Buchdahl limit can be violated without making the pressure goes to infinity.", "In this work, we investigate the form of pressure equation $p^{\\prime }$ with negative sign much further.", "This paper is organized as follows.", "In section , we briefly revisit the models (Refs.", "[1] and [2]).", "We discuss the numerical scheme in Appendix (page ), which is crucial to justify our results in the following section .", "In section  part A, we discuss the model in more detail with non-negative pressure and energy density assumption, and we use a simple linear EoS.", "We look at the effect from the semi-classical terms compared to the usual TOV equation.", "In section  part B, we discuss the numerical results.", "Finally, we summarize our work in Section ." ], [ "COMPACT STARS IN THE THEORY OF SEMI-CLASSICAL GRAVITY", "In ref.", "[1] a new type of TOV equations can obtained by using the renormalized energy-momentum tensor and solving the modified Einstein field equation (EFE) $G_{\\mu \\nu }=\\frac{8\\pi G}{c^4} T_{\\mu \\nu } + 96\\pi ^2 l_p^2 \\langle \\hat{T}_{\\mu \\nu } \\rangle ,$ where $l_p$ is the so-called renormalized Planck length.", "This quantity is related to the usual Planck length $L_\\text{Pl}=\\sqrt{\\hbar G/c^3}$ by $l_p = \\sqrt{N\\over 12\\pi }L_\\text{Pl},$ where $N\\gg 1$ is the number of particle fields.", "From now on, we shall use the natural units ($c=1$ ).", "The renormalized stress-energy tensor (RSET) describes the quantum vacuum polarization of $N\\gg 1$ matter fields.", "We shall symbolize the usual Planck length as $L_\\text{Pl}$ so that there is no ambiguity with the normalized one $l_p$ , which will be treated as an adjustable coupling constant.", "Similar to the derivation of the usual TOV equation, the space-time metric $g_{\\mu \\nu }$ is the static spherically symmetric metric $ds^2&=&ds^2_{(2)}+r^2(d\\theta ^2+\\sin ^2\\theta d\\varphi ^2),\\\\ds^2_{(2)}&=&-e^{\\nu (r)}dt^2+{dr^2\\over 1-2Gm(r)/r},$ which has a time symmetry denoted mathematically by a timelike Killing vector $\\xi =\\partial _t$ , and the energy-momentum tensor can be expressed by the usual perfect fluid $T^{\\mu \\nu }=(\\rho +p)u^\\mu u^\\nu +pg^{\\mu \\nu },$ where $u^\\mu $ is the 4-velocity of the fluid, which is timelike and normalized ($u^\\mu u_\\mu =-1$ ), and $g^{\\mu \\nu }$ is the inverse of the metric.", "The RSET is given by the so-called s-wave Polyakov approximation (see Ref.", "[21] page 216) $\\langle \\hat{T}_{\\mu \\nu } \\rangle = {\\delta ^a_\\mu \\delta ^b_\\nu \\over 4\\pi r^2} \\langle \\hat{T}_{ab}^{(2)}\\rangle ,$ where indices $a$ and $b$ denote 2-dimensional space-time coordinates $ds^2_{(2)}$ where $\\langle \\hat{T}_{ab}^{(2)}\\rangle =\\langle 0|\\hat{T}_{ab}^{(2)}|0\\rangle $ is evaluated.", "The Boulware state $|0\\rangle $ is associated with the Killing vector $\\xi $ from the metric, i.e., $\\xi ^\\mu =\\delta ^\\mu _t$ which implies $\\xi _\\mu =-e^{\\nu }\\delta _{\\mu t}$ .", "It is a usual practice to use null coordinate to obtain $\\langle \\hat{T}_{ab}^{(2)}\\rangle $ and then transform back to $(t,r)$ coordinate.", "It was shown in [22] that there is a shortcut equivalent to the usual practice, i.e., $\\langle \\hat{T}_{ab}^{(2)}\\rangle ={1\\over 48\\pi }\\left(R^{(2)}g_{ab}+A_{ab}-{1\\over 2}g_{ab}A,\\right)$ where $R^{(2)}$ is the Ricci scalar from the 2-dimensional metric and $A_{ab}$ is related to the Killing vector $\\xi $ (in this case $|\\xi |=e^{\\nu /2}$ ) by $A_{ab}={4\\over |\\xi |}\\nabla _a\\nabla _b|\\xi |.$ From Wald's axioms [9], the contracted Bianchi identity $\\nabla _\\mu \\langle \\hat{T}^{\\mu \\nu }\\rangle =0$ should be satisfied.", "This can be checked by substituting the components of the RSET the identity.", "Then from the contracted Bianchi identity $\\nabla _\\mu T^{\\mu \\nu }=0$ and the above modified EFE (REF ) we obtain three equations of motion $-p^{\\prime }-{(p+\\rho )\\over 2}{\\nu ^{\\prime }}&=&0,\\\\{\\nu ^{\\prime }\\over r}-{2Gm\\over r^3(1-2Gm/r)}&=&{8\\pi Gp\\over 1-2Gm/r}-{l_P^2\\over 4}\\left({\\nu ^{\\prime }\\over r}\\right)^2,\\\\{2Gm^{\\prime }\\over r^2}&=&8\\pi G \\rho + {l_P^2\\over r^2}\\left[\\left(1-{2Gm\\over r}\\right)\\left(\\nu ^{\\prime \\prime }+\\left(\\nu ^{\\prime }\\right)^2\\right)\\right.\\nonumber \\\\&&\\left.-\\left({Gm^{\\prime }\\over r}-{Gm\\over r^2}\\right){\\nu ^{\\prime }}-{3\\over 4}\\left(1-{2Gm\\over r}\\right)\\left(\\nu ^{\\prime }\\right)^2\\right].$ Here the prime denotes differentiation with respect to $r$ .", "There are two roots of the metric solution $\\nu ^{\\prime }$ from Eq.", "(): $\\nu ^{\\prime }=-{2r\\over l_P^2}\\left(1\\pm \\sqrt{1+{l_p^2\\over r^2}{2Gm\\over r}{(1+4\\pi r^3 p/m)\\over (1-2Gm/r)}}\\right).$ It leads to two different expressions for both of $p^{\\prime }$ and $m^{\\prime }$ .", "The negative and positive signs on the right-hand side will be named as negative and (positive) branches.", "In the limit of $l_p\\rightarrow 0$ , the negative branch approaches the standard GR-TOV equation while the positive branch does not.", "In the next section, we focus on this negative branch.", "Regarding the validity of our numerical calculation, we provide the detailed discussions in Appendix in page ." ], [ "Modified TOV Equations from Semi-classical Gravity", "Here we investigate the model in Ref.", "[1] with phenomenological point of view, i.e, we only consider non-negative $p$ .", "Since in general $(l_p/r)^2X$ is not constant, where $X={2Gm\\over r}{(1+4\\pi r^3 p/m)\\over (1-2Gm/r)},$ there was a hope for a more general EoS than (REF ), that is by relaxing the requirement $\\lambda =$ constant so that $\\lambda $ is a function of $r$ .", "By inspecting equation $p^{\\prime }={(p+\\rho )r\\over l_P^2}\\left(1+k\\sqrt{1+{l_p^2\\over r^2} X}\\right)$ where $k=\\pm 1$ , we discuss it in three cases: (1) ${l_p^2\\over r^2} X<1$ , (2) ${l_p^2\\over r^2} X>1$ and (3) ${l_p^2\\over r^2} X=1$ .", "Clearly $X>0$ , since $X\\le 0$ must be from $2Gm/r\\ge 1$ .", "From case (1), by binomial expansion with respect to ${l_p^2\\over r^2} X$ to only a leading order, we obtain $p^{\\prime }={\\left\\lbrace \\begin{array}{ll}{\\rho +p\\over 2}\\left({4r\\over l_p^2 X}+{X\\over r}\\right), & k=+1,\\\\-{\\rho +p\\over 2}{X\\over r}, & k=-1.\\end{array}\\right.", "}$ Since for any reasonable perfect fluid demand positive pressure and surface of the star demand zero pressure, we need $p^{\\prime }<0$ , which is not satisfied by $p^{\\prime }$ from $k=+1$ while $k=-1$ is just standard TOV equation.", "From case (2), again by binomial expansion with respect to ${r^2/(l_p^2 X)}$ , this leads us to $p^{\\prime }=k {(\\rho +p)\\sqrt{X}\\over l_p}.$ This expression eliminates $k=+1$ .", "From the case (3), this leads us to $p^{\\prime }={(p+\\rho )r\\over l_P^2}\\left(1+k\\sqrt{2}\\right),$ which again imply that we should choose $k=-1$ .", "Hence from now we shall only investigate $p^{\\prime }$ generated from $k=-1$ so that both $p_c>0$ and $p(R)=0$ are satisfied.", "Notice that Eq.", "() has $m^{\\prime }$ on both sides, the full expression of $m^{\\prime }$ is rather lengthy.", "Since we choose $k=-1$ , we use $p^{\\prime }={(p+\\rho )r\\over l_P^2}\\left(1-\\sqrt{1+{l_p^2\\over r^2} X}\\right),$ then we have $m^{\\prime }={4\\pi \\rho r^2} \\left({1+\\sum _{i=1}^7 A_i\\over 1+\\sum _{i=1}^4 B_i}\\right),$ where the nominator consists of $A_1&=&\\frac{3 l_p^2 p}{\\rho r^2 \\sqrt{1+{l_p^2\\over r^2} X}},\\\\A_2&=&-\\frac{3 l_p^2 m (1+4\\pi r^3 p/m)}{4\\pi \\rho r^5 \\sqrt{1+{l_p^2\\over r^2} X}},\\\\A_3&=&-\\frac{ G l_p^2 m^2 (1+4\\pi r^3 p/m)}{2\\pi \\rho r^6 (1-2Gm/r) \\sqrt{1+{l_p^2\\over r^2} X}},\\\\A_4&=&-\\frac{ m \\left(1-\\sqrt{1+{l_p^2\\over r^2} X}\\right)}{4\\pi \\rho r^3},\\\\A_5&=&-\\frac{ \\left(1-2Gm/r\\right) \\left(1-\\sqrt{1+{l_p^2\\over r^2} X}\\right)}{4\\pi G\\rho r^2},\\\\A_6&=&\\frac{\\left(1-2Gm/r\\right) \\left(1-\\sqrt{1+{l_p^2\\over r^2} X}\\right)^2}{8\\pi G\\rho l_p^2},\\\\A_7&=&\\frac{ l_p^2 }{ \\rho r \\sqrt{1+{l_p^2\\over r^2} X}}p^{\\prime },$ and the denominator consists of $B_1&=&\\frac{4 \\pi l_p^2 pr}{ m \\sqrt{1+{l_p^2\\over r^2} X}},\\\\B_2&=&-\\frac{ l_p^2 \\left(1+4\\pi r^3 p/m\\right)}{ r^2 \\sqrt{1+{l_p^2\\over r^2} X}},\\\\B_3&=&-\\frac{2 G l_p^2 m \\left(1+4\\pi r^3 p/m\\right)}{ r^3 \\left(1-2Gm/r\\right) \\sqrt{1+{l_p^2\\over r^2} X}},\\\\B_4&=&-{\\left(1-\\sqrt{1+{l_p^2\\over r^2} X}\\right)}.$ Since $m^{\\prime }$ is complicated, we need to be careful of fixing the constants and the initial data of pressure.", "Before solving those equations numerically, let us investigate both $p^{\\prime }$ and $m^{\\prime }$ at limit $l_p\\rightarrow 0$ .", "Expanding $p^{\\prime }$ with respect to $l_p$ , we have $p^{\\prime }=-\\frac{G \\left(m+4 \\pi r^3 p\\right) (\\rho +p)}{r (r-2 G m)}+\\frac{G^2 l_p^2 \\left(m+4 \\pi r^3 p\\right)^2 (\\rho +p)}{2 r^3 (r-2 G m)^2}+\\mathcal {O}(l_p^3).$ Let us expand these $A_i$ and $B_i$ w.r.t.", "$l_p$ : $A_1&=&\\frac{l_p^2 }{r^2 }\\frac{3 p}{\\rho }+\\mathcal {O}(l_p^3)>0,\\\\A_2&=&-\\frac{l_p^2 }{r^2}\\frac{3 m\\left(1+4 \\pi r^3 p/m\\right)}{4 \\pi r^3 \\rho }+\\mathcal {O}(l_p^3)<0,\\\\A_3&=&-\\frac{l_p^2}{r^2}\\frac{G m^2\\left(1+4 \\pi r^3 p/m\\right)}{2 \\pi r^4 \\rho (1-2 G m/r)}+\\mathcal {O}(l_p^3)<0,\\\\A_4&=&\\frac{l_p^2}{r^2}\\frac{G m^2 \\left(1+4 \\pi r^3 p/m\\right)}{4 \\pi r^4 \\rho ( 1-2 G m/r)}+\\mathcal {O}(l_p^3)>0,\\\\A_5&=&\\frac{l_p^2}{r^2}\\frac{ m\\left(1+4 \\pi r^3 p/m\\right)}{4 \\pi r^3 \\rho }+\\mathcal {O}(l_p^3)>0,\\\\A_6&=&\\frac{l_p^2}{r^2}\\frac{Gm^2 \\left( 1+4 \\pi r^3 p/m\\right)^2}{8 \\pi r^4 \\rho ( 1-2 G m/r)}+\\mathcal {O}(l_p^3)>0,\\\\A_7&=&-\\frac{l_p^2}{r^2}\\frac{Gm (1+p/\\rho ) \\left( 1+4 \\pi r^3 p/m\\right)}{r (1-2G m/r)}+\\mathcal {O}(l_p^3)<0,$ and $B_1&=&\\frac{l_p^2}{r^2}\\frac{4 \\pi r^3 p}{ m}+\\mathcal {O}(l_p^3)>0,\\\\B_2&=&-\\frac{l_p^2}{r^2} \\left(1+\\frac{4 \\pi r^3 p}{ m}\\right)+\\mathcal {O}(l_p^3)<0,\\\\B_3&=&-\\frac{l_p^2}{r^2} \\frac{2 G m\\left(1+4 \\pi r^3 p/m\\right)}{r(1-2 G m/r)}+\\mathcal {O}(l_p^3)<0,\\\\B_4&=&\\frac{l_p^2}{r^2} \\frac{Gm \\left( 1+4 \\pi r^3 p/m\\right)}{r (1-2 G m/r)}+\\mathcal {O}(l_p^3)>0,$ Since we know that $(Gm/r),(r^3p/m),(r^3\\rho /m)$ and $(p/\\rho )$ are dimensionless, notice that no singularity occur on each $A_i$ and $B_i$ as $r=r_c\\sim 0$ .", "Summing them up, $\\sum _{i=1}^7 A_i &=& \\frac{l_p^2 }{r^2 }\\left[ \\frac{3 p}{\\rho }-2\\frac{ m\\left(1+4 \\pi r^3 p/m\\right)}{4 \\pi r^3 \\rho }-\\frac{G m^2 \\left(1+4 \\pi r^3 p/m\\right)}{4 \\pi r^4 \\rho ( 1-2 G m/r)}\\right.\\nonumber \\\\&&\\left.+\\frac{Gm^2 \\left( 1+4 \\pi r^3 p/m\\right)^2}{8 \\pi r^4 \\rho ( 1-2 G m/r)}-\\frac{Gm (1+p/\\rho ) \\left( 1+4 \\pi r^3 p/m\\right)}{r (1-2G m/r)}\\right]+\\mathcal {O}(l_p^3),\\\\\\sum _{i=1}^4 B_i &=&\\frac{l_p^2 }{r^2 }\\left[-1-\\frac{Gm \\left(1+4 \\pi r^3 p/m\\right)}{r(1-2 G m/r)}\\right]+\\mathcal {O}(l_p^3).$ The value of $\\rho _0$ plays a significant role.", "On the surface of the object, $p=0$ , $\\rho =\\rho _0$ , $r=R$ and $m=M$ , so we have $p^{\\prime }(R)=-\\frac{G M \\rho _0}{R^2(1-2 G M /R)}+\\frac{G^2 l_p^2 M^2 \\rho _0}{2 R^5 (1-2 G M/R)^2}+\\mathcal {O}(l_p^3)$ and $\\sum _{i=1}^7 A_i &=& \\frac{l_p^2 }{R^2 }\\left[-2\\frac{ M}{4 \\pi R^3 \\rho _0}-\\frac{G M^2 }{4 \\pi R^4 \\rho _0 ( 1-2 G M/R)}\\right.\\nonumber \\\\&&\\left.+\\frac{G M^2 }{8 \\pi R^4 \\rho _0 ( 1-2 G M/R)}-\\frac{G M }{R (1-2G M/R)}\\right]+\\mathcal {O}(l_p^3),\\\\\\sum _{i=1}^4 B_i &=&\\frac{l_p^2 }{R^2 }\\left[-1-\\frac{G M }{R(1-2 G M/R)}\\right]+\\mathcal {O}(l_p^3),$ hence $m^{\\prime }(R)$ finite since $\\rho _0\\ne 0$ .", "It seems that if the value of $\\rho _0$ goes to zero, the changes of both $R$ and $M$ might be significant.", "Now in the center $r=r_c\\sim 0$ we have $\\sum _{i=1}^7 A_i &=& \\frac{l_p^2 }{r_c^2 }\\left[ \\frac{3 p_c}{\\rho _c}-2\\frac{\\left(1+3p_c/\\rho _c\\right)}{3}-\\frac{G m_c^2 \\left(1+3p_c/\\rho _c\\right)}{4 \\pi r_c^4 \\rho _c ( 1-2 G m_c/r_c)}\\right.\\nonumber \\\\&&\\left.+\\frac{G m_c^2 \\left( 1+3p_c/\\rho _c\\right)^2}{8 \\pi r_c^4 \\rho _c ( 1-2 G m_c/r_c)}-\\frac{G m_c (1+p_c/\\rho _c) \\left( 1+3p_c/\\rho _c\\right)}{r_c (1-2G m_c/r_c)}\\right]+\\mathcal {O}(l_p^3)\\nonumber \\\\&=&\\frac{l_p^2 }{r_c^2 }\\left[\\frac{p_c}{\\rho _c}-\\frac{2}{3}\\right]+\\mathcal {O}(r_c^2),\\\\\\sum _{i=1}^4 B_i &=&\\frac{l_p^2 }{r_c^2 }\\left[-1-\\frac{G m_c \\left(1+3p_c/\\rho _c\\right)}{r_c(1-2 G m_c/r_c)}\\right]+\\mathcal {O}(l_p^3)= -\\frac{l_p^2 }{r_c^2 }+\\mathcal {O}(r_c^2),$ where we have substitute the mass in this limit $m_c=(4/3)\\pi \\rho _cr_c^3$ .", "Then we have $p^{\\prime }(r_c)\\sim -\\frac{4}{3} \\pi G r_c (\\rho _c+p_c) (\\rho _c+3 p_c)+\\frac{8}{9} \\pi ^2 G^2 l_p^2 r_c (\\rho _c+p_c) (\\rho _c+3 p_c)^2$ and $m^{\\prime }(r_c)\\sim 4\\pi r_c^2\\rho _c\\left(\\frac{1+(l_p/r_c)^2(3p_c/\\rho _c-2/3)}{1-(l_p/r_c)^2}\\right).$ These equations are valid only when $r$ near $r_c$ and clearly $m^{\\prime }(r_c)$ is singular when $r_c=l_p$ .", "Suppose that we have at the center $r_c=\\alpha l_p$ where $\\alpha >0$ .", "Then this means that $\\alpha \\ne 1$ .", "Notice that Eq.", "(REF ) can be rewritten as $p^{\\prime }(r_c)\\simeq -\\frac{4}{3} \\pi G \\rho _c ^2 \\left(1+\\frac{p_c}{\\rho _c}\\right) \\left(1+3 \\frac{p_c}{\\rho _c}\\right)\\alpha l_p\\left[1-\\frac{2}{3} \\pi G \\rho _c \\left(1+3 \\frac{p_c}{\\rho _c}\\right) l_p^2\\right].$ The second term in the square bracket can be bigger than the first term if $\\alpha l_p^3$ is large enough than $\\alpha l_p$ .", "Hence, we conclude that $\\alpha l_p^3$ should be sufficiently small such that $p^{\\prime }(r)<0$ .", "Looking at Eq.", "(REF ), then the higher-order terms should have the pattern $\\mathcal {O}\\left(\\alpha l_p^{2n+1}\\right)$ , with $n$ an integer.", "This means that we should hold $l_p$ first then adjusting $\\alpha $ such that the second term in Eq.", "(REF ) does not dominate the first one, i.e., $l_p<\\sqrt{\\frac{3}{2\\pi G (\\rho _c+3p_c)}}\\text{, or equivalently, }N<\\frac{18}{L_\\text{Pl}^2 G (\\rho _c+3p_c)}.$ Hence the upper bound of $l_p$ depends on both $p_c$ and the EoS, which should satisfy the strong energy condition.", "By defining an arbitrary positive valued constant $\\alpha =r_c/l_p$ , Eq.", "(REF ) becomes $m^{\\prime }(r_c)\\sim 4\\pi r_c^2\\rho _c\\left[1-\\frac{3p_c/\\rho _c+1/3}{1-\\alpha ^2}\\right].$ Logically the mass should grow from center to the surface, so $m^{\\prime }(r)>0$ ($0<r\\le R$ ).", "Notice that if $\\alpha >1$ then $m^{\\prime }(r_c)>0$ is trivially satisfied.", "On the other hand, if $\\alpha <1$ then $m^{\\prime }(r_c)>0$ can happen if the second term inside the square bracket is less than unity.", "This implies $3p_c<(2/3-\\alpha ^2)\\rho _c.$ If $\\alpha =\\sqrt{2/3}$ then $p_c<0$ .", "This negative center pressure is contradicting with our previous assumption that $p>0$ and $\\rho >0$ inside the star.", "If $\\alpha >\\sqrt{2/3}$ then $\\rho _c<-\\left({3p_c\\over \\alpha ^2-2/3}\\right)<0.$ This expression is also a contradiction with our previous assumption.", "If $\\alpha <\\sqrt{2/3}$ then $\\rho _c>\\left({3p_c\\over 2/3-\\alpha ^2}\\right)>4.5p_c,$ which imply speed of sound squared $dp/d\\rho =w<2/9$ , lower than the upper bound from QCD and causality, where $w\\le 1/3$ and $w\\le 1$ , respectively [23].", "According to Ref.", "[3], the maximum compactness produced by a linear equation of state (EoS) $\\rho =p/w+\\rho _0$ is $\\left({2GM\\over R}\\right)_\\text{max}\\sim \\frac{8}{9 \\left(\\frac{0.51 w +0.77}{w (w +4.18)}+1\\right)},$ which is a monotonically increasing function of $w$ as $w$ grow but cannot go beyond the Buchdahl limit $2GM/R=8/9$ .", "So $w<2/9$ will produce less compact stars than $w=1/3$ .", "Also notice that this condition $m^{\\prime }(r)$ applies only at $r=r_c$ , so it may not be true for $r>r_c$ .", "The case of $\\alpha <1$ may still be investigated with negative valued $m_c$ , which is possible according to Ref [24].", "From Eq.", "() then we can see that singularity of $m^{\\prime }(r)$ may not be there anymore for $\\alpha =1$ .", "If we demand $\\sum _i B_i>0$ then $m_c<\\frac{\\rho _c r_c}{G \\rho _c-3 G p_c}\\equiv m_{c,max} $ and since $m_c<0$ we obtain another restriction for the EoS $w>1/3.$ Notice that $m_{c,max}$ can be made close to zero so we can use limit $m_c\\rightarrow 0^-$ .", "Eq.", "(REF ) in the limit of $m_c\\rightarrow 0^-$ becomes $p^{\\prime }(r_c)<0$ so $p$ will decrease from $p_c$ .", "In this limit, Eq.", "(REF ) becomes $\\sum _i A_i\\sim \\alpha ^{-2}\\left[{p_c\\over \\rho _c}-{2\\over 3}\\right].$ Since we demand $\\sum _i B_i>0$ and $m^{\\prime }(r)>0$ , then $w>2/3.$ This is from considering $m_c\\rightarrow 0^-$ .", "If on the other hand we consider $m_c\\rightarrow -\\infty $ , notice that Eq.", "(REF ) set the minimum value of $m_c$ , i.e., $0>m_c>-4\\pi r_c^3 p_c\\equiv m_{c,min}.", "$ Notice that $m_{c,min}$ can be very close to zero so the range of $m_c$ is quite small.", "Thus for the case of $m_c<0$ , we obtain that both $w$ and $m_c$ are restricted to $w>2/3$ and $m_{c,min}<m_c<m_{c,max}$ .", "Due to the tight restriction of $m_c<0$ , we do not focus on $m_c<0$ case in the following sections but rather on the usual positive valued $m_c=4\\pi r_c^3\\rho _c/3$ .", "Figure: Here we have profiles of pp and mm in the upper panels from l p =1l_p=1 mm and α=20\\alpha =20 and we vary the speed of sound squared ww.", "The metric function exp(ν)\\exp (\\nu ) and 1-2Gm/r1-2Gm/r are also shown for each ww in the lower panels.Figure: Here we show the M-R curve from varying ww.", "The semi-classical correction does not produce significant discrepancy compared to TOV from GR.Figure: Here we show the M-R curve from varying BB.", "Decreasing BB produces higher mass and larger radius.Figure: Here we show the M-R curve from varying α\\alpha .", "The choice of α\\alpha does not show any difference in the result.Figure: Here we show the M-R curve from varying l p l_p.", "Increasing l p l_p also does not significantly shift the M-R curve." ], [ "Numerical Results", "The EoS used here have the following linear form $\\rho (p)=p/w+\\rho _0,$ where $w$ is the speed of sound squared from thermodynamics and $\\rho _0$ is a positive valued constant.", "The constant $\\rho _0$ cannot be zero since it implies no solution and cannot be negative, or it will violate weak energy condition [3].", "The constant $w$ is restricted to at least two conditions [23]: (1) from causality, we have $0<w\\le 1$ and (2) from QCD and other known theories, we have $0<w\\le 1/3$ .", "Here we focus on compact stars with largest compactness possible, so we shall choose $1/3\\le w\\le 1$ and $\\alpha >1$ .", "To integrate both $p^{\\prime }$ and $m^{\\prime }$ numerically we need to fix $l_p, \\alpha , \\rho _0$ , and $p_c$ carefully such that all boundary conditions are satisfied.", "Here we shall use natural units for those four constants following Ref. [25].", "In this set of units called “NS” units, $r$ is in metres, both $p$ and $\\rho $ are in MeV/fm$^3$ and $m$ is in MeV m$^3$ /fm$^3$ .", "The compactness is defined as $C=GM/R$ .", "We use $\\rho _0=4B$ , where $B$ is the so-called bag constant from the MIT bag model [18].", "The upper bound of $l_p$ can now be determined.", "It is usual to have both $p_c$ and $\\rho _0$ at most $\\sim 10^{3}$ MeV/fm$^3$ , then by Eq.", "(REF ), we have $l_p\\lesssim 10$ km.", "Since the observed neutron stars have $R\\sim 10$ km, we have $l_p<R$ , which is trivial if we consider $l_p$ to be related to the Planck length.", "This finding is equivalent to fixing the upper bound for the dimensionless parameter $N$ , which is $N<10^{79}$ .", "This number is a huge quantity since the RSET should satisfy (in SI units using $\\hbar c=3.162 \\times 10^{-27}$ kg m$^3$ s$^{-2}$ ) $|\\langle T_{\\mu \\nu } \\rangle | < \\frac{|T_{\\mu \\nu }|}{3.162 \\times 10^{52} \\text{ kg m$^3$ s$^{-2}$}}.$ Since the smallness of $l_p$ often make calculators cannot detect the second term in the square root in Eq.", "(REF ), this form tends to make $p^{\\prime }=0$ in some $r$ .", "To evade this, we use an equivalent form of Eq.", "(REF ), i.e., $p^{\\prime }=-\\frac{(p+\\rho )X}{r\\left(1+\\sqrt{1+(l_p/r)^2 X}\\right)}.$ The following numerical results show that the contribution of RSET does not significantly affect the maximum mass.", "An example for profiles from varying $w$ are shown in Fig.", "REF .", "We see that the contribution of $w$ does make higher mass and larger radius resulting in an ultra-compact star, but we will not discuss its echo property since this compactness can also be obtained from the standard TOV equation in GR.", "Varying $B$ , $\\alpha $ and $l_p$ lead to similar profiles.", "The M-R curves from varying $w$ , $B$ , $\\alpha $ and $l_p$ are shown in Figs.", "REF , REF , REF and REF , respectively.", "Here the semi-classical correction does not produce significant discrepancy from arbitrary choice of $p_c$ compared to TOV from GR.", "We also had tried using $\\alpha <1$ , but we cannot find the numerical solution.", "The mass always goes down to a negative value, even though it increases a little in the beginning.", "It is interesting to note that for the similar case considered in Ref.", "[2] their integration method breaks down at $r=l_p$ .", "They use constant energy density, and they integrate the equations from the surface to the core." ], [ "Conclusions", "In this work, we analyze the semi-classical theory of gravity proposed in Ref. [1].", "Note that the theory has two different sets of equations characterized by the sign in $p^{\\prime }$ .", "This happened since the Einstein equation for metric solution $\\exp (\\nu )$ in (REF ) has two roots and $p^{\\prime }$ is related to $\\nu ^{\\prime }$ by Eq.", "(REF ).", "From the positive branch, defining a constant $\\lambda $ is done to obtain analytic solutions.", "By definition of $\\lambda $ , $p$ is related to energy density $\\rho $ and mass $m$ .", "The solution's character $p$ is as follows: starting from the negative value of $p$ at $r\\rightarrow 0$ and increase the $p$ to zero.", "This solution resembles a combination of the known gravastars or black star models.", "We show that the numerical solution can be obtained by integrating from its surface to its core.", "From the negative branch, which goes to the usual TOV equation at the formal limit $l_p\\rightarrow 0$ , we had reproduced the results from [2] for the case of constant energy density.", "In this paper, we analyze this negative branch further by applying linear EoS $\\rho =p/w+\\rho _0$ with $w$ =1/3 and $w$ =1, respectively.", "The range of $l_p$ is dependent on the choice of $r_c$ .", "The reason is as follows.", "Since the equation $m^{\\prime }(r)$ is much more complicated than usual TOV equation (see Eqs.", "(REF )-()), it has terms that can make $m^{\\prime }(r)<0$ , which may imply negative $m$ since at the center $m_c\\sim 0$ .", "These terms are dependent on the values of both $l_p$ and $r_c$ such that $m^{\\prime }(r_c)$ is singular when $l_p=r_c$ .", "To investigate $m^{\\prime }(r)$ around this singularity, we fix a relation $r_c=\\alpha l_p$ with $\\alpha \\ne 1$ .", "We also demand $m^{\\prime }(r)>0$ for $0<r<R$ since it is usual practice to expect the mass increase from the center to surface.", "We found that, with some approximation steps, $0<\\alpha <\\sqrt{2/3}$ demands a very restricted set of EoS with a speed of sound squared $w<2/9$ .", "This value implies the compactness lowered, so we are not discussing this aspect.", "For $\\sqrt{2/3}\\le \\alpha <1$ , this demands either $p_c<0$ or $\\rho _c<0$ which contradict our starting assumption that both pressure and energy density should not be negative.", "For the case of $\\alpha >1$ , we can use any EoS.", "Thus our analysis agrees with Ref.", "[2], i.e., their integration breaks down at $r=l_p$ .", "The value of $l_p$ has an upper bound-constrained by the pressure at the center $p_c$ .", "From our approximation, we found from the dimensional analysis that $l_p<10$ km, which is of order $R$ .", "This upper bound implies $N<10^{79}$ .", "This fact implies that $l_p$ can be much larger than Planck length.", "Through varying all four parameters $w$ , $B$ , $\\alpha $ , and $l_p$ , our numerical results show that the M-R curves are indistinguishable compared to the TOV equation results in GR.", "Hence the parameter $l_p$ , though not in the limit $l_p\\rightarrow L_\\text{Pl}$ , has no significant signature compared to the standard TOV equation.", "Compactness reaching the ultra-compact range can be achieved by adjusting both $w$ and $B$ in the TOV equation system with and without semi-classical correction.", "Moreover, the authors in Ref.", "[2] showed that the Buchdahl limit could be violated using constant energy density, which means setting $w$ equal to infinity.", "Hence, it seems that we need to reach somehow much higher compactness larger than the Buchdahl limit to see the significance of the semi-classical correction.", "This way might be done either by adjusting decreasing $B$ to near zero or increasing $w$ such that $w>1$ , which violates causality.", "Hence we conclude this section with the following points.", "(1) The structure of the Eqs.", "(REF )-() restricts the range of $\\alpha $ and $w$ in some ways that turn out to give results not significant if compared to the usual TOV equation system.", "Therefore, similar to the conclusion in Ref.", "[3] with GR theory that the ultra-compact objects from the negative branch of semi-classical gravity with a linear equation of state are unable to generate demanding gravitational echoes.", "(2) Our numerical results are consistent with both Ref.", "[3] and Ref. [2].", "See the detailed discussion in Appendix.", "(3) The positive branch is still open for more detailed numerical analysis.", "We expect that the quantum effect within semi-classical approximation is only significant for exotic compact objects, considered the solution related to the positive branch.", "This work is funded by Publikasi Terindeks Internasional (PUTI) Doktor 2020, No.", "NKB-614/UN2.RST/HKP.05.00/2020.", "*" ], [ "Discussions Regarding Numerical Method", "In this appendix, we discuss the integration schemes and their results.", "As a disclaimer, in this paper, all the calculations are done using Mathematica 10.0.", "The schemes are named as forward (backward, resp.)", "integration, corresponding to integrating from center to surface (surface to center).", "It is known that for the standard TOV equation, we can do both forward and backward integration.", "Here we discuss both forward, and backward integration to obtain numerical solutions for (1) negative branch obtained previously by Ho and Matsuo [2] in the case of constant energy density $\\rho $ and (2) the negative branch obtained previously by Carballo-Rubio [1] using constant $\\lambda $ trick.", "We emphasize these forward and backward integration schemes because we use forward integration in the next section, but both [1] and [2] imply backward integration.", "We show that for the negative branch, we get consistent results from both backward and forward integration.", "This evidence justifies our numerical method for the negative branch system that we discussed in the section III onwards." ], [ "Forward and backward integration solutions from negative branch", "Suppose we redefine the $rr$ component of the metric by $F(r)&=&\\sqrt{C(r)\\left(1-{2Gm(r)\\over r}\\right)},\\\\C(r)&=&e^{\\nu (r)},$ and the constants by $\\alpha =l_p^2$ and $\\kappa =8\\pi G$ .", "Then the equations () and () are equivalent to equations (5.7) and (5.8) in [2], $0&=&-\\frac{1}{8 r^2 C(r)^2}\\left[-3 \\alpha F(r)^2 C^{\\prime }(r)^2+4 r C(r)^2 F(r) F^{\\prime }(r)\\right.\\nonumber \\\\&&\\left.+2 C(r) F(r) \\left(\\alpha C^{\\prime }(r) F^{\\prime }(r)+F(r) \\left(\\alpha C^{\\prime \\prime }(r)-2 r C^{\\prime }(r)\\right)\\right)\\right.\\nonumber \\\\&&\\left.+2 \\kappa r^2 C(r)^3 (p+\\rho )\\right],\\\\0&=&\\frac{1}{4 r^2 C(r)^2}\\left[\\alpha F(r)^2 C^{\\prime }(r)^2-\\alpha C(r) F(r) \\left(F(r) C^{\\prime \\prime }(r)+C^{\\prime }(r) F^{\\prime }(r)\\right)\\right.\\nonumber \\\\&&\\left.-2 C(r)^2 F(r) \\left(r F^{\\prime }(r)+F(r)\\right)+C(r)^3 \\left(2-\\kappa r^2 (\\rho -p)\\right)\\right]$ To avoid confusion, we keep our symbol for energy density and metric function as $\\rho $ and $\\nu $ , respectively, while in [2] the authors use $m$ and $\\rho $ .", "In this paper, we assign $m$ as the mass function.", "Setting the energy density as constant $\\rho =\\rho _0$ , we can see from (REF ) that the pressure becomes $p(r)=-\\rho _0+p_0 e^{-\\nu (r)/2}.$ Here $p_0$ is a constant that should satisfy the boundary condition at the surface $p(R)=0$ , i.e., $p_0=\\rho _0 \\sqrt{1-2GM/R}$ with $M=m(R)$ .", "Only in this subsection, the integrations are done for the tortoise coordinate $r_*$ which is defined as $r_*=\\int {dr\\over F(r)}.$ This then give us the equations for $r(r_*)$ and $\\nu (r_*)$ as $0&=&-\\frac{1}{8 r(r_*)^2 }\\left[2\\left\\lbrace \\alpha \\nu ^{\\prime \\prime }(r_*)-2r(r_*)\\nu ^{\\prime }(r_*) r^{\\prime }(r_*)\\right\\rbrace \\right.\\nonumber \\\\&&\\left.+4r(r_*)r^{\\prime \\prime }(r_*)+2\\kappa r(r_*)^2 p_0 e^{\\nu (r_*)/2}-\\alpha [\\nu ^{\\prime }(r_*)]^2\\right],\\\\0&=&\\frac{1}{4 r(r_*)^2}\\left[-\\alpha \\nu ^{\\prime \\prime }(r_*)-2(r(r_*) r^{\\prime \\prime }(r_*)+[r^{\\prime }(r_*)]^2)\\right.\\nonumber \\\\&&\\left.+ e^{\\nu (r_*)} [2-\\kappa r(r_*)^2(2\\rho _0-p_0 e^{-\\nu (r_*)/2})]\\right].$ From substracting (REF ) with () we can see there are two roots of $\\eta ^{\\prime }(r_*)$ , corresponding to the negative and positive branch we had discussed before.", "Since in this subsection we focus on the negative branch, which goes to TOV equation in the limit $\\alpha \\rightarrow 0$ $\\nu ^{\\prime }(r_*)=-\\frac{2}{\\alpha }\\left(r(r_*) r^{\\prime }(r_*)-\\sqrt{r(r_*)^2 \\left(\\alpha \\kappa p_0 e^{\\frac{\\nu (r_*)}{2}}+r^{\\prime }(r_*)^2-\\alpha \\kappa \\rho _0 e^{\\nu (r_*)}\\right)+\\alpha \\left(e^{\\nu (r_*)}-r^{\\prime }(r_*)^2\\right)}\\right).", "$ This corresponds to the second equation $r^{\\prime \\prime }(r_*)&=& \\frac{1}{4 \\left(\\alpha -r(r_*)^2\\right) r^{\\prime }(r_*)}\\left[\\kappa r(r_*)^2 e^{\\frac{\\nu (r_*)}{2}} \\left(2 \\rho _0 e^{\\frac{\\nu (r_*)}{2}}-p_0\\right)\\right.\\nonumber \\\\&&\\left.\\times \\left(2 \\sqrt{r(r_*)^2 \\left(\\alpha \\kappa p_0 e^{\\frac{\\nu (r_*)}{2}}+r^{\\prime }(r_*)^2-\\alpha \\kappa \\rho _0 e^{\\nu (r_*)}\\right)+\\alpha \\left(e^{\\nu (r_*)}-r^{\\prime }(r_*)^2\\right)}-\\alpha \\nu ^{\\prime }(r_*)\\right)\\right.\\nonumber \\\\&&\\left.-\\left(2 \\sqrt{r(r_*)^2 \\left(\\alpha \\kappa p_0 e^{\\frac{\\nu (r_*)}{2}}+r^{\\prime }(r_*)^2-\\alpha \\kappa \\rho _0 e^{\\nu (r_*)}\\right)+\\alpha \\left(e^{\\nu (r_*)}-r^{\\prime }(r_*)^2\\right)}-\\alpha \\nu ^{\\prime }(r_*)\\right)\\right.\\nonumber \\\\&&\\left.\\times 2 e^{\\nu (r_*)} +4 r(r_*) r^{\\prime }(r_*) \\left(\\alpha \\kappa p_0 e^{\\frac{\\nu (r_*)}{2}}+r^{\\prime }(r_*)^2-\\alpha \\kappa \\rho _0 e^{\\nu (r_*)}\\right)\\right] $ So we have a first order equation for $\\nu (r_*)$ and second order equation for $r(r_*)$ .", "These depends on three parameters $M$ , $R_*=r_*(R)$ , $\\rho _0$ and two coupling constants $\\alpha $ and $\\kappa $ .", "The initial conditions at $r_*=R_*$ are $R_*&=&R+2GM \\ln (R/2GM -1),\\\\\\nu (R_*)&=&\\ln (\\sqrt{1-2GM/R}),\\\\r(R_*)&=&R,\\\\r^{\\prime }(R_*)&=&1-2GM/R.$ Then we do the backward integration.", "To fix the three input parameters $R, M,$ and $\\rho _0$ , we follow recipe in [2]: fix $M$ first, then fix $R$ such that $R>2GM$ , then explore the value of $\\rho _0$ .", "To every value of $\\rho _0$ , we have different shooting parameter values $k$ related to $R$ by $R=2GM/k$ , so it is restricted to $0<k<1$ .", "It is also carefully chosen so that at the center the slope of $p$ is not steep ($|p^{\\prime }(r_*\\rightarrow -\\infty )|<\\infty $ ).", "This result has two reasons.", "First, assigning $p^{\\prime }(r_*\\rightarrow -\\infty )\\sim 0$ is done so that when we integrate the equation in the opposite direction (forward), we can get the same curves.", "Second, the negative branch goes to the TOV equation in the small $\\alpha $ limit and it is known that, in the TOV system, $p^{\\prime }(r=0)\\sim 0$ so the negative branch should also has this property for sufficiently small $\\alpha $ .", "We assume that $r_*$ is linear to $r$ in the region near $r=0$ so that $p^{\\prime }(r_*\\rightarrow -\\infty )\\sim 0$ imply $|p^{\\prime }(r\\rightarrow 0)|<\\infty $ .", "Figure: The backward and forward integration on the negative branch gives same numerical curves.", "Here we integrate them with respect to tortoise coordinate r * r_*.", "Here kk is the shooting parameter to determine RR by R=2GM/kR=2GM/k so that p ' (r * =0)∼0p^{\\prime }(r_*=0)\\sim 0.", "Here e (ν) e^{(\\nu )} goes really close to zero as r * →-∞r_*\\rightarrow -\\infty but never reach zero.The results from doing backward integration and calculating again using forward integration are shown in Fig.", "REF .", "We obtain that the forward integration produces the same curves as the backward one.", "The pressure, in the region $r_*\\rightarrow -\\infty $ , is indeed does not go to Planck scale for the macroscopic size of $R$ ." ], [ "Forward and backward integration solutions from positive branch", "The author in Ref.", "[1] had shown that the positive branch have exact solutions whose properties are very similar to a mixture of gravastars and black stars, i.e., the negative pressure in the interior $p={-1+\\mathcal {O}(l_p^2/r^2)\\over 8\\pi G r^2 R^2}\\left[R^2-r^2e^{\\frac{(\\lambda +1) (r^2-R^2)}{l_p^2}}\\right],$ with $\\lambda >1$ a constant, but its energy density is still positive valued $\\rho (r)={1+\\mathcal {O}(l_p^2/r^2)\\over 8\\pi G r^2 R^2}\\left[R^2+r^2e^{\\frac{(\\lambda +1) (r^2-R^2)}{l_p^2}}\\right].", "$ Notice that $p(0)<0$ and $\\rho (0)>0$ .", "Moreover, both satisfy weak, dominant, and null energy conditions.", "The strong energy conditions are assumed unnecessary since the existence of Casimir energy in experiments violate it.", "The key point of the derivation of Eqs.", "(REF ) and (REF ) is the introduction of a constant $\\lambda >1$ that is defined as $\\lambda \\equiv \\sqrt{1+{l_p^2\\over r^2}{2Gm\\over r}{(1+4\\pi r^3 p/m)\\over (1-2Gm/r)}},$ where it implies a fixed form of EoS and mass profile $\\rho &=& -p+\\frac{l_p^2}{(\\lambda +1) r}p^{\\prime },\\\\m&=&\\frac{r^3 \\left(-8 \\pi G l_p^2 p+\\lambda ^2-1\\right)}{2 G \\left(l_p^2+\\left(\\lambda ^2-1\\right) r^2\\right)}.$ Since $p=0$ at $r=R$ , we have the compactness determined by $R, l_p$ and $\\lambda $ as $2C={2GM\\over R}=\\frac{1}{ \\left(l_p^2/\\left[R^2\\left(\\lambda ^2-1\\right)\\right]+1\\right)},$ with $M=m(R)$ .", "The resulting exact solution for metric is $\\nu (r)=\\nu (R)+{\\frac{(\\lambda +1) (R^2-r^2)}{l_p^2}},\\\\$ with $\\nu (R)=\\ln \\left(1-2GM/R+\\mathcal {O}(l_p^2/R^2)\\right)$ .", "The solutions can lead to arbitrary compactness depending on the value of $\\lambda $ and are stable by curvature and boundary conditions arguments.", "After some algebra, one can arrive at the following equation of motion $p^{\\prime }(r)&=&g(r)+p(r) h(r),\\\\g(r)&=&\\frac{(\\lambda +1)^2 \\left(l_p^4+(\\lambda -2) l_p^2 r^2-\\left(\\lambda ^2-1\\right) r^4\\right)}{4 \\pi G l_p^2 r \\left(l_p^2-(\\lambda +1) r^2\\right) \\left(l_p^2+\\left(\\lambda ^2-1\\right) r^2\\right)},\\\\h(r)&=&\\frac{2 (\\lambda +1) r \\left((2 \\lambda +1) l_p^4+\\left(\\lambda ^3+\\lambda ^2-2 \\lambda -2\\right) l_p^2 r^2-(\\lambda -1) (\\lambda +1)^2 r^4\\right)}{l_p^2 \\left(l_p^2-(\\lambda +1) r^2\\right) \\left(l_p^2+\\left(\\lambda ^2-1\\right) r^2\\right)}.$ We integrate by backward integration starting from $r=R$ and $p(R)=0$ inward.", "Since this is a first order equation for $p$ , we can see that $R$ is already tied with $M$ so there is no shooting parameter $k$ like in the previous subsection, where $R$ can be set at an arbitrary value as long as $R>2GM$ and $p^{\\prime }(r=0)\\sim 0$ .", "From $g(r)$ , we can see that the equation has singularity at $r=0$ so we expect uncontrolled behavior of $p$ near $r=0$ .", "Figure: Profiles of p(r)p(r) from the positive branch from backward integration (top panel), forward integration (middle panel), and the comparison between each numerical results (bottom panel).Here we calculate first using backward integration, then the values from backward integration is used for forward integration.", "In the most bottom panel, we show the real and imaginary part of log(p/p c )\\log (p/p_c) since for the forward integration, p/p c =0p/p_c=0 at r=r f ∼5r=r_f\\sim 5 m and after this point p/p c <0p/p_c<0 when r>r f r>r_f and thus log(p/p c )∈ℂ\\log (p/p_c)\\in \\mathbb {C}.", "Hence both scheme does not give similar result.We show the backward integration solutions in top panel of Fig.", "REF and it is clear that it violates $|p^{\\prime }(r\\rightarrow 0)|<\\infty $ .", "We tried to use the smallest possible $r=r_c$ as the initial position and its corresponding $p(r_c)$ for starting the forward integration.", "The resulting curve in the middle panel, while the comparison is in the bottom panel.", "The results from backward and forward integration are different.", "The real and imaginary words in the most bottom panel is the real and imaginary part of $\\log (p/p_c)$ from both backward and forward integration.", "We show them here since the pressure $p/p_c$ from forward integration suddenly goes to zero at $r=r_f$ much earlier than $R$ from the backward integration, where after this point $p/p_c<0$ and thus $\\log (p/p_c)\\in \\mathbb {C}$ for forward integration.", "In Fig.", "REF , $r_f\\sim 5$ m while $R=10$ km.", "We still do not yet know how to remedy this, but the numerical result from backward integration perfectly fits the analytic solution (REF )." ], [ "In this subsubsection, we intend to check the validity of our numerical calculation by comparing the typical result of section 6.1.3, density for regular geometry in Ref.", "[2] with the same EOS (constant energy density EOS) input but calculating by using our code where we restricted the calculation only for the interior of the star.Note that we perform a calculation by using backward and forward integrations to check the consistency of the numerical calculation.", "It means that the geometry is regular for 0 $\\le $ r $\\le $ R, and pressure is finite and positive inside the star.", "Note that we do not investigate the too small density and too large density cases of the Ho-Matsuo model wherein both latter cases $r_*$ do not always monotonically increase when $r$ increased.", "We had reproduced relatively similar plots with Fig.", "14 and Fig.", "15 of Ref.", "[2], and the trend of the plot satisfies the condition discussed in section 6.1.3 of Ref. [2].", "The results are shown in Fig.", "REF .", "However, we unable to reproduce exactly the quantitative result as shown in Fig.", "14 and Fig.", "15 of Ref.", "[2] since the authors of Ref.", "[2] did not specify the units they are using.", "Therefore, we only guess the initial conditions until we obtain similar behavior of both $r(r_*)$ and $\\nu (r_*)$ .", "We choose $R$ and $\\rho _0$ suitably (with $2GM$ fixed) such that the profile $p(r_*)$ decrease monotonically as $r_*$ increases.", "It turns out that this choice gives us $r$ that is increasing monotonically as $r_*$ increases as the one shows in Ref.", "[2], so in principle, we can interchange $r$ and $r_*$ .", "If we replot all functions in Fig.", "REF as functions of $r$ instead of $r_*$ , we obtain Fig.", "REF .", "In Fig.", "REF , we show the numerical calculation result by integrating the equations (REF )-(REF ) from center to surface.", "We compare it with the usual TOV equation in GR.", "We can notice that $p$ becomes very steep as $r\\rightarrow 0$ since $r$ (as a function of $r_*$ ) drop really quick in that area.", "We also notice that $m_c$ (mass near the center) is negative valued and quite large in magnitude.", "We suspect that this negative $m_c$ may be why the compactness can be very close to the black hole limit ($2GM/R=1$ ).", "We suspect that this $m_c<0$ is the reason for such high compactness.", "However, this value is indeed outside the range from our lower and upper bound ($m_{c,min}$ and $m_{c,max}$ from Eq.", "(REF ) and Eq.", "(REF )) which is $-10^{-3} M_\\odot \\lesssim m_c \\lesssim -10^{-7} M_\\odot $ .", "Therefore, it seems that our analytical upper and lower bound estimation of $m_c<0$ may not be to justify for constant energy density EOS.", "Figure: The solid line in these plots are the same as in Fig.", "but the dashed line is from TOV GR system.", "Both data are integrated from center to surface and notice the large compactness 2GM/R2GM/R at the right bottom panel.", "Notice that pp increases dramatically as r→0r\\rightarrow 0.", "Since in SCGrav r>l p r>l_p, then pp will never go to infinity.", "But in TOV GR, rr can be arbitrarily close to zero so pp will go to infinity as r→0r\\rightarrow 0.", "Notice also that the mass at the center (m c m_c) is negative valued.", "This may be the reason why the compactness can be very close to black hole limit (2GM/R=12GM/R=1) for the case of constant energy density EOS.To obtain such high compactness, we also observe that the system should have very large density $\\rho _0$ and central pressure $p_c$ .", "In our units, we estimate that $\\rho _0 \\sim 10^{12}$ MeV fm$^{-3}$ .", "This is about nine orders of magnitude larger than the bag constants $B$ that we used.", "Notice also from Fig.", "REF that $p_c \\sim 10^{16}$ MeV fm$^{-3}$ .", "Now we input $\\rho _0$ and $p_c$ around this estimation, but with $m_c=0$ , into our equations (REF )-() and integrate them from $r=\\alpha l_p$ to $r=R$ .", "The result is shown in Fig.", "REF .", "Note here that $r_*$ is also monotonically increasing with respect to $r$ .", "We also have the compactness slightly above the Buchdahl limit (BL) $2GM/R=8/9$ .", "Figure: Here we use linear EOS with w=∞w=\\infty , B∼10 12 B \\sim 10^{12} MeV fm -3 ^{-3}, and p c ∼10 16 p_c \\sim 10^{16} MeV fm -3 ^{-3} while m c =0m_c=0.", "Unlike Fig.", ", the compactness is still far from black hole limit, although is larger than Buchdahl limit (BL).To verify our expectation that a very high uniform density and central pressure can give us a star with compactness beyond BL, we calculate similar profiles as Fig.", "REF with variations of $w$ , $B$ , and $p_c$ .", "We show the results in Table REF .", "In the top three rows, we vary $w$ .", "In the middle three rows, we vary $p_c$ .", "Moreover, in the top three rows, we vary $B$ .", "We can see that increasing $w$ will increase the compactness.", "However, to obtain compactness over the Buchdahl limit, we need to increase $B$ to at least eight orders of magnitude and the pressure to at least thirteen orders of magnitude.", "The entries with bold fonts are from Fig.", "REF .", "We also compare them with the result from TOV GR and find the compactness quite close to SCGrav.", "Of course, the results from TOV GR that violate the Buchdahl limit will have $p(r\\rightarrow 0)\\rightarrow \\infty $ .", "Of course, the results from TOV GR that violate the Buchdahl limit will have $p(r\\rightarrow 0)\\rightarrow \\infty $ , which is not correct for the semiclassical model since it should satisfy $r>l_p>0$ condition.", "We verify this expectation by showing Table REF .", "Table: Here is the comparison of compactness with variations of ww, BB, and p c p_c, with m c =0m_c=0 in our semiclassical model (SCGrav) using constant density EOS." ] ]
2105.11691
[ [ "Environmental Kuznets Curve & Effectiveness of International Policies:\n Evidence from Cross Country Carbon Emission Analysis" ], [ "Abstract In this article, we are presenting the relationship between environmental pollution and the income level of the selected twenty-four countries.", "We implemented a data-based research analysis where, for each country, we analyzed the related data for fifty-six years, from 1960 to 2016, to assess the relationship between the carbon emission and income level.", "After performing the related data analysis for each country, we concluded whether the results for that country were in line with the Environmental Kuznets Curve (EKC) hypothesis.", "The EKC hypothesis suggests that the carbon emission per capita starts a declining trend when the country-specific high level of income is reached.", "The results of our data analyses show that the EKC hypothesis is valid for high-income countries and the declining trends of carbon emission are clearly observed when the income level reaches a specific high enough level.", "On the other hand, for the non-high income countries, our analysis results show that it is too early to make an assessment at this growth stage of their economies because they have not reached their related high-enough income per capita levels yet.", "Furthermore, we performed two more additional analyses on high-income countries.", "First, we analyzed the related starting years of their carbon emission declining trends.", "The big variance in the starting years of the carbon emission declining trends shows that the international policies are clearly ineffective in initiating the declining trend in carbon emission.", "In addition, for the high-income countries, we explained the differences in their carbon emission per capita levels in 2014 with their SGI indices and their dependence on high-carbon emission energy production." ], [ "[display] 18pt 1em 1em 0pt-50pt0.3cm 4cm2.5cm2.5cm2.5cm figurechapter tablechapter footnotechapter equationchapter 0pt Environmental Kuznets Curve & Effectiveness of International Policies: Evidence from Cross Country Carbon Emission Analysis Elvan Ece Satıcı & Bayram Cakir May 2021 In this article, we are presenting the relationship between the environmental pollution and the income level of the selected twenty-four countries.", "We implemented a data based research analysis where, for each country, we analyzed the related data for fifty-six years, from 1960 to 2016, to assess the relationship between the carbon emission and the income level.", "After performing the related data analysis for each country, we concluded whether the results for that country were in line with the Environmental Kuznets Curve (“EKC”) hypothesis.", "The EKC hypothesis suggests that the carbon emission per capita starts a declining trend when the country specific high-level of income is reached.", "The results of our data analyses show that the EKC hypothesis is valid for high-income countries and the declining trends of the carbon emission are clearly observed when the income level reaches a specific high enough level.", "On the other hand, for the non-high income countries our analysis results show that it is too early to make an assessment at this growth stage of their economies because they have not reached their related high-enough income per capita levels yet.", "Furthermore, we performed two more additional analysis on high-income countries.", "First, we analyzed the related starting years of their carbon emission declining trends.", "The big variance in the starting years of the carbon emission declining trends show that the international policies are clearly ineffective in initiating the declining trend in carbon emission.", "In addition, for the high-income countries, we explained the differences in their carbon emission per capita levels in 2014 with their SGI indices and their dependence on high-carbon emission energy production.", "CHAPTER 1: INTRODUCTION 1         CHAPTER 2: LITERATURE REVIEW 5         CHAPTER 3: DATA 8         CHAPTER 4: D. ANALYSIS OF THE SELECTED COUNTRIES 10         CHAPTER 5: THE START OF THE CARBON EMISSION DECLINE 10         CHAPTER 6: HIGH-INCOME COUNTRIES' CARBON EMISSION LEVELS 10         CHAPTER 7: CONCLUSION 25         REFERENCES 27         In this article, we are presenting the relationship between the environmental pollution and the income level of the selected twenty-four countries.", "In assessing the relationship between the carbon emission and the income level, we implemented a data based research analysis where, for each country, we analyzed the related data for fifty-six years, from 1960 to 2016.", "In section B, we are explaining the Environmental Kuznets Curve hypothesis and we are providing a brief literature review on the topic.", "In section C, we are defining the data used in this research as the base of our analyses.", "In section D, we analyze each one of the selected twenty-four countries in order to understand the relationships between their income levels and their carbon emission levels.", "We classify these countries under two categories as “high-income countries” and “non-high income countries” based on the World Bank’s data on “GDP per capita”.", "The selected countries with GDP per capita levels of more than 20,000 USD are categorized as the “high-income” countries whereas the other countries with GDP per capita below 12,000 USD are referred to as the “non-high income” countries.", "For each one of the selected twenty-four countries, income per capita and carbon emission per capita data for fifty-six years, ranging from 1960-2016, is used in the analysis to present the environmental pollution trend with respect to the income level.", "At the end of the data analysis for each country, we conclude whether the carbon emission trend in the related country was in line with the Environmental Kuznets Curve (“EKC”) hypothesis.", "In section E, we concentrate on the high-income countries and analyze the starting year of the carbon emission per capita declining trend with respect to the related GDP per capita.", "If the international policies were effective in starting the carbon emission per capita declining trend in these high-income countries, we would expect to observe a convergence around the specific years when these policies were put in effect.", "In addition to analyzing the effectiveness of the international policies, we also analyze the effect of the national efforts in the initiation of the carbon emission per capita decline trend by using the SGI scores of the related countries.", "Briefly, SGI score states the country’s overall diligence on the environment with respect to the effectiveness of the policies and regulations enforced by their governments.", "In section F, we present further information on high-income countries and focus on their levels of carbon emission per capita in 2014.", "We explain the reasons behind the differences in carbon emission per capita levels in 2014 with the countries’ related SGI indices and their dependence on high-carbon emission energy production.", "Due to the steady increase in the air pollution in the past decades, researches have started to focus on understanding the dynamics between the income levels of the countries and their environmental degradation levels.", "In fact, the concept of “Environmental Kuznets curve” (“EKC”) was mentioned first in 1991 by Grossman and Krueger in their study of North American Free Trade Agreement (NAFTA) (Hervieux and Mahieu 2014).", "Grossman and Krueger stated in their paper titled “Environmental Impacts of a North American Free Trade Agreement” that “We find for two pollutants (sulfur dioxide and \"smoke\") that concentrations increase with per capita GDP at low levels of national income, but decrease with GDP growth at higher levels of income” (1991).", "In 1992, the World Bank published its report titled “World Development Report 1992: Development and the Environment” which argued that economic growth would enable improved environmental conditions if policies and programs were put in place: The main message of this year's report is the need to integrate environmental considerations into development policymaking.", "The report argues that continued, and even accelerated, economic and human development is sustainable and can be consistent with improving environmental conditions, but that this will require major policy, program, and institutional shifts.", "A twofold strategy is required.", "First, the positive links between efficient income growth and the environment need to be aggressively exploited.", "Second, strong policies and institutions need to be put in place which cause decision makers to adopt less damaging forms of behavior.", "(World Bank 1992) Grossman and Krueger named the Environmental Kuznets Curve (“EKC”) after its resemblance to Kuznet’s s (1955) inverted U-shaped relationship between income inequality and development (Dasgupta et al.", "2002).", "In their article titled “Confronting the Environmental Kuznets Curve”, Dasgupta, Laplante, Wang and Wheeler explained the EKC hypothesis as follows: In the first stage of industrialization, pollution in the environmental Kuznets curve world grows rapidly because people are more interested in jobs and income than clean air and water, communities are too poor to pay for abatement, and environmental regulation is correspondingly weak.", "The balance shifts as income rises.", "Leading industrial sectors become cleaner, people value the environment more highly, and regulatory institutions become more effective.", "Along the curve, pollution levels off in the middle-income range and then falls toward pre-industrial levels in wealthy societies.", "(Dasgupta et al.", "2002) Following the publication of this World Bank report in 1992, many researchers have conducted numerous researches on this subject globally, using different techniques and various pollutants as variables.", "In this article, Gross Domestic Product (“GDP”) per capita (constant 2010 USD) data is used as a reference to indicate the income level of each of the selected countries.", "GDP per capita (constant 2010 USD) data for the selected countries is obtained from the World Bank database.", "GDP per capita (constant 2010 USD) is defined as the gross domestic product of the related country divided by its midyear population.", "The GDP is calculated as the sum of gross value added by all resident producers in the economy plus any product taxes and minus any subsidies not included in the value of the products, without making any deductions for depreciation of fabricated assets or for depletion and degradation of natural resources (“GDP per capita (Constant 2010 USD)” 2021).", "GDP per capita data used in this article is “in constant 2010 U.S. dollars” to ensure comparability among the years in the related period, which is from 1960 to 2016.", "In this article, the countries with GDP per capita above 20,000 USD in 2016 are considered as high-income countries whereas the countries with GDP per capita below 12,000 USD in 2016 are referred as “non-high income” countries.", "The threshold income levels used in this article for the categorization of the high-income and non-high income countries are aligned with the World Bank definitions, stating that high-income countries are the ones with Gross National Income per capita of 12,275 USD or more in 2010 (Beer and Prydz 2019).", "Analogous to the data for GDP per capita, the carbon dioxide emissions data for the related period is also obtained from the World Bank database.", "The World Bank defines carbon emission per capita as the carbon dioxide emissions from the “burning of fossil fuels and the manufacture of cement,” where the “carbon dioxide produced during consumption of solid, liquid and gas fuels and gas flaring” is also included (“CO2 Emissions (Metric Tons per Capita)” 2021).", "The data is available from 1960 to 2016 for most of the countries except for Germany where the data obtained starts from 1991, and France and Italy for which the data is only available until 2014.", "The Environmental Report of the Sustainable Governance Indicators measures the effectiveness of the policies and regulations enforced by the governments.", "The qualitative measures addressed by this report are the ambition levels of the environmental targets the governments seek to fulfill, the impact of the implementation, and the integration of the policies in relevant sectors (Stiftung 2020).", "In this report, 41 OECD and the EU countries are ranked according to their environmental performance on a scale from 1 to 10 (Stiftung 2020).", "Higher index scores indicate better environmental performance, based on the criteria of the report.", "For example, 1 is for the countries in which “Environmental concerns have been largely abandoned,” and 10 is given to countries whose “Environmental policy goals are ambitious and effectively implemented as well as monitored within and across most relevant policy sectors that account for the largest share of resource use and emissions” (Stiftung 2020).", "In this article, SGI scores are utilized in two sections.", "First, it is used in section E where we analyze the starting periods of the related decreasing trends of the CO2 emissions per capita.", "Furthermore, it is also used in section F where we compare and explain the carbon emission per capita levels of the countries in 2014.", "In this section, we analyze the carbon emission per capita emission trends of twenty-four selected countries to assess whether the Environmental Kuznets Curve (“EKC”) theory is applicable.", "In order to assess the validity of the EKC in these countries, we plot the carbon emission per capita level of each country together with its income level, measured by GDP per capita, for the time-period between 1960-2016.", "Out of these twenty-four countries, fifteen of them are high-income countries and the other nine countries are referred to as non-high income countries in this article.", "The related GDP per capita of the selected high-income countries are all over 20,000 USD.", "The fifteen selected high-income countries that are analyzed in section (i) below in this article are Switzerland, Denmark, Australia, USA, Canada, Austria, Japan, Finland, Germany, Belgium, France, UK, Italy, Spain and Portugal, respectively.", "On the other hand, the related GDP per capita of the nine selected non-high income countries are all below 12,000 USD.", "The nine selected non high-income countries that are analyzed in section (ii) below in this article are Brazil, Malaysia, South Africa, Colombia, China, Peru, Egypt, India and Pakistan, respectively, i.", "High income countries : The graphs showing the carbon emission per capita vs GDP per capita trends for the fifteen high-income countries are presented in Figures 1-15.", "The time-period for these graphics are mostly from 1990 to 2016 unless the information is not available for a specific country for this period in the World Bank’s database.", "The information is unavailable only for a few of these selected countries, either at the beginning or at the end of the time-period.", "The graphs for the high-income countries are ordered by the GDP per capita level of each country in 2016 in descending order.", "When all the graphs for the selected high-income countries are analyzed, clearly, in all of these high income countries carbon emission per capita is increasing together with the increase in GDP per capita until a specific high level of GDP per capita is reached in the related country.", "When the specific high level of GDP per capita is reached, then, the carbon emission per capita levels off and starts to decline even though the GDP per capita continues to increase, forming an inverted U-shaped curve as defined by the EKC theory.", "For each one of the selected fifteen high-income countries, we present a brief analysis below: 1) Switzerland: In 2016, Switzerland had the highest GDP per capita (77,026 USD) among the selected countries whereas its carbon emission per capita emission level was rather low at 4.12 metric tons per capita.", "When we analyze Switzerland’s carbon emission per capita vs GDP per capita graph (Figure 1), we see that carbon emission per capita reached its peak as early as in 1973 at 7.33 metric tons per capita when the GDP per capita was high at 53,777 USD.", "Carbon emission per capita started to decline after this peak point at 7.33 metric tons per capita in 1973 until it dropped down to 4.12 metric tons per capita in 2016.", "The inverted U-shape curve for Switzerland in Figure 1 validates the EKC theory as its carbon emission per capita is at an increasing trend as GDP per capita was increasing and then it turned into a declining trend after reaching its peak point when the GDP per capita reached 53,777 USD in 1973.", "2) Denmark: Denmark had the second highest GDP per capita after Switzerland among the selected countries in 2016 at 61,878 USD.", "Even though Denmark’s carbon emission per capita emission level was higher than Switzerland’s in 2016, it was still rather low when compared to the emissions of the other selected countries at 5.55 metric tons per capita.", "In Denmark, carbon emission per capita emission reached its peak in 1996 at 13.71 metric tons per capita when the GDP per capita was high at 50,262 USD.", "The trend of carbon emission per capita started to decline after this peak point and it reached 5.55 metric tons per capita in 2016.", "In Figure 2, Denmark’s carbon emission per capita graph clearly has an inverted U- shape that validates the EKC theory.", "3) Australia: Australia had the third highest GDP per capita after Switzerland and Denmark among the selected countries in 2016 at 55,729 USD.", "However, carbon emission per capita level of Australia in 2016 (15.54 metric tons per capita) is substantially higher when compared to the levels of Switzerland (4.12 metric tons per capita) and Denmark (5.55 metric tons per capita).", "Australia’s EKC curve presented an increasing trend of carbon emission per capita as the GDP per capita increased from 1960 to 2010.", "In 2010, carbon emission per capita reached its peak point at 17.74 metric tons per capita where the GDP per capita level was 52,022 USD.", "From 2011 onwards, even though the GDP per capita of the country continued to increase, carbon emission per capita started its declining trend where it fell to 15.54 metric tons per capita level in 2016 when the GDP per capita was at 55,729 USD.", "It is important to note that, the trend of Australia’s carbon emission per capita trend is in line with those of the Switzerland and Denmark in the context that carbon emission per capita starts to drop when the country reaches its country specific high GDP per capita level, which is over 50,000 USD.", "However, it is also important to note that overall carbon emission per capita level of Australia (15.54 metric tons per capita) is relatively much higher than Switzerland’s (4.12 metric tons per capita) and Denmark’s (5.55 metric tons per capita).", "Despite the significant difference in the overall carbon emission per capita levels with these countries, Australia still went through a similar pattern of carbon emission per capita with Switzerland and Denmark where the CO2 per capita emission started to decline when GDP per capita reached over 50,000 USD.", "Australia’s carbon emission per capita emission curve in Figure 3 clearly presents the typical characteristics of the EKC theory with an inverted U-shaped curve.", "4) USA: USA had the fourth highest GDP per capita among the selected countries in 2016 at 52,556 USD.", "However, USA’s carbon emission per capita level in 2016 is very high (15.50 metric tons per capita), similar to the level of Australia.", "USA’s carbon emission per capita vs GDP per capita graph is presented in Figure 4.", "Carbon emission per capita reached its peak point at 22.5 metric tons per capita in 1973 when GDP per capita was at 25,794 USD.", "However, the shape of the curve was volatile afterwards and carbon emission per capita started its consistent decreasing trend only after 2001 when GDP per capita reached its specific high enough level at 44,729 USD.", "In 2000, carbon emission per capita was at 20.2 metric tons per capita before it started its declining trend in 2001.", "However, despite the declining curve trend, carbon emission per capita of the USA was still relatively high when compared with other countries even at its lowest point (15.50 metric tons per capita) in 2016 when GDP per capita was at 52,556 USD.", "As shown in Figure 4, USA’s graph also has the typical characteristics of the EKC.", "The shape of the curve is an inverted U-shape and it validates the EKC theory.", "5) Canada: Canada had the fifth highest GDP per capita among the selected countries in 2016 at 50,193 USD and, furthermore, its carbon emission per capita was rather high like Australia and USA at 15.09 metric tons per capita.", "The carbon emission per capita trend for Canada (Figure 5) has two main similarities with the USA’s graph.", "First similarity is the long time lag between the peak level of carbon emission per capita and the start of the consistent decreasing trend afterwards.", "Carbon emission per capita reached its peak point in 1980 at 18.08 metric tons per capita when GDP per capita was at 29,357 USD.", "However, the consistent decreasing trend started only after twenty-seven years in 2007 when the carbon emission per capita level was at 17.39 metric tons per capita and the GDP per capita was at 48,534 USD.", "The second similarity with the USA’s carbon emission trend is their relatively high carbon emission per capita levels in 2016.", "In fact, Canada’s lowest carbon emission per capita was in 2016 at 15.09 metric tons per capita level, similar to the level of USA, which was also high at 15.50 metric tons per capita in the same year.", "The shape of Canada’s carbon emission per capita trend is slightly different from the other high-income countries.", "Although Canada’s carbon emission per capita curve shape is also an inverted U-shape similar to the other high-income countries’ curves, its inverted U-shape is slightly different from the rest.", "This difference in shape is a result of Canada’s dependence on petroleum, natural gas, and hydroelectricity for its economic development.", "Canada’s economy is relatively more energy intensive when compared to the other high- income countries.", "It ranks fourth among the top energy producers of petroleum and total liquids in the world, behind only the United States, Saudi Arabia, and Russia and, furthermore, it exports a significant amount of the produced energy to the United States (US Energy Information Administration 2020).", "As shown in Figure 5, Canada’s carbon emission per capita graph also has an inverted U-shape and it validates the EKC theory.", "Since Canada’s economy largely depends on energy production, we see a slightly different inverted U-shaped curve in its graph.", "6) Austria: Austria has one of the highest GDP per capita levels within the selected European countries in this article.", "In 2016, the GDP per capita of Austria was at 48,260 USD and the carbon emission per capita was at 7.03 metric tons per capita.", "In Austria, carbon emission per capita increased together with the increase in GDP per capita until it reached its peak year in 2005 at 8.97 metric tons per capita when the GDP per capita was 44,638 USD.", "The carbon emission per capita started to decline after this peak point.", "In 2016, carbon emission per capita level dropped to 7.03 metric tons per capita when the GDP per capita level was at 48,260 USD.", "In 2016, Austria’s carbon emission per capita level was much less than Australia’s, USA’s and Canada’s but higher than Switzerland’s and Denmark’s.", "As shown in Figure 6, Austria’s graph also has the typical characteristics of the EKC.", "The shape of the curve is an inverted U-shape and it validates the EKC theory.", "7) Japan: Japan has the highest GDP per capita level within the selected Asian countries and it ranks as the seventh highest in income level within all the selected countries in this article.", "In 2014, the GDP per capita of Japan was at 47,403 USD and the carbon emission per capita was at 8.94 metric tons per capita.", "At first, Japan’s carbon emission per capita also increased with the increase in its GDP per capita, as suggested by the EKC hypothesis.", "However, when it reached its peak levels, it stayed at these levels for a very long time (forty years) as shown in Figure 7.", "Japan’s carbon emission per capita level reached 8.47 metric tons per capita in 1973 and stayed at these high levels for forty years until it reached its peak point in 2004 at 9.9 metric tons per capita when GDP per capita was at 43,672 USD.", "After reaching this peak point in 2004, carbon emission per capita started to decrease in line with the relative decrease in GDP per capita.", "In 2013, GDP per capita reached a higher level when compared with the past 10 years at 46,249 USD and, accordingly, carbon emission per capita increased to 9.76 metric tons per capita.", "In 2014, however, a consistent declining trend of carbon emission per capita started even though GDP per capita (46,484 USD) was increasing.", "Carbon emission per capita dropped to 8.9 metric tons per capita level where GDP per capita was at 47,403 USD in 2016.", "Japan’s carbon emission per capita curve (Figure 7) shape also supports the EKC theory because the carbon emission per capita increased together with the increase in GDP per capita until the country specific high level of GDP per capita was reached in 2013 and, then, the carbon emission per capita started to decline from 2014 onwards.", "However, Japan’s EKC trend is slightly different from the majority of the high-income countries’ curves because of the long time lag between the year when the peak level of carbon emission per capita was reached (2004) and the year when the decreasing trend of carbon emission per capita started (2014).", "One of the reasons for this long time lag between the peak year and the year of the start of the declining trend is the GDP per capita trend within this period.", "The GDP per capita also dropped and went up again within this period, affecting the inverted U-shape of the carbon emission per capita curve, accordingly.", "In fact, there is a similarity between Japan’s and Canada’s carbon emission per capita curves.", "In both of these countries, carbon emission per capita levels stayed at high levels for a long time before starting to decline.", "The carbon emission level per capita level stayed high in Canada for twenty-seven years whereas this period lasted forty years in Japan.", "There are two main reasons for Japan’s very long, sticky period at high levels of carbon emission per capita.", "The first reason is Japan’s high energy usage.", "According to 2019 statistics of the US Energy Information Administration, Japan is a major energy importer in the world as the fifth-largest oil consumer.", "Furthermore, it is the largest liquefied natural gas (LNG) importer, the fourth-largest crude oil importer and the third-largest coal importer behind China and India.", "Japan historically used to have higher shares of energy production in nuclear energy until the Fukushima nuclear energy accident in 2011.", "After the nuclear accident, energy fuel mix has shifted to natural gas, oil and renewable energy.", "Although the share of oil in energy production declined from about 80% in the 1970’s to 40% in 2019 as a result of declining and aging population, high energy efficiency measures, and an expanding fleet of hybrid and electric vehicles, it is still the major source of energy production.", "Coal still has 26% share in total energy consumption whereas the increasing share of natural gas has reached 21% of total primary consumption.", "Before the 2011 earthquake, Japan was the third-largest nuclear power producer after USA and France.", "Almost 13% of Japan’s total energy production was produced from nuclear energy in 2010 whereas this share decreased sharply to 3% in 2010 after the earthquake.", "The share of nuclear energy will be increasing again in the near future as per the government’s plan to reduce energy imports.", "Renewable energy is about 10% of Japan’s energy consumption and its share is expected to increase (US Energy Information Administration 2020).", "The second reason for Japan’s long period of high CO2 per capita emission level is the “lost decade” which started with the economic stagnation caused by the burst of the asset price bubble in early 1990’s (Callen and Ostry 2013).", "This economic stagnation slowed down the GDP per capita growth of Japan when compared with other countries and resulted in lengthening the period where the CO2 per capita emission stayed high.", "Despite this long period of high carbon emission per capita, the shape of the CO2 per capita curve of Japan still validates the EKC theory.", "8) Finland: Finland is also one of the selected high-income European countries in this article.", "In 2016, the GDP per capita of Finland was at 46,750 USD and the carbon emission per capita was at 8.35 metric tons per capita.", "Finland’s carbon emission per capita curve trend is shown in Figure 8.", "In Finland’s curve, at the beginning of the related time-period carbon emission per capita increased together with the GDP per capita, as it is also stated in the EKC theory.", "In 1980, carbon emission per capita reached its high level of 12.19 metric tons per capita level when GDP per capita was at 25,512 USD.", "Carbon emission per capita stayed around this level until it reached its peak level at 13.17 metric tons per capita when GDP per capita was at 42,707 USD in 2003.", "In line with the EKC theory, the decreasing trend started after this peak point was reached in 2003.", "In 2016, carbon emission per capita level dropped to 8.35 metric tons per capita level when GDP per capita was at 46,750 USD.", "Finland’s carbon emission per capita emission curve shape presented in Figure 8 is an inverted U-shape and it clearly validates the EKC theory.", "9) Germany: In 2016, the GDP per capita of Germany was at 45,960 USD and the carbon emission per capita was at 8.84 metric tons per capita.", "In Figure 9, Germany’s carbon emission per capita vs GDP per capita graph is presented.", "The carbon emission curve is graphed between the years of 1991 to 2016 because Germany’s carbon emission per capita information is available only after 1991 due to its political history.", "In 1945, after the Second World War, the country was divided into East Germany and West Germany and German reunification did not take place until 1990.", "As a result, the carbon emission per capita data is available only from 1991 onwards after the reunification.", "In 1991, the reunified Germany’s GDP per capita was at 33,836 USD and its CO2 per capita was at 11.62 metric tons per capita.", "Germany’s carbon emission per capita curve presented a declining trend since 1991 since its GDP per capita was already high enough in 1991 for the start of the declining trend.", "Germany’s CO2 per capita dropped to 8.84 metric tons per capita in 2016 from 11.62 in 1991.", "Germany’s carbon emission curve also supports the EKC theory, which states that when a high-enough GDP per capita is reached, a declining trend of CO2 per capita is observed even though GDP per capita continues to increase.", "10) Belgium: In 2016, the GDP per capita of Belgium was at 45,943 USD and the carbon emission per capita was at 8.55 metric tons per capita.", "In Belgium’s carbon emission per capita graph (Figure 10), carbon emission per capita had increased from 1960 to 1973.", "In 1973, carbon emission per capita reached its peak level at 14.26 metric tons per capita when the GDP per capita of Belgium was at 22,868 USD.", "In 1979, when GDP per capita reached 25,946 USD, carbon emission per capita was very close to its peak level again at 14.24 metric tons per capita and, after this year, from 1980 onwards, carbon emission per capita followed a declining trend.", "In 2016, carbon emission per capita dropped to 8.55 metric tons per capita whereas GDP per capita increased to 45,943 USD.", "Belgium’s CO2 per capita vs GDP per capita graph clearly has the characteristics stated in the EKC theory where carbon emission per capita had increased together with the GDP per capita until the peak level was reached and, then, a declining trend of CO2 per capita was observed despite the increase in GDP per capita.", "11) France: In 2014, the GDP per capita of France was at 41,481 USD and the carbon emission per capita was at 4.57 metric tons per capita.", "Since the carbon emission per capita data in the World Bank’s database for France and Italy is only up to 2014, France’s graph in Figure 11 presents the carbon emission per capita curve from 1960 to the last available data in 2014.", "In 1960, the carbon emission per capita level in France was at 5.82 metric tons per capita whereas its GDP per capita was at 12,744 USD.", "However, in 1973, carbon emission per capita reached its peak level at 9.71 metric tons per capita when the GDP per capita was at 22,843 USD.", "The declining trend started after 1979 when the carbon emission per capita reached its peak level at 9.6 metric tons per capita and the GDP per capita was at 26,578 USD.", "After a sharp declining trend, in 2014 the CO2 per capita dropped to 4.57 metric tons per capita despite the increase in GDP per capita to 41,481 USD.", "Similar to the graphs of the other European countries that are analyzed above in this article, France’s carbon emission per capita vs GDP per capita graph in Figure 11 also presents an inverted U-shaped curve and supports the hypothesis of the EKC theory.", "12) United Kingdom (“UK”): In 2016, the GDP per capita of the United Kingdom was at 42,500 USD and the carbon emission per capita was at 5.78 metric tons per capita.", "The carbon emission per capita vs GDP per capita curve for the United Kingdom is presented in Figure 12.", "Carbon emission per capita in the United Kingdom slightly increased from the 11.15 metric tons per capita level in 1960 (when the GDP per capita was at 13,934 USD) to 11.82 metric tons per capita at its peak level in 1971 as the GDP per capita increased to 18,474 USD.", "However, starting from 1974 onwards, carbon emission per capita in the United Kingdom started to decline and, in 2016, it dropped to 5.78 metric tons per capita, approximately half of its peak level in 1971.", "In summary, UK’s carbon emission per capita curve also clearly supports the hypothesis of the EKC theory which states that carbon emission per capita starts to decline after reaching its peak level at a certain high GDP per capita level.", "13) Italy: Similar to France, the carbon emission per capita data in the World Bank’s database for Italy is available also only up to 2014.", "In 2014, the GDP per capita of Italy was at 33,667 USD and the carbon emission per capita was at 5.27 metric tons per capita.", "In Italy, carbon emission per capita was at 2.18 metric tons per capita in 1960 and the GDP per capita was at 10,879 USD.", "Carbon emission per capita reached its peak point (8.22 metric tons per capita) when GDP per capita more than tripled from its level in 1960 to 37,227 USD in 2004.", "As shown in Figure 13, from 2005 onwards, carbon emission per capita started to decline and in 2014 it dropped down to the 5.27 metric tons per capita level.", "Italy’s carbon emission per capita vs GDP per capita graph also clearly presents an inverted U-shaped curve as shown in Figure 13.", "As a result, the trend in Italy also supports the hypothesis of the EKC theory.", "14) Spain: In 2016, the GDP per capita of Spain was at 31,449 USD and the carbon emission per capita was at 5.25 metric tons per capita.", "In 1960, the carbon emission per capita in Spain was low at 1.61 metric tons per capita and GDP per capita was at 7,376 USD.", "In the coming years, the carbon emission per capita increased together with the increase in the GDP per capita until it reached its peak level in 2005.", "In 2005, GDP per capita was at 31,029 USD and carbon emission per capita was at its peak at 8.10 metric tons per capita.", "From 2006 onwards, carbon emission per capita trend showed a consistent declining trend and it dropped down to 5.25 metric tons per capita level in 2016.", "As shown in Figure 14, Spain’s carbon emission per capita vs GDP per capita graph also presents an inverted U-shaped curve, similar to the other high-income countries’ graphs that are analyzed above in this article.", "Therefore, Spain’s carbon emission per capita curve also has the related characteristics of a typical EKC and supports the hypothesis of this theory.", "15) Portugal: In 2016, the GDP per capita of Spain was at 22,534 USD and the carbon emission per capita was at 5.25 metric tons per capita.", "In 1960, Portugal’s GDP per capita was at 4,501 USD and carbon emission per capita was at 0.93 metric tons per capita.", "Portugal’s carbon emission per capita level increased together with the increase in its GDP per capita until it reached its peak point in 1999 at 6.31 metric tons per capita, approximately 6.8 times its level in 1960.", "When the carbon emission per capita level was at its peak in 1960, the GDP per capita was at 20,853 USD level, which was approximately 4.6 times its level in 1960.", "Carbon emission per capita stayed around this peak level for 6 years until 2005 and it started to decline only after 2006 onwards.", "In 2016, Portugal’s GDP per capita increased to 22,534 USD whereas its carbon emission per capita dropped to 4.72 metric tons per capita.", "As shown in Figure 15, Portugal’s carbon emission per capita trend also presents the characteristic inverted U-shaped curve of the EKC hypothesis, similar to all the other high-income countries that were analyzed in this article.", "16) Conclusion for the analysis on high-income countries: As the analyses of the long-term carbon emission per capita trend for all the selected fifteen countries consistently showed, the declining trend of the carbon emission curve starts only after the related, country specific high GDP per capita level is reached in each of these countries.", "Furthermore, we observe that the specific high level of GDP per capita year differs from country to country.", "Therefore, we conclude that the environmental protection policies issued by the international agencies are ineffective in initiating a decrease in these high-income countries’ carbon emission levels.", "If the policies of the international institutions were successful in decreasing the carbon emission levels, then, the carbon emission per capita levels in these countries would have started to drop around the same year, in coherence with the timing of the effectiveness of these policies.", "However, when we analyze the carbon emission per capita emission trends of these selected fifteen high-income countries, it is clear that there is not a correlation between the decreasing trend of the carbon emission and the timing of these international institutions policies.", "The CO2 per capita vs GDP per capita graphs clearly show that the carbon emission starts to decline in different years in these selected high-income countries.", "The results of the analyses of these fifteen selected high-income countries show clearly that the major determinant for the initiation of the declining carbon emission trend is reaching the related specific high-income level per capita relevant for each country.", "All the graphs of these selected high-income countries are in accordance with the inverted U-shaped curve shape hypothesis of the EKC theory.", "Also, the results of the analyses in this section also indicates that, unfortunately, international policies are ineffective in ensuring a decrease in the carbon emission levels of these high-income countries.", "Figure: NO_CAPTIONFigure: NO_CAPTIONii.", "Non - high income countries : Carbon emission per capita vs GDP per capita graph of the nine non-high income countries are analyzed in this section.", "The nine non-high income countries are: Brazil, Malaysia, South Africa, Colombia, Peru, China, Egypt, India and Pakistan.", "The carbon emission curves and the GDP per capita trends of these countries are presented in Figures 16-24.", "In general, the GDP per capita of these selected non-high income countries are relatively very low when compared to the high-income countries.", "In fact, the average GDP per capita for these nine countries in 2016 was only 7,076 USD.", "As a result, most of these countries still have an increasing trend of carbon emission because their economies are still growing and their GDP per capita is still increasing, accordingly.", "The EKC theory states that carbon emission per capita will start to decline only when the GDP per capita reaches a certain country-specific high level.", "Since the GDP per capita levels of these non-high income countries are still in the increasing trend stage, carbon emission per capita levels are also still at the increase stage and the peak points are not reached yet The graphs of these nine non-high income countries are analyzed below: 1) Brazil: In 2016, the GDP per capita of Brazil was at 10,966 USD and the carbon emission per capita was at 2.24 metric tons per capita.", "Although Brazil has the highest GDP per capita among these selected nine low-income countries, carbon emission per capita is still in the steady increase trend since the level of GDP per capita has not reached its country specific high level yet.", "For example, the GDP per capita for Portugal, the country with the lowest GDP per capita among the high-income countries, is 22,534 USD in 2016, which is approximately the double of Brazil’s GDP per capita.", "The carbon emission level for Brazil increases from its 0.65 metric tons per capita level in 1960 (when GDP per capita was 3,417 USD), to 2.24 metric tons per capita level in 2016 when GDP per capita was at 10,966 USD.", "Brazil’s CO2 per capita vs GDP per capita graph is presented in Figure 16.", "The carbon emission per capita is still in the increasing phase, in line with its increasing GDP per capita trend.", "Therefore, for Brazil, it is too early at this stage to state whether its carbon emission curve will be in line with the hypothesis of the EKC theory.", "However, the increasing trend phase of the curve is in line with the first part of the EKC theory, which states that the carbon emission per capita is expected to increase until it reaches a country-specific high-income per capita level.", "2) Malaysia: In 2016, the GDP per capita of Brazil was at 11.244 USD and the CO2 per capita emission was at 8.09 metric tons per capita.", "Malaysia’s carbon emission per capita curve in Figure 17 presents a strong increasing trend for CO2 per capita as its GDP per capita is increasing.", "It would be fair to say that, it is too early at this GDP per capita level to state whether Malaysia’a carbon emission per capita trend will support the hypothesis of the EKC theory because the GDP per capita is still at a relatively low level (11,244 USD) in 2016.", "However, as it is in the case for Brazil, the increasing trend phase of the curve is in line with the first part of the EKC theory, which states that the carbon emission per capita is expected to increase until it reaches a country-specific high-income per capita level.", "3) South Africa: The income per capita level of South Africa is still at a rather low level in 2016, at 7,477 USD.", "In fact, the GDP per capita level of South Africa is at almost one-third of the income per capita level of Portugal (22,534 USD), which has the lowest GDP per capita among the selected high-income countries.", "South Africa’s carbon emission per capita level is at 8.48 metric tons per capita.", "South Africa is the largest coal producer in its region.", "It has the tenth-largest recoverable coal reserves in the world and 75% of the total coal reserves in Africa.", "Energy consumption is increasing in South Africa as the economy is growing and the government is building more coal-fired power stations to meet the demand.", "However, similar to Canada, South Africa is also an energy exporter.", "It is the fifth-largest coal exporter in the world and exports 30% of its coal production to countries in Asia, especially to India.", "In brief, South Africa is a developing country where GDP per capita is still rather low and the demand for energy is increasing due to its growing economy.", "Furthermore, South Africa is also exporting a significant amount of produced energy to other countries and its economic growth is dependent on these exports.", "As a result, carbon emission per capita in South Africa is still in an increasing trend as shown in Figure 18.", "It is still too early at this stage to state whether the carbon emission per capita curve of South Africa will be in line with the EKC hypothesis because its GDP per capita level is still at rather low levels and it is still a developing country.", "However, the increasing trend phase of the curve is in line with the first part of the EKC theory, which states that the carbon emission per capita is expected to increase until it reaches a country-specific high-income per capita level.", "4) Colombia: Similar to South Africa, the income per capita level of Colombia is still rather low at 7,634 USD in 2016.", "However, Colombia’s carbon emission per capita level is at a much lower level, (2.03 metric tons per capita) when compared to the one of South Africa (8.48 metric tons per capita).", "According to 2019 statistics of the US Energy Information Administration, Colombia is South America’s largest coal producer and third-largest oil producer.", "However, even though it is such a major coal and oil producer, Colombia uses hydropower for most of its domestic electricity needs and exports coal and oil mainly to USA.", "Therefore, its carbon emission per capita is much lower when compared to South Africa’s carbon emission.", "As shown in Figure 19, Colombia’s carbon emission per capita is increasing as its GDP per capita is increasing.", "However, as it was also the case in the other non-high income countries that are analyzed above, it is still too early at this stage to state whether the carbon emission per capita curve of Colombia will be in line with the EKC hypothesis because its GDP per capita level is still at rather low levels.", "However, the increasing trend phase of the curve is in line with the first part of the EKC theory, which states that the carbon emission per capita is expected to increase until it reaches a country-specific high-income per capita level.", "5) China: In Figure 20, China’s carbon emission per capita trend is presented together with its GDP per capita trend.", "In 2016, GDP per capita of China is still rather low at 6,908 USD and carbon emission per capita level is at 7.18 metric tons per capita level.", "In China’s graph, carbon emission per capita trend is increasing as its GDP per capita trend is increasing.", "In 1960, carbon emission per capita was at 1.17 and GDP per capita was at only 192 USD.", "In 2016, carbon emission per capita increased 6.1 times and reached 7.18, whereas GDP per capita increased almost 36 times to 6,908 USD.", "As it is the case in the other non-high income countries analyzed above, it is still too early at this stage to state whether the carbon emission per capita curve of China will be in line with the EKC hypothesis because its GDP per capita level is still at rather low levels.", "However, the increasing trend phase of the curve is in line with the first part of the EKC theory, which states that the carbon emission per capita is expected to increase until it reaches a country specific high-income per capita level.", "6) Peru: In 2016, the income per capita level of Peru is still rather low at 6,262 USD and its carbon emission is at 1.86 metric tons per capita.", "Peru is also similar to South Africa and Colombia in exporting energy sources despite its growing domestic needs.", "Peru has rich hydrocarbons, oil, natural gas and coal reserves (US Energy Information Administration 2020).", "In 1960, Peru’s carbon emission was at 0.80 metric tons per capita and it increased 2.3 times to 1.86 metric tons per capita in 2016.", "As shown in Figure 21, Peru’s carbon emission per capita is increasing as its GDP per capita is increasing.", "However, as it was also the case in the other non-high income countries that are analyzed above, it is still too early at this stage to state whether the carbon emission per capita curve of Peru will be in line with the EKC hypothesis because its GDP per capita level is still at rather low levels.", "However, the increasing trend phase of the curve is in line with the first part of the EKC theory, which states that the carbon emission per capita is expected to increase until it reaches a country-specific high-income per capita level.", "7) Egypt: In 2016, GDP per capita of Egypt is still very low at 2,761 USD and carbon emission per capita level is at 2.53 level.", "In Figure 22, Egypt’s carbon emission per capita trend is presented together with its GDP per capita trend.", "Egypt’s carbon emission curve is in an increasing trend as its GDP per capita is increasing.", "In 1960, carbon emission per capita was at 0.60 and GDP per capita was at only 578 USD.", "Although GDP per capita of Egypt is much higher than China’s (192 USD) in 1960, it is still rather low.", "In 2016, carbon emission per capita increased approximately 4.2 times and reached 2.53 whereas GDP per capita increased 4.8 times to 2,761 USD.", "Despite the high increase rate in its GDP per capita, it is still too early at this stage to state whether the carbon emission per capita curve of Egypt will be in line with the EKC hypothesis because its GDP per capita level is still at rather low levels.", "However, the increasing trend phase of the curve is in line with the first part of the EKC theory, which states that the carbon emission per capita is expected to increase until it reaches a country specific high-income per capita level.", "8) India: In 2016, GDP per capita of India is rather low at 1,876 USD and CO2 per capita level is also low at 1.82 level.", "As it is presented in Figure 23, India’s CO2 per capita trend is increasing together with its GDP per capita.", "In 1960, CO2 per capita was at 0.27 and GDP per capita was at only 330 USD.", "In 2016, CO2 per capita increased approximately 6.7 times and reached 1.82 whereas GDP per capita increased 5.7 times to 1,876 USD.", "Despite the high increase rate in its GDP per capita, its level in 2016 is still low at 1,876 USD.", "Therefore, it is still too early at this stage to state whether the carbon emission per capita curve of India will be in line with the EKC hypothesis because its GDP per capita level is still at rather low levels.", "However, the increasing trend phase of the curve is in line with the first part of the EKC theory, which states that the carbon emission per capita is expected to increase until it reaches a country specific high-income per capita level.", "9) Pakistan: In 2016, GDP per capita of Pakistan is rather low at 1,118 USD and CO2 per capita level is also low at 0.99 level.", "As it is presented in Figure 24, Pakistan’s CO2 per capita trend is also increasing together with its GDP per capita, similar to the rest of the selected non-high income countries.", "In 1960, CO2 per capita was at 0.31 and GDP per capita was at only 302 USD.", "In fact, both the carbon emission per capita and the income per capita of Pakistan is very similar to that of India in 1960.", "In 2016, CO2 per capita increased approximately 3.2 times and reached 0.99 whereas GDP per capita increased 3.7 times to 1,118 USD.", "For the period of 1960-2016, carbon emission and income growth rates in Pakistan are high but relatively lower than India’s.", "Despite the substantial increase rate in its GDP per capita, its level in 2016 is still very low at 1,118 USD.", "As it was also the case in all of the other selected non-high income countries, it is still too early at this stage to state whether the carbon emission per capita curve of Pakistan will be in line with the EKC hypothesis because its GDP per capita level is still at rather low levels.", "However, the increasing trend phase of the curve is in line with the first part of the EKC theory, which states that the carbon emission per capita is expected to increase until it reaches a country specific high-income per capita level.", "10) Conclusion for the analysis on non-high income countries: Nine selected non-high income countries (Brazil, Malaysia, South Africa, Colombia, China, Peru, Egypt, India, Pakistan) are analyzed in the section above.", "The carbon emission per capita of all these countries are still in an increasing trend because their GDP per capita levels are still at rather low levels.", "In summary, it is still too early at this stage to state whether the carbon emission per capita curve of these selected non-high income countries will be in line with the EKC hypothesis due to their still rather low GDP per capita levels.", "However, the increasing trend phase of their carbon emission curves are in line with the first part of the EKC theory, which states that the carbon emission per capita is expected to increase until it reaches a country-specific high-income per capita level.", "Figure: NO_CAPTIONFigure: NO_CAPTIONIn section D(i), the carbon emission trends of the fifteen selected high-income countries were analyzed and it was observed that for the high-income countries, carbon emission per capita level of each country started to decline only after the country specific high-income level, measured by GDP per capita, was reached.", "Furthermore, it was observed that the decline in the carbon emission levels started in different years and, as a result, it was inferred that the policies of the international institutions were ineffective in initiating the decline trends in carbon emission levels.", "In this section, the year of the start of the declining trend for the carbon emission per capita for these selected high-income countries is further analyzed.", "In the table below, the year of the start of the decreasing trend, the related GDP of that year, the level of CO2 per capita and the SGI of 2018 for the high income countries (except for Germany that shows only a decreasing trend due to data unavailability) are summarized: Figure: NO_CAPTIONThe table above clearly shows that the year of the start of the declining trend is very different for the high-income countries.", "This difference of the year in which the carbon emission declining trend starts proves that the policies of the international institutions are unfortunately ineffective in initiating the start of the declining trend in these countries.", "If the policies of the international policies were effective, then, the declining trend of the carbon emission curve would have started around the same years.", "In the table above, we can see that the starting year of the decline in the carbon emission trend has a wide range from 1973 to 2013.", "The country with the earliest carbon emission per capita decreasing trend is Switzerland.", "Switzerland’s carbon emission per capita started to decrease after 1973 when its GDP per capita was at 53,777 USD.", "Furthermore, Switzerland is a very successful country in environmental protection with a SGI index score of 9.", "This awareness and responsiveness to carbon emission together with its high GDP per capita explains the reason for Switzerland’s earliest start of the decreasing trend in its carbon emission curve.", "On the other hand, Australia’s decreasing trend of carbon emission per capita started in 2009 when its GDP per capita was at 51,767 USD.", "GDP per capita level of Australia was similar to that of Switzerland when its carbon emission per capita started to decrease.", "However, SGI for Australia was quite low (4) when compared to the high score of Switzerland (9).", "Furthermore, carbon emission per capita levels of these countries were also quite different at the time when their decreasing trends started.", "Carbon emission per capita for Switzerland was at 7.33 metric tons per capita whereas that of Australia was very high at 18.20 metric tons per capita.", "Therefore, it can be inferred that despite the differences in carbon emission sensitivity and performance of these countries, the carbon emission per capita started to decline only when the country specific high-income per capita was reached.", "Denmark and Finland were both European countries with very close levels of carbon emission per capita levels at the start of the decreasing trend of their carbon emission curves.", "Denmark’s carbon emission level was at 13.71 metric tons per capita and Finland’s was at 13.17 metric tons per capita.", "Both of these countries also have high SGI index scores where Denmark’s score is at 9 and Finland’s score is at 8.", "However, despite these similarities in their carbon emission trends, sensitivities and performances, the starting year of their decreasing carbon emission trends were very different.", "Denmark’s decreasing trend started relatively earlier in 1996 whereas Finland’s decline in the carbon emission per capita curve started 7 years later in 2003.", "It is clear that the GDP per capita is the major differentiating factor for the time difference in the start of the declining carbon emission curves of these countries.", "Denmark’s GDP per capita reached 50,262 USD in 1996 whereas Finland’s was at still 32,963 USD in that year.", "In fact, carbon emission per capita for Finland started to drop only in 2003 only when its GDP per capita reached 42,707 USD which was its country specific high income level as per the hypotheses of EKC theory.", "The declining trends of the carbon emission curves of Italy and Spain start in 2004 and in 2005, respectively.", "Their GDP per capita levels are both above 30,000 USD.", "Italy’s GDP per capita is at 37,227 USD in 2004 and that of Spain is at 31,029 USD in 2005.", "Furthermore, their carbon emission per capita levels are similar, as well.", "Carbon emission per capita for Italy is 8.22 metric tons per capita at the start of the decreasing trend in 2004 whereas that of Spain is at 8.10 metric tons per capita in 2005.", "Their SGI index scores are both 7.", "Since their SGI scores are the same and their carbon emission per capita are close, the only other determinant that is different in these countries are their GDP per capita’s.", "The one for Italy is higher (37,227 USD in 2004) than the one in Spain (31,029 USD in 2005), however, they are still close to each other.", "Therefore, in this case, we see only one year of difference in the starting year of the declining carbon curve.", "USA and Canada, on the other hand, have very high carbon emission per capita levels when compared to the most of the other high-income countries because petroleum and oil plays a major role in their economy.", "The decreasing trend started in the USA in 1978 whereas it started in 1979 in Canada.", "Their GDP per capita levels are around 29,000 USD level when this first sharp decrease started.", "Then, there is another decrease point in Canada in 2006 when GDP per capita reached 45,858 and CO2 per capita level was at 17.56 metric tons per capita.", "In the USA, there was also a second decrease trend in 2004 when the GDP per capita was at 47,288 USD and carbon emission per capita was at 19.66 metric tons per capita.", "In brief, in Canada and in the USA, the dependence of the economies on petroleum and oil interrupted the decreasing trends of their CO2 per capita curves for a period but, later, the decline continued nevertheless with the further increase in their GDP per capita levels.", "Furthermore, the level of CO2 per capita was still very high when compared to the other high-income countries in 2014, as shown in Figure 25 in the next section, due to the dependence in their economies on oil and petroleum.", "France and Belgium also had similar GDP per capita levels as the USA and Canada in 1979 when the decreasing trends on their CO2 per capita curves started.", "GDP per capita of France was at 26,578 USD in 1979 whereas that of Belgium was at 25,946 USD.", "The start of the declining trend of their carbon emission curves is the same for France and Belgium, 1974.", "The carbon emission per capita for France (9.64 metric tons per capita) and Belgium (14.24 metric tons per capita) were lower than the ones for the USA and Canada at the start of the decreasing trend because the economies of France and Belgium are less dependent on oil and petroleum when compared to the USA and Canada.", "Furthermore, the decreasing trend in France and Belgium were not interrupted like it was the case for the USA and Canada due to less dependence in their economies on oil and petroleum.", "The SGI index for France is 7 whereas that of Belgium is 6.", "Therefore, it can be said that since their GDP per capita levels were almost the same in 1979, the start of their carbon emission curves was 1979 for both of these countries despite the different scores on SGI.", "In brief, the EKC theory is valid for high-income countries that reached a certain country specific high level of GDP per capita.", "When the carbon emission per capita vs GDP per capita of high-income countries are analyzed, it is clear that the GDP per capital is the main determinant for the carbon emission per capita trend for these selected countries.", "It is also clear that the international institutions policies are not effective in starting the declining trend of the carbon emission per capita because the trend starts in a wide range of different years in these countries.", "The main factor for initiating the start of the decreasing trend is reaching the country specific high level of GDP per capita, rather than the international institutions policies.", "It is clear that reaching the country specific high level of GDP per capita is the main determinant for initiating the start of the declining trend of the carbon emission curve.", "The final level of carbon emission per capita that was reached and the time lag between the peak and the start of the declining curve are affected by a country’s economic dependence on petroleum/oil production and its SGI index score performance.", "If the economy is highly dependent on petroleum/oil/coal like Canada, the USA and Japan, the decreasing trend of carbon emission per capita is not as strong as the other high-income countries like Belgium, France and the UK.", "In summary, in this section, it is proven that the declining trend of the carbon emission trend starts only after the country specific high GDP per capita level is reached in these high-income countries.", "Furthermore, the effects of the economic dependence on oil and petroleum and the performance on SGI score on carbon emission is also presented.", "In this section, the role of the government benevolence and the effect of the energy production dependency of the economy on carbon emission is analyzed.", "In Figure 25, the degradation levels as of 2014 for the selected high-income countries are presented.", "The 2014 data is used in this analysis because the latest CO2 per capita data available for France and Italy is for 2014 in the World Bank’s database.", "Figure: Fig 25:Degradation levels of countries in 2014In this graph, most of the countries’ carbon emission per capita levels are below 10 except for Canada, USA and Australia.", "Even though the EKC theory is also valid for these exceptional countries, their carbon emission per capita levels are at significantly higher levels than the rest.", "The reasons for these high levels of carbon emissions can be explained by their economic dependence on energy production and government benevolence.", "SGI index shows the effectiveness of the environmental policies in countries.", "The SGI index scores for Switzerland (9) and Denmark (9) are the highest.", "Accordingly, these countries have relatively low carbon emission per capita emission levels as shown in Figure 27.", "The SGI score of 9 shows that “Environmental policy goals are ambitious and effectively implemented as well as monitored within and across most relevant policy sectors that account for the largest share of resource use and emissions”(Stiftung, Bertelsmann 2020).", "As a result, it can be said that ambitious and effective environmental policies resulted in the relatively lower level of carbon emission per capita in Switzerland and Denmark.", "On the other hand, USA and Australia have lower SGI scores, 4 for both of them.", "This shows that “Environmental policy goals are neither particularly ambitious nor are they effectively implemented and coordinated across relevant policy sectors” (Stiftung, Bertelsmann 2020).", "As a result, it can be stated that the insufficiency of the environmental policies result in USA’s and Australia’s high carbon emission per capita level.", "Canada, however, has a higher SGI score than USA and Australia.", "Canada’s SGI score is 7 which means that “Environmental policy goals are mainly ambitious and effectively implemented and are monitored within and across some of the relevant policy sectors that account for the largest share of resource use and emissions.”(Stiftung, Bertelsmann 2020).", "Another important factor for Canada’s high carbon emission per capita level is the economy’s energy export dependency.", "Canada has to export energy to the USA (which means that they have to produce more energy than their citizens use) in order to maintain their economic wealth which results in a higher carbon emission per capita for the country when compared to the other selected high-income countries.", "In summary, it is clear that the SGI score index of the country and the dependency of the economy on energy production with high carbon emission (coal, petroleum and oil) have a major role in the level of the CO2 per capita of the countries.", "In this article, we analyze the carbon emission trends of twenty-four selected countries.", "Our data analysis shows clearly that in all of the fifteen high-income countries that were selected, EKC hypothesis is valid and carbon emission starts to decline in high-income countries when a certain country specific high-income level is reached.", "On the other hand, it is too early to infer whether EKC hypothesis would be applicable to non-high income countries since their income levels are still at very low levels in the selected timeframe, which is from 1960-2016.", "Furthermore, our analysis showed clearly that the international policies were, unfortunately, ineffective in initiating a declining trend for carbon emission in high-income countries.", "We also incorporated SGI scores in our analysis to assess the effect of the national efforts and policies on the start year of the carbon emission trend.", "In the last section, we focused on the carbon emission levels of all the selected high-income countries in 2014 and explained the relationship between their government benevolence represented by the SGI scores and their levels of carbon emission.", "Our analysis showed that there is a strong relationship between the overall SGI scores of these countries and their carbon emission levels.", "Furthermore, we also concluded that the dependence of the economy on energy production with high carbon emission (coal, petroleum and oil), also, have a major role in the level of the CO2 per capita of these countries.", "Beer, Espen, and Prydz Divyanshi Wadhwa.", "“Classifying Countries by Income.The World Bank IBRD IRA, World Bank Group, 9 Sept. 2019, datatopics.worldbank.org/ world-development-indicators/stories/the-classification-of-countries-by-income.html.", "Callen, Tim, and Jonathan D. Ostry.", "“Japan's Lost Decade — Policies for Economic Revival.” International Monetary Fund, International Monetary Fund, 13 Feb. 2013, www.imf.org/external/pubs/nft/2003/japan/index.htm.", "CO2 Emissions (Metric Tons per Capita).” The World Bank IBRD IDA CO2 emissions (metric tons per capita) Data,Carbon Dioxide Information Analysis Center, Environmental Sciences Division, Oak Ridge National Laboratory, Tennessee, United States., 2021, data.worldbank.org/indicator/EN.ATM.CO2E.PC.", "Dasgupta, Susmita, et al.", "“Confronting the Environmental Kuznets Curve.", "Journal of Economic Perspectives, vol.", "16, no.", "1, 2002, pp.", "147–168., doi:10.1257/0895330027157.", "“GDP per Capita (Constant 2010 USD).” textitThe World Bank IBRD IDA GDP per Capita (Constant 2010 USD) | Data,World Bank National Accounts Data, OECD National Accounts Data Files, 2021, donnees.banquemondiale.org/ Grossman, Gene M., and Alan B. Krueger.“Environmental Impacts of a North American Free Trade Agreement.” NBER.", "National Bureau of Economic Research, Nov. 1991, www.nber.org/papers/w3914.", "Hervieux, Marie-Sophie, and Pierre-Alexandre Mahieu.", "“A Detailed Systematic Review of the Recent Literature on Environmental Kuznets Curve Dealing with CO2.” HAL Archives-Ouvertes.fr, Laboratoire d’Economie Et De Management Nantes-Atlantique Université De Nantes , 2014, hal.archives-ouvertes.fr/hal-01010243/document.", "Kuznets, Simon.", "“Economic Growth and Income Inequality.” The American Economic Review, XLV , no.", "1, Mar.", "1955, pp.", "1–30.", "Stiftung, Bertelsmann.", "“SGI 2020: Downloads.” SGI 2020 | Downloads, Sustainable Governance Indicators, 2020, www.sgi-network.org/2020/Downloads.", "US Energy Information Administration.", "U.S. Energy Information Administration - EIA - Independent Statistics and Analysis 2 Dec. 2020, www.eia.gov/international/analysis/country/JPN.", "World Bank.", "“World Development Report 1992.” Open Knowledge Repository, New York: Oxford University Press © World Bank, 1992, openknowledge.", "worldbank.org/handle/10986/5975." ] ]
2105.11756
[ [ "SGD with Coordinate Sampling: Theory and Practice" ], [ "Abstract While classical forms of stochastic gradient descent algorithm treat the different coordinates in the same way, a framework allowing for adaptive (non uniform) coordinate sampling is developed to leverage structure in data.", "In a non-convex setting and including zeroth order gradient estimate, almost sure convergence as well as non-asymptotic bounds are established.", "Within the proposed framework, we develop an algorithm, MUSKETEER, based on a reinforcement strategy: after collecting information on the noisy gradients, it samples the most promising coordinate (all for one); then it moves along the one direction yielding an important decrease of the objective (one for all).", "Numerical experiments on both synthetic and real data examples confirm the effectiveness of MUSKETEER in large scale problems." ], [ "Introduction", "Coordinate Descent (CD) algorithms have become unavoidable in modern machine learning because they are tractable [1] and competitive to other methods when dealing with key problems such as support vector machines, logistic regression, LASSO regression and other $\\ell _1$ -regularized learning problems [2], [3].", "They are applied in a wide variety of problems ranging from linear systems [4], [5] to finite sum optimization [6], [7] and composite functions [8] with parallel [9], [10], distributed [11], [12] and dual [13], [14], [15] variants.", "In many contributions [16], [17], [18], [19], [20], [21], the choice of the coordinate sampling policy is conducted through some optimality criterion estimated along the algorithm.", "On the one hand, efficient forms of CD methods rely on a deterministic procedure [22] which adapts to the underlying structure in data at the expense of higher calculation and thus, may be costly.", "On the other hand, stochastic gradient descent (SGD) methods are computationally efficient but often treat all coordinates equally and thus, may be sub-optimal.", "In the spirit of adaptive schemes, we tend to bridge the gap between the best of both worlds by developing, within a noisy gradient framework, a general stochastic coordinate descent method with a particular selection strategy.", "We are interested in solving unconstrained optimization problems of the form $\\min _{\\theta \\in \\mathbb {R}^p} f(\\theta )$ , where the objective function $f$ may be either known exactly or accessed through noisy observations.", "When $f$ is differentiable, a common appproach is to rely on the gradient of $f$ .", "However, in many scenarios and particularly in large-scale learning, the gradient may be hard to evaluate or even intractable.", "Hence, one usually approximates the gradient using zeroth or first order estimates [23], [24].", "The former constructs pseudo-gradients by sampling some perturbed points or using finite differences [25], [26], [27], [28] (see [29] for a recent survey and numerous references) leading to biased gradient estimates while the latter often relies on data sampling techniques [30], [31] to obtain unbiased gradient estimates.", "In both cases, a random gradient estimate is available at a cheap computing cost and the method consists in moving along this estimate at each iteration.", "Early seminal works on such stochastic algorithms include [32], [33] and an excellent review dealing with large scale learning problems is given in [34].", "Starting from an initial point $\\theta _0 \\in \\mathbb {R}^p$ , the SGD algorithm is defined by the update rule $\\forall t \\ge 0, \\quad \\theta _{t+1} = \\theta _{t} - \\gamma _{t+1} g_t$ where $g_t\\in \\mathbb {R}^p $ is a gradient estimate at $\\theta _t$ (possibly biased) and $(\\gamma _t)_{t\\ge 1}$ is some learning rate sequence that should decrease throughout the algorithm.", "While the computation of $g_t$ may be cheap, it still requires the computation of a vector of size $p$ which may be a critical issue in high-dimensional problems.", "To address this difficulty, we rely on sampling well-chosen coordinates of the gradient estimate at each iteration.", "We consider the framework of stochastic coordinate gradient descent (SCGD) which modifies standard stochastic gradient descent methods by adding a selection step to perform random coordinate descent.", "The SCGD algorithm of is defined by the following iteration $\\left\\lbrace \\begin{array}{ll}\\theta _{t+1}^{(k)} = \\theta _{t}^{(k)} &\\text{ if } k \\ne \\zeta _{t+1} \\\\\\theta _{t+1}^{(k)} = \\theta _{t}^{(k)} - \\gamma _{t+1} g_t^{(k)} &\\text{ if } k=\\zeta _{t+1}\\end{array}\\right.$ where $ \\zeta _{t+1}$ is a random variable valued in $\\llbracket 1,p \\rrbracket $ which selects a coordinate of the gradient estimate.", "The distribution of $\\zeta _t$ is called the coordinate sampling policy.", "Note that the SCGD framework is very general as it contains as many methods as there are ways to generate both the gradient estimate $g_t$ and the random variables $\\zeta _t$ .", "Contributions.", "(i)(Theory) We show the almost-sure convergence of the SCGD iterates $(\\theta _t)_{t \\in \\mathbb {N}}$ towards the minimizer $\\theta ^\\star $ of $f$ as well as non-asymptotic bounds on the optimality gap $\\operatorname{\\mathbb {E}}[f(\\theta _t)-f(\\theta ^\\star )]$ .", "The working conditions are relatively weak as the function $f$ is only required to be $L$ -smooth (classical in non-convex problems) and the stochastic gradients are possibly biased with unbounded variance (using a growth condition related to expected smoothness [35]).", "(ii)(Practice) We develop a new algorithm, called MUSKETEER, for MUltivariate Stochastic Knowledge Extraction Through Exploration Exploitation Reinforcement.", "In the image of the motto 'all for one and one for all', this procedure belongs to the SCGD framework with a particular design for the coordinate sampling policy.", "It compares the value of all past gradient estimates $g_t$ to select a descent direction (all for one) and then moves the current iterate according to the chosen direction (one for all).", "The heuristic is the one of reinforcement learning in the sense that large gradient coordinates represent large decrease of the objective and can be seen as high rewards.", "The resulting directions should be favored compared to the path associated to small gradient coordinates.", "By updating the coordinate sampling policy, the algorithm is able to detect when a direction becomes rewarding and when another one stops being engaging.", "Related Work.", "The authors of [22] investigate the deterministic Gauss-Southwell rule which consists of picking the coordinate with maximum gradient value.", "In trusting large gradients, this rule looks like the one of MUSKETEER except that no stochastic noise -neither in the gradient evaluation nor in the coordinate selection- is present in their algorithm.", "In that aspect, our method differs from all the previous CD studies [16], [17], [18], [19], [20], [21] which rely on $\\nabla f$ .", "Among the SGD literature, sparsification methods [36], [37] were developed for communication efficiency.", "They use a gradient estimate $g$ which is sparsified using probability weights to reach an unbiased estimate of the gradient.", "In contrast, the SCGD framework allows the gradient to be biased as no importance re-weighting is performed.", "Note also that, to cover zeroth order methods, the gradient estimate itself $g_t$ is allowed to be biased as for instance in the recent study [38].", "Finally, the non-asymptotic bounds are inspired from [39] where the authors provide a non-asymptotic analysis for standard SGD.", "Outline.", "Section introduces the mathematical framework with the different sampling strategies and Section contains our main theoretical results.", "Section is dedicated to MUSKETEER algorithm and a numerical analysis is performed in Section .", "Proofs, technical details and additional experiments may be found in the appendix.", "Notations.", "Denote by $(e_1,\\ldots ,e_p)$ the canonical basis of $\\mathbb {R}^p$ and for $k \\in \\llbracket 1,p \\rrbracket $ , $D(k)=e_k^{} e_k^{T} \\in \\lbrace 0,1\\rbrace ^{p \\times p}$ is a diagonal matrix with a 1 in position $k$ .", "$\\Vert \\cdot \\Vert _2$ and $\\Vert \\cdot \\Vert _\\infty $ are respectively the Euclidian and infinity norm.", "For any $u \\in \\mathbb {R}^p$ , $u^{(k)}$ is the k-th coordinate of $u$ ; ${1}_A$ is the indicator function of the event $A$ , i.e., ${1}_A=1$ is $A$ is true and ${1}_A=0$ otherwise.", "Denote by $\\mathcal {U(}\\llbracket 1,p \\rrbracket )$ the uniform distribution over $\\llbracket 1,p \\rrbracket $ .", "For a vector of probability weights $d=(d^{(1)},\\ldots ,d^{(p)})$ with $\\sum _{k=1}^p d^{(k)}=1$ , denote by $Q(d)$ the associated categorical distribution." ], [ "Problem Set-up, Data Sampling and Biased Gradients", "Consider the classical stochastic optimization problem $\\min _{\\theta \\in \\mathbb {R}^p} \\left\\lbrace f(\\theta ) = \\operatorname{\\mathbb {E}}_{\\xi }[f(\\theta ,\\xi )] \\right\\rbrace ,$ where $\\xi $ is a random variable.", "In many scenarios, e.g.", "empirical risk minimization or reinforcement learning, the gradient $\\nabla f$ cannot be computed in a reasonable time and only a stochastic version, possibly biased, is available.", "The distribution of $\\xi $ is called the data sampling policy as it refers to the sampling mechanism in the empirical risk minimization (ERM) framework.", "This running example is presented below and shall be considered throughout the paper.", "Other classical optimization problems where stochastic gradients are available include adaptive importance sampling [40], policy gradient methods [41] and optimal transport [42].", "Running Example (ERM).", "Given some observed data $z_1 ,\\ldots ,z_n\\subset \\mathcal {Z}$ and a loss function $\\ell : \\mathbb {R}^p \\times \\mathcal {Z} \\rightarrow \\mathbb {R}$ , the objective function $f$ approximates the risk $\\operatorname{\\mathbb {E}}_{z}[\\ell (\\theta ,z)]$ by the so-called empirical risk defined as $ \\forall \\theta \\in \\mathbb {R}^p, \\quad f(\\theta ) = \\frac{1}{n} \\sum _{i=1}^n \\ell (\\theta ,z_i).", "$ Evaluating $f$ or its gradient is prohibitive in large scale machine learning as it requires seeing all the samples in the dataset.", "Instead, after picking at random an index $ j = \\xi $ , uniformly distributed over $\\llbracket 1, n\\rrbracket $ , the $k$ -th coordinate of the gradient estimate may be computed as $ (\\ell (\\theta + h e_k,z_j)-\\ell (\\theta ,z_j)) / {h}$ .", "When differentiation is possible, another gradient estimate is offered by $\\nabla _\\theta \\ell (\\theta , z_j) $ .", "These two gradient estimates are of a different nature: the first one, often referred to as zeroth order estimate, is biased whereas the second one, often referred to as first order estimate, is unbiased.", "$\\Box $" ], [ "Adaptive Data Sampling Policy in ERM.", "There are many other methods to generate gradients.", "An extension is to average the previous estimates over random sets of size $m$ composed of pairwise distinct indexes in $\\llbracket 1,n \\rrbracket $ .", "This method is usually referred to as mini-batching [35].", "Recently, some authors focused on the data sampling policy and developed adaptive sampling strategies along with theoretical guarantees.", "Such methods include adaptive non-uniform sampling [31], selective sampling [30], [43] and survey sampling [44].", "Other works such as [45] build optimal sampling distribution in the sense of variance reduction.", "Since the present paper deals with the coordinate descent framework, the study of the data sampling policy is beyond the scope of our work.", "We refer to the numerous contributions mentioned above for an in-depth analysis.", "Throughout the paper, the gradient generator is denoted by $g_h(\\cdot , \\xi )$ where the parameter $h\\ge 0$ represents the underlying bias as claimed in the next assumption.", "This level of generality allows to include zeroth order estimate as discussed right after the assumption.", "Assumption 1 (Biased gradient) There exists a constant $c\\ge 0$ such that: $\\forall h>0 ,\\, \\forall \\theta \\in \\mathbb {R}^p,\\quad \\Vert \\operatorname{\\mathbb {E}}_{\\xi }[ g_h(\\theta ,\\xi ) ] - \\nabla f(\\theta )\\Vert _2 \\le c h.$ We can immediately provide two well-spread zeroth order estimates for which Assumption REF is satisfied.", "The smoothed gradient estimate [27] is given for all $\\theta \\in \\mathbb {R}^p$ by $g_{h}(\\theta ,\\xi ) = h^{-1} [f(\\theta + h U,\\xi )-f(\\theta ,\\xi )]U$ where $U $ is a standard Gaussian vector (independent from $\\xi $ ).", "An alternative version consists in taking $U$ uniformly distributed over the unit sphere.", "The finite difference gradient estimate is given for all $\\theta \\in \\mathbb {R}^p$ by $g_{h}(\\theta ,\\xi ) = \\sum _{k=1}^p g_{h}(\\theta ,\\xi )^{(k)}e_k$ where for all $k=1,\\ldots ,p$ the coordinates are $g_{h}(\\theta ,\\xi )^{(k)} = h^{-1}[f(\\theta + h e_k,\\xi )-f(\\theta ,\\xi )] $ .", "Both previous examples share the following general property.", "There exists a probability measure $\\nu $ satisfying $\\int _{\\mathbb {R}^p} x x^T \\nu (\\mathrm {d}x) = I_p$ such that, $\\forall h>0,\\theta \\in \\mathbb {R}^p, \\quad \\operatorname{\\mathbb {E}}_{\\xi }[ g_h (\\theta ,\\xi ) ] = \\int _{\\mathbb {R}^p} x \\left\\lbrace \\frac{ f(\\theta + h x ) - f(\\theta ) }{h} \\right\\rbrace \\nu (\\mathrm {d}x).$ The smoothed gradient estimate is recovered when $\\nu $ is the standard Gaussian measure and taking $\\nu = \\sum _{k=1}^p \\delta _{e_k}/p $ covers the finite differences estimate.", "As detailed in the next subsection, an interesting framework is to use measures $\\nu $ that evolve through time and put different weights on the different directions.", "As stated in the following proposition, when the function $f$ is $L$ -smooth, i.e., $\\nabla f$ is $L$ -Lipschitz, the bias of the gradient estimate (REF ) is of order $ h$ and thus satisfies Assumption REF .", "Proposition 1 Under Eq.", "(REF ), if $f$ is $L$ -smooth, then Assumption REF holds true with $c = \\sqrt{C} L / 2$ where $ C = \\int _{\\mathbb {R}^p} \\Vert x\\Vert _2 ^6 \\nu (\\mathrm {d}x) <\\infty $ .", "The previous proposition allows us to cover the two methods: smoothing and finite difference.", "Note that for the latter, the constant $C$ is equal to 1.", "Note finally that Assumption REF enables to work with classical unbiased gradient by taking $c=0$ ." ], [ "Coordinate Sampling Policy", "Let $(\\xi _t)_{t\\ge 1} $ be a sequence of independent and identically distributed random variables.", "Let $(\\gamma _t)_{t\\ge 1} $ be a sequence of positive numbers called learning rates.", "Let $(h_t)_{t\\ge 1} $ be a sequence of positive numbers called smoothing parameters.", "Denote by $ g_t = g _{h_{t+1} } ( \\theta _t, \\xi _{t+1} )$ the gradient estimate at time $t$ .", "The classical SGD update rule is given by $\\theta _{t+1} = \\theta _{t} - \\gamma _{t+1} g_t ,\\quad t\\ge 0,$ For any $t \\in \\mathbb {N}, \\mathcal {F}_t = \\sigma ( \\theta _0, \\theta _1,\\ldots , \\theta _t)$ is the $\\sigma $ -field associated to the sequence of iterates $(\\theta _t)_{t \\in \\mathbb {N}}$ .", "The framework of SCGD is introduced thanks to random coordinate sampling.", "At each step, only one coordinate of the parameter of interest is updated.", "This coordinate is selected at random according to a distribution valued in $\\llbracket 1,p \\rrbracket $ which is allowed to evolve during the algorithm.", "The iteration of the coordinate sampling algorithm is given coordinate-wise by $\\left\\lbrace \\begin{array}{ll}\\theta _{t+1}^{(k)} = \\theta _{t}^{(k)} &\\text{ if } k \\ne \\zeta _{t+1} \\\\\\theta _{t+1}^{(k)} = \\theta _{t}^{(k)} - \\gamma _{t+1} g_t^{(k)} &\\text{ if } k=\\zeta _{t+1}\\end{array}\\right.$ where $ \\zeta _{t+1}$ is a random variable valued in $\\llbracket 1,p \\rrbracket $ .", "Hence $\\zeta _{t+1}$ selects the coordinate along which the $t$ -th descent shall proceed.", "The distribution of $\\zeta _{t+1}$ is called the coordinate sampling policy as opposed to the data sampling policy governed by the random variable $\\xi _{t+1}$ .", "The distribution of $\\zeta _{t+1}$ is characterized by the probability weights vector $d_{t} = (d_{t}^{(1)},\\ldots ,d_{t}^{(p)})$ defined by $d_{t}^{(k)} = \\mathbb {P} ( \\zeta _{t+1} = k |\\mathcal {F}_{t}), \\quad k\\in \\llbracket 1,p \\rrbracket .$ The categorical distribution on $\\llbracket 1,p \\rrbracket $ associated to $d_t$ is denoted by $Q(d_t)$ , i.e., conditionally to $\\mathcal {F}_t$ , we have: $\\forall t \\ge 0, \\quad \\zeta _{t+1} \\sim Q (d_t) \\quad \\text{ with } \\quad d_{t} = (d_{t}^{(1)},\\ldots ,d_{t}^{(p)}).$ Running Example (ERM).", "The CD algorithm defined by Equation (REF ) can easily be applied in the ERM framework.", "The coordinate sampling strategy $\\zeta \\sim Q(d_t)$ combined with the uniform data sampling $\\xi \\sim \\mathcal {U}(\\llbracket 1,n \\rrbracket )$ leads to $\\theta _{t+1}^{(\\zeta )} = \\theta _{t}^{(\\zeta )} - (\\gamma _{t+1} / h_{t+1} ) (\\ell (\\theta _t + h_{t+1} e_\\zeta , z_\\xi )-\\ell (\\theta _t , z_\\xi )) $ (zeroth order) and $\\theta _{t+1}^{(\\zeta )} = \\theta _{t}^{(\\zeta )} - \\gamma _{t+1} \\partial _{\\theta _{\\zeta }} \\ell (\\theta _t, z_\\xi )$ (first order).", "$\\Box $ Given the past, the data sampling and coordinate sampling draws should not be related.", "Assumption 2 (Conditional Independence) $\\zeta _{t+1}$ is independent from $\\xi _{t+1} $ conditionally on $\\mathcal {F}_t$ .", "This assumption is natural in the ERM context as in most cases there is no particular link between the sample indexes and the coordinates.", "Futhermore, the independence property plays an important role in our proofs.", "The SCGD algorithm defined in (REF ) is simply written with matrix notation as $\\theta _ {t+1} = \\theta _ {t} - \\gamma _{t+1} D(\\zeta _{t+1}) g _t,$ where $D(k) = e_{k}^{} e_{k}^T \\in \\mathbb {R}^{p\\times p}$ has its entries equal to 0 except the $(k,k)$ which is 1.", "Observe that the distribution of the random matrix $D(\\zeta _{t+1})$ is fully characterized by the matrix $D_t = \\operatorname{\\mathbb {E}}[ D(\\zeta _{t+1})|\\mathcal {F}_{t}] = Diag(d_{t}^{(1)},\\ldots ,d_{t}^{(p)}).$ Note that under Assumptions REF and REF , , the average move of SCGD follows a biased gradient direction.", "For instance, when $c=0$ the average move of SCGD is given by $\\operatorname{\\mathbb {E}}[\\theta _ {t+1} - \\theta _ {t} |\\mathcal {F}_t] = -\\gamma _{t+1} D_t \\nabla f (\\theta _t ) $ which bears resemblance to the Conditioned-SGD iteration [34].", "Such preprocessing is meant to refine the gradient direction through a matrix mulitplication for a better understanding of the underlying structure of the data.", "A natural question rises on the choice of the matrix $D_t$ among all the possible coordinate sampling distributions.", "The SCGD framework is efficient as soon as one can compute each coordinate of the gradient estimate.", "This is the case for ZO optimization with finite differences where the full gradient estimate uses $p$ partial derivatives, each of them requiring two queries of the objective function.", "SCGD reduces this cost to a single coordinate update.", "Remark 1 (Batch coordinates) A natural extension is to consider subsets of coordinates, a.k.a.", "block-coordinate descent.", "Note that this framework is covered by our approach as the proofs can be extended by summing different matrices $D(\\zeta )$ .", "Similarly to mini-batching [35], one can consider multiple draws for the coordinates that are to be updated.", "The selecting random matrix $D(\\zeta _{t+1})$ may be replaced by a diagonal matrix with $m(<p)$ non-zero coefficients.", "For that matter, it is enough to have multiple draws from the categorical distribution $Q(d_t)$ .", "Remark 2 (Parallelization) Several families of communication-reduction methods such as quantization [36], gradient sparsification [37], [46] or local-SGD [47] have been proposed to reduce the overheads of distribution.", "The SCGD framework can benefit from such data parallelization techniques.", "When a fixed number $m$ of machines is available, it is then possible to gain computational acceleration by drawing $m$ times the coordinate distribution $Q(d_t)$ on the different machines and then transmit the batch of selected coordinates to the workers." ], [ "Adaptive and Unbiased Policies", "To understand more clearly the differences between SGD and SCGD, we shall rely on a more general iteration scheme.", "This framework is useful to compare different algorithms in terms of adaptive policies and unbiased estimates.", "Consider the following general update rule $ \\theta _ {t+1} = \\theta _ {t} - \\gamma _{t+1} h ( \\theta _t, \\omega _{t+1} ), \\quad t \\ge 0$ where $h$ is a gradient generator and $(\\omega _t)_{t\\ge 1}$ is a sequence of random variables which are not necessarily independent nor identically distributed.", "Observe that both frameworks, SGD and SCGD, are instances of (REF ).", "For example, the randomness of SCGD can be expressed through $\\omega _t = (\\xi _t,\\zeta _t)$ .", "Definition 1 (Policy) Denote by $P_t$ the distribution of $\\omega _{t+1}$ given $\\mathcal {F}_t$ .", "The sequence $(P_{t})_{t \\ge 0} $ is called the policy of the stochastic algorithm.", "The policy of a stochastic algorithm is an important tool as it determines the randomness introduced over time.", "On the one hand, it provides insights on the expected behavior of the algorithm.", "On the other hand, it measures the ability to adapt through the iterations.", "Definition 2 (Unbiased and Adaptive) A policy $(P_{t})_{t \\ge 0} $ is called \"unbiased\" if: $\\forall \\theta \\in \\mathbb {R}^p,t\\ge 0$ , $\\int h ( \\theta , \\omega ) P_t(\\mathrm {d} \\omega ) \\propto \\nabla f (\\theta ) $ .", "It is called \"naive\" if $P_t$ does not change with $t$ , otherwise it is adaptive.", "With these definitions in mind, it is clear that the SGD policy (REF ) under Assumption REF with $c=0$ is unbiased and naive, and so does the policy induced by first order gradient in ERM.", "Within the framework of SCGD, a policy cannot be unbiased and adaptive as claimed in the next proposition.", "Proposition 2 (Unbiased coordinate policy) Under Assumption REF with $c=0$ , if $ {\\operatorname{Span}} \\lbrace \\nabla f(\\theta ) \\, : \\, \\theta \\in \\mathbb {R}^p \\rbrace $ is dense in $\\mathbb {R}^p $ , then the only unbiased coordinate sampling policy is $D_t = I_p / p$ .", "It corresponds to uniform coordinate sampling.", "When working under Assumption REF with $c=0$ , SCGD with uniform coordinate sampling is unbiased and hence similar to SGD.", "This is confirmed in the numerical experiments (Appendix ,).", "However, a uniform sampling does not use any available information to favor coordinates among others.", "Thus, the approach promoted in the paper is different: past gradient values are used to update the probability weights of $D_t$ .", "The resulting method is an adaptive algorithm which is biased.", "Remark 3 (Importance Coordinate Sampling) Note that the general framework defined above includes the particular case where the coordinates are selected according to $\\zeta $ then reweighted as proposed in [37].", "This corresponds to the choice $h(\\theta , \\omega _{t+1} ) = D^{-1}_t D(\\zeta _{t+1}) g(\\theta , \\xi _{t+1}).$ Even though such a policy is adaptive and unbiased, it turns out -from our numerical experiments (Appendix )- that it behaves similarly to the uniform version and is therefore sub-optimal." ], [ "Main Theoretical Results", "In a general non-convex setting, we investigate the almost sure convergence of SCGD algorithms as well as non-asymptotic bounds.", "The following two assumptions on the objective function $f$ are classical among the SGD literature.", "Assumption 3 (Smoothness and Coercivity) • $f:\\mathbb {R}^{p} \\rightarrow \\mathbb {R}$ is twicely continuously differentiable and $L$ -smooth.", "• $f$ is coercive, i.e., $\\lim _{\\Vert \\theta \\Vert \\rightarrow + \\infty } f(\\theta ) = +\\infty $ and the equation $\\nabla f(\\theta ) = 0$ has a unique solution $\\theta ^\\star $ .", "When dealing with stochastic algorithms, the stochastic noise associated to the gradient estimates is the keystone for the theoretical analysis.", "To treat this term, we consider a weak growth condition, related to the notion of expected smoothness as introduced in [35] (see also [48], [49]).", "Assumption 4 (Growth condition) With probability 1, there exist $0 \\le \\mathcal {L},\\sigma ^2 < \\infty $ such that for all $\\theta \\in \\mathbb {R}^p$ and $h> 0$ , we have: $\\operatorname{\\mathbb {E}}\\left[\\Vert g_h(\\theta ,\\xi )\\Vert _{2}^2 \\right] \\le 2 \\mathcal {L} \\left( f(\\theta ) - f(\\theta ^\\star )\\right) + \\sigma ^2.$ This bound on the stochastic noise $ \\operatorname{\\mathbb {E}}\\left[ \\Vert g (\\theta ,\\xi ) \\Vert _{2}^2 \\right] $ is the key to prove the almost sure convergence of the algorithm.", "Note that Assumption REF is weak as it allows the noise to be large when the iterate is far away from the optimal point.", "In that aspect, it contrasts with uniform bounds of the form $\\operatorname{\\mathbb {E}}\\left[\\Vert g(\\theta ,\\xi )\\Vert _2^2 \\right] \\le \\sigma ^2$ for some deterministic $\\sigma ^2 >0$ [50], [51], [52].", "Observe that such uniform bound is recovered by taking $\\mathcal {L}=0$ in Assumption REF but cannot hold when the objective function $f$ is strongly convex [53].", "The standard Robbins-Monro condition, $\\sum _{t \\ge 1} \\gamma _{t} = +\\infty $ and $\\sum _{t \\ge 1} \\gamma _t ^2 < +\\infty $ is required in the next theorem.", "Theorem 1 (Almost sure convergence of SGD) Suppose that Assumptions REF to REF are fulfilled.", "If the learning rates satisfy the standard Robins-Monro and $h_t^2 = O( \\gamma _t) $ , then the sequence of iterates $(\\theta _t)_{t \\in \\mathbb {N}}$ defined in (REF ) converges almost surely towards the minimizer $ \\theta ^\\star $ , i.e., $\\theta _t \\rightarrow \\theta ^\\star $ as $t \\rightarrow +\\infty $ .", "The success of the proposed approach relies on the following restrictions between the learning rates sequence $(\\gamma _t)_{t \\in \\mathbb {N}}$ and the weights of the coordinate policy.", "This is formally stated in the following assumption, referred to as the extended Robbins-Monro condition.", "Denote by $\\beta _{t+1}$ the smallest probability weight at time $t$ , i.e., $\\beta _{t+1} = \\min _{1 \\le k \\le p} d_{t}^{(k)} .$ Assumption 5 (Extended Robbins-Monro condition) $ \\sum _{t \\ge 1} \\gamma _{t} \\beta _{t} = +\\infty $ and $ \\sum _{t \\ge 1} \\gamma _{t}^2 < +\\infty .$ From a practical point of view, those are not restrictive as they can always be implemented by the user.", "In the case $D_t = I_p$ , this is simply the standard Robbins-Monro condition.", "Theorem 2 (Almost sure convergence of CSGD) Suppose that Assumptions REF to REF are fulfilled.", "If the learning rates satisfy Assumption REF and $h_t^2 = O( \\gamma _t) $ , then the sequence of iterates $(\\theta _t)_{t \\in \\mathbb {N}}$ defined in (REF ) converges almost surely towards the minimizer $ \\theta ^\\star $ , i.e., $\\theta _t \\rightarrow \\theta ^\\star $ as $t \\rightarrow +\\infty $ .", "For a non-asymptotic analysis, we place ourselves under the Polyak–Łojasiewicz (PL) condition [54] which does not assume convexity of $f$ but retains many properties of strong convexity, e.g.", "the fact that every stationary point is a global minimum.", "Assumption 6 (PL inequality) There exists $\\mu >0$ s.t.", ": $\\forall \\theta \\in \\mathbb {R}^p, \\Vert \\nabla f(\\theta ) \\Vert _2^2 \\ge 2\\mu \\left( f(\\theta ) - f(\\theta ^\\star ) \\right).$ Similarly to [39], we introduce $\\varphi _\\alpha :\\mathbb {R}_+^\\star \\rightarrow \\mathbb {R}, \\varphi _\\alpha (t) = \\alpha ^{-1}(t^\\alpha - 1)$ if $\\alpha \\ne 0$ and $\\varphi _\\alpha (t) =\\log (t)$ if $\\alpha =0$ .", "Denoting $\\delta _t = \\operatorname{\\mathbb {E}}\\left[ f(\\theta _t) - f(\\theta ^\\star ) \\right]$ and assuming that $\\beta _{t+1} \\ge \\beta >0$ , one can obtain the recursion equation: $\\delta _t \\le \\left( 1 - 2\\mu \\beta \\gamma _t + L \\mathcal {L} \\gamma _t^2 \\right) \\delta _{t-1} + \\gamma _{t}^2 (\\sigma ^2 L + c^2)/2$ , leading to the following Theorem on non-asymptotic bounds for SCGD methods.", "Theorem 3 (Non-asymptotic bounds) Suppose that Assumptions REF to REF are fulfilled and let $(\\theta _t)_{t \\in \\mathbb {N}}$ defined in (REF ) with $\\gamma _t = \\gamma t^{-\\alpha }$ and $h_t = \\sqrt{\\gamma _t}$ .", "Denote by $\\delta _t = \\operatorname{\\mathbb {E}}\\left[ f(\\theta _t) - f(\\theta ^\\star ) \\right]$ and assume that there exists $\\beta >0$ such that $\\beta _{t+1} \\ge \\beta >0$ .", "We have for $\\alpha \\in [0,1]$ : • If $0 \\le \\alpha < 1$ then $\\delta _t \\le 2 \\exp \\left( 2 L \\mathcal {L} \\gamma ^2 \\varphi _{1-2\\alpha }(t)\\right) \\exp \\left(-\\frac{\\mu \\beta \\gamma }{4} t^{1-\\alpha }\\right) \\left( \\delta _0 + \\frac{\\sigma ^2+2c^2}{2 \\mathcal {L}} \\right) + \\frac{\\gamma (\\sigma ^2 L + 2c^2) }{\\mu \\beta } t^{-\\alpha }$ • If $\\alpha = 1$ then $\\delta _t \\le 2 \\exp \\left( L \\mathcal {L} \\gamma ^2\\right) \\left( \\delta _0 + \\frac{\\sigma ^2 + 2c^2}{2 \\mathcal {L}} \\right) t^{-\\mu \\beta \\gamma } + \\left(\\frac{\\sigma ^2L}{2}+c^2\\right) \\gamma ^2 \\varphi _{\\mu \\beta \\gamma /2 - 1}(t) t^{-\\mu \\beta \\gamma /2}$ Remark 4 (Importance weights) The conclusion of Theorem REF remains valid for the update rule $\\theta _ {t+1} = \\theta _ {t} - \\gamma _{t+1} W_t D(\\zeta _{t+1}) g _t$ where $W_t $ is a diagonal matrix with coefficients $ (w_{t}^{(1)},\\ldots , w_{t}^{(p)})$ such that $\\beta _{t+1} = \\min _{1 \\le k \\le p} w_{t}^{(k)}d_{t}^{(k)} $ .", "Remark 5 (Norms and constants) A quick inspection of the proof reveals that Assumptions REF and REF may be replaced respectively by: $\\forall \\theta \\in \\mathbb {R}^p,h>0$ , $\\Vert \\mathbb {E} _\\xi [ g_h (\\theta , \\xi ) ] - \\nabla f(\\theta ) \\Vert _\\infty \\le c h$ and $\\max _{k= 1,\\ldots , p} \\operatorname{\\mathbb {E}}[ g_h^{(k)} (\\theta ,\\xi ) ^2] \\le 2 \\mathcal {L} \\left( f(\\theta ) - f(\\theta ^\\star )\\right) + \\sigma ^2$ .", "Since $\\Vert \\cdot \\Vert _{\\infty }\\le \\Vert \\cdot \\Vert _2 \\le \\sqrt{p} \\Vert \\cdot \\Vert _{\\infty }$ , the above constant scales more efficiently with the dimension.", "Remark 6 (Rates) The optimal convergence rate in Theorem REF is of order $O(1/t)$ , obtained with $\\alpha =1$ under the condition $ \\mu \\beta \\gamma >2$ .", "Such rate matches optimal asymptotic minimax rate for stochastic approximation [55] and recovers the rate of [38] for SGD with biased gradients." ], [ "MUSKETEER Algorithm", "This section is dedicated to the algorithm MUSKETEER which performs an adaptive reweighting of the coordinate sampling probabilities to leverage the data structure.", "Note that this procedure is general and may be applied on top of any stochastic optimization algorithm as soon as one has acces to coordinates of a gradient estimate.", "The algorithm of interest alternates between two elementary blocks: one for the exploration phase and another one for the exploitation phase.", "Exploration phase.", "The goal of this phase is twofold: perform stochastic coordinate gradient descent and collect information about the noisy directions of the gradient.", "The former task is done using the current coordinate sampling distribution $Q(d_n)$ which is fixed during this phase whereas the latter is computed through cumulative gains.", "Exploitation phase.", "This phase is the cornerstone of the probability updates since it exploits the knowledge of the cumulative gains to update the coordinate sampling probability vector $d_{n}$ in order to sample more often the relevant directions of the optimization problem.", "[H] MUSKETEER linenodelimiter=.", "[1] $\\theta _0 \\in \\mathbb {R}^p$ , $N,T \\in \\mathbb {N}$ , $(\\gamma _t)_{t \\ge 0}, \\ (\\lambda _n)_{n \\ge 0}, \\ \\eta >0$ .", "Initialize probability weights $d_0 = (1/p,\\ldots ,1/p)$ blue// start with uniform sampling Initialize cumulative gains $G_0 = (0,\\ldots ,0)$ $n=0,\\ldots ,N-1$ Initialize current gain $\\widetilde{G}_0 = (0,\\ldots ,0)$ Run Explore$(T,d_n)$                             blue// to compute current gain $\\widetilde{G}_T$ Run Exploit$(G_n, \\widetilde{G}_T,\\lambda _n ,\\eta )$                    blue// to update weights $d_{n+1}$ Return final point $\\theta _N$ Consider a fixed iteration $n \\in \\mathbb {N}$ of MUSKETEER's main loop.", "The exploration phase may be seen as a multi-armed bandit problem [56] where the arms are the gradient coordinates for $k\\in \\llbracket 1,p \\rrbracket $ .", "At each time step $t\\in \\llbracket 1,T \\rrbracket $ , a coordinate $\\zeta $ is drawn according to $Q(d_n)$ and the relative gradient $g_t^{(\\zeta )}/d_{n}^{(\\zeta )}$ , representing the reward, is observed.", "Note that an importance sampling strategy is used to produce an unbiased estimate of the gradient when dealing with first order methods.", "The rewards are then used to build cumulative gains $\\widetilde{G}_{T}$ which can be written in a vectorized form as an empirical sum of the visited gradients during the exploration phase $ \\forall k \\in \\llbracket 1,p \\rrbracket , \\quad \\widetilde{G}_{T}^{(k)} = \\frac{1}{T} \\sum _{t=1}^{T} \\frac{g_t^{(k)}}{d_{n}^{(k)}} {1}_{\\lbrace \\zeta _{t+1} = k \\rbrace }, \\quad i.e.", "\\quad \\widetilde{G}_T = \\frac{1}{T} \\sum _{t=1}^{T} D_n^{-1} D(\\zeta _{t+1})g(\\theta _t,\\xi _{t+1}).$ This average reduces the noise induced by the gradient estimates but may be sign-dependent.", "Thus, one may rely on the following cumulative gains which are also considered in the experiments, $ \\widetilde{G}_T = \\frac{1}{T} \\sum _{t=1}^{T} D_n^{-1} D(\\zeta _{t+1}) |g(\\theta _t,\\xi _{t+1})| \\quad \\text{or} \\quad \\widetilde{G}_T = \\frac{1}{T} \\sum _{t=1}^{T} D_n^{-1} D(\\zeta _{t+1})g(\\theta _t,\\xi _{t+1})^2.$ Starting from $G_0=(0,\\ldots ,0)$ , the total gain $G_n$ is updated in a online manner during the exploitation phase using the update rule $G_{n+1} = G_{n} + (\\widetilde{G}_T-G_n)/(n+1).$ Once the average cumulative gains are computed, one needs to normalize them to obtain probability weights.", "Such normalization can be done by a natural $\\ell _1$ -reweighting or a softmax operator with a parameter $\\eta >0$ .", "To cover both cases, consider the normalizing function $\\varphi :\\mathbb {R}^p \\rightarrow \\mathbb {R}^p$ defined by $\\varphi (x)^{(k)} = |x^{(k)}|/\\sum _{j=1}^p |x^{(j)}|$ or $\\varphi (x)^{(k)} = \\exp (\\eta x^{(k)})/\\sum _{j=1}^p \\exp (\\eta x^{(j)}).$ Following the sequential approach of the EXP3 algorithm [56], [57], the probability weights are updated through a mixture between the normalized average cumulative gains $\\varphi (G_n)$ and a uniform distribution.", "The former term takes into account the knowledge of the gains by exploiting the rewards while the latter ensures exploration.", "Given a sequence $(\\lambda _n) \\in [0,1]^{\\mathbb {N}}$ , we have for all $k\\in \\llbracket 1,p \\rrbracket $ , $ d_{n+1}^{(k)} = (1-\\lambda _n) \\varphi (G_n)^{(k)} + \\lambda _n \\frac{1}{p}\\cdot $ [H] Explore$(T,d_n)$ linenodelimiter=.", "[1] $t=1,\\ldots ,T$ Sample coordinate $\\zeta \\sim Q(d_n)$ and data $\\xi $ Move iterate: $\\theta _{t+1}^{(\\zeta )} = \\theta _{t}^{(\\zeta )} - \\gamma _{t+1} g_h^{(\\zeta )}(\\theta _t,\\xi )$ Update gain $\\widetilde{G}_{t+1}^{(\\zeta )}$ using (REF ) or (REF ) Return vector of gains $\\widetilde{G}_{T}$ [H] Exploit$(G_n,\\widetilde{G}_{T},\\lambda _n, \\eta )$ linenodelimiter=.", "[1] Update total average gain $G_n$ in an online manner Compute normalized gains $\\varphi (G_n)$ with $\\ell _1$ -weights or softmax Update probability weights $d_{n+1}$ with the mixture of Eq.", "(REF ) As a corollary of Theorem REF , the sequence of iterates $(\\theta _t)_{t \\in \\mathbb {N}}$ obtained by MUSKETEER converges almost surely, i.e.", "$\\theta _t \\rightarrow \\theta ^\\star $ as $t \\rightarrow +\\infty $ .", "Since $\\nabla f$ is continuous, the gradients $\\nabla f(\\theta _t)$ get smaller through the iterations and the softmax weights get closer to $1/p$ .", "Thus, in the asymptotic regime, there is no favorable directions among all the possible gradient directions.", "Hence, near the optimum, the coordinate sampling policy of MUSKETEER is likely to treat all the coordinates equally.", "Theorem 4 (Weak convergence) Suppose that Assumptions REF to REF are fulfilled and that the learning rates satisfy the standard Robins-Monro condition.", "Then MUSKETEER's coordinate policy $(Q(d_n))_{n \\in \\mathbb {N}}$ with softmax normalization converges weakly to the uniform distribution, i.e., $Q(d_n) \\leadsto \\mathcal {U}(\\llbracket 1,p \\rrbracket )$ as $n \\rightarrow +\\infty $ .", "Remark 7 (On the choice of $\\lambda _n$ and $\\eta $ ) The uniform term in Equation (REF ) ensures that all coordinates are eventually visited.", "Taking $\\lambda _n \\rightarrow 0$ at a specific rate (which can be derived from the proof) gives more importance to the cumulative gains.", "The parameter $\\eta $ is fixed during the algorithm and may be tuned through an analysis of the regret [56].", "Remark 8 (Choice of Exploration Size $T$ ) Choosing the value of $T$ is a central question known as the exploration-exploitation dilemma in reinforcement learning.", "As $T$ gets large, the exploration phase gathers more information leading to fewer but more accurate updates.", "Conversely, with a small value of $T$ , the probabilities get updated more often, at the price of less collected information.", "Setting $T=p$ ensures that, in average, all the coordinates are visited once during the exploration phase.", "Nevertheless, a smaller value $T=\\lfloor \\sqrt{p} \\rfloor $ is taken in the experiments and lead to great performance.", "Remark 9 (Asymptotic behavior) The previous results highlight two main features of MUSKETEER: the sequence of iterates converges almost surely and the coordinate policy converges weakly.", "The latter point suggests that, in the long run, MUSKETEER is similar to the uniform coordinate version of SCGD.", "However, the weak convergence of the rescaled process $(\\theta _t - \\theta ^\\star ) / \\sqrt{ \\gamma _t}$ remains an open question.", "In light of the link between SCGD and Conditioned-SGD, discussed in Section REF , we conjecture that the behavior of MUSKETEER is asymptotically equivalent to SCGD with uniform policy.", "This is in line with the continuity property obtained in [58] within the Conditioned-SGD framework and relates to the convergence of stochastic Newton algorithms [59]." ], [ "Numerical Experiments", "In this section, we empirically validate the SCGD framework by running MUSKETEER and competitors on synthetic and real datasets.", "First, we focus on regularized regression problems adopting the data generation process of [21] in which the covariates exhibit a certain block structure.", "Second, MUSKETEER is employed to train different neural networks models on real datasets for multi-label classification task.", "For ease of reproducibility, the code, technical details and additional results (with different data settings, normalization and hyperparameters) are available in the appendix.", "Methods in competition.", "The set of methods is restricted to ZO methods.", "This choice leads to an honest comparison based on the number of function queries.", "MUSKETEER is implemented according to Section with $T = \\lfloor \\sqrt{p} \\rfloor $ , softmax and $\\ell _1$ normalization for the simulated and real data respectively.", "The different cumulative gains of Eq.", "(REF ) are considered, namely AVG, SQR and ABS for the gradients, their squares or their absolute value respectively.", "The method FULL is the finite difference gradient estimate computed over all coordinates and UNIFORM stands for the uniform coordinate sampling policy.", "NESTEROV implements the gaussian smoothing of [27].", "In all cases, $\\theta _0 = (0,\\ldots , 0)^T \\in \\mathbb {R}^p$ and the optimal SGD learning rate of the form $\\gamma _k = \\gamma /(k+k_0)$ is used.", "Regularized linear models.", "Consider the objective $f(\\theta ) = (1/n) \\sum _{i=1}^n f_i(\\theta ) + \\mu \\Vert \\theta \\Vert ^2$ .", "Given a data matrix $X=(x_{i,j}) \\in \\mathbb {R}^{n \\times p}$ and labels $y \\in \\mathbb {R}^n$ or $\\lbrace -1,+1\\rbrace ^n$ , the Ridge regression problem and $\\ell _2$-regularized logistic regression are respectively given by $f_i(\\theta ) = (y_i - \\sum _{j=1}^p x_{i,j} \\theta _j)^2$ and $f_i(\\theta ) = \\log (1+\\exp (-y_i \\sum _{j=1}^p x_{i,j} \\theta _j)$ .", "Similarly to [21], we endow the data matrix $X$ with a block structure.", "The columns are drawn as $X[:,k] \\sim \\mathcal {N}(0,\\sigma _k^2 I_n)$ with $\\sigma _k^2 = k^{-\\alpha }$ for all $k\\in \\llbracket 1,p \\rrbracket $ .", "The parameters are set to $n=10,000$ samples in dimension $p=250$ with an exploration size equal to $T = \\lfloor \\sqrt{p} \\rfloor = 15$ .", "The regularization parameter is set to the classical value $\\mu =1/n$ .", "Figure REF provides the graphs of the optimaliy gap $t\\mapsto f(\\theta _t)-f(\\theta ^\\star )$ averaged over 20 independent simulations for different values of $\\alpha \\in \\lbrace 2;5;10\\rbrace $ .", "First, note that the uniform sampling strategy shows similar performance to the classical full gradient estimate.", "Besides, MUSKETEER with average or absolute gains shows the best performance in all configurations.", "Greater values of $\\alpha $ , i.e.", "stronger block structure, improve our relative performance with respect to the other methods as shown by Figures REF and REF .", "Neural Networks.", "We focus on the training of neural networks within the framework of multi-label classification.", "The datasets in the experiments are popular publicly available deep learning datasets: MNIST [60] and Fashion-MNIST [61].", "Given an image, the goal is to predict its label among ten classes.", "The neural architecture is based on linear layers in dimension $p=55,050$ with $T = 234$ .", "Figure REF shows the means and standard deviations of the training losses of the different ZO methods averaged over 10 independent runs.", "Interestingly, the performance of MUSKETEER also benefit from the adaptive structure in terms on accuracy of the test set (see Figures REF and REF ).", "This allows to quantify the statistical gain brought by MUSKETEER over standard ZO methods.", "Figure: Logistic α=5\\alpha =5Figure: Evolution of test accuracy.Appendix: SGD with Coordinate Sampling: Theory and Practice Appendix collects the technical proofs of the main results and Appendix presents additional results.", "Appendix shows some illustrative 2d-examples.", "Appendix gathers additional details about the experimental protocols and the code.", "Appendix presents experiments in which stochastic first order estimates are used.", "Finally, Appendices and present additional experiments in various settings for zeroth order and stochastic first order methods respectively.", "[sections] [sections]l1" ], [ "Proof of Proposition ", "Under Eq.", "(REF ), using Jensen inequality, we find $\\Vert \\operatorname{\\mathbb {E}}_{\\xi }[ g_\\mu (\\theta ,\\xi ) ] - \\nabla f (\\theta ) \\Vert _2^2& = \\left\\Vert \\int _{\\mathbb {R}^p} x \\left( \\frac{f(\\theta + \\mu x ) - f(\\theta )}{\\mu } - x^T \\nabla f (\\theta ) \\right) \\nu (\\mathrm {d}x)\\right\\Vert _2^2\\\\&\\le \\int _{\\mathbb {R}^p} \\Vert x\\Vert ^2_2 \\left( \\frac{f(\\theta + \\mu x ) - f(\\theta )}{\\mu } - x^T \\nabla f (\\theta ) \\right)^2 \\nu (\\mathrm {d}x)\\\\&= \\mu ^{-2} \\int _{\\mathbb {R}^p} \\Vert x\\Vert ^2_2 \\left( {f(\\theta + \\mu x ) - f(\\theta )} - \\mu x^T \\nabla f (\\theta ) \\right)^2 \\nu (\\mathrm {d}x)$ Using the quadratic bound of $L$ -smooth functions, we obtain $\\Vert \\operatorname{\\mathbb {E}}_{\\xi }[ g_\\mu (\\theta ,\\xi ) ] - \\nabla f (\\theta ) \\Vert _2^2& \\le \\mu ^{-2} \\frac{L^2}{4} \\int \\Vert x\\Vert _2^2 \\Vert \\mu x \\Vert _2^4\\nu (\\mathrm {d}x)= \\mu ^{2} \\frac{L^2}{4} \\int \\Vert x\\Vert _2^6 \\nu (\\mathrm {d}x).$" ], [ "Proof of Theorem ", "We classically rely on the Robbins-Siegmund Theorem (Theorem REF in Section REF ).", "Since $\\theta \\mapsto f(\\theta )$ is $L$ -smooth, we have the quadratic bound $f(\\eta ) \\le f(\\theta ) + \\langle \\nabla f(\\theta ),\\eta -\\theta \\rangle + \\frac{L}{2} \\Vert \\eta -\\theta \\Vert _2^2$ .", "Using the update rule $\\theta _{t+1} = \\theta _{t} - \\gamma _{t+1} g_t$ , we get $f(\\theta _{t+1})&\\le f(\\theta _{t}) + \\langle \\nabla f(\\theta _{t}),\\theta _{t+1}-\\theta _{t} \\rangle + \\frac{L}{2}\\Vert \\theta _{t+1}-\\theta _{t}\\Vert _2^2 \\\\&= f(\\theta _{t}) - \\gamma _{t+1} \\langle \\nabla f(\\theta _{t}), g_t \\rangle + \\frac{L}{2} \\gamma _{t+1}^2\\Vert g_t\\Vert _2^2.$ Using that $2\\langle a , b \\rangle = \\Vert a\\Vert _2 ^2 + \\Vert b\\Vert _2 ^2 - \\Vert a-b\\Vert _2 ^2 &\\ge \\Vert a\\Vert _2 ^2 - \\Vert a-b\\Vert _2 ^2$ and taking the conditional expectation, we get $&\\operatorname{\\mathbb {E}}_t\\left[f(\\theta _{t+1})-f(\\theta ^\\star )\\right] \\\\&\\le f(\\theta _{t}) -f(\\theta ^\\star ) - \\gamma _{t+1}\\langle \\nabla f(\\theta _{t}), \\operatorname{\\mathbb {E}}_t[ g_t] \\rangle + \\frac{L}{2} \\gamma _{t+1}^2\\operatorname{\\mathbb {E}}_t[\\Vert g_t\\Vert _2^2]\\\\& \\le f(\\theta _{t}) -f(\\theta ^\\star ) - \\frac{\\gamma _{t+1}}{2} \\Vert \\nabla f(\\theta _{t})\\Vert _2^2 + \\frac{\\gamma _{t+1}}{2} \\Vert \\nabla f(\\theta _{t}) - \\operatorname{\\mathbb {E}}_t[ g_t] \\Vert _2^2 + \\frac{L}{2} \\gamma _{t+1}^2\\operatorname{\\mathbb {E}}_t[\\Vert g_t\\Vert _2^2]$ On the one hand, using Assumption REF , we obtain $\\Vert \\nabla f(\\theta _{t}) - \\operatorname{\\mathbb {E}}_t[ g_t] \\Vert _2^2 &\\le h_{t+1}^{2} c^2$ On the other hand, using Assumption REF , there exist $0 \\le \\mathcal {L},\\sigma ^2 <\\infty $ such that almost surely $\\forall t \\in \\mathbb {N}, \\quad \\operatorname{\\mathbb {E}}_t\\left[\\Vert g_t\\Vert _2^2\\right] =\\operatorname{\\mathbb {E}}_\\xi \\left[\\Vert g(\\theta _t, \\xi ) \\Vert _2^2 \\right] \\le 2 \\mathcal {L} \\left( f(\\theta _{t}) - f(\\theta ^{\\star })\\right) + \\sigma ^2.$ It follows that $&\\operatorname{\\mathbb {E}}_t\\left[f(\\theta _{t+1})-f(\\theta ^\\star )\\right] \\\\&\\le (1 + L \\mathcal {L} \\gamma _{t+1}^2) ( f(\\theta _{t}) -f(\\theta ^\\star )) - \\frac{\\gamma _{t+1}}{2}\\Vert \\nabla f(\\theta _{t})\\Vert _2^2 +\\gamma _{t+1} h_{t+1} ^2 c^2 + \\frac{L}{2} \\gamma _{t+1}^2 \\sigma ^2$ Introduce $V_t = f(\\theta _{t})-f(\\theta ^{\\star }), W_t = \\gamma _{t+1} \\Vert \\nabla f(\\theta _{t})\\Vert _2^2 /2 $ , $a_t = L \\mathcal {L} \\gamma _{t+1}^2$ and $b_t = c^2 h_{t+1} ^2 \\gamma _{t+1} + (L/2) \\gamma _{t+1} ^2 \\sigma ^2 $ .", "These four random sequences are non-negative $\\mathcal {F}_t$ -measurable sequences with $\\sum _t a_t <\\infty $ and $\\sum _t b_t <\\infty $ almost surely.", "Moreover we have $\\forall t \\in \\mathbb {N}, \\quad \\operatorname{\\mathbb {E}}\\left[V_{t+1}|\\mathcal {F}_t\\right] \\le (1+a_t) V_{t} - W_t + b_t.$ We can apply Robbins-Siegmund Theorem to have $(a) \\ \\sum _{t \\ge 0} W_t <\\infty \\ a.s. \\qquad (b) \\ V_{t} \\stackrel{a.s.}{\\longrightarrow } V_{\\infty }, \\operatorname{\\mathbb {E}}\\left[V_{\\infty }\\right]<\\infty .", "\\qquad (c) \\ \\sup _{t \\ge 0} \\operatorname{\\mathbb {E}}\\left[V_{t}\\right]<\\infty .$ Therefore we have a.s. that $(f(\\theta _{t}))$ converges to a finite value $f_{\\infty } \\in L^1$ and $\\sum _{t \\ge 0} \\gamma _{t+1} \\Vert \\nabla f(\\theta _{t})\\Vert _2^2 <+\\infty $ .", "There exists an event $\\Omega _0 \\subset \\Omega $ such that, $\\operatorname{\\mathbb {P}}(\\Omega _0)=1$ and for every $\\omega \\in \\Omega _0$ , $\\limsup _{t} f(\\theta _t(\\omega )) < \\infty $ and the series $\\sum _t \\gamma _{t+1} \\Vert \\nabla f(\\theta _t(\\omega )) \\Vert _2^2$ converges.", "Since $\\lim _{\\Vert \\theta \\Vert \\rightarrow \\infty } f(\\theta ) = \\infty $ , we deduce that for every $\\omega \\in \\Omega _0$ , the sequence $(\\theta _t(\\omega ))_{t \\ge 0}$ is bounded in $\\mathbb {R}^p$ .", "Therefore the limit set $\\chi _{\\infty }(\\omega )$ (set of accumulation points) of the sequence $(\\theta _t(\\omega ))$ is non-empty.", "The convergence of the series $\\sum _t \\gamma _{t+1} \\Vert \\nabla f(\\theta _t(\\omega )) \\Vert _2^2 < \\infty $ along with the condition $\\sum _t \\gamma _{t+1} = +\\infty $ only implie that $\\liminf _{t \\rightarrow \\infty } \\Vert \\nabla f(\\theta _{t}(\\omega ))\\Vert _2^2 = 0, \\quad \\operatorname{\\mathbb {P}}-a.s.$ Hence, since $\\theta \\mapsto \\nabla f(\\theta )$ is continuous, there exits a limit point $\\theta _{\\infty }(\\omega ) \\in \\chi _{\\infty }(\\omega )$ such that $\\Vert \\nabla f(\\theta _{\\infty }(\\omega ))\\Vert _2^2 = 0$ , i.e., $\\nabla f(\\theta _{\\infty }(\\omega ))=0$ .", "Because the set of solutions $\\lbrace \\theta \\in \\mathbb {R}^p, \\nabla f(\\theta ) = 0 \\rbrace $ is reduced to the singleton $\\lbrace \\theta ^{\\star } \\rbrace $ , we have $\\theta _{\\infty }(\\omega ) = \\theta ^\\star $ .", "Since $(f(\\theta _t(\\omega )))$ converges, it implies that $\\lim _t f(\\theta _t(\\omega )) = f(\\theta ^{\\star })$ and for every limit point $x \\in \\chi _{\\infty }(\\omega )$ , we have $f(\\theta ) = f(\\theta ^{\\star })$ .", "Since the set $\\lbrace \\theta \\in \\mathbb {R}^p, f(\\theta ) = f(\\theta ^{\\star }) \\rbrace $ is equal to $\\lbrace \\theta ^\\star \\rbrace $ , the limit set $\\chi _{\\infty }(\\omega )$ is also reduced to $\\lbrace \\theta ^{\\star } \\rbrace $ ." ], [ "Proof of Theorem ", "We start by following the proof of Theorem REF .", "We write $f(\\theta _{t+1})&\\le f(\\theta _{t}) - \\gamma _{t+1} \\langle \\nabla f(\\theta _{t}), D(\\zeta _{t+1}) g_t \\rangle + \\frac{L}{2} \\gamma _{t+1}^2\\Vert D(\\zeta _{t+1}) g_t\\Vert _2^2\\\\& = f(\\theta _{t}) - \\gamma _{t+1} \\nabla _{\\zeta _{t+1}} f(\\theta _{t}) g_t^{(\\zeta _{t+1})} + \\frac{L}{2} \\gamma _{t+1}^2 g_t^{(\\zeta _{t+1})2}$ Taking the expectation with respect to $\\xi _{t+1}$ and using Assumption REF , we find $\\mathbb {E}_ { \\xi _{t+1}} [ f(\\theta _{t+1}) - f(\\theta ^\\star ) ]& \\le f(\\theta _{t}) - f(\\theta ^\\star ) - \\gamma _{t+1} \\nabla _{\\zeta _{t+1}} f(\\theta _{t}) \\tilde{g}_t ^{(\\zeta _{t+1})} + \\frac{L}{2} \\gamma _{t+1}^2 \\mathbb {E}_ { \\xi _{t+1}} [ g_t^{(\\zeta _{t+1})2}]$ where $\\tilde{g}_t = E_ { \\xi } [g _{h_{t+1}} ( \\theta _t, \\xi ) ] $ .", "We use the inequality $2ab \\ge a^2 - (a-b)^2 $ and Assumption REF to get $2 \\nabla _{\\zeta _{t+1}} f(\\theta _{t}) \\tilde{g}_t^{(\\zeta _{t+1})}&\\ge \\nabla _{\\zeta _{t+1}} f(\\theta _{t}) ^2- ( \\nabla _{\\zeta _{t+1}} f(\\theta _{t}) - \\tilde{g}_t^{(\\zeta _{t+1})} )^2\\\\&\\ge \\nabla _{\\zeta _{t+1}} f(\\theta _{t}) ^2- \\max _{k=1,\\ldots , p} ( \\nabla _{k} f(\\theta _{t}) - \\tilde{g}_t^{(k)} )^2\\\\& \\ge \\nabla _{\\zeta _{t+1}} f(\\theta _{t}) ^2-c^2 h_{t+1} ^2$ We also have, invoking Assumption REF , that $ \\mathbb {E}_ { \\xi _{t+1}} [ g_t^{(\\zeta _{t+1})2}] \\le \\max _{k=1,\\ldots , p} \\mathbb {E}_ { \\xi _{t+1}} [ g_t^{(k)2}] \\le 2 \\mathcal {L} ( f(\\theta _{t}) - f(\\theta ^\\star ) ) + \\sigma ^2.", "$ We finally obtain that $&\\mathbb {E}_ { \\xi _{t+1}} [ f(\\theta _{t+1}) - f(\\theta ^\\star ) ] \\\\& \\le (1 + L \\mathcal {L} \\gamma _{t+1}^2) ( f(\\theta _{t}) - f(\\theta ^\\star )) - \\gamma _{t+1} \\nabla _{\\zeta _{t+1}} f(\\theta _{t}) ^2 /2 + c^2 \\gamma _{t+1} h_{t+1} ^2 /2+\\frac{L}{2} \\gamma _{t+1}^2 \\sigma ^2$ It remains to take the expectation with respect to $\\zeta _{t+1}$ to get $&\\mathbb {E}_ { {t}} [ f(\\theta _{t+1}) - f(\\theta ^\\star ) ] \\\\& \\le (1 + L \\mathcal {L} \\gamma _{t+1}^2) ( f(\\theta _{t}) - f(\\theta ^\\star )) - \\gamma _{t+1} \\sum _{k=1} ^p d_{t,k} \\nabla _{k} f(\\theta _{t}) ^2 /2 + c^2 \\gamma _{t+1} h_{t+1} ^2 /2+\\frac{L}{2} \\gamma _{t+1}^2 \\sigma ^2 \\\\&\\le (1 + L \\mathcal {L} \\gamma _{t+1}^2) ( f(\\theta _{t}) - f(\\theta ^\\star )) - \\gamma _{t+1}\\beta _{t+1} \\Vert \\nabla f(\\theta _t) \\Vert _2^2 /2 + c^2 \\gamma _{t+1} h_{t+1} ^2 /2+\\frac{L}{2} \\gamma _{t+1}^2 \\sigma ^2$ In a similar way as in the proof of Theorem REF , since $\\sum _{t \\ge 0} \\gamma _{t+1}\\beta _{t+1} = +\\infty $ , we conclude making use of the Robbins-Siegmund Theorem." ], [ "Proof of Theorem ", "From the proof of Theorem REF and using $\\beta $ as a uniform lower bound on $\\beta _{t+1}$ , we have $\\operatorname{\\mathbb {E}}_t\\left[f(\\theta _{t+1})-f(\\theta ^{\\star })\\right] \\le \\left(1 + L \\mathcal {L} \\gamma _{t+1}^2 \\right)\\left[f(\\theta _{t})-f(\\theta ^{\\star })\\right] - \\gamma _{t+1} \\beta \\Vert \\nabla f(\\theta _{t})\\Vert _2^2 + \\frac{\\sigma ^2 L + c^2}{2} \\gamma _{t+1}^2.$ Inject the PL inequality $\\Vert \\nabla f(\\theta _{t})\\Vert _2^2\\ge 2\\mu (f(\\theta _t) - f(\\theta ^*) )$ from Assumption REF to have $\\operatorname{\\mathbb {E}}_t\\left[f(\\theta _{t+1})-f(\\theta ^{\\star })\\right] \\le \\left(1 - 2\\mu \\beta \\gamma _{t+1} + L \\mathcal {L} \\gamma _{t+1}^2 \\right)\\left[f(\\theta _{t})-f(\\theta ^{\\star })\\right] + \\frac{\\sigma ^2 L + c^2}{2} \\gamma _{t+1}^2.$ Define $\\delta _t = \\operatorname{\\mathbb {E}}\\left[ f(\\theta _t) - f(\\theta ^\\star ) \\right]$ to finally obtain the recursion equation $\\delta _t \\le \\left( 1 - 2\\mu \\beta \\gamma _t + L \\mathcal {L} \\gamma _t^2 \\right) \\delta _{t-1} + \\frac{\\sigma ^2 L + c^2}{2} \\gamma _{t}^2$ Applying the same result from [39] with the family of functions $\\varphi _\\alpha $ defined by $\\varphi _\\alpha (t) = \\alpha ^{-1}(t^\\alpha - 1)$ if $\\alpha \\ne 0$ and $\\varphi _\\alpha (t) =\\log (t)$ if $\\alpha =0$ along with the learning rates $\\gamma _t = \\gamma t^{-\\alpha }$ .", "$\\delta _t \\le \\left\\lbrace \\begin{array}{ll}2 \\exp \\left( 2 L \\mathcal {L} \\gamma ^2 \\varphi _{1-2\\alpha }(t)\\right) \\exp \\left(-\\frac{\\mu \\beta \\gamma }{4} t^{1-\\alpha }\\right) \\left( \\delta _0 + \\frac{\\sigma ^2+2c^2}{2 \\mathcal {L}} \\right) + \\frac{\\gamma (\\sigma ^2 L + 2c^2) }{\\mu \\beta } t^{-\\alpha } &\\text{ if } \\alpha < 1 \\\\2 \\exp \\left( L \\mathcal {L} \\gamma ^2\\right) \\left( \\delta _0 + \\frac{\\sigma ^2 + 2c^2}{2 \\mathcal {L}} \\right) t^{-\\mu \\beta \\gamma } + \\left(\\frac{\\sigma ^2L}{2}+c^2\\right) \\gamma ^2 \\varphi _{\\mu \\beta \\gamma /2 - 1}(t) t^{-\\mu \\beta \\gamma /2} &\\text{ if } \\alpha =1\\end{array}\\right.$" ], [ "Proof of Theorem ", "Starting from $G_0=(0,\\ldots ,0)$ , the total average gain $G_n$ is updated in a online manner during the exploitation phase and collects all the empirical sums of the gradient gradient estimates as $ G_n = \\frac{1}{nT}\\sum _{t=1}^{nT} D_t^{-1} D(\\zeta _{t+1})g(\\theta _t,\\xi _{t+1}), \\qquad \\operatorname{\\mathbb {E}}\\left[G_n \\right] = \\frac{1}{nT}\\sum _{t=1}^{nT} \\nabla f(\\theta _t).$ The goal is to show that $G_n \\rightarrow 0$ using martingale properties.", "Thanks to Theorem REF , we have the almost sure convergence $\\theta _t \\rightarrow \\theta ^*$ which gives, since $\\theta \\mapsto \\nabla f(\\theta )$ is continuous, that $\\nabla f(\\theta _t) \\rightarrow 0$ almost surely.", "Applying Cesaro's Lemma, it holds that $\\operatorname{\\mathbb {E}}\\left[G_n \\right] \\rightarrow 0$ .", "It is enough to consider the difference $\\left(G_n^{(k)} - \\operatorname{\\mathbb {E}}\\left[G_n^{(k)} \\right] \\right)$ for each $k\\in \\llbracket 1,p \\rrbracket $ .", "Introducing the martingale increments $\\Delta _ {t+1}^{(k)} = \\frac{ g (\\theta _t , \\xi _{t+1} )^{(k)} }{d_t^{(k)}} 1 _{\\lbrace \\zeta _{t+1} = k\\rbrace } - \\partial _ k f(\\theta _t) , \\qquad \\operatorname{\\mathbb {E}}\\left[\\Delta _ {t+1}^{(k)} | \\mathcal {F}_t \\right]=0.$ It remains to show that, with probability 1, $G_n^{(k)} - \\operatorname{\\mathbb {E}}\\left[G_n ^{(k)} \\right] = \\frac{1}{nT} \\sum _{t=1}^{nT} \\Delta _ {t+1} ^{(k)} \\rightarrow 0 .$ Or equivalently, that, for each coordinate $k\\in \\llbracket 1,p \\rrbracket $ $\\sum _{t=1}^{nT} \\Delta _ {t+1}^{(k)} = o (n) .$ The latter being a sum of martingale increments, we are in position to apply the strong law of large numbers for martingales which can be find as Assertion 2 of Theorem 1.18 in [62].", "Using Assumption REF , there exist $0 \\le \\mathcal {L},\\sigma ^2 <\\infty $ such that almost surely $\\forall t \\in \\mathbb {N}, \\quad \\operatorname{\\mathbb {E}}\\left[ (g(\\theta _t,\\xi _{t+1} )^{(k)} ) ^2 | \\mathcal {F}_t \\right]\\le 2 \\mathcal {L} \\left( f(\\theta _{t}) - f(\\theta ^{\\star })\\right) + \\sigma ^2.$ Using the almost sure convergence $\\theta _t \\rightarrow \\theta ^\\star $ , we deduce that there is exist a compact set $K$ which contains the sequence of iterates $(\\theta _t)_{t \\in \\mathbb {N}}$ and using that $f$ is continuous gives the upper bound $\\forall k\\in \\llbracket 1,p \\rrbracket \\quad \\operatorname{\\mathbb {E}}\\left[ (g(\\theta _t,\\xi _{t+1})^{(k)})^2 | \\mathcal {F}_t \\right] \\le M = 2\\mathcal {L} \\sup _{\\theta \\in K} (f(\\theta ) - f(\\theta ^*) ) + \\sigma ^2.$ Hence, the quadratic variation is bounded as follows $\\sum _{t=1}^{nT} \\operatorname{\\mathbb {E}}\\left[ (\\Delta _ {t+1}^{(k)})^2 | \\mathcal {F}_{t} \\right]&\\le \\sum _{t=1}^{nT} \\operatorname{\\mathbb {E}}\\left[ \\left( \\frac{ g(\\theta _t,\\xi _{t+1})^{(k)} }{ d_t^{(k)} }\\right) ^2 | \\mathcal {F}_{t} \\right] \\\\&\\le (p / \\lambda )^2 \\sum _{t=1}^{nT} \\operatorname{\\mathbb {E}}[ (g(\\theta _t,\\xi _{t+1})^{(k)})^2 | \\mathcal {F}_{t} ] \\\\&\\le (p / \\lambda )^2 {nT} M .$ Equation (REF ) follows from applying the previously mentioned law of large number." ], [ "Almost sure convergence of MUSKETEER ", "By definition, we have for all $k\\in \\llbracket 1,p \\rrbracket $ , $d_{t+1}^{(k)} = (1-\\lambda _t) \\varphi (G_t)^{(k)} + \\lambda _t \\frac{1}{p}$ implying that $ \\beta _{t+1} = \\min _{k\\in \\llbracket 1,p \\rrbracket } d_{t}^{(k)} \\ge \\lambda _t / p$ .", "As a consequence, as soon as $\\sum _{t\\ge 1} \\lambda _t \\gamma _t = +\\infty $ , the assumption $\\sum _{t\\ge 1} \\beta _t \\gamma _t = +\\infty $ is satisfied.", "Applying Theorem REF we obtain the almost sure convergence of MUSKETEER.", "The condition $\\sum _{t\\ge 1} \\lambda _t \\gamma _t = +\\infty $ is easily satisfied with a fixed value $\\lambda _t \\equiv \\lambda $ in the mixture update and one can also use a slowly decreasing sequence, e.g.", "$\\lambda _t = 1/\\log (t)$ ." ], [ "Regret analysis in the convex case", "Assume that the objective $f$ is convex and consider the average estimate $\\bar{\\theta }_T = \\frac{1}{T} \\sum _{t=1}^T \\theta _t$ .", "To analyze the benefits of using MUSKETEER over uniform coordinate sampling, we rely on a regret analysis with the following quantity: $S(f,\\hat{\\theta }) = \\operatorname{\\mathbb {E}}[f(\\hat{\\theta })] - f(\\theta ^\\star )$ .", "Using convexity we have on the one hand $f(\\theta _t)-f(\\theta ^\\star ) \\le \\langle \\theta _t - \\theta ^\\star , \\nabla f(\\theta _t) \\rangle $ and on the other hand $f(\\bar{\\theta }_T)-f(\\theta ^\\star ) \\le \\frac{1}{T} \\sum _{t=1}^T \\left(f(\\theta _t)-f(\\theta ^\\star ) \\right)$ which give together the following upper bound $f(\\bar{\\theta }_T)-f(\\theta ^\\star ) \\le \\frac{1}{T} \\sum _{t=1}^T \\langle \\theta _t - \\theta ^\\star , \\nabla f(\\theta _t) \\rangle .$ Using an unbiased gradient estimate $v_t$ , i.e.", "$\\operatorname{\\mathbb {E}}_t[v_t]=\\nabla f(\\theta _t)$ , we can write $\\operatorname{\\mathbb {E}}[f(\\bar{\\theta }_T)]-f(\\theta ^\\star ) \\le \\operatorname{\\mathbb {E}}\\left[\\frac{1}{T} \\sum _{t=1}^T \\langle \\theta _t - \\theta ^\\star , \\operatorname{\\mathbb {E}}_t[v_t]) \\rangle \\right].$ The term in the expectation is bounded using Lemma REF with $v_t = D_t^{-1} D(\\zeta _{t+1})g_t$ as $\\frac{1}{T}\\sum _{t=1}^T \\langle \\theta _t-\\theta ^\\star ,v_t \\rangle \\le \\frac{\\Vert \\theta ^\\star \\Vert ^2}{2 \\gamma T} + \\frac{\\gamma }{2T} \\sum _{t=1}^T \\Vert D_t^{-1} D(\\zeta _{t+1})g_t\\Vert ^2.$ Take the expectation on both side to control the regret as $S(f,\\bar{\\theta }_T) \\le \\frac{\\Vert \\theta ^\\star \\Vert ^2}{2 \\gamma T} + \\frac{\\gamma }{2T} \\sum _{t=1}^T \\operatorname{\\mathbb {E}}\\left[ \\sum _{k=1}^p \\frac{|\\partial _k f(\\theta _t)|^2}{d_{t}^{(k)}}\\right].$ The term in expectation should be minimized with respect to the probability weights $d_{t}^{(k)}$ .", "Intuitively, in order to maintain the overall sum as small as possible, the large gradient coordinates should be sampled more often, i.e.", "we would like to have $d_{t}^{(k)}$ large whenever $|\\partial _k f(\\theta _t)|^2$ is large.", "(Uniform Coordinate Sampling) For all $k \\in \\llbracket 1,p \\rrbracket $ , we have $d_{t}^{(k)} = 1/p$ so that $\\frac{1}{T} \\sum _{t=1}^T \\operatorname{\\mathbb {E}}\\left[ \\sum _{k=1}^p \\frac{|\\partial _k f(\\theta _t)|^2}{d_{t}^{(k)}}\\right] = \\frac{p}{T} \\sum _{t=1}^T \\operatorname{\\mathbb {E}}\\left[ \\sum _{k=1}^p |\\partial _k f(\\theta _t)|^2\\right] = \\frac{p}{T} \\sum _{t=1}^T \\operatorname{\\mathbb {E}}\\left[ \\Vert \\nabla f(\\theta _t)\\Vert ^2\\right].$ (MUSKETEER) For all $k \\in \\llbracket 1,p \\rrbracket $ , we have $d_{t}^{(k)} = (1-\\lambda _{t-1})\\varphi (G_{t-1})^{(k)} + \\lambda _{t-1}/p$ so that $\\frac{1}{T} \\sum _{t=1}^T \\operatorname{\\mathbb {E}}\\left[ \\sum _{k=1}^p \\frac{|\\partial _k f(\\theta _t)|^2}{d_{t}^{(k)}}\\right] = \\frac{p}{T} \\sum _{t=1}^T \\operatorname{\\mathbb {E}}\\left[ \\sum _{k=1}^p \\frac{|\\partial _k f(\\theta _t)|^2}{(1-\\lambda _{t-1})p\\varphi (G_{t-1})^{(k)} + \\lambda _{t-1}}\\right],$ where the denominator is stricly larger than 1 for all the coordinates associated to large gains.", "Indeed, let $k \\in \\llbracket 1,p \\rrbracket $ the index of such coordinate.", "Since it is a rewarding coordinate, the normalizing step implies that $\\varphi (G_{t-1})^{(k)} > 1/p$ and $(1-\\lambda _{t-1})p\\varphi (G_{t-1})^{(k)} + \\lambda _{t-1} > 1$ .", "This property translates the adaptive nature of the probability weights used in the MUSKETEER strategy." ], [ "Auxiliary Results", "Theorem 5 [63] Consider a filtration $\\left(\\mathcal {F}_{n}\\right)_{n \\ge 0}$ and four sequences of random variables$\\left(V_{n}\\right)_{n \\ge 0},\\left(W_{n}\\right)_{n \\ge 0}, \\left(a_{n}\\right)_{n \\ge 0}$ and $\\left(b_{n}\\right)_{n \\ge 0}$ that are adapted and non-negative.", "Assume that almost surely $\\sum _{k} a_k <\\infty $ and $\\sum _{k} b_k <\\infty $ .", "Assume moreover that $\\operatorname{\\mathbb {E}}\\left[V_0\\right] < \\infty $ and $\\forall n \\in \\mathbb {N}: \\operatorname{\\mathbb {E}}[V_{n+1} | \\mathcal {F}_n ] \\le (1 + a_n) V_{n} - W_{n} + b_{n}.$ Then it holds $(a) \\ \\sum _{k} W_k <\\infty \\ a.s. \\qquad (b) \\ V_{n} \\stackrel{a.s.}{\\longrightarrow } V_{\\infty }, \\operatorname{\\mathbb {E}}\\left[V_{\\infty }\\right]<\\infty .", "\\qquad (c) \\ \\sup _{n \\ge 0} \\operatorname{\\mathbb {E}}\\left[V_{n}\\right]<\\infty .$ The idea of the proof is to build a non-negative super-martingale to obtain the almost sure convergence towards an $L^1$ random variable.", "Introduce $\\pi _n = \\prod _{k=1}^n (1+a_k)^{-1}, \\pi _0 = 1$ and let us prove that $(\\pi _n)$ converges almost surely to $\\pi _{\\infty } \\in (0,1]$ .", "By definition, the sequence $(\\pi _n)$ is decreasing and in virtue of $1+x \\le \\exp (x)$ we have that $\\log (\\pi _n) \\ge - \\sum _{k=1}^n a_k \\ge -\\sum _{k=1}^{\\infty } a_k$ so $(\\pi _n)$ is lower bounded, hence converges.", "Since $\\exp \\left(-\\sum _{k=1}^{\\infty } a_k\\right) \\le \\pi _n \\le 1$ , we have $\\pi _{\\infty } \\in (0,1]$ .", "We define the modified random variables $\\widetilde{V}_n = \\pi _{n-1} V_n, \\quad \\widetilde{b}_n = \\pi _n b_n, \\quad \\widetilde{W}_n = \\pi _n W_n, \\quad S_n = \\widetilde{V}_n + \\sum _{k=0}^{n-1} \\widetilde{W}_k + \\sum _{k=n}^{\\infty } \\widetilde{b}_k,$ with $S_0 = \\widetilde{V}_0 + \\sum _{k=0}^{\\infty } \\widetilde{b}_k$ .", "We prove that $(S_n)$ converges almost surely towards a positive $S_{\\infty } \\in L^1$ .", "First note that $(S_n)$ is a non-negative process because $(V_n),(W_n)$ and $(b_n)$ are non-negative.", "Then, for all $n \\in \\mathbb {N}$ we have $\\operatorname{\\mathbb {E}}\\left[S_{n+1}|\\mathcal {F}_n\\right] \\le \\pi _n \\operatorname{\\mathbb {E}}\\left[V_{n+1}|\\mathcal {F}_n\\right] + \\sum _{k=0}^{n} \\widetilde{W}_k + \\sum _{k=n+1}^{\\infty } \\widetilde{b}_k = \\pi _{n-1} V_n + \\sum _{k=0}^{n-1} \\widetilde{W}_k + \\sum _{k=n}^{\\infty } \\widetilde{b}_k \\le S_n.$ $(S_n)$ is a non-negative super-martingale hence it converges $S_n \\stackrel{a.s.}{\\longrightarrow } S_{\\infty }$ with the upper bound $\\operatorname{\\mathbb {E}}\\left[S_{\\infty }\\right]\\le \\operatorname{\\mathbb {E}}\\left[S_{0}\\right] = \\operatorname{\\mathbb {E}}\\left[V_0 \\right]+ \\sum _{k=0}^{\\infty } \\operatorname{\\mathbb {E}}\\left[\\widetilde{b}_k\\right]$ .", "Since $\\sum _{k=0}^{\\infty } \\widetilde{b}_k = \\sum _{k=0}^{\\infty } \\pi _k b_k \\le \\sum _{k=0}^{\\infty } b_k < \\infty $ a.s., the last inequality shows that $\\operatorname{\\mathbb {E}}\\left[S_{\\infty }\\right] < \\infty $ so that $S_{\\infty }$ is almost surely finite.", "Besides, we have $\\sum _{k=0}^{n-1} \\widetilde{W}_k \\le S_n$ so the series $\\sum _k \\widetilde{W}_k$ is an upper bounded positive series: it converges almost surely.", "Since $\\lim _n \\pi _n = \\pi _{\\infty } \\in (0,1]$ , we have the almost sure convergence of $\\sum _k W_k$ in virtue of $\\forall n \\le m, \\quad \\sum _{k=n}^m W_k \\le \\pi _m^{-1} \\sum _{k=n}^m \\pi _k W_k = \\pi _m^{-1} \\sum _{k=n}^m \\widetilde{W}_k,$ which shows (a).", "Since $\\sum _{k} b_k < \\infty $ a.s., we have the almost sure convergence of $\\sum _{k} \\widetilde{b}_k$ .", "Therefore the sequence $\\widetilde{V}_n = S_n - \\sum _{k=0}^{n-1} \\widetilde{W}_k - \\sum _{k=n}^{\\infty } \\widetilde{b}_k$ converges almost surely.", "Because $V_n = \\pi _n \\widetilde{V}_n$ and $\\lim _n \\pi _n = \\pi _{\\infty } >0$ , we also have the convergence of $(V_n)$ towards $V_{\\infty }$ which gives (b).", "Finally, the inequality $ \\pi _{n-1} V_n= \\widetilde{V}_n \\le S_n$ gives $\\operatorname{\\mathbb {E}}\\left[V_n \\right] \\le \\pi _{\\infty }^{-1} \\operatorname{\\mathbb {E}}\\left[S_0 \\right]$ and proves (c).", "Lemma 1 Let $\\theta _1,\\ldots ,\\theta _T$ be an arbitrary sequence of vectors.", "Any algorithm with initialization $\\theta _1=0$ and update rule $\\theta _{t+1} = \\theta _t - \\gamma v_t$ satisfies $\\sum _{t=1}^T \\langle \\theta _t-\\theta ^\\star ,v_t \\rangle \\le \\frac{\\Vert \\theta ^\\star \\Vert ^2}{2 \\gamma } + \\frac{\\gamma }{2} \\sum _{t=1}^T \\Vert v_t\\Vert ^2.$ In particular, for $B,\\rho >0$ , if we have $\\Vert v_t\\Vert \\le \\rho $ and we set $\\gamma =\\sqrt{B^2/(\\rho ^2 T)}$ then for every $\\theta ^\\star $ with $\\Vert \\theta ^\\star \\Vert \\le B$ , we have $T^{-1} \\sum _{t=1}^T \\langle \\theta _t-\\theta ^\\star ,v_t \\rangle \\le B \\rho /\\sqrt{T}.$ Proof Using algebraic manipulations (completing the square), we obtain: $\\left\\langle \\theta _{t}-\\theta ^{\\star }, v_{t}\\right\\rangle &=\\frac{1}{\\gamma }\\left\\langle \\theta _{t}-\\theta ^{\\star }, \\gamma v_{t}\\right\\rangle \\\\&=\\frac{1}{2 \\gamma }\\left(-\\left\\Vert \\theta _{t}-\\theta ^{\\star }-\\gamma v_{t}\\right\\Vert ^{2}+\\left\\Vert \\theta _{t}-\\theta ^{\\star }\\right\\Vert ^{2}+\\gamma ^{2}\\left\\Vert v_{t}\\right\\Vert ^{2}\\right) \\\\&=\\frac{1}{2 \\gamma }\\left(-\\left\\Vert \\theta _{t+1}-\\theta ^{\\star }\\right\\Vert ^{2}+\\left\\Vert \\theta _{t}-\\theta ^{\\star }\\right\\Vert ^{2}\\right)+\\frac{\\gamma }{2}\\left\\Vert v_{t}\\right\\Vert ^{2}$ where the last equality follows from the definition of the update rule.", "Summing the equality over $t,$ we have $\\sum _{t=1}^{T}\\left\\langle \\theta _{t}-\\theta ^{\\star }, v_{t}\\right\\rangle =\\frac{1}{2 \\gamma } \\sum _{t=1}^{T}\\left(-\\left\\Vert \\theta _{t+1}-\\theta ^{\\star }\\right\\Vert ^{2}+\\left\\Vert \\theta _{t}-\\theta ^{\\star }\\right\\Vert ^{2}\\right)+\\frac{\\gamma }{2} \\sum _{t=1}^{T}\\left\\Vert v_{t}\\right\\Vert ^{2}$ The first sum on the right-hand side is a telescopic sum that collapses to $\\left\\Vert \\theta _{1}-\\theta ^{\\star }\\right\\Vert ^{2}-\\left\\Vert \\theta _{T+1}-\\theta ^{\\star }\\right\\Vert ^{2}$ .", "Then we have $\\sum _{t=1}^{T}\\left\\langle \\theta _{t}-\\theta ^{\\star }, v_{t}\\right\\rangle &=\\frac{1}{2 \\gamma }\\left(\\left\\Vert \\theta _{1}-\\theta ^{\\star }\\right\\Vert ^{2}-\\left\\Vert \\theta _{T+1}-\\theta ^{\\star }\\right\\Vert ^{2}\\right)+\\frac{\\gamma }{2} \\sum _{t=1}^{T}\\left\\Vert v_{t}\\right\\Vert ^{2} \\\\& \\le \\frac{1}{2 \\gamma }\\left\\Vert \\theta _{1}-\\theta ^{\\star }\\right\\Vert ^{2}+\\frac{\\gamma }{2} \\sum _{t=1}^{T}\\left\\Vert v_{t}\\right\\Vert ^{2} \\\\&=\\frac{1}{2 \\gamma }\\left\\Vert \\theta ^{\\star }\\right\\Vert ^{2}+\\frac{\\gamma }{2} \\sum _{t=1}^{T}\\left\\Vert v_{t}\\right\\Vert ^{2}$ where the last equality is due to the definition $\\theta _{1}=0 .$ This proves the first part of the lemma.", "The second part follows by upper bounding $\\left\\Vert \\theta ^{\\star }\\right\\Vert $ by $B,\\left\\Vert v_{t}\\right\\Vert $ by $\\rho ,$ dividing by $T,$ and plugging in the value of $\\gamma $" ], [ "Illustrative Example (stochastic first order)", "We perform a comparison on a simple example in dimension $p=2$ with the functions $f(x,y) = (x^2 + y^2)/2$ and $h(x,y) = x^2/2$ .", "Note that the function $h$ only depends on the first coordinate and an adaptive coordinate descent method should favor this direction.", "Figure REF presents the optimization paths of the different methods: SGD, Uniform and MUKSTEER.", "With the function $f$ which does not present any particular design or favorable descent direction, the Uniform and Musketeer policies perform as good as classical SGD.", "More interestingly, when dealing with the function $h$ , our method MUSKETEER (red) finds that the horizontal direction associated to axis $(Ox)$ is the relevant one for optimization.", "After collecting some information during the exploration phase, the probability weights got updated to favor the horizontal direction, leading to a faster convergence.", "Figure: Comparison of SGD/Uniform/Musketeer on simple 2D-examples" ], [ "Regularized linear models", "We consider the ERM paradigm with linear models, namely regularized regression problems with objectives of the form $f(\\theta ) = (1/n) \\sum _{i=1}^n f_i(\\theta ) + \\mu \\Vert \\theta \\Vert ^2$ .", "Similarly to [21], we endow the data matrix $X$ with a block structure.", "The columns are drawn as $X[:,k] \\sim \\mathcal {N}(0,\\sigma _k^2 I_n)$ with $\\sigma _k^2 = k^{-\\alpha }$ for all $k\\in \\llbracket 1,p \\rrbracket $ .", "The parameters are set to $n=10,000$ samples in dimension $p=250$ with an exploration size equal to $T = \\lfloor \\sqrt{p} \\rfloor = 15$ .", "The regularization parameter is set to the classical value $\\mu =1/n$ .", "We update the parameter vector with the optimal learning rate $\\gamma _k = \\gamma /(k+k_0)$ in the experiments.", "Other learning rates in the framework of stochastic first order methods are considered in Appendix .", "• (zeroth order) For the Ridge regression, we set $\\gamma =3,k_0=10$ and for the logistic regession $\\gamma =10,k_0=5$ .", "The gradient estimate $g$ is computed using queries of a function $f_i$ where $i \\sim \\mathcal {U}(\\llbracket 1,n \\rrbracket )$ .", "We use the $\\ell _1$ -reweighting with $\\lambda _t = 1/\\log (t)$ or softmax with $\\lambda _n \\equiv 0.5$ , which both satisfy Assumption REF .", "• (first order) The learning rate is equal to $\\gamma _k = 1/k$ ($\\gamma =1,k_0=0$ ).", "The gradient estimate $g$ is computed using mini-batches of size 8.", "The weighting parameter $\\eta >0$ in the softmax part of the probability weights is set to $\\eta =1$ and the parameter $\\lambda $ in Equation (REF ) is chosen as $\\lambda _t = 1/\\log (t)$ which satisfies the extended Robbins-Monro condition REF ." ], [ "Neural Networks", "Dataset description and parameter configuration.", "The three datasets in the experiments are popular publicly available deep learning datasets.", "The underlying machine learning task is the one of multi-label classification.", "• MNIST [60]: a database of handwritten digits with a training set of 60,000 examples and a test set of 10,000 examples.", "The digits have been size-normalized and centered in a fixed-size image.", "The original black and white (bilevel) images from NIST were size normalized to fit in a 20x20 pixel box while preserving their aspect ratio.", "The resulting images contain grey levels as a result of the anti-aliasing technique used by the normalization algorithm.", "The images were centered in a 28x28 image by computing the center of mass of the pixels, and translating the image so as to position this point at the center of the 28x28 field.", "Each training and test example is assigned to the corresponding handwritten digit between 0 and 9.", "• Fashion-MNIST [61]: a dataset of Zalando's article images, composed of a training set of 60,000 examples and a test set of 10,000 examples.", "Each example is a 28x28 grayscale image, associated with a label from 10 classes.", "It shares the same image size and structure of training and testing splits as the MNIST database.", "Each training and test example is assigned to one of the following labels: T-shirt/top (0); Trouser (1); Pullover (2); Dress (3); Coat (4); Sandal (5); Shirt (6); Sneaker (7); Bag (8); Ankle boot (9).", "• Kuzushiji-MNIST: Kthis dataset is a drop-in replacement for the MNIST dataset (28x28 grayscale, 70,000 images), provided in the original MNIST format as well as a NumPy format.", "Since MNIST is restricted to 10 classes, one character here represents each of the 10 rows of Hiragana when creating Kuzushiji-MNIST.", "• CIFAR10 [64]: The CIFAR-10 dataset consists of $60,000$ $32 \\times 32$ colour images in 10 classes, with $6,000$ images per class.", "There are $50,000$ training images and $10,000$ test images.", "The dataset is divided into five training batches and one test batch, each with $10,000$ images.", "The test batch contains exactly $1,000$ randomly-selected images from each class.", "The training batches contain the remaining images in random order, but some training batches may contain more images from one class than another.", "Between them, the training batches contain exactly $5,000$ images from each class.", "Each training and test example is assigned to one of the following labels: airplane (0); automobile (1); bird (2); cat (3); deer (4); dog (5); frog (6); horse (7); ship (8); truck (9).", "Figure: K-MNISTTwo different neural networks are used in the experiments: one with linear layers for MNIST, Fashion-MNIST, K-MNIST another one with convolutional layers for CIFAR10.", "For the first network, the total number of parameters is $p=55,050$ .", "For the second network,the dimension is $p=64,862$ .", "In both cases, the exploration size is $T = \\lfloor \\sqrt{p} \\rfloor $ .", "In the experiments with stochastic first order methods, we use batches of coordinates with $m=p/10$ ." ], [ "Networks architecture.", "The code -available upon request- is written in Python3 within the PyTorch library.", "The architectures are given below for the sake of completeness.", "def __init__(self,input_size,hidden_size,output_size):         super(Net, self).__init__()         self.input_size  = input_size         self.hidden_size = hidden_size         self.output_size = output_size         self.linear1 = nn.Linear(self.input_size,self.hidden_size)         self.linear2 = nn.Linear(self.hidden_size,self.hidden_size)         self.linear3 = nn.Linear(self.hidden_size,self.output_size)    def forward_pass(self, x):         x = self.linear1(x)         x = torch.sigmoid(x)         x = self.linear2(x)         x = torch.sigmoid(x)         x = self.linear3(x)         x = torch.log_softmax(x, dim=0)         return x    def __init__(self):         super(Net, self).__init__()         self.conv1 = nn.Conv2d(3, 12, 5)         self.pool  = nn.MaxPool2d(2, 2)         self.conv2 = nn.Conv2d(12, 16, 5)         self.fc1   = nn.Linear(16 * 5 * 5, 120)         self.fc2   = nn.Linear(120, 84)         self.fc3   = nn.Linear(84, 10)    def forward_pass(self, x):         x = self.pool(F.relu(self.conv1(x)))         x = self.pool(F.relu(self.conv2(x)))         x = x.view(-1, 16 * 5 * 5)         x = F.relu(self.fc1(x))         x = F.relu(self.fc2(x))         x = self.fc3(x)         return x" ], [ "Hyperparameters and Hardware.", "Hyperparameters.", "When training neural networks with linear layers, we use: batch_size = 32; input_size = 28*28; hidden_size = 32; output_size = 64, along with the parameters • (zeroth order) $\\gamma $ = 10 (Mnist and Fashion-Mnist) $\\gamma $ =15 (Kmnist); h = 0.01; $\\ell _1$ normalization with $\\lambda _n = 1/\\log (n)$ ; softmax normalization with $\\lambda _n \\equiv 0.2$ and $\\eta =5$ .", "• (first order) $\\gamma $ = 0.01 (Mnist,Fashion-Mnist,Cifar10); normalization = softmax with $\\eta \\in \\lbrace 1,2,10\\rbrace $ ; $\\lambda _t = 0$ (only exponential weights).", "Hardware.", "The experiments of linear models are run using a processor Intel Core i7-10510U CPU 1.80GHz $\\times $ 8; the neural networks are trained using GPU from Google Colab (GPU: Nvidia K80 / T4; GPU Memory: 12GB/16GB; GPU Memory Clock: 0.82GHz/1.59GHz; Performance: 4.1 TFLOPS / 8.1 TFLOPS) ZO Neural Networks with $\\ell _1$ normalization.", "Figure: KMNISTZO Neural Networks, Comparison of $\\ell _1$ and Softmax normalizations.", "Figure: Fashion-Exp" ], [ "Numerical Experiments with stochastic first order methods", "In this section, we empirically validate the SCGD framework by running MUSKETEER and competitors on synthetic and real datasets problems with stochastic first order methods.", "First, we focus on ridge regression and regularized logistic regression problems adopting the data generation process of [21] in which the covariates exhibit a certain block structure.", "Second, MUSKETEER is employed to train different neural networks models on real datasets for multi-label classification task.", "From a practical point of view, the optimization procedure is implemented through a PyTorch optimizer which allows an easy deployment and integration.", "Methods in competition.", "The set of methods in competition is restricted to stochastic coordinate-based methods along with standard SGD playing the role of the baseline.", "This choice allows an honest comparison as the parameter tuning can be the same for all methods.", "MUSKETEER is implemented according to Section with an exploration size $T = \\lfloor \\sqrt{p} \\rfloor $ and different values of $\\eta $ are used to feed the discussion on the adaptiveness.", "The method UNIFORM stands for the uniform coordinate sampling policy in SCGD.", "The method ADAPTIVE is the importance sampling based method described in Remark REF .", "This method is no longer part of the SCGD framework and corresponds to the one developed in [37].", "Among the different methods, MUSKETEER is the only one exhibiting a bias when generating gradients.", "In all cases, $\\theta _0 = (0,\\ldots , 0)^T \\in \\mathbb {R}^p$ and the optimal SGD learning rate $\\gamma _k = 1/k$ is used.", "For a fair comparison of SGD against SCGD, we normalize the number of passes over the coordinates: one SGD step updates the $p$ coordinates of $\\theta $ so we allow to take $p$ steps for the coordinate-based methods in the mean time." ], [ "Linear models.", "We apply ERM to regularized regression and classification problems.", "Given a data matrix $X=(x_{i,j}) \\in \\mathbb {R}^{n \\times p}$ with labels $y \\in \\mathbb {R}^n$ and a regularization parameter $\\mu >0$ , the Ridge regression objective is defined by $f(\\theta )=\\frac{1}{2n} \\sum _{i=1}^n (y_i - \\sum _{j=1}^p x_{i,j} \\theta _j)^2 + \\frac{\\mu }{2} \\Vert \\theta \\Vert _2^2$ and the $\\ell _2$-regularized logistic regression is given by $f(\\theta )= \\frac{1}{n} \\sum _{i=1}^n \\log (1+\\exp (-y_i \\sum _{j=1}^p x_{i,j} \\theta _j)) + \\mu \\Vert \\theta \\Vert _2^2$ where $\\mu $ is set to the classical value $\\mu =1/n$ .", "Similarly to [21], we endow the data matrix $X$ with a block structure.", "The columns are drawn as $X[:,k] \\sim \\mathcal {N}(0,\\sigma _k^2 I_n)$ with $\\sigma _k^2 = k^{-\\alpha }$ for $k \\in \\llbracket 1,p \\rrbracket $ .", "The parameters are set to $n=10,000$ samples in dimension $p=250$ and $T = 15$ .", "Figure: Logistic α=5\\alpha =5Figure REF provides the graphs of the optimaliy gap $t\\mapsto f(\\theta _t)-f(\\theta ^\\star )$ averaged over 20 independent simulations for different values of $\\alpha \\in \\lbrace 2;5;10\\rbrace $ .", "First, note that the uniform sampling strategy shows similar performance to the classical SGD and that the (unbiased) importance sampling version ADAPTIVE is also of the same order.", "Besides, the clear winner is MUSKETEER as it offers the best performance in all configurations.", "Greater values of $\\alpha $ (stronger block structure) improve our relative performance with respect to the other methods as shown by Figures REF and REF ." ], [ "Neural Networks.", "To asses the practical performance of MUSKETEER, we focus on the training of neural networks within the framework of multi-label classification.", "The datasets in the experiments are popular publicly available deep learning datasets: MNIST [60], Fashion-MNIST [61] and CIFAR10 [64].", "Given an image, the goal is to predict its label among ten classes.", "Two different neural networks are used in the experiments: one with linear layers for MNIST and Fashion-MNIST ($p=55,050$ and $T = 234$ ) , another one with convolutional layers for CIFAR10 ($p=64,862$ and $T = 254$ ).", "Figure: CIFAR-10Figure REF compares the evolution of the training loss of SGD against MUSKETEER averaged over 10 independent simulations with different values of $\\eta $ .", "A great value of this parameter strengthens the adaptive scheme as it gives more importance to the weights in Equation (REF ), leading to stronger decrease of the objective function.", "Interestingly, the performance of MUSKETEER also benefit from such adaptive structure in terms on accuracy of the test set (see Table REF ).", "This allows to quantify the statistical gain brought by MUSKETEER over SGD.", "Table: Test Accuracy (in %)." ], [ "Ridge Regression ($\\ell _1$ -reweighting) with different settings of {{formula:a501b000-2e0a-4b93-8575-4785cfe8c2af}}", "We consider the Ridge regression problem with the classical regularization parameter value $\\mu =1/n$ and run several experiments in various settings of $(n,p)$ .", "We endow the data matrix $X$ with a block structure.", "The columns are drawn as $X[:,kB+1:kB+B] \\sim \\mathcal {N}(0,\\sigma _k^2 I_n)$ with $\\sigma _k^2 = k^{-\\alpha }$ for all $k\\in \\llbracket 0,(p/B)-1 \\rrbracket $ .", "The parameter $B$ is the block-size and is set to $B=10$ for the Ridge regression.", "The parameter $\\alpha $ represents the block structure and is set to $\\alpha =5$ .", "The different Figures below present the evolution of the optimality gap $t \\mapsto [f(\\theta _t)-f(\\theta ^\\star )]$ averaged over 20 independent runs.", "The learning rates is the same for all methods, fixed to $\\gamma _k = 1/(k+10)$ .", "The different settings are: number of samples $n \\in \\lbrace 1,000;2,000;5,000\\rbrace $ and dimension $p \\in \\lbrace 20;50;100;200\\rbrace $ .", "We use the $\\ell _1$ normalization in Equation (REF ) with $\\lambda _n = 1/\\log (n)$ .", "Figure: [f(θ t )-f(θ ☆ )][f(\\theta _t)-f(\\theta ^\\star )] for Ridge Regression with n=5000n=5000 and p=20,50,100,200p=20,50,100,200" ], [ "Ridge Regression (softmax reweighting) with different settings of $(n,p)$", "We consider the Ridge regression problem with the classical regularization parameter value $\\mu =1/n$ and run several experiments in various settings of $(n,p)$ .", "We endow the data matrix $X$ with a block structure.", "The columns are drawn as $X[:,kB+1:kB+B] \\sim \\mathcal {N}(0,\\sigma _k^2 I_n)$ with $\\sigma _k^2 = k^{-\\alpha }$ for all $k\\in \\llbracket 0,(p/B)-1 \\rrbracket $ .", "The parameter $B$ is the block-size and is set to $B=10$ for the Ridge regression.", "The parameter $\\alpha $ represents the block structure and is set to $\\alpha =5$ .", "The different Figures below present the evolution of the optimality gap $t \\mapsto [f(\\theta _t)-f(\\theta ^\\star )]$ averaged over 20 independent runs.", "The learning rates is the same for all methods, fixed to $\\gamma _k = 1/(k+10)$ .", "The different settings are: number of samples $n \\in \\lbrace 1,000;2,000;5,000\\rbrace $ and dimension $p \\in \\lbrace 20;50;100;200\\rbrace $ .", "We use the softmax normalization in Equation (REF ) with $\\lambda _n \\equiv 0.5$ and $\\eta =1$ .", "Figure: [f(θ t )-f(θ ☆ )][f(\\theta _t)-f(\\theta ^\\star )] for Ridge Regression with n=5000n=5000 and p=20,50,100,200p=20,50,100,200" ], [ "Logistic Regression ($\\ell _1$ -reweighting) with different settings of {{formula:7915b714-7799-4ec5-9548-c4174f39b950}}", "We consider the $\\ell _2$ -Logistic regression problem with the classical regularization parameter value $\\mu =1/n$ and run several experiments in various settings of $(n,p)$ .", "We endow the data matrix $X$ with a block structure.", "The columns are drawn as $X[:,kB+1:kB+B] \\sim \\mathcal {N}(0,\\sigma _k^2 I_n)$ with $\\sigma _k^2 = k^{-\\alpha }$ for all $k\\in \\llbracket 1,(p/B)-1 \\rrbracket $ .", "The parameter $B$ is the block-size and is set to $B=5$ for the Logistic regression.", "The parameter $\\alpha $ represents the block structure and is set to $\\alpha =5$ .", "The different Figures below present the evolution of the optimality gap $t \\mapsto [f(\\theta _t)-f(\\theta ^\\star )]$ averaged over 20 independent runs.", "The learning rates is the same for all methods, fixed to $\\gamma _k = 10/(k+5)$ .", "The different settings are: number of samples $n \\in \\lbrace 1,000;2,000;5,000\\rbrace $ and dimension $p \\in \\lbrace 20;50;100;200\\rbrace $ .", "We use the $\\ell _1$ normalization in Equation (REF ) with $\\lambda _n = 1/\\log (n)$ .", "Figure: [f(θ t )-f(θ ☆ )][f(\\theta _t)-f(\\theta ^\\star )] for logistic Regression with n=5000n=5000 and p=20,50,100,200p=20,50,100,200" ], [ "Logistic Regression (softmax reweighting) with different settings of $(n,p)$", "We consider the $\\ell _2$ -Logistic regression problem with the classical regularization parameter value $\\mu =1/n$ and run several experiments in various settings of $(n,p)$ .", "We endow the data matrix $X$ with a block structure.", "The columns are drawn as $X[:,kB+1:kB+B] \\sim \\mathcal {N}(0,\\sigma _k^2 I_n)$ with $\\sigma _k^2 = k^{-\\alpha }$ for all $k\\in \\llbracket 1,(p/B)-1 \\rrbracket $ .", "The parameter $B$ is the block-size and is set to $B=5$ for the Logistic regression.", "The parameter $\\alpha $ represents the block structure and is set to $\\alpha =5$ .", "The different Figures below present the evolution of the optimality gap $t \\mapsto [f(\\theta _t)-f(\\theta ^\\star )]$ averaged over 20 independent runs.", "The learning rates is the same for all methods, fixed to $\\gamma _k = 10/(k+5)$ .", "The different settings are: number of samples $n \\in \\lbrace 1,000;2,000;5,000\\rbrace $ and dimension $p \\in \\lbrace 20;50;100;200\\rbrace $ .", "We use the softmax normalization in Equation (REF ) with $\\lambda _n \\equiv 0.5$ and $\\eta =1$ .", "Figure: [f(θ t )-f(θ ☆ )][f(\\theta _t)-f(\\theta ^\\star )] for logistic Regression with n=5000n=5000 and p=20,50,100,200p=20,50,100,200" ], [ "Effect of Importance Sampling (IS) on Ridge Regression", "We consider the same setting as in Subsection REF and study the effect of using importance sampling weights in the update rule of MUSKETEER.", "Indeed, MUSKETEER update rule is defined with the following biased gradient estimate $\\theta _{t+1} = \\theta _{t} - \\gamma _{t+1}D(\\zeta _{t+1})g_t,$ and the importance sampling (IS) strategy consists in adding $D_t^{-1}$ to reach an unbiased estimate $\\theta _{t+1} = \\theta _{t} - \\gamma _{t+1}D_{t}^{-1}D(\\zeta _{t+1})g_t.$ For the different configurations, we compare the MUSKETEER methods with their importance sampling counterparts.", "The Figures below show that the importance sampling methods perform similarly to the uniform coordinate sampling strategy and are therefore sub-optimal.", "Figure: [f(θ t )-f(θ ☆ )][f(\\theta _t)-f(\\theta ^\\star )] for Ridge Regression with n=5000n=5000 and p=50,200p=50,200" ], [ "Effect of Importance Sampling (IS) on Logistic Regression", "We consider the same setting as in Subsection REF and study the effect of using importance sampling weights in the update rule of MUSKETEER.", "Indeed, MUSKETEER update rule is defined with the following biased gradient estimate $\\theta _{t+1} = \\theta _{t} - \\gamma _{t+1}D(\\zeta _{t+1})g_t,$ and the importance sampling (IS) strategy consists in adding $D_t^{-1}$ to reach an unbiased estimate $\\theta _{t+1} = \\theta _{t} - \\gamma _{t+1}D_{t}^{-1}D(\\zeta _{t+1})g_t.$ For the different configurations, we compare the MUSKETEER methods with their importance sampling counterparts.", "The Figures below show that the importance sampling methods perform similarly to the uniform coordinate sampling strategy and are therefore sub-optimal.", "Figure: [f(θ t )-f(θ ☆ )][f(\\theta _t)-f(\\theta ^\\star )] for Logistic Regression with n=5000n=5000 and p=50,200p=50,200" ], [ "Comparing learning rates", "While we make no claim that MUSKETEER outperforms any gradient based method, we demonstrated both theoretical and empirical evidence of working with our method rather than naive SGD.", "This section investigates the effect of different learning rates $\\gamma _k = \\gamma /k$ and reveals a safe behavior of MUSKETEER as it performs better than the other methods in all configurations with a stronger difference when dealing with small values of $\\gamma $ .", "We consider the Ridge regression problem with the classical regularization parameter value $\\mu =1/n$ and run several experiments in the setting $n=5,000$ samples and dimension $p \\in \\lbrace 20;100;200\\rbrace $ .", "We endow the data matrix $X$ with a block structure.", "The columns are drawn as $X[:,k] \\sim \\mathcal {N}(0,\\sigma _k^2 I_n)$ with $\\sigma _k^2 = k^{-\\alpha }$ for all $k\\in \\llbracket 1,p \\rrbracket $ .", "The parameter $\\alpha $ of block structure is $\\alpha =8$ .", "The gradient estimate $g$ is computed using mini-batches of size 4.", "The different Figures below present the evolution of the optimality gap $t \\mapsto [f(\\theta _t)-f(\\theta ^\\star )]$ averaged over 20 independent runs for $N=100$ iterations with normalized passes over coordinates.", "The learning rates are set to $\\gamma _k = \\gamma /k$ with $\\gamma \\in \\lbrace 0.5; 1; 1.5; 2\\rbrace $ .", "Figure: [f(θ t )-f(θ ☆ )][f(\\theta _t)-f(\\theta ^\\star )] for Ridge Regression with p=200p=200 and γ∈{0.5;1;1.5;2}\\gamma \\in \\lbrace 0.5; 1; 1.5; 2\\rbrace" ], [ "Ridge Regression with different settings of $(n,p)$", "We consider the Ridge regression problem with the classical regularization parameter value $\\mu =1/n$ and run several experiments in various settings of $(n,p)$ .", "We endow the data matrix $X$ with a block structure.", "The columns are drawn as $X[:,kB+1:kB+B] \\sim \\mathcal {N}(0,\\sigma _k^2 I_n)$ with $\\sigma _k^2 = k^{-\\alpha }$ for all $k\\in \\llbracket 0,(p/B)-1 \\rrbracket $ .", "The parameter $B$ is the block-size and is set to $B=5$ for the Ridge regression.", "The parameter $\\alpha $ represents the block structure and is set to $\\alpha =10$ .", "The data sampling process $\\xi $ of gradient estimate $g$ is computed using mini-batches of size 8.", "The different Figures below present the evolution of the optimality gap $t \\mapsto [f(\\theta _t)-f(\\theta ^\\star )]$ averaged over 20 independent runs for $N=1000$ iterations with normalized passes over coordinates.", "The learning rates is the same for all methods, fixed to $\\gamma _k = 1/k$ .", "The different settings are: number of samples $n \\in \\lbrace 1,000;2,000;5,000\\rbrace $ and dimension $p \\in \\lbrace 20;50;100;200\\rbrace $ .", "Figure: [f(θ t )-f(θ ☆ )][f(\\theta _t)-f(\\theta ^\\star )] for Ridge Regression with n=5000n=5000 and p=20,50,100,200p=20,50,100,200" ], [ "Logistic Regression with different settings of $(n,p)$", "We consider the $\\ell _2$ -Logistic regression problem with the classical regularization parameter value $\\mu =1/n$ and run several experiments in various settings of $(n,p)$ .", "We endow the data matrix $X$ with a block structure.", "The columns are drawn as $X[:,kB+1:kB+B] \\sim \\mathcal {N}(0,\\sigma _k^2 I_n)$ with $\\sigma _k^2 = k^{-\\alpha }$ for all $k\\in \\llbracket 1,(p/B)-1 \\rrbracket $ .", "The parameter $B$ is the block-size and is set to $B=2$ for the Logistic regression.", "The parameter $\\alpha $ represents the block structure and is set to $\\alpha =5$ .", "The data sampling process $\\xi $ of gradient estimate $g$ is computed using mini-batches of size 32.", "The different Figures below present the evolution of the optimality gap $t \\mapsto [f(\\theta _t)-f(\\theta ^\\star )]$ averaged over 20 independent runs for $N=1000$ iterations with normalized passes over coordinates.", "The learning rates is the same for all methods, fixed to $\\gamma _k = 1/k$ .", "The different settings are: number of samples $n \\in \\lbrace 1,000;2,000;5,000\\rbrace $ and dimension $p \\in \\lbrace 20;50;100;200\\rbrace $ .", "Figure: [f(θ t )-f(θ ☆ )][f(\\theta _t)-f(\\theta ^\\star )] for logistic Regression with n=5000n=5000 and p=20,50,100,200p=20,50,100,200" ] ]
2105.11818
[ [ "Obscuring digital route choice information prevents delay-induced\n congestion" ], [ "Abstract Although routing applications increasingly affect individual mobility choices, their impact on collective traffic dynamics remains largely unknown.", "Smart communication technologies provide accurate traffic data for choosing one route over other alternatives, yet inherent delays undermine the potential usefulness of such information.", "Here we introduce and analyze a simple model of collective traffic dynamics which result from route choice relying on outdated traffic information.", "We find for sufficiently small information delays that traffic flows are stable against perturbations.", "However, delays beyond a bifurcation point induce self-organized flow oscillations of increasing amplitude -- congestion arises.", "Providing delayed information averaged over sufficiently long periods of time or, more intriguingly, reducing the number of vehicles adhering to the route recommendations may prevent such delay-induced congestion.", "We reveal the fundamental mechanisms underlying these phenomena in a minimal two-road model and demonstrate their generality in microscopic, agent-based simulations of a road network system.", "Our findings provide a way to conceptually understand system-wide traffic dynamics caused by broadly used non-instantaneous routing information and suggest how resulting unintended collective traffic states could be avoided." ], [ "Introduction", "Digitization, real-time collection of data and access to information have transformed everyday life.", "For example, drivers nowadays often base their route choices on traffic information accessed en-route in almost real-time.", "In principle, choosing and adapting the travel route based on current travel times might lead to more efficient usage of the street infrastructure and help prevent congestion.", "However, this notion is controversial as providing information about the travel time is not necessarily beneficial for the overall state of the system and may even increase congestion [1], [2], [3], [4], [5], [6], for example, if the available information is imperfect [7].", "Specifically, it has been shown from a systemic perspective that selfish routing may lead to non-optimal collective states in which the travel time averaged across all vehicles is higher than the theoretical optimum [8], [9], [10], [11], [6].", "Furthermore, unpleasant side-effects exist, such as increased usage of low-capacity roads through residential areas, use of complicated routes with higher accident risk, and increased noise and air pollution [12], [13], [14].", "In addition to unintended negative effects of wide-spread usage of routing apps, the travel time information that drivers rely on to select their route is inherently outdated [15], [16].", "The information only reflects temporary traffic states and relevant decisions of other drivers are not taken into account.", "Once all drivers choose the route with the predicted shortest travel time based on outdated information, it might not be the fastest route and unforeseen congested states may arise by the time they are en-route.", "This kind of collective behavior is termed as overreaction [1] and may induce oscillations in route utilization [17], [18], [19], [20], [21].", "Here, we analyze a simple macroscopic traffic flow model, describing how individual route choice decisions based on delayed information may impact the collective traffic dynamics.", "We investigate the emergence of traffic flow instabilities due to delayed information and how to prevent the resulting congestion.", "Studying the mechanism underlying such instabilities for a two-road system reveals a bifurcation where the stability of the equilibrium flow changes: above a critical delay of travel time information, street load oscillations increase in amplitude, eventually causing a traffic jam throughout the system.", "Inspired by findings that averaging information prevents destabilizing feedback loops, for example in decentral smart power grids [22] and information routing in optical fiber networks [23], we demonstrate how congestion induced by delayed traffic information may be prevented by offering time-averaged rather than instantaneous travel time predictions.", "We confirm the robustness of our analytical insights with microscopic agent-based simulations in a larger network.", "We furthermore show that the same mechanism of avoiding overreaction prevents congestion if only a fraction of all drivers has access to the information – a counterintuitive finding given that providing travel time information is supposed to decrease driving times.", "The macroscopic state of traffic along a single street at time $t$ is described by the number or density of vehicles $N(t)$ which changes as vehicles enter and leave the street, d Nd t = in-out(N) , where $\\nu _{\\text{in}}$ denotes the rate of incoming vehicles and $\\nu _{\\text{out}}(N)$ denotes the rate of vehicles leaving the system depending on the current traffic conditions.", "On a street with a maximal capacity $N_0$  , with an increasing street load, the distance between two vehicles is lowered, which forces drivers to slow down.", "Thus, the average speed of vehicles on a street decreases, which is equivalent to an increase of the travel time compared to the time $t_0$ a vehicle needs to drive through the empty street, ttravel(N)=t0 (NN0 )-1 ((NN0)-1)    , i. e. the travel time grows exponentially with the number of vehicles.", "Such an exponential dependence has been suggested in early studies of traffic flow [24] and correctly captures the qualitative dependence of travel time.", "For small densities $N/N_0 \\rightarrow 0$ , this function resembles the most common travel time function used [25] .", "In this macroscopic description, the instantaneous rate of outflow is given as the number of vehicles currently on the street divided by the current travel time out(N)= Nttravel(N) =N2t0 N0[(NN0)-1]-1    .", "This function qualitatively resembles the form of density-flow-diagrams (Fig.", "REF a) found empirically[26].", "For low street loads, the outflow increases approximately linearly with the number of vehicles on the street, until a maximal outflow is reached.", "Above this maximum, the outrate decreases again – congestion arises.", "Note that the general impact of outdated information does not depend on the exact curve given by Eq.", "(REF ) but rather on its qualitative form.", "Our choice for equation (REF ) thus does not put any limits on the generality of our findings.", "If drivers have various route options $i$ to reach their destination, the inflow rate $\\nu _{\\text{in},i}$ of each of the routes $i$ depends on the drivers' route choices.", "Given that travel times $t_{\\text{signal},i}$ for all streets $i$ are provided to the drivers, it has been found empirically that they choose one route $i$ out of a set of route options indexed by $j$ probabilistically, following a Boltzmann distribution as described by the multinomial logit model [27], [28] in,i = in0 (e-tsignal,ij e-tsignal,j) , where $\\nu _{\\text{in}}^0 = \\sum _i \\nu _{\\text{in},i}$ is a constant denoting the total rate of vehicles entering the road-system and $\\beta $ is a parameter which governs the impact of information on the individual decision: $\\beta =0$ corresponds to uniformly random choices, $\\beta \\rightarrow \\infty $ describes deterministic best response dynamics.", "Within the following, we set an intermediate value $\\beta =1$  , such that drivers react sufficiently to travel time differences but route choices are not fully deterministic.", "Within the first part of the paper, we study a two-road network, $i \\in \\lbrace 1,2\\rbrace $ , with identical normalized length and capacities, $N_{0,i}=t_{0,i}=1$  , and interpret $N_i$ as the density of vehicles on street $i$ .", "If drivers are provided with up-to-date information on the travel times for both routes, their route choice is a reaction to current imbalances in street loads.", "Hence, the two-road system will only become congested if the inflow rate of vehicles is higher than the outflow rate.", "The maximal in-rate of a single street that does not lead to congestion, $\\nu _\\text{max}$  , is thus given by the maximum of the out-rate $\\nu _\\text{max} = \\max _N \\nu _\\text{out}(N)$ (Fig.", "REF a).", "So, for a two road system, traffic will flow freely for in-rates smaller than $2\\nu _\\text{max}$ if no information delay exists (Fig.", "REF a).", "As demonstrated by the plot of street load with time in Fig.", "REF b, the density of vehicles on both route options will settle at a stable equilibrium, which is given by the fixed point $N^\\ast _\\text{low}$ marked in Fig.", "REF a.", "Once the inflow exceeds the critical in-rate of the two-street network, the traffic dynamics undergo a phase change and become congested." ], [ "Unstable traffic flow due to outdated travel time information", "The general effect of route choice based on delayed information is a change in stability of the two-road network dynamics, as demonstrated in Fig.", "REF a.", "While for small delays the maximal in-rate is still only bounded by the street capacities, with increasing delay congestion sets in already for in-rates which are below the maximal values.", "If drivers receive information about travel times based on measurements at a an earlier time $t-\\tau $  , drivers keep choosing a route which was optimal at the moment of measurement but might not be the fastest route anymore.", "As a result, the road that seemed better at measurement time becomes even more crowded than the formerly worse alternative.", "Only when this change in relative street loads becomes visible in the delayed information, drivers adapt their route choice behavior.", "As long as the information delay $\\tau $ is smaller than a critical delay $\\tau _c(\\nu _\\text{in}^0)$ the load differences between the two roads stay small such that some drivers keep choosing the seemingly worse option.", "Moreover, the rate of outgoing vehicles is sufficiently high even for maximal loads to counterbalance the extreme values of street load, as shown in Fig.", "REF c : the number of vehicles on each street oscillates around the stable fixed point $N^\\ast _\\text{low}$ with decreasing amplitude.", "If the information delay $\\tau $ increases above the critical value, the fixed point $N_\\text{low}^\\ast $ becomes unstable.", "Drivers choose one route preferentially for longer before the travel time information updates.", "The larger travel time differences cause them to overreact even stronger in the other direction, reinforcing the dynamics and leading to oscillations of the street load with growing amplitude, as illustrated in Fig.", "REF d. Finally, once the number of vehicles on one of the two streets exceeds the second, unstable fixed point $N_\\text{high}^\\ast $  , the outflow of vehicles exceeds the inflow and congestion arises.", "As a consequence, every incoming driver now chooses the alternative road, which will thus eventually also congest.", "Fig.", "REF e depicts this long-term effect of large information delays.", "The critical delay $\\tau _c(\\nu _\\text{in}^0)$ for a given in-rate derived from stability analysis of the delay-differential equation [29] that governs the dynamics of street loads, d Nid t = in,i[N1 (t-), N2(t-)]-out,i [N1(t), N2(t)] , agrees well with the stability regimes obtained from the direct numerical integration of the delay-differential equation (REF ) (see Fig.", "REF a ; for details see Appendix )." ], [ "Averaging over outdated travel-time information prevents congestion", "Outdated information induces congestion due to an overreaction of drivers to the travel time information they get.", "To prevent this fatal overreaction, we reduce the differences in the signalled travel times of the two route options by providing travel time information based on the average load, tsignal,i=ttravel(Ni(t-)Tav) , where $\\langle N_i(t-\\tau )\\rangle _{T_\\text{av}} =\\frac{1}{T_\\text{av}}\\int _{t-\\tau -T_\\text{av}}^{t-\\tau } N_i(t^{\\prime }) \\, \\mathrm {d} t^{\\prime }$ and $T_\\text{av}$ is the length of the time span within which data for the averaged information is collected.", "Providing information averaged over past data to drivers, the traffic flow dynamics changes as shown in the bifurcation diagram REF a.", "For a small delay, i.e.", "for almost up-to-date information, congestion arises for smaller in-rates compared to no averaging, marked by the red region in Fig.", "REF a.", "This points to the fact that once we average, we effectively introduce additional delays which cause overreactions instead of preventing them.", "However, in the region of intermediate and high delay, averaging reinstates stability for some in-rates (see the blue region in Fig.", "REF a).", "By averaging over data from a larger time span, extreme values of street loads impact route choice less compared to providing information from just one time point.", "This stabilizing effect of averaged information depends on the averaging time window $T_\\text{av}$ (Fig.", "REF b).", "While for small delays ($\\tau =1$ ) averaging always leads to higher instability and thus to lower critical in-rates, for larger delays ($\\tau =5$ and $\\tau =10$ ), averaging prevents congestion for sufficiently long averaging time spans." ], [ "Traffic on networks", "To demonstrate the robustness of the analysis from the macroscopic traffic model, i.e.", "the minimal two-road system, we analyze the impact of delayed information for larger networks with microscopic agent-based simulations of individual vehicles in a 5x5-square grid.", "Within the simulation, we set the capacity measure $N_{0,e}=10$ for all streets $e$ , leaving all other parameters in equations Eq.", "(REF ) and Eq.", "(REF ) unchanged.", "Vehicles enter the system following a Poisson process with a rate $\\nu _\\text{in}^0$ with origin and destination sampled from a uniform distribution.", "When a vehicle enters the system, the driver chooses a path $i$ from one of the possible shortest paths between their origin and destination based on the available (possibly outdated) information, where the travel time along the route $i$ is the sum of the travel times along all streets on this route.", "As before, route choices depend on the signalled travel times as given by Eq.", "(REF ) ." ], [ "Classifying congested traffic", "Microscopic, agent-based simulations of drivers relying on delayed travel time information for their routing decisions on non-trivial street networks pose several challenges.", "First, stochastic fluctuations of the number of inflowing vehicles may induce spontaneous and persistent congestion [30].", "Second, as the routes of vehicles typically consist of more than one street segment, a natural additional information delay occurs if drivers make their route choice at the beginning of their trip without later adaptation.", "Third, classifying a state as congested becomes more subtle in such a stochastic system.", "Sudden peaks in the street loads can occur randomly without leading to permanent congestion.", "To circumvent wrong characterizations due to random fluctuations in our simulation, we increase the threshold at which we consider the system as congested from $N_\\text{max} := N(\\nu _\\text{max})\\approx 16$ , which we find from Eq.", "(REF ), to $N_\\text{threshold}=100$ vehicles.", "We furthermore choose a comparatively long simulation time of 400 time steps to ensure that no eventually congested state is overlooked." ], [ "Averaging prevents congestion in complex street networks", "The impact of information delay on the collective traffic dynamics on larger street networks is comparable to the impact observed in the minimal two-road setting.", "The higher the delay, the lower is the critical in-rate for which congestion occurs.", "This general change in traffic flow dynamics is summarized in the bifurcation diagram REF a , in which stochastic effects are accounted for by showing the fraction of 100 simulation runs with equal parameters that end in a congested state.", "As in the two-road scenario, an increased delay leads to higher average loads on the streets in the network.", "The already existing imbalance between highly frequented central edges and less frequented boundary regions, visible in Fig.", "REF b , is reinforced if information is delayed, as demonstrated for $\\tau =5$ in Fig.", "REF c .", "Only beyond a critical delay the street loads increase above $N_\\text{max}$ , i.e.", "congestion arises (Fig.", "REF d).", "Interestingly, the street loads of two alternative routes (marked by red and blue paths with arrows in Fig.", "REF d ) undergo oscillations that grow in amplitude, similar to those occurring in the two-road model for high delays (see Fig.", "REF e).", "Thus, also in non-trivial street networks, the overreaction of drivers induces unbalanced, alternating street loads which eventually lead to a congested state.", "When providing averaged information, the same qualitative changes of the traffic flow dynamics occur for larger grid networks as for the small two-road system.", "The differences in ratio of congested states between simulations with and without averaging plotted in Fig.", "REF confirm that while for nearly up-to-date information, averaging leads to higher instability, it increases the bound of stable in-rates for high information-delay." ], [ "Informing only a fraction of drivers prevents congestion", "In a real-world setting, not all drivers rely on (the same) traffic information when choosing their route.", "To find whether the impact of delay on traffic flow dynamics stays the same for partially informed drivers, we provide only a fraction $f$ of drivers with travel time information, while the remaining fraction $1-f$ decide uniformly randomly for one of the alternative route options.", "If information is almost up-to-date, ($\\tau = 1$ ), overreaction is small and hardly impacts the stability of the free flow state (see Fig.", "REF a).", "On the contrary, the provided information assists drivers to react to current load imbalances and thus prevents congestion.", "Hence, a reduction of the fraction $f$ of informed drivers as in Fig.", "REF always leads to a decrease of critical in-rates, i.e.", "an earlier occurrence of instability.", "If information is delayed ($\\tau = 5,15$ ), reducing the fraction $f$ of informed drivers increases the critical in-rate, i.e.", "the system supports a higher amount of traffic (Fig.", "REF ).", "This effect can be understood in the same way as averaging outdated information.", "By collectively choosing a route which drivers are told to be optimal, they cause load imbalance and, eventually, congested states.", "In a setting where only a fraction of drivers actually have access to the information, this overreaction is damped.", "Nevertheless, if too many route choices are made randomly, important information on the load distribution within the network is lost, as e.g.", "the fact that central streets are typically more loaded than outer ones (Fig.", "REF b).", "If this general tendency of load imbalance is not known to more than a critical percentage of the drivers, congestion will arise already for lower in-rates compared to the case in which all drivers use the available information.", "We have introduced and analyzed a macroscopic model of how delayed travel-time information may affect traffic flow dynamics.", "Information delays may lead to oscillations on alternative routes, both in a simple two-road system and in more complex street networks.", "Our finding of oscillations due to time delay is in line with previous results modelling traffic flow in a two-street network with agent-based simulations where information is provided with a delay that depends on the street load [18].", "Additionally, linear stability analysis of the corresponding delay differential equations has revealed that once the delay exceeds a critical value, the load imbalances diverge and congestion arises on both routes.", "We have proposed two possible strategies to prevent overreaction and congestion due to outdated information.", "First, if travel time information averaged over past observations is provided instead of a value measured at a single time point, the capacity of the system, i.e., the critical in-rate at which congestion occurs, increases for intermediate or high information delays.", "Thus, this strategy prevents traffic congestion in a similar way in which it prevents destabilizing feedback loops in decentral smart power grids [22] and reduces blocking probabilities in optical fiber networks [23].", "Second, we found that the collective overreaction is less severe if only a fraction of drivers receives the delayed information.", "This finding stands in contrast to the naïve expectation that providing available information to all individuals may optimally improve system performance.", "Indeed, our results show that random route choices of the uninformed drivers lead to the same increase of the critical in-rate at which congestion occurs as observed for averaged information, similar to previous work that investigated this effect on selfish routing without explicitly discussing the role of information delay [31], [32], [14].", "The combination of analytical insights in the minimal two street network and the observations in networks suggests that our results may robustly transfer to more complex settings as well.", "Determining the magnitude of the reported effects in larger empirical street networks is left for future work.", "However, as delays are inherent in providing information in any real world scenario, for example due to updating times and unpredictable feedback mechanisms [15], [16], we expect this effect to be relevant in a broad range of settings.", "In summary, our results suggest that providing partially or completely obscured information about travel times may prevent congestion induced by information delays when the delay exceeds a critical value.", "In the near future, these findings may be especially relevant for autonomous vehicle traffic [33].", "As traffic flow stability will not be limited by human imperfections such as random braking [26], [34], the collective dynamics of vehicle fleets may play a dominant role in the emergence and prevention of congestion.", "We thank Dirk Witthaut for valuable discussions.", "The authors thank the German Academic Scholarship Foundation for its role in initiating the project.", "V.K.", "thanks the Evangelisches Studienwerk.", "M.F.B.", "thanks the International Max Planck Research School for Intelligent Systems (IMPRS-IS).", "This research was supported through the Center for Advancing Electronics Dresden (cfaed)." ], [ "Competing interests", "The authors declare no competing interests.", "The code that supports the findings of this study will be made available online upon publication." ], [ "Linear stability analysis ", "With delayed information, the differential equation governing the street load dynamics is d Nid t= Ni(t) = Fi(Ni, { Nj(t-)}j) =in,i( {tsignalj[Nj(t-)]}j) - Ni(t)N0Ni(t)(Ni(t)/N0)-1 , where $j$ indexes all street segments, the rate of inflow $\\nu _{\\text{in},i}$ into a street $i$ is defined according to Eq.", "(REF ) and the signalled travel time for road $i$ , tsignali[Ni(t-)]= (Ni( t-)/N0)-1Ni(t-)/N0    , depends on the street load at time $t-\\tau $  .", "To find whether an equilibrium point $N_1^\\ast =N_2^\\ast =N^\\ast $ , defined by $\\frac{\\mathrm {d} N_i}{\\mathrm {d} t} = 0$ , is stable, we consider a small perturbation from equilibrium[29], [35] Ni(t) = Ni+ Ni(t)    , linearize the differential equation and insert the exponential ansatz Ni(t) = Ai et    .", "The possible values for the exponent $\\lambda $ are determined by the characteristic equation det(J0 +e- J-1 ) = 0   , with the entries J0, ij = .", "( Nj(t) Fi(t) ) Ni,j(t)=Ni,j(t-)=N J, ij =.", "( Nj(t-) Fi(t) ) Ni,j(t)=Ni,j(t-)=N    of the two Jacobi matrices.", "The equilibrium $N^\\ast $ is unstable if $\\mathfrak {Re}(\\lambda ) > 0$ for any solution $\\lambda $ of equation () .", "Thus, to determine the stability boundary shown in Fig.", "REF , we numerically calculate for each delay the critical in-rate $\\nu _\\text{in}^\\text{crit}$ for which $\\mathrm {max}\\left[\\mathfrak {Re}(\\lambda )\\right] = 0$ (see Fig.", "REF ).", "To analyze the stability of the system when providing averaged information, we simplify the integral in equation (REF ) by adding a second differential equation for the averaged street loads $N_{i,\\text{av}}=\\langle N_i(t-\\tau )\\rangle _{T_\\text{av}}$ with two explicit delays $\\tau $ and $\\tau + T_\\text{av}$ .", "The dynamical system is thus described by two delay differential equations for each street $i$ , d Nid t = Gi(Ni,{Nj,av}j) =in,i( {tsignalj[Nj,av]}j) -Ni(t)N0 Ni(t)(Ni(t)/N0)-1   , d Ni,avd t = 1Tav (Ni(t-) - Ni(t--Tav) ) .", "and yields the characteristic equation det(J0 +e- J1 +e-(+Tav)J2-1 ) = 0   , where the Jacobians are now 4x4 matrices with entries J0, = x(t)x(t) J1, = x(t)x(t-) J2, = x(t)x(t--Tav)   , for $x_\\alpha , x_\\beta \\,\\in \\, \\lbrace N_1, N_2, N_{1,\\text{av}}, N_{2,\\text{av}} \\rbrace $  .", "Again, we find the critical in-rate $\\nu _\\text{in}^\\text{crit}$ by determining for each delay $\\tau $ (and averaging time window $T_\\mathrm {av}$ ) the $\\nu _\\text{in}$ for which a solution $\\lambda $ to Eq.", "() with zero real part exists.", "Figure: Linear stability analysis predicts critical in-rate.", "We determine the stability boundary illustrated in Fig.", "and by finding the critical in-rate ν in crit \\nu _\\text{in}^\\mathrm {crit} for each information delay τ\\tau above which the traffic flow becomes unstable.", "This instability is characterized by a solution λ\\lambda with a positive real part of the characteristic equation () (or equivalently Eq.", "() when providing averaged information)." ] ]
2105.11790
[ [ "Affine Transport for Sim-to-Real Domain Adaptation" ], [ "Abstract Sample-efficient domain adaptation is an open problem in robotics.", "In this paper, we present affine transport -- a variant of optimal transport, which models the mapping between state transition distributions between the source and target domains with an affine transformation.", "First, we derive the affine transport framework; then, we extend the basic framework with Procrustes alignment to model arbitrary affine transformations.", "We evaluate the method in a number of OpenAI Gym sim-to-sim experiments with simulation environments, as well as on a sim-to-real domain adaptation task of a robot hitting a hockeypuck such that it slides and stops at a target position.", "In each experiment, we evaluate the results when transferring between each pair of dynamics domains.", "The results show that affine transport can significantly reduce the model adaptation error in comparison to using the original, non-adapted dynamics model." ], [ "Background", "Optimal Transport.", "Let $(X,d)$ be a metric space equipped with a lower semi-continuous cost function $c:X\\times X \\rightarrow \\mathbb {R}_{\\ge 0}$ , e.g the Euclidean distance $c(x,y) = \\Vert x-y\\Vert $ .", "Then, the optimal transport (OT) problem between two probability measures $\\nu _0, \\nu _1 \\in \\mathcal {P}(X)$ is given by the Kantorovich problem $\\mathrm {OT}_c(\\nu _0, \\nu _1) = \\min _{\\gamma \\in \\mathrm {ADM}(\\nu _0,\\nu _1)}\\mathbb {E}_\\gamma [c],$ where $\\mathrm {ADM}(\\nu _0,\\nu _1)$ is the set of joint probabilities with marginals $\\nu _0$ and $\\nu _1$ , and $\\mathbb {E}_\\nu [f]$ denotes the expected value of $f$ under $\\nu $ .", "The optimal $\\gamma $ minimizing (REF ) is called the OT plan.", "Denote by $\\mathcal {L}(X)$ the law of a random variable $X$ .", "Then, the OT problem extends to random variables $X,Y$ as $\\mathrm {OT}_c(X,Y) \\overset{\\Delta }{=}\\mathrm {OT}_c(\\mathcal {L}(X), \\mathcal {L}(Y)).$ Assuming that either of the considered measures are absolutely continuous, then the Kantorovich problem is equivalent to the Monge problem $\\mathrm {OT}_c(\\mu ,\\nu ) = \\min \\limits _{T: T_\\#\\mu = \\nu } \\mathbb {E}_{X\\sim \\mu }[c(X,T(X))],$ where the minimizing $T$ is called the OT map, and $T_\\#\\mu $ denotes the push-forward measure, which is equivalent to the law of $T(X)$ , where $X\\sim \\mu $ .", "Wasserstein distance.", "Let $X$ be a random variable over $\\mathbb {R}^d$ satisfying $\\mathbb {E}[\\Vert X-x_0\\Vert ^2]<\\infty $ for some $x_0\\in \\mathbb {R}^d$ , and thus for any $x\\in \\mathbb {R}^d$ .", "We denote this class of random variables by $\\mathcal {P}_2(\\mathbb {R}^d)$ .", "Then, the 2-Wasserstein distance $W_2$ between $X,Y\\in \\mathcal {P}_2(\\mathbb {R}^d)$ is defined as $W_2(X,Y) = \\mathrm {OT}_{d^2}(X, Y)^{\\frac{1}{2}},$ where $d(x,y) = \\Vert x-y\\Vert $ .", "The OT maps in this case are characterized by the following theorem Theorem 1 (Knott-Smith Optimality Criterion [11]) Let $X\\in \\mathcal {P}_2(\\mathbb {R}^d)$ .", "Furthermore, assume $T(x) = \\nabla \\phi (x)$ for a convex function and that $T(X)\\in \\mathcal {P}_2(\\mathbb {R}^d)$ .", "Then $T$ is the unique OT map between $\\mu $ and $T_\\#\\mu $ .", "When working with the 2-Wasserstein case, we can without loss of generality assume centered distributions, as $\\begin{aligned}W_2^2(X,Y) =& W_2^2(X-\\mathbb {E}[X], Y-\\mathbb {E}[Y])\\\\&+\\Vert \\mathbb {E}[X]-\\mathbb {E}[Y]\\Vert ^2,\\end{aligned}$ splitting the 2-Wasserstein distance into two independent terms concerning the $L^2$ distance between the means and the 2-Wasserstein distance between the centered measures.", "Furthermore, if we have an OT map $T^{\\prime }$ between $X-\\mathbb {E}[X]$ and $Y-\\mathbb {E}[Y]$ , then $T(x) = T^{\\prime }(x-\\mathbb {E}[X]) + \\mathbb {E}[Y],$ is the OT map between $X$ and $Y$ .", "One of the rare cases where the 2-Wasserstein distance admits a closed form solution, is between two multivariate Gaussian distributions $\\nu _i=\\mathcal {N}(\\mu _i,\\Sigma _i)$ , $i=1,2$ which is given by [12], [13], [14], [15] $\\begin{aligned}W_2^2(\\nu _1, \\nu _2) =& \\Vert \\mu _1-\\mu _2\\Vert ^2 + \\mathrm {Tr}(\\Sigma _1) + \\mathrm {Tr}(\\Sigma _2)\\\\&- 2 \\mathrm {Tr}\\left(\\Sigma _2^\\frac{1}{2} \\Sigma _1 \\Sigma _2^\\frac{1}{2}\\right)^\\frac{1}{2}.\\end{aligned}$ Furthermore, the OT map between $\\nu _1$ and $\\nu _2$ is given by the affine map $\\begin{aligned}F(x) = Ax + b,\\quad A &= \\Sigma _2^\\frac{1}{2}\\left(\\Sigma _2^\\frac{1}{2}\\Sigma _1\\Sigma _2^\\frac{1}{2}\\right)^{-\\frac{1}{2}}\\Sigma _2^\\frac{1}{2},\\\\b &= \\mu _2 - A\\mu _1,\\end{aligned}$ where $\\Sigma ^\\frac{1}{2}$ denotes the matrix square-root.", "When restricted to the set of Gaussians, the 2-Wasserstein distance results from a Riemannian metric [16].", "The Gaussians are special to the 2-Wasserstein distance also due to the following theorem.", "Theorem 2 (Gelbrich bound [17]) Let $X,Y\\in \\mathcal {P}_2(\\mathbb {R}^d)$ and $N_X,N_Y$ be their normal approximations sharing the same means and covariances.", "Then, $W_2(N_X,N_Y) \\le W_2(X,Y).$ Sim-to-Real Transfer.", "Assume a source domain with a dynamics model $g_s:\\mathcal {S}\\times \\mathcal {A}\\rightarrow \\mathcal {S}$ , mapping a state-action pair to the next state, and a corresponding target domain with a dynamics model $g_t$ .", "In a typical setting, the source domain would be given by simulation, and the target domain by real world data.", "Then, assume that there exists an underlying transfer map $T^*$ , so that $T^*(s, a, g_s(s,a)) = g_t(s,a)$ .", "Now, given state and action pairs $(s_i,a_i)$ , $i=1,...,N$ , data is collected from both domains as collections of triplets $X_s = \\lbrace (s_i,a_i,g_s(s_i,a_i)\\rbrace _{i=1}^N$ , and similarly for $X_t$ .", "The goal is then to infer $T^*$ using these two datasets.", "Limitations of Optimal Transport.", "As we apply OT to infer the map $T$ (in practice between the empirical measures defined by $X_t$ and $X_s$ ), some restrictions arise due to the nature of OT, formalized by the Brenier factorization theorem: Theorem 3 (Brenier Factorization [18]) Let $\\Omega \\subset \\mathbb {R}^n$ be bounded smooth domain, $T:\\Omega \\rightarrow \\mathbb {R}^n$ a Borel map which does not map positive volume into zero volume.", "Then, $T$ uniquely decomposes as $T=t \\circ u,$ where $u:\\Omega \\rightarrow \\Omega $ is volume preserving and $t=\\nabla \\psi $ is the gradient of a convex function $\\psi :\\mathbb {R}^n \\rightarrow \\mathbb {R}$ .", "In the linear world, this translates into the polar decomposition of a matrix: given an invertible matrix $A$ , we can write it uniquely as $A=PU$ , where $P$ is symmetric positive definite and $U$ orthogonal.", "According to Theorem REF , for the 2-Wasserstein distance, OT maps are precisely the gradient of convex maps.", "Therefore, we can only learn $f$ modulo volume preserving functions with OT." ], [ "Affine Transport", "In the following, we define the affine transport (AT) framework, a simplification of OT, which we will apply to the Sim-to-Real problem.", "It is motivated by cheap, closed-form computations that are simple to implement, and allows for theoretical guarantees.", "Affine Transport.", "Denote by $N_X$ the normal approximation of a random variable $X$ .", "Then, the affine transport map (AT map) $T_\\mathrm {aff}$ between a source $X$ and a target $Y$ is given by the OT map between $N_X$ and $N_Y$ , that is, the affine transformation $\\begin{aligned}T_\\mathrm {aff}(x) &= Ax + b\\\\A &= \\Sigma (Y)^\\frac{1}{2}\\left(\\Sigma (Y)^\\frac{1}{2} \\Sigma (X) \\Sigma (Y)^\\frac{1}{2}\\right)^{-\\frac{1}{2}}\\Sigma (Y)^\\frac{1}{2}\\\\b &= \\mu (Y)-A\\mu (X),\\end{aligned}$ where $\\Sigma (X)$ is the covariance matrix and $\\mu (X)$ the mean of $X$ .", "If needed, we will denote the AT map by $T_\\mathrm {aff}[X,Y]$ to emphasize the source and target.", "Connections with Optimal Transport.", "Naturally if $X$ and $Y$ are already Gaussian, the AT and OT maps are exactly the same.", "But are there other cases where AT and OT coincide?", "Yes, whenever $Y$ is a positive semi-definite affine transform of $X$ .", "In the following, we consider centered distributions and linear maps, but due to (REF ) and (REF ), the results can be extended to the non-centered case by replacing linear maps with affine maps.", "We start with a corollary of the Knott-Smith optimality criterion.", "Corollary 1 Let $X,Y\\in \\mathcal {P}_2(\\mathbb {R}^d)$ be centered, and assume that $Y=TX$ , where $T$ is a positive semi-definite matrix.", "Then, $T$ is the optimal transport map from $X$ to $Y$ .", "Apply Theorem REF to the convex $\\phi (x)= x^T T x$ , where $T$ is positive semi-definite.", "The next result then states when AT and OT coincide.", "Furthermore, it allows computing the 2-Wasserstein distance between arbitrary $X$ and $Y$ in closed-form using their normal approximations, given that $X$ and $Y$ differ by an affine transformation.", "Theorem 4 Let $X,Y\\in \\mathcal {P}_2(\\mathbb {R}^d)$ be centered and $Y = TX$ for a positive definite matrix $T$ .", "Then $T$ is also the OT map between $N_X$ and $N_Y$ , and $W_2(N_X,N_Y) = W_2(X,Y)$ .", "That is, $T = T_\\mathrm {aff}[X,Y]$ .", "Corollary REF states that $T$ is an OT map, and $\\Sigma (T N_X) = T\\Sigma (X)T = \\Sigma (Y).$ Therefore, $T N_X = N_Y$ , and by Theorem REF , $T$ is the OT map between $N_X$ and $N_Y$ .", "Finally, we compute $\\begin{aligned}W^2_2(N_X,N_Y) =& \\mathrm {Tr}[\\Sigma (X)] + \\mathrm {Tr}[T\\Sigma (X)T]\\\\&- 2\\mathrm {Tr}[T^\\frac{1}{2}\\Sigma (X) T^\\frac{1}{2}]\\\\=& \\operatornamewithlimits{arg\\,min}\\limits _{T:T(X) = Y}\\mathbb {E}_X[\\Vert X-T(X)\\Vert ^2]\\\\=& W_2^2(X,Y).\\end{aligned}$ Error Bounds for Affine Transport.", "If $X$ and $Y$ differ by a more complicated transformation than an affine one, how well does $T_\\mathrm {aff}X$ match $Y$ ?", "We answer this below, by first bounding the difference between the 2-Wasserstein distance of $X,Y$ and the 2-Wasserstein distance between their normal approximations.", "Proposition 1 Let $X,Y\\in \\mathcal {P}_2(\\mathbb {R}^d)$ and $N_X, N_Y$ be their normal approximations.", "Then, we have the bound $\\left| W_2(N_X, N_Y) - W_2(X,Y)\\right| \\le \\frac{2\\mathrm {Tr}\\left[\\left(\\Sigma (X)\\Sigma (Y)\\right)^\\frac{1}{2}\\right]}{\\sqrt{\\mathrm {Tr}[\\Sigma (X)] + \\mathrm {Tr}[\\Sigma (Y)]}}.$ By Theorem REF , we have $W_2(N_X,N_Y) \\le W_2(X,Y)$ .", "On the other hand, $\\begin{aligned}W^2_2(X,Y) &= \\min \\limits _{\\gamma \\in \\mathrm {ADM}(X,Y)} \\int _{\\mathbb {R}^d \\times \\mathbb {R}^d} \\Vert x-y\\Vert ^2 d\\gamma (x,y)\\\\&\\le \\int _{\\mathbb {R}^d \\times \\mathbb {R}^d}\\left(\\Vert x\\Vert ^2 + \\Vert y\\Vert ^2\\right) d\\gamma (x,y)\\\\&= \\mathrm {Tr}[\\Sigma (X)] + \\mathrm {Tr}[\\Sigma (Y)].\\end{aligned}$ Combining the above inequalities, we get $\\begin{aligned}&\\left|W_2(N_X,N_Y) - W_2(X,Y)\\right|\\\\\\le & \\left| \\sqrt{\\mathrm {Tr}[\\Sigma (X)] + \\mathrm {Tr}[\\Sigma (Y)]} - W_2(N_X,N_Y)\\right|.\\end{aligned}$ Let $a = \\mathrm {Tr}[\\Sigma (X)] + \\mathrm {Tr}[\\Sigma (Y)]$ , and so $W^2_2(N_X,N_Y) = a - b$ , where $b = 2\\mathrm {Tr}\\left[\\left(\\Sigma (X)\\Sigma (Y)\\right)^\\frac{1}{2}\\right]$ .", "Then the RHS of (REF ) can be written as $\\left|\\sqrt{a} - \\sqrt{a-b}\\right| = \\frac{|a - (a-b)|}{\\sqrt{a} + \\sqrt{a-b}} \\le \\frac{b}{\\sqrt{a}},$ where the inequality follows from positivity of $W_2(N_X,N_Y)=\\sqrt{a-b}$ .", "This implies the claim.", "Proposition REF implies the following 2-Wasserstein bound for the normal approximation of a random variable.", "Corollary 2 Let $X\\in \\mathcal {P}_2(\\mathbb {R}^d)$ and $N_X$ be its normal approximation.", "Then $W_2(N_X,X) \\le \\sqrt{2}\\mathrm {Tr}[\\Sigma (X)]^\\frac{1}{2}.$ Apply Proposition REF with $X=X$ and $Y=N_X$ .", "Finally, we show the following error bound for AT.", "which only depends on the target.", "Proposition 2 Let $X,Y\\in \\mathcal {P}_2(\\mathbb {R}^d)$ and $T_\\mathrm {aff}$ be the AT map from $X$ to $Y$ .", "Then, $W_2(T_\\mathrm {aff}X,Y) \\le \\sqrt{2}\\mathrm {Tr}\\left[\\Sigma (Y)\\right]^\\frac{1}{2}.$ By a direct computation $\\begin{aligned}W_2^2(T_\\mathrm {aff}X, Y) &\\le \\mathrm {Tr}[\\Sigma (T_\\mathrm {aff}X)] + \\mathrm {Tr}[\\Sigma (Y)]\\\\&= 2\\mathrm {Tr}[\\Sigma (Y)].\\end{aligned}$ The bound given in Proposition REF lets us define the following affinity score $\\rho _\\mathrm {aff}(X,Y) = 1 - \\frac{W_2(T_\\mathrm {aff}X,Y)}{\\sqrt{2}\\mathrm {Tr}[\\Sigma (Y)]^\\frac{1}{2}},$ describing how much $Y$ differs from being a positive-definite affine transformation of $X$ , as $0\\le \\rho _\\mathrm {aff}(X,Y) \\le 1$ , and $\\rho _\\mathrm {aff}(X,Y)=1$ when $X$ and $Y$ differ by an affine transformation, and $\\rho _\\mathrm {aff}(X,Y)=0$ when the OT plan between $T_\\mathrm {aff}X$ and $Y$ is the independent distribution.", "Figure: Sim2Real transfer illustrated for the blue puck simulated with isotr_low_offc settings.Affine Transport for Domain Adaptation.", "Now, given source and target data $X_s$ and $X_t$ , respectively, we can simply apply affine transport and use $T_\\mathrm {aff}[X_s, X_t]$ as the transfer map.", "However, as discussed in Sec.", ", this way we can only learn the underlying transfer map up to a volume preserving measure.", "More concretely, if $X_t=T^*(X_s)$ , where $T^*(x) = A^*x + b^*$ is affine, we can write the polar decomposition $A^*=PU$ .", "If the orthogonal $U$ is the identity, then AT recovers the underlying transfer map $A^*$ .", "Otherwise, we need a way to take into account the orthogonal part.", "This motivates us to 'preprocess' the data by applying Procrustes alignment [19]: given matrices $A,B$ , find the orthogonal matrix $R$ such that $R = \\operatornamewithlimits{arg\\,min}\\limits _{R:R^TR=I} \\Vert RA-B\\Vert .$ In practice $R$ is easy to compute.", "Let $M=BA^T$ with the singular-value decomposition $M=V_1 D V_2^T$ , then $R = V_1V_2^T$ .", "The transfer learning method resulting from applying Procrustes alignment and AT is summarized in Algorithm .", "Note that although we do recover a pair of a positive definite and an orthogonal matrix, there are no quarantees that $(T_\\mathrm {aff}, R) = (P,U)$ , i.e, we do not necessarily recover the polar decomposition for the underlying transfer map.", "InputInput OutputOutput Transfer map $T$ .", "$V_1, D , V_2^T \\leftarrow \\mathrm {SVD}(X_t,X_s^T)$ $R \\leftarrow V_1V_2^T$ $T^{\\prime } \\leftarrow T_\\mathrm {aff}[RX_s, X_t]$ $T \\leftarrow T^{\\prime } \\circ R$ Sim-to-Learn Transfer with AT" ], [ "Gym environments", "In order to assess the method's suitability for domain adaptation, we used the Hopper and HalfCheetah environment from OpenAI Gym [20].", "In these environments, we randomize the mass and length of each link, randomly invert the direction of each joint, and disable random joints.", "The state space in Hopper is 11-dimensional and contains the positions and velocities of each degree of freedom (the rotation angles of each of the 3 joints, the rotation of the whole system, and 2 translations), and the action space is 3-dimensional and consists of torques applied to each joint.", "The Ant state space is similar in structure, with 27-dimensional state space and 8-dimensional action space.", "Both state spaces ignore the absolute position of the agent in the world (the $x$ position in case of Hopper and $x$ and $y$ in Ant), but include the velocity along these axes.", "Domain adaptation is done through applying randomizations to various environment parameters; more specifically, we randomize link masses, joint length (as shown in Figure REF ), and by disabling or inverting the direction of certain joints.", "The randomization ranges are presented in Table REF .", "The results of this evaluation (using the Hopper and Ant environments are presented in Figures REF and REF .", "Based on these results, we can observe that using affine transport, despite its simplicity, significantly improves the state prediction performance, with the largest relative reduction apparent for the inverted domain.", "Figure: The Ant (fig:smallant, fig:medant, fig:bigant) and Hopper (fig:smallhop, fig:medhop, fig:bighop) environments with different joint lengths used for evaluationTable: Randomization ranges for the MuJoCo environmentsFigure: Results in the Ant environment: before adaptation (fig:antprior), the ρ aff \\rho _{\\mathrm {aff}} values  (fig:antrhos), and the errors after adaptation (fig:anterrors)Figure: Results in the Hopper environment: before adaptation (fig:hopperprior), the ρ aff \\rho _{\\mathrm {aff}} values  (fig:hopperrhos), and the errors after adaptation (fig:hoppererrors)" ], [ "Hockey Puck", "We experiment with the data describing a robot hand hitting a puck with a stick, trying to reach a specific point with the puck.", "We have simulated data $X_\\mathrm {sim}$ and real data $X_\\mathrm {real}$ consisting of 4-tuples $(x,y,z_0,z_1)$ , where $(x,y)$ is the end location of the puck and $(z_0,z_1)$ are latent variables encoding the action, which are the same for the simulated and real cases.", "In the real environment, we use two different pucks, a blue one and a red one.", "In the simulated case, we use pucks with 11 different isotropies.", "In Table REF we present the prior 2-Wasserstein distance between the simulated and real data, the 2-Wasserstein distance between the affinely transferred data and real data, and the $\\rho _\\mathrm {aff}(X_\\mathrm {sim},X_\\mathrm {real})$ score.", "The transfer is illustrated in Figure REF .", "In Figure REF , we vary the amount of real and simulated data points, sharing the same actions, used to learn the Sim2Real transport map $T_\\mathrm {aff}$ .", "Table: Pointwise error for the predicted state for Sim2Real and Simulation, as well as the estimated affinity score ρ aff (X,Y)\\rho _\\mathrm {aff}(X,Y) between the simulated XX and real YY datasets.Figure: Pointwise transfer error of AT for different simulations as we vary the amount of real data points used to compute the AT map.", "The shaded area represent the 2σ2\\sigma deviations." ], [ "Conclusion", "In this paper, we introduced affine transport and showed that it can be used to improve dynamics model performance through domain adaptation in robotics tasks, both in simulation and from simulation to reality.", "The method works well in low-dimensional state and action spaces, such as in case of hockey puck; it, however, struggles to generalize to new data in higher dimensional spaces, such as in the case of Hopper.", "Overall, the method can serve as a baseline for further research in domain adaptation through optimal transport.", "While the method successfully adapts the dynamic model, the local character of the fit makes it potentially problematic for use with reinforcement learning.", "In that scenario, a transport map fit in on an offline dataset, a potential distribution shift could pull the reinforcement learning algorithm towards solutions which may look good under under the model, but only so because these transition were not observed in the adaptation dataset.", "The simplest solution would be to update the transport map online; this, however, would attempt to fit a global affine model, which may be too simple to capture the true relations between source and target domains in all but the simplest cases.", "More intricate solution would attempt to prevent distribution shift from occuring in the first place; in that case, methods studied in the context of offline reinforcement learning could prove to be useful [21], [22]." ], [ "Acknowledgment", "This work was supported by the Academy of Finland (Flagship programme: Finnish Center for Artificial Intelligence FCAI and grants 294238, 319264, 292334, 334600, 328400).", "We acknowledge the computational resources provided by Aalto Science-IT project." ] ]
2105.11739
[ [ "Efficient Bayesian model selection for coupled hidden Markov models with\n application to infectious diseases" ], [ "Abstract Performing model selection for coupled hidden Markov models (CHMMs) is highly challenging, owing to the large dimension of the hidden state process.", "Whilst in principle the hidden state process can be marginalized out via forward filtering, in practice the computational cost of doing so increases exponentially with the number of coupled Markov chains, making this approach infeasible in most applications.", "Monte Carlo methods can be utilized, but despite many remarkable developments in model selection methodology, generic approaches continue to be ill-suited for such high-dimensional problems.", "Here we develop specialized solutions for CHMMs with weak inter-chain dependencies.", "Specifically we construct effective proposal distributions for the hidden state process that remain computationally viable as the number of chains increases, and that require little user input or tuning.", "This methodology is particularly applicable to individual-level infectious disease models characterized as CHMMs, in which each chain represents an individual, and the coupling represents contact between individuals.", "Since the only significant contacts are between susceptible and infectious individuals, and since multiple infection pathways are often possible, the resulting CHMMs naturally have low inter-chain dependencies.", "We demonstrate the utility of our methodology with an application to a study of highly pathogenic avian influenza in chickens." ], [ "Introduction", "Hidden Markov models (HMMs) are frequently utilized in the analysis of data that arise from some hidden time-varying state process.", "In a HMM this hidden state process is modeled as a Markov chain, and each observation is assumed to only relate to the state of the chain at the associated observation time.", "Coupled HMMs build on the HMM framework in order to model multiple interacting processes.", "Each interacting process is modeled as a Markov chain, the dynamics of which, at any given time, depend on the states of every chain at that time.", "CHMMs tend to exhibit nonlinear dynamics and unpredictable trajectories, and small changes to the model structure can give rise to radically different outcomes.", "Where multiple model specifications can be considered valid, there is a frequent need to select between competing models based on the available data.", "There has been a lot of interest within the Bayesian statistics community in developing methods for model selection , , , , .", "The focus of these approaches is on obtaining Bayes factors, which indicate the strength of evidence between pairs of models.", "This is commonly achieved by estimating the model evidence of each model, requiring the integration over all model parameters and hidden states, and from which the Bayes factors can be evaluated.", "Despite the many advances in Bayesian model selection methodology, a generic solution for high-dimensional models remains elusive.", "However, for certain classes of high-dimensional models it may be possible to construct specialized solutions.", "CHMMs typically have relatively few model parameters, which then impose a structure on the evolution of the hidden state process through time.", "The challenge for undertaking model selection in CHMMs stems from the need to reconstruct this high-dimensional hidden state process.", "In this paper we develop methodology to select between, or average over, competing CHMMs for which the hidden state process is governed by weak inter-chain dependencies.", "We construct proposal distributions for the hidden state process, which are then used in an importance sampling algorithm to determine the model evidence of each model.", "The key features of our proposal distributions are that they remain computationally viable as the number of individuals being modeled increases, and that they are constructed automatically from the model specification and data, requiring little input from the user.", "This allows them to be applied to different models and data sets with ease.", "CHMMs are particularly well suited for individual-level infectious disease models: we are rarely able to detect when an individual becomes infected or recovers, and so modeling the infection status of individuals as a hidden state process is necessary; by coupling chains we can model disease transmission between individuals; and diagnostic tests can return false positives or false negatives, which can be accounted for in CHMMs.", "Furthermore, inter-chain dependencies tend to be weak as the probability of a susceptible individual becoming infected does not depend on the exact states of all other individuals, but rather some lower dimensional summary providing the infection pressure.", "The methodology presented here can be a useful tool for public health professionals to compare the appropriateness of competing models in the light of real-world data.", "This can offer crucial insight into disease transmission characteristics, the potential number of future infections, and the effectiveness of possible intervention strategies.", "The paper structure is as follows.", "In Section we describe coupled hidden Markov models and discuss instances of weak inter-chain dependence.", "In Section we present our methodology for constructing proposal distributions of the hidden state process and discuss implementation considerations.", "In Section we introduce our application data set and provide results.", "We finish with a discussion in Section ." ], [ "Coupled hidden Markov models", "HMMs comprise a hidden state process $X_{1:T}$ that indicates the state of the system over some discrete set of times $t= 1,..,T$ , and a visible series of observations $Y_{1:T}$ that provide imperfect information about the hidden state process.", "Elements of the hidden state process are assumed to have a discrete finite state space, such that $X_t \\in \\mathcal {X}$ , where $\\mathcal {X}= \\lbrace 1,...,N_X \\rbrace $ .", "The hidden state process is initialized at time $t=1$ with the set of probabilities $P_i = \\mathbb {P}(X_{1} = i)$ , and trajectories of the hidden state process are governed by a Markov transition kernel $P_{ij} = \\mathbb {P}(X_{t} = j \\mid X_{t-1} = i)$ .", "Both $P_i$ and $P_{ij}$ may additionally depend on model parameters.", "The observations may have a discrete or continuous state space, and depend only on the hidden state process at time $t$ , i.e.", "$p(Y_t \\mid X_{1:T}) = p(Y_t \\mid X_{t})$ .", "CHMMs are a collection of HMMs in which the hidden state processes are linked .", "We denote $X_{t}^{1:K}$ as the states of $K$ chains at time $t$ .", "The state of chain $k$ at time $t$ depends on the states of all chains at time $t-1$ , so that the trajectories of the hidden state process are governed by Markov transition kernels ${P^k_{ij} = \\mathbb {P}(X^k_{t} = j \\mid X^k_{t-1} = i, X^{-k}_{t-1})}$ .", "An observation for chain $k$ at time $t$ is conditionally independent of the states of other chains given the hidden state $X^k_{t}$ , i.e.", "$p(Y^k_t \\mid X^{1:K}_{t}) = p(Y^k_t \\mid X^k_{t})$ .", "In many applications, the Markov transition kernels depend only on a set of summary statistics for the states of the other chains, i.e., ${P^k_{ij} = \\mathbb {P}(X^k_{t} = j \\mid X^k_{t-1} = i, \\mathcal {S}(X^{-k}_{t-1}))}$ , as shown in Figure REF .", "This is frequently the case when CHMMs are used to model infectious diseases where individuals have been observed longitudinally.", "Here, each chain $X^k_{1:T}$ represents the hidden state process of individual $k$ , and the state space represents the possible set of infection states through which individuals can traverse.", "Commonly used infection states include susceptible, infectious, and removed, but others may be used depending on the etiology of the disease.", "The more contacts a susceptible individual has with infected individuals, the greater the probability this individual has of becoming infected.", "This transmission process between individuals is represented by the coupling of the chains, but is highly abstracted.", "An extreme example is when homogeneous population mixing is assumed, in which case the probability of a susceptible individual becoming infected does not depend on which individuals are infectious, only the total number of infectious individuals.", "CHMMs with such dependency structures typically exhibit weak inter-chain dependencies, which can be exploited in order to undertake inference using methods that would typically perform poorly in high dimensional problems.", "Figure: Left: An example of a coupled hidden Markov model with three chains.", "X t k X^k_t represent values of the hidden state process, and Y t k Y^k_t represent observations.", "Superscripts indicate the chain number, and subscripts indicate the time index.", "Arrows show dependencies between the different variables.", "Right: An example of a CHMM with weak inter-chain dependencies.", "With all other values fixed, the value of X t 1 X^1_t depends only on X t-1 1 X^1_{t-1}, X t+1 1 X^1_{t+1}, Y t 1 Y^1_t, and summary statistics for X t-1 2:K X^{2:K}_{t-1} (S t-1 S_{t-1}) and the transition probabilities of the other chains between times tt and t+1t+1 (S t:t+1 S_{t:t+1}).", "Such CHMMs are commonly used as individual-level infectious disease models." ], [ "Methods", "When undertaking Bayesian inference the usual goal is to evaluate the posterior distribution of all unknown parameters and hidden states conditioned on the available data.", "The posterior distribution is given by Bayes' Theorem: $p(\\theta , X_{1:T}^{1:K} \\mid Y_{1:T}^{1:K}) \\propto p(Y_{1:T}^{1:K} \\mid X_{1:T}^{1:K}, \\theta ) p (X_{1:T}^{1:K} \\mid \\theta ) p (\\theta ),$ where $p (\\theta )$ is a user defined prior distribution, $p (X_{1:T}^{1:K} \\mid \\theta )$ is the product of transition probabilities of the hidden state process, and $p(Y_{1:T}^{1:K} \\mid X_{1:T}^{1:K}, \\theta )$ is the density induced by the observation model.", "In non-trivial situations the posterior distribution is intractable, and so is often numerically approximated using Monte Carlo methods such as MCMC.", "When undertaking model selection we are also interested in evaluating the normalizing constant to Equation (REF ), termed the model evidence.", "To evaluate the model evidence we are required to integrate over all model parameters and hidden states: $p(Y_{1:T}^{1:K}) = \\int \\int p(Y_{1:T}^{1:K} \\mid X_{1:T}^{1:K}, \\theta ) p (X_{1:T}^{1:K} \\mid \\theta ) p (\\theta ) \\; \\rm {d}X_{1:T}^{1:K} \\; \\rm {d}\\theta .$ Taking the ratio of the model evidence terms from two models provides the Bayes factor, which indicates the strength of evidence between the models .", "MCMC does not estimate the model evidence, but many other Monte Carlo approaches have been developed with this purpose in mind.", "Some commonly used approaches include importance sampling , SMC , Chib's method , , , power posteriors , and nested sampling .", "Such Monte Carlo algorithms rarely perform well when applied to problems with a large number of hidden states.", "There are exceptions when the full conditional distribution $p(X_{1:T}^{1:K} \\mid Y_{1:T}^{1:K}, \\theta )$ is tractable, or if the (marginal) likelihood $p(Y_{1:T}^{1:K} \\mid \\theta )$ is otherwise available.", "For instance, in the full conditional distribution of the hidden state process is incorporated into the proposal distribution in an importance sampling algorithm, and demonstrates that importance sampling can be viable even with a sizable number of hidden states.", "When the full conditional distribution $p(X_{1:T}^{1:K} \\mid Y_{1:T}^{1:K}, \\theta )$ is intractable, there are no Monte Carlo algorithms that can reliably estimate the model evidence for general CHMMs.", "However, for CHMMs with weak inter-chain dependency, we can design effective Monte Carlo solutions to this problem.", "The aim of this paper is to provide simple and effective approaches for estimating the model evidence via importance sampling." ], [ "Importance sampling with hidden states", "For importance sampling we require two proposal distributions, $q(\\theta )$ and $q(X^{1:K}_{1:T} \\mid \\theta )$ , satisfying $q(\\theta ) q(X^{1:K}_{1:T} \\mid \\theta ) >0$ whenever $p(\\theta , X^{1:K}_{1:T} \\mid Y^{1:K}_{1:T})>0$ .", "Then, for $i=1,...,N$ , $\\theta ^{(i)}$ and $X^{1:K;(i)}_{1:T}$ are sampled from the proposal distributions and assigned importance weights $w^{(i)} = \\frac{p\\left(Y^{1:K}_{1:T} \\mid X^{1:K;(i)}_{1:T}, \\theta ^{(i)} \\right) p\\left( X^{1:K;(i)}_{1:T} \\mid \\theta ^{(i)} \\right) p\\left( \\theta ^{(i)} \\right)}{ q\\left( X^{1:K;(i)}_{1:T} \\mid \\theta ^{(i)} \\right) q\\left( \\theta ^{(i)} \\right) }.$ Averaging the importance weights provides an unbiased estimate of the model evidence.", "The efficiency of importance sampling is dramatically affected by the choice of proposal distributions, with the optimal choices being $p(\\theta \\mid Y^{1:K}_{1:T})$ and $p(X^{1:K}_{1:T} \\mid Y^{1:K}_{1:T}, \\theta )$ .", "In this case the importance weights equal the model evidence, since we are sampling from the target distribution.", "In most situations these distributions are not available, and should instead be approximated.", "We discuss suitable approximations for each proposal distribution below." ], [ "Choice of $q(\\theta )$", "An efficient proposal distribution $q(\\theta )$ will be a good approximation of $p(\\theta \\mid Y^{1:K}_{1:T})$ .", "Given the success of MCMC algorithms at generating samples from the posterior distribution even when we have hidden states, one possibility is to design proposal distributions based on the output of an MCMC algorithm targeting $p(\\theta \\mid Y^{1:K}_{1:T})$ .", "suggests using a multivariate t-distribution, where the location and scale parameters can be estimated from the output of an MCMC algorithm.", "Alternatively, suggests the use of defense mixtures, for example we could use $q(\\theta ) = \\lambda \\mathcal {N}(\\theta ; \\mu ,\\Sigma ) + (1-\\lambda ) p(\\theta )$ , where $\\lambda $ is a mixing proportion and $\\mathcal {N}(\\theta ; \\mu ,\\Sigma )$ is a multivariate Gaussian distribution with mean and covariance being estimated from MCMC output.", "When the posterior distribution is unimodal, both selections tend to perform well , although the best choice is problem specific.", "If the posterior distribution is multi-modal, or otherwise poorly approximated by these distributions, it is possible to fit a mixture of distributions to design the proposal distribution.", "The optimal proposal for the hidden state process is the full conditional distribution $p(X^{1:K}_{1:T} \\mid Y^{1:K}_{1:T}, \\theta )$ , in which case the importance weights simplify to $w^{(i)} = \\frac{p\\left(Y^{1:K}_{1:T} \\mid \\theta ^{(i)} \\right) p\\left( \\theta ^{(i)} \\right)}{ q\\left( \\theta ^{(i)} \\right) }.$ In other words, the hidden state process is marginalized out and we are effectively performing importance sampling only in the parameter space.", "Since the dimension of the model parameters is usually small compared to that of the hidden state process, the variance of the model evidence estimate will be much lower.", "In principle, if the hidden state process takes discrete values then the full conditional distribution can be evaluated and sampled from by using the forward-filtering backward-sampling algorithm (FFBS).", "FFBS uses a forward recursion to evaluate the sequence of filtering distributions $p(X^{1:K}_t \\mid Y^{1:K}_{1:t}, \\theta )$ for $t=1,...,T$ , followed by a backward recursion to sample from the sequence of distributions $p(X^{1:K}_t \\mid X^{1:K}_{t+1}, Y^{1:K}_{1:T}, \\theta )$ .", "With each $X^{1:K}_t$ taking $N_X$ possible states, the computational cost of FFBS is $\\mathcal {O}(N_X^{2}T)$ , leading to poor computational scalability in CHMMs as the number of chains increases.", "Specifically, if we assume that each chain has a common state space, such that $X_t^k \\in \\mathcal {S}$ , where $\\mathcal {S} = \\left\\lbrace 1,...,S \\right\\rbrace $ , then the number of possible states across all chains is $N_X = S^K$ , and the computational cost of FFBS becomes $\\mathcal {O}(S^{2K}T)$ .", "Since the computational cost scales exponentially with $K$ , FFBS can only be used in CHMMs with very few chains.", "When the computational cost of performing FFBS is prohibitive, alternative proposal distributions must be considered.", "Due to the large dimension of the hidden state process, approximating $p(X^{1:K}_{1:T} \\mid Y^{1:K}_{1:T}, \\theta )$ is highly challenging.", "Here, we consider adapting methods that have been utilized in MCMC to update the hidden state process conditional on a set of parameter values.", "Many such methods, for example single-site updates and block updates , are computationally fast, but can cause slow mixing in MCMC.", "Recently the Individual FFBS algorithm was developed, which is computationally scalable in the number of chains, and maintains strong mixing in MCMC for CHMMs with weak inter-chain dependencies.", "IFFBS updates the hidden state process using Gibbs sampling on each chain.", "That is, sampling from the distributions ${p\\left(X^{k}_{1:T} \\mid X^{-k}_{1:T}, Y^{k}_{1:T}, \\theta \\right).", "}$ This is achieved by sequentially performing a modified version of FFBS on each chain, conditional on the states of all other chains through time.", "The computational cost of IFFBS is $\\mathcal {O}(S^3 K T)$ , which is linear in number of chains and a significant improvement over FFBS.", "In MCMC, IFFBS can cause poor mixing when chains are highly correlated, but this tends to be uncommon for CHMMs with weak inter-chain dependencies.", "Here we focus on further developing IFFBS for use in importance sampling." ], [ "Using IFFBS to design importance proposal distributions", "We develop two proposal distributions that utilize IFFBS.", "For each proposed parameter $\\theta ^{(i)}$ , we obtain a set of samples $\\widehat{X}_{1:T}^{1:K;(j)}$ , $j=1,\\dots ,N$ distributed as $p(X^{1:K}_{1:T} \\mid Y^{1:K}_{1:T}, \\theta )$ using IFFBS.", "Our proposal distributions are then constructed based on these guiding samples." ], [ "Direct application of IFFBS", "The simplest approach is to use IFFBS itself as a proposal distribution, initiated using the guiding samples.", "We call this direct IFFBS (DIFFBS).", "We determine a high posterior probability realization of the hidden state process from the guiding samples, denoted $\\widetilde{X}^{1:K}_{1:T}$ , and then propose from $q(X^{1:K}_{1:T} \\mid \\theta ^{(i)}) = p(X^1_{1:T} \\mid \\widetilde{X}^{2:K}_{1:T}, Y^{1}_{1:T}, \\theta ^{(i)}) \\prod _{k=2}^{K-1} p(X^k_{1:T} \\mid X^{1:k-1}_{1:T}, \\widetilde{X}^{k+1:K}_{1:T}, Y^{k}_{1:T}, \\theta ^{(i)}) \\\\\\times p(X^K_{1:T} \\mid X^{1:K-1}_{1:T}, Y^{K}_{1:T}, \\theta ^{(i)}).$ Here, each term in the product is an IFFBS update step for a single chain.", "Proposing one realization of the hidden state process in this manner is computationally expensive, as obtaining the guiding samples requires $N$ iterations of IFFBS over every chain.", "Instead, we can propose multiple realizations of the hidden state process for each $\\theta ^{(i)}$ based on the same guiding samples and then use the following importance weight: $w^{(i)} = \\frac{p\\left( \\theta ^{(i)} \\right)}{ q\\left( \\theta ^{(i)} \\right) } \\frac{1}{L} \\sum _{l=1}^{L} \\frac{p\\left(Y^{1:K}_{1:T} \\mid X^{1:K;(i,l)}_{1:T}, \\theta ^{(i)} \\right) p\\left( X^{1:K;(i,l)}_{1:T} \\mid \\theta ^{(i)} \\right) }{ q\\left( X^{1:K;(i,l)}_{1:T} \\mid \\theta ^{(i)} \\right) },$ This can be performed using the same realization $\\widetilde{X}^{1:K}_{1:T}$ for each of the $L$ proposal distributions, or alternatively we could use $L$ different realizations from the guiding samples.", "Whilst DIFFBS has the benefit of being a simple adaptation of IFFBS, it does not necessarily provide a good approximation of the optimal proposal distribution.", "Consider decomposing the optimal proposal distribution by chain, $p(X^{1:K}_{1:T} \\mid Y^{1:K}_{1:T}, \\theta ^{(i)}) = p(X^1_{1:T} \\mid Y^{1:K}_{1:T}, \\theta ^{(i)}) \\prod _{k=2}^{K} p(X^k_{1:T} \\mid X^{1:k-1}_{1:T},Y^{k:K}_{1:T}, \\theta ^{(i)}),$ and then comparing this with Equation (REF ).", "It is clear that we are proposing from a series of conditional distributions in place of marginal distributions, i.e.", "a comparison between ${p(X^k_{1:T} \\mid X^{1:k-1}_{1:T}, \\widetilde{X}^{k+1:K}_{1:T}, \\theta ^{(i)}, Y^{k}_{1:T})}$ and ${p(X^k_{1:T} \\mid X^{1:k-1}_{1:T}, \\theta ^{(i)},Y^{k:K}_{1:T})}$ shows that we are conditioning on values for $X^{k+1:K}_{1:T}$ instead of marginalizing them out.", "This will likely lead to under-dispersed proposal distributions, and high variance model evidence estimates.", "Here we construct a second algorithm based on the decomposition of the optimal proposal distribution given in Equation (REF ).", "The basic principle is to use the guiding samples to obtain Monte Carlo estimates of each marginal distribution using IFFBS.", "As such, we refer to this algorithm as marginal IFFBS, or MIFFBS.", "For the first term, $p(X^{1}_{1:T} \\mid Y^{1:K}_{1:T} , \\theta )$ , we decompose in a similar manner as FFBS, $p( X^{1}_{1:T} \\mid Y^{1:K}_{1:T} , \\theta ) = p(X^{1}_T \\mid Y^{1:K}_{1:T}, \\theta ) \\prod _{t=1}^{T-1} p(X^{1}_t \\mid X^{1}_{t+1:T}, Y^{1:K}_{1:T}, \\theta ).$ Note that we no longer have the Markov property, as we are marginalizing out the other chains.", "The first term on the right hand side of Equation (REF ) can be expressed as $ p(X^{1}_T \\mid Y^{1:K}_{1:T}, \\theta ) = \\int p(X^{1}_T \\mid X_{1:T}^{2:K}, Y^{1}_{1:T}, \\theta ) p(X_{1:T}^{2:K} \\mid Y^{1:K}_{1:T}, \\theta ) \\rm {d}X_{1:T}^{2:K},$ for which we can obtain a Monte Carlo approximation by sampling from ${p(X_{1:T}^{2:K} \\mid Y^{1:K}_{1:T}, \\theta )}$ and giving each sample a weight proportional to $p(X^{1}_T \\mid X_{1:T}^{2:K}, Y^1_{1:T}, \\theta )$ .", "To obtain such a sample, we can take our guiding samples and discard the values for the first chain, leaving $\\widehat{X}_{1:T}^{2:K;(j)}$ , $j=1,...,N$ .", "An approximation to $p(X^{1}_T \\mid Y^{1:K}_{1:T}, \\theta )$ is then given by $ \\widehat{p}(X^{1}_T \\mid Y^{1:K}_{1:T}, \\theta ) = \\frac{1}{N}\\sum _{j=1}^{N} p(X^{1}_T \\mid \\widehat{X}_{1:T}^{2:K;(j)}, Y^1_{1:T}, \\theta ).", "$ Each component of the sum is obtained from the forward recursion of IFFBS conditioned on $\\widehat{X}_{1:T}^{2:K;(j)}$ .", "We then propose $x^{1}_{T} \\sim \\widehat{p}(X^{1}_T \\mid Y^{1:K}_{1:T}, \\theta )$ and initialize the backward sampling recursion of MIFFBS for the first chain.", "In the backward recursion we need to approximate the terms in the product of Equation (REF ), which are conditioned on parts of the hidden state process that have already been proposed.", "These terms are again marginal distributions that can be expressed as $ p(X^{1}_t \\mid x^{1}_{t+1:T}, Y^{1:K}_{1:T}, \\theta ) = \\int p(X^{1}_t \\mid x^{1}_{t+1:T}, X^{2:K}_{1:T}, Y^{1}_{1:t}, \\theta ) p(X^{2:K}_{1:T} \\mid x^{1}_{t+1:T}, Y^{1:K}_{1:T}, \\theta ) \\rm {d}X^{2:K}_{1:T} .", "$ In order to construct Monte Carlo approximations as before, we could consider obtaining samples from ${p(X^{2:K}_{1:T} \\mid x^{1}_{t+1:T}, Y^{1:K}_{1:T}, \\theta )}$ and giving each sample a weight proportional to ${p(X^{1}_t \\mid x^{1}_{t+1:T}, X^{2:K}_{1:T}, Y^{1}_{1:t}, \\theta )}$ .", "However, this will be computationally expensive to undertake at every time step.", "Instead, note that $ p(X^{2:K}_{1:T} \\mid x^{1}_{t+1:T}, Y^{1:K}_{1:T}, \\theta ) \\propto p(x^{1}_{t+1:T} \\mid X^{2:K}_{1:T}, Y^{1:K}_{1:T}, \\theta ) p(X^{2:K}_{1:T} \\mid Y^{1:K}_{1:T}, \\theta ), $ suggesting that we can make use of the existing guiding samples, which are distributed according to $p(X^{2:K}_{1:T} \\mid Y^{1:K}_{1:T}, \\theta )$ .", "Hence we can use the Monte Carlo approximation $ \\widehat{p}(X^{1}_t \\mid x^{1}_{t+1:T}, Y^{1:K}_{1:T}, \\theta ) \\propto \\sum _{j=1}^{N} p(X^{1}_t \\mid x^{1}_{t+1:T}, \\widehat{X}_{1:T}^{2:K;(j)}, Y^{1}_{1:t}, \\theta ) p(x^{1}_{t+1:T} \\mid \\widehat{X}_{1:T}^{2:K;(j)}, Y^{1}_{1:T}, \\theta ), $ in order to propose $x^{1}_{t}$ .", "Since we are approximating a probability mass function on a finite state-space, we can easily self-normalize.", "The term ${p(X^{1}_t \\mid x^{1}_{t+1:T}, \\widehat{X}_{1:T}^{2:K;(j)}, Y^{1}_{1:t}, \\theta )}$ is obtained by performing a backward step of IFFBS between $t+1$ and $t$ .", "The term $p(x^{1}_{t+1:T} \\mid \\widehat{X}_{1:T}^{2:K;(j)}, Y^{1}_{1:T}, \\theta )$ can be decomposed as $p(x^{1}_{t+1:T} \\mid \\widehat{X}_{1:T}^{2:K;(j)}, Y^{1}_{1:T}, \\theta ) = \\\\ p(x^{1}_T \\mid \\widehat{X}_{1:T}^{2:K;(j)}, Y^{1}_{1:T}, \\theta ) \\prod _{u=t+1}^{T-1} p(x^{1}_{u} \\mid x^{1}_{u+1:T}, \\widehat{X}_{1:T}^{2:K;(j)}, Y^{1}_{1:u}, \\theta ),$ the components of which have already been calculated as part of the backwards recursion.", "Repeating this process backwards through time gives a set of proposed values $x^{1}_{1:T}$ approximately distributed according to $p(X^{1}_{1:T} \\mid Y^{1:K}_{1:T} , \\theta ) $ .", "Having proposed values for the first chain, we now wish to propose values for the remaining chains sequentially.", "That is, we wish to approximate $ p( X^k_{1:T} \\mid x^{1:k-1}_{1:T}, Y^{k:K}_{1:T} , \\theta ) = p( X^k_{T} \\mid x^{1:k-1}_{1:T}, Y^{k:K}_{1:T} , \\theta ) \\prod _{t=1}^{T} p( X^k_{t} \\mid X^k_{t+1:T}, x^{1:k-1}_{1:T}, Y^{k:K}_{1:t} , \\theta ).", "$ Again, each of these terms are marginal distributions, and so for $k<K$ $p( X^k_{T} \\mid x^{1:k-1}_{1:T}, Y^{k:K}_{1:T} , \\theta ) = \\\\ \\int p( X^k_{T} \\mid x^{1:k-1}_{1:T}, X^{k+1:K}_{1:T}, Y^k_{1:T} , \\theta ) p( X^{k+1:K}_{1:T} \\mid x^{1:k-1}_{1:T}, Y^{k:K}_{1:T}, \\theta ) \\rm {d}X_{1:T}^{k+1:K}.$ Considering a Monte Carlo approximation, one approach would be to obtain samples ${p( X^{k+1:K}_{1:T} \\mid x^{1:k-1}_{1:T}, Y^{k:K}_{1:T}, \\theta )}$ that are given weights $p( X^k_{T} \\mid x^{1:k-1}_{1:T}, X^{k+1:K}_{1:T}, Y^k_{1:T} , \\theta ).$ However, at this stage we have a weighted sample $\\widehat{X}_{1:T}^{k:K}$ approximating $p( X^{k:K}_{1:T} \\mid x^{1:k-1}_{1:T}, Y^{k:K}_{1:T}, \\theta )$ .", "It is computationally cheaper to simply discard all samples for chain $k$ , and treat the remainder of the weighted sample as being representative of $p( X^{k+1:K}_{1:T} \\mid x^{1:k-1}_{1:T}, Y^{k:K}_{1:T}, \\theta )$ .", "Proposing values for $x^k_{1:T}$ then proceeds as in the case for chain 1.", "Finally, for $k=K$ we can sample from the full conditional distribution $p( X^K_{1:T} \\mid x^{1:K-1}_{1:T}, Y^K_{1:T} , \\theta )$ using IFFBS.", "As with many other algorithms that use sequential weighted approximations to characterize distributions of interest, degeneracy issues may occur.", "Similar to SMC, this can be tracked by monitoring the effective sample size (ESS) of the guiding samples.", "It makes sense to regenerate the guiding samples when the ESS falls below some threshold, say $\\frac{N}{2}$ , as this indicates that the approximation is being dominated by a small number of samples.", "To do so we generate a new set of guiding samples from scratch using IFFBS conditioned on any values of the hidden state process we have proposed thus far.", "Since we only need to obtain guiding samples for chains for which we do not have proposed values, this regeneration step becomes computationally cheaper as the algorithm progresses.", "The full algorithm is presented in Algorithm 1.", "[p] MIFFBS algorithm.", "Obtain sample $\\widehat{X}_{1:T}^{1:K;(1:N)}$ distributed as $p(X_{1:T}^{1:K} \\mid Y_{1:T}^{1:K}, \\theta )$ using IFFBS, and set weights $W^{(n)} = \\frac{1}{N}$ .", "$k=1,...,K$ - If $\\mbox{ESS}<\\frac{N}{2}$ obtain a new sample $\\widehat{X}_{1:T}^{k:K;(1:N)}$ distributed as $p(X_{1:T}^{k:K} \\mid x_{1:T}^{1:k-1}, Y_{1:T}^{k:K}, \\theta )$ using IFFBS, and set weights $W^{(n)} = \\frac{1}{N}$ .", "$t=1,..,T$ - Calculate modified forward filtered probabilities $p(X_t^{k} \\mid x_{1:t+1}^{1:k-1},\\widehat{X}_{1:t+1}^{k+1:K;(n)}, Y_{1:t}^{k}, \\theta )$ for $X_t^{k} \\in \\mathcal {S}$ and $n=1,...,N$ (see Algorithm REF ).", "- Calculate weighted average of the final modified forward filter probabilities for $X_T^{k} \\in \\mathcal {S}$ $\\widehat{p} (X_T^{k} \\mid x_{1:T}^{1:k-1}, Y_{1:T}^{k:K}, \\theta ) = \\sum _{n=1}^{N} W^{(n)} p(X_T^{k} \\mid x_{1:T}^{1:k-1},\\widehat{X}_{1:T}^{k+1:K;(n)}, Y_{1:T}^{k}, \\theta ).$ - Sample $x_T^{k}$ from $\\widehat{p} (X_T^{k} \\mid x_{1:T}^{1:k-1}, Y_{1:T}^{k:K}, \\theta )$ and for $n=1,...,N$ set weight $\\mbox{ $ W^{(n)} \\propto W^{(n)} p(x_T^{k} \\mid x_{1:T}^{1:k-1},\\widehat{X}_{1:T}^{k+1:K;(n)}, Y_{1:T}^{k}, \\theta ).", "$ } $ $t=T-1,...,1$ - Calculate the backward smoothing probabilities for $X_t^{k} \\in \\mathcal {S}$ and $n=1,...,N$ $p(X_t^{k} \\mid x_{1:T}^{1:k-1}, x_{t+1:T}^{k}, \\widehat{X}_{1:T}^{k+1:K;(n)}, Y_{1:T}^{k}, \\theta ) = \\\\\\frac{p(X_t^{k} \\mid x_{1:t+1}^{1:k-1},\\widehat{X}_{1:t+1}^{k+1:K;(n)}, Y_{1:t}^{k}, \\theta ) p(x_{t+1}^{k} \\mid x_{t}^{1:k-1},X_{t}^{k},\\widehat{X}_{t}^{k+1:K;(n)},\\theta ) }{p(x_{t+1}^{k} \\mid x_{1:t+1}^{1:k-1},\\widehat{X}_{1:t+1}^{k+1:K;(n)}, Y_{1:t}^{k}, \\theta )}$ - Calculate weighted average of the backward smoothing probabilities $X_t^{k} \\in \\mathcal {S}$ $\\widehat{p} (X_t^{k} \\mid x_{1:T}^{1:k-1}, x_{t+1:T}^{k}, Y_{1:T}^{k:K}, \\theta ) = \\sum _{n=1}^{N} W^{(n)} p(X_t^{k} \\mid x_{1:T}^{1:k-1}, x_{t+1:T}^{k},\\widehat{X}_{1:T}^{k+1:K;(n)}, Y_{1:T}^{k}, \\theta ).$ - Sample $x_t^{k}$ from $\\widehat{p} (X_t^{k} \\mid x_{1:T}^{1:k-1}, x_{t+1:T}^{k}, Y_{1:T}^{k:K}, \\theta )$ and for $n=1,...,N$ set weight $\\mbox{ $ W^{(n)} \\propto W^{(n)} p(x_t^{k} \\mid x_{1:T}^{1:k-1}, x_{t+1:T}^{k},\\widehat{X}_{1:T}^{k+1:K;(n)}, Y_{1:T}^{k}, \\theta ).", "$ } $ [t] Calculate modified forward filtered probabilities for individual $k$ and sample $n$ in MIFFBS.", "$t=1$ - Set initial probabilities $P^k_1(X_1^{k}) = p(X_1^{k} \\mid \\theta )$ for $X_1^{k} \\in \\mathcal {S}$ .", "- Calculate predictive probabilities for $X_t^{k} \\in \\mathcal {S}$ $ P^k_t(X_t^{k}) = \\sum _{s=1}^{S} p(X_t^{k} \\mid x_{t-1}^{1:k-1},X_{t-1}^{k}=s,\\widehat{X}_{t-1}^{k+1:K;(n)},\\theta ) p(X_{t-1}^{k}=s \\mid x_{1:t}^{1:k-1},\\widehat{X}_{1:t}^{k+1:K;(n)}, Y_{1:t-1}^{k}, \\theta ).", "$ $t<T$ - Calculate the product of the transition probabilities of remaining individuals for $X_t^{k} \\in \\mathcal {S}$ $ Q_{t}^k(X_t^{k}) = p(x_{t+1}^{1:k-1},\\widehat{X}_{t+1}^{k+1:K;(n)} \\mid x_{t}^{1:k-1}, X_t^{k}, \\widehat{X}_{t}^{k+1:K;(n)}, \\theta ) $ - Set $Q^k_T(X_T^{k}) = 1$ for $X_T^{k} \\in \\mathcal {S}$ .", "- Calculate modified forward filtered probabilities for $X_t^{k} \\in \\mathcal {S}$ $ p(X_t^{k} \\mid x_{1:t+1}^{1:k-1},\\widehat{X}_{1:t+1}^{k+1:K;(n)}, Y_{1:t}^{k}, \\theta ) = p(Y_t^k \\mid X_t^{k}, \\theta ) Q_{t}^k(X_t^{k}) P^k_t(X_t^{k}).", "$" ], [ "Considerations for implementation", "IFFBS can be coded to give scaling $\\mathcal {O}(S^3 K T)$ or $\\mathcal {O}(S K^2 T)$ .", "In this paper we are primarily interested in the scaling properties as the number of chains increases, and so use the former construction.", "This requires tracking appropriate summary statistics, which in general would be the number of chains in each state at any time, in order to avoid repeatedly summing over every chain.", "For applications with a small number of chains, the second construction may be more efficient.", "For some models the suggested proposal distributions will be invalid if they lack full posterior support.", "In infectious disease models, for example, DIFFBS will provide a valid proposal distribution if there is an exterior source of infection pressure, but might not otherwise.", "In particular, consider DIFFBS being initialized on a realization of the hidden state process in which individual 1 is the only infected individual over some time interval $\\mathcal {T}$ , after which others are infected.", "Proposed values for individual 1 will always include a period of infection over $\\mathcal {T}$ , as without this there is zero probability of other individuals becoming infected at later times.", "For MIFFBS the probability of constructing an invalid proposal distribution decreases as the number of guiding samples increases." ], [ "Application: Avian influenza in chickens", "Our motivating example is a study of highly pathogenic avian influenza (HPAI) in chickens .", "Chickens were genetically modified (transgenic chickens) to expresses a short-hairpin RNA that interferes with virus propagation, providing resistance to HPAI.", "In order to test the efficacy of the modification, a series of transmission experiments were undertaken.", "A crossed experimental design was employed with four independent experiments.", "In each experiment there are five challenge chickens (inoculated with HPAI at the start of the experiment), and twelve uninfected in-contact chickens.", "Group 1 consisted of five non-transgenic challenge chickens and twelve non-transgenic in-contact chickens, group 2 consisted of five non-transgenic challenge chickens and twelve transgenic in-contact chickens, group 3 consisted of five transgenic challenge chickens and twelve non-transgenic in-contact chickens, and group 4 consisted of five transgenic challenge chickens and twelve transgenic in-contact chickens.", "Each experiment was undertaken over 10 days, with the chickens being observed twice daily.", "Chickens were removed from the experiment for a variety of reasons: dying naturally, removed for being clinically sick, and removed at random for immunohistological studies.", "The individual-level data can be ascertained in Figure REF ." ], [ "Aims", "The experiments demonstrate a clear difference between transgenic and non-transgenic chickens, with non-transgenic chickens having a larger mortality rate than transgenic chickens.", "However, it is not immediately clear if this difference results from differences in transmission potential, susceptibility to infection, or both.", "endeavors to resolve this ambiguity by using Bayesian model selection with population-level data, specifically the number of removed chickens of each type per half day.", "Visibly sick (moribund) birds are artificially removed from the experiment, but are assumed to have died naturally within the next half day.", "The authors use a technique called Approximate Bayesian Computation (ABC), which targets an approximate posterior distribution $\\pi _{\\epsilon } \\left( \\theta \\mid Y_{1:T} \\right)$ and evaluates an approximate model evidence $\\pi _{\\epsilon } \\left( Y_{1:T} \\right)$ .", "Broadly speaking, ABC involves generating a set of simulated data $Y^{sim}_{1:T}$ from the model whenever a parameter value is proposed, and accepting the proposal whenever the distance between simulated and observed data is smaller than some threshold, $\\epsilon $ .", "By implementing an ABC version of the alive particle filter (APF) within particle marginal Metropolis Hastings (PMMH) and constructing novel efficiency saving methods, achieves a small threshold, where at each observation the cumulative number of observed deaths can differ by at most 1.", "The resulting Bayes factors suggest that the transmission rate differs between transgenic and non-transgenic chickens, but that the initial infection probability, death rate, and susceptibility are the same.", "Our goal here is to improve on the analysis of by using individual-level models and targeting the correct posterior distribution.", "By using individual-level models, we are able to make full use of the available data rather than condensing the data into population-level summaries.", "This may be more informative for our inference.", "Likewise, we may obtain more reliable conclusions by targeting the correct posterior distribution rather than an approximation." ], [ "Models", "The models under consideration in are continuous-time SIR models, where at any point in time each chicken may be susceptible (S), infected (I), or removed (R).", "The simplest model treats transgenic and non-transgenic chickens identically.", "Challenge birds are infected on day 0 with probability $p$ .", "The probability of an $S \\rightarrow I$ transition over some small time interval $[t, t+dt)$ is $ P (S \\rightarrow I \\mbox{ in } [t, t+dt)) = \\frac{\\beta S I}{N} dt + o(dt), $ where $S$ denotes the number of susceptible chickens, $I$ denotes the number of infected chickens, $N$ denotes the total number of chickens, and $\\beta $ is the transmission parameter.", "Likewise the probability of an $I \\rightarrow R$ transition over some small time interval $[t, t+dt)$ is $ P (I \\rightarrow R \\mbox{ in } [t, t+dt)) = \\gamma I dt + o(dt), $ where $\\gamma $ is the removal rate.", "The alternative models under consideration contain different parameters for transgenic and non-transgenic chickens.", "For instance, the probability of infection on day 0 for challenge birds may be $p^{N}$ for non-transgenic chickens and $p^{T}$ for transgenic chickens.", "Likewise there may be two transmission parameters ($\\beta ^{N}$ and $\\beta ^{T}$ ), two removal rates ($\\gamma ^{N}$ and $\\gamma ^{T}$ ), an additional susceptibility term ($\\nu ^{N}$ ), or any combination of these.", "The parameterizations for all 16 models are shown in Table REF .", "In order to apply our methods we need discrete-time individual-level model equivalents to the continuous-time population-level models presented in .", "To do this, we start with the continuous-time model and make the assumption that for each half-day period the transition rates remain constant.", "In other words, if a chicken becomes infected within a half day period, it does not become infectious until the start of the next half-day period.", "We can then directly evaluate the transition probability matrix for each chicken over each half day.", "For example, in the case of the simplest SIR model the transition probability matrix for any chicken from time $t$ to $t+0.5$ is $\\mathbb {P}_t = \\left( \\begin{array}{c c c} E_{SI} & R_g \\left( E_{IR} - E_{SI} \\right) & 1 + \\left( R_g - 1 \\right) E_{SI} - R_g E_{IR} \\\\0 & E_{IR} & 1 - E_{IR} \\\\ 0 & 0 & 1 \\end{array} \\right),$ where $E_{SI} = \\exp \\left( - 0.5 \\frac{\\beta I_t}{N} \\right), \\; \\; \\; E_{IR} = \\exp \\left( - 0.5 \\gamma \\right), \\; \\; \\; R_g = \\frac{\\frac{\\beta I}{N}}{\\frac{\\beta I}{N} - \\gamma }, \\\\$ and $I_t$ is the number of infected chickens at time $t$ .", "Following , our prior distributions are $\\mathcal {U}(0,1)$ for $p^N$ and $p^T$ , and $\\mbox{Exp}(1)$ for $\\beta ^N$ , $\\beta ^T$ , $\\nu ^N$ , $\\gamma ^N$ , and $\\gamma ^T$ .", "Since our primary goal in this section is to compare Bayes factor estimates from our analyses with those presented in , and that our focus is on the methodology for constructing proposal distributions for the hidden state process, we do not consider the issue of sensitivity to the prior distributions here.", "However, we do stress that Bayes factors are sensitive to the choice of prior distributions, and model selection analyses should be mindful of this.", "Table: The number of parameters in each of the 16 models.", "For the initial infection (pp), transmission (β\\beta ), and removal parameters (γ\\gamma ), a single parameter indicates that there is no distinction between transgenic and non-transgenic birds and two parameters indicates a distinction.", "Likewise for the susceptibility parameter (ν\\nu ), zero parameters indicates no distinction and one parameter indicates a distinction." ], [ "Simulated data", "We compare the bias, precision, and scaling of DIFFBS, MIFFBS, Chib's method and a particle filter on simulated data.", "Since the primary challenge in obtaining model evidence estimates lies in integrating over the hidden state process, we restrict this part of the investigation to determining the marginal likelihood conditioned on a fixed set of parameter values.", "We generate 5 sets of simulated data from model 16, each using the same cross experimental design as the real data, but with varying numbers of chickens.", "Specifically, we simulate data with 4, 8, 16, 32, and 64 chickens per pen, with 1, 2, 5, 10, and 19 challenge birds respectively.", "The parameters for simulating the data are $p^N = 0.9$ , $p^T = 0.8$ , $\\nu ^N = 1.2$ , $\\beta ^N = 2.3$ , $\\beta ^T = 1.4$ , $\\gamma ^N = 0.5$ , and $\\gamma ^T = 0.3$ , which are then fixed for the purpose of estimating the marginal likelihood.", "We censor some chickens in the 12 hours preceding a death, to mimic removals from finding a moribund bird.", "In particular, whenever we have simulated a death, the chicken is removed early with probability 0.5.", "We tune each of the methods to generate 1000 estimates of the marginal likelihood in one hour for each data set.", "For DIFFBS we first obtain 100 guiding samples, from which we determine a high posterior realization $\\widetilde{X}_{1:T}$ on which to condition our proposal.", "We repeatedly propose from this same realization for a set number of replicates, which acts as our tuning variable to control the computational cost.", "The average of the replicated estimates then provides one estimate of the marginal likelihood.", "For MIFFBS the tuning variable is the number of guiding samples used to construct the proposal distribution.", "The guiding samples are regenerated whenever the effective sample size has fallen to half the initial sample size.", "For convenience, we only perform this regeneration whenever we start proposing values for a new individual.", "In order to apply Chib's method, we need to define parameter blocks for which we have the full conditional distributions.", "We define each block as an individual update, as we do in DIFFBS and MIFFBS.", "Chib's method approximates the full conditional distribution for the hidden state process via a series of mixture distributions.", "Similar to MIFFBS, Chib's method is guided by samples of the hidden state process, the number of which again acts as our tuning variable.", "In any realization of the hidden state process can be used as an input into the approximation for a marginal likelihood estimate, but here we sample from the approximation as an importance proposal.", "Chib's method shares the same possible bias issues as DIFFBS and MIFFBS, but as with MIFFBS the probability of this occurring decreases as the number of guiding samples increases.", "The particle filter operates sequentially through time, and requires the specification of proposal distributions that are ideally conditioned on upcoming observations.", "Simply using observations of whether chickens are dead or alive leads to particle degeneracy, especially with larger numbers of chickens.", "The reason for this is the removal of moribund birds.", "Moribund birds are observed as alive upon being extracted from the pen, and so may be sampled as susceptible.", "However, an extracted chicken can not then transition to the observed $R$ state as it can no longer become infected.", "It is clear that moribund birds must be infected at these extraction times, and we must condition our proposal distributions accordingly.", "The particle filter provides unbiased estimates of the marginal likelihood, and the tuning variable is the number of particles.", "Finally, when we have a small number of chickens (4 or 8 chickens per pen), we can compare our estimates with the true marginal likelihood, which can be obtained using the forward filter.", "This takes 0.04 seconds with 4 chickens per pen, and 262 seconds with 8 chickens per pen.", "The computational cost scales by a factor of 9 per additional chicken, and so the cost with 16 chickens per pen would be 358 years, demonstrating the poor scalability of the forward filter when applied to individual-level models.", "Table: Log marginal likelihood estimates in the simulation study for DIFFBS, Chib's method, MIFFBS, and the particle filter.", "For 4 and 8 chickens per pen we have also included the true log marginal likelihood obtained from the forward filter.", "The integer component of the log marginal likelihood is shown to the left of the methods.", "The mean estimate is the log of the average marginal likelihood from 1000 estimates.", "The range adds/subtracts 3 standard errors.The results are presented in Table REF , which includes the mean marginal likelihood estimates and ranges indicated by $\\pm 3$ standard errors.", "Comparing the mean estimates to the true values, DIFFBS clearly has a small bias.", "With both 4 and 8 chickens per pen, the true values does not lie within 3 standard errors of the mean.", "The remaining methods seem to perform well, with the mean estimates being close to the true value.", "Even though Chib's method and MIFFBS have the potential to be biased, there seem to enough guiding samples to prevent this issue.", "Comparing the different methods, the particle filter provides the most precise estimates in the smaller data sets (4 and 8 chickens per pen), and MIFFBS provides the most precise estimates in the larger data sets (16, 32, and 64 chickens per pen).", "This is most notable with 64 chickens per pen, where the range of the particle filter estimates is $\\sim 0.1$ on the log scale, vs $\\sim 0.01$ for MIFFBS.", "Chib's method is more precise than the particle filter in the largest dataset, but less precise than both MIFFBS and the particle filter otherwise.", "Finally, DIFFBS maintains a relatively good precision, but as stated provides biased estimates.", "Broadly speaking, DIFFBS and the particle filter scale linearly in the number of individuals, meaning that the tuning variables need to be halved for a doubling of the population size in order to maintain the same computational cost.", "Some factors do impact this, including computations in the algorithms that are not impacted by the tuning variable, and run times also have a stochastic element.", "DIFFBS uses 8346 replicates in the smallest data set and 645 in the largest, and the particle filter uses 189 405 particles in the smallest data set and 11 900 in the largest.", "Chib's method scales quadratically in the number of individuals, and so in the largest data set has only 18 samples constructing the proposal distribution for any individual, down from 1096 in the smallest.", "The scaling of MIFFBS depends on the frequency of the guiding sample regeneration.", "If the number of regeneration steps is proportional to the number of individuals, then MIFFBS will scale quadratically in the number of individuals.", "If, on the other hand, no regeneration is required, MIFFBS scales linearly in the number of individuals.", "With adaptive regeneration scheme it is also possible for the scaling to land somewhere between these two extremes.", "MIFFBS used 4695 guiding samples in the smallest data set, and 428 in the largest.", "This is similar to the scaling observed in DIFFBS, suggesting near-linear scaling for this investigation." ], [ "HPAI data", "We apply MIFFBS within importance sampling to estimate the model evidence for all 16 models fitted to the individual-level HPAI data.", "For the proposal distribution on the model parameters we construct a defense mixture, which was shown to perform well in .", "For each model we run an MCMC algorithm targeting the joint posterior distribution $\\pi (\\theta , X_{1:T} \\mid Y_{1:T})$ .", "We obtain 10000 samples in each case, which takes approximately 30 seconds and does not significantly contribute to the cost of the model selection algorithm.", "We then transform the model parameters as $\\log (-\\log (p^N))$ , $\\log (-\\log (p^T))$ , $\\log (\\nu ^N)$ , $\\log (\\beta ^N)$ , $\\log (\\beta ^T)$ , $\\log (\\gamma ^N)$ , and $\\log (\\gamma ^T)$ , to which we fit a multivariate Gaussian distribution.", "The defense mixture is then a weighted mixture distribution of the prior and the fitted Gaussian, with weights 0.05 and 0.95 respectively.", "Each time a new set of parameters are proposed we generate 1000 guiding samples of the hidden state process using IFFBS.", "We use an adaptive regeneration scheme for the guiding samples, which is triggered whenever the effective samples size is smaller than 500 as we initiate a proposal for a new individual.", "Table: Comparison of model evidence estimates and rankings for the MIFFBS importance sampling algorithm on the individual level model and the ABC alive particle filter on the population level model .", "The mean column gives the log of the model evidence estimate, and range indicates the log model evidence with ±3\\pm 3 standard errors.", "*** Indicates the best model, ** indicates models with Bayes between 1 and 1/3.21 / 3.2 with respect to the best model, and * indicates models with Bayes between 1/3.21 / 3.2 and 1/101 / 10.The model evidence estimates are shown in Table REF compared with those provided in .", "The model evidence estimates do not directly compare, as uses population-level models and data, whereas we use individual-level models and data.", "However, we can compare the rankings, and the strength of evidence between models indicated by the Bayes factors.", "Model 3 is favored in both analyses, which indicates two transmission parameters, one initial infection parameter, one removal parameter, and zero susceptibility parameters.", "Considering a threshold for the Bayes factor of 3.2 for substantial evidence in favor of the superior model , both analyses find models 4, 7, 8, and 11 as having significant support in comparison to model 3, although the exact ordering changes slightly.", "We also find significant support for model 12.", "With a threshold of 10 to indicate strong evidence in favor of the superior model, our analysis is more discriminating.", "We find strong evidence against six models, whereas finds strong evidence against four.", "The ABC algorithm failed to estimate a model evidence for model 14, due to requiring an excessively large number of model simulations.", "We have no issues with this, but do determine it to be the weakest model.", "The model averaged posterior distributions are shown in Figure REF , compared with those in .", "There is significant overlap in the marginal posterior distribution for each parameter, but we estimate larger values for the initial infection probabilities $p^N$ and $p^T$ , and smaller values for the removal parameters $\\gamma ^N$ and $\\gamma ^T$ .", "Figure: Model averaged posterior distributions for the model parameters.", "Dashed lines are those presented in , and solid lines are those obtained from the individual-level analysis using MIFFBS.An advantage of using individual-level models and data is that we can infer information about each individual through time.", "The model averaged marginal state posterior distributions are shown in Figure REF , along with the observed removal times.", "Figure: Marginal posterior daily infection probabilities for each chicken.", "Shaded circles indicate the infection probabilities (white indicates probability zero).Squares indicates a known death either via direct observation (no strike) or implied via removal of moribund birds (with strike)." ], [ "Discussion", "In this paper we have introduced effective algorithms for proposing values for the hidden state process of CHMMs with weak inter-chain dependencies, which we refer to as DIFFBS and MIFFBS.", "These proposal distributions are based on a chain-level decomposition, which naturally takes advantage of the dependency structure, providing algorithms that remain computationally viable as the number of chains increases and that require little user input.", "We demonstrated the utility of these methods in both simulation studies and by undertaking a model selection experiment for highly pathogenic avian influenza.", "This data set has previously been investigated in at the population level with an approximate inference scheme.", "We were able to improve on this study by better utilizing the data with individual-level models, and by targeting the correct posterior distribution.", "The key biological finding from is strong evidence for a difference in transmission between transgenic potential and non-transgenic chickens, and our results give refined, but consistent conclusions.", "There are several possible extensions to the proposed methodology.", "DIFFBS was shown to be biased in our application, and MIFFBS can potentially be biased if a small number of guiding samples are used.", "There are potential solutions to explore for this issue.", "For example, the transition probabilities in the proposal distributions could be modified to ensure full posterior support, or else DIFFBS and MIFFBS could be included as part of a mixture distribution.", "Since any such modification will lead to a worse approximation of the full conditional distribution, these choices are likely to lead to bias-variance trade-offs.", "DIFFBS and MIFFBS will lose efficiency when the chains are strongly correlated.", "If correlated groups of chains can be determined, for instance by picking out households in an infectious disease model, then block updates may be used in place of individual-level updates.", "Here a modified forward filter can be used to estimate the full conditional distribution of the hidden state process for the block of chains, conditional on the remainder.", "However, the computational cost scales exponentially with the size of the largest block.", "Focusing on MIFFBS, here we have only considered regenerating the guiding samples upon initializing a proposal for a new chain.", "This simplifies the implementation of the algorithm, but could be further generalized.", "This will be important in problems with long time horizons, where we can expect degeneracy to occur in the weights.", "Whilst we have focused on model selection, MIFFBS could also be useful in estimating model parameters.", "For example, in infectious disease models, parameter values and the hidden state process are frequently strongly correlated, and so MCMC implementations that iterate between updating the model parameters and hidden states do not mix effectively.", "MIFFBS could overcome such difficulties if used within a pseudo-marginal scheme to update both components jointly.", "In conclusion, we have presented novel methods for sampling the hidden state process of CHMMs with weak inter-chain dependencies, with favorable scaling properties as the number of chains increases.", "These sampling distributions are constructed automatically from the model specification and data, requiring little user input or tuning.", "This is a particularly useful feature when performing model selection, as the algorithms do not need to be modified to target different models.", "These properties have been illustrated in a model selection investigation of highly pathogenic avian influenza data, and the methods are readily implementable to many other infectious disease models and wider-ranging applications." ] ]
2105.11807
[ [ "Fourier majorants that match norms" ], [ "Abstract Denote the coefficients in the complex form of the Fourier series of a function $f$ on the interval $[-\\pi, \\pi)$ by $\\hat f(n)$.", "It is known that if $p = 2j/(2j-1)$ for some integer $j>0$, then for each function $f$ in $L^p$ there exists another function $F$ in $L^p$ that majorizes $f$ in the sense that $\\hat F(n) \\ge |\\hat f(n)|$ for all $n$, but that also satisfies $\\|F\\|_p \\le \\|f\\|_p$.", "Rescaling $F$ suitably then gives a majorant with the same $L^p$ norm as $f$.", "We show how that majorant comes from a variant in $L^{2j}$ of the notion of exact majorant in $L^2$." ], [ "Introduction", "Thus $\\hat{f}(n) = \\int _{-\\pi }^\\pi f(\\theta )e^{-in\\theta }\\,d\\theta /2\\pi $ .", "Call $F$ a majorant of $f$ , and $f$ a minorant of $F$ , when $|\\hat{f}(n)| \\le \\hat{F}(n)$ for all integers $n$ .", "In that case, $\\Vert f\\Vert _2 \\le \\Vert F\\Vert _2$ ; also, if $j$ is an integer greater than 1, then $F^j$ majorizes $f^j$ , and hence $(\\Vert f\\Vert _{2j})^{2j} = (\\Vert f^j\\Vert _2)^2 \\le (\\Vert F^j\\Vert _2)^2 =(\\Vert F\\Vert _{2j})^{2j}.$ Finally, $\\Vert f\\Vert _\\infty \\le \\Vert F\\Vert _\\infty $ when $F$ majorizes $f$ .", "This pattern does not persist for other exponents.", "Hardy and Littlewood [9] considered the upper majorant property, asserting that there is a constant $U(p)$ so that $\\Vert f\\Vert _p \\le U(p) \\Vert F\\Vert _p$ whenever $F$ majorizes $f$ .", "They gave an example showing that if this property holds in $L^3$ then the constant $U(3)$ must be strictly larger than 1.", "Other work [2], [1] revealed that the property fails for the exponents $p$ in the interval $(0, \\infty )$ that are not even integers.", "See [13], [8], [7],[12], [3] and [5] for refinements of these results, complements to them and connections with other questions.", "Here we consider the lower majorant property, also introduced in [9].", "It holds when there is a constant $L(p)$ so that each function $f$ in $L^p$ has a majorant $F$ with $\\Vert F\\Vert _p \\le L(p)\\Vert f\\Vert _p.$ This is clearly true when $p=2$ with $L(2) = 1$ , with $F$ equal to the exact majorant of $f$ given by $\\hat{F} = |\\hat{f}|$ .", "It also holds when $p = 1$ , since one can write a given function $f$ in $L^1$ as the square of a function $g$ in $L^2$ , form the exact majorant $G$ of $g$ , and let $F = G^2$ .", "Then $\\Vert F\\Vert _1 = \\frac{1}{2\\pi }\\int _{-\\pi }^\\pi |G(t)|^2\\,dt= (\\Vert G\\Vert _2)^2\\\\=(\\Vert g\\Vert _2)^2= \\frac{1}{2\\pi }\\int _{-\\pi }^\\pi |g(t)|^2\\,dt= \\Vert f\\Vert _1.$ The sequence $\\hat{F}$ is equal to the convolution of two copies of the sequence $\\hat{G}$ , and similarly for $\\hat{f}$ and $\\hat{g}$ .", "Since there is no cancelation in the convolution giving $F$ , the fact that $\\hat{G}=|\\hat{g}|$ implies that $\\hat{F} \\ge |\\hat{f}|$ .", "When $p \\in [1, \\infty )$ , a simple duality argument [2] shows that if $L^p$ has the lower majorant property, then its dual space $L^{p^{\\prime }}$ has the upper majorant property with $U(p^{\\prime }) \\le L(p)$ .", "By the work cited earlier, this can only happen when $p^{\\prime } = \\infty $ or $p^{\\prime }$ is an even integer, thus ruling out all exponents $p$ in the interval $(1, \\infty )$ except for $p = 2$ and the special exponents with $p=2j/(2j-1)$ , where $j$ is an integer strictly greater than 1.", "When $1 < p < \\infty $ , a less simple duality argument [9], [1] shows that the upper majorant property for $L^{p^{\\prime }}$ implies the lower majorant property for $L^p$ , with $L(p) \\le U(p^{\\prime })$ .", "In particular, it holds, with $L(p)=1$ , for the special exponents.", "That duality proof and alternatives [14], [4], [11] to it do not include a description of a suitable majorant of a given function for these values of $p$ .", "For various good reasons, those arguments covered general exponents $p$ and $p^{\\prime }$ , but the duality is now known to mainly have impact for the special values of $p$ .", "In [6], this hindsight was exploited by showing in those cases that a certain product of $2j-1$ functions in $L^{2j}$ gives a majorant with minimal norm in $L^{2j/(2j-1)}$ .", "Suitably rescaling that product yields a majorant with the same $L^{2j/(2j-1)}$ norm as the original function.", "In this paper, we offer a more direct description of that particular majorant.", "We state our conclusions about it in the next section, and prove those conclusions in the following section.", "Then we reformulate them as statements about convolution on the integers, and comment on how they extend to all discrete groups, abelian or not.", "In an appendix, we discuss the modifications needed on groups like $\\operatorname{\\mathbb {R}}^n$ ." ], [ "Dual maximizers", "We suppose in the rest of this paper that $1 < p < \\infty $ .", "In this paragraph, we summarize some of the reasoning in [9].", "First, if $f \\in L^p$ and $g \\in L^{p^{\\prime }}$ , then the series $\\sum _n \\hat{f}(n)\\overline{\\hat{g}(n)}$ converges to$\\int _{-\\pi }^\\pi f(t)\\overline{g(t)}dt/2\\pi $ .", "By Hölder's inequality, $\\left|\\sum _n \\hat{f}(n)\\overline{\\hat{g}(n)}\\right|\\le \\Vert f\\Vert _p\\Vert g\\Vert _{p^{\\prime }}.$ In the special case where $p^{\\prime }$ is even and $\\hat{G} \\ge 0$ , it is also true that $\\sum _n |\\hat{f}(n)|\\hat{G}(n) \\le \\Vert f\\Vert _p\\Vert G\\Vert _{p^{\\prime }},$ because each term $|\\hat{f}(n)|\\hat{G}(n)$ can be rewritten as $\\hat{f}(n)\\overline{\\hat{g}(n)}$ for a function $g$ that is majorized by $G$ .", "Recall that a complex function-valued function $f$ factors as $|f|\\operatorname{sgn}(f)$ , where $\\operatorname{sgn}(f)$ vanishes off the support of $f$ .", "If $f \\in L^p$ , then letting $g = |f|^{p-1}\\operatorname{sgn}(f)$ puts $g$ in  $L^{p^{\\prime }}$ , and makes $f = |g|^{p^{\\prime }-1}\\operatorname{sgn}(g)$ .", "In the special cases where $p^{\\prime } = 2j$ for some positive integer $j$ , this simplifies to $f = |g|^{2j-1}\\operatorname{sgn}(g)= |g|^{2j-2}|g|\\operatorname{sgn}(g)\\\\=|g|^{2(j-1)}g = (g{\\overline{g}})^{j-1}g.$ Moreover, $|f|^{2j/(2j-1)} = |g|^{2j}$ , and $\\Vert f\\Vert _{2j/(2j-1)}= \\Vert g\\Vert _{2j}^{2j-1}$ .", "Denote the set of functions $G$ in $L^{2j}$ for which $\\hat{G} \\ge 0$ by $PD^{2j}$ .", "Also consider its closed subset $PD_f^{2j}$ where $\\hat{G}$ vanishes off the support of $\\hat{f}$ , and the closed subset $PD^{2j}_{(g, f)}$ of $PD_f^{2j}$ where $G$ has the same norm in $L^{2j}$ as $g$ .", "Let $\\Phi _f$ be the function on $PD^{2j}$ that maps $G$ to $\\sum _n |\\hat{f}(n)|\\hat{G}(n)$ , which is finite by inequality (REF ).", "Theorem 2.1 There is a unique function $G$ in the set $PD^{2j}(f, g)$ for which $\\Phi _f(G)$ is maximal in that set.", "The product $F := (G\\overline{G})^{j-1}G$ then has the same norm in $L^{2j/(2j-1)}$ as $f$ , and $F$ majorizes $f$ .", "Remark 2.2 When $j=1$ , the two functions $f$ and $g$ coincide, as do $G$ and $F$ , which are their exact majorants.", "When $j > 1$ , it may be reasonable to say that the pair $(G, F)$ specified above is the exact majorant in $L^{2j}\\times L^{2j/(2j-1)}$ of the pair $(g, f)$ .", "Remark 2.3 As in [1], the function $f$ above has a unique majorant, $F^{\\prime }$ say, of minimal $L^{2j/(2j-1)}$ norm.", "That majorant coincides with the one in Theorem REF if and only if $\\Vert F^{\\prime }\\Vert _{2j/(2j-1)} = \\Vert f\\Vert _{2j/(2j-1)}$ .", "It is shown in [6] that those two norms agree if and only if forming the exact majorant in $L^2$ of the function $g$ above does not change its $L^{2j}$ norm.", "Remark 2.4 These are the cases where the maximizing function $G$ above is the exact majorant of $g$ in $L^2$ .", "In particular, $\\hat{G} = |\\hat{g}|$ if the support of $\\hat{g}$ is a Sidon $B_{j}$ set, that is each number in the sum of $j$ copies of the set has a unique representation modulo permutation of the $j$ summands.", "Remark 2.5 We will comment at end of the next section on the cases where $\\Vert F^{\\prime }\\Vert _{2j/(2j-1)} < \\Vert f\\Vert _{2j/(2j-1)}$ ." ], [ "More details", "Let $L^{2j}_f$ comprise all functions $h$ in $L^{2j}$ whose coefficients vanish off the support of $\\hat{f}$ .", "When $f$ is a trigonometric polynomial, $L^{2j}_f$ is finite dimensional, and its subset $PD^{2j}_{(g,f)}$ is compact.", "Now $\\Phi _f$ is obviously continuous on $PD^{2j}_{(g,f)}$ in this case; so there is a function $G$ in that set at which $\\Phi _f(G)$ is maximal.", "The existence of such a maximizer $G$ for other functions $f$ will be established at the end of this proof.", "Let $M = \\Phi _f(G)$ .", "If $\\Phi _f(H) = M$ for another function $H$ in $PD^{2j}_{(g, f)}$ , let $G^{\\prime } = (G+H)/2$ .", "Then $\\Phi _f(G^{\\prime }) = M$ too, but $\\Vert G^{\\prime }\\Vert _{2j} < \\Vert g\\Vert _{2j}$ by the strict convexity of the $L^{2j}$ norm.", "Rescaling $G^{\\prime }$ to have the same $L^{2j}$ norm as $g$ would then give a function in $PD^{2j}(g,f)$ where $\\Phi _f$ takes a value larger than $M$ , contrary to the maximality of $M$ .", "So $G$ is unique.", "As noted earlier, $\\Vert f\\Vert _{2j/(2j-1)}= \\Vert g\\Vert _{2j}^{2j-1}$ .", "The corresponding statement about $F$ and $G$ holds for the same reasons.", "It then follows that $\\Vert F\\Vert _{2j/(2j-1)} = \\Vert f\\Vert _{2j(2j-1)}$ because $\\Vert G\\Vert _{2j} = \\Vert g\\Vert _{2j}$ .", "By inequality (REF ) and the fact that $\\Vert F\\Vert _{2j/(2j-1)}= \\Vert G\\Vert _{2j}^{2j-1}$ , $\\Phi _f(G) \\le (\\Vert G\\Vert _{2j})^{2j}.$ Given an integer $n$ in the support of $\\hat{f}$ , let $z^n$ be the function mapping $\\theta $ to $e^{in\\theta }$ .", "Another useful property of $(\\Vert \\cdot \\Vert _{2j})^{2j}$ is that, for a real parameter $t$ , $\\left[\\frac{d}{dt}\\left(\\Vert G + tz^n\\Vert _{2j}\\right)^{2j}\\right]_{t=0}= 2j\\hat{F}(n).$ As in [9], this follows from differentiation inside an integral sign, with $2j$ replaced by any exponent in the interval $(1, \\infty )$ .", "For the $L^{2j}$ norm, however, it is enough to expand $(G + tz^n)^j(\\overline{G + tz^n})^j$ as a polynomial in powers of $t$ , integrate each term with respect to $\\theta $ , and examine the resulting coefficient of $t$ .", "The conclusions of the theorem are trivial when $\\Vert f\\Vert _{2j/(2j-1)} = 0$ .", "In the remaining cases, as $H$ runs through $PD^{2j}(g,f)$ , the $L^{2j}$ norm of $H$ is constant, and $\\Phi _f(H) \\ne 0$ .", "So the fraction $[(\\Vert H\\Vert _{2j})/\\Phi _f(H)]^{2j}$ is minimized when $H = G$ .", "That fraction extends to the nontrivial part of $PD^{2j}_f$ , that is to the union of rays of the form $\\lbrace sH: s > 0, H \\in ~PD^{2j}(g,f)\\rbrace .$ The fraction is constant on each such ray, and minimal on the one where $H = G$ .", "Form the quotient $Q_n(t) := \\frac{(\\Vert G + tz^n\\Vert _{2j})^{2j}}{\\Phi _f(G + tz^n)^{2j}}$ when $t \\ge 0$ .", "The derivative of $Q_n(t)$ at $t=0$ exists and is equal to $\\frac{2j\\Phi _f(G)^{2j-1}}{\\Phi _f(G)^{4j}}\\left[\\Phi _f(G)\\hat{F}(n) - \\left(\\Vert G\\Vert _{2j}\\right)^{2j}\\left|\\hat{f}(n)\\right|\\right].$ Since $Q_n(t)$ is minimal at $t=0$ , this derivative is nonnegative.", "So $\\hat{F}(n) \\ge \\frac{\\left(\\Vert G\\Vert _{2j}\\right)^{2j}}{\\Phi _f(G)}\\left|\\hat{f}(n)\\right| \\ge \\left|\\hat{f}(n)\\right|,$ by inequality (REF ).", "Now let $f$ be any function in $L^{2j/(2j-1)}$ .", "By the density of trigonometric polynomials in $L^{2j/(2j-1)}$ , one can write $f$ as an infinite sum of trigonometric polynomials $f_k$ , where $\\sum _k\\Vert f_k\\Vert _{2j/(2j-1)} \\le (1 + \\varepsilon )\\Vert f\\Vert _{2j/(2j-1)}.$ By inequality (REF ), the series $\\sum _k\\Phi _{f_k}$ converges uniformly to $\\Phi _f$ on bounded subsets of $PD^{2j}$ .", "It follows that $\\Phi _f$ is continuous on $PD^{2j}$ and that $\\Phi _f(H) \\le \\Vert f\\Vert _{2j/(2j-1)}\\Vert H\\Vert _{2j}$ for all $H$ in $PD^{2j}$ .", "Since the function $\\Phi _f$ is bounded on the set $PD^{2j}_{(g, f)}$ , there is a sequence $(G_k)$ in $PD^{2j}_{(g, f)}$ for which $\\Phi _f(G_k)$ converges to the supremum of the values of $\\Phi _f$ in that set.", "If $(G_k)$ is a Cauchy sequence, then by the continuity of $\\Phi _f$ , that supremum is attained at $G = \\lim _{k\\rightarrow \\infty }G_k$ .", "By the uniform convexity of the $L^{2j}$ norm, for each positive number $\\varepsilon $ there is a number $R$ in the interval $[0, 1)$ with the following property.", "If $H$ and $K$ both belong to $PD^{2j}(g, f)$ , and if $\\Vert (H+K)/2\\Vert _{2j} > R\\Vert g\\Vert _{2j}$ , then $\\Vert H-K\\Vert _{2j} < \\varepsilon $ .", "This reduces the task of proving that $(G_k)$ is a Cauchy sequence to checking when $R \\in [0, 1)$ that if $k$ and $k^{\\prime }$ are large enough, then $\\Vert (G_k + G_{k^{\\prime }})/2\\Vert _{2j} > R\\Vert g\\Vert _{2j}$ .", "Let $M$ be the supremum of $\\Phi _f(H)$ as $H$ runs through $PD^{2j}_{(g, f)}$ .", "When $k$ and $k^{\\prime }$ are large enough, $\\Phi _f(G_k) > RM$ , and $\\Phi _f(G_{k^{\\prime }}) > RM$ .", "It follows that $\\Phi _f((G_k + G_{k^{\\prime }})/2) > RM$ too.", "Rename $(G_k + G_{k^{\\prime }})/2$ as $G^{\\prime }$ , and let $H = \\frac{\\Vert g\\Vert _{2j}}{\\Vert G^{\\prime }\\Vert _{2j}}G^{\\prime }.$ This rescaling puts $H$ in $PD^{2j}_{(g, f)}$ , so that $\\Phi _f(H) \\le M$ .", "On the other hand, $\\Phi _f(H) = \\frac{\\Vert g\\Vert _{2j}\\Phi _f(G^{\\prime })}{\\Vert G^{\\prime }\\Vert _{2j}}.$ Therefore, $RM < \\Phi _f(G^{\\prime }) = \\frac{\\Vert G^{\\prime }\\Vert _{2j}}{\\Vert g\\Vert _{2j}}\\Phi _f(H)\\le \\frac{\\Vert G^{\\prime }\\Vert _{2j}}{\\Vert g\\Vert _{2j}}M,$ and $\\Vert G^{\\prime }\\Vert _{2j} > R\\Vert g\\Vert _{2j}$ as required.", "Remark 3.1 Another way to describe $G$ is that letting $H=G$ minimizes $\\Vert H\\Vert _{2j}$ in the part of $PD^{2j}_f$ where $\\Phi _f(H) = M$ .", "In Appendix  below, this approach is used in $L^{2j}(\\operatorname{\\mathbb {R}}^n)$ .", "Remark 3.2 Putting $H = G/M$ minimizes $\\Phi _f(H)$ subject to the constraint that $\\Phi _f(H) = 1$ .", "It is shown in [6] that $\\Vert G/M\\Vert _{2j}$ must then be the reciprocal of the norm of the minimal majorant, $F^{\\prime }$ say, of $f$ in $L^{2j/(2j-1)}$ .", "On the other hand, $\\frac{\\Vert F\\Vert _{2j/(2j-1)}}{\\Vert F^{\\prime }\\Vert _{2j/(2j-1)}}= \\frac{(\\Vert G\\Vert _{2j})^{2j-1}}{M/\\Vert G\\Vert _{2j}}= \\frac{(\\Vert G\\Vert _{2j})^{2j}}{M}=\\frac{\\left(\\Vert G\\Vert _{2j}\\right)^{2j}}{\\Phi _f(G)}.$ Denote the last fraction above by $r$ , and rewrite the first inequality in line (REF ) as the statement that $\\hat{F} \\ge r|\\hat{f}|$ .", "Then $F/r$ also majorizes $f$ , while $\\Vert F/r\\Vert _{2j/(2j-1)} = \\Vert F^{\\prime }\\Vert _{2j/(2j-1)}$ by equation (REF ).", "The uniqueness of the minimal majorant yields that $F/r = F^{\\prime }$ and $F = rF^{\\prime }$ .", "Remark 3.3 When $r > 1$ , the majorant $F$ does not have minimal norm in $L^{2j/(2j-1)}$ , and $f$ has other majorants with the same norm as $f$ .", "Indeed, for any nontrivial function, $H$ say, in $L^{2j/(2j-1)}$ with nonnegative coefficients, adding a suitably rescaled copy of $H$ to $F^{\\prime }$ gives a majorant for $f$ with the same norm as $f$ .", "That majorant is not a rescaled copy of $F^{\\prime }$ unless $H$ is.", "Remark 3.4 In the description of $G$ above, it is not necessary to insist a priori that $\\hat{G}$ vanish off the support of $\\hat{f}$ .", "Consider minimizing $\\Vert G\\Vert _{2j}$ with the constraints that $\\hat{G} \\ge 0$ and $\\Phi _f(G) = M$ , for instance.", "If the solution had some strictly positive coefficient outside the support of $\\hat{f}$ , then replacing that coefficient with 0 would decrease the norm of the solution in $L^{2j}$ without changing the value of $\\Phi _f$ ." ], [ "Convolution on the integers", "We reformulate part of Theorem REF .", "Theorem 4.1 Let $j$ be a positive integer.", "For each function $g$ in $L^{2j}$ , there is another function $G$ in $L^{2j}$ with the following properties.", "$\\hat{G} \\ge 0$ .", "$\\hat{G} = 0$ off the support of the Fourier coefficients of $(g\\overline{g})^{j-1}g$ .", "$\\Vert G\\Vert _{2j} = \\Vert g\\Vert _{2j}$ .", "$(G\\overline{G})^{j-1}G$ majorizes $(g\\overline{g})^{j-1}g$ .", "In particular, this holds when $g$ is a trigonometric polynomial, and then $G$ is a trigonometric polynomial too.", "Use the term light version of Theorem REF for that case.", "Easy approximation arguments using this version of the theorem yield that every function in $L^{2j/(2j-1)}$ has a majorant with no larger $L^{2j/(2j-1)}$ norm.", "Given a trigonometric polynomial $g$ , let $a = \\hat{g}$ , and let $\\tilde{a}$ be the sequence with $\\tilde{a}(n) = \\overline{a(-n)}$ for all $n$ ; these are the coefficients of $\\overline{g}$ .", "Fix an integer $j > 1$ , and let $(\\tilde{a}*a)^{*j}$ denote the $j$ -th convolution power of the convolution product $\\tilde{a}*a$ .", "The Fourier coefficients of $|g|^{2j}$ are given by the sequence $(\\tilde{a}*a)^{*j}$ , while those of $g^j(\\overline{g})^{j-1}$ are given by $(a*\\tilde{a})^{*(j-1)}*a$ .", "Since $|g|^{2j} \\ge 0$ , its integral with respect to the measure $d\\theta /2\\pi $ is equal to its 0-th Fourier coefficient, that is to the value $[(\\tilde{a}*a)^{*j}](0)$ of the sequence $(\\tilde{a}*a)^{*j}$ at the index 0.", "Similar comments apply to the trigonometric polynomial $G$ , with coefficients $b$ say.", "So the light version of Theorem REF is equivalent to the following statement about convolution on the integers.", "Theorem 4.2 Let $j$ be a positive integer.", "Given a finitely-supported function $a$ on the integers, let $c = a*(\\tilde{a}*a)^{*(j-1)}$ .", "Then there is a function $b$ on the integers with the following properties.", "$b \\ge 0$ .", "$b$ vanishes off the support of $c$ .", "$[(\\tilde{b}*b)^{*j}](0)= [(\\tilde{a}*a)^{*j}](0)$ .", "$(b*\\tilde{b})^{*(j-1)}*b\\ge |c|$ .", "This follows from Theorem REF , and can also be proved directly by choosing the function $b$ to maximize the sum $\\sum _n b(n)|c(n)|$ subject to the first three conditions enumerated above.", "Remark 4.3 The convolution version of equation (REF ) runs as follows.", "Let $\\delta _n$ be the sequence that takes value 1 at $n$ , and that vanishes otherwise.", "Given a real number $t$ , let $d = b + t\\delta _n$ , and form $[(\\tilde{d}*d)^{*j}](0)$ .", "Then the derivative at $t=0$ of the latter is $2j[(b*\\tilde{b})^{*(j-1)}*b](n)$ .", "It is easy to check this by expanding $[(\\tilde{d}*d)^{*j}](0)$ in powers of $t$ .", "Remark 4.4 In this context, inequality (REF ) is only required when $g$ is a trigonometric polynomial, and $f$ is a suitable product of such polynomials.", "The corresponding statement for convolution is that if $a$ and $k$ are finitely supported functions on the integers, then $\\left|\\left[\\tilde{k}*(a*\\tilde{a})^{*(j-1)}*a\\right](0)\\right| \\le [(\\tilde{k}*k)^{*j}(0)]^{1/2j}\\left[(\\tilde{a}*a)^{*j}(0)\\right]^{(2j-1)/2j}.$ This follows from Hölder's inequality on the unit circle.", "It can also be verified in the style of the proof of Theorem REF .", "Just rescale in nontrivial cases to make $(\\tilde{a}*a)^{*j}(0)$ equal to 1, and then minimize $\\frac{(\\tilde{k}*k)^{*j}(0)}{\\left|\\left[\\tilde{k}*(a*\\tilde{a})^{*(j-1)}*a\\right](0)\\right|^{2j}}.$ The fact that positive bounded operators on Hilbert spaces have unique positive $j$ -th roots is useful here.", "Remark 4.5 Theorem REF and the two remarks above extend to all discrete groups, abelian or not.", "Hölder's inequality extends to noncommutative $L^p$ spaces as in [17].", "The counterpart of the instance in Remark REF can also be proved by the method outlined there.", "Remark 4.6 Similar comments apply in the context of trace ideals [16]." ], [ "Majorization\non\nnondiscrete\nduals\n", "We now explain how some of our methods extend to the spaces $L^p(\\operatorname{\\mathbb {R}}^n)$ , where comparisons between transforms are made on a dual copy of $\\operatorname{\\mathbb {R}}^n$ , which is not discrete.", "When $1 < p \\le 2$ , transforms of $L^p$ functions on $\\operatorname{\\mathbb {R}}^n$ can be identified with (equivalence classes of) functions on the dual copy.", "When $p^{\\prime } > 2$ , functions in $L^{p^{\\prime }}(\\operatorname{\\mathbb {R}}^n)$ have transforms in the sense of tempered distributions, but in many cases those transforms are not functions.", "Duality arguments in [15], [10] used summability on the group $\\operatorname{\\mathbb {R}}^n$ and its dual copy to reduce the study of majorant properties in $L^p$ spaces to instances where $f$ and $\\hat{f}$ are both functions.", "Again [10], [15] the upper majorant property only holds when $p$ is infinite or even, and the lower majorant property only holds when $p = 1$ or $p^{\\prime }$ is an even integer.", "Here, we offer a different proof that the upper majorant property implies the lower majorant property for the dual exponent when $p$ is even; again, we get more information about the forms of some majorants.", "Recall that a distribution is said to be nonnegative if it maps each nonnegative test function to a nonnegative number, and then the distribution acts by integration against a nonnegative Borel measure.", "When the distributions acts by integration against a function, the distribution is nonnegative if and only if that function is nonnegative almost everywhere.", "When two functions $f$ and $F$ both belong to $L^p$ , where $1 \\le p \\le 2$ , call $F$ a majorant of $f$ if $\\hat{F} \\ge |\\hat{f}|$ almost everywhere on the dual copy of $\\operatorname{\\mathbb {R}}^n$ .", "More generally, for two distributions $\\Phi $ and $\\Psi $ , call $\\Phi $ a majorant of $\\Psi $ if the transform $\\hat{\\Phi }$ is represented by a nonnegative measure, while $\\hat{\\Psi }$ is represented by a measure with the property that $|\\hat{\\Psi }(S)| \\le \\hat{\\Phi }(S)$ for all Borel sets $S$ .", "If $p$ is even, then the upper majorant property holds with constant 1.", "That is, if $\\Phi \\in L^p$ , then $\\Psi \\in L^p$ and $\\Vert \\Psi \\Vert _p \\le \\Vert \\Phi \\Vert _p$ .", "The only cases of this needed here are those where $\\hat{\\Phi }$ and $\\hat{\\Psi }$ are represented by functions in $L^1\\cap L^\\infty $ .", "Then those transforms also belong to $L^2$ , and it follows that $\\Phi $ and $\\Psi $ belong to $L^2\\cap L^\\infty $ .", "One can then compare the $L^{2j}$ norms of $\\Phi $ and $\\Psi $ by considering the $j$ -th convolution powers of $\\hat{\\Phi }$ and $\\hat{\\Psi }$ and comparing their $L^2$ norms.", "Also recall that a distribution is said to vanish on an open set if the distribution annihilates every test function whose support is a compact subset of the open set.", "Then there is a largest such open set, and the support of the distribution is defined to be the complement of that largest open set.", "When the distribution acts by integration against a function, that support is the smallest closed set outside which the function takes the value 0 almost everywhere.", "Theorem 1.1 Let $p = 2j/(2j-1)$ for some integer $j>1$ , let $f$ be a function in $L^p(\\operatorname{\\mathbb {R}}^n)$ , and let $S$ be the support of the distribution $\\hat{f}$ .", "Then $f$ has a majorant of the form $G^j(\\overline{G})^{j-1}$ , where $G \\in L^{p^{\\prime }}(\\operatorname{\\mathbb {R}}^n)$ , and $\\hat{G}$ is a nonnegative distribution whose support is included in $S$ .", "The majorant of minimal $L^p$ norm has this form, and $\\Vert G\\Vert _{p^{\\prime }} \\le (\\Vert f\\Vert _p)^{1/(2j-1)}$ in that case.", "The support of the transform of the minimal majorant is included in the closure of the algebraic sum of $j$ copies of $S$ and $j-1$ copies of $-S$ .", "Modify the approach in Remark REF , replacing sums with integrals, and replacing pointwise inequalities with ones that hold almost everywhere.", "For now, only require that $1 < p < 2$ .", "Given a measurable function $c$ in $L^{p^{\\prime }}$ on the dual copy of $\\operatorname{\\mathbb {R}}^n$ , let $R(c)$ be the set of distributions $w$ for which $\\hat{w}$ can be identified with a nonnegative, bounded, measurable function with bounded support on the dual copy of $\\operatorname{\\mathbb {R}}^n$ , and for which $\\int _{\\operatorname{\\mathbb {R}}^n} |c|\\hat{w} = 1.$ Then $R(c)$ is nonempty when $c$ is nontrivial, because there are bounded sets of positive measure on which the values of $|c|$ are bounded away from 0 and $\\infty $ .", "Suitably rescaling the indicator function of such a set gives a function $\\hat{w}$ satisfying condition (REF ).", "The inverse transform of that function belongs to both $L^\\infty (\\operatorname{\\mathbb {R}}^n)$ and $L^2(\\operatorname{\\mathbb {R}}^n)$ , and hence to $L^{p^{\\prime }}(\\operatorname{\\mathbb {R}}^n)$ .", "Let $K_p(c)$ be the infimum of $L^{p^{\\prime }}$ norms of members of the nonempty convex set $R(c)$ .", "Since the $L^{p^{\\prime }}$ norm on is uniformly convex, the closure of $R(c)$ in $L^{p^{\\prime }}$ has a unique element of smallest norm, which must be $K_p(c)$ .", "Say that a function $F$ in $L^p(\\operatorname{\\mathbb {R}}^n)$ is a partial majorant of $\\check{c}$ if $\\hat{F} \\ge |c|$ almost everywhere in the set where $c \\ne 0$ .", "The set of partial majorants is convex, and it is closed in $L^{p}$ .", "If this set is nonempty, then it has a unique element of minimal $L^{p}$ norm, by uniform convexity again.", "Lemma 1.2 Let $1 < p < 2$ , and let $c \\in L^{p^{\\prime }}(\\operatorname{\\mathbb {R}}^n)$ .", "If $\\Vert c\\Vert _{p^{\\prime }} \\ne 0$ , then $\\check{c}$ has a partial majorant in $L^p$ if and only if $K_p(c) > 0$ .", "In that case, the minimal $L^p$ norm of partial majorants of $\\check{c}$ is equal to $1/K_p(c)$ .", "The partial majorant of minimal norm is a rescaled copy of $h|h|^{p^{\\prime }-2}$ , where $h$ has minimal $L^{p^{\\prime }}$ norm in the closure of $R(c)$ in $L^{p^{\\prime }}$ .", "Finally, the distributional support of the transform of $h$ is included in the distributional support of $c$ .", "To prove this, start with the fact that if $\\check{c}$ has a partial majorant $F$ in $L^p$ , and if $w \\in R(c)$ , then $1 = \\int _{\\operatorname{\\mathbb {R}}^n} \\hat{w}|c|\\le \\int _{\\operatorname{\\mathbb {R}}^n} \\hat{w}\\hat{F}= \\int _{\\operatorname{\\mathbb {R}}^n} \\overline{w} F\\le \\Vert w\\Vert _{p^{\\prime }}\\Vert F\\Vert _p.$ The second equality above is the instance of the Parseval relation that makes $\\int _{\\operatorname{\\mathbb {R}}^n} \\overline{\\widehat{h^{\\prime }}}\\hat{f}= \\int _{\\operatorname{\\mathbb {R}}^n} \\overline{h^{\\prime }} f$ when $f \\in L^p$ and $h$ is the inverse transform of a bounded function with bounded support.", "Inequality ((REF )) makes $\\Vert w\\Vert _{p^{\\prime }} \\ge 1/\\Vert F\\Vert _p$ for every $w$ in $R(c)$ , so that $K_p(c) \\ge 1/\\Vert F\\Vert _p$ .", "In particular, $\\Vert F\\Vert _p \\ge 1$ when $K_p(c) = 1$ .", "Rescale $c$ to reduce matters to the latter case, and then let $h$ be as specified in the statement of the lemma.", "Let $k = h|h|^{p^{\\prime }-2}$ .", "Fix a function $w$ in the set $R(c)$ , and let $\\phi (t)$ be equal to $(\\Vert h + tw\\Vert _{p^{\\prime }})^{p^{\\prime }}$ .", "This has derivative $p^{\\prime }\\int \\hat{k}\\hat{w}$ at $t=0$ .", "The quotients $(h + tw)/(1 + t\\int |c|\\hat{w})$ belong to the closure of $R(c)$ in $L^{p^{\\prime }}$ when $t \\ge 0$ .", "Take $p^{\\prime }$ -th powers of the $L^{p^{\\prime }}$ norms of these quotients, and require that the derivatives with respect to $t$ of these powers be nonnegative at $t=0$ .", "The outcome is that $\\int (\\hat{k} - |c|)\\hat{w} \\ge 0$ for all $w$ in $R(c)$ .", "It follows that $\\hat{k} \\ge |c|$ almost everywhere on the set where $c \\ne 0$ .", "On the other hand, $k$ has $L^p$ norm equal to 1, which is a lower bound for the the norms of partial majorants of $\\check{c}$ in $L^p$ .", "So $k$ must be the partial majorant of minimal $L^p$ norm.", "By definition, all functions in the set $R(c)$ have transforms whose distributional supports are included in the distributional support of $c$ .", "Then $h$ has this property too.", "This completes the proof of the lemma.", "To deduce the theorem, first check that if $p = 2j/(2j-1)$ for some integer $j>1$ , and $f \\in L^p(\\operatorname{\\mathbb {R}}^n)$ , then $K_p(\\hat{f}) \\ge 1/\\Vert f\\Vert _p$ .", "To that end, write $\\hat{f} = \\varepsilon |\\hat{f}|$ for a measurable function $\\varepsilon $ with absolute-value 1.", "Now let $h$ be any function in $R(|\\hat{f}|)$ , and let $h^{\\prime }$ be the inverse transform of the product $\\varepsilon h$ .", "Then $1 = \\int \\hat{h}|\\hat{f}|= \\int \\widehat{h^{\\prime }}\\overline{\\hat{f}}= \\int h^{\\prime }\\overline{f}.$ By the upper majorant property, $h^{\\prime } \\in L^{2j}$ with $\\Vert h^{\\prime }\\Vert _{2j} \\le \\Vert h\\Vert _{2j}$ .", "Hölder's inequality then yields that $1 \\le \\Vert h^{\\prime }\\Vert _{2j}\\Vert f\\Vert _p\\le \\Vert h\\Vert _{2j}\\Vert f\\Vert _p.$ This makes $1/\\Vert f\\Vert _p$ a lower bound for the $L^{p^{\\prime }}$ norm of every such function $h$ , and hence for $K_p(\\hat{f})$ .", "In particular, $K_p(\\hat{f}) > 0$ , and $f$ has a minimal partial majorant, $F$ say, in $L^p$ .", "To see that $\\Vert F\\Vert _p \\le ~\\Vert f\\Vert _p$ , again rescale to the case where $K_p(\\hat{f}) = 1$ .", "In that case, write $h$ as a limit in $L^{p^{\\prime }}$ norm of a sequence $(h_n)$ of members of the set $R(c)$ .", "The transforms of the functions $h_n$ are nonnegative bounded functions with bounded supports.", "The functions $k_n := (h_n)^j(\\overline{h_n})^{j-1}$ belong to $L^p$ , and the sequence $(k_n)$ converges in $L^p$ to $k$ , so that $\\left(\\widehat{k_n}\\right)$ converges in $L^{p^{\\prime }}$ to $\\hat{k}$ .", "The transform of $k_n$ is equal to the convolution of $j$ copies of $\\widehat{h_n}$ with $j-1$ copies of $\\widehat{\\overline{h_n}}$ .", "It follows that $\\widehat{k_n}$ is nonnegative almost everywhere.", "Moreover, if a test function $\\psi $ vanishes on the closure of the algebraic sum of $j$ copies of $S$ and $j-1$ copies of $-S$ , then $\\int k_n\\psi = 0$ for all $n$ .", "So $\\int k\\psi = 0$ , and the support of $k$ is included in the closure of that sum of sets.", "Since $\\hat{k}$ is nonnegative almost everywhere, it is a full majorant of $\\check{c}$ .", "It must be the one of minimal norm because it has minimal norm among partial majorants.", "Remark 1.3 Suitably rescaling the minimal majorant $G$ again gives one with the same $L^p$ norm as $f$ .", "Proving the existence of such a majorant by the method used to prove Theorem REF seems harder here, because it is not clear that a suitable counterpart of the function $\\Phi _f$ is continuous." ] ]
2105.11642
[ [ "Unified description of cuprate superconductors using four-band $d$-$p$\n model" ], [ "Abstract In the 35 years since the discovery of cuprate superconductors, we have not yet reached a unified understanding of their properties, including their material dependence of the superconducting transition temperature $T_{\\text{c}}$.", "The preceding theoretical and experimental studies have provided an overall picture of the phase diagram, and some important parameters for the $T_{\\text{c}}$, such as the contribution of the Cu $d_{z^2}$ orbital to the Fermi surface and the site-energy difference $\\Delta_{dp}$ between the Cu $d_{x^2-y^2}$ and O $p$ orbitals.", "However, they are somewhat empirical and limited in scope, always including exceptions, and do not provide a comprehensive view of the series of cuprates.", "Here we propose a four-band $d$-$p$ model as a minimal model to study material dependence in cuprates.", "Using the variational Monte Carlo method, we theoretically investigate the phase diagram for the La$_2$CuO$_4$ and HgBa$_2$CuO$_4$ systems and the correlation between the key parameters and the superconductivity.", "Our results comprehensively account for the empirical correlation between $T_{\\text{c}}$ and model parameters, and thus can provide a guideline for new material design.", "We also show that the effect of the nearest-neighbor $d$-$d$ Coulomb interaction $V_{dd}$ is actually quite important for the stability of superconductivity and phase competition." ], [ "INTRODUCTION", "The discovery of superconductivity in cuprates has brought about the significant progress in strongly-correlated electron systems [1].", "Along with the high superconducting transition temperature $T_{\\text{c}}$ , cuprates show anomalous phases and phenomena such as the Mott metal-insulator transition, pseudogap phenomena, and antiferromagnetic (AF) and charge-density-wave (or stripe) phases [2], [3].", "Recently, nematic order [4] and ferromagnetic fluctuation [5], [6] have also been observed experimentally.", "These newly observed experiments being constantly reported with the underlying, possibly exotic, physics have continued to attract many researchers' interests over 35 years since the first high-$T_{\\text{c}}$ cuprate superconductor was synthesized.", "It has been widely believed that the strong AF spin fluctuation characteristic to the two-dimensional square lattice structure triggers the various anomalous phenomena in cuprates [7], [8], [9], [10].", "On the basis of this picture, the effective one-band Hubbard model [11] has been studied with various numerical methods extensively [12], [13].", "It succeeded to describe the important physics in cuprates not only for the ground state but also for the excited state.", "The angle-resolved photoemission spectroscopy (ARPES) experiments show that the Fermi surface consists of only one energy band mainly derived from the Cu $d_{x^2-y^2}$ orbital [14] and thus the one-band models are justified as long as the low-energy physics is concerned.", "However, recent theoretical and experimental development sheds light on the importance of the orbital degree of freedom in cuprates.", "For example, material dependence of $T_{\\text{c}}$ is difficult to discuss within the one-band models.", "The previous studies of the one-band models generally discussed the material dependence via the shape of different Fermi surfaces originated by different hopping integrals, e.g., the nearest-neighbor hopping $t$ and the next nearest-neighbor hopping $t^{\\prime }$ .", "Indeed, such treatment allowed us to capture the overall feature for the difference between hole-doped and electron-doped systems [15].", "However, the material dependence of $T_{\\text{c}}$ for the hole-doped systems has not been able to be explained properly.", "Experimentally, $T_{\\text{c}}$ becomes higher with larger $|t^{\\prime }/t|$ , i.e., more rounded Fermi surface [16].", "On the contrary, it is predicted theoretically that in most one-band models, $T_{\\text{c}}$ is reduced because of poor nesting properties in rounded Fermi surfaces [17], [18].", "A clue to resolving this contradiction can be found by seriously considering the orbital degrees of freedom, especially, the Cu $d_{z^2}$ orbital [18], [19], [20].", "The study for the two-orbital model shows that the contribution of the $d_{z^2}$ orbital to the Fermi surface, which is strong in low $T_{\\text{c}}$ materials, works against the $d_{x^2-y^2}$ -wave superconductivity [18].", "This picture can explain the material dependence of $T_{\\text{c}}$ .", "Indeed, a recent ARPES experiment directly observed the hybridization gap between the $d_{z^2}$ and $d_{x^2-y^2}$ orbitals and pointed out the importance of the multiorbital effect on superconductivity [21].", "Another extension to a multiorbital model is the introduction of the O $p$ orbitals.", "A minimal model that includes the Cu $d_{x^2-y^2}$ , O $p_x$ , and O $p_y$ orbitals in the CuO$_2$ plane is known as the (three-band) $d$ -$p$ model or the Emery model [22].", "The three-band $d$ -$p$ model has been studied with various numerical methods [23], [24], [25], [26], [27], [29], [28], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47] as much as the one-band models.", "In these studies, the site-energy difference $\\Delta _{dp}$ between the Cu $d_{x^2-y^2}$ and O $p_{x/y}$ orbitals is found to be an important parameter for understanding the material dependence of $T_{\\text{c}}$ .", "For example, the three-band $d$ -$p$ model can successfully reproduce the negative correlation between $\\Delta _{dp}$ and $T_{\\text{c}}$  [34], namely, a smaller $\\Delta _{dp}$ leads to a higher $T_{\\text{c}}$ .", "Furthermore, the role of the O $p$ orbitals in the superconducting pairing [37], [41], [43], [44], [47], stripe order [26], [38], loop current [29], [30], [33], and nematic order [33], [36], [39], [40] has been discussed with the three-band $d$ -$p$ model.", "However, it is still difficult to correctly capture the correlation between Fermi surface topology and $T_{\\text{c}}$ within the three-band $d$ -$p$ model [28], [34].", "In this paper, we study a four-band $d$ -$p$ model composed of the Cu $d_{x^2-y^2}$ and $d_{z^2}$ orbitals and the O $p_x$ and $p_y$ orbitals, which can be expected to resolve the problems mentioned above.", "As typical examples of single-layer hole-doped cuprates, we consider La$_2$ CuO$_4$ and HgBa$_2$ CuO$_4$ systems.", "We construct the tight-binding model for these systems based on the first-principles calculation and examine the effect of Coulomb interaction by the variational Monte Carlo (VMC) method.", "We show that the material dependence of $T_{\\text{c}}$ is well explained with two key parameters, the site energy $\\varepsilon _{d_{z^2}}$ of the Cu $d_{z^2}$ orbital and the site-energy difference $\\Delta _{dp}$ between the Cu $d_{x^2-y^2}$ and O $p_{x/y}$ orbitals, which is consistent with the empirical relation.", "We thus propose that the present four-band $d$ -$p$ model is a minimal model that can properly describe the material dependence of cuprate superconductors.", "Furthermore, we also study the effect of the nearest-neighbor $d$ -$d$ Coulomb interaction $V_{dd}$ , which has not been discussed in detail previously.", "We show that $V_{dd}$ substantially affects the stability of superconductivity and the phase competition among various phases, suggesting an important parameter for the effective model of cuprates.", "In addition, we find that $V_{dd}$ induces a crossover from a Slater insulator to a Mott insulator at the undoped limit.", "The rest of this paper is organized as follows.", "In Sec.", ", a four-band $d$ -$p$ model on the two-dimensional square lattice is introduced.", "The VMC method and the variational wave functions are also explained in Sec. .", "The numerical results are then provided in Sec. .", "The tight-binding energy bands for the La$_2$ CuO$_4$ and HgBa$_2$ CuO$_4$ systems, obtained on the basis of the first-principles calculation, are first shown in Sec.", "REF .", "The material and doping dependences of a superconducting correlation function are then examined and the role of two key parameters $\\varepsilon _{d_{z^2}}$ and $\\Delta _{dp}$ are clarified in Sec.", "REF .", "The effect of the nearest-neighbor $d$ -$d$ Coulomb interaction $V_{dd}$ is investigated in Sec.", "REF .", "The phase competition among superconductivity and other phases is briefly discussed in Sec.", "REF .", "Finally, the paper concludes with a summary in Sec. .", "The details of the variational wave functions are described in Appendix.", "As an effective low-energy model of cuprates, we consider a four-band $d$ -$p$ model on the two-dimensional square lattice (see Fig.", "REF ) defined by the following Hamiltonian: $H=H_{\\text{kin}}+H_{\\text{int}}-H_{\\text{dc}}.$ Here, the kinetic term $H_{\\text{kin}}$ is described by Hkin=i,j,,tijcicj =k,m Em(k)akmakm, where Eq.", "(REF ) is the kinetic term in an orbital representation and Eq.", "(REF ) is in a band representation.", "$c^{\\dagger }_{i\\alpha \\sigma }$ ($c_{i\\alpha \\sigma }$ ) is a creation (annihilation) operator of an electron at site $i$ with spin $\\sigma \\,(=\\uparrow ,\\downarrow )$ and orbital $\\alpha \\, (=1,2,3,4)$ corresponding to ($d_{x^2-y^2}$ , $d_{z^2}$ , $p_x$ , $p_y$ ), respectively.", "$t^{\\alpha \\beta }_{ij}$ denotes a hopping integral between orbital $\\alpha $ at site $i$ and orbital $\\beta $ at site $j$ .", "$t^{\\alpha \\alpha }_{ii}$ is a site energy $\\varepsilon _{\\alpha }$ for orbital $\\alpha $ at site $i$ .", "Equation (REF ) is obtained by diagonalizing Eq.", "(REF ), and the energy eigenvalue $E_m(\\textbf {k})$ is characterized by the wave vector $\\textbf {k}$ and the energy band index $m\\,(=1,2,3,4)$ .", "$a^{\\dagger }_{\\textbf {k}m\\sigma }$ ($a_{\\textbf {k}m\\sigma }$ ) is a creation (annihilation) operator of the corresponding energy band with spin $\\sigma $ .", "The undoped parent compounds of cuprates correspond to one hole (i.e., seven electrons) per unit cell in this model, which is conventionally referred to as half filling, and hereafter we denote the carrier density as the number $\\delta $ of holes per unit cell that are introduced into the system at half filling.", "Figure: (a) Schematic lattice structure of the four-band dd-pp model, forming the two-dimensional square lattice.Each Cu site contains d x 2 -y 2 d_{x^2-y^2} and d z 2 d_{z^2} orbitals, while there is either p x p_x or p y p_y orbital on O sites.The lattice constant between the nearest-neighbor Cu sites is set to be one, i.e., primitive translation vectors |𝐞 x |=|𝐞 y |=1|\\textbf {e}_x|=|\\textbf {e}_y|=1.", "(b) Phase convention for the Cu d x 2 -y 2 d_{x^2-y^2} orbital and the O p x p_x and p y p_y orbitals.Solid (open) ovals indicate the positive (negative) phase.The hopping integral between the d x 2 -y 2 d_{x^2-y^2} and p x/y p_{x/y} orbitals, t 1 t_1, is shown with the sign.The definitions of other hopping integrals (t 2 -t 6 t_2-t_6) are described in Appendix .The Coulomb interaction term $H_{\\text{int}}$ is composed of eight terms, Hint=Udi(nd1ind1i+nd2ind2i) +(U'd-J2)ind1ind2i-2JiSd1iSd2i -J'i(ci1ci1ci2ci2+ci2ci2ci1ci1) +Upi(npxinpxi+npyinpyi)+Vdp<i,j>ndinpx/yj +Vpp<i,j>npxinpyj+Vdd<i,j>ndindj.", "Here, $n^{\\alpha }_i=n^{\\alpha }_{i\\uparrow }+n^{\\alpha }_{i\\downarrow }$ with $n^{\\alpha }_{i\\sigma }=c^{\\dagger }_{i\\alpha \\sigma }c_{i\\alpha \\sigma }$ is the number operator and $\\textbf {S}^{\\alpha }_i$ is the spin angular momentum operator at site $i$ with orbital $\\alpha $ .", "$d_1$ and $d_2$ are abbreviations for $d_{x^2-y^2}$ and $d_{z^2}$ orbitals, respectively, and $n^d_i=n^{d_1}_i+n^{d_2}_i$ .", "$U_d,U_d^{\\prime },J,$ and $J^{\\prime }$ represent on-site intraorbital, interorbital, Hund's coupling, and pair-hopping interactions between $d$ orbitals, respectively.", "In this study, we set $J^{\\prime }=J$ and $U_d=U^{\\prime }_d+2J$  [48].", "The on-site intraorbital Coulomb interaction within $p$ orbitals, $U_p$ , is also introduced.", "The last three terms in $H_{\\text{int}}$ take into account the intersite Coulomb interactions between nearest-neighbor orbitals, $V_{dp},V_{pp},$ and $V_{dd}$ , as shown in Fig.", "REF , where the sum $\\sum _{\\left<i,j\\right>}$ runs over all pairs of nearest-neighbor orbitals located at site $i$ and $j$ .", "Figure: The Coulomb interaction parameters between nearest-neighbor orbitals.Solid (open) circles represent Cu (O) atoms.In addition, the following double counting correction term $H_{\\text{dc}}$ is introduced, $H_{\\text{dc}}=\\biggl [\\left\\lbrace U_d+2\\left(U^{\\prime }_d-\\frac{J}{2}\\right)+16V_{dd}\\right\\rbrace \\langle n^d\\rangle _0\\ \\\\{+8V_{dp}\\left<n^p\\right>_0\\biggr ]\\sum _in^d_i} \\\\+\\left\\lbrace (U_p+8V_{pp})\\left<n^p\\right>_0+8V_{dp}\\langle n^d\\rangle _0\\right\\rbrace \\sum _i (n^{p_x}_i+n^{p_y}_i),$ where $\\langle n^d\\rangle _0=\\frac{1}{N_{\\text{S}}}\\sum _i\\langle n^d_i\\rangle $ and $\\left<n^p\\right>_0=\\frac{1}{N_{\\text{S}}}\\sum _i\\langle (n^{p_x}_i+n^{p_y}_i)\\rangle $ are the average electron density of the $d$ and $p$ orbitals in the noninteracting limit and $N_{\\text{S}}$ is the total number of unit cells.", "When we apply a many-body calculation method to a multiorbital model, the site energy of each orbital is shifted due to the interaction effect.", "However, such energy shifts have already been included in the energy band of the tight-binding model constructed from the first-principles calculation.", "This is a so-called double counting problem and should be treated with care especially in the $d$ -$p$ model [49].", "Here, we subtract the term $H_{\\text{dc}}$ from the Hamiltonian to correct the energy shift.", "This is one of the reasonable treatments to avoid the double counting.", "Table: The tight-binding parameters for the La 2 _2CuO 4 _4 and HgBa 2 _2CuO 4 _4 systems in eV units estimated on the basis of maximally localized Wannier orbitals from the first-principles LDA calculation.For comparison, the tight-binding parameters for the La 2 _2CuO 4 _4 system with reference to the QSGW method are also shown (denoted as “revised”).The definitions of t i t_i and ε α \\varepsilon _{\\alpha } are described in Appendix .Δ dp =ε d x 2 -y 2 -ε p \\Delta _{dp}=\\varepsilon _{d_{x^2-y^2}}-\\varepsilon _{p}, i.e., the site-energy difference between the Cu d x 2 -y 2 d_{x^2-y^2} and O p x/y p_{x/y} orbitals.Figure: The energy dispersions and projected density of states onto the d x 2 -y 2 d_{x^2-y^2}, d z 2 d_{z^2}, and p x/y p_{x/y} orbitals for the noninteracting tight-binding models for (a) La 2 _2CuO 4 _4, (b) La 2 _2CuO 4 _4(revised), and (c) HgBa 2 _2CuO 4 _4 systems.The tight-binding parameters are given in Table .Fermi energy (defined as zero energy) is set to be the case of 15% hole doping (δ\\delta =0.15).The high symmetric momenta are indicated as Γ=(0,0)\\Gamma =(0,0), M=(π,π)\\text{M}=(\\pi ,\\pi ), and X=(π,0)\\text{X}=(\\pi ,0)." ], [ "VMC method", "The effect of Coulomb interaction is treated using a VMC method [50], [51], [52].", "The trial wave function considered here is a Gutzwiller-Jastrow type composed of four parts, $\\left|\\Psi \\right>=P^{(2)}_{\\text{G}}P_{\\text{J}_{\\text{c}}}P_{\\text{J}_{\\text{s}}}\\left|\\Phi \\right>.$ $\\left|\\Phi \\right>$ is a one-body part constructed by diagonalizing the one-body Hamiltonian including the off-diagonal elements $\\lbrace \\rho ^{\\alpha }_i\\rbrace $ , $\\lbrace m^{\\alpha }_i\\rbrace $ , and $\\lbrace \\Delta ^{\\alpha \\beta }_{ij}\\rbrace $ , which induce long-range ordering of charge, spin, and superconductivity, respectively.", "The renormalized hopping integrals $\\lbrace \\tilde{t}^{\\alpha \\beta }_{ij}\\rbrace $ are also included in $\\left|\\Phi \\right>$ as variational parameters.", "The explicit forms of them are described in Appendix.", "The Gutzwiller factor $P^{(2)}_{\\text{G}}=\\prod _{i,\\gamma }\\bigl [\\text{e}^{-g_{\\gamma }}\\left|\\gamma \\right>\\left<\\gamma \\right|_i\\bigr ]$ is extended to two-orbital systems [53], [54].", "In $P^{(2)}_\\text{G}$ , possible 16 patterns of charge and spin configuration of the $d_{x^2-y^2}$ and $d_{z^2}$ orbitals at each site $\\left| \\gamma \\right>$ , i.e., $\\left|0\\right>=\\left|0\\;0\\right>$ , $\\left|1\\right>=\\left|0\\uparrow \\right>$ , $\\cdots $ , $\\left|15\\right>=\\left|\\uparrow \\downarrow \\;\\uparrow \\downarrow \\right>$ , are differently weighted with $\\text{e}^{-g_{\\gamma }}$ and $\\lbrace g_{\\gamma }\\rbrace $ are optimized as variational parameters.", "The remaining operators $P_{\\text{J}_{\\text{c}}}=\\exp \\Bigl [-\\sum _{i,j}\\sum _{\\alpha ,\\beta }v^{\\text{c}}_{ij\\alpha \\beta }n^{\\alpha }_in^{\\beta }_j\\Bigr ]$ and $P_{\\text{J}_{\\text{s}}}=\\exp \\Bigl [-\\sum _{i,j}\\sum _{\\alpha ,\\beta }v^{\\text{s}}_{ij\\alpha \\beta }s^z_{i\\alpha }s^z_{j\\beta }\\Bigr ]$ are charge and spin Jastrow factors, which control long-range charge and spin correlations, respectively.", "$s^z_{i\\alpha }$ is the $z$ component of the spin angular momentum operator at site $i$ with orbital $\\alpha $ .", "We set $v^{\\text{c}}_{ii\\alpha \\beta }=v^{\\text{s}}_{ii\\alpha \\beta }=0$ for $\\alpha ,\\beta =d_1,d_2$ because the on-site correlation between the $d_{x^2-y^2}$ and $d_{z^2}$ orbitals are already taken into account in $P^{(2)}_{\\text{G}}$ .", "In this paper, we focus mainly on the superconducting correlation functions to examine where the superconductivity appears in the phase diagram.", "The detailed studies on other competing orders will be discussed elsewhere.", "The variational parameters in $\\left|\\Psi \\right>$ are therefore $\\lbrace \\tilde{t}^{\\alpha \\beta }_{ij}\\rbrace $ , $\\lbrace \\Delta ^{\\alpha \\beta }_{ij}\\rbrace $ , $\\lbrace g_{\\gamma }\\rbrace $ , $\\lbrace v^{\\text{c}}_{ij\\alpha \\beta }\\rbrace $ , and $\\lbrace v^{\\text{s}}_{ij\\alpha \\beta }\\rbrace $ .", "They are simultaneously optimized using stochastic reconfiguration method [55].", "We show results for $N_{\\text{S}}$ =24$\\times $ 24=576 unit cells (and thus 576$\\times $ 4=2304 orbitals in total), which is large enough to avoid finite size effects.", "The antiperiodic boundary conditions are set for both $x$ and $y$ directions of the primitive lattice vectors." ], [ "Band structures of La$_2$ CuO{{formula:8b318172-e5b1-42fe-ac93-8691b9b258a3}} and HgBa{{formula:c70efd68-34eb-420d-850f-118fa5eba766}} CuO{{formula:9c768307-d374-47af-bf18-a90c43e63a53}}", "First, we discuss the material dependence of the band structure.", "As a typical example of single-layer hole-doped cuprates, we study the La$_2$ CuO$_4$ and HgBa$_2$ CuO$_4$ systems.", "We construct maximally localized Wannier orbitals [56], [57] from the first-principles calculation in the local-density approximation (LDA) with ecalj package [58] and fit them with the hopping integrals $t_i$ ($i=1-6$ ) and the site energy of each orbital $\\varepsilon _{\\alpha }$ .", "The parameter sets determined for these systems are listed in Table REF and the explicit form of the tight-binding model is described in Appendix REF .", "Note that the estimated site energy of the Cu $d_{z^2}$ orbital, $\\varepsilon _{d_{z^2}}$ , and the hybridization between the Cu $d_{z^2}$ and O $p_{x/y}$ orbitals, $t_4$ , depend significantly on the method of first-principles calculation.", "In fact, the estimated $\\varepsilon _{d_{z^2}}$ is much lower in the quasiparticle self-consistent $GW$ (QSGW) method [61], [62], [63] than in the conventional LDA calculation [64].", "To clarify the effect, we also consider another parameter set of La system with reference to the QSGW band structure (labeled as “revised”), where the site energy $\\varepsilon _{d_{z^2}}$ is lower and $t_4$ is also slightly smaller than those estimated on the basis of the LDA calculation (See Table REF ).", "Figure REF shows the noninteracting tight-binding energy bands for La and Hg systems.", "We can notice the clear difference among them: (i) The density of states (DOS) of the $d_{z^2}$ component is extended from 0 to -2 eV in the La system [Fig.", "REF (a)], while it is almost localized around -2 eV in the Hg system [Fig.", "REF (c)].", "This is because the $d_{z^2}$ orbital is hybridized with the $p_{x/y}$ orbital in the La system much more strongly than in the Hg system.", "The $d_{z^2}$ electrons obtain the itinerancy through the hybridization and therefore the $d_{z^2}$ component of the La system is more extended than that of the Hg system.", "The revised version of the La system is located somewhere in between [Fig.", "REF (b)].", "(ii) The site-energy difference between the $d_{x^2-y^2}$ and $p_{x/y}$ orbitals, $\\Delta _{dp}=\\varepsilon _{d_{x^2-y^2}}-\\varepsilon _{p}$ , is larger in the La system than in the Hg system.", "This affects the occupancy of each orbital and thus the strength of the electron correlation.", "For example, when $\\Delta _{dp}\\, (>0)$ is small, the energy band crossing the Fermi energy contains more component of the $p_{x/y}$ orbital, in which the intraorbital Coulomb interaction is smaller.", "Starting from these energy band structures, we shall investigate the ground state property of the La and Hg systems using the VMC method.", "We assume that the tight-binding parameters remain unchanged with hole doping.", "The Coulomb interaction parameters are set as $(U_d,U^{\\prime }_d,J,U_p,V_{dp},V_{pp})=(8.0, 6.4, 0.8, 4.0, 2.0, 1.6)\\;t_1$ for both La and Hg systems with reference to Ref. [65].", "Note that the values for the La system are larger in eV units because of the larger $t_1$ .", "In the following, we set $t_1$ as a unit of energy.", "We first set $V_{dd}=0$ and then discuss the effect of finite $V_{dd}$ .", "As shown in Sec.", "REF , we find that even small $V_{dd}$ can substantially affect the property of the system.", "To discuss the material dependence of superconductivity, we calculate the superconducting correlation function defined as $P^{dd}(\\textbf {r})=\\frac{1}{N_{\\text{S}}}\\sum _i\\sum _{\\tau ,\\tau ^{\\prime }}f^{(dd)}_{\\tau \\tau ^{\\prime }}\\bigl <\\Delta ^{\\dagger }_{\\tau }(\\textbf {R}_i) \\Delta _{\\tau ^{\\prime }}(\\textbf {R}_i+\\textbf {r})\\bigr >,$ where $\\Delta ^{\\dagger }_{\\tau }(\\textbf {R}_i)$ is a creation operator of singlet pairs between nearest-neighbor $d_{x^2-y^2}$ orbitals, $\\Delta ^{\\dagger }_{\\tau }(\\textbf {R}_i)=\\frac{1}{\\sqrt{2}}(c^{\\dagger }_{i1\\uparrow }c^{\\dagger }_{i+\\tau 1\\downarrow }+c^{\\dagger }_{i+\\tau 1\\uparrow }c^{\\dagger }_{i1\\downarrow }),$ and $\\tau $ runs over four nearest-neighbor Cu sites ($\\tau =\\pm \\mathbf {e}_x,\\pm \\mathbf {e}_y$ ).", "$f^{(dd)}_{\\tau \\tau ^{\\prime }}$ is a form factor of a superconducting gap function with $d_{x^2-y^2}$ symmetry, namely, $f^{(dd)}_{\\tau \\tau ^{\\prime }}=1$ for $\\tau \\parallel \\tau ^{\\prime }$ and $-1$ for $\\tau \\perp \\tau ^{\\prime }$ .", "$\\langle \\cdots \\rangle $ denotes $\\langle \\Psi |\\cdots |\\Psi \\rangle / \\langle \\Psi |\\Psi \\rangle $ for the optimized variational wave function $|\\Psi \\rangle $ .", "If $P^{dd}(\\textbf {r})$ is saturated to a finite value for $r=|\\mathbf {r}|\\rightarrow \\infty $ , superconducting long-range order exists.", "Figure: (a) Superconducting correlation function P dd (r)P^{dd}(r) and P dp (r)P^{dp}(r) for the Hg system at a hole doping rate δ=0.153\\delta =0.153.The sign change of (b) dd-dd and (c) dd-pp pairings.Cu (O) sites are indicated by solid (open) circles in (b) and (c).Figure REF (a) shows the behavior of $P^{dd}(r=|\\textbf {r}|)$ for the Hg system at a hole doping rate $\\delta =0.153$ .", "It shows good convergence for $r \\gtrsim 4$ and reveals that the superconducting long-range order certainly exists.", "The sign of the $d$ -$d$ pairing in a real space is shown in Fig.", "REF (b).", "It is positive in the $x$ direction and negative in the $y$ direction, reflecting the $d_{x^2-y^2}$ -wave symmetry expected in cuprate superconductors.", "We also calculate the superconducting correlation function for pairing formed between the $d_{x^2-y^2}$ and $p_{x/y}$ orbitals $P^{dp}(r)$ , which is defined in the same way as $P^{dd}(r)$ except that $c^{\\dagger }_{i+\\tau 1\\sigma }$ in Eq.", "(REF ) is replaced with $c^{\\dagger }_{i+\\frac{\\tau }{2}3(4)\\sigma }$ for $\\tau =\\pm \\textbf {e}_{x(y)}$ .", "Although the value is one order of magnitude smaller than $P^{dd}(r)$ [see Fig.", "REF (a)], $P^{dp}(r)$ is also saturated to a finite value, indicative of the long-range order of $d$ -$p$ pairing.", "Note that the sign of the $d$ -$p$ pairing changes alternatively along both $x$ and $y$ directions and thus shows $p$ -like symmetry as shown in Fig.", "REF (c).", "This is due to the phase convention of $p_x$ and $p_y$ orbitals adopted here [see Fig.", "REF (b)] and is consistent with the $d_{x^2-y^2}$ pairing symmetry [66].", "It is also compatible with the preceding study of the three-band $d$ -$p$ model [45].", "This kind of real space or orbital representation is useful for the analysis of cuprate superconductors [43], [44], [47] because the pairing length is expected to be very short [67].", "Figure: P dd P^{dd} as a function of the hole doping rate δ\\delta for the three systems." ], [ "Material dependence: Effect of $d_{z^2}$ orbital", "Let us now examine the material and doping dependence of the superconducting correlation function.", "We take the converged value of $P^{dd}(r\\rightarrow \\infty )$ as a strength of superconductivity $P^{dd}$ .", "Figure REF shows the doping dependence of $P^{dd}$ for the La, La(revised), and Hg systems.", "For all cases, $P^{dd}$ displays a dome shape as a function of the hole doping rate $\\delta $ .", "At $\\delta =0$ , the system is insulating due to the strong correlation effect and thus $P^{dd}=0$ .", "As $\\delta $ increases, mobile carriers are introduced into the system and the mobility of the Cooper pair increases.", "On the other hand, the strength of the $d$ -$d$ pairing itself is reduced by doping because the electron correlation effect is also reduced.", "The balance between these two factors results in the dome-shaped behavior of $P^{dd}$ .", "This picture is expected to be universal for all hole-doped cuprate superconductors.", "Figure: Total and α\\alpha orbital components of momentum distribution function for the La system at (a) δ=0.153\\delta =0.153 (superconducting phase) and (b) δ=0.181\\delta =0.181 (paramagnetic metallic phase).The high symmetric momenta are indicated as Γ=(0,0)\\Gamma =(0,0), M=(π,π)\\text{M}=(\\pi ,\\pi ), and X=(π,0)\\text{X}=(\\pi ,0).Next, let us study the importance of the orbital character near the Fermi energy in the material dependence of superconductivity.", "Generally, a large DOS at the Fermi energy is favorable for superconductivity due to the large energy gain of gap opening.", "However, a detailed structure of the DOS, namely, the orbital character and $\\mathbf {k}$ dependence should be carefully investigated.", "For this purpose, we calculate the following momentum distribution function of holes, $n^{\\alpha }(\\textbf {k})=\\frac{1}{2}\\sum _{\\sigma } \\langle c_{\\textbf {k}\\alpha \\sigma }c^{\\dagger }_{\\textbf {k}\\alpha \\sigma } \\rangle ,$ where $c^{\\dagger }_{\\textbf {k}\\alpha \\sigma }$ ($c_{\\textbf {k}\\alpha \\sigma }$ ) is a Fourier transform of $c^{\\dagger }_{i\\alpha \\sigma }$ ($c_{i\\alpha \\sigma }$ ) in Eq.", "(REF ).", "Figure REF shows $n^{\\alpha }(\\textbf {k})$ for the La system at the superconducting phase ($\\delta =0.153$ ) and the paramagnetic metallic phase ($\\delta =0.181$ ).", "The discontinuities around the X point and in the middle of the $\\Gamma $ -M line in Fig.", "REF (b) for the paramagnetic metallic phase indicate the existence of Fermi surface.", "Since the node of the superconducting gap runs along the $\\Gamma $ -M line, a discontinuity also exists in the superconducting phase, as shown in Fig.", "REF (a).", "We can observe that the $d_{z^2}$ component has large weight around the X point, exhibiting a peak structure in $n^{\\alpha }(\\textbf {k})$ , where the superconducting gap becomes the largest (so-called “hot spot”).", "This feature is expected to be unfavorable for superconductivity because the AF spin fluctuation, which promotes the $d_{x^2-y^2}$ -wave pairing, is suppressed when the Cu $d_{z^2}$ orbital contributes to the formation of the Fermi surface [68].", "Figure: (a) δ\\delta dependence of the ratio RR of the d z 2 d_{z^2} component to the total at the X point for the three systems.Solid (open) symbols represent the superconducting (paramagnetic metallic) phase.", "(b) δ\\delta dependence of the quasiparticle renormalization factor zz for the three systems.Symbols are the same with those in (a).To investigate the effect of the $d_{z^2}$ component, the ratio of the $d_{z^2}$ component to the total at the X point, $R=n^{d_{z^2}}(\\text{X})/n^{\\text{tot}}(\\text{X})$ , is calculated in Fig REF (a).", "For the La system, the ratio $R$ increases with increasing $\\delta $ and shows rapid enhancement for $\\delta >0.16$ .", "It coincides with the sudden disappearance of $P^{dd}$ for $\\delta >0.16$ shown in Fig.", "REF .", "On the other hand, $R$ for the La(revised) system is much smaller than that of the La system, because $\\varepsilon _{d_{z^2}}$ is lower and the hybridization between the $d_{z^2}$ and $d_{x^2-y^2}$ orbitals via the $p_{x/y}$ orbitals is smaller.", "As a result, the superconducting phase is extended to a larger value of $\\delta $ and a smooth dome shape is observed in $P^{dd}$ vs. $\\delta $ as shown in Fig.", "REF .", "For the Hg system, $R$ is much more suppressed because the $d_{z^2}$ -orbital based band is almost localized and detached from the $d_{x^2-y^2}$ -orbital based band [see Fig.", "REF (c)].", "This is an ideal condition for superconductivity [18] and therefore $P^{dd}$ becomes largest for the Hg system.", "These results conclude that superconductivity is more enhanced when the $d_{z^2}$ -orbital based band is deeply sinking and its contribution to the low-energy physics is small.", "Therefore, the material dependence of superconductivity is understood only by incorporating the $d_{z^2}$ orbital explicitly into a model such as our model, which is a remarkable advantage over the usual one-band Hubbard and $t$ -$J$ models, and even the three-band $d$ -$p$ model.", "$P^{dd}$ calculated here corresponds to the square of the superconducting order parameter and is closely related to the superconducting transition temperature $T_{\\text{c}}$ .", "The critical doping rate $\\delta _{\\text{c}}$ for the La system, where $P^{dd}$ becomes zero, seems to be too small ($\\sim 0.16$ ) compared with the experimental value ($\\delta _{\\text{c}}=0.25-0.3$ ).", "Furthermore, the sudden disappearance of $P^{dd}$ is also unrealistic.", "It can be inferred that the actual value of $\\varepsilon _{d_{z^2}}$ is much lower than the value estimated from the LDA calculation.", "Indeed, as shown in Table REF , the value of $\\varepsilon _{d_{z^2}}$ with reference to the QSGW calculation is much lower.", "Furthermore, the QSGW band structure [64] can well explain the resonant inelastic X-ray scattering experiment [69].", "We expect that the revised band structure shown in Fig.", "REF (b) properly includes the correction for the LDA calculation and gives more realistic result for the La system." ], [ "Material dependence: Effect of apical oxygen height", "From the viewpoint of the actual lattice structure, $\\varepsilon _{d_{z^2}}$ is governed by the apical oxygen height, i.e., the distance between the apical oxygen and the copper: The larger the apical oxygen height is, the lower $\\varepsilon _{d_{z^2}}$ is with respect to $\\varepsilon _{d_{x^2-y^2}}$ , because of the crystal field effect [70], [68].", "In addition, a larger apical oxygen height leads to a lower site energy $\\varepsilon _{p_z}$ of the apical oxygen due to the decrease of a crystal field effect, which in turn lowers the $\\varepsilon _{d_{z^2}}$ through the hybridization between $p_z$ and $d_{z^2}$ orbitals despite the increase of the distance between apical oxygen and the copper.", "Therefore, our result suggests that a larger apical oxygen height leads to a higher $T_{\\text{c}}$ through a lower $\\varepsilon _{d_{z^2}}$ .", "Indeed, the experimentally observed apical oxygen height of the Hg system is larger than that of the La system.", "This tendency is also consistent with the so-called Maekawa's plot [71], where a lower $\\varepsilon _{p_z}$ is related to a higher $T_{\\text{c}}$ .", "Although the model itself does not explicitly include the $p_z$ orbital of the apical oxygen, the present four-band $d$ -$p$ model properly incorporates the effect of the apical oxygen height via adjusting the site energy $\\varepsilon _{d_{z^2}}$ ." ], [ "Material dependence: Effect of $\\Delta _{dp}$", "We also show in Fig.", "REF (b) the quasiparticle renormalization factor $z$ estimated from the jumps in the total momentum distribution function $n^{\\text{tot}}(\\textbf {k})$ along the nodal direction of the $d_{x^2-y^2}$ -wave superconducting gap.", "At $\\delta =0$ , the system is insulating and thus $z=0$ .", "With increasing $\\delta $ , $z$ increases according to the decrease of the electron correlation effect.", "We find that $z$ for the La system is smaller than that for the Hg system, indicating that the electron correlation effect is stronger in the La system.", "This can be attributed to the larger $\\Delta _{dp}=\\varepsilon _{d_{x^2-y^2}}-\\varepsilon _{p}$ for the La system, which results in the larger $d$ -orbital occupancy of holes when it is doped.", "The electron correlation has a dual effect: One is to enhance the superconducting pairing and the other is to reduce the mobility of Cooper pairs.", "In the present case, the latter effect would be dominant and thus $P^{dd}$ for the La system is suppressed compared with the Hg system.", "This tendency is consistent with the negative correlation between $\\Delta _{dp}$ and $T_{\\text{c}}$ as mentioned in Sec. .", "Figure: Ground state phase diagram for the Hg system at δ=0\\delta =0 (a) within the paramagnetic phase and (b) with the AF phase.The dotted curve in (b) corresponds to the solid curve in (a).Green stars in (a) and (b) represent the parameters used in Fig.", "." ], [ "Effect of $V_{dd}$", "Now we discuss the effect of the Coulomb interaction between nearest-neighbor $d$ orbitals, $V_{dd}$ .", "$V_{dd}$ is expected to be smaller than other Coulomb interactions, $U_d, U^{\\prime }_d, U_p, V_{dp}$ , and $V_{pp}$  [65].", "However, $V_{dd}$ directly affects the charge and spin correlations between nearest-neighbor $d$ electrons, which dominate the properties of cuprate superconductors.", "As discussed in this section, we verify that the superconductivity in the model studied here is more sensitive to the value of $V_{dd}$ than other Coulomb interactions.", "Hereafter, we treat only $U_d$ and $V_{dd}$ as independent parameters, and set other Coulomb interactions as $(U^{\\prime }_d,J,U_p,V_{dp},V_{pp})=(0.8, 0.1, 0.5, 0.25, 0.2)\\;U_d$ with reference to Ref.", "[65] Figure: P dd P^{dd} as a function of the hole doping rate δ\\delta at U d /t 1 =8U_d/t_1=8 and V dd /t 1 =0,0.3,0.6V_{dd}/t_1=0, 0.3, 0.6 for the Hg system.Figure REF (a) shows the ground state phase diagram for the Hg system at $\\delta =0$ within the paramagnetic phase.", "When $V_{dd}=0$ , a metal-insulator transition occurs at $U_d/t_1\\sim 6.3$ , assuming the paramagnetic phase.", "The metal-insulator transition is detected by monitoring the jump in the total momentum distribution function $n^{\\text{tot}}(\\textbf {k})$ as well as the long-range behavior of the charge Jastrow factor $P_{\\text{J}_{\\text{c}}}$  [72].", "This transition is a Mott metal-insulator transition because $\\delta =0$ corresponds to the case with one hole per unit cell, $n_{\\text{hole}}=1$ , where the system cannot be a band insulator.", "With the introduction of small but finite $V_{dd}$ (one order of magnitude smaller than $U_d$ ), the metallic region is substantially enlarged.", "This is understood because the densities of empty sites and doubly-occupied sites increase to reduce the energy loss of $V_{dd}$ , thus effectively weakening $U_d$ by $V_{dd}$ , and then the insulating phase is destabilized.", "It is noteworthy that the AF insulator is always more stable than the paramagnetic insulator in the present parameter space.", "With increasing $U_d$ , the system undergoes the phase transition from the metallic phase to a Slater-type AF insulator [a blue line in Fig.", "REF (b)], followed by the crossover to a Mott-type AF insulator [a red dotted line in Fig.", "REF (b)].", "Here, a Slater-type AF insulator is an insulator that becomes metallic without AF order, while a Mott-type AF insulator is an insulator that remains insulating without AF order.", "The crossover line in Fig.", "REF (b) is hence identical with the paramagnetic metal-insulator transition line in Fig.", "REF (a).", "As explained next, it is crucially important for the appearance of high-$T_{\\text{c}}$ superconductivity whether the AF insulator in the parent compounds ($\\delta $ =0) is Slater-type or Mott-type [31], [32], [73], [72].", "Next we study the effect of $V_{dd}$ on superconductivity.", "Figure REF shows the superconducting correlation function $P^{dd}$ at $U_d/t_1=8$ and $V_{dd}/t_1=0,0.3,$ and 0.6 for the Hg system.", "As $V_{dd}$ increases, $P^{dd}$ is suppressed and the peak position is moved to a smaller value of $\\delta $ .", "In particular, a significant suppression is observed when $V_{dd}/t_1=0.6$ .", "In this case, $P^{dd}$ vs. $\\delta $ does not show a dome-shaped behavior but instead an almost monotonic decrease, which rather reminds us of electron-doped cuprates [74], [75].", "This observation of $P^{dd}$ vs. $\\delta $ corresponds to the fact that the system is out of the Mott insulator region at $\\delta =0$ (see Fig.", "REF ).", "The view of a “doped Mott insulator” is thus no longer valid.", "Similar claims have been made in the study of one-band Hubbard models [76], [77].", "Our result suggests that it is essential to start from the Mott insulator region at $\\delta =0$ to reproduce the dome-shaped behavior observed experimentally in hole-doped cuprates.", "This can be considered as a good criterion for choosing the reasonable Coulomb interaction parameters of an effective model for cuprates." ], [ "Phase competition", "Finally, we briefly discuss the competition between superconductivity and other phases.", "The energy comparison among various phases in the ground state is a subtle problem and depends significantly on the numerical methods.", "The VMC method used here tends to overestimate the magnetic long-range ordered phases, although it is much improved as compared with a mean-field type approximation.", "In fact, we find that the AF phase and the stripe phase with both spin and charge modulations have lower variational energies than $d_{x^2-y^2}$ -wave superconductivity for $\\delta <0.3$ .", "Nevertheless, we believe that the present results capture the essence of the material dependence of cuprate superconductors and the conclusion is unchanged, when the improved wave functions incorporating the quantum fluctuations suppress these overestimated competing orders.", "We also note that the effect of $V_{dd}$ is also important for the phase competition.", "This is because most of the competing phases including the phase separation are governed by the correlation between nearest-neighbor $d$ electrons.", "This is also the case in the one-band Hubbard model [78], [79].", "The detailed ground state phase diagram, including various competing phases, for the four-band $d$ -$p$ model is left for a future study.", "To obtain the unified description of cuprate superconductors, we have studied the four-band $d$ -$p$ model for the La$_2$ CuO$_4$ and HgBa$_2$ CuO$_4$ systems.", "We have shown that a lower $\\varepsilon _{d_{z^2}}$ with respect to $\\varepsilon _{d_{x^2-y^2}}$ and a smaller $\\Delta _{dp}(>0)$ lead to a higher $T_{\\text{c}}$ .", "The former results in a more localized $d_{z^2}$ -orbital based band that do not interfere the superconductivity.", "The latter results in a larger $z$ , namely, a weaker electron correlation effect, which promotes the itinerancy of mobile carriers and thus enhances superconductivity.", "The present four-band $d$ -$p$ model covers these two factors, beyond the usual one-band and even three-band $d$ -$p$ models.", "Therefore, this model is considered to be a minimal model that can properly describe the material dependence of cuprate superconductors, and thus it can also provide a valuable guideline to design new materials with a higher $T_{\\text{c}}$ .", "The effect of $V_{dd}$ has also been investigated.", "Although the value of $V_{dd}$ is small compared with other Coulomb interaction parameters, it substantially affects the ground state property of the system.", "$V_{dd}$ weakens the effective $U_d$ and induces the paramagnetic metal-insulator transition, or the crossover from a Slater insulator to a Mott insulator.", "The stability of superconductivity is also affected by $V_{dd}$ .", "Considering the doping dependence, we have to start from the Mott insulator region at $\\delta =0$ to obtain the stable superconductivity and the dome-shaped dependence of $P^{dd}$ as a function of $\\delta $ .", "Therefore, the appropriate estimation of $V_{dd}$ is important for the modelling of cuprates, as in other strongly-correlated electron systems where various phases compete.", "The authors thank K. Kuroki for useful discussions.", "The computation has been done using the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo and the supercomputer system HOKUSAI in RIKEN.", "This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas “Quantum Liquid Crystals” (KAKENHI Grant No.", "JP19H05825) from JSPS of Japan, and also supported by JSPS KAKENHI (Grant Nos.", "JP21H04446, JP20K03847, JP19K23433, JP19H01842, and JP18H01183).", "*" ], [ "Construction of the trial wave function", "Here, we describe the details of the trial wave function together with the noninteracting tight-binding model obtained on the basis of the first-principles calculations in Sec.", "REF .", "The construction of the trial wave function is the most important part for the VMC method.", "Depending on the trial state, both real and $\\textbf {k}$ space representations are used." ], [ "Noninteracting energy band", "First, we describe how to construct the noninteracting tight-binding energy band discussed in Sec.", "REF .", "The noninteracting energy band is obtained by diagonalizing the following one-body Hamiltonian: Hkin=k, ( ck1, ck2, ck3, ck4) t11 t*21 t*31 t*41 t21 t22 t*32 t*42 t31 t32 t33 t*43 t41 t42 t43 t44 ck1 ck2 ck3 ck4 =k,m Em(k)akmakm with the hopping matrix elements given as t11=dx2-y2, t21=0, t22=dz2-2t5(kx+ky), t31=2it112kx, t32=-2it412kx, t33=px+2t3kx+2t6[(kx+ky)+(kx-ky)], t41=-2it112ky, t42=-2it412ky, t43=2t2[(12kx+12ky)-(12kx-12ky)], t44=py+2t3ky+2t6[(kx+ky)+(kx-ky)], where $c^{\\dagger }_{\\textbf {k}\\alpha \\sigma }$ ($c_{\\textbf {k}\\alpha \\sigma }$ ) is a creation (annihilation) operator of an electron with momentum $\\textbf {k}$ , spin $\\sigma \\,(=\\uparrow ,\\downarrow )$ , and orbital $\\alpha \\, (=1,2,3,4)$ corresponding to ($d_{x^2-y^2}$ , $d_{z^2}$ , $p_x$ , $p_y$ ), respectively.", "The bonds between nearest-neighbor Cu sites are set as unit vectors ($|\\textbf {e}_x|=|\\textbf {e}_y|=1$ ) and the bonds between nearest-neighbor Cu and O sites are one-half of them (see Fig.", "REF ).", "The hopping integrals $t_i$ ($i=1-6$ ) and the site energy of each orbital $\\varepsilon _{\\alpha }$ are determined by fitting the band structures that are obtained by the LDA or QSGW calculation.", "The specific values for the La, La(revised), and Hg systems are listed in Table REF .", "Equation (REF ) is obtained by diagonalizing the Hamiltonian matrix in Eq.", "(REF ) and is the same with Eq.", "(REF ) in Sec.", "REF .", "$E_m(\\textbf {k})$ is the noninteracting energy band characterized by the wave vector $\\textbf {k}$ and the energy band index $m\\,(=1,2,3,4)$ with $a^{\\dagger }_{\\textbf {k}m\\sigma }$ ($a_{\\textbf {k}m\\sigma }$ ) being a creation (annihilation) operator of the corresponding energy band with spin $\\sigma $ .", "To construct the trial wave function for superconductivity, we employ the Bogoliubov de-Gennes (BdG) type Hamiltonian in real space [80], i.e., $H_{\\text{BdG}}=\\sum _{i,j}\\sum _{\\alpha ,\\beta }\\left(c^{\\dagger }_{i\\alpha \\uparrow }, c_{i\\alpha \\downarrow }\\right)\\begin{pmatrix}T^{\\alpha \\beta }_{ij\\uparrow } & \\Delta ^{\\alpha \\beta }_{ij} \\\\\\Delta ^{\\alpha \\beta *}_{ji} & -T^{\\alpha \\beta }_{ji\\downarrow }\\end{pmatrix}\\begin{pmatrix}c_{j\\beta \\uparrow } \\\\c^{\\dagger }_{j\\beta \\downarrow }\\end{pmatrix}.$ Here, $T^{\\alpha \\beta }_{ij\\sigma }$ is obtained from the Fourier transform of the matrix in Eq.", "(REF ) with renormalized hopping integrals $\\tilde{t}_i$ and also includes the chemical potential term.", "The chemical potential $\\mu $ is set to the Fermi energy in the noninteracting limit.", "$\\Delta ^{\\alpha \\beta }_{ij}$ corresponds to an anomalous part that describes the superconducting pairing in real space.", "Therefore, the variational parameters to be optimized in $\\left|\\Phi \\right>$ are $\\tilde{t}_i$ ($i=2-6$ ) and $\\lbrace \\Delta ^{\\alpha \\beta }_{ij}\\rbrace $ with $\\tilde{t}_1=t_1$ being fixed as a unit of energy.", "In this study, we consider the pairing between nearest-neighbor orbitals, $d$ -$d$ , $d$ -$p_x$ , $d$ -$p_y$ , $p_x$ -$p_x$ , and $p_y$ -$p_y$ , where $d$ denotes the $d_{x^2-y^2}$ orbital.", "In the paramagnetic phase, we simply set $\\Delta ^{\\alpha \\beta }_{ij}=0$ .", "We can also construct the trial wave function with the band ($\\textbf {k}$ space) representation, where $\\Delta ^{mn}(\\textbf {k})\\propto \\bigl < a_{\\textbf {k}m\\uparrow }a_{-\\textbf {k}n\\downarrow }\\bigr >$ is the variational parameter.", "However, we find that the trial wave function with the real space representation always gives lower (i.e., better) energy than that with the band representation, especially, for large Coulomb interaction parameters.", "This is because the Coulomb interaction depends on the orbital, not on the band, and thus the trial wave function with the real space (orbital) representation gives the better result." ], [ "Uniform spin AF and stripe phases", "As mentioned in Sec.", "REF , various long-range orderings of charge and spin can be described by introducing $\\lbrace \\rho ^{\\alpha }_i\\rbrace $ and $\\lbrace m^{\\alpha }_i\\rbrace $ .", "A uniform spin AF phase with A and B sublattices for the orbital $\\alpha $ is expressed with the staggered potential $m^{\\alpha }_i={\\left\\lbrace \\begin{array}{ll}-s_{\\sigma } m^{\\alpha } & \\text{for A sublattice} \\\\+s_{\\sigma } m^{\\alpha } & \\text{for B sublattice}\\end{array}\\right.", "}$ where $s_{\\sigma }=1(-1)$ for $\\sigma =\\uparrow (\\downarrow )$ .", "For a stripe phase with charge and spin periodicities $\\lambda ^{\\alpha }_{\\text{c}}=2\\pi /q^{\\alpha }_{\\text{c}}$ and $\\lambda ^{\\alpha }_{\\text{s}}=2\\pi /q^{\\alpha }_{\\text{s}}$ , respectively, the following potentials with spatial modulation in the $x$ direction should be introduced: $\\rho ^{\\alpha }_i=\\rho ^{\\alpha }\\cos [q^{\\alpha }_{\\text{c}}(x_i-x^{\\alpha }_{\\text{c}})]$ and $m^{\\alpha }_i=(-1)^{x_i+y_i}m^{\\alpha }\\sin [q^{\\alpha }_{\\text{s}}(x_i-x^{\\alpha }_{\\text{s}})],$ where $x^{\\alpha }_{\\text{c}}$ and $x^{\\alpha }_{\\text{s}}$ control the relative phases of charge and spin orderings, respectively.", "For example, the stripe phase observed around $\\delta =1/8$ in several cuprate superconductors corresponds to $\\lambda _{\\text{c}}=4$ and $\\lambda _{\\text{s}}=8$ , although the orbital dependence and relative phase are still under debate." ] ]
2105.11664
[ [ "Deep High-Resolution Representation Learning for Cross-Resolution Person\n Re-identification" ], [ "Abstract Person re-identification (re-ID) tackles the problem of matching person images with the same identity from different cameras.", "In practical applications, due to the differences in camera performance and distance between cameras and persons of interest, captured person images usually have various resolutions.", "We name this problem as Cross-Resolution Person Re-identification which brings a great challenge for matching correctly.", "In this paper, we propose a Deep High-Resolution Pseudo-Siamese Framework (PS-HRNet) to solve the above problem.", "Specifically, in order to restore the resolution of low-resolution images and make reasonable use of different channel information of feature maps, we introduce and innovate VDSR module with channel attention (CA) mechanism, named as VDSR-CA.", "Then we reform the HRNet by designing a novel representation head to extract discriminating features, named as HRNet-ReID.", "In addition, a pseudo-siamese framework is constructed to reduce the difference of feature distributions between low-resolution images and high-resolution images.", "The experimental results on five cross-resolution person datasets verify the effectiveness of our proposed approach.", "Compared with the state-of-the-art methods, our proposed PS-HRNet improves 3.4\\%, 6.2\\%, 2.5\\%,1.1\\% and 4.2\\% at Rank-1 on MLR-Market-1501, MLR-CUHK03, MLR-VIPeR, MLR-DukeMTMC-reID, and CAVIAR datasets, respectively.", "Our code is available at \\url{https://github.com/zhguoqing}." ], [ "Introduction", "Person re-identification (re-ID) intends to match person images with the same identity across images captured by various cameras.", "Re-ID has become the spotlight in the field of machine learning and computer vision owing to its wide practicability in recent years [1], [2], [3], [4], [5], [6], [7], [8], [9].", "Driven by recent advances of deep learning, existing researches of re-ID focus on designing deep feature extraction networks to improve the matching accuracy of re-ID [10], [11], [12], [13].", "Although these approaches have achieved satisfactory performance and alleviated the influence of person pose changes, background clutters or part occlusions to a certain extent, these methods are usually on the basis of the prerequisite that the gallery images and the query images possess the same resolution and sufficient fine-grained details.", "However, such prerequisite is difficult to guarantee in practical applications.", "The problem of matching the same person images with different resolutions is named as Cross-Resolution Person Re-identification [13], [14], [15], [16].", "Figure: Illustration of the difference between (a) traditional person re-ID task in the ideal scenario and (b) cross-resolution person re-ID task.", "Compared with high-resolution (HR) query images and gallery images, low-resolution (LR) query images contain less fine-grained details, which causes a significant reduction in recognition accuracy and brings great challenges to the work of matching.Fig.", "REF shows the difference between (a) the person re-ID task in an ideal condition and (b) the cross-resolution person re-ID task.", "Ideally, query images maintain the same high-resolution (HR) as gallery images.", "However, due to the differences in camera performance and distance between probes and target pedestrians, captured query images often possess lower resolution than gallery images.", "The lack of image information makes the traditional re-ID methods incapable of effectively extracting the discriminant features of images for matching, which has become a stumbling block to the development of re-ID task.", "In order to settle the above problem, many cross-resolution person re-ID algorithms have been put forward in recent years.", "In early work, the main idea plans to explore the common feature representation space of HR and LR images by using metric learning or dictionary learning methods, such as [14], [13], [17], [18].", "However, the performance of these methods are restricted due to the incapability of recovering the information lost in LR images.", "Later, some researchers try to introduce super-resolution (SR) technology into cross-resolution person re-ID.", "SING [16] first applies SRCNN [19] as the resolution recovery module and jointly trains the SRCNN sub-network and the re-ID sub-network.", "Since then, different SR networks, such as SRGAN [20] and FFSR [21], are introduced as the resolution recovery module to further optimize the framework.", "Recently, some new methods represented by INTACT [40] have been proposed, and more novel and effective mechanisms have been applied to raise the detection accuracy to a new level.", "These methods have achieved significant performance improvement, but it is still far below the practical application standard.", "Through detailed comparison and analysis of numerous recent cross-resolution person re-ID methods, we gradually discovered some commonalities contained in them.", "The most conspicuous is that the existing approaches almost all use convolutional neural networks with the property of down-sampling such as ResNet [53] as feature extraction networks.", "We believe that this is the most detrimental factor in the existing methods.", "Using such networks as the feature extraction backbone will inevitably cause further loss of fine-grained information from low-resolution images.", "Besides, excessive emphasis on low-resolution image reconstruction seems to have formed a stereotyped thinking pattern.", "In fact, through experiments, we found that complex super-resolution networks may not perform better than a simple one in cross-resolution person re-ID task under certain circumstances.", "More energy should be devoted to the study of deep semantic information and feature information extracted from low-resolution images.", "In this paper, we propose the PS-HRNet to solve the limitations analyzed above.", "Firstly, we further improve the super-resolution capability of VDSR [22] in terms of deep semantic information learning by adding the channel attention mechanism, and name the modified SR module as VDSR-CA.", "Besides, based on the finding that the unique parallel architecture of HRNet [23] is helpful to alleviate the impact of resolution difference, we utilize the HRNet as the feature extraction network.", "Here we propose the HRNet-ReID to capture multi-resolution features of person images by introducing a novel representation head to HRNet, which can adapt HRNet to the person re-ID problem.", "In addition, our PS-HRNet adopts a pseudo-siamese framework [65], [66] so as to further decrease the distribution difference between LR image features and HR image features.", "The training strategy of the whole network is divided into two phases.", "In the first phase, only the HRNet-ReID module in HR branch of pseudo-siamese framework is trained on traditional HR person re-ID datasets.", "In the second phase, we use joint training strategy to train both the VDSR-CA module and two HRNet-ReID modules simultaneously on cross-resolution person re-ID datasets.", "The outstanding contributions of our work are summarized in the following three points: $\\bullet $ We put forward a feature extraction network named as HRNet-ReID, which combines native HRNet-W32 backbone with a novel representation head designed by us to adapt HRNet to the specific person re-ID mission, overcoming the flaw caused by conventional feature extraction networks in existing methods.", "$\\bullet $ We construct a pseudo-siamese framework named as PS-HRNet which combines our proposed VDSR-CA and HRNet-ReID to further explore the feature space at a deeper level and successfully reduce the distribution difference between LR image features and HR image features, providing an original solution to the cross-resolution person re-ID problem.", "$\\bullet $ We have carried out extensive experiments on five cross-resolution person re-ID datasets, and all of them have achieved the highest level in the industry.", "Compared with the state-of-the-arts, our proposed PS-HRNet improves 3.4%, 6.2%, 2.5%, 1.1% and 4.2% at Rank-1 on MLR-Market-1501, MLR-CUHK03, MLR-VIPeR, MLR-DukeMTMC-reID, and CAVIAR datasets, respectively.", "Figure: The unified architecture of Deep High-Resolution Pseudo-Siamese Framework (PS-HRNet) proposed by us for cross-resolution person re-ID task, which is a pseudo-siamese framework consisting of double HRNet-ReID feature extraction networks and one VDSR-CA module.", "The VDSR-CA is utilized to restore the resolution of input LR images.", "The HRNet-ReID is designed to adapt the HRNet to the person re-ID task, and extracts discriminating features from restored images.", "The right side gives a brief sketch of the two types of modules.", "⊗\\otimes and ⊕\\oplus denote element-wise product and element-wise add, respectively.", "As the main structure in PS-HRNet, the pseudo-siamese framework is adopted to close the feature distributions of LR and HR images.", "Our PS-HRNet is trained alternatively in two phases: (1) Update the single HRNet-ReID-H with the loss ℒ ID {{\\cal L}_{ID}} (Eq.", "(10)); (2) Update the joint multi-task learning loss with the loss ℒ TOTAL {{\\cal L}_{TOTAL}} (Eq.", "(12)).", "The above two phases are marked with blue and black arrows, respectively.", "(best viewed in color)." ], [ "RELATED WORK", "In this section, we will roughly introduce the related works concerned with traditional person re-ID and cross-resolution person re-ID, and revisit two utilized core modules and effective pseudo-siamese framework." ], [ "Person re-ID", "Person re-ID has achieved rapid development in the past decades.", "A series of methods have been proposed to extract more robust and discriminating feature representations, and overcome the difficulties brought by person pose changes, background clutters or part occlusions.", "Specifically, to solve pose changes, Liu et al.", "[26] design a pose-transferable GAN which aims to produce person images with multiple poses for data enhancement.", "To address background clutter, some methods based on attention mechanism or semantic parsing are proposed.", "Li et al.", "[27] apply spatial and channel attention to make network focus on more informative parts.", "Kalayeh et al.", "[28] adopt semantic parsing to segment the foreground information and background information to reduce the interference of background.", "Besides, extensive methods have achieved great progress in occluded re-ID [29], [30], unsupervised re-ID [31], [32], cross-modality re-ID [33], [34], and so on.", "However, most of existing methods neglect the resolution mismatch problem which is a common situation in practical scenarios." ], [ "Cross-Resolution Person re-ID", "To solve the resolution inconsistency problem, a few methods have been proposed recently.", "Previous traditional methods [35], [36] mainly focus on dictionary learning and metric learning, which achieve limited performance due to the lack of detail features in LR images.", "Encouraged by the flourishing development of convolutional neural networks (CNN) [37] and super-resolution (SR) technology, some SR-based methods are proposed and greatly improve the matching accuracy.", "For instance, Jiao et al.", "[16] make the first attempt to combine the SRCNN and re-ID network into one framework, and propose a jointly training strategy.", "Besides, some methods adopt GANs to further improve the framework.", "Specifically, Wang et al.", "[38] adopt SR-GAN repeatedly to build a cascaded structure.", "Li et al.", "[39] restore image resolutions and learn the resolution-invariant representations.", "Recently, Cheng et al.", "[40] optimize SR-reID joint framework from the perspective of training strategy and achieve the best performance, which enhances the compatibility between two sub-networks by utilizing the underlying association knowledge between SR and re-ID.", "Most existing deep learning methods based on SR attach their importance to reconstruct SR images and make the generated images visually closer to the original HR images.", "However, these methods ignore the distribution differences between the LR image features and HR image features extracted by the feature extraction network separately." ], [ "Revisit VDSR and HRNet", "As a high-performance super-resolution method, VDSR [22] applies a deeper network to gain in-depth image information and further ameliorates the structure of SRCNN [19].", "Motivated by the prevalence and development of residual network (ResNet) , VDSR adopts the residual connection to overcome the difficulty in convergence of deep networks.", "Therefore, the capacity of VDSR on image reconstruction surpasses SRCNN strikingly.", "HRNet [23] is first proposed to deal with the Human Pose Estimation task, and then surpasses all predecessors in other fields such as key point detection, pose estimation and multi-person pose estimation [41], [42].", "The core structure of HRNet contains four parallel streams, which is logically presented as follow: $\\begin{aligned}\\begin{array}{r}{{\\cal N}_{11}} \\rightarrow {{\\cal N}_{21}} \\rightarrow {{\\cal N}_{31}} \\rightarrow {{\\cal N}_{41}} \\\\{\\rm { }} \\searrow {{\\cal N}_{22}} \\rightarrow {{\\cal N}_{32}} \\rightarrow {{\\cal N}_{42}} \\\\{\\rm { }} \\searrow {{\\cal N}_{33}} \\rightarrow {{\\cal N}_{43}} \\\\{\\rm { }} \\searrow {{\\cal N}_{44}},\\end{array}\\end{aligned}$ where ${{\\cal N}_{sr}}$ is a sub-stream in the $s$ -th phase and $r$ is a resolution index.", "In order to maintain the high resolution of the same branch while obtaining information from other branches, layers at the junction of different ${\\cal N}$ in the same $s$ -th can propagate all of its information to every sub-stream ${{\\cal N}}$ in the next $s$ -th phase.", "Such unique parallel-cross structure enables the module to acquire and transmit information between networks of different scales.", "Besides, the structure has the ability to maintain high-resolution feature map for each branch and each phase." ], [ "Pseudo-Siamese Framework", "The siamese neural network architecture was first proposed to verify the signature of a check at the end of the last century, and achieved satisfactory expectations [24].", "It feeds inputs into two identical neural networks that share weights with each other to map the inputs to a brand new feature space and then measure the difference between the inputs under the new feature representation [25].", "Inspired by the success of siamese neural network architecture, the pseudo-siamese framework is proposed for various detection and recognition tasks [66].", "Different from the weights sharing mechanism of the siamese neural network architecture, the network contained in the latter does not need to share weights, so it can be composed of two identical or different sub-networks, which makes the pseudo-siamese framework possess higher degrees of flexibility and a wider range of application scenarios.", "The proposal of the pseudo-siamese network brings a novel thinking to traditional classification and comparison tasks.", "On the basis of the above existing works, in our method, we add the channel attention (CA) mechanism [43] to VDSR, which makes the network perceive the more informative channels in the process of image reconstruction.", "In addition, we design a brand-new representation head of HRNet to make full utilization of person image features.", "Finally, we structure the pseudo-siamese framework, composed of two different sub-networks, by which high-resolution and low-resolution pedestrian images received respectively to explore the similarity and feature distribution of cross-resolution images in new feature representation." ], [ "PROPOSED METHOD", "In this section, the proposed PS-HRNet is introduced for the cross-resolution person re-ID problem by giving an overview of its unified architecture first, followed by the details of its main components and the pseudo-siamese framework." ], [ "Framework Overview", "As illustrated in Fig.", "REF , our proposed PS-HRNet adopts a pseudo-siamese framework as the global structure, and contains two main modules, i.e., HRNet-ReID and VDSR-CA.", "In order to clearly clarify the following formulas, we first define the notations of datasets which will be used in this paper.", "In the training phase, we define $N$ high-resolution (HR) images with associated labels as ${{\\cal D}_h} = \\lbrace x_h^i,y^i\\rbrace _{i = 1}^N$ , where $x_h^i \\in {\\mathbb {R} ^{H \\times W \\times 3}}$ .", "We down-sample each HR image with the down-sample rate $r \\in \\left\\lbrace {2,3,4} \\right\\rbrace $ (i.e.", "the spatial size of a down-sampled image becomes $\\frac{H}{r} \\times \\frac{W}{r}$ ).", "The generated corresponding low-resolution (LR) images are denoted as $ {{\\cal D}_l} = \\lbrace x_l^i,y^i\\rbrace _{i = 1}^N$ .", "Our proposed PS-HRNet method has two main objectives.", "Just as most SR-based methods, one objective is to use the super-resolution reconstruction module to restore the missing detail information in LR images and reduce the visual difference between LR and HR images.", "In our method, the original VDSR [22] is improved by adding a channel attention block and named as VDSR-CA, which is applied to generate a HR version for each LR image.", "The other is to minimize the discrepancies in feature distribution between LR and HR images and enhance the matching accuracy by constructing a pseudo-siamese framework.", "For each pair of input images, we utilize HRNet-ReID as the feature extraction network for each branch to learn HR and LR feature representations, and use losses to promote the reduction in distribution differences of these features.", "These modules will be illustrated meticulously in following subsections." ], [ "VDSR with Channel Attention", "In most traditional CNN-based super-resolution modules, each channel of the feature map is treated equally in the process of information transmission between every two feature maps.", "However, the reality is that the image features contained in different channels of the feature map are various, and these diversities contribute to the recovery of high-frequency features in super-resolution task to a different extent.", "So it is very meaningful to assign different weights to the channels of feature map to embody the difference between the channels.", "Therefore, we adopt a channel attention (CA) mechanism proposed in RCAN [43] to redistribute the characteristics of different channels of the feature map to enhance the recovery ability of SR networks.", "The formulaic representation of the CA mechanism is as follows: $\\begin{aligned}{\\hat{f}_c} = {s_c} \\cdot {f_c},\\end{aligned}$ where ${f_c}$ and ${s_c}$ denote the feature map and scaling factor in the same $c$ -th channel of input image.", "The complete expression of $s$ is $\\begin{aligned}s = S\\left( {{W_U}\\delta \\left( {{W_D}z} \\right)} \\right),\\end{aligned}$ where ${W_U}$ and ${W_D}$ denote the channel-upscaling layer and channel-downscaling layer with the ratio $r$ as the sampling rate, respectively.", "$S$ and $\\delta $ denote Sigmoid gating [44] and the activation function ReLU [45], respectively.", "The channel-wise statistic $z \\in \\mathbb {R}{^C}$ is obtained from the feature map and the $c$ -th element of $z$ is represented by the following formula: $\\begin{aligned}{z_c} = {H_{GP}}({f_c}) = \\frac{1}{{H \\times W}}\\sum \\limits _{i = 1}^H {\\sum \\limits _{j = 1}^W {{f_c}\\left( {i,j} \\right)} },\\end{aligned}$ where $H_{GP}$ denotes global pooling function, and ${{f_c}\\left( {i,j} \\right)}$ denotes the pixel value at position $({i,j})$ of $c$ -th feature ${f_c}$ .", "On the basis of RCAN's contribution, we adopt the VDSR as the super-resolution (SR) module which is fused with the channel attention (CA) mechanism to further enhance the performance.", "We add CA layer between every two layers of the original VDSR network.", "The fused SR module is named as VDSR-CA.", "The simplest Manhattan distance is utilized to train the VDSR-CA module, and the loss ${{\\cal L}_{SR}}$ is calculated as: $\\begin{aligned}{{\\cal L}_{SR}} = \\sum \\limits _{i = 1}^{P \\times K} {||{\\mathbb {E}}{_{x_l^i\\sim {D_l},x_h^i\\sim {D_h}}}[{\\cal G}\\left( {x_l^i} \\right) - x_h^i]|{|_1}},\\end{aligned}$ where ${\\cal G}$ denotes VDSR-CA module.", "$P$ and $K$ denote the number of selected persons and the number of corresponding images of each selected person, respectively." ], [ "Innovation of High-Resolution Network", "HRNet aims to fully extract features of input person images for retrieving and matching.", "Existing representation head of HRNet, such as HRNet-W32-C cannot perform satisfactorily in re-ID task.", "For this reason, we design a new representation head of HRNet which is adapted to re-ID task.", "We name the improved HRNet as HRNet-ReID.", "Figure: The detailed process of the last feature map in each branch of HRNet-ReID.", "Both Adaptive Average Pooling(AAP) and Adaptive Max Pooling(AMP) are utilized to extract the channel features from the output of each branch.", "Then the two obtained features are added together, and further reshaped into a feature sequence.Fig.", "REF illustrates the elaborate processing of the last feature map in each branch.", "For the reason that feature maps with higher resolutions may contain more pixel space information, we design a multi-resolution feature fusion strategy in HRNet.", "Adaptive Average Pooling (AAP) and Adaptive Max Pooling (AMP) operations are used to compress and refine the feature map information.", "For each branch, the two feature maps extracted by pooling layers are added into one feature map.", "Then the final sequence of each branch is obtained by reshaping the generated feature map.", "Specifically, the output sequence $Se{q^{(n)}}$ is defined as: $\\begin{aligned}Se{q^{(n)}} &= {\\cal P}({f^{(n)}},n;AAP,AMP,{\\cal V})\\\\&= {\\cal V}(AAP({f^{(n)}},n,n) + {\\lambda _n}AMP({f^{(n)}},n,n)),\\end{aligned}$ where $\\cal P$ denotes the mapping from feature map to feature sequence.", "${f^{(n)}}$ denotes the final feature map of HRNet in each branch.", "$n \\in \\lbrace 1,2,3,4\\rbrace $ denotes the index of branch in HRNet, and ${\\cal V}$ denotes the reshape operation which can transfer a feature map into a feature sequence.", "In particular, ${f^{(n)}}$ also represents the feature map containing the $n{\\rm { - th}}$ highest resolution in the four branches, and the parameter $n$ also controls the output size of adaptive pooling layers.", "Guided by the above designment, we use map ${\\cal P}$ to serialize the four feature maps of HRNet and reconstruct the feature representations.", "Fig.", "REF shows the architecture of the entire output representation which contains $Se{q^{(1\\sim 4)}}$ obtained from map ${\\cal P}$ and $Se{q^{(5)}}$ concatenated by $Se{q^{(1\\sim 4)}}$ .", "We exploit a simple classification module to further process $Se{q^{(5)}}$ which consists of two fully connected layers denoted as $F{C_1}$ and $F{C_2}$ .", "The outputs of them are represented as ${\\ell _f}$ and ${\\ell _c}$ respectively.", "Figure: Illustration of the proposed representation head in HRNet-ReID.", "For each branch, the output is first processed by the mapping 𝒫{\\cal P} which is detailed introduced in Fig.", ".", "The extracted feature sequences Seq (1∼4) Se{q^{(1\\sim 4)}} represented by four short columns of different gray levels are concatenated together and denoted as the longest Seq (5) Se{q^{(5)}}.", "Then we obtain the feature representation ℓ f {\\ell _f} and the classification output ℓ c {\\ell _c} through two full connection layers FC 1 F{C_1} and FC 2 F{C_2}." ], [ "HRNet-ReID Training", "In recent years, a unified learning strategy which combines metric learning and representation learning has been widely applied in the person re-ID task and achieved great performance.", "In the training phase, the extracted $Se{q^{(1\\sim 5)}}$ in HRNet-ReID participate in metric learning, and the classification output ${\\ell _c}$ participates in the representation learning.", "In the testing phase, we concatenate $Se{q^{(5)}}$ and ${\\ell _f}$ as the final feature representation for evaluation.", "A representative loss function in metric learning called triplet loss is usually used for fine-grained recognition at the individual level.", "Here, we take advantage of the batch hard triplet loss ${{\\cal L}_{BH}}$ , which is calculated as: $\\begin{aligned}\\begin{array}{l}{{\\cal L}_{BH}} = \\sum \\limits _{i = 1}^P {\\sum \\limits _{a = 1}^K {\\sum \\limits _{t = 1}^5 {[m + {{\\max }_{p = 1...K}}||Seq_{a,i}^{(t)} - Seq_{p,i}^{(t)}|{|_2}} }}\\\\\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad - {\\min _{\\scriptstyle j = 1...P\\hfill \\atop {\\scriptstyle n = 1...K\\hfill \\atop \\scriptstyle j \\ne a\\hfill }}}||Seq_{a,i}^{(t)} - Seq_{n,j}^{(t)}|{|_2}{]_ + },\\end{array}\\end{aligned}$ where $a$ denotes an anchor image, $p$ denotes a positive sample image, $n$ denotes a negative sample image, and $m$ represents a margin parameter to control the differences between intra and inter distances.", "Moreover, we adopt the cross entropy label smooth loss that combines a label smoothing mechanism as the classification loss ${{\\cal L}_{CE}}$ for representation learning: $\\begin{aligned}{{\\cal L}_{CE}} = \\sum \\limits _{n = 1}^{P \\times K} {[ - \\sum \\limits _{y = 1}^M {\\log (p(y))q(y)} ]},\\end{aligned}$ where $M$ represents the number of person labels involved in training set, and $p(y)$ denotes the probability that predicted label is $y$ .", "Besides, the definition of $q(y)$ is: $\\begin{aligned}q(y) = \\left\\lbrace \\begin{array}{l}1 - \\frac{{M - 1}}{M}\\varepsilon \\quad \\quad if\\quad y = {y_{truth}}\\\\\\\\\\frac{\\varepsilon }{M} \\quad \\quad \\quad \\quad \\quad \\quad others\\end{array} \\right.\\end{aligned}$ where ${y_{truth}}$ is the ground-truth label of the input image and $\\varepsilon $ is a parameter of ${{\\cal L}_{CE}}$ .", "Based on the two loss functions mentioned above (Eq.", "(7) and (8)), for each batch of training set, we compute the HRNet-ReID loss ${{\\cal L}_{ID}}$ by: $\\begin{aligned}{{\\cal L}_{ID}} = {\\lambda _{CE}}{{\\cal L}_{CE}} + {\\lambda _{BH}}{{\\cal L}_{BH}},\\end{aligned}$ where ${\\lambda _{CE}}$ and ${\\lambda _{BH}}$ are parameters to control the importance of ${{\\cal L}_{CE}}$ and ${{\\cal L}_{BH}}$ , respectively.", "It is worth emphasizing that, in the whole architecture of our PS-HRNet, HRNet-ReID as a feature extraction network, which plays different roles in different phases.", "As shown in Fig.", "REF , HRNet-ReID undertakes the task of effectively extracting discriminating features from input HR images in the first phase of the blue arrow representation.", "In the second phase of the black arrow representation, HRNet-ReID of second phase both participates in multi-task joint learning and forms a pseudo-siamese framework with the first phase HRNet-ReID.", "For the convenience of representation and distinction, when the resolution of the input images is HR or LR, we define HRNet-ReID as HRNet-ReID-H and HRNet-ReID-L, respectively." ], [ "Multi-Task Learning Under Pseudo-Siamese Framework", "The design and application of the pseudo-siamese frame-work make the training mode of PS-HRNet different from most other cross-resolution person re-ID methods.", "To build a joint multi-task learning structure as the black arrow shown in Fig.", "2, we specially design a set of training strategies.", "In the first phase, single HRNet-ReID is trained individually with the HR images from ${\\cal D}_h$ .", "By minimizing the loss ${\\cal L}_{ID}$ and observing the indicators of HRNet-ReID in the testing set, we obtain a high-performance HRNet-ReID module defined as HRNet-ReID-H for subsequent joint learning and construction of the pseudo-siamese framework.", "In the second phase, we first concatenate VDSR-CA with the newly defined HRNet-ReID-L, and then construct the pseudo-siamese framework with the obtained HRNet-ReID-H from the first phase and HRNet-ReID-L from the concatenated structure via the Manhattan distance loss ${\\cal L_{PS}}$ .", "The pseudo-siamese framework is adopted to measure the similarity of two inputs.", "Compared with the siamese framework, the pseudo-siamese framework is more suitable for the situation where two inputs have a certain difference.", "We first define the ordered set ${A} = \\left\\lbrace {Seq^{(1)},Seq^{(2)},Seq^{(3)},Seq^{(4)},Seq^{(5)},{\\ell _{{c}}}} \\right\\rbrace $ .", "Then the loss ${\\cal L_{PS}}$ is defined as: $\\begin{aligned}{\\cal L_{PS}} = \\sum \\limits _{i = 1}^m {||C_h^{(i)} - C_l^{(i)}} |{|_1},\\end{aligned}$ where ordered set ${C} \\subseteq {A}$ and ${C} \\ne \\varnothing $ , denotes a combination of elements in ordered set $A$ .", "$C_h$ and $C_l$ denote the set of elements from HRNet-ReID-H and HRNet-ReID-L under the same combination, respectively.", "$m$ denotes the amount of elements in ordered set $C$ .", "The process of joint multi-task learning depends on the training set from both ${\\cal D}_h$ and ${\\cal D}_l$ .", "The VDSR-CA module first reads the LR images from ${\\cal D}_l$ for super-resolution reconstruction, and outputs the restored images with the same resolution as the HR images.", "The restored images are read by HRNet-ReID-H to train itself with the loss ${{\\cal L}_{ID}}$ (Eq.", "(10)), and at the same time, they participate in the calculation of ${{\\cal L}_{SR}}$ (Eq.", "(5)) together with the corresponding HR images from ${\\cal D}_h$ .", "Finally, the guidance of HRNet-ReID-H to HRNet-ReID-L is realized by loss ${\\cal L_{PS}}$ (Eq.", "(11)).", "To sum up, we combine the ID loss (Eq.", "(10)), the SR loss (Eq.", "(5)) and the PS loss (Eq.", "(11)) into the joint multi-task learning loss ${{\\cal L}_{TOTAL}}$ , defined as: $\\begin{aligned}{{\\cal L}_{TOTAL}} = {{\\cal L}_{I{D}}} + {\\lambda _{SR}}{{\\cal L}_{{\\rm {SR}}}}{\\rm { + }}{\\lambda _{PS}}{{\\cal L}_{PS}},\\end{aligned}$ where ${\\lambda _{SR}}$ and ${\\lambda _{PS}}$ are two parameters to control the importance of ${{\\cal L}_{{\\text{SR}}}}$ and ${{\\cal L}_{PS}}$ , respectively.", "Algorithm 1 gives a summary of the entire training process.", "When in the testing phase, we uniformly input the gallery and query images into the trained network composed of VDSR-CA and HRNet-ReID-L.", "The generated $Se{q^{(5)}}$ and ${\\ell _f}$ are concatenated as the ultimate feature representation for matching and evaluation.", "The entire testing process is executed end-to-end without additional operations." ], [ "EXPERIMENTS", "In this part, we first give detailed descriptions of the datasets for evaluation, the experimental settings, and the specific implementation details.", "Then, a number of experiments are carried out to prove the validity of our proposed method through the comparison with existing methods and ablation studies." ], [ "Datasets", "Our experiment involves nine datasets, including four high-resolution re-ID datasets for traditional re-ID task: VIPeR [46], CUHK03 [47], DukeMTMC-reID [48], Market-1501 [49], and four synthetic cross-resolution datasets named as MLR datasets which are constructed from traditional versions: MLR-VIPeR, MLR-CUHK03, MLR-DukeMTMC-reID, MLR-Market1501, as well as one native cross-resolution dataset sampled in real world: CAVIAR [50].", "The five cross-resolution datasets involved in our experiments are described as follows: The MLR-VIPeR includes 632 person-image pairs captured from 2 cameras, a total of 1264 pictures.", "Following [16], we randomly down-sample all the pictures captured by one of the cameras by a down-sampling rate $r \\in \\lbrace 2,3,4\\rbrace $ and the images collected from another camera remain unchanged.", "Here we use the standard 316/316 training/testing identity split." ], [ "MLR-CUHK03", "The MLR-CUHK03 dataset is generated from images taken by 10 (5 pairs) different cameras.", "It contains 14097 images of 1467 individuals.", "As [16], for each pair of cameras, we down-sample the images captured from one camera by randomly selecting a down-sampling rate $r \\in \\lbrace 2,3,4\\rbrace $ , while the resolution of the images collected by the other cameras are unchanged.", "Here we use the 1,367/100 training/testing identity split." ], [ "MLR-DukeMTMC-reID", "The MLR-DukeMTMC dataset includes 36,411 images of 1,404 identities captured from 8 cameras.", "Following [39], we randomly select one camera, and down-sample the images by the same down-sampling rate while the image resolution of other cameras remains unchanged.", "We use the standard 702/702 training/testing identity split." ], [ "MLR-Market-1501", "The MLR-Market-1501 dataset is composed of 32,668 pictures of 1,501 persons captured by 6 cameras.", "Following [39], we preprocess images of one camera with the same down-sampling rate, while other image resolutions remain unchanged.", "According to the person ID label, the dataset is separated into a training set containing 751 pedestrians and a testing set containing 750 pedestrians." ], [ "CAVIAR", "The CAVIAR is a dataset collected in real world.", "It consists of 1,220 pictures of 72 persons collected from 2 cameras.", "Following [16], 22 persons are discard by us with only HR images and we randomly split the dataset into two halves based on 25 identities labels for training and testing, respectively.", "Table: Experimental results of cross-resolution person re-ID (%).", "The bold and underlined numbers indicate top two results, respectively." ], [ "Experimental Settings", "In the training phase, both the four traditional re-ID datasets and the corresponding cross-resolution datasets are applied to train the module.", "The training process is divided into two phases: In the first phase, we only train the HRNet-ReID-H on the traditional high-resolution re-ID datasets, and obtain a high-performance module; In the second phase, we combine the VDSR-CA module with the HRNet-ReID-L guided by HRNet-ReID-H and jointly train them on both traditional datasets and cross-resolution datasets.", "In the testing phase, we evaluate the performance of our proposed PS-HRNet on five cross-resolution datasets, where the query sets contain LR images and the gallery sets contain HR images.", "In particular, since CAVIAR is a genuine cross-resolution dataset, we follow the experimental setting in [16] for training and testing.", "We adopt the standard single-shot person re-ID settings, and apply the average cumulative match characteristic (CMC) to quantify the performance and report the results of ranks 1, 5 and 10." ], [ "Implementation details", "In the VDSR-CA module, we retain the entire structure of the original VDSR network, and embed the Channel Attention (CA) mechanism proposed in the RCAN [43].", "The internal parameter $r$ of CA mechanism is set to 4.", "With respect to HRNet-ReID module, we select the HRNet-W32-C pretrained with the ImageNet dataset as the backbone for feature extraction.", "Here we redesign the classifier to adapt to the re-ID task as shown in Fig.", "REF .", "The lengths of $Se{q^{(1\\sim 5)}}$ are set to 2048, 1024, 2048, 1024 and 6144, respectively.", "Besides, the dimension of ${\\ell _f}$ is set to 512, and the dimension of ${\\ell _c}$ is equal to the number of target categories.", "Before training, all images are resized to $256\\times 128\\times 3$ .", "A mini-batch contains 24 pairs of images of $P = 4$ persons, and each person has $K = 6$ pairs of HR and LR images.", "We choose SGD to optimize our module with weight decay $5\\times {10^{{\\rm { - }}4}}$ .", "The learning rates for training HRNet-ReID and VDSR-CA are set to $8.5\\times {10^{{\\rm { - 3}}}}$ and $8.5\\times {10^{{\\rm { - 4}}}}$ , respectively, which are decreased by 0.1 every 30 epochs.", "Our module is trained for 70 epochs in total.", "The hyper-parameters $m$ , ${\\lambda _{CE}}$ , ${\\lambda _{BH}}$ , ${\\lambda _{SR}}$ and ${\\lambda _{PS}}$ are set to 0.1, 1.15, 0.2, 0.5 and 0.5, respectively.", "In ${{\\cal L}_{PS}}$ , we select $\\left\\lbrace {Se{q^{(1)}},Se{q^{(4)}},Se{q^{(5)}},{\\ell _c}} \\right\\rbrace $ as a combination to participate in the operation.", "Some data augmentation tricks are utilized, such as random flipping, padding and random cropping.", "We perform our experiments with PyTorch of version 1.6 on single 11GB NVIDIA RTX 2080Ti GPU." ], [ "Comparisons to State-of-the-Art Methods", "We compare our PS-HRNet with a series of far-ranging state-of-the-art person re-ID methods, which can be roughly separated into two main categories.", "(1) Conventional methods designed for traditional person re-ID task: PCB [51], DenseNet-121 [52], ResNet-50 [53], SE-ResNet-50 [54], SPreID [28], Part Aligned [55], CamStyle [56] and FD-GAN [58]; (2) Pointed methods designed for cross-resolution person re-ID task: JUDEA [14], SDF [59], SLD$^2$ L [13], SING [16], CSR-GAN [38], FFSR [60], RIFE [60], FFSR+RIFE [60], RAIN [15], CAD-Net [39], CAD-Net++ [61], PRI [62], PCB+PRI [62] and PyrNet+PRI [62].", "These methods in the comparison almost cover all the current methods in the cross-resolution re-ID field.", "The comparison results of the above approaches on five datasets are listed in Table REF .", "We can evidently observe that: $\\bullet $ Our PS-HRNet obtains state-of-the-art performance on all five datasets, and its Rank-1 outperforms the best competitor by 6.1% on the MLR-CUHK03 dataset.", "$\\bullet $ Compared with the conventional person re-ID methods, our method outperforms their best result by 11.9% on the MLR-Market-1501 dataset, which indicates that the information hided in LR images cannot be extracted and utilized effectively by conventional methods when processing cross-resolution person images.", "Besides, it also demonstrates that the super-resolution reconstruction module with channel attention mechanism plays a significant role in the cross-resolution re-ID task.", "$\\bullet $ Compared with the pointed methods designed for solution of cross-resolution person re-ID problem, PS-HRNet outperforms all existing methods, which reflects the importance of the tailor-made high-resolution feature extraction network and pseudo-siamese framework.", "Table: Evaluating ours loss components on MLR-Market1501.", "ID: identity classification loss (Eq.", "(10)), SR: super-resolution loss (Eq.", "(5)), PS: our pseudo-Siamese framework loss (Eq.", "(11))." ], [ "Analysis of Loss Functions", "Our PS-HRNet jointly trains image SR module, feature extraction network and pseudo-siamese framework with several loss functions (cf.", "Eq.", "(12) etc.).", "We use the same research strategy as described in INTACT [40] to study the effectiveness of different losses in PS-HRNet on the MLR-Market-1501 dataset.", "Table REF reports the ablation results, which reflect that: $\\bullet $ When only ID loss is included, compared with Table REF , it can be obviously found that the performance of our method surpasses almost all existing methods except INTACT [40] and PCB+PRI [62] on the Rank-1, and even reaches the same performance level of single PRI [62].", "The results reflect the great feature extraction ability of HRNet on cross-resolution person images which should be owed to its unique high-resolution parallel structure.", "This also demonstrates the necessity and rationality of applying HRNet to process low-resolution images.", "$\\bullet $ With the addition of SR loss, the performance is further improved by 2.9%, which can prove that the image reconstruction function provided by VDSR-CA module has a positive effect on cross-resolution person re-ID.", "$\\bullet $ The addition of pseudo-siamese framework loss ultimately increases the Rank-1 by 3.9%, which verifies the positive effect of the pseudo-siamese framework.", "Furthermore, it proves the necessity of reducing the discrepancies of feature space between HR and LR images, which has been overlooked all along.", "Table: Performance comparison test of super-resolution reconstruction and cross-resolution person reID on the MLR-CUHK03 test set.$\\bullet $ Following INTACT [40], experiments on loss are conducted to explore the influence of different loss combinations on image restoration quality and whether it will affect the restored images after using the pseudo-siamese framework.", "We use PSNR and SSIM which are two quantitative indicators that reflect the quality of image restoration to evaluation.", "We compare our method with CycleGAN [63], SING [16], CSR-GAN [38] and CAD-Net [39].", "According to the comparative results in Table REF , the solo SR module achieves the best performance in image super-resolution reconstruction which also confirms the effectiveness of VDSR-CA.", "After using SR+ID, the PSNR and SSIM are slightly reduced, but the recognition accuracy is greatly improved.", "Furthermore, after adding the pseudo-siamese framework, the value of PSNR and SSIM have no changed, which indicates that the pseudo-siamese framework does not affect the quality of the restored images on visual level.", "Figure: Visual comparisons of restored HR test images from the MLR-CUHK03 dataset.", "Given input images of 3 different low-resolutions r∈2,3,4r \\in {2,3,4}, our PS-HRNet outputs the corresponding restored HR images of solo SR module, SR+ID module and SR+ID+PS module.$\\bullet $ Following INTACT [40], Fig.", "REF shows some person images restored by our method.", "These images are selected from the testing set of MLR-CUHK03.", "Although our PS-HRNet is superior to the existing baseline method in recognition accuracy, there is still a gap between the recovery quality of low-resolution images and the ground truth on visual level.", "Such outcome does make perfect sense.", "As we mentioned earlier, the focus is not on the restoration of image quality but on the examination of deep-level feature and semantic information.", "Blindly using visual sensory experience and related indicators as a standard for image restoration is of little significance because the way the neural network observes the image is completely different from the human visual mechanism.", "This also gives a reasonable interpretation on the variation of PSNR and SSIM in Table REF .", "Therefore, we do not recommend subsequent studies on cross-resolution person re-identification to conduct excessive super-resolution reconstruction experiments at the aspect of human vision.", "Table: Recognition accuracy (%) of different SR modules on MLR-Market1501." ], [ "Analysis of VDSR-CA", "In order to further explore the effectiveness of the VDSR-CA, we adopt ResNet-50 [53] as a unified feature extraction network on the MLR-Market1501 dataset, and test the effect of using cubic interpolation, SRCNN [19], VDSR-CA and RCAN [43] as super-resolution modules for joint learning.", "The recognition accuracy of different SR modules on MLR-Market1501 are presented in Table REF , which reflects that the performance of our VDSR-CA module goes far beyond the classic SRCNN network and is better than the original VDSR network.", "Although the Rank indicators are slightly lower than the result of joint learning with RCAN, VDSR-CA has a lighter architecture than RCAN which means more practical in real-world applications.", "Considering the above factors, our VDSR-CA is the best choice for super-resolution modules.", "Table: Recognition accuracy (%) of different pooling method on Market1501 and DukeMTMC-reID.Table: Recognition accuracy (%) of different feature extraction networks on Market1501 and DukeMTMC-reID." ], [ "Analysis of HRNet-ReID", "The function of HRNet-ReID aims to fully extract features of input person images for retrieving and matching.", "In order to make a large number of high-dimensional feature map data extracted by the previous HRNet-W32 can be effectively processed by the classification layer at the end of HRNet-ReID, we introduce adaptive average pooling and adaptive max pooling as important means of feature compression.", "Table REF demonstrates the effect of different pooling options on the recognition accuracy.", "Obviously, under the same experimental conditions, only the simultaneous application of AAP and AMP can obtain higher detection accuracy.", "Theoretically, maximum pooling and average pooling play specific roles in extracting feature texture information and global background information respectively.", "Therefore, the combination of average pooling and adaptive max pooling is valid.", "For further investigating the performance of HRNet-ReID as the backbone in the person re-ID task, we conduct experiments on the following conventional person re-ID network combined with ResNet-50, PCB and DenseNet as the comparisons with HRNet-ReID on the Market-1501 and DukeMTMC-reID datasets: DPFL [21], PCB [51], FD-GAN [58], CamStyle [56], DenseNet121 [52].", "In addition, in order to validate the effectiveness of our designed HRNet representation head, we use the original version of HRNet-W32-C [64] as reference.", "The representation head of HRNet-W32-C is proposed to solve the problem of image classification on the ImageNet dataset and achieves good performance.", "As shown in Table REF , the experimental results reflect that our HRNet-ReID significantly outperforms the original HRNet-W32-C and other methods on both datasets which confirms the effectiveness of our modified representation head.", "Table: Recognition accuracy (%) of different sequence combinations on MLR-Market1501." ], [ "Analysis of pseudo-siamese framework", "For the purpose of exploring the impact of different training strategies under the pseudo-siamese framework, we select multiple sets of $Se{q^{(n)}}$ and ${\\ell _c}$ to participate in the training of pseudo-siamese framework loss and perform testing on the MLR-Market-1501 dataset.", "According to the combinatorial mathematics, there are 84 combinations for the calculation of Eq.", "(11) in theory.", "Here we select three typical combinations for evaluation.", "The experimental results are listed in Table REF .", "We can clearly observe that different selections of sequences bring different performances.", "Here we adopt the combination of $\\left\\lbrace {Se{q^{(1)}},Se{q^{(4)}},Se{q^{(5)}},{\\ell _c}} \\right\\rbrace $ in our training strategy.", "Limited by time, there may be a better combination that further improves the performance of our PS-HRNet." ], [ "Conclusion", "In this article, we design a novel approach named Deep High-Resolution Pseudo-Siamese Framework (PS-HRNet) to significantly alleviate the the resolution mismatch problem and improve recognition accuracy in cross-resolution person re-ID task.", "Our framework utilizes VDSR-CA as the super-resolution module, HRNet-ReID as the feature extraction network.", "The former integrates channel attention mechanism into VDSR, which can reasonably utilize the valuable high frequency components contained in different channels of feature map and restore the missing discriminative information in LR images effectively.", "The latter possesses a novel representation head designed by us which can effectively extract fine-grained details from cross-resolution person images.", "What's more, the pseudo-siamese framework is adopted and plays a significant component in reducing the distribution difference in feature information between LR and HR images.", "With extensive experiments, the results confirm that our PS-HRNet can extract discriminating and robust feature representations from cross-resolution images, and achieves the state-of-the-art performance in existing five benchmarks." ] ]
2105.11722
[ [ "Divisor class groups of double covers over projective spaces" ], [ "Abstract In this paper, we prove that the divisor class group of a double cover of the complex projective space $\\mathbb{P}^n$ is generated by divisorial sheaves whose direct images split into direct sums of two invertible sheaves on $\\mathbb{P}^n$.", "This result shows that any locally free sheaf of rank two on $\\mathbb{P}^n$ is generated by direct sums of line bundles on $\\mathbb{P}^n$ via some double cover.", "Moreover, we give a condition for an irreducible divisor on $\\mathbb{P}^n$ to be a splitting divisor for a double cover whose divisor class group is finitely generated." ], [ "Introduction", "In this paper, a morphism $\\phi :X\\rightarrow Y$ (or $X$ ) is called a double cover of $Y$ if $\\phi $ is a finite morphism of degree 2 from a normal variety $X$ to a smooth variety $Y$ over $\\mathbb {C}$ .", "A double cover $\\phi :X\\rightarrow Y$ is determined by the branch divisor $B_\\phi $ and a divisor $L$ on $Y$ satisfying $B_\\phi \\sim 2L$ .", "If $\\mathop {\\rm Pic}\\nolimits (Y)$ has no 2-torsion element, then a double cover $\\phi :X\\rightarrow Y$ is determined by only the branch divisor $B_\\phi $ .", "Double covers appear in various situations.", "In the study of algebraic surfaces, many mathematicians applied double covers (cf.", "[3], [5], [8], [11], [14] and so on).", "In particular, Horikawa [8] gave a method for resolving the singularities of $X$ , which enable us to compute several invariants of $X$ (or the resolution of $X$ ) from data of $Y$ and $B_\\phi $ .", "However, it seems that there are few studies about the relation between the divisor class group $\\mathrm {Cl}(X)$ and the branch divisor $B_\\phi $ for a double cover $\\phi :X\\rightarrow Y$ (cf.", "[4], [17]).", "In this paper, we prove that $\\mathrm {Cl}(X)$ for a double cover $\\phi :X\\rightarrow \\mathbb {P}^n$ of the projective space $\\mathbb {P}^n$ is generated by irreducible divisors corresponding to divisorial sheaves $\\mathcal {L}$ whose direct images $\\phi _\\ast \\mathcal {L}$ split into direct sums of line bundles on $\\mathbb {P}^n$ .", "In the study of the embedded topology of plane curves (started from Zariski's result [19]), Tokunaga [18] introduced splitting curves for double covers of smooth surfaces.", "To distinguish the embedded topology of plane curves, it is effective to compute whether an irreducible curve on $\\mathbb {P}^2$ is a splitting curve for a double cover $\\phi :X\\rightarrow \\mathbb {P}^2$ (cf.", "[1], [15], [16]).", "For the double cover $\\phi :X\\rightarrow \\mathbb {P}^2$ branched at a smooth conic, Bannai and the author [2] gave a criterion whether a nodal curve on $\\mathbb {P}^2$ is a splitting curve for $\\phi $ by using the structure of $\\mathop {\\rm Pic}\\nolimits (X)\\cong \\mathbb {Z}^2$ .", "This result seems to show the importance of the structure of $\\mathrm {Cl}(X)$ for the study of the embedded topology.", "In the study of locally free sheaves of rank two (say 2-bundles for short), Schwarzenberger [12] proved that any 2-bundle ${\\mathcal {E}}$ on a smooth surface $Y$ is isomorphic to the direct image of an invertible sheaf on a non-singular double cover.", "This result is generalized to any dimension in [17] as follows; any 2-bundle on a smooth variety $Y$ is isomorphic to the direct image $\\phi _\\ast \\mathcal {L}$ of a divisorial sheaf $\\mathcal {L}$ on $X$ for some double cover $\\phi :X\\rightarrow Y$ .", "For the smooth locus $X^\\circ $ of a double cover $X$ , the group structure of $\\mathop {\\rm Pic}\\nolimits (X^\\circ )$ is interpreted in terms of transition functions of direct images of invertible sheaves on $X^\\circ $ in [17].", "Hence, for a smooth variety $Y$ , the structures of $\\mathrm {Cl}(X)$ for double covers $X$ of $Y$ relate to 2-bundles on $Y$ .", "Let us state the main result of this paper.", "For a double cover $\\phi :X\\rightarrow Y$ , let $\\mathrm {Cl}(X)$ be the divisor class group of $X$ .", "Since $X$ is normal, there is a natural one-to-one correspondence between $\\mathrm {Cl}(X)$ and the set of divisorial sheaves on $X$ (cf.", "[13]).", "Let $\\mathop \\mathrm {sCl}\\nolimits _\\phi (X)$ be the subgroup of $\\mathrm {Cl}(X)$ generated by divisorial sheaves $\\mathcal {L}$ whose direct images $\\phi _\\ast \\mathcal {L}$ split into direct sums of invertible sheaves; $ \\mathop \\mathrm {sCl}\\nolimits _\\phi (X):=\\left\\langle \\mathcal {L}\\ \\big | \\ \\phi _\\ast \\mathcal {L}\\cong \\mathop {\\mathcal {O}}\\nolimits _Y(D_1)\\oplus \\mathop {\\mathcal {O}}\\nolimits _Y(D_2) \\mbox{ for ${}^\\exists D_1, D_2\\in \\mathop {\\rm Div}\\nolimits (Y)$} \\right\\rangle .", "$ Remark 1.1 In the case where $X$ is smooth, $\\mathop \\mathrm {sCl}\\nolimits _\\phi (X)$ is denoted by $\\mathop \\mathrm {sPic}\\nolimits _\\phi (X)$ in [17].", "The main theorem of this paper is that the subgroup $\\mathop \\mathrm {sCl}\\nolimits _\\phi (X)$ coincides with $\\mathrm {Cl}(X)$ for any double cover $\\phi :X\\rightarrow \\mathbb {P}^n$ of the projective space $\\mathbb {P}^n$ , which proves [17] for double covers of $\\mathbb {P}^n$ .", "Theorem 1.2 Let $\\phi :X\\rightarrow \\mathbb {P}^n$ be a double cover of $\\mathbb {P}^n$ , and let $F$ be a homogeneous polynomial defining the branch divisor $B_\\phi $ .", "$\\mathrm {Cl}(X)=\\mathop \\mathrm {sCl}\\nolimits _\\phi (X)$ ; $\\mathrm {Cl}(X)$ is generated by $\\phi ^\\ast \\mathop {\\mathcal {O}}\\nolimits _{\\mathbb {P}^n}(1)$ and divisorial sheaves $\\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C})$ for irreducible components $\\widetilde{C}$ of $\\phi ^\\ast C$ , where $C$ runs irreducible divisors on $\\mathbb {P}^n$ defined by $f=0$ such that $F=h^2+fg$ for some homogeneous polynomials $g,h$ .", "We call an irreducible divisor $C$ on $Y$ in REF an SPS-divisor for $\\phi $ (see Definition REF ).", "By [17], any 2-bundle on $\\mathbb {P}^n$ is generated by direct sums of invertible sheaves via some double cover as follows.", "Corollary 1.3 Let ${\\mathcal {E}}$ be a 2-bundle on $\\mathbb {P}^n$ .", "Then there exist a double cover $\\phi :X\\rightarrow \\mathbb {P}^n$ , irreducible divisors $\\widetilde{C}_1,\\dots ,\\widetilde{C}_m$ on $X$ and $k_1,\\dots ,k_m\\in \\mathbb {Z}$ such that $\\phi _\\ast \\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C}_j)$ splits for $j=1,\\dots ,m$ , and ${\\mathcal {E}}\\cong \\phi _\\ast \\left(\\left( \\mathop {\\mathcal {O}}\\nolimits _X(k_1 \\widetilde{C}_1)\\otimes \\dots \\otimes \\mathop {\\mathcal {O}}\\nolimits _X(k_m \\widetilde{C}_m) \\right)^{\\vee \\vee }\\right),$ where $\\mathcal {L}^{\\vee }$ is the dual sheaf $\\mathop {\\rm Hom}\\nolimits _{\\mathop {\\mathcal {O}}\\nolimits _X}(\\mathcal {L},\\mathop {\\mathcal {O}}\\nolimits _X)$ for a coherent sheaf $\\mathcal {L}$ on $X$ .", "Remark 1.4 In [4], for divisorial sheaves $\\mathcal {L}_1,\\mathcal {L}_2$ on a double cover $X$ , $\\phi _\\ast ((\\mathcal {L}_1\\otimes \\mathcal {L}_2)^{\\vee \\vee })$ and $\\phi _\\ast (\\mathcal {L}_1^{-1})$ are described using an exact sequence.", "In [17], for divisorial sheaves $\\mathcal {L}_i$ on $X$ , $\\phi _\\ast ((\\mathcal {L}_1^{k_1}\\otimes \\dots \\otimes \\mathcal {L}_m^{k_m})^{\\vee \\vee })$ is described in terms of transition functions of $\\phi _\\ast \\mathcal {L}_i$ .", "This paper is organized as follows: In §, we discuss about divisors on a smooth variety and a double cover as the preliminary to proving the main results.", "In §, we prove Theorem REF and Corollary REF .", "In §, we give a condition for an irreducible divisor on $\\mathbb {P}^n$ to be a splitting divisor for a double cover $\\phi :X\\rightarrow \\mathbb {P}^n$ in the case where $\\mathrm {Cl}(X)$ is finitely generated (Theorem REF )." ], [ "Preliminary", "In this paper, a Weil divisor on a normal variety is called simply a divisor.", "For a divisor $D$ on a normal variety $X$ , the sheaf $\\mathop {\\mathcal {O}}\\nolimits _X(D)$ associated to $D$ is defined by $ \\Gamma (V,\\mathop {\\mathcal {O}}\\nolimits _X(D)):=\\lbrace f\\in \\mathbb {C}(X)^\\ast \\mid (f)|_V+D|_V\\ge 0\\rbrace \\cup \\lbrace 0\\rbrace $ for each open subset $V\\subset X$ , where $(f)$ is the principal divisor given by a rational function $f\\in \\mathbb {C}(X)^\\ast $ .", "Note that $\\mathop {\\mathcal {O}}\\nolimits _X(D)$ is a reflexive sheaf of rank 1, called a divisorial sheaf.", "See [7] for basic properties of reflexive sheaves, and [9], [6] or [13] for the correspondence between divisors and divisorial sheaves on a normal variety.", "In this section, we introduce notation and some lemmas about divisors and the splitting of divisorial sheaves on a double cover under the direct image." ], [ "Divisors on smooth varieties", "We consider a smooth variety $Y$ .", "For an open affine subset $U_0$ of $Y$ , let $\\mathop {\\rm Pic}\\nolimits (Y,U_0)$ be the subgroup of $\\mathop {\\rm Pic}\\nolimits (Y)$ generated by irreducible divisors whose support is contained in $Y\\setminus U_0$ : $\\mathop {\\rm Pic}\\nolimits (Y,U_0):=\\left\\langle \\mathop {\\mathcal {O}}\\nolimits _Y(H) \\ \\big | \\ H\\subset Y\\setminus U_0 : \\mbox{an irreducible divisor} \\right\\rangle $ Since $Y\\setminus U_0$ contains at most finitely many irreducible divisors on $Y$ , $\\mathop {\\rm Pic}\\nolimits (Y,U_0)$ is finitely generated.", "When we fix an open affine subset $U_0\\subset Y$ , we also fix irreducible divisors $H_1,\\dots , H_m\\subset Y\\setminus U_0$ which generate $\\mathop {\\rm Pic}\\nolimits (Y,U_0)$ : $\\mathop {\\rm Pic}\\nolimits (Y,U_0)=\\big \\langle \\mathop {\\mathcal {O}}\\nolimits _Y(H_1),\\dots ,\\mathop {\\mathcal {O}}\\nolimits _Y(H_m)\\big \\rangle .", "$ In this case, put $\\mathbf {H}:=(H_1,\\dots ,H_m)$ .", "Let $\\lbrace U_i\\rbrace _{i\\in I}$ be an open affine covering of $Y$ such that $H_j$ are principal on each $U_i$ , i.e., $H_j|_{U_i}=(x_{ij})|_{U_i}$ for a rational function $x_{ij}\\in \\mathbb {C}(Y)^\\ast $ .", "In other words, $\\lbrace (U_i,x_{ij})\\rbrace $ is a Cartier divisor on $Y$ defining $H_j$ .", "In this case, $x_{ij}\\in \\Gamma (U_i,\\mathop {\\mathcal {O}}\\nolimits _Y)$ since $H_j$ is effective.", "We may assume that $x_{0j}=1$ since $U_0\\cap H_j=\\emptyset $ .", "For $\\mathbf {a}=(a_1,\\dots ,a_m)\\in \\mathbb {Z}^m$ , we put $ \\mathbf {a}\\mathbf {H}:=a_1H_1+\\dots +a_mH_m, \\qquad \\mathbf {x}_i^{\\mathbf {a}}:=x_{i1}^{a_1}\\dots x_{im}^{a_m}.", "\\qquad $ We identify $\\mathop {\\mathcal {O}}\\nolimits _Y(\\mathbf {a}\\mathbf {H})$ as an $\\mathop {\\mathcal {O}}\\nolimits _Y$ -submodule of the constant sheaf $\\mathbb {C}(Y)$ via the multiplication map $\\mathop {\\mathcal {O}}\\nolimits _{U_i}\\hookrightarrow \\mathbb {C}(Y)$ by $\\mathbf {x}_i^{-\\mathbf {a}}$ for $i\\in I$ : $ \\mathop {\\mathcal {O}}\\nolimits _{U_i}\\underset{\\mathbf {x}_{i}^{-\\mathbf {a}}}{\\overset{\\sim }{\\longrightarrow }} \\mathop {\\mathcal {O}}\\nolimits _Y(\\mathbf {a}\\mathbf {H})|_{U_i}\\subset \\mathbb {C}(Y).", "\\ $ For a global section $f\\in \\operatorname{H}^0(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(\\mathbf {aH}))$ , we regard the restriction $f|_{U_i}$ as an element $\\Gamma (U_i,\\mathop {\\mathcal {O}}\\nolimits _Y)$ via the above identification.", "Hence we have $ f|_{U_i}=\\mathbf {x}_i^{\\mathbf {a}}f|_{U_0} $ in $\\Gamma (U_0\\cap U_i,\\mathop {\\mathcal {O}}\\nolimits _Y)\\subset \\mathbb {C}(Y)$ since we assume that $x_{0j}=1$ for $j=1,\\dots ,m$ .", "Lemma 2.1 Let $Y$ be a smooth variety, and let $D\\subset Y$ be a divisor on $Y$ .", "Let $U_0$ be an open affine subset of $Y$ .", "If $D$ is principal on $U_0$ , then $\\mathop {\\mathcal {O}}\\nolimits _Y(D)\\in \\mathop {\\rm Pic}\\nolimits (Y,U_0)$ .", "Let $\\lbrace (U_i,f_i)\\rbrace $ be a Cartier divisor on $Y$ defining $D$ , where $\\lbrace U_i\\rbrace $ is an open affine covering containing $U_0$ .", "Then the Cartier divisor $\\lbrace (U_i,f_i/f_0)\\rbrace $ defines a divisor $D^{\\prime }$ on $Y$ such that $D^{\\prime }\\sim D$ and $\\mathop {\\rm Supp}\\nolimits (D^{\\prime })\\subset Y\\setminus U_0$ .", "Hence we obtain $\\mathop {\\mathcal {O}}\\nolimits _Y(D)\\in \\mathop {\\rm Pic}\\nolimits (Y,U_0)$ .", "Lemma 2.2 Let $Y$ be a smooth variety.", "The followings are equivalent: $\\mathop {\\rm Pic}\\nolimits (Y)$ is finitely generated; there exists an open affine subset $U_0\\subset Y$ whose coordinate ring is a UFD.", "Furthermore, in the case of REF , $\\mathop {\\rm Pic}\\nolimits (Y)=\\mathop {\\rm Pic}\\nolimits (Y,U_0)$ .", "We first suppose that $\\mathop {\\rm Pic}\\nolimits (Y)$ is generated by a finite number of irreducible divisors $H_1,\\dots ,H_m$ on $Y$ .", "For $j=1,\\dots ,m$ , let $\\lbrace (U_i,x_{ij})\\rbrace _{i\\in I}$ be a Cartier divisor on $Y$ defining $H_j$ , where $\\lbrace U_i\\rbrace _{i\\in I}$ is an open affine covering of $Y$ .", "We may assume that $U_0\\cap (H_1\\cup \\dots \\cup H_m)=\\emptyset $ for some $0\\in I$ , and that $x_{0j}=1$ for $j=1,\\dots ,m$ .", "Let $C_0$ be an irreducible divisor on $U_0$ , and let $C$ be the closure of $C_0$ in $Y$ .", "By the assumption, $C$ is linearly equivalent to $\\mathbf {a}\\mathbf {H}$ for some $\\mathbf {a}\\in \\mathbb {Z}^m$ .", "Then we obtain $\\mathop {\\mathcal {O}}\\nolimits _{U_0}(C_0)\\cong \\mathop {\\mathcal {O}}\\nolimits _Y(\\mathbf {aH})|_{U_0}\\cong \\mathop {\\mathcal {O}}\\nolimits _{U_0}$ .", "Thus $\\mathop {\\rm Pic}\\nolimits (U_0)=0$ , and the coordinate ring of $U_0$ is a UFD (cf.", "[10]).", "Conversely, we suppose REF .", "Let $C$ be an irreducible divisor on $Y$ .", "Since $\\Gamma (U_0,\\mathop {\\mathcal {O}}\\nolimits _Y)$ is a UFD, $C$ is defined on $U_0$ by $f_0=0$ for some $f_0\\in \\Gamma (U_0,\\mathop {\\mathcal {O}}\\nolimits _Y)$ .", "By Lemma REF , we obtain $\\mathop {\\mathcal {O}}\\nolimits _Y(C)\\in \\mathop {\\rm Pic}\\nolimits (Y,U_0)$ .", "Hence $\\mathop {\\rm Pic}\\nolimits (Y)=\\mathop {\\rm Pic}\\nolimits (Y,U_0)$ holds.", "Since $Y\\setminus U_0$ contains at most finitely many irreducible divisors, $\\mathop {\\rm Pic}\\nolimits (Y)$ is finitely generated." ], [ "Double covers", "Let $\\phi :X\\rightarrow Y$ be a double cover, and let $\\iota :X\\rightarrow X$ denote the covering transformation of $\\phi $ .", "Let $B_\\phi $ , $L$ and $R_\\phi $ denote following divisors: $B_\\phi &: \\mbox{the branch divisor on $Y$ of $\\phi $}; \\\\L \\ &: \\mbox{a divisor on $Y$ with $B_\\phi \\sim 2L$ and $\\phi _\\ast \\mathop {\\mathcal {O}}\\nolimits _X\\cong \\mathop {\\mathcal {O}}\\nolimits _Y\\oplus \\mathop {\\mathcal {O}}\\nolimits _Y(-L)$}; \\mbox{ and}\\\\R_\\phi &: \\mbox{the ramification divisor on $X$ of $\\phi $}.$ Let $F\\in \\operatorname{H}^0(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(B_\\phi ))$ be a global section defining $B_\\phi $ .", "Lemma 2.3 The divisorial sheaf $\\mathop {\\mathcal {O}}\\nolimits _X(R_\\phi )$ is isomorphic to $\\phi ^\\ast \\mathop {\\mathcal {O}}\\nolimits _Y(L)$ .", "Let $\\lbrace (U_i,f_i)\\rbrace $ be a Cartier divisor on $Y$ defining $L$ .", "Then $\\lbrace f_if_j^{-1}\\rbrace $ is a system of transition functions of the line bundle $p:\\mathbb {L}\\rightarrow Y$ associated to $L$ .", "Namely, $\\mathbb {L}$ is given by the gluing maps $U_j\\times \\mathbb {C}\\rightarrow U_i\\times \\mathbb {C}$ over $U_i\\cap U_j$ defined by $(P,\\xi _j)\\mapsto (P,f_if_j^{-1}\\xi _j)$ for $P\\in U_i\\cap U_j$ .", "Since $B_\\phi \\sim 2L$ , $\\mathop {\\mathcal {O}}\\nolimits _Y(B_\\phi )$ can be identified with an $\\mathop {\\mathcal {O}}\\nolimits _Y$ -submodule of $\\mathbb {C}(Y)$ via the multiplication by $f_i^{-2}$ , $\\mathop {\\mathcal {O}}\\nolimits _{U_i}\\rightarrow \\mathbb {C}(Y)$ .", "We regard $F_i:=F|_{U_i}$ as an element of $\\Gamma (U_i,\\mathop {\\mathcal {O}}\\nolimits _Y)$ via the above identification.", "Hence $F_i=f_i^2f_j^{-2}F_j$ .", "Then the double cover $X$ is defined as the subvariety of $\\mathbb {L}$ locally given by $\\xi _i^2=F_i$ in $p^{-1}(U_i)$ , and $\\phi =p|_X$ (cf.", "[8]).", "Moreover, $R_\\phi $ is defined on $V_i:=\\phi ^{-1}(U_i)$ by $t_i=0$ , where $t_i:=\\xi _i|_{V_i}$ .", "Hence $\\lbrace (V_i,t_i)\\rbrace $ is a Cartier divisor on $X$ defining $R_\\phi $ , and satisfies $t_i=f_if_j^{-1}t_j$ for each $i,j$ .", "Therefore $\\mathop {\\mathcal {O}}\\nolimits _X(R_\\phi )\\cong \\phi ^\\ast \\mathop {\\mathcal {O}}\\nolimits _Y(L)$ .", "We define splitting divisors for double covers as splitting curves defined in [18].", "Definition 2.4 Let $\\phi :X\\rightarrow Y$ be a double cover of a smooth variety $Y$ , and let $C$ be an irreducible divisor on $Y$ .", "We call $C$ a pre-splitting divisor for $\\phi :X\\rightarrow Y$ if $\\phi ^\\ast C=\\widetilde{C}+\\iota ^\\ast \\widetilde{C}$ for a divisor $\\widetilde{C}$ on $X$ .", "In addition, if $\\widetilde{C}\\ne \\iota ^\\ast \\widetilde{C}$ , $C$ is called a splitting divisor for $\\phi $ .", "We call $C$ a simple pre-splitting divisor (say an SPS-divisor for short) for $\\phi :X\\rightarrow Y$ if a defining equation $f\\in \\operatorname{H}^0(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(C))$ of $C$ satisfies $F=h^2+fg$ for some $g\\in \\operatorname{H}^0(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(2L-C))$ and $h\\in \\operatorname{H}^0(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(L))$ .", "In addition, if $C\\lnot \\subset B_\\phi $ , $C$ is called a simple splitting divisor for $\\phi $ .", "Note that irreducible components of $B_\\phi $ are SPS-divisors for a double cover $\\phi :X\\rightarrow Y$ , and an SPS-divisor $C$ for $\\phi :X\\rightarrow Y$ is a pre-splitting divisor for $\\phi $ (cf.", "[17]).", "Next we see that, for a divisorial sheaf $\\mathcal {L}$ on a double cover $X$ of $Y$ , it is enough for splitting of $\\phi _\\ast \\mathcal {L}$ to consider the restriction of $\\mathcal {L}$ to the smooth locus of $X$ .", "Let $X^\\circ $ be the smooth locus of $X$ , and let $Y^\\circ $ be the image $\\phi (X^\\circ )$ of $X^\\circ $ under $\\phi $ .", "Put $\\phi ^\\circ :=\\phi |_{X^\\circ }:X^\\circ \\rightarrow Y^\\circ $ .", "The inclusion $\\eta :X^\\circ \\hookrightarrow X$ induces an isomorphism $\\eta ^\\ast :\\mathrm {Cl}(X)\\rightarrow \\mathop {\\rm Pic}\\nolimits (X^\\circ )$ since $X$ is normal.", "This isomorphism $\\eta ^\\ast $ gives an isomorphism from $\\mathop \\mathrm {sCl}\\nolimits _\\phi (X)$ to $\\mathop \\mathrm {sPic}\\nolimits _{\\phi ^\\circ }(X^\\circ )$ .", "Lemma 2.5 Let $\\widetilde{C}$ be an irreducible divisor on $X$ , and let $C$ be the divisor on $Y$ such that $\\phi ^\\ast C=\\widetilde{C}+\\iota ^\\ast \\widetilde{C}$ .", "Then $\\phi _\\ast \\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C})$ splits if and only if $\\phi ^\\circ _\\ast (\\eta ^\\ast \\mathop {\\mathcal {O}}\\nolimits _{X}(\\widetilde{C}))$ splits.", "In particular, the isomorphism $\\eta ^\\ast :\\mathrm {Cl}(X)\\rightarrow \\mathop {\\rm Pic}\\nolimits (X^\\circ )$ induces an isomorphism $\\eta ^\\ast |_{\\mathop \\mathrm {sCl}\\nolimits _\\phi (X)}:\\mathop \\mathrm {sCl}\\nolimits _\\phi (X)\\rightarrow \\mathop \\mathrm {sPic}\\nolimits _{\\phi ^\\circ }(X^\\circ )$ .", "Since $\\eta $ is isomorphic over $Y^\\circ $ , we have $\\phi _\\ast \\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C})|_{Y^\\circ }\\cong \\phi _\\ast ^\\circ (\\eta ^\\ast \\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C}))$ .", "Hence if $\\phi _\\ast \\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C})$ splits, then $\\phi ^\\circ _\\ast (\\eta ^\\ast \\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C}))$ splits.", "Conversely, we suppose $\\phi _\\ast ^\\circ (\\eta ^\\ast \\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C}))\\cong \\mathop {\\mathcal {O}}\\nolimits _{Y^\\circ }(D_1^\\circ )\\oplus \\mathop {\\mathcal {O}}\\nolimits _{Y^\\circ }(D_2^\\circ )$ for two divisors $D_1^\\circ , D_2^\\circ $ on $Y^\\circ $ .", "Let $D_1,D_2$ be the closures of $D_1^\\circ , D_2^\\circ $ in $Y$ , respectively.", "Then we have $\\phi _\\ast \\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C})|_{Y^\\circ }\\cong (\\mathop {\\mathcal {O}}\\nolimits _Y(D_1)\\oplus \\mathop {\\mathcal {O}}\\nolimits _Y(D_2))|_{Y^\\circ }$ .", "Therefore $\\phi _\\ast \\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C})\\cong \\mathop {\\mathcal {O}}\\nolimits _Y(D_1)\\oplus \\mathop {\\mathcal {O}}\\nolimits _Y(D_2)$ since both sheaves are reflexive, and $\\mathop {\\rm codim}\\nolimits _Y(Y\\setminus Y^\\circ )\\ge 2$ .", "Remark 2.6 Let $C$ be an SPS-divisor for a double cover $\\phi :X\\rightarrow Y$ .", "Then, for an irreducible component $\\widetilde{C}$ of $\\phi ^\\ast C$ , $\\phi _\\ast \\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C})$ splits by Lemma REF and [17].", "Hence $\\mathop \\mathrm {sCl}\\nolimits _\\phi (X)$ is generated by $\\phi ^\\ast \\mathop {\\rm Pic}\\nolimits (Y)$ and irreducible components of $\\phi ^\\ast C$ for all SPS-divisors $C$ for a double cover $\\phi :X\\rightarrow Y$ by [17]." ], [ "Divisor class groups of double covers", "In this section, we prove Theorem REF and Corollary REF .", "We first consider a double cover $\\phi :X\\rightarrow Y$ of a smooth variety $Y$ and an irreducible divisor $\\widetilde{C}$ on $X$ .", "For an open affine subset $V\\subset X$ , let $\\mathrm {Cl}(X,V)$ be the subgroup of $\\mathrm {Cl}(X)$ generated by irreducible divisors with support in $X\\setminus V$ ; $ \\mathrm {Cl}(X,V):=\\left\\langle \\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{D})\\ | \\ \\widetilde{D}\\subset X\\setminus V : \\mbox{ an irreducible divisor on $X$} \\right\\rangle .", "$ Then the following lemma holds.", "Lemma 3.1 Let $\\phi :X\\rightarrow Y$ be a double cover of a smooth variety $Y$ , and let $\\widetilde{C}$ be an irreducible divisor on $X$ with $\\widetilde{C}\\ne \\iota ^\\ast \\widetilde{C}$ .", "Put $C$ as the divisor on $Y$ such that $\\phi ^\\ast C=\\widetilde{C}+\\iota ^\\ast \\widetilde{C}$ .", "Let $U_0\\subset Y$ be an open affine subset with $C\\cap U_0\\ne \\emptyset $ , and put $V_0:=\\phi ^{-1}(U_0)$ .", "Assume that $B_\\phi $ and $L$ are principal on $U_0$ , and that $\\mathop {\\rm Pic}\\nolimits (Y,U_0)$ is generated by irreducible divisors $H_1,\\dots , H_m\\subset Y\\setminus U_0$ .", "If $\\widetilde{C}$ is principal on $V_0$ , then there exist $\\mathbf {d}\\in \\mathbb {Z}^m$ and an effective divisor $\\widetilde{\\mathbf {\\mathcal {D}}}$ on $X$ with support in $X\\setminus V_0$ satisfying the following conditions: $\\widetilde{C}+\\widetilde{\\mathbf {\\mathcal {D}}}\\sim \\phi ^\\ast (\\mathbf {d H})$ , hence $\\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C})\\in \\mathrm {Cl}(X,V_0)$ ; if an irreducible component $D$ of $\\mathbf {\\mathcal {D}}$ satisfies $\\operatorname{H}^1(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(L-D))=0$ , then $D$ is an SPS-divisor for $\\phi $ , where $\\mathbf {\\mathcal {D}}$ is the divisor on $Y$ with $\\phi ^\\ast \\mathbf {\\mathcal {D}}=\\widetilde{\\mathbf {\\mathcal {D}}}+\\iota ^\\ast \\widetilde{\\mathbf {\\mathcal {D}}}$ .", "In particular, if $\\operatorname{H}^1(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(L-D))=0$ for any irreducible component $D$ of $\\mathbf {\\mathcal {D}}$ , then $\\widetilde{C}\\in \\mathop \\mathrm {sCl}\\nolimits _\\phi (X)$ .", "We take an open affine covering $\\lbrace U_i\\rbrace _{i\\in I}$ of $Y$ containing $U_0$ such that $B_\\phi $ , $L$ and $H_1,\\dots , H_m$ are principal on each $U_i$ .", "Fix Cartier divisors $\\lbrace (U_i,x_{ij})\\rbrace $ defining $H_j$ for $j=1,\\dots ,m$ .", "We may assume that $x_{0j}=1$ for $j=1,\\dots ,m$ .", "By Lemma REF , we have $L\\sim \\mathbf {\\ell H}$ for some $\\mathbf {\\ell }=(\\ell _1,\\dots ,\\ell _m)\\in \\mathbb {Z}^m$ .", "Then $B_\\phi \\sim 2\\mathbf {\\ell H}$ and $R_\\phi \\sim \\phi ^\\ast (\\mathbf {\\ell H})$ .", "Let $F\\in \\operatorname{H}^0(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(2\\mathbf {\\ell H}))$ and $t\\in \\operatorname{H}^0(X,\\phi ^\\ast \\mathop {\\mathcal {O}}\\nolimits _Y(\\mathbf {\\ell H}))$ be sections defining $B_\\phi $ and $R_\\phi $ , respectively, such that $t^2=F$ as elements in $\\mathbb {C}(X)=\\mathbb {C}(Y)\\oplus t\\,\\mathbb {C}(Y)$ .", "Put $F_i:=F|_{U_i}\\in \\Gamma (U_i,\\mathop {\\mathcal {O}}\\nolimits _Y)$ via $\\mathop {\\mathcal {O}}\\nolimits _{U_i}\\overset{\\sim }{\\rightarrow }\\mathop {\\mathcal {O}}\\nolimits _{Y}(2\\mathbf {\\ell H})|_{U_i}\\subset \\mathbb {C}(Y)$ , and $t_i:=t|_{V_i}$ via the multiplication by $\\mathbf {x}_i^{-\\mathbf {\\ell }}$ , $\\mathop {\\mathcal {O}}\\nolimits _{V_i}\\overset{\\sim }{\\rightarrow }\\phi ^\\ast \\mathop {\\mathcal {O}}\\nolimits _Y(\\mathbf {\\ell H})|_{V_i}\\subset \\mathbb {C}(X)$ .", "Then we have $ F_i=\\mathbf {x}_i^{2\\mathbf {\\ell }}F_0, \\qquad \\qquad t_i=\\mathbf {x}_i^{\\mathbf {\\ell }}t_0.", "$ Since $\\widetilde{C}$ is principal on $V_0$ , there are $p_0,q_0\\in \\Gamma (U_0,\\mathop {\\mathcal {O}}\\nolimits _Y)$ such that $\\widetilde{C}$ is defined on $V_0$ by $p_0+q_0t_0=0$ .", "Note that $q_0\\ne 0$ since $\\iota ^\\ast \\widetilde{C}\\ne \\widetilde{C}$ .", "Since $H_1,\\dots ,H_m$ generate $\\mathop {\\rm Pic}\\nolimits (Y,U_0)$ , after multiplying $p_0+q_0t_0$ by a unit of $\\Gamma (U_0,\\mathop {\\mathcal {O}}\\nolimits _Y)$ if necessary, we may assume that $p_0$ and $q_0$ have no pole on $Y\\setminus (H_1\\cup \\dots \\cup H_m)$ , and that the principal divisors $(p_0)$ and $(q_0)$ on $Y$ have no common component except for $H_1,\\dots ,H_m$ .", "Hence we obtain $(p_0)=D_p-\\mathbf {aH}$ and $(q_0)=D_q-\\mathbf {bH}$ for $\\mathbf {a}=(a_1,\\dots ,a_m),\\mathbf {b}=(b_1,\\dots ,b_m)\\in \\mathbb {Z}^m$ , where $D_p$ and $D_q$ are effective divisors on $Y$ consisting of components except for $H_1,\\dots ,H_m$ which are defined on $U_0$ by $p_0=0$ and $q_0=0$ , respectively.", "Put $ p_i:=\\mathbf {x}_i^{\\mathbf {a}}p_0, \\qquad \\qquad q_i:=\\mathbf {x}_i^{\\mathbf {b}}q_0 $ for $i\\in I$ .", "Note that $\\lbrace (U_i,p_i)\\rbrace $ and $\\lbrace (U_i,q_i)\\rbrace $ are Cartier divisors on $Y$ defining $D_p$ and $D_q$ , respectively.", "Put $d_j:=\\max \\lbrace a_j,b_j+\\ell _j\\rbrace $ for $j=1,\\dots ,m$ , and $\\mathbf {d}:=(d_1,\\dots ,d_m)$ .", "Then $ {D}:=\\big \\lbrace \\left(V_i, \\ \\mathbf {x}_i^{\\mathbf {d}-\\mathbf {a}}p_i+\\mathbf {x}_i^{\\mathbf {d}-\\mathbf {b}-\\mathbf {\\ell }}q_it_i\\right)\\big \\rbrace $ is an effective Cartier divisor on $X$ which is linearly equivalent to $\\phi ^\\ast (\\mathbf {dH})$ .", "Since $\\widetilde{C}$ is defined on $V_0$ by $p_0+q_0t_0=0$ , ${{D}}$ defines an effective divisor $\\widetilde{C}+\\widetilde{\\mathbf {\\mathcal {D}}}$ such that $\\mathop {\\rm Supp}\\nolimits (\\widetilde{\\mathbf {\\mathcal {D}}})\\subset X\\setminus V_0$ .", "Hence $\\mathbf {d}$ and $\\widetilde{\\mathbf {\\mathcal {D}}}$ satisfy REF .", "Note that $ \\Big \\lbrace \\left(U_i, \\ \\mathbf {x}_i^{2(\\mathbf {d}-\\mathbf {a})}p_i^2-\\mathbf {x}_i^{2(\\mathbf {d}-\\mathbf {b}-\\mathbf {\\ell })}q_i^2 F_i\\right)\\Big \\rbrace =\\Big \\lbrace \\left(U_i, \\ \\mathbf {x}_i^{2\\mathbf {d}}\\big (p_0^2-q_0^2 F_0\\big )\\right)\\Big \\rbrace $ is a Cartier divisor on $Y$ defining $C+\\mathbf {\\mathcal {D}}$ .", "Let $D$ be an irreducible component of $\\mathbf {\\mathcal {D}}$ defined by $f\\in \\operatorname{H}^0(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(D))$ which satisfies $\\operatorname{H}^1(\\mathop {\\mathcal {O}}\\nolimits _Y(L-D))=0$ .", "On $ U_i\\cap D$ , we have $ \\mathbf {x}_i^{2(\\mathbf {d}-\\mathbf {a})}p_i^2=\\mathbf {x}_i^{2(\\mathbf {d}-\\mathbf {b}-\\mathbf {\\ell })}q_i^2 F_i.", "$ If $(\\mathbf {x}_i^{\\mathbf {d}-\\mathbf {a}}p_i)|_{U_i\\cap D}= 0$ as a rational function of $D$ , then $D$ is a component of $B_\\phi $ by the choices of $p_0$ , $q_0$ and $\\mathbf {d}$ .", "Hence $F=f g$ for some $g\\in \\operatorname{H}^0(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(B_\\phi -D))$ , and $D$ is an SPS-divisor for $\\phi $ .", "Suppose that $(\\mathbf {x}_i^{\\mathbf {d}-\\mathbf {a}}p_i)|_{U_i\\cap D}\\ne 0$ for $i\\in I$ .", "Then $(\\mathbf {x}_i^{\\mathbf {d}-\\mathbf {b}-\\mathbf {\\ell }}q_i)|_{U_i\\cap D}\\ne 0$ , and $\\lbrace (U_i\\cap D,\\ (\\mathbf {x}_i^{\\mathbf {\\ell }-\\mathbf {a}+\\mathbf {b}}p_iq_i^{-1})|_{U_i\\cap D}) \\rbrace $ is an effective Cartier divisor on $D$ linearly equivalent to $\\mathbf {\\ell }\\mathbf {H}|_{D}$ .", "Since $\\operatorname{H}^1(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(\\mathbf {\\ell }\\mathbf {H}-D))=0$ , we have the following exact sequence $ 0\\rightarrow \\operatorname{H}^0(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(\\mathbf {\\ell }\\mathbf {H}-D))\\overset{\\cdot f}{\\rightarrow } \\operatorname{H}^0(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(\\mathbf {\\ell }\\mathbf {H}))\\rightarrow \\operatorname{H}^0(D,\\mathop {\\mathcal {O}}\\nolimits _{D}(\\mathbf {\\ell }\\mathbf {H}))\\rightarrow 0.", "$ Thus there is a global section $h\\in \\operatorname{H}^0(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(\\mathbf {\\ell H}))$ such that $ h|_{D\\cap U_i}=\\left(\\mathbf {x}_i^{\\mathbf {\\ell }-\\mathbf {a}+\\mathbf {b}}p_iq_i^{-1})\\right|_{D\\cap U_i}=\\left(\\mathbf {x}_i^{\\mathbf {\\ell }}p_0q_0^{-1})\\right|_{D\\cap U_i} $ for $i\\in I$ .", "We consider the following exact sequence $ 0\\rightarrow \\operatorname{H}^0(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(2\\mathbf {\\ell }\\mathbf {H}-D))\\overset{\\cdot f}{\\rightarrow } \\operatorname{H}^0(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(2\\mathbf {\\ell }\\mathbf {H}))\\overset{}{\\rightarrow } \\operatorname{H}^0(D,\\mathop {\\mathcal {O}}\\nolimits _{D}(2\\mathbf {\\ell }\\mathbf {H})).", "$ Since $(F-h^2)|_D=0$ , there exists $g\\in \\operatorname{H}^0(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(2\\mathbf {\\ell }\\mathbf {H}-D))$ such that $F=h^2+fg$ .", "Hence $D$ is an SPS-divisor, and REF holds.", "If $\\operatorname{H}^1(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(L-D))=0$ for all irreducible component $D$ of $\\mathbf {\\mathcal {D}}$ , then we obtain $\\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C})\\cong (\\mathop {\\mathcal {O}}\\nolimits (-\\widetilde{\\mathbf {\\mathcal {D}}})\\otimes \\phi ^\\ast \\mathop {\\mathcal {O}}\\nolimits _Y(\\mathbf {d}\\mathbf {H}))^{\\vee \\vee }\\in \\mathop \\mathrm {sCl}\\nolimits _\\phi (X)$ by Remark REF .", "Remark 3.2 The assumption $\\operatorname{H}^1(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(L-D))=0$ in REF seems strong for an irreducible component $D$ of $\\mathbf {\\mathcal {D}}$ to be an SPS-divisor for $\\phi $ .", "We may be able to replace $\\operatorname{H}^1(Y,\\mathop {\\mathcal {O}}\\nolimits _Y(L-D))=0$ with a weaker condition in REF .", "As a corollary of Lemma REF , we obtain Theorem REF and Corollary REF .", "[Proof of Theorem REF ] For any double cover $\\phi :X\\rightarrow \\mathbb {P}^1$ of the projective line $\\mathbb {P}^1$ , REF holds (cf.", "[17]).", "Assume that $n\\ge 2$ , and let $\\phi :X\\rightarrow \\mathbb {P}^n$ be a double cover.", "Since $\\mathop {\\rm Pic}\\nolimits (\\mathbb {P}^n)$ is generated by a hyperplane $H\\subset \\mathbb {P}^n$ , and since $\\operatorname{H}^1(\\mathbb {P}^n,\\mathop {\\mathcal {O}}\\nolimits _{\\mathbb {P}^n}(m))=0$ for any $m\\in \\mathbb {Z}$ , we obtain $\\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C})\\in \\mathop \\mathrm {sCl}\\nolimits _\\phi (X)$ for any irreducible divisor $\\widetilde{C}$ on $X$ with $\\iota ^\\ast \\widetilde{C}\\ne \\widetilde{C}$ by Lemma REF .", "Therefore we have $\\mathop \\mathrm {sCl}\\nolimits _\\phi (X)=\\mathrm {Cl}(X)$ since $\\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C})=\\phi ^\\ast \\mathop {\\mathcal {O}}\\nolimits _{\\mathbb {P}^n}(m)$ for some $m\\in \\mathbb {Z}$ if $\\iota ^\\ast \\widetilde{C}=\\widetilde{C}$ .", "Assertion REF follows from Remark REF .", "[Proof of Corollary REF ] Let ${\\mathcal {E}}$ be a 2-bundle on $\\mathbb {P}^n$ .", "By [17], there exist a double cover $\\phi :X\\rightarrow \\mathbb {P}^n$ and a divisorial sheaf $\\mathcal {L}$ on $X$ such that $\\phi _\\ast \\mathcal {L}\\cong {\\mathcal {E}}$ .", "By Theorem REF , we have $\\mathcal {L}\\in \\mathop \\mathrm {sCl}\\nolimits _\\phi (X)$ .", "Hence the assertion holds.", "To search generators of $\\mathrm {Cl}(X)$ for a double cover $\\phi :X\\rightarrow \\mathbb {P}^n$ , it is enough to consider irreducible divisors $C$ on $\\mathbb {P}^n$ with $\\deg C\\le \\deg L$ .", "Corollary 3.3 Let $\\phi :X\\rightarrow \\mathbb {P}^n$ be a double cover.", "Then $\\mathrm {Cl}(X)$ is generated by $\\phi ^\\ast \\mathop {\\mathcal {O}}\\nolimits _{\\mathbb {P}^n}(1)$ and irreducible components of $\\phi ^\\ast C$ for SPS-divisors $C$ with $\\deg (C)\\le l:=\\deg (L)$ .", "By Remark REF and [17], $\\mathrm {Cl}(X)$ is generated by $\\phi ^\\ast \\mathop {\\mathcal {O}}\\nolimits _{\\mathbb {P}^n}(1)$ and irreducible components $\\widetilde{C}$ of $\\phi ^\\ast C$ with $\\phi _\\ast \\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C})\\cong \\mathop {\\mathcal {O}}\\nolimits _{\\mathbb {P}^n}(-d)\\oplus \\mathop {\\mathcal {O}}\\nolimits _{\\mathbb {P}^n}$ for SPS-divisors $C$ and $d\\ge 0$ .", "If $\\widetilde{C}$ is an irreducible divisor on $X$ such that $\\phi _\\ast \\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C})\\cong \\mathop {\\mathcal {O}}\\nolimits _{\\mathbb {P}^n}(-d)\\oplus \\mathop {\\mathcal {O}}\\nolimits _{\\mathbb {P}^n}$ with $d\\ge 0$ , and let $C$ be the divisor on $\\mathbb {P}^n$ with $\\phi ^\\ast C=\\widetilde{C}+\\iota ^\\ast \\widetilde{C}$ .", "By [17], $\\deg (C)=l-d\\le l$ .", "Corollary 3.4 Let $\\phi :X\\rightarrow \\mathbb {P}^n$ be a double cover branched at a smooth divisor $B_\\phi $ .", "If $n\\ge 3$ , then $\\mathop {\\rm Pic}\\nolimits (X)\\cong \\mathbb {Z}$ .", "Since $B_\\phi $ is smooth, the double cover $X$ is also smooth.", "Let $C$ be an SPS-divisor for $\\phi $ defined by $f=0$ with $F=h^2+fg$ .", "Note that the algebraic subset $S$ defined by the ideal $\\langle f,g,h\\rangle $ has at most codimension 3 in $\\mathbb {P}^n$ .", "If $h\\ne 0$ , then $g\\ne 0$ since $B_\\phi $ is reduced, and $B_\\phi $ is singular along $S$ , which is a contradiction.", "Hence we have $h=0$ .", "Since $B_\\phi $ is irreducible, we obtain $f=F$ and $g=1$ up to unit.", "Thus $C=B_\\phi $ .", "By Theorem REF and Lemma REF , we obtain $\\mathop {\\rm Pic}\\nolimits (X)\\cong \\mathop {\\rm Pic}\\nolimits (\\mathbb {P}^n)\\cong \\mathbb {Z}$ .", "For a double cover $\\phi :X\\rightarrow Y$ , we have a condition for $\\mathrm {Cl}(X)$ to be finitely generated by using Lemma REF .", "Proposition 3.5 Let $\\phi :X\\rightarrow Y$ be a double cover.", "The group $\\mathrm {Cl}(X)$ is finitely generated if and only if there exists an open affine subset $U_0\\subset Y$ such that the coordinate ring of $V_0:=\\phi ^{-1}(U_0)$ is a UFD.", "Moreover, in this case, $\\mathrm {Cl}(X)=\\mathrm {Cl}(X,V_0)$ .", "Let $X^\\circ $ be the smooth locus of $X$ , and let $\\eta :X^\\circ \\hookrightarrow X$ be the inclusion.", "Note that $\\eta $ naturally induces an isomorphism $\\eta ^\\ast :\\mathrm {Cl}(X)\\rightarrow \\mathop {\\rm Pic}\\nolimits (X^\\circ )$ since $X$ is normal.", "By Lemma REF , $\\mathop {\\rm Pic}\\nolimits (X^\\circ )$ is finitely generated if and only if there exists an open affine subset $V_0^{\\prime }$ of $X^\\circ $ whose coordinate ring $\\Gamma (V_0^{\\prime },\\mathop {\\mathcal {O}}\\nolimits _X)$ is a UFD.", "If $\\mathrm {Cl}(X)\\cong \\mathop {\\rm Pic}\\nolimits (X^\\circ )$ is finitely generated, then let $U_0\\subset \\phi (V_0^{\\prime }\\cap \\iota (V_0^{\\prime }))$ be an open affine subset of $Y$ ; the coordinate ring of $V_0:=\\phi ^{-1}(U_0)$ is a UFD since it is a localization of the UFD $\\Gamma (V_0^{\\prime },\\mathop {\\mathcal {O}}\\nolimits _X)$ .", "Conversely, if there is an open affine subset $U_0$ of $Y$ such that the coordinate ring of $V_0$ is a UFD, then by Lemma REF , $\\mathrm {Cl}(X)$ is generated by irreducible divisors on $X$ with support in $X\\setminus V_0$ ." ], [ "A condition for splitting divosors", "In this section, we give a condition for an irreducible divisor on $\\mathbb {P}^n$ to be a splitting divisor for a double cover of $\\mathbb {P}^n$ whose divisor class group is finitely generated.", "Proposition 4.1 Let $\\phi :X\\rightarrow \\mathbb {P}^n$ be a double cover of $\\mathbb {P}^n$ , and let $H$ be a hyperplane in $\\mathbb {P}^n$ .", "Let $D_1,\\dots ,D_m$ be SPS-divisors for $\\phi :X\\rightarrow \\mathbb {P}^n$ .", "Put $U:=\\mathbb {P}^n\\setminus (H\\cup D_1\\cup \\dots \\cup D_m)$ and $V:=\\phi ^{-1}(V)$ .", "Then $\\mathrm {Cl}(X)=\\mathrm {Cl}(X,V)$ if and only if $V$ is an open affine subset of $X$ whose coordinate ring is a UFD.", "Since $U$ is affine, $V$ is also affine.", "Suppose that $\\mathrm {Cl}(X)$ is generated by irreducible components of $\\phi ^\\ast (H+D_1+\\dots +D_m)$ .", "Let $\\widetilde{C}_V$ be an irreducible divisor on $V$ , and let $\\widetilde{C}$ be the closure of $\\widetilde{C}_V$ in $X$ .", "Then $\\widetilde{C}$ is linearly equivalent to $\\widetilde{\\mathbf {\\mathcal {D}}}$ with support in $X\\setminus V$ , and $\\mathop {\\mathcal {O}}\\nolimits _V(\\widetilde{C}_V)\\cong \\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{\\mathbf {\\mathcal {D}}})|_V\\cong \\mathop {\\mathcal {O}}\\nolimits _V$ .", "Therefore $\\mathrm {Cl}(V)=0$ , and the coordinate ring of $V$ is a UFD.", "Conversely, suppose that the coordinate ring of $V$ is a UFD.", "Let $\\widetilde{C}$ be an irreducible divisor on $X$ .", "Then $\\widetilde{C}$ is principal on $V$ since $\\mathrm {Cl}(V)=0$ .", "By Lemma REF  REF , we obtain $\\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{C})\\in \\mathrm {Cl}(X,V)$ .", "Remark 4.2 If there is an SPS-divisor $D$ with $\\deg D=\\deg L$ for a double cover $\\phi :X\\rightarrow \\mathbb {P}^n$ , then there are infinite SPS-divisors.", "Indeed, for an irreducible component $\\widetilde{D}$ of $\\phi ^\\ast D$ , we have $\\phi _\\ast \\mathop {\\mathcal {O}}\\nolimits _X(\\widetilde{D})\\cong \\mathop {\\mathcal {O}}\\nolimits _{\\mathbb {P}^n}^{\\oplus 2}$ , and the image $\\phi (\\widetilde{D}^{\\prime })$ of any divisor $\\widetilde{D}^{\\prime }$ which is linearly equivalent to $\\widetilde{D}$ is an SPS-divisor for $\\phi $ by [17].", "It is easy to see that an SPS-divisor is a splitting divisor for a double cover.", "Hence we consider a condition for a non-SPS-divisor to be a splitting divisor.", "The next theorem gives a condition for a non-SPS-divisor on $\\mathbb {P}^n$ to be a splitting divisor for a double cover $\\phi :X\\rightarrow \\mathbb {P}^n$ in the case where $\\mathrm {Cl}(X)$ is finitely generated.", "Let $H\\subset \\mathbb {P}^n$ be a hyperplane, and put $l:=\\deg L$ .", "Theorem 4.3 Let $\\phi :X\\rightarrow \\mathbb {P}^n$ be a double cover, and let $C$ be an irreducible divisor on $\\mathbb {P}^n$ defined by $f\\in \\operatorname{H}^0(\\mathbb {P}^n,\\mathop {\\mathcal {O}}\\nolimits _{\\mathbb {P}^n}(C))$ such that $C\\lnot \\subset B_\\phi $ .", "Assume that there are SPS-divisors for $\\phi $ , $D_1,\\dots ,D_m$ such that $\\mathrm {Cl}(X)$ is generated by $\\phi ^\\ast H$ and irreducible components of $\\phi ^\\ast (D_1+\\dots +D_m)$ , and that $C\\ne D_j$ for $j=1,\\dots ,m$ .", "Let $g_j\\in \\operatorname{H}^0(\\mathbb {P}^n,\\mathop {\\mathcal {O}}\\nolimits _{\\mathbb {P}^n}(D_j))$ be a defining equation of $D_j$ for $j=1,\\dots ,m$ .", "Then the followings are equivalent: $C$ is a splitting divisor for $\\phi $ ; there exist $\\mathbf {a}=(a_1,\\dots ,a_m)\\in \\mathbb {Z}_{\\ge 0}^m$ with $\\deg (C+a_1D_1+\\dots +a_mD_m)=2k$ , $p\\in \\operatorname{H}^0(\\mathbb {P}^n,\\mathop {\\mathcal {O}}\\nolimits _{\\mathbb {P}^n}(k))$ and $q\\in \\operatorname{H}^0(\\mathbb {P}^n,\\mathop {\\mathcal {O}}\\nolimits _{\\mathbb {P}^n}(k-l))$ for some $k\\in \\mathbb {Z}$ such that $ p^2-q^2F=fg_1^{a_1}\\dots g_m^{a_m}.", "$ We may assume that $H\\ne C$ and $H\\ne D_j$ for $j=1,\\dots ,m$ .", "Put $U:=\\mathbb {P}^n\\setminus (H\\cup D_1\\cup \\dots \\cup D_m)$ and $V:=\\phi ^{-1}(U)$ .", "Then we have $U\\cap C\\ne \\emptyset $ .", "We first suppose that REF holds.", "In this case, $\\phi ^\\ast (C+a_1D_1+\\dots +a_mD_m)=\\widetilde{\\mathbf {\\mathcal {D}}}+\\iota ^\\ast \\widetilde{\\mathbf {\\mathcal {D}}}$ , where $\\widetilde{\\mathbf {\\mathcal {D}}}$ is defined by $p+qt=0$ .", "Here $t\\in \\operatorname{H}^0(X,\\phi ^\\ast \\mathop {\\mathcal {O}}\\nolimits _{\\mathbb {P}^n}(l))$ defines the ramification divisor $R_\\phi $ on $X$ satisfying $t^2=F$ .", "Let $\\widetilde{C}$ be a common component of $\\phi ^\\ast C$ and $\\widetilde{\\mathbf {\\mathcal {D}}}$ .", "Since $C\\lnot \\subset B_\\phi $ , $\\phi ^\\ast C$ is reduced.", "Hence we have $\\widetilde{C}\\ne \\iota ^\\ast \\widetilde{C}$ , and $\\phi ^\\ast C=\\widetilde{C}+\\iota ^\\ast \\widetilde{C}$ since $\\phi ^\\ast C|_V=(\\widetilde{\\mathbf {\\mathcal {D}}}+\\iota ^\\ast \\widetilde{\\mathbf {\\mathcal {D}}})|_V$ .", "Next, suppose that $\\phi ^\\ast C=\\widetilde{C}+\\iota ^\\ast \\widetilde{C}$ for an irreducible divisor $\\widetilde{C}$ on $X$ with $\\iota ^\\ast \\widetilde{C}\\ne \\widetilde{C}$ .", "By Proposition REF , $V$ is an open affine subset of $X$ whose coordinate ring is a UFD.", "Hence $\\widetilde{C}$ is principal on $V$ .", "By Lemma REF , there exist $d\\in \\mathbb {Z}$ and an effective divisor $\\widetilde{\\mathbf {\\mathcal {D}}}\\subset X\\setminus V$ on $X$ such that $\\widetilde{C}+\\widetilde{\\mathbf {\\mathcal {D}}}\\sim \\phi ^\\ast \\mathop {\\mathcal {O}}\\nolimits _{\\mathbb {P}^n}(d)$ .", "Let $\\mathbf {\\mathcal {D}}$ be the effective divisor on $\\mathbb {P}^n$ with $\\phi ^\\ast \\mathbf {\\mathcal {D}}=\\widetilde{\\mathbf {\\mathcal {D}}}+\\iota ^\\ast \\widetilde{\\mathbf {\\mathcal {D}}}$ .", "Since all irreducible components of $\\mathbf {\\mathcal {D}}$ are SPS-divisors by Lemma REF  REF , $\\mathbf {\\mathcal {D}}=a_1 D_1+\\dots +a_m D_m$ for $a_1,\\dots ,a_m\\in \\mathbb {Z}_{\\ge 0}$ .", "Furthermore, the proof of Lemma REF implies that $C+\\mathbf {\\mathcal {D}}$ is defined by $p^2-q^2F=0$ for some homogeneous polynomials $p, q$ .", "Therefore, we have $p^2-q^2F=fg_1^{a_1}\\dots g_m^{a_m}$ .", "It is clear that $\\deg (f g_1^{a_1}\\dots g_m^{a_m})=2k$ , $\\deg (p)=k$ and $\\deg (q)=k-l$ for some $k\\in \\mathbb {Z}$ .", "Example 4.4 (cf.", "[2]) Let $\\phi :X\\rightarrow \\mathbb {P}^2$ be a double cover branched at a smooth conic $B_\\phi $ .", "In this case, $X\\cong \\mathbb {P}^1\\times \\mathbb {P}^1$ and $\\mathop {\\rm Pic}\\nolimits (X)\\cong \\mathbb {Z}^2$ .", "Any tangent line of $B_\\phi $ is an SPS-splitting divisor for $\\phi $ .", "For a tangent line $H$ of $B_\\phi $ , the irreducible components of $\\phi ^\\ast H$ generate $\\mathop {\\rm Pic}\\nolimits (X)\\cong \\mathbb {Z}^2$ .", "By Theorem REF , a plane curve $C\\lnot \\subset B_\\phi $ defined by $f=0$ is a splitting curve for $\\phi $ if and only if there are homogeneous polynomials $p,q$ and $a\\in \\mathbb {Z}_{\\ge 0}$ such that $p^2-q^2F=f h^a$ , where $h$ is a defining equation of $H$ .", "Example 4.5 Let $\\phi :X\\rightarrow \\mathbb {P}^2$ be a double cover branched at a smooth quartic $B_\\phi $ .", "In this case, $X$ is isomorphic to a blowing-up of $X^{\\prime }:=\\mathbb {P}^2$ at 7 points $P_1,\\dots ,P_7\\in X^{\\prime }$ in general position, and the covering map $\\phi $ is induced by the linear system $\\Lambda $ on $X^{\\prime }$ consisting of cubic curves passing through $P_1,\\dots ,P_7$ .", "$\\begin{tikzpicture}\\node (P1) at (-1.1,1) {X^{\\prime }:=\\mathbb {P}^2};\\node (X) at (1,1) {X};\\node (P2) at (1,0) {\\mathbb {P}^2};[->] (X) -- node[above,scale=0.7] {blow-up} (P1);[->] (X) -- node[right,scale=0.7] {\\phi } (P2);[dashed,->] (P1) -- node[below,scale=0.7] {\\Phi _\\Lambda } (P2);\\end{tikzpicture}$ Let $\\widetilde{D}_1^{\\prime }$ (resp.", "$\\widetilde{D}_2^{\\prime }$ ) be a line (resp.", "a smooth conic) on $X^{\\prime }$ passing through just one point (resp.", "4 points) of $P_1,\\dots ,P_7$ .", "Let $\\widetilde{D}_1, \\widetilde{D}_2\\subset X$ be the strict transform of $\\widetilde{D}_1^{\\prime }$ and $\\widetilde{D}_2^{\\prime }$ under the blowing-up, respectively.", "Then we can check that the images $\\phi (\\widetilde{D}_1)$ and $\\phi (\\widetilde{D}_2)$ are SPS-divisors of degree 2 for $\\phi :X\\rightarrow \\mathbb {P}^2$ .", "This implies that $\\phi :X\\rightarrow \\mathbb {P}^2$ has infinite SPS-divisors.", "On the other hand, $\\mathop {\\rm Pic}\\nolimits (X)$ is generated by the exceptional divisors $\\widetilde{E}_1,\\dots ,\\widetilde{E}_7$ and the strict transform $\\widetilde{E}_0$ of a line on $X^{\\prime }$ passing through two of $P_1,\\dots ,P_7$ under the blowing-up.", "Let $H_0,\\dots ,H_7$ be the images of $\\widetilde{E}_0,\\dots ,\\widetilde{E}_7$ by $\\phi $ .", "Then $H_j$ are bitangent lines of $B_\\phi $ .", "By Theorem REF , a plane curve $C\\lnot \\subset B_\\phi $ defined by $f=0$ is a splitting curve for $\\phi $ if and only if there are homogeneous polynomials $p,q$ and $a_0,\\dots ,a_7\\in \\mathbb {Z}_{\\ge 0}$ such that $ p^2-q^2F=f h_0^{a_0}\\dots h_7^{a_7}, $ where $h_j$ is a defining equation of $H_j$ for $j=0,\\dots ,7$ .", "To use Theorem REF , SPS-divisors $D_1,\\dots ,D_m\\subset \\mathbb {P}^n$ such that irreducible components of $\\phi ^\\ast (D_1+\\dots +D_m)$ generate $\\mathrm {Cl}(X)$ are important.", "It is a problem to compute such SPS-divisors $D_1,\\dots ,D_m$ for a double cover $\\phi :X\\rightarrow \\mathbb {P}^n$ .", "Problem 4.6 Let $\\phi :X\\rightarrow \\mathbb {P}^n$ be a double cover.", "Give a method for computing SPS-divisors $D_1,\\dots ,D_m$ such that $\\mathrm {Cl}(X)$ is generated by $\\phi ^\\ast H$ and irreducible components of $\\phi ^\\ast (D_1+\\dots +D_m)$ .", "Taketo SHIRANE Department of Mathematical Sciences, Faculty of Science and Technology, Tokushima University, 2-1 Minamijyousanjima-cho, Tokushima 770-8506, JAPAN.", "E-mail: [email protected], ORCID iD: 0000-0002-4531-472X." ] ]
2105.11797
[ [ "Improving Few-shot Learning with Weakly-supervised Object Localization" ], [ "Abstract Few-shot learning often involves metric learning-based classifiers, which predict the image label by comparing the distance between the extracted feature vector and class representations.", "However, applying global pooling in the backend of the feature extractor may not produce an embedding that correctly focuses on the class object.", "In this work, we propose a novel framework that generates class representations by extracting features from class-relevant regions of the images.", "Given only a few exemplary images with image-level labels, our framework first localizes the class objects by spatially decomposing the similarity between the images and their class prototypes.", "Then, enhanced class representations are achieved from the localization results.", "We also propose a loss function to enhance distinctions of the refined features.", "Our method outperforms the baseline few-shot model in miniImageNet and tieredImageNet benchmarks." ], [ "Introduction", "Deep learning models based on Convolutional Neural Networks (CNNs) have shown remarkable performance in the image classification task [1], [2], [3].", "These models are usually trained in a supervised manner, relying on a large-scale dataset that provides sufficient labeled image samples.", "However, in practice, collecting and labeling thousands of target class images would be expensive, sometimes impossible.", "Few-shot learning addresses the classification problems in the limited conditions, where the information of each class is given with only a few exemplary (or support) images.", "In a few-shot setting, the scarcity of sample data makes conventional classifiers with a fully-connected layer vulnerable to overfitting.", "Hence metric learning-based classifiers are preferred instead, where a query image is mapped to feature space and then classified by the distances to the class representations acquired from support images [4], [5], [6], [7], [8], [9], [10].", "The Siamese Network architecture [4] is the first to validate the importance of feature embedding in one-shot learning.", "Vinyals et al.", "[5] propose a training scheme repeating “episodes” of evaluation scenarios as a form of meta-learning.", "Prototypical Networks [6] introduce a simple classifier using the Euclidean distance to the class prototypes, defined as the centroids of each class's support feature vectors.", "More recent works explore different class representations and distance metrics.", "Oreshkin et al.", "[7] suggest a task-dependent adaptive metric (TADAM).", "Zhao et al.", "[8] propose a variational Bayesian framework that represents each class as a distribution.", "It uses a probabilistic-based metric, which belongs to weighted Euclidean distances.", "Deep subspace networks (DSN) [9] utilize subspace representations and classifies a query based on the shortest distance from the query to its projections onto subspaces.", "Figure: A support image may contain class-irrelevant instances.", "Our method localizes the class object and acquires the region of interest (RoI) feature.", "(a) An image of a man and a dog labeled `dog.'", "(b) The t-SNE plot of features of the whole image and the RoI, colored in blue and green respectively.", "(`×\\times ' denotes the average feature.", ")Figure: Proposed framework.", "Using the class prototype, we localize the class object and extract a class-relevant feature from each support image to compute refined class representation.", "⊙\\odot denotes the similarity mapping operation in Eq.", ".In metric learning, embedding quality plays a crucial role [11].", "An image representation vector is conventionally achieved with a CNN encoder followed by a global average pooling (GAP) layer.", "However, GAP does not take into account the spatial information of where the feature is originated.", "As a result, the embedding may contain features of class-irrelevant instances or background.", "For instance, Fig.", "REF (a) shows an exemplary image labeled as `dog,' but the extracted feature represents the overall scene of a man with a dog.", "Figure REF (b) shows that this causes an undesirable bias from the feature that focuses on the foreground object.", "A class representation obtained from such impure support features may not express the class specifically.", "To tackle this problem, we propose a framework that localizes the class object and extracts only relevant features at the region of interest (RoI).", "Our method does not require ground-truth bounding box annotations and localizes the class objects with weak supervision.", "A naive approach to address a weakly supervised object localization (WSOL) problem is to use class activation mapping (CAM) [12].", "It uses the linear (i.e., fully connected) classifier's weights to visualize activations focusing on the most discriminative parts in the images, consequently highlighting the class objects.", "Unfortunately, CAM cannot be directly employed for WSOL in few-shot realm due to the overfitting issue of the linear classifier.", "As a countermeasure, we utilize similarity mapping [13], which is originally proposed to visualize the similar regions between two images.", "Our framework spatially decomposes the similarity between the image features and the class prototypes to localize foreground objects.", "More class-relevant features are obtained from the predicted bounding boxes via RoIAlign-based feature propagation, resulting in refined class representations.", "We also introduce a novel loss function that regards our class representations to improve the model's discriminative power.", "Our method outperforms prior metric learning-based classifiers that do not consider class object locality in miniImageNet [5] and tieredImageNet [14] datasets.", "The proposed framework also produces good localization results." ], [ "Preliminaries", "An $N$ -way $K$ -shot problem is a few-shot classification task, where a small support set $S=\\lbrace (\\textrm {\\textbf {x}}_i, y_i)\\rbrace _{i=1}^{K}$ is given each for $N$ classes.", "Here, each $\\textrm {\\textbf {x}}_i \\in \\mathbb {R}^{H \\times W \\times 3}$ is an RGB image with height $H$ and width $W$ , and $y_i \\in \\lbrace 1, \\cdots , N\\rbrace $ is the corresponding label.", "$S_c$ denotes the support set labeled with class $c$ , while $\\textrm {\\textbf {x}}_i^c$ denotes its support images.", "Prototypical network (ProtoNet) [6] computes a $D$ -dimensional vector $p_c$ as the prototype for each class using an embedding function $f_\\phi : \\mathbb {R}^{H \\times W \\times 3} \\rightarrow \\mathbb {R}^{D}$ with learnable parameters $\\phi $ .", "A prototype is the mean vector of the embeddings belonging to its class: $ p_c = \\frac{1}{|S_c|}\\sum _{\\textrm {\\textbf {x}}_i^c \\in S_c} f_\\phi (\\textrm {\\textbf {x}}_i^c).$ ProtoNet classifies a query image $\\textrm {\\textbf {x}}^q$ as the class of the nearest neighboring prototype in the Euclidean distance metric: $ \\hat{y}^q = \\min _c \\Vert f_\\phi (\\textrm {\\textbf {x}}^q) - p_c \\Vert .$" ], [ "Proposed framework", "Figure REF shows an overview of our framework.", "We adopt the ProtoNet architecture to produce `prototype' class representations.", "During the embedding process, a feature map before back-end spatial pooling layers $\\textrm {FM}^{\\textrm {\\textbf {x}}} \\in \\mathbb {R}^{h \\times w \\times D}$ is retrieved for each image $\\textrm {\\textbf {x}}$ , where $h$ and $w$ denote the channels' height and width.", "Class objects are commonly visible among the support images.", "Thus, each support image would have regions highly correlated with the class representation.", "After the prototypes are derived, we can compute the similarity map $\\textrm {SM}^{\\textrm {\\textbf {x}}, c} \\in \\mathbb {R}^{h \\times w \\times 1}$ to highlight the class-relevant regions of each image by projecting its class prototype to every feature map coordinate: $ \\textrm {SM}^{\\textrm {\\textbf {x}}, c}_{(i,j)} = \\frac{p_c}{\\Vert p_c\\Vert } \\cdot \\textrm {FM}^{\\textrm {\\textbf {x}}}_{(i,j)} && \\textrm {for } (i,j) = (1,1), \\cdots , (h,w).$ $\\textrm {SM}^{\\textrm {\\textbf {x}}, c}$ is a scaled spatial decomposition of the cosine similarity between the image embedding and the prototype [13].", "Next, we upsample similarity maps to the original image size using bilinear interpolation and segment the regions of which the value is above a relative threshold $\\tau $ of the max value of each similarity map.", "Then we take a bounding box that covers the largest connected component.", "For every support image, we extract the RoI feature vector $f_\\phi ^c (\\textrm {\\textbf {x}}_i^c)$ from the bounding box of its corresponding class via RoIAlign [15] and pooling operations.", "We define a novel class representation $r_c$ as the mean vector of the RoI features belonging to its class.", "Our framework follows the classifying strategy in Eq.", "REF , replacing $p_c$ with our new representation $r_c$ ." ], [ "Loss function", "In ProtoNet, the logits are defined with the squared Euclidean distance to the prototypes and a temperature parameter $T$ .", "The embedding function is optimized with a cross-entropy loss: $L_{base} (y^q=c | \\textrm {\\textbf {x}}^q) = -\\log \\frac{\\exp {(-\\Vert f_\\phi (\\textrm {\\textbf {x}}^q) - p_c\\Vert ^2 / T)}}{\\sum _{k=1}^{N} \\exp {(-\\Vert f_\\phi (\\textrm {\\textbf {x}}^q) - p_k\\Vert ^2 / T)}} .$ Similarly, we propose a loss term $L_{roi}$ that considers our novel class representations instead of the prototypes: $L_{roi} (y^q=c | \\textrm {\\textbf {x}}^q) = -\\log \\frac{\\exp {(-\\Vert f_\\phi (\\textrm {\\textbf {x}}^q) - r_c\\Vert ^2 / T)}}{\\sum _{k=1}^{N} \\exp {(-\\Vert f_\\phi (\\textrm {\\textbf {x}}^q) - r_k\\Vert ^2 / T)}}.$ $L_{roi}$ has an explicit objective of distinguishing RoI features and allows the model to have more discriminative ability.", "Our final loss function is as follows: $L_{ours} = L_{base} + \\lambda _{roi} L_{roi},$ where $\\lambda _{roi}$ is a balancing parameter." ], [ "Setups", "We evaluated our framework on popular subsets of ImageNet [16] in few-shot learning studies; miniImageNet [5] and tieredImageNet [14] datasets.", "MiniImageNet includes 600 images for all 100 classes (i.e.", "total 60,000 images).", "We followed the conventional split by [17] and used 64 classes for training, 16 classes for validation, and 20 classes for testing.", "TieredImageNet is a large-scale dataset which considers the high-level category separation between training and testing classes.", "TieredImageNet consists of 351, 97, and 160 classes for training, validation, and testing, respectively.", "For both datasets, we evaluated our approach on 5-way 1-shot and 5-way 5-shot scenarios.", "We report the average classification accuracy and the 95% confidence interval after performing evaluation on 10,000 sampled tasks.", "Each task tests 15 query images per class (i.e.", "75 queries in total)." ], [ "Implementation details", "Our framework is built on the premise that the prototypes already have reliable representative power, so we trained the ProtoNet to its extent.", "We selected the ResNet-12 encoder used in [18] as our embedding function and initialized it following the pre-training strategy suggested in [19], [20].", "We appended a single layer classifier and trained it for the whole train-split classes using cross-entropy loss to initialize the embedding network's weights.", "Then we further optimized the feature extractor for 200 epochs with $L_{base}$ using the episodic training scheme [5].", "To be specific, we repeated random batches of few-shot classification tasks (the same scenarios used in evaluation) and trained parameters using SGD optimizer with the learning rate of 0.0001.", "The temperature parameter $T$ was set to 64, except the case in 5-shot scenario on tieredImageNet where we used $T$ = 32.", "Our framework proposes the class object bounding box given $\\tau $ = 0.5.", "We further optimized the encoder's parameters with $L_{ours}$ , setting $\\lambda _{roi}$ = 1.0, 0.5 for 1-shot and 5-shot scenarios, respectively.", "We used the $\\tau $ value of 0.7 instead of 0.5 while training 5-shot scenarios.", "Additional training takes 100 epochs, with the other hyperparameters unchanged." ], [ "Classification results comparison", "We compared our method with metric learning-based models that use different class representations, all using the ResNet-12 backbone structure.", "ProtoNet [6], our baseline work, shares the same weights with our framework.", "We also compared with cross attention network (CAN) [21], which shares a similar motivation to ours.", "CAN extracts a feature that attends to the target object by inspecting the spatial correlation between the query and support images.", "Table REF shows the classification performance comparison of our method on the miniImageNet and tieredImageNet.", "Our framework produces more discriminative class representations than the prototypes.", "Further optimization using the proposed loss function also enhanced the classification performances for both ProtoNet and our framework.", "This implies that training with the consideration of distinguishing RoI features improves the overall embedding quality.", "Our method outperformed prior approaches, except the case in the 5-way 1-shot task on tieredImageNet.", "CAN exploits the query image to identify the class object, and thus have strength in 1-shot scenarios.", "Our method produces a query-independent class representation that explicitly localizes and focuses on the target object." ], [ "Localization results analysis", " Figure: Localization the class objects from query images in a miniImageNet 5-way 5-shot scenario.Apart from the classification performance, we were concerned about our framework's localizing ability.", "Our framework produces WSOL predictions as intermediate outcomes.", "Once we predict a query label, we can also locate the most probable region of the class object using the similarity map between the query feature map and the representation vector of the predicted class.", "Since neither miniImageNet nor tieredImageNet provide ground-truth bounding box annotations, we were not able to assess the localization quality numerically.", "However, qualitative results in Fig.", "REF show that our framework can successfully localize the class objects even when there are multiple salient objects.", "Figure: Localization of foreground objects in support images in a miniImageNet 5-way 1-shot scenario.We further examined if our framework can localize the class object even when a single support image is given.", "The class object can be inferred as the common object among the support images that share the same label.", "In 1-shot scenarios, however, we cannot know which object the label indicates without additional information.", "Nevertheless, our model finds the region most highly correlated to its overall feature and gives plausible predictions when the foreground objects are large and significant.", "Figure REF shows some of the success cases of the 1-shot foreground object localization." ], [ "Conclusion", "In this paper, we investigate the representation learning in the few-shot realm and point out that average pooling of the whole image features can result in impure representations.", "To overcome this limitation, we propose a framework that localizes the class-relevant regions of the images with weak supervision using class prototypes and extracts a refined class representation for each class.", "We also introduce a loss function that directly addresses the refined features.", "Our approach outperforms the baseline work in various benchmarks.", "It also shows reasonable localization results without bounding box annotations." ] ]
2105.11715
[ [ "High-Frequency aware Perceptual Image Enhancement" ], [ "Abstract In this paper, we introduce a novel deep neural network suitable for multi-scale analysis and propose efficient model-agnostic methods that help the network extract information from high-frequency domains to reconstruct clearer images.", "Our model can be applied to multi-scale image enhancement problems including denoising, deblurring and single image super-resolution.", "Experiments on SIDD, Flickr2K, DIV2K, and REDS datasets show that our method achieves state-of-the-art performance on each task.", "Furthermore, we show that our model can overcome the over-smoothing problem commonly observed in existing PSNR-oriented methods and generate more natural high-resolution images by applying adversarial training." ], [ "Introduction", "Most learning-based methods utilize the high capacity of deep neural networks with remarkable ability to understand the content and style of the image that they have shown in visual recognition tasks, including image classification and object detection.", "Using these high capacities and analytic powers of deep neural networks, learning-based methods have been successfully adapted to the field of image enhancement and have shown better performances compared to traditional model-based methods in laboratory environments.", "When applied to real-world problems, however, most learning-based methods have failed to produce such good results while model-based methods are more flexible and applicable to low-resolution images with various kinds of blur and noises.", "This is because learning-based methods learn how to enhance the quality of images only by analyzing relations between given pairs of low-resolution images and their corresponding high-resolution ones in the training phase.", "However, in real-world problems, only low-resolution images are given and their high-resolution pairs are unknown.", "This means that the models have to infer new relations that they have never learned, which often leads to huge performance degradation when they solve real-world problems.", "Another problem called the “ill-posed problem” also makes solving real-world problems more challenging; there are countless high-resolution image candidates in solution spaces corresponding to a given low-resolution image, while the number of high-resolution outputs human viewers perceive natural is very small or unique.", "The ill-posed problem makes it very difficult for deep neural networks to derive natural high-resolution outputs when solving real-world problems.", "Research on mathematical ways to reduce the solution spaces in unsupervised environments has been recently proposed to deal with the problem.", "Many studies on the architectural design of deep neural networks have been proposed over the years and have shown great performances.", "However, they have recently reached the limit; little progress has been made except for marginal improvements on performances.", "This is because deep neural networks are originally optimized for understanding the content of images based on the high capacity of deeply stacked layers, so they are less capable of interpreting and restoring detailed information of corrupted images.", "Accordingly, recent studies are more focused on conveying mathematical properties of images to the existing models rather than designing deeper networks.", "In keeping with this trend, we not only propose novel architectures of deep neural network for image enhancement problems but also introduce some state-of-the-art model-agnostic methods to make networks capable of producing sharper and more realistic images by providing abstract characteristics and high-frequency components of images with a little modification in the structure of existing models." ], [ "Related work", "In recent years, many studies have been proposed to solve the SISR problem using different deep-learning techniques.", "In 2015, Dong et al.", "[1] introduced deep learning methods into the SISR problem, proposing SRCNN that is a fully convolutional neural network that enables end-to-end mapping between input and output images.", "In 2016, Kim et al.", "[2] proposed VDSR that utilizes contextual information spread over large patches of images using large receptive fields to convolutional layers.", "In 2017, Tai et al.", "[3] proposed a very deep network structure consisting of 52 convolutional layers called DRRN by designing a recursive block with a multi-path structure while Ledig et al.", "[4] proposed SRResNet with 16 blocks of deep ResNet and also introduced GAN-based SRGAN which is optimized for perceptual loss calculated on feature maps of the VGG [5] network.", "In 2017, Lim et al.", "[6] proposed a novel model named EDSR.", "They removed every batch normalization from their network and stacked 16 residual blocks, which extracts high-frequency information from low-resolution images.", "In the same year, Tong et al.", "[7] proposed SRDenseNet, which consists of 8 dense blocks [8] and skip connections that combine feature maps from different levels.", "In 2018, Zhang et al.", "[9] introduced a residual dense block that allows direct connections from preceding blocks, leading to a continuous memory mechanism.", "Zhang et al.", "[10] also proposed a novel model called RCAN, which added channel attention to EDSR and introduced a Residual in Residual module to construct a 10 times deeper network.", "They used skip connections with various lengths to help their model separately extract abundant low-frequency features and scarce but important high-frequency information from low-resolution images.", "Until 2019, studies have mainly focused on modifying networks' architectural design by introducing or combining various kinds of neural blocks.", "However, as the neural networks became sufficiently deep and wide, structural modifications alone could expect nothing but only small marginal improvements.", "To overcome such issues, researchers have recently focused on the intrinsic limitations of the SISR problem or attempted to combine their neural networks with traditional model-based methods.", "In 2020, Guo et al.", "[11] introduced cycle consistency to their network to solve the intrinsic ill-posed problem; there are infinite high-resolution images that can be downsampled to the given low-resolution input images.", "They reconstructed the RCAB proposed by RCAN [10] into a UNet [12] structure.", "In this process, they also produced images with $1/2$ and $1/4$ size of the target resolution from the low-resolution inputs, and then compared them with downscaled output images.", "Through this process, which is named dual regression, they could maintain cycle consistency and enable their networks trained with unlabeled data at the same time.", "Pan et al.", "[13] constrained their network with input information by utilizing a pixel substitution scheme from low-resolution images.", "They added degraded image blurred by known blur kernel to the input image and forwarded them iteratively into the deblurring network.", "From this process, they tried to convert a given difficult blind kernel problem to an easy non-blind problem so that their model can restore sharp images more easily.", "Instead of solving the ill-posed problem by giving cycle consistency to the network with constraint from input information, several attempts have been proposed to create a human interpretable network structure by applying meaningful kernel to the convolutional layers of the network.", "Huang et al.", "[14] introduced a Multi-Scale Hessian Filtering (MSHF) consisting of kernels that extract edges from multi-scale, leading their model to approach the high-frequency information of images from different angles and scales.", "On the other hand, Shang et al.", "[15] uses rectangular-shaped receptive fields such as $1\\times 3$ or $3\\times 1$ in parallel rather than randomly initializing $3\\times 3$ convolutional kernels.", "In this way, their model, named RFB-ESRGAN, becomes human interpretable and could adaptively analyze both horizontal and vertical information of images.", "Figure: Our proposed networkA study has also been proposed to apply knowledge distillation to the SISR problem to enable models to use the rich information in high-resolution images during the training phase.", "Lee et al.", "[16] forward the encoded feature of HR images to the teacher network, which shares the same structure as the student network, allowing the teacher network to use privileged information to obtain better outcomes.", "The student network then used variational information distillation [17] technique that allows the teacher network to distill their encoded features to the student network so it can learn how to extract privileged information, allowing the model to extract more meaningful features from a given low-resolution input.", "As EDSR [6] and RCAN [10] separately extract shallow and deep features from the image on RGB color space, a study that tried to take a step further from color domain to frequency domain and decompose high frequency and low-frequency information has been proposed.", "Pang et al.", "[18] split input images into high, medium, and low frequencies and passed them to the network individually, and then aggregated each convoluted feature map adaptively to generate high-resolution images.", "However, instead of using mathematical methods such as FFT or DWT, they simply divided the frequency domain using three convolutional layers, which is easy to fail to extract valid and meaningful frequency information." ], [ "Multi-scale Edge Filtering", "Successful super-resolution requires the understanding of structure of images.", "In particular, we need to separate high-frequency and low-frequency regions and make adaptively appropriate analyses for each region to successfully detach the noise map from the original image.", "This is because the distribution of pixel value appears different in each region; pixels in high-frequency regions often have large variance while smaller variances are more observed in low-frequency areas.", "We propose a module that extracts edges from given images to obtain information about high-frequency areas.", "The obtained information is transferred to the network and used to increase restoring performance by focusing more on high-frequency regions that are difficult to reconstruct.", "The module consists of convolutional layers initialized with pre-defined filters, making the back-propagation scheme possible and enabling end-to-end optimization when the network is training the data." ], [ "Feature Attention Module", "RCAN [10] achieved better results by adding channel attention to residual blocks from EDSR [6].", "Figure REF (a) illustrates the concept of channel attention.", "The channel attention takes a vector pooled from feature maps as input and feed-forward it through a series of convolutional layers.", "Here, the layers give us weights for each channel by operating dot products for local channel-wise regions from the average pooled vector.", "This process allows the network to determine the importance between channels in the feature map and focus on channels with more information.", "Figure: Feature Attention ModuleEDSR and RCAN restore images using feature maps obtained by summing the shallow features and deep features.", "However, as they simply added two features, they have failed to consider the relative importance of shallow and deep features.", "Since shallow and deep features contain different kinds of information, such as low and high-frequency, their importance cannot be the same.", "Also, the characteristic of given image changes which feature contains more information.", "Therefore, it is necessary to introduce a module that identifies the characteristics of given images and determines each importance before adding feature maps with different information.", "To solve this problem, we introduce the Feature Attention Module.", "Before feature maps are added, the relative weights of importance are estimated by our Feature Attention Module.", "We calculated weight of importance in a vector form with the dimension of the channel in feature maps, considering that each channel has different importance.", "Figure REF shows the structure of our feature attention module.", "First, we concatenate vectors from each feature map by using the global average pooling layer.", "Then we pad the stack of vectors and feed them into convolution in the feature-wise direction rather than the channel-wise way.", "We padded the vectors to maintain the output dimension and make the convolution to compute every feature evenly.", "By multiplying each feature map in an element-wise way, we could finally obtain a weighted sum of features depending on their importance." ], [ "High-Pass Filtering Loss", "The concept of perceptual loss was first introduced by Johnson et al.", "[19] who tried to solve the image transformation problem by comparing content and style discrepancies between two images.", "They used VGG-16 [5] pre-trained for image classification as the loss network and measured perceptual differences of output and ground-truth images.", "Motivated by their method, we propose a loss function that compares feature differences in the high-frequency domain instead of comparing per-pixel differences in a color space.", "The commonly used perceptual loss uses the VGG-16 network pre-trained on ImageNet dataset.", "However, the network trained on image classification is optimized to extract feature representation that contains information about the class of objects in images.", "The objective of network is to figure out what objects are in the image, not to extract detailed patterns or complex high-frequency information.", "What we need, however, is neural networks that extract such detailed patterns and high-frequency information and feature representations of the networks that are needed to extract such information.", "Because the commonly used perceptual loss is not appropriate to our problem solving, we have trained a neural network that is optimized to high-frequency extraction.", "We first extracted high-frequency signals by applying a high-pass filter to the image transformed into the frequency domain via Fast Fourier Transform.", "Then, we trained a simple three-layered CNN, or high-pass filtering network, which takes images as input and generates high-pass filtered signals.", "Figure REF shows an example of high-frequency signals extracted by a traditional high-pass filter using FFT and our high-pass filtering network.", "Figure REF shows a visualization of feature maps produced by intermediate layers of the network during extracting high-frequency signals.", "We utilized this high-pass filtering network as the loss network and defined the high-pass filtering loss function as following: $\\mathcal {L}_{hf}(I_{SR}, I_{HR}) = \\mathcal {L}_{0}^{\\phi }(I_{SR}, I_{HR}) + \\mathcal {L}_{1}^{\\phi }(I_{SR}, I_{HR})$ where $\\phi $ denotes the high-pass filtering network.", "The loss network $\\phi $ analyzes images from various perspectives to generate high-frequency signals where intermediate layers give us abstract feature maps, including edges.", "We measure the feature difference of $I_{SR}$ and $I_{HR}$ by feed-forwarding two images to fixed $\\phi $ where the feature difference is trainable by back-propagation as it is generated through convolutional layers.", "By minimizing the high-pass filtering loss, high-frequency features are added to the $I_{SR}$ , allowing us to obtain sharper images." ], [ "Soft Gradient Magnitude Similarity Map Masking", "The local perceptual quality of output images often varies by region.", "In general, the lower perceptual quality is more observed in areas containing detailed and irregular patterns, but these results depend on which model is used.", "To learn models that perform evenly, we need to know which part of the resulting image is poor, and therefore more training is needed.", "Here, we adopted the Gradient Magnitude Similarity (GMS) map [20] to evaluate the local quality of images.", "The gradient magnitude of given image $I$ is computed as follows: $GM(I) = \\sqrt{(I\\ast G_x)^2 + (I\\ast G_y)^2} $ where $G_x$ and $G_y$ denote prewitt filters.", "With the gradient magnitudes of $I_{HR}$ and $I_{SR}$ , we compute the GMS map as follows: $GMS(I_{HR}, I_{SR}) = 1 - \\frac{2 GM(I_{HR}) GM(I_{SR}) + c}{GM(I_{HR})^2 + GM(I_{SR})^2 + c} $ where we set $c=170$ for pixel values in $[0,255]$ .", "Note that the value of the GMS map is closer to zero where two images are similar while it is closer to one where two images are different.", "To give information about which area is more damaged and thus training should be weighted to the loss function, we multiply $I_{HR}$ and $I_{SR}$ with the GMS map before we put them into our loss functions.", "However, since the GMS map is calculated pixel-wise, it can be computed high for some lucky locations with similar pixel values even where they are contained in severely corrupted regions.", "So we first binarized the GMS map and then remove tiny or trivial regions using the opening which is defined as erosion followed by dilation.", "In the mathematical morphology, the opening of a binary image $A$ by the structuring element $B$ is expressed as follows: $\\mbox{\\textit {Erosion:}}\\quad A \\ominus B &= \\bigcap _{b\\in B}A_{-b} \\\\ \\mbox{\\textit {Dilation:}}\\quad A \\oplus B &= \\bigcup _{b\\in B}A_{b} \\\\\\mbox{\\textit {Opening:}}\\quad A \\circ B &= (A\\ominus B)\\oplus B $ where $A_b$ denotes the translation of $A$ by $b$ .", "The opening is often applied to coarse images to remove pixel-wise outliers and make them locally smooth.", "Here, adopting the opening to the coarse GMS map allows us to eliminate pixel noise and acquire more smooth labels while maintaining information about the locally damaged area inside the image.", "Two images on left side of Figure REF shows an visual example of applying the opening to GMS map.", "We can obseved that the map distinguishes between well-reconstructed and poorly-reconstructed area smoother when the opening followed by thresholding is applied to the coarse GMS map.", "Here, we use the opened-binarized GMS map, or the hard GMS map, to mask images to let our network re-train only on poorly-reconstructed areas.", "The hard GMS map assigns each pixel a hard label whether to train or not.", "In practice, however, it is more reasonable to express with score or probability of how much pixel should be trained.", "Therefore, we transform the discretized hard GMS map into soft GMS map so it represent the pixel-wise score.", "To soften the hard GMS map, we smoothed the boundaries between different regions within the hard GMS map by applying blurring with isotropic Gaussian kernel and additional image opening to remove outliers in an iterative manner.", "In the soft GMS map, pixels at the center of well or poorly-reconstructed area have more confident score close to 0 or 1, respectively, while scores close to 0.5 are assigned to pixels if they are close to boundaries.", "Figure REF show examples of masked results using the hard and the soft GMS maps." ], [ "Network Architecture", "Figure REF shows the structure of our proposed network.", "Our deblurring model first resizes image into three different scales, then extracts low-level and high-level features through its head and body for each scale, respectively.", "The heads of the network consist of one convolutional layer each, like our denoising model, while the bodies are composed of a different number of Residual Channel Attention Blocks depending of the scale; from largest to smallest scale, each body consists of 4, 16, and 64 blocks, respectively.", "Here, we let bodies on smaller scales have more blocks with deeper layers because they use less GPU memory as their computation is relatively lower.", "To take full advantage of deeply stacked bodies on smaller scale, we combined low-level features from smaller scales with those from larger scales through our feature attention module before we put them into each body.", "This allows deeper bodies for smaller scales take more diverse features and generate richer high-level features.", "After the bodies extract high-level features, we combine high-level features with low-level and multi-scale edge filtering module for each scale by feature attention.", "Here, we upscale and combine high-level features from smaller scale to larger scales, which allows the tails for larger scale take more features from diverse scales.", "To send features to different scales, we use strided convolution for downscale and pixel shuffle to upscale the features." ], [ "Training Detail", "In the training phase, we trained our model for 800 epochs for small image patches and 20 epochs for large image patches with Adam optimizer and initial learning rate $10^{-4}$ with learning rate decay by $0.99$ by every $1,000$ steps.", "For each iteration, 16 batches with $192\\times 192$ sized cropped patches from large images were used for epochs with small images where one batch with $320\\times 180$ sized input image and $1280\\times 720$ sized target image was used for epochs with large images.", "Lastly, L1 function was used for the loss function." ], [ "Experimental Result", "In this section, we introduce the results of our model applied to Image Denoising, Image Deblurring, and Single Image Super-Resolution." ], [ "Image Denoising", "We first applied our model to a synthetic noisy dataset generated by adding white Gaussian noise with $\\sigma =10$ and 30 to DIV2K dataset, respectively.", "In the training phase, we optimized our model for every variance of the noise at once.", "Using different kinds of noise together, our model has become flexible to more diverse noise levels.", "Table REF shows a comparison of the results of our model and other learning-based models using PSNR and SSIM scores.", "Our proposed model proved the best performance in most cases.", "Furthermore, we used the Smartphone Image Denoising Dataset, which is often called SIDD [21] to evaluate our model on real noisy images.", "This dataset consists of pairs of real noisy images taken under various conditions using smartphone cameras and ground-truth images, of which defective pixels are corrected manually.", "We showed that our model could solve real-world denoising problems by training and evaluating our model on the SIDD dataset.", "Table REF shows a comparison of the results of our model and other learning-based models on the SIDD dataset.", "Figure: Our Image Denoising results.Table: Comparison of denoising results in PSNR and SSIM scores on DIV2K + AWGN dataset.", "Best scores marked in bold.Table: Comparison of denoising results in PSNR and SSIM scores on SIDD dataset.", "Best scores marked in bold.Figure: Our Image Deblurring results.Figure: Visual comparison of PSNR-oriented and Perceptual methods" ], [ "Image Deblurring", "We first trained and evaluated our deblurring model on Flickr2K dataset [6].", "While REDS consists of similar images from several daily videos, Flickr2K contains different images with various objects and detailed patterns.", "Therefore, it is suitable for an extensive experiment to show that our model can be applied to images with more diverse information.", "To create blurry images in various conditions, we applied randomly chosen blur kernels from set of isotropic and anisotropic Gaussian kernels of various sizes and angles to randomly cropped and rotated patches.", "By augmenting the blurry images, our model could observe and learn the various blurring conditions on the limited images.", "Table REF shows our model achieves the state-of-the-art results on Flickr2K dataset.", "Table REF shows our experimental results on the REDS dataset from “NTIRE 2021 Image Deblurring Challenge - Track2.", "JPEG Artifacts”.", "We also provide the results of ablation studies that evaluate the effect of our Multi-Scale Edge Filtering and Feature Attention Module.", "Ablations studies show that our proposed model achieves top scores at PSNR, and SSIM.", "Considering that PSNR measures absolute errors and SSIM measures the perceived change in structural information, based on luminance and contrast of images, it can be inferred that the Feature Attention Module helps the model understand the structural information.", "Figure REF shows some selected deblurring results of our proposed model on REDS dataset.", "Our model successfully reconstruct objects that are difficult to identify from the blurry image to identifiable levels.", "Table: Comparison of deblurring results in PSNR and SSIM scores on Flickr2K dataset.", "Best scores marked in bold.Table: Comparison of deblurring results in PSNR and SSIM scores on REDS - JPEG dataset.", "Best scores marked in bold." ], [ "Single Image Super-Resolution", "Compared to other blind SISR methods, our proposed net achieves higher PSNR and SSIMS scores while our proposed GAN produces perceptually more natural results.", "Figure REF shows detailed results of our models.", "Table: Comparison of SISR results on DIV2K dataset.", "Best scores marked in bold.Table: Comparison of SISR results on REDS dataset.", "Best scores marked in bold." ], [ "Conclusion", "This paper introduces multi-scale edge filtering that extracts high-frequency information from noisy images.", "This helps our model perform statistical analysis and reconstruction suitable for each region of the image.", "We also add feature attention modules to enable the network to determine feature maps containing more important information.", "Also, we introduce a high-pass filtering loss function that compares feature maps generated from a high-pass filtering network computing the high-frequency information of the results and ground truth images.", "Finally, our soft GMS masking helps the model identify which areas of the resulting image are more compromised and need to be more focused on additional training processes.", "Experimental results show that our model can achieve state-of-the-art PSNR and SSIM scores compared to other learning-based methods.", "However, when visualizing the results, over-smoothing problems have been observed as in other PSNR-oriented methods.", "Adversarial training was applied to pre-trained models using a discriminator that distinguishes real and synthetic images, allowing the model to generate much more natural images.", "In future research, we will study learning-based methods that achieve superior scores in both PSNR and LPIPS by enabling the model to extract sufficient information from low-resolution images." ] ]
2105.11711
[ [ "On Enhancing Ground Surface Detection from Sparse Lidar Point Cloud" ], [ "Abstract Ground surface detection in point cloud is widely used as a key module in autonomous driving systems.", "Different from previous approaches which are mostly developed for lidars with high beam resolution, e.g.", "Velodyne HDL-64, this paper proposes ground detection techniques applicable to much sparser point cloud captured by lidars with low beam resolution, e.g.", "Velodyne VLP-16.", "The approach is based on the RANSAC scheme of plane fitting.", "Inlier verification for plane hypotheses is enhanced by exploiting the point-wise tangent, which is a local feature available to compute regardless of the density of lidar beams.", "Ground surface which is not perfectly planar is fitted by multiple (specifically 4 in our implementation) disjoint plane regions.", "By assuming these plane regions to be rectanglar and exploiting the integral image technique, our approach approximately finds the optimal region partition and plane hypotheses under the RANSAC scheme with real-time computational complexity." ], [ "INTRODUCTION", "This template provides authors with most of the formatting specifications needed for preparing electronic versions of their papers.", "All standard paper components have been specified for three reasons: (1) ease of use when formatting individual papers, (2) automatic compliance to electronic requirements that facilitate the concurrent or later production of electronic products, and (3) conformity of style throughout a conference proceedings.", "Margins, column widths, line spacing, and type styles are built-in; examples of the type styles are provided throughout this document and are identified in italic type, within parentheses, following the example.", "Some components, such as multi-leveled equations, graphics, and tables are not prescribed, although the various table text styles are provided.", "The formatter will need to create these components, 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2105.11649
[ [ "Discovery of two infrared objects with strong ice absorption in the\n AKARI slit-less spectroscopic survey of the Galactic Plane" ], [ "Abstract We discover two infrared objects that show deep absorption features of H2O, CO2, and CO ices in the AKARI/Infrared Camera (IRC) slit-less spectroscopic survey of the Galactic plane in 2.5--13 micron.", "Both objects are located neither in known star-forming regions nor in known dense clouds.", "For one of the objects, Object 1, we successfully extract a spectrum from 2.5 to 13 micron, which also shows several absorption features in 5--13 micron, including deep silicate absorption at 10 micron.", "For the other object, Object 2, only a spectrum from 3.1 to 5 micron is reliably extracted due to the presence of nearby overlapping objects and faint nebulosity.", "Both objects show warm (>100 K) CO gas absorption in addition to the ice absorption features, suggesting that they are embedded young stellar objects (YSOs).", "On the other hand, both objects have spectral energy distributions (SEDs) that peak at around 5 micron and decrease towards longer wavelengths.", "These characteristics of the SEDs and the presence of deep absorption features cannot easily be accounted for by standard YSO models.", "They may be explained as background stars behind dense clouds.", "We discuss possible nature of the objects and implications of the present discovery." ], [ "Introduction", "In cold, dense regions, various kinds of ice species are formed and play significant roles in the interstellar chemistry as well as in the formation of stars and planetary systems [12].", "While Infrared Space Observatory (ISO) provided an extensive spectroscopic database of ice absorption features for massive young stellar objects [44], [30] and CO$_2$ ice properties in the Taurus dark cloud [102], [65], Spitzer and AKARI as well as ground-based spectroscopy extended the study of ices to low-mass young stellar objects (LYSO) and the various ice species in several dense clouds [13], [14], [15], [9], [57], [76], [77], [66], [1], [63], [64].", "Theoretical and experimental studies suggest that ices are formed on the grain surface via diffusive surface reactions or energetic processes such as photolysis or radiolysis [49], [26], [61].", "Although the formation and evolution processes of ice species in dense regions are not yet fully understood, the profiles of absorption features in the infrared provide us with valuable information on the thermal and energetic processes imposed upon ices as well as on the nature of the objects associated with ice absorption [75], [66], [63], [12].", "The presence of ice absorption is also thought to be a reliable indicator for the identification of young stellar objects [98], [85], [86], [83], which is difficult to be made unambiguously solely by infrared photometric observations because some galaxies and dusty evolved stars have photometric characteristics similar to YSOs [47], [46], [7], [55], [58].", "H$_2$ O, CO$_2$ , CO, and CH$_3$ OH ices are known to be the major ice species in YSOs and dense clouds, and they have the major bands at 3.0, 6.0, 13.1 (H$_2$ O), 4.67 (CO), 4.26, 15.2 $\\mu $ m (CO$_2$ ), 3.53, 6.76, 8.9, and 9.75 $\\mu $ m (CH$_3$ OH) [43], [44].", "Other complex ice species also have characteristic bands in 4.6–10 $\\mu $ m [44], [12].", "Therefore, the near-infrared (NIR) and mid-infrared (MIR) spectroscopy is an efficient means to study the properties of interstellar ices as well as the nature of objects associated with ice absorption.", "In this paper, we report discovery of two interesting infrared objects that show strong absorption bands of H$_2$ O ice at 3.0 $\\mu $ m, CO$_2$ ice at 4.26 $\\mu $ m, and CO ice at 4.67 $\\mu $ m based on a NIR to MIR slit-less spectroscopic survey (2.5–13 $\\mu $ m) of the Galactic plane carried out with the Infrared Camera (IRC) onboard the AKARI satellite [70].", "Both objects are located neither in known star-forming regions nor in known dense clouds.", "They could be run-away YSOs [33], [78], which elude past YSO surveys, or located behind unknown compact, dense clouds.", "In §, the observations and data reduction are described, and the results are presented in §.", "Analysis of the spectra is given in §.", "Possible identification of the nature of the objects is discussed in § and a summary is given in §." ], [ "Observations and data reduction", "The present observations were carried out in the slit-less spectroscopic survey mode with the grisms of the IRC, which had a field-of-view of $10\\times 10$ and obtained spectra of point sources for 2.5–13 $\\mu $ m [67], as part of the program of the Interstellar Medium in our Galaxy and Nearby galaxies [54].", "Spectra of 2.5–5 $\\mu $ m were taken with the NIR channel of the IRC in the NIR Grism (NG) mode, while those of 5–13 $\\mu $ m were obtained with the MIR-S channel in the Short-MIR Grism 1 (SG1, 5.0–8.2 $\\mu $ m) and Short-MIR Grism 2 (SG2, 7.6–13.0 $\\mu $ m) modes.", "The spectral resolutions are about 0.03, 0.12, and 0.21 $\\mu $ m for the NG, SG1, and SG2 modes, respectively [67].", "We surveyed nine regions in the Carina arm, eight in the Crux arm, and five in the Perseus arm.", "Table  summarizes the positions, IDs, and dates of the observations.", "lclrl Observation log of the slit-less spectroscopic survey of the Galactic plane 0pt Target name Observation ID 2cPositiona Observation date 2cR.A.", "(J2000.0) Dec. CAR-079_O001 1402304.1 09$^{\\mathrm {h}}$ 53$^{\\mathrm {m}}$ 29$^{\\mathrm {s}}$ -56$$ 14$$ 00$$ 2007 June 29 CAR-079TO002 1400201.1 09$^{\\mathrm {h}}$ 55$^{\\mathrm {m}}$ 42$^{\\mathrm {s}}$ -55$$ 50$$ 29$$ 2006 December 29 CAR-079TS004 1400203.1 09$^{\\mathrm {h}}$ 57$^{\\mathrm {m}}$ 53$^{\\mathrm {s}}$ -55$$ 26$$ 49$$ 2006 December 29 CAR-079TO003 1400197.1 10$^{\\mathrm {h}}$ 01$^{\\mathrm {m}}$ 24$^{\\mathrm {s}}$ -56$$ 27$$ 04$$ 2006 December 31 CAR-079_S002 1402301.1 10$^{\\mathrm {h}}$ 04$^{\\mathrm {m}}$ 19$^{\\mathrm {s}}$ -56$$ 44$$ 58$$ 2007 July 02 CAR-057TO005 1400229.1 13$^{\\mathrm {h}}$ 13$^{\\mathrm {m}}$ 55$^{\\mathrm {s}}$ -62$$ 45$$ 34$$ 2007 February 06 CAR-057TO003 1400225.1 13$^{\\mathrm {h}}$ 14$^{\\mathrm {m}}$ 18$^{\\mathrm {s}}$ -63$$ 15$$ 27$$ 2007 February 07 CAR-057TT001 1400231.1 13$^{\\mathrm {h}}$ 20$^{\\mathrm {m}}$ 44$^{\\mathrm {s}}$ -65$$ 11$$ 49$$ 2007 February 06 CAR-057TS002 1400215.1 13$^{\\mathrm {h}}$ 22$^{\\mathrm {m}}$ 35$^{\\mathrm {s}}$ -62$$ 39$$ 19$$ 2007 February 10 CRU-048TO003 1400259.1 14$^{\\mathrm {h}}$ 03$^{\\mathrm {m}}$ 25$^{\\mathrm {s}}$ -61$$ 10$$ 09$$ 2007 February 12 CRU-048TO002 1400257.1 14$^{\\mathrm {h}}$ 04$^{\\mathrm {m}}$ 34$^{\\mathrm {s}}$ -61$$ 38$$ 59$$ 2007 February 13 CRU-048TO004 1400261.1 14$^{\\mathrm {h}}$ 07$^{\\mathrm {m}}$ 23$^{\\mathrm {s}}$ -61$$ 01$$ 38$$ 2007 February 13 CRU-048TS001 1400247.1 14$^{\\mathrm {h}}$ 09$^{\\mathrm {m}}$ 57$^{\\mathrm {s}}$ -61$$ 59$$ 24$$ 2007 February 14 CRU-048TS003 1400251.1 14$^{\\mathrm {h}}$ 13$^{\\mathrm {m}}$ 53$^{\\mathrm {s}}$ -61$$ 49$$ 49$$ 2007 February 15 CRU+032_SP05 1401027.1 18$^{\\mathrm {h}}$ 48$^{\\mathrm {m}}$ 41$^{\\mathrm {s}}$ -00$$ 14$$ 45$$ 2007 April 02 CRU+032_S003 1401033.1 18$^{\\mathrm {h}}$ 59$^{\\mathrm {m}}$ 30$^{\\mathrm {s}}$ -01$$ 44$$ 43$$ 2007 April 06 CRU+032_SP01 1401025.1 18$^{\\mathrm {h}}$ 59$^{\\mathrm {m}}$ 32$^{\\mathrm {s}}$ -01$$ 36$$ 49$$ 2007 April 06 PER+070_S001 1401019.1 20$^{\\mathrm {h}}$ 04$^{\\mathrm {m}}$ 51$^{\\mathrm {s}}$ 29$$ 12$$ 08$$ 2007 May 02 PER+070_O001 1401017.1 20$^{\\mathrm {h}}$ 10$^{\\mathrm {m}}$ 45$^{\\mathrm {s}}$ 32$$ 35$$ 26$$ 2007 May 06 PER+070_S002 1402365.1 20$^{\\mathrm {h}}$ 12$^{\\mathrm {m}}$ 59$^{\\mathrm {s}}$ 33$$ 30$$ 10$$ 2007 May 08 PER+070_O003 1402370.1 20$^{\\mathrm {h}}$ 14$^{\\mathrm {m}}$ 02$^{\\mathrm {s}}$ 32$$ 43$$ 58$$ 2007 May 08 PER+070_O002 1402369.1 20$^{\\mathrm {h}}$ 15$^{\\mathrm {m}}$ 21$^{\\mathrm {s}}$ 33$$ 08$$ 57$$ 2007 May 08 aThe intended center position of the field-of-view ccc 2MASS ID, position, and photometric data of the objectsa 0pt Properties/Band Object 1 Object 2 2MASS ID J14041323-6112401 J14042016-6115495 R.A. (J2000.0) DEC 14$^{\\mathrm {h}}$ 04$^{\\mathrm {m}}$ 13.2$^{\\mathrm {s}}$   -61$$ 12$$ 40$$ 1 14$^{\\mathrm {h}}$ 04$^{\\mathrm {m}}$ 20.2$^{\\mathrm {s}}$   -61$$ 15$$ 49$$ 5 $J$ $> 16.712$ $> 17.548 $ $H$ $14.914 \\pm 0.11$ $> 16.880$ $K_\\mathrm {s}$ $10.346 \\pm 0.023$ $11.730 \\pm 0.026$ $W1$ $8.755 \\pm 0.023$ $9.600 \\pm 0.024$ $W2$ $6.828 \\pm 0.020$ $7.903 \\pm 0.020$ $W3$ $6.699 \\pm 0.034$ $7.216 \\pm 0.040$ $W4$ $5.772 \\pm 0.066$ $> 4.615$ b $[3.6]$ $7.676 \\pm 0.041$ $8.493 \\pm 0.040$ $[4.5]$ $6.592 \\pm 0.056 $ $7.888 \\pm 0.037$ $[5.8]$ $6.050 \\pm 0.036$ $7.151 \\pm 0.037$ $[8]$ $5.925 \\pm 0.030$ $7.170 \\pm 0.029$ $[24]$ $4.65 \\pm 0.02$ $6.00 \\pm 0.33$ AKARI 9 $\\mu $ m (mJy) $119.3 \\pm 14.6$ $< 50$ aPhotometric data are given in magnitude except for AKARI 9 $\\mu $ m, which is in units of mJy.", "The WISE data are taken from the ALLWISE catalog and the IRAC and MIPS data are from the GLIMPSE and MIPSGAL catalogs.", "The AKARI data are taken from the JAXA/ISAS server (see text).", "bWe set this value as a lower limit because of the presence of faint nebulosity (see §).", "Figure: Locations of the two objects that show strong ice absorption features in the AKARI slit-less survey.The background is the WISE band 3 (12 μ\\mu m) image.The yellow dashed rectangle shows the field-of-view of the IRC observations (10×1010\\times 10), while the white arrow indicates the upward dispersion directionin the slit-less spectrum images shown in Figure .", "The yellow and red circles show the locations of Object 1 and Object 2, respectively.The yellow plus indicates an AGB star candidate , the red pluses show H2 regions , and the blue pluses displaystar-forming regions .Figure: Slit-less images of the present observations.", "(a): NG image (2.5–5 μ\\mu m), (b): MIR-SG1 image (5.0–8.2 μ\\mu m), and (c): MIR-SG2 image (7.6–13.0 μ\\mu m).The yellow and red arrows indicate Object 1 and Object 2, respectively.The wavelength decreases upward in the NG image, while increases in the SG1 and SG2 images.The two black stripes seen in the NG spectra of the two objects in (a) correspond to the absorption bands of CO 2 _2 ice at 4.26 μ\\mu m and CO ice at 4.67 μ\\mu m. For Object 1,the broad absorption of H 2 _2O ice at 3.0 μ\\mu m is also visible, while that of Object 2 is not clearly seen due to the nearby bright source.Two objects are found in one of the Crux arm regions (target name of CRU-048TO003), which show deep absorption of CO$_2$ ice at 4.26 $\\mu $ m and CO ice at 4.67 $\\mu $ m in their slit-less spectra of the NIR channel by visual inspection.", "We call the two objects Object 1 and Object 2 in the following.", "No similar objects that show strong ice absorption were found in the other 22 regions by visual inspection.", "Their locations are indicated in Figure REF on the WISE band 3 (12 $\\mu $ m) image together with nearby H2 regions (red pluses), star-forming regions (blue pluses), and an Asymptotic Giant Branch (AGB) star candidate (yellow plus).", "Figure REF shows that Object 1 is located in a moderately dark region in the MIR andin a region that is associated neither with bright infrared sources nor with strong diffuse emission in the MIR.", "No appreciable dark clouds are seen in the optical extinction map [37], while Object 2 is located in a region associated with faint diffuse emission.", "Nearby star-forming activities are located at $> 4$ south of the two objects.", "The slit-less spectrum images are shown in Figure REF .", "The NG image (Figure REF a) shows the presence of two black stripes that correspond to the CO$_2$ ice and CO ice absorption bands in both objects.", "For Object 1, the broad H$_2$ O ice absorption at 3.0 $\\mu $ m is also recognized.", "Both objects are listed in the point source catalogs of the Two Micron All Sky Survey (2MASS) [87], the Wide-field Infrared Survey Explorer (WISE) [104], the Spitzer Galactic Legacy Infrared Mid-Plane Survey Extraordinaire (GLIMPSE) survey [8], [22], and the Spitzer MIPS Galactic (MIPSGAL) survey [19].", "Object 1 is also listed in the point source catalog of AKARI MIR all-sky survey [53], while Object 2 is not.", "We confirmed that Object 2 had been observed in the AKARI MIR all-sky survey and we gave the nominal upper limit of 50 mJy for it at 9 $\\mu $ m [53].", "In the following, we denote WISE band 1–4 data as $W1$ –$W4$ , IRAC band 1–4 data as [3.6], [4.5], [5.8], and [8], and MIPS 24 $\\mu $ m data as [24] in magnitude.", "The positions, 2MASS IDs, and photometric data of the two objects are summarized in Table , where the WISE data are taken from the ALLWISE catalog and the IRAC and MIPS data are from the GLIMPSE and MIPSGAL catalogs in the NASA/IPAC Infrared Science Archive (IRSA)https://irsa.ipac.caltech.edu/frontpage/.", "The AKARI data are taken from the JAXA/ISAS serverhttps://www.ir.isas.jaxa.jp/AKARI/Archive/Catalogues/PSC/.", "The photometric data are given in magnitude except for AKARI 9 $\\mu $ m, which is in units of mJy.", "Both objects are located near the Galactic plane ($b \\sim 04$ ).", "Because of the presence of nearby sources, they do not satisfy the criteria of the automatic selection and are not included in the AKARI MIR slit-less spectroscopic catalog [107].", "We applied a careful data reduction procedure to avoid contamination from nearby sources and extracted the spectra of both objects from the IRC slit-less spectral images of the NIR and MIR-S channels by using the IRC spectroscopic data reduction toolkit of version 20150331https:///www.ir.isas.jaxa..jp/AKARI/Observation/support/IRC#software and the latest wavelength calibration for the NIR spectrum was applied [6].", "In the catalog production process, MIR spectra are extracted for 7 pixels in the spatial direction to not lose the source flux and obtain an optimal signal-to-noise ratio [107].", "We needed to apply a narrower spatial extraction window to avoid contamination in the spectrum extraction of the present data and the aperture corrections were applied based on the point spread function for the narrow extraction window [67].", "The sky background was subtracted using the sky data near the object.", "Object 1 is located in a relatively empty region of the sky on the Galactic plane, and there is an infrared source located at a distance of about 8 from Object 1 in the direction vertical to the dispersion (Figure REF ).", "The image quality (FWHM) of the NIR channel is better than 43 [70].", "The NG spectrum (2.5–5 $\\mu $ m) of Object 1 was extracted for 4 pixels ($= 584$ ) to avoid the contamination from the nearby source.", "We, then, applied a two-Gaussian component fit in the spatial direction developed by [81] to estimate the contamination of nearby sources [63].", "We confirmed that the spectrum extracted by the toolkit agreed with the two-Gaussian component fit within the uncertainty and the contamination was insignificant.", "We extracted the SG1 and SG2 spectra (5–13 $\\mu $ m) for 3 pixels ($= 702$ ).", "The image quality (FWHM) of the MIR-S channel is better than 51 [70].", "The nearby source is fainter in the MIR (Figure REF ) and does not make a contamination to the SG1 and SG2 spectra of Object 1.", "The spectra of SG1 and SG2 are smoothly connected at 8 $\\mu $ m and we use the SG1 spectrum for wavelengths shorter than 8 $\\mu $ m and the SG2 spectrum for those longer than 8 $\\mu $ m. Around Object 2, there are a nearby source with a similar brightness at a distance of 5 and a source brighter than Object 2 at a distance of 8 in the direction vertical to the dispersion in the NG image (Figure REF a).", "We applied a two-Gaussian component fit in the spatial direction to estimate the contamination from the nearby sources in the NG spectrum and adopted a narrow extraction window of 2 pixels ($=292$ ), which reduced the signal-to-noise ratio but enabled to avoid the contamination for the spectral region 3.1–5.0 $\\mu $ m. We found that it was not possible to reliably recover the spectrum for 2.5–3.1 $\\mu $ m due to the faint continuum with the strong H$_2$ O ice absorption at 3 $\\mu $ m. Object 2 is also located in a region with faint nebulosity in the MIR (Figure REF ) and is very faint in the SG2 image.", "In the SG1 image, it was impossible to avoid contamination from the nebulosity completely and obtain a reliable spectrum even with a narrow extraction window.", "It was also difficult to select a reliable sky area for the background subtraction.", "To recover spectral information as much as possible, therefore, the same window size of 3 pixels as for Object 1 was adopted.", "The spectrum for 5–8 $\\mu $ m taken with the SG1 mode was able to be extracted with low reliability, while the spectrum taken with the SG2 mode (8–13 $\\mu $ m) was very faint and could not be extracted.", "Therefore, Object 2 has a reliable spectrum extracted only for 3.1–5.0 $\\mu $ m. During the slit-less observations, a 3 $\\mu $ m image and a 9 $\\mu $ m image were taken to provide the origin of the wavelength scale in the slit-less images [67].", "The absolute wavelength in the spectrum extraction process is estimated to be accurate better than a half pixel, which corresponds to 0.005, 0.03, and 0.05 $\\mu $ m for the NG, SG1, and SG2 spectra, respectively.", "Lastly, the spectrum between 4.95–5.0 $\\mu $ m in the NG was removed because of the potential contamination of the second-order light [6]." ], [ "Results", "The extracted spectra of Object 1 and Object 2 are shown in Figure REF .", "For Object 2, the gray lines indicate the regions, where the spectra are affected by nearby sources and faint nebulosity, and are not reliable.", "There is a small jump between the NG and SG1 spectra of Object 1 at 5 $\\mu $ m. We did not make any correction to stitch the spectra.", "Photometric data of 2MASS, Spitzer/IRAC and MIPS, WISE, and AKARI are also plotted together.", "The red circles show the original catalog values, while the purple squares indicate the color-corrected ones, for which the spectral information for color correction is estimated from the observed spectrum.", "For Object 1 (Figure REF a), the color corrections are generally small except for those at the deep absorption features ($W1$ (3.4 $\\mu $ m), IRAC band 4 (8 $\\mu $ m), and AKARI 9 $\\mu $ m).", "For Object 2 (Figure REF b), the color correction is possible only for IRAC band 2 (4.5 $\\mu $ m).", "The photometric data are generally in agreement with the observed spectra, suggesting that the spectra are extracted properly.", "For Object 1, deep absorption features of H$_2$ O ice at 3 $\\mu $ m and silicate at 10 $\\mu $ m are apparent, which can be noticed even in the photometric data.", "The absorption features of CO$_2$ ice at 4.26 $\\mu $ m and CO ice at 4.67 $\\mu $ m are clearly seen.", "In addition, several absorption bands are present in 5–8 $\\mu $ m. They are discussed in the next section.", "In the 20 $\\mu $ m region, the $W4$ (22 $\\mu $ m) and MIPS 24 $\\mu $ m show discrepancy.", "Its cause is not clear at the present.", "Both data, however, suggest that the flux is decreasing towards longer wavelengths.", "For Object 2, the spectrum at around 3 $\\mu $ m is very faint and is not reliably extracted as described in §.", "However, the presence of deep absorption is suggested by the faintness of the flux at around 3 $\\mu $ m and by the decreasing trend of the spectrum from $\\sim 3.3$  $\\mu $ m. The spectrum of 3.1–5.0 $\\mu $ m shows several absorption features, including CO$_2$ ice at 4.26 $\\mu $ m and CO ice at 4.67 $\\mu $ m. The $W4$ (22 $\\mu $ m) data are appreciably brighter than the MIPS 24 $\\mu $ m data.", "Object 2 is associated with faint nebulosity in the MIR region (Figure REF ).", "We attribute the difference to the larger beam of WISE, which is affected by the nebulosity, and take the WISE data point as an upper limit.", "The nebulosity also affects detection of AKARI 9 $\\mu $ m. Inspection of the original data confirms 50 mJy as a conservative upper limit [53], which may suggest the presence of silicate absorption.", "The spectrum of Object 2 at wavelengths longer than 20 $\\mu $ m is not well constrained, and only a decreasing trend similar to Object 1 is suggested.", "In addition to the NIR and MIR photometry, we searched for far-infrared (FIR) data of both objects in the Herschel Infrared Galactic Plane Survey data [62].", "There are no objects listed at the object positions in the HIGAL catalogs.https://tools.ssdc.asi.it/HiGAL.jsp Upper limits of the detection depend on the position of the sky.", "From the fluxes of objects detected within 2 of the target positions and the completeness limits of the survey [62], we conservatively estimate upper limits as 0.8, 3, 9, 10, and 10 Jy at 70, 160, 250, 350, and 500 $\\mu $ m, respectively." ], [ "Absorption features", "Because of the small jump between NIR and MIR spectra, we define the continuum separately for the NIR and MIR regions.", "For 2.5–5 $\\mu $ m, we fit a quadratic polynomial to the logarithmic of the flux at the continuum regions, 2.58–2.65, 3.86–4.15, and 4.73–4.82 $\\mu $ m for Object 1, and 3.51–4.13 and 4.78–4.93 $\\mu $ m for Object 2.", "These continua fit the continuum regions adequately.", "For wavelengths longer than 5 $\\mu $ m, there are no reliable continuum points and we simply fit a 1000K blackbody for Object 1 [44] to the photometry points of IRAC bands 3 and 4, and MIPS 24 $\\mu $ m. No continuum is estimated for Object 2 for wavelengths longer than 5 $\\mu $ m. The assumed continua are shown by the blue solid lines in Figure REF .", "The optical depth is estimated using these continua.", "In the following, we discuss the spectral features in 2.5–4.0, 4.0–5.0, 5.0–8.0, and 8–13 $\\mu $ m, separately.", "For Object 2, only the features in the reliable spectrum of 4.0–5.0 $\\mu $ m are discussed." ], [ "2.5–4.0 $\\mu $ m", "Figure REF shows the spectrum of Object 1 in the 2.5–4 $\\mu $ m region.", "In addition to the deep absorption of H$_2$ O ice at 3.0 $\\mu $ m, there are weak features seen at $\\sim 3.4$ and 3.54 $\\mu $ m. They can be attributed to O-H stretching modes of CH$_3$ OH ice [35].", "CH$_3$ OH ice also has a stronger band at 3.07 $\\mu $ m. We first fit the features in 3.3–3.6 $\\mu $ m with CH$_3$ OH ice and estimate its contribution to the 3.0 $\\mu $ m band.", "A linear baseline (solid purple line in Figure REF a) is subtracted from the observed spectrum and the spectrum in 3.3–3.6 $\\mu $ m is fitted with the absorbance data of pure CH$_3$ OH ice at 15 K [41] taken from the Leiden laboratory ice database.https://icedb.strw.leidenuniv.nl The band profile hardly changes below 100 K and the choice of different temperature data does not affect the fit result.", "Pure CH$_3$ OH ice data fit the observed spectrum reasonably well (Figure REF b).", "From the fit, we estimate the column density of CH$_3$ OH ice as $(2.6 \\pm 0.6) \\times 10^{17}$  cm$^{-2}$ , assuming that the band strength of the 3.54 $\\mu $ m band is $7.6 \\times 10^{-18}$  cm molecule$^{-1}$ [35].", "With the estimated column density, the contribution of CH$_3$ OH ice to the 3 $\\mu $ m absorption is found to be insignificant (blue dotted line in Figure REF a).", "Figure: Optical depth spectrumc of Object 1 in the 2.5–4.0 μ\\mu m region.", "The observed spectrum is shown by the black solid lines withthe error bars.", "(a) The solid red line indicates the fit with the amorphous H 2 _2O ice at 60 K, while the green long-dashed line and the light-blue short-dashed lineshow the fits with the amorphous H 2 _2O ices at 15 K and 100 K, respectively .", "The blue dotted line indicates the contribution of CH 3 _3OH iceestimated from the feature at 3.54 μ\\mu m (see Figure b and text).", "The solid purple line shows the assumed linear baseline for the extraction of theCH 3 _3OH feature shown in b.", "(b) A close-up view of the 3.25–3.70 μ\\mu m region after subtraction ofthe baseline.", "The red solid line indicates thefit with the spectrum of CH 3 _3OH ice at 15K .Figure REF b also suggests the presence of weak features at around 3.4 and 3.47 $\\mu $ m. The 3.47 $\\mu $ m feature has been seen in MYSOs, LYSOs, and background stars [84].", "It has been attributed to either a CH vibration mode of hydrogen atoms bonded to tertiary carbon atoms [2] or to ammonia hydrate formed in NH$_3$ :H$_2$ O mixture ice [31], [32].", "The 3.4 $\\mu $ m feature has often been seen in diffuse clouds and could be a sign of hydrocarbons [82], [73], [20].", "The laboratory data of CH$_3$ OH fit the 3.4 $\\mu $ m feature fairly well, and there is no need for a 3.4 $\\mu $ m hydrocarbon feature.", "It should, however, be noted that the fit is also dependent on the assumed continuum.", "Further observations with a higher signal-to-noise ratio are needed to confirm their presence.", "A strong extended red wing on the 3.0 $\\mu $ m H$_2$ O ice absorption band is clearly seen in Figure REF a.", "While a correlation of the red wing with the column density of H$_2$ O ice is suggested for YSOs [93], there are objects that show the 3.0 $\\mu $ m band without the red wing and those with a very strong wing [63].", "The origins of the red wing are still in debate.", "The red wing and its references are discussed in more detail in [63] and [12].", "The profile of the red wing of Object 1 is similar to Type 1 in [63], being typical of YSOs and background stars.", "In this paper, we do not attempt to fit the red wing.", "We also avoid the region around the peak of the 3.0 $\\mu $ m absorption in the fit because of the low signal-to-noise ratio.", "We fit the spectrum of the regions of 2.7–2.9 $\\mu $ m and 3.1–3.2 $\\mu $ m with the spectrum of amorphous H$_2$ O ice.", "We adopt the optical constants of amorphous H$_2$ O ice measured at various temperatures between 20 and 150 K provided by [60].", "Because the 3.0 $\\mu $ m band is strong and broad, it is necessary to take account of the particle shape effect [39], [63].", "We assume a continuous distribution of ellipsoids (CDE) for the shape.", "The original CDE assumes a flat distribution of the shape, giving equal probabilities to extreme shapes, i.e., infinitely narrow needles or thin disks [11].", "In this paper, we assume a shape distribution with a peak at a sphere [71] as a more realistic distribution.", "The difference in the assumed distribution is small and does not affect the following results.", "The amorphous H$_2$ O ice at 60 K is found to show the best fit to the observed spectrum (red solid line in Figure REF a).", "The amorphous H$_2$ O ice at 15 K has a peak at a shorter wavelength and cannot fit the longer-wavelength side of the observed profile very well (green long-dashed line), while the one at 100 K shows a sharper peak due to partial crystallization, which does not fit the observation either (light-blue short-dashed line).", "Note that the fits have a larger weight on the 2.7–2.9 $\\mu $ m because of the higher signal-to-noise ratio than in 3.1–3.2 $\\mu $ m. The present analysis suggests that the H$_2$ O ice towards Object 1 is accounted for by amorphous ice, which is thermally processed to some extent.", "The column density of H$_2$ O ice is estimated from the integrated band strength as $(54.9 \\pm 1.1) \\times 10^{17}$  cm$^{-2}$ , assuming that the band strength is given as $2.1 \\times 10^{-16}$  cm molecule$^{-1}$ for the amorphous H$_2$ O ice at 60 K [60].", "For Object 2, the spectrum between 3.1 and 4.0 $\\mu $ m is extracted without contamination.", "However, this spectral range of Object 2 is much fainter than that of Object 1 (Figure REF ) and thus noisy.", "No features are clearly seen in this spectral range and we do not discuss it in this paper." ], [ "4.0–5.0 $\\mu $ m", "Figure REF shows the 4.0–5.0 $\\mu $ m spectra of the optical depth for Object 1 and Object 2, which indicate the presence of several absorption features.", "H$_2$ O ice is known to have a broad, shallow feature of a combination mode at around 4.5 $\\mu $ m [48].", "Its contribution is estimated from the column density determined from the absorption at 3.0 $\\mu $ m (brown dot-dashed line) and taken into account in the following fit for Object 1.", "The feature is very broad and does not affect the other features seen in this spectral range except that it adds additional continuum.", "For Object 2, no information is available from the 3 $\\mu $ m region and we assume the same column density of H$_2$ O ice as in Object 1.", "Inclusion of the H$_2$ O ice component reduces the column densities of other components by about 10% for Object 2 because it changes only the continuum level, but does not affect the fit and the following discussion.", "Deep absorption features are clearly seen at $\\sim 4.26$ and $\\sim 4.67$  $\\mu $ m in both objects and they are attributed to CO$_2$ and CO ices, respectively.", "The intrinsic widths of the two bands are narrower than the IRC spectral resolution ($\\sim 0.03$  $\\mu $ m) and the ice properties that are extracted from the details of the band profiles cannot be discussed [75].", "In the following fit, we assume that these two bands are approximated by a single Gaussian.", "Figure: Optical depth spectra at 4.0–5.0 μ\\mu m for Object 1 (a) and Object 2 (b).", "The observed spectra are indicated by theblack solid lines with the error bars and the best fit results are shown by the red solid lines.", "The green short-dashed lines and the purple dottedlines indicate the CO 2 _2 and CO ice components, respectively.The gray double-dotted dashed lines show the CO gas component and the blue long dashed lines show the unidentified4.4 μ\\mu m component.", "The light blue solid line indicates the XCN component forObject 1, which is not included for Object 2 (see text).The brown dot-dashed lines show the contribution from H 2 _2O ice, which is estimated from the3.0 μ\\mu m absorption for Object 1.", "For Object 2, we assume the same contribution as in Object 1 (see text).Around the CO ice absorption, a distinct shoulder is seen at $\\sim 4.6$  $\\mu $ m for both objects.", "A broad wing is also present at the longer wavelength side.", "These can be attributed to a CO gas component, whose $P$ and $R$ branches account for the features at the longer and shorter wavelength sides, respectively [75], [1].", "The XCN feature is also present at around 4.6 $\\mu $ m and can contribute to the blue shoulder [95].", "We calculate the CO gas at temperatures between 50 to 500 K with an interval of 50 K and search for the best fit.", "For Object 1, the blue shoulder is very strong compared to the red wing, and thus the XCN component is required to account for the blue shoulder.", "CO gas is needed to fit the red wing of the 4.67 $\\mu $ m feature.", "The CO gas temperature is, however, not well constrained for Object 1 because of the presence of the overlapping XCN feature in the blue side.", "CO gas with temperatures between 150 to 500 K does not make a significant difference in the fit if we adjust the XCN component.", "For Object 2, the red wing component is relatively strong.", "The CO gas can account for both the blue shoulder and red wing without XCN component.", "Therefore, the XCN component is not included in the fit for Object 2.", "The best fit of the CO gas is obtained with 150 K. We assume the same temperature (150 K) for the CO gas in Object 1.", "The XCN feature consists of two components [95], but they are not resolved with the present spectral resolution.", "We simply assume a single Gaussian for the XCN component in the fit for Object 1.", "In addition to these components, there is a broad feature at around 4.4 $\\mu $ m. Similar features have been seen towards several YSOs [86], but their origin is not known.", "The number of of the objects that show the feature is still small, and any correlation with other features is not recognized.", "The feature is not seen in AKARI spectra of background stars [63].", "We call it 4.4 $\\mu $ m component and approximate it by a Gaussian.", "Also there are small dips at around 4.5 $\\mu $ m and at a shorter wavelength side ($\\sim 4.1$  $\\mu $ m) of the CO$_2$ feature seen in the spectrum of Object 2.", "Their origins are not known at the present either.", "We do not include these components in the fit since they are small features and do not affect the following results.", "In the following fits, the peak wavelength and the band width are set as free parameters except for the 4.4 $\\mu $ m component for Object 2, for which we fix the peak wavelength and width as being the same as those for Object 1.", "The best fit results are shown by the red solid lines in Figure REF .", "The column densities and the peak wavelengths in the best fits are summarized in Table REF .", "For CO$_2$ ice, the peak wavelength is found to be shorter than 4.26 $\\mu $ m for both objects (Table REF ).", "While the 4.4 $\\mu $ m broad component shifts the peak wavelength of the CO$_2$ component slightly, the observed peaks are always seen at a wavelength of 4.26 $\\mu $ m or shorter.", "Therefore, the peak is clearly shorter than the peak position of apolar CO$_2$ ice of $> 4.26$  $\\mu $ m, suggesting that a non-negligible fraction of CO$_2$ ice is either polar or pure [39].", "For CO ice, the absorption feature is observed at 4.66–4.67 $\\mu $ m, which agrees with pure CO ice.", "Polar CO ice has a peak at a wavelength longer than 4.68 $\\mu $ m and does not match with the observed spectra [39].", "The differences in the peak wavelengths are smaller than the spectral resolution ($\\sim 0.03$  $\\mu $ m), but comparable with the resolution per pixel ($\\sim 0.01$  $\\mu $ m).", "They are larger than the uncertainty in the wavelength ($\\sim 0.005$  $\\mu $ m, see §).", "High-spectral resolution observations are needed to confirm the properties of the CO and CO$_2$ ices accurately.", "The column densities of CO and CO$_2$ ices are estimated from the absorption features at 4.67 and 4.26 $\\mu $ m, assuming that the band strengths are 1.1 and $7.6 \\times 10^{-17}$  cm molecules$^{-1}$ , respectively [43].", "Since both features are deep, and narrower than the spectral resolution, we need to take account of the saturation effect of the low spectral resolution.", "We simulate the features in a way similar to [86], assuming that the band width (FWHM) is 18 and 9.71 cm$^{-1}$ for the CO$_2$ and CO features, respectively [44], and estimate the correction factors.", "The correction factors are found to be 1.19 and 1.16 for the CO$_2$ absorption feature for Object 1 and Object 2, respectively, and 1.19 and 1.27 for the CO feature for Object 1 and Object 2, respectively.", "Using these correction factors, we estimate the column densities of CO and CO$_2$ ices as shown in Table REF .", "The XCN feature is known to consist of two components, 4.60 and 4.62 $\\mu $ m. The 4.62 $\\mu $ m component is securely assigned to OCN$^-$ [95], while the carrier of the 4.60 $\\mu $ m component is unknown.", "The present data show a peak at 4.60 $\\mu $ m. However, the peak wavelength depends on the assumed CO gas temperature to some extent.", "With a higher CO gas temperature, the peak shifts to a longer wavelength.", "Since the present spectral resolution is not enough to resolve the XCN feature and the CO gas ro-vibrational transitions, it is not possible to discuss the details of the XCN component.", "We assume the same band strengths for both components as $1.3 \\times 10^{-16}$  cm molecule$^{-1}$ [94], [66] and estimate the column density.", "Since the feature is shallow and broad, we do not apply any correction for the saturation effect.", "Fits with the CO gas of temperatures different from 150 K always provide a higher XCN abundance.", "Thus, the estimated XCN abundance should be taken as a lower limit.", "The results are summarized in Table REF .", "For the 4.4 $\\mu $ m feature, the integrated band intensities ($=\\int \\tau $ d$\\nu $ ) in units of cm$^{-1}$ are given in Table REF ." ], [ "5.0–8.0 $\\mu $ m", "Figure REF shows the optical depth spectrum of Object 1 in the region 5.0–8.0 $\\mu $ m, which shows absorption features at around 6 and 6.8 $\\mu $ m. [14] decompose the ice features in this spectral range into five distinct components: C1 (5.84 $\\mu $ m), C2 (6.18 $\\mu $ m), C3 (6.755 $\\mu $ m), C4 (6.943 $\\mu $ m), and C5 (a broad component covering 5.8–8 $\\mu $ m).", "Possible carriers are discussed in detail in [56], [14], and [66].", "The C1 component has been attributed to H$_2$ O, H$_2$ CO, and HCOOH ices.", "We assume the H$_2$ O ice column density from the 3.0 $\\mu $ m fit and estimate its contribution (brown dot-dashed line).", "The band of H$_2$ O ice shifts to a longer wavelength and the peak intensity increases, when H$_2$ O ice is diluted by CO$_2$ [57].", "To be consistent with the fit of the 3.0 $\\mu $ m band, however, we use the same pure amorphous H$_2$ O ice data at 60 K in the fit.", "There is also a weak feature of CH$_3$ OH ice at around 6.8 $\\mu $ m. Its contribution is estimated from the 3.54 $\\mu $ m band (Figure REF ), which is found to be insignificant compared to the observed feature (blue dotted line).", "After subtracting only the contribution from H$_2$ O ice as in [14], we fit the spectrum with the C1, C2, C3, and C4 components simply assuming that they are approximated by Gaussians with the central peak wavelengths and the FWHMs given by [14] convolved with the IRC spectral resolution (0.12 $\\mu $ m).", "Because of the low spectral resolution of the present spectrum, detailed band profiles do not affect the fit results.", "The four-component fit is shown by the red solid line, which reproduces the observed spectrum fairly well.", "The present fit does not require the C5 component, but it is difficult to confirm the presence or absence of the C5 component from the present spectrum because of the uncertainty in the assumed continuum.", "The peak optical depths of the four components in the best fit are shown in Table REF .", "The contribution of amorphous H$_2$ O ice is estimated to be about a half ($\\sim 54\\%$ ) of the 6.0 $\\mu $ m feature.", "There seems to be also a feature at around 7.3–7.5 $\\mu $ m. It is weak and its presence has to be confirmed by further observations.", "Figure: Optical depth spectrum of Object 1 in 5.0–8.0 μ\\mu m. The observed spectrum is indicated by theblack solid line with the error bars.", "The contribution of H 2 _2O ice estimated from the 3.0 μ\\mu m band and that ofCH 3 _3OH ice estimated from the 3.54 μ\\mu m band are indicated by the browndot-dashed line and the blue dotted line, respectively.The red solid line shows the best fit result with the summation of the C1, C2, C3, and C4 components for the spectra, from which the contribution fromH 2 _2O ice has been removed, but that of CH 3 _3OH has not(see text).", "The purple, gray, orange, and green solid lines indicate the C1, C2, C3, and C4 components, respectively." ], [ "8.0–13.0 $\\mu $ m", "Figure REF a shows the optical depth spectrum of Object 1 for 8.0–13.0 $\\mu $ m together with the contributions from H$_2$ O ice (brown dot-dashed line) and CH$_3$ OH ice (blue dotted line) estimated from the 3.0 and 3.53 $\\mu $ m bands, respectively.", "The contribution from CH$_3$ OH ice is negligible, while H$_2$ O ice adds broad absorption at wavelengths longer than 10 $\\mu $ m. The spectrum shows deep absorption at around 10 $\\mu $ m attributable to amorphous silicate.", "We estimate the optical depth at 9.7 $\\mu $ m, $\\tau _{9.7}$ , as $2.9 \\pm 0.3$ after subtracting the contributions from H$_2$ O and CH$_3$ OH ices.", "In addition, there seems some excess absorption at around 11 $\\mu $ m on the smooth absorption feature of amorphous silicate.", "[103] report that similar excess is seen towards YSOs and in the interstellar medium (ISM), attributing it to crystalline forsterite.", "[36] make a thorough study of this feature in the various lines-of-sight and show the ubiquitous presence of the feature in YSOs and even in the diffuse ISM.", "They discuss several possible origins of the feature in detail, concluding that it arises from crystalline silicate.", "[36] use the spectral regions 9.9–10.2 and 12.0–13.0 $\\mu $ m to estimate the amorphous component and extract excess absorption.", "Since the spectral region of 9.9–10.2 $\\mu $ m of the present spectrum is very noisy, we fit instead the spectral regions 8.9–10.3 and 12.0–13.0 $\\mu $ m by a quadratic equation and extract the excess absorption (Figure REF b).", "The equation fits the spectrum in 9.9–10.2 $\\mu $ m reasonably well, and the different choice of the spectral region for the fit does not affect the extracted excess and the following discussion.", "The red line in Figure REF indicates the average excess profile derived in [36].", "Note that the actual profile has some asymmetry, which varies from object to object.", "In Figure REF , a Gaussian with the average peak wavelength (11.08 $\\mu $ m) and the average FWHM (0.76 $\\mu $ m) is simply plotted to show an approximate profile of the average excess.", "The amplitude is scaled to that of Object 1.", "The excess of Object 1 peaks at around 11.3–11.5 $\\mu $ m, which is slightly longer than the peaks derived by [36], particularly compared to those found in the ISM ($\\sim 11.1$  $\\mu $ m).", "Note that some YSOs show peaks at longer wavelengths (up to $\\sim 11.2$  $\\mu $ m).", "The ratio of the excess to the silicate absorption at 9.7 $\\mu $ m ($ 0.13 \\pm 0.06$ ) is also larger, but is roughly in agreement with those found in YSOs ($\\sim 0.05$ ) and the diffuse ISM [36] within the uncertainty.", "Figure: (a) Optical depth spectrum of Object 1 in 8.0–13.0 μ\\mu m. The observed spectrum is indicated by theblack solid line with the error bars.", "The contribution of H 2 _2O ice estimated from the 3.0 μ\\mu m band is shown by the browndot-dashed line, while that of CH 3 _3OH ice from the 3.53 μ\\mu m band is indicated by the blue dotted line.The red solid line shows the optical depth after subtracting the contributions from H 2 _2O ice and CH 3 _3OH ice.", "The green dashed line shows a polynomialfit to the red solid line (see text).", "(b) Excess optical depth after subtracting the polynomial fit to the smooth component (green dashed line in (a)).The red line indicates the average excess profile at 11 μ\\mu m derived in .", "Note that the actual profiles have some asymmetry, whichvaries from object to object and is not included in the plot (see text).", "The amplitude is scaled to that of Object 1.cCCCCC Column densities and peak wavelengths of ice species in the best fit 0pt 3cObject 1 2cObject 2 Column densities Abundancea Peak wavelength Column densities Peak wavelength Species ($\\times 10^{17}$  cm$^{-2}$ ) (%) ($\\mu $ m) ($\\times 10^{17}$  cm$^{-2}$ ) ($\\mu $ m) H$_2$ O ice 54.9 1.1 100 3.06 CO$_2$ ice 6.5 0.2 11.8 0.4 4.25 5.3 0.3 4.26 CO ice 12.7 1.0 23.1 2.0 4.66 19.5 1.5 4.67 CO gas 28.0 5.3 51.0 9.6 109 4 XCN 0.9 0.1 1.7 0.2 4.60 CH$_3$ OH ice 2.6 0.6 4.8 1.0 3.54 4.4 $\\mu $ m feature 37.8 1.4c 4.38 63.11.8c 4.38 C1 0.25 0.04d 5.84 C2 0.23 0.02d 6.18 C3 0.29 0.03d 6.755 C4 0.39 0.03d 6.943 silicate 2.9 0.3 d 9.7 aRelative abundance to H$_2$ O ice bUpper limit because of the possible contribution from H$_2$ CO ice (see text) cIntended band intensity ($=\\int \\tau $ d$\\nu $ ) in units of cm$^{-1}$ dPeak optical depth" ], [ "Discussion", "We found two intriguing objects that show deep ice absorption features in the AKARI/IRC spectroscopic survey of the Galactic plane.", "To investigate the nature of the objects and the location of the ice species, we discuss their ice properties and infrared spectral energy distributions (SEDs) in the following sections." ], [ "Properties of absorption features", "Ice absorption features have been observed towards YSOs and used as a good indicator for the identification of YSOs [98], [85], [86], [83].", "On the other hand, ice absorption is also observed towards background stars sitting behind quiescent, dense clouds [57], [101], [15], [21], [63].", "Therefore, they are not secure evidence for the identification of YSO nature of the present objects.", "In this section, we discuss possible evidence for thermal processing on ices, which is not expected for the features in background stars and thus supports the YSO identification of the objects, based on their spectra.", "The 3.0 $\\mu $ m H$_2$ O ice absorption feature of Object 1 peaks at 3.06 $\\mu $ m, which is longer than amorphous H$_2$ O ice at 15 K ($\\sim 3.02$  $\\mu $ m, Figure REF a).", "The difference is small, but is larger than the uncertainty in the wavelength ($\\sim 0.005$  $\\mu $ m).", "The absorption profile of 15 K H$_2$ O ice does not fit the absorption profile at the longer wavelength side well, which requires a contribution from warm amorphous H$_2$ O ice of thermally processed.", "The shape effect does not account for the difference and the low spectral resolution does not affect the characteristics of the band profile either.", "[44] analyze the 3.0 $\\mu $ m of H$_2$ O ice for a number of MYSOs and show that some of them require a warm ($ \\ge 50$  K) ice component in addition to the 10 K component.", "[15] presented spectra of the H$_2$ O ice absorption at 3 $\\mu $ m for several background stars, suggesting no variations in the band profile.", "The 3 $\\mu $ m absorption features of background stars have a peak between 3.0 and 3.1 $\\mu $ m, similar to Object 1.", "Figure REF shows a comparison of the 3 $\\mu $ m optical depth spectrum of Object 1 with those of a YSO [44] and a background star [15].", "The background star shows a wider profile in the blue side than the YSO and Object 1.", "Other background stars in [15] show similar characteristics.", "At the red side, no difference is seen between the YSO and the background star, and Object 1 shows a slightly narrower width.", "The wider width at the blue side of the background star may be attributable to the presence of cold, amorphous ice.", "The H$_2$ O ice profile of Object 1 is in better agreement with the YSO spectrum, but the difference in the profile between the YSO and the background star is not very large.", "It should also be noted that large H$_2$ O ice dust shifts the peak to a longer wavelength [88].", "While it cannot be ruled out that the longer wavelength of peak absorption of the 3 $\\mu $ m feature could be accounted for by large dust, the 60 K amorphous ice of a CDE fits the observed spectrum reasonably well, suggesting that the observed 3.0 $\\mu $ m band profile can also be attributed to thermally processed H$_2$ O ice towards Object 1.", "Figure: Comparison of the 3 μ\\mu m H 2 _2O ice optical depth spectrum of Object 1(black line) with those of a YSO and of a background star .", "The optical depth spectra are normalized to unity at the maximum.The spectra of 4–5 $\\mu $ m contain ample information on the properties of ice species.", "The abundance of CO$_2$ ice (relative to H$_2$ O ice) is $11.8 \\pm 0.4$ % for Object 1.", "It is in the range for MYSOs, and at the lowest end of the abundance distribution of LYSOs and lower than the range of background stars [12].", "Therefore, this abundance ratio suggests that Object 1 may be a MYSO, although this is not definitive evidence, taking account of the distribution of the abundance and the uncertainty in the abundance estimation.", "On the other hand, CO ice abundance is large ($23.1 \\pm 2.0$ %) for Object 1 and its column density is large compared to CO$_2$ ice for Object 2.", "This is a secure result since the observed spectra of both objects show very deep absorption at 4.67 $\\mu $ m despite the low spectral resolution.", "The CO ice abundance is expected to decrease in a higher temperature environment because of its low sublimation temperature ($\\sim 20$  K).", "The median abundance of CO ice in MYSOs is smaller than that in LYSO and background stars, but the CO ice abundance has a wider distribution than that of CO$_2$ ice [12].", "The observed abundance is in a typical range for LYSOs and background stars, while it is still in the range of the abundance distribution of MYSOs [12].", "The band peak positions of CO ice suggests that it is pure CO ice, while that of CO$_2$ ice suggests that it is either pure or polar.", "At an early phase of ice formation, CO$_2$ ice is thought to be formed concurrently with H$_2$ O ice and thus it should be polar.", "At some point, most of frozen-out CO is no longer converted into CO$_2$ ice and the abundance of CO ice increases [66].", "The observed large abundance of CO ice suggests that both objects may be in this evolutionary stage.", "The suggested properties of pure CO and polar CO$_2$ ices are compatible with this interpretation, although the present spectra do not have a sufficient spectral resolution to discuss their profiles in detail.", "The presence of warm CO gas component is a strong indicator for the presence of an embedded heating source.", "Warm CO gas ($> 50$  K) has been observed in absorption towards embedded YSOs [75], [1].", "In dense clouds, the gas temperature is supposed to be as low as 20 K and warm CO gas has not been observed in background stars [63].", "For Object 1, the red wing of the 4.67 $\\mu $ m band can be best accounted for by warm CO gas ($> 100$  K).", "For Object 2, the strong blue shoulder cannot be attributed to XCN, since it would require a very large column density for the XCN component, suggesting that the presence of warm CO gas ($\\sim 150$  K) is a secure conclusion.", "CH$_3$ OH ice is believed to form on grain surfaces and several different formation processes are proposed by laboratory experiments; i.e., ultraviolet photolysis, radiolysis, and CO hydrogenation [51], [99].", "Large amounts of CH$_3$ OH ice have been observed only towards YSOs and very dense cores that are likely to form stars [15], and it could be a good indicator of YSOs [4], [3].", "However, the CH$_3$ OH ice abundance of Object 1 is not very large and is in the range of YSOs and background stars [15], [100].", "Thus, it is not a secure indicator of the identification as a YSO for Object 1.", "However, the CH$_3$ OH ice abundance of Object 1 is in the range of YSOs and such background stars, although a factor of $\\sim 5$ less than the most extreme YSO cases.", "Thus, it does not seem to be a secure indicator of the identification as a YSO for Object 1.", "The 4.62 $\\mu $ m XCN feature has also been thought to be an indicator of energetic processing.", "However, the relatively large abundance observed in LYSOs suggests that the carrier can be produced through purely thermal acid-base reactions in ices [94], [95].", "It should be noted that the XCN feature has not been observed towards background stars.", "The estimated abundance of XCN in Object 1 is quite large ($1.7 \\pm 0.2$ %), being in the range for MYSOs and much larger than the upper limit for background stars [66], [12].", "The XCN abundance in Object 1 depends on the assumed temperature of CO gas, and the present result provides a lower limit (§REF ).", "Higher spectral resolution data that resolve ro-vibrational bands of the CO gas lines and XCN feature are needed to estimate the the properties of the CO gas and the XCN abundance accurately [75].", "The features in 5.0–8.0 $\\mu $ m (C1–C5) can also be used to study the thermal processing of ices [15].", "There is no clear difference in the distribution of each of the C1–C4 feature strengths among MYSOs, LYSOs, and background stars, and the presence of these features can neither support nor rule out the YSO nature of the objects unambiguously [14], [66].", "The C1–C4 components are present in all classes of objects (MYSOs, LYSOs, and background stars) and their strengths typically correlate well with the H$_2$ O column density.", "The mere presence of these feature therefore can neither support nor rule out the YSO nature of the objects unambiguously [14], [66].", "However, the ratio of the C4 to C3 components is large towards some YSOs, shifting the 6.85 $\\mu $ m feature to a longer wavelength, compared to background stars [56], [14], [15], [77].", "Figure REF plots the H$_2$ O ice column density normalized by the peak optical depth of the silicate band (a measure of the H$_2$ O ice abundance) against the ratio of the integrated optical depths of the C4 to C3 components, $\\tau (C4)/\\tau (C3)$ , for Object 1 together with those of the YSO and background star samples taken from [14], [15].", "Object 1 shows a relatively low H$_2$ O ice abundance and the ratio of the C4 to C3 components larger than the majority of background star samples (blue triangles).", "It is located in the region occupied mostly by YSOs.", "It should be noted that the C5 component has not been observed towards background stars [66].", "The present spectrum is not sufficient to confirm the presence of the C5 component.", "Figure: H 2 _2O ice column density normalized by the peak optical depth of the silicate absorption at 9.7 μ\\mu m (a measure of H 2 _2O abundance)against the ratio of the integrated optical depths of the C4 to C3 components.", "The data point of Object 1 is shown by the red circle.", "The green squares and blue triangles indicate thedata for YSOs and background stars, respectively, taken from , .The observed silicate absorption at 10 $\\mu $ m in Object 1 shows excess absorption at around 11.3 $\\mu $ m and it is best ascribed to crystalline silicate.", "The excess is detected in YSOs as well as in the diffuse ISM.", "In the ISM, crystalline silicates are thought to be gradually amorphized by cosmic-ray hits [16].", "Therefore, the presence of the excess itself cannot distinguish YSOs from background stars.", "A larger amount of excess is sometimes seen towards embedded YSOs, which can be attributed to thermal processing of amorphous silicate by the radiation from the YSO [36].", "The observed relatively large excess may suggest MYSO origin.", "Further observations of the 10 $\\mu $ m band are needed to estimate the amount of the excess accurately.", "Figure REF shows a correlation of the peak optical depth at 9.7 $\\mu $ m, $\\tau _{9.7}$ , against that at 3 $\\mu $ m, $\\tau _{3.0}$ for the YSO and background star samples taken also from [14], [15].", "The correlation line for the sample of background stars is also indicated by the black solid line [15].", "Most of the background stars lie near the correlation line, while there are several YSOs above the correlation line, in particular for $\\tau _{3.0} > 3$ .", "Object 1 is located well above the correlation line at a large $\\tau _{3.0} $ (= 4.32), suggesting that Object 1 may be a YSO.", "Note that the background stars located well above the correlation line at around $\\tau _{3.0} \\sim 0.6-1.0$ are those towards the core L 328, which may trace diffuse ISM rather than dense clouds [15].", "Figure: Peak optical depth at 9.7 μ\\mu m τ 9.7 \\tau _{9.7} against that at 3 μ\\mu m τ 3.0 \\tau _{3.0}.The black solid line shows a correlation line for the data of background stars .Object 1 is shown by the red circle.The green squares and blue triangles indicate thedata for YSOs and background stars, respectively, taken from , .The difference in abundance of various ice species among MYSOs, LMYSOs, background stars is not very large and their abundance distributions have overlapping ranges [66], [15], [12].", "Therefore, it is difficult to draw a definite conclusion on the nature of the two objects from the abundance of ice species.", "On the other hand, there is no evidence against the YSO identification for both objects.", "Several lines of evidence indicate that Object 1 may be a (M)YSO.", "The presence of warm CO gas and XCN feature and the relatively large C4 to C3 component ratio in Object 1 support the YSO nature together with the skewed profile of the 3.0 $\\mu $ m H$_2$ O ice.", "Object 2 has less evidence because of the limited range of the reliable spectrum, but the strong blue shoulder of the CO ice absorption feature suggests the presence of a large amount of warm CO gas towards Object 2, supporting that Object 2 is also a YSO." ], [ "Spectral energy distribution", "The SEDs of both objects peak at around 5 $\\mu $ m (Figure REF ).", "The SED of embedded YSOs generally increases towards longer wavelengths [44], [14], while that of background stars peaks at wavelengths shorter than 4 $\\mu $ m [15], [63].", "There are, however, a few YSOs whose SED has a peak at around 5 $\\mu $ m and decreases towards longer wavelengths, resembling the SEDs of Object 1 and Object 2 [63].", "Figure: Spitzer IRAC and MIPS [3.6]-[5.8] vs. [8.0]-[24] two-color diagram of YSOs in the NGC 1333 region .The different color symbols indicate different evolutionary stages of YSOs (black: circles: Class I*, blue squares: Class I, light blue triangles: Class II; purplereverse triangles: Class II/III).The areas of YSOs in the Cygnus X region are also indicated by the dashed lines.", "The thick and light blue dashed lines roughly enclose the areas ofClass I YSOs and Class II YSOs, respectively, in the Cygnus X region.", "The positions of Object 1 and Object 2 are indicated by the red circles labeledas Obj 1 and Obj 2, respectively.The green arrow indicates the reddening vector of A K A_\\mathrm {K}=5.Figure REF shows a two-color diagram of Spitzer IRAC and MIPS [3.6]-[5.8] versus [8.0]-[24] for YSOs in the NGC 1333 region taken from [47].", "The distribution of YSOs in the Cygnus X region [7] is also indicated by the dashed lines.", "Object 1 and Object 2 are located outside of the YSO regions in the two-color diagram.", "They are too blue in 8–24 $\\mu $ m compared to standard YSOs.", "The reddening vector shown by the green arrow suggests that they are rather background stars with large extinction.", "Taking the relation $A_\\mathrm {v}/\\tau _{9.7} = 18.5 \\pm 2.0$ [38], the visual extinction $A_\\mathrm {v}$ is estimated as $54 \\pm 8$ .", "The WISE colors also confirm the background characteristics of both objects.", "The WISE colors of Object 1, $W1 - W2 = 1.927 \\pm 0.030$ and $ W3 - W4 = 0.927 \\pm 0.074$ , are located well outside of the YSO region in the classification scheme of [58].", "The color $W1 - W2$ is very red, but $W3-W4$ is not, suggesting that it is a background star with large extinction.", "Object 2 has $W1 - W2 = 1.697 \\pm 0.031$ and $ W3 - W4 $ ($=2.601 \\pm 0.057$ ), which place it in the YSO region.", "However, as described in §, $W4$ is much brighter than MIPS 24 $\\mu $ m data (Figure REF ).", "It may have to be taken as an upper limit because of the presence of nebulosity.", "If we use MIPS 24 $\\mu $ m data as a replacement of $W4$ , then Object 2 will be placed outside of the YSO region.", "The visual extinction towards the regions of both objects is, however, estimated to be less than 10 based on the optical data [37] and no thick CO clouds have been detected towards them [29], [18].", "No dark clouds are listed at the positions of the two objects in the Spitzer dark cloud catalog either [74].", "Faint nebulosity is present around Object 2 (§), whose surface brightness is estimated as 10 MJy sr$^{-1}$ at 24 $\\mu $ m. According to the ISM dust emission model of [23], this brightness corresponds to the hydrogen column density of $\\sim 5 \\times 10^{22}$  cm$^{-2}$ or $A_\\mathrm {V}$ of about 25, if the incident radiation field intensity $U$ is similar to the solar neighborhood.", "If the radiation field is stronger, $A_\\mathrm {V}$ becomes smaller.", "The geometry of the nebulosity relative to Object 2 is not known and it is not clear if ice species can be formed in the nebulosity.", "No nebulosity is seen around Object 1 (Figure REF ).", "While the nebulosity could be the origin of the extinction for Object 2 in part, it seems unlikely that Object 1 is a background star with large extinction unless there is an unknown very small, but thick cloud on the line-of-sight.", "The blue nature of the objects is also confirmed by the comparison with YSO SED models by [79].", "No YSO models fit the observations of both objects satisfactorily.", "The best results are obtained with background stars with large extinction ($A_\\mathrm {v} \\sim 50$ ) .", "Since the models by [79] do not include ice absorption, we compare the present observations with edge-on disk models, in which ice absorption is included [25], to understand the origin of the discrepancy.", "Detailed modeling is not the purpose of this paper and only comparison of the SED of Object 1 with some example models is shown in Figure REF .", "The same discussion can be applied for Object 2.", "In Figure REF , the edge-on disk models of the inclination angle of 45 with different envelope masses [25] are shown by the solid lines together with the observations (black circles).", "Models with other inclination angles show a similar trend.", "The blue, red, and green solid lines show examples of the models with the envelope masses of 0.55, 1.5 and 4 M$_\\odot $ , respectively, normalized at 4.6 $\\mu $ m. The observed depth of ice absorption can be best reproduced by the model with the envelope mass of 1.5 M$_\\odot $ (red line) among them and it fits the observations up to 12 $\\mu $ m generally well.", "However, for wavelengths longer than 20 $\\mu $ m, the model SED starts deviating from the observations.", "The model SED is either flat (blue) or increasing (red and green), while the observed SED decreases.", "Note that both objects are not detected in the HIGAL survey [62] and the plotted upper limits are very conservative.", "If we decrease the envelope mass (blue line), it slightly reduces the discrepancy, but the ice absorption features become shallower and does not match the observations very well.", "Models with smaller inclination angles show shallower ice absorption and do not reproduce the observations.", "Figure: SED of Object 1 and examples of edge-on disk models in units of νF ν \\nu F_\\nu .", "The black circles show the observed photometric points.Note that the data at 1.2, 70, and 160 μ\\mu m are upper limits.", "The edge-on models of the inclination angle of 45 are scaled at 4.6 μ\\mu m and shown by the solid linesfor the envelope masses of 0.55 (blue), 1.5 (red), and 4 M ⊙ _\\odot (green).The edge-on disk with a large inclination angle is optically thick at wavelengths shorter than 5 $\\mu $ m, but becomes optically thin at longer wavelengths.", "The absorbed energy in the disk is emitted at wavelengths longer than 20 $\\mu $ m. Therefore, it is a natural consequence that edge-on disk models with large inclination angles and deep absorption show strong MIR and FIR emission relative to the NIR.", "Absorption occurs on the line-of-sight, while emission comes from the entire envelope.", "If the absorbing envelope is very clumpy and located just on the line-of-sight, the MIR to FIR emission is reduced, which could be reconciled with the observed SEDs.", "Asymmetric, clumpy distributions of dust are sometimes seen in proto-planetary disks and transition disks [42], [97] and thus may not be unusual for YSOs.", "If the outer radius is truncated, it could also reduce the FIR emission.", "Further investigations on modeling are needed to understand the SEDs of the present objects." ], [ "Nature of the objects and implications", "Both objects show features at both the blue and red sides of the CO ice feature at 4.67 $\\mu $ m and they are attributed to the absorption of warm CO gas.", "Warm CO gas is not expected to be present in quiescent, dense clouds and thus is strong evidence for the YSO identification [75], [1], [63], [69].", "The relatively large abundance of XCN estimated for Object 1 also supports the YSO characteristics since XCN has never been observed in background stars and the estimated abundance is rather high compared to the upper limits for background stars [66], [12].", "Further observations with high spectral resolution are needed to resolve the CO gas and XCN features and estimate their properties unambiguously.", "There are other pieces of evidence to support the YSO identification for Object 1, including an indication of thermally processed amorphous H$_2$ O ice and the abundance of CO$_2$ ice, but none of them is very secure evidence since similar characteristics are also observed in background stars.", "There is no strong evidence against the YSO identifications for the both objects in the characteristics of ice absorption features.", "On the other hand, the SEDs of both objects are rather blue in 8–24 $\\mu $ m, which put them outside of the standard YSO region in the two-color diagram (Figure REF ).", "The blue nature and non-detection at FIR suggest that they are background stars with large extinction.", "Since there is no evidence for the presence of dense clouds in optical and CO observations, those clouds must be very compact.", "The presence of such isolated dense, compact clouds is not known.", "If the two objects are background stars, it will make an impact on our view of the dense gas distribution in our Galaxy and low-temperature chemistry in the ISM.", "H$_2$ O ice has also been observed in O-rich AGB stars with high mass-loss rates (OH/IR stars) and post-AGB stars with dense, cooled envelopes [45], [68], [92].", "Their SEDs have a peak at around 10 $\\mu $ m and decreasing towards longer wavelengths [68], [34].", "The WISE color $W2 -W3$ of Object 1 and Object 2 is $0.129 \\pm 0.039$ and $0.687 \\pm 0.044$ , respectively, which is very blue compared to the $W1-W2$ color, and only a few evolved stars are located in this region of the two-color diagram [58].", "The 3 $\\mu $ m H$_2$ O ice features in those O-rich evolved stars show a narrower profile than those of YSOs [45] or have peaks at 3.1 $\\mu $ m, indicating that a significant fraction of H$_2$ O ice is crystalline [59].", "While recent models show that complex organic ices can be formed in C-rich evolved stars [96], features of other ice species have not been detected in O-rich evolved stars with H$_2$ O ice absorption probably because of carbon-poor environments [92].", "Therefore, it is not very likely that the present objects are evolved stars.", "Absorption features of various ice species are also detected in starburst galaxies and dust-enshrouded AGNs [89], [90], [91], [52], [106], [105].", "They are usually associated with the 3.3 $\\mu $ m emission of aromatic species and/or emission of hydrogen recombination lines.", "Since the present objects are located on the Galactic plane and do not show these emission features, it is not likely that they are galaxies either.", "If the background star is M-type, it could account for the observed absorption of CO gas.", "However, M-type giants show a much broader CO absorption feature starting from 4.3 $\\mu $ m and extending up to 4.9 $\\mu $ m [50] due to the high temperature of the CO gas in their atmosphere ($\\gtrsim 3000$  K).", "M-type dwarfs show the dominance of higher-level transitions in the first overtone of CO absorption in the 2 $\\mu $ m region [27] and are expected to have a broad fundamental transition feature of CO absorption from 4.3 up to $\\sim 5$  $\\mu $ m [72].", "Brown dwarfs also show a very broad CO absorption at around 4.5 $\\mu $ m [108].", "In the spectral region of 7–8 $\\mu $ m, M-type giants have a broad absorption feature of the fundamental transitions of SiO gas [50], while M-type dwarfs show no strong features [28].", "SiO gas absorption could be responsible for part of the observed features at 7–8 $\\mu $ m in Object 1 (Figure REF ), but SiO gas has much stronger, broad absorption of the first overtone at around 4.2 $\\mu $ m in the spectra of M-type giants [50], which is not observed in the present spectrum.", "These expected characteristics do not match with the observed spectra of both objects, suggesting it unlikely that they are background M-type stars.", "Both objects are not located in known star-forming regions.", "Figure REF indicates that the nearby star-forming activities are located in the southern part of the sky of the two objects, where intense diffuse MIR emission is observed.", "According to the catalog of the star-forming regions of [5], the nearest star-forming region is IRAS 14004-6104 for Object 1 (6$$ 8 separation) and IRAS 14004-6104 and RAGL 4188 for Object 2 (both have a 4$$ 2 separation).", "The nearest H2 region is G311.540+00.319 for both objects and the separations are 5$$ 99 and 4$$ 22 for Object 1 and Object 2, respectively [10].", "Such separations may not be unexpected from the drift motion of YSOs from star-forming regions [40], [24] and the two objects could be run-away YSOs, although the distance to the two objects and their evolutionary stages are not known.", "An isolated YSO is also reported to be present, which may have been formed in situ [17].", "YSO population not associated with star-forming regions has an impact on the understanding of star-forming activities and history in our Galaxy [33], [78].", "If they are YSOs, the present result suggests that similar objects have eluded past photometric surveys of YSOs [47], [46], [7], [55], [58].", "This makes significant implications on the population of YSOs in our Galaxy and nearby galaxies.", "If Object 1 is a MYSO as suggested from the low CO$_2$ ice abundance, it could further make implications on the formation scenario of massive stars, which is not well understood yet, and on the galaxy evolution [109], .", "Their SEDs may be accounted for by clumpy distribution of absorbing materials around a YSO.", "Further investigations on modeling are strongly encouraged to understand their nature." ], [ "Summary", "We have discovered two intriguing infrared objects in the AKARI/IRC slit-less spectroscopic survey of the Galactic Plane.", "For Object 1, a full spectrum from 2.5 to 13 $\\mu $ m was successfully extracted, while only a spectrum of 3.1–5 $\\mu $ m was reliably extracted for Object 2 due to the presence of nearby objets and faint nebulosity.", "Both objects show deep absorption features of H$_2$ O, CO$_2$ , and CO ices.", "They also show warm ($>100$  K) CO gas absorption, suggesting that they are embedded YSOs.", "The spectrum of Object 1 also indicates strong XCN absorption, further supporting the YSO identification.", "Other indicators for the thermal processing of ices are also suggested, though they are also compatible with the ice properties seen in background stars.", "They are not definite evidence for the YSO identification, while there are no indications against it in their spectra.", "On the other hand, their SEDs peak at around 5 $\\mu $ m and decrease towards longer wavelengths.", "They are not detected in the HIGAL survey.", "These characteristics are better reproduced by background stars with large ($A_\\mathrm {V} \\sim 50$ ) extinction.", "Both objects are located neither in known star-forming regions nor in known dense clouds.", "Although the observed SEDs may be explained if the absorbing ice species are located in clumpy concentrations just on the line-of-sight in the edge-on disk surrounding a YSO, their true nature remains uncertain based on the currently available data.", "If they are background stars with large extinction, there must be unknown compact, dense clouds in the lines-of-sight towards them.", "The presence and the formation of such isolated dense clouds are not known.", "If ices are formed in these environments, it will have an impact on the chemical processes in the ISM.", "If they are truly YSOs, their blue color in the MIR suggests that similar kinds of objects have eluded in past photometric surveys, which have significant implications on our understanding of star-formation process and the distribution of star-forming activities in our Galaxy and nearby galaxies.", "It is important to identify true nature of the objects by further observations.", "This work is based on observations with AKARI, a JAXA project with the participation of ESA.", "The authors thank all the members of the AKARI project for their continuous support.", "They also thank A. C. Adwin Boogert for providing us with the data of the YSO and background star samples and for useful comments, and Issei Yamamura for helpful comments on the infrared spectrum of M-type stars.", "This work is based in part on observations made with the Spitzer Space Telescope, which was operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA.", "This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation, and those from the Wide field Infrared Survey Explorer, which is a joint project of the Unviersity of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.", "Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.", "This work has also made use of the SIMBAD database, operated at CDS, Strasbourg, France.", "The authors thank the Unviersity of Tokyo Research Internship Program (UTRIP) student Ingrid Koch for her contribution at the initial stage of this work.", "This work was supported by JSPS KAKENHI Grant Number JP16H00934, JP18K03691, JP19H05067, and JP20H05845.", "AKARI (IRC), Spitzer (IRAC and MIPS), WISE, Herschel (PACS and SPIRE)" ] ]
2105.11660
[ [ "FILTRA: Rethinking Steerable CNN by Filter Transform" ], [ "Abstract Steerable CNN imposes the prior knowledge of transformation invariance or equivariance in the network architecture to enhance the the network robustness on geometry transformation of data and reduce overfitting.", "It has been an intuitive and widely used technique to construct a steerable filter by augmenting a filter with its transformed copies in the past decades, which is named as filter transform in this paper.", "Recently, the problem of steerable CNN has been studied from aspect of group representation theory, which reveals the function space structure of a steerable kernel function.", "However, it is not yet clear on how this theory is related to the filter transform technique.", "In this paper, we show that kernel constructed by filter transform can also be interpreted in the group representation theory.", "This interpretation help complete the puzzle of steerable CNN theory and provides a novel and simple approach to implement steerable convolution operators.", "Experiments are executed on multiple datasets to verify the feasibility of the proposed approach." ], [ "Verification of Lemma ", "() can be verified to follow Lemma  as: $&\\mathsf {K}^{\\mathrm {C}_N}_{k \\rightarrow \\text{reg}}(\\phi + \\theta _{i_1})\\\\&= \\operatorname{diag}\\big (P(i_1) \\mathsf {K}\\big ) \\beta _k, \\quad \\text{c.f.", "(\\ref {eq:cn-trivial-proof2})}\\\\&= P(i_1) \\operatorname{diag}(\\mathsf {K}) P(i_1)^{-1} \\beta _k\\\\&= \\rho ^{\\mathrm {C}_N}_\\text{reg}(g) \\mathsf {K}^{\\mathrm {C}_N}_{k \\rightarrow \\text{reg}}\\psi _{0, k}(g)^{-1}, \\quad \\text{c.f.", "(\\ref {eq:beta-rotate0}).", "}$ We can also verify this for $\\overline{\\mathsf {K}}^{\\mathrm {C}_N}_{k \\rightarrow \\text{reg}}= \\operatorname{diag}(\\overline{\\mathsf {K}}) \\beta _k.$" ], [ "Verification of Lemma ", "First note it is easy to verify that for $i_0 = 0$ , i.e.", "$g = (0, i_1)$ , the Lemma  holds in the same way as (REF ), $\\mathsf {K}^{\\mathrm {D}_N}_{j, k \\rightarrow \\text{reg}}(\\phi + \\theta ) &= \\rho ^{\\mathrm {D}_N}_\\text{reg}(g) \\mathsf {K}^{\\mathrm {D}_N}_{j, k \\rightarrow \\text{reg}}{\\psi _{j, k}(g)}^{-1}.$ We then generalize () on $\\mathsf {K}^{\\mathrm {C}_N}_{k \\rightarrow \\text{reg}}$ and $\\overline{\\mathsf {K}}^{\\mathrm {C}_N}_{k \\rightarrow \\text{reg}}$ given a reflected action $g = (1, i_1)$ : $&\\mathsf {K}^{\\mathrm {C}_N}_{k \\rightarrow \\text{reg}}(-\\phi + \\theta _{i_1})\\\\&= \\operatorname{diag}\\big (\\mathsf {K}(-\\phi + \\theta _{i_1})\\big ) \\beta _k\\\\&= B(i_1) \\operatorname{diag}(\\overline{\\mathsf {K}}) B(i_1)^{-1} \\beta _k, \\quad \\text{c.f.", "(\\ref {eq:beta-rotate1})}\\\\&= B(i_1) \\operatorname{diag}(\\overline{\\mathsf {K}}) \\beta _k \\psi _{0, k}(g)^{-1}\\\\&= -B(i_1) \\operatorname{diag}(\\overline{\\mathsf {K}}^{\\mathrm {C}_N}) \\beta _k \\psi _{1, k}(g)^{-1}\\\\&= B(i_1) \\operatorname{diag}(\\mathsf {K}^{\\mathrm {C}_N}) \\beta _k \\psi _{0, k}(g)^{-1}\\\\&= -B(i_1) \\operatorname{diag}(\\mathsf {K}^{\\mathrm {C}_N}) \\beta _k \\psi _{1, k}(g)^{-1}.$ Note that (REF ) and () both have two equivalent forms denoted with $\\psi _{0, k}(g)$ and $\\psi _{1, k}(g)$ respectively.", "Now we can show $\\mathsf {K}^{\\mathrm {D}_N}_{j, k \\rightarrow \\text{reg}}$ follows Lemma  for $j=0$ , $i_0=1$ , i.e.", "$g = (1, i_1)$ as: $&\\mathsf {K}^{\\mathrm {D}_N}_{j, k \\rightarrow \\text{reg}}(-\\phi + \\theta _{i_1})\\\\&=\\begin{bmatrix}{\\mathsf {K}^{\\mathrm {C}_N}_{k \\rightarrow \\text{reg}}(-\\phi + \\theta _{i_1})}^\\top & {\\overline{\\mathsf {K}}^{\\mathrm {C}_N}_{k \\rightarrow \\text{reg}}}(-\\phi + \\theta _{i_1})^\\top \\end{bmatrix}^\\top \\\\\\begin{split}&= \\left[\\begin{matrix}B(i_1) \\operatorname{diag}(\\overline{\\mathsf {K}}^{\\mathrm {C}_N}) \\beta _k \\psi _{0, k}(g)^{-1} \\end{matrix}\\right.\\\\&\\qquad \\qquad \\left.\\begin{matrix} B(i_1) \\operatorname{diag}(\\mathsf {K}^{\\mathrm {C}_N}) \\beta _k \\psi _{0, k}(g)^{-1}\\end{matrix}\\right]^\\top ,\\\\&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\text{c.f.", "(\\ref {eq:irrep-exchange})}\\\\\\end{split}\\\\\\begin{split}&= \\rho ^{\\mathrm {D}_N}_\\text{reg}(g) \\begin{bmatrix}{\\mathsf {K}^{\\mathrm {C}_N}_{k \\rightarrow \\text{reg}}}^\\top & {\\overline{\\mathsf {K}}^{\\mathrm {C}_N}_{k \\rightarrow \\text{reg}}}^\\top \\end{bmatrix}^\\top \\psi _{0, k}(g)^{-1},\\\\&\\quad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\text{c.f.", "(\\ref {eq:irrep-cn-kernel}), (\\ref {eq:irrep-cn-kernel-conj})}\\\\\\end{split}\\\\&= \\rho ^{\\mathrm {D}_N}_\\text{reg}(g) \\mathsf {K}^{\\mathrm {D}_N}_{j, k \\rightarrow \\text{reg}}\\psi _{0, k}(g)^{-1}.$ The verification is similar for $j=1$ , $i_0=1$ , i.e.", "$g = (1, i_1)$ : $&\\mathsf {K}^{\\mathrm {D}_N}_{j, k \\rightarrow \\text{reg}}(-\\phi + \\theta _{i_1})\\\\&=\\begin{bmatrix}{\\mathsf {K}^{\\mathrm {C}_N}_{k \\rightarrow \\text{reg}}(-\\phi + \\theta _{i_1})}^\\top & -{\\overline{\\mathsf {K}}^{\\mathrm {C}_N}_{k \\rightarrow \\text{reg}}}(-\\phi + \\theta _{i_1})^\\top \\end{bmatrix}^\\top \\\\\\begin{split}&= \\left[\\begin{matrix} -B(i_1) \\operatorname{diag}(\\overline{\\mathsf {K}}^{\\mathrm {C}_N}) \\beta _k \\psi _{1, k}(g)^{-1} \\end{matrix}\\right.\\\\&\\qquad \\qquad \\left.\\begin{matrix} B(i_1) \\operatorname{diag}(\\mathsf {K}^{\\mathrm {C}_N}) \\beta _k \\psi _{1, k}(g)^{-1}\\end{matrix}\\right]^\\top , \\\\&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\text{c.f.", "(\\ref {eq:irrep-exchange})}\\\\\\end{split}\\\\\\begin{split}&= \\rho ^{\\mathrm {D}_N}_\\text{reg}(g) \\begin{bmatrix}{\\mathsf {K}^{\\mathrm {C}_N}_{k \\rightarrow \\text{reg}}}^\\top & -{\\overline{\\mathsf {K}}^{\\mathrm {C}_N}_{k \\rightarrow \\text{reg}}}^\\top \\end{bmatrix}^\\top \\psi _{0, k}(g)^{-1},\\\\&\\quad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\text{c.f.", "(\\ref {eq:irrep-cn-kernel}), (\\ref {eq:irrep-cn-kernel-conj})}\\\\\\end{split}\\\\&= \\rho ^{\\mathrm {D}_N}_\\text{reg}(g) \\mathsf {K}^{\\mathrm {D}_N}_{j, k \\rightarrow \\text{reg}}\\psi _{0, k}(g)^{-1}.$" ], [ "Verification of Lemma ", "This kernel can be verified as follows for $g = (0, i_1)$ : $&\\mathsf {K}^{\\mathrm {C}_N}_{\\text{reg} \\rightarrow \\text{reg}}(\\phi + \\theta _{i_1})\\\\&=\\begin{bmatrix} \\mathsf {K}^{\\mathrm {C}_N}_{0 \\rightarrow \\text{reg}}(\\phi + \\theta _{i_1}) \\cdots \\mathsf {K}^{\\mathrm {C}_N}_{{\\frac{N}{2}} \\rightarrow \\text{reg}}(\\phi + \\theta _{i_1}) \\end{bmatrix} V^{-1}\\\\\\begin{split}&= \\left[\\begin{matrix} \\rho ^{\\mathrm {C}_N}_\\text{reg}(g) \\mathsf {K}^{\\mathrm {C}_N}_{0 \\rightarrow \\text{reg}}\\psi _{0, 0}(g)^{-1}, \\cdots ,\\end{matrix}\\right.\\\\&\\qquad \\qquad \\left.\\begin{matrix}\\rho ^{\\mathrm {C}_N}_\\text{reg}(g) \\mathsf {K}^{\\mathrm {C}_N}_{{\\frac{N}{2}} \\rightarrow \\text{reg}}\\psi _{0, {\\frac{N}{2}}}(g)^{-1} \\end{matrix}\\right] V^{-1},\\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\text{c.f.", "(\\ref {eq:cn-irrep-lemma1})}\\\\\\end{split}\\\\&= \\rho ^{\\mathrm {C}_N}_\\text{reg}(g) \\begin{bmatrix} \\mathsf {K}^{\\mathrm {C}_N}_{0 \\rightarrow \\text{reg}}\\cdots \\mathsf {K}^{\\mathrm {C}_N}_{{\\frac{N}{2}} \\rightarrow \\text{reg}}\\end{bmatrix} D^{\\mathrm {C}_N} V^{-1}\\\\&= \\rho ^{\\mathrm {C}_N}_\\text{reg}(g) \\begin{bmatrix} \\mathsf {K}^{\\mathrm {C}_N}_{0 \\rightarrow \\text{reg}}\\cdots \\mathsf {K}^{\\mathrm {C}_N}_{{\\frac{N}{2}} \\rightarrow \\text{reg}}\\end{bmatrix} V^{-1} V D^{\\mathrm {C}_N} V^{-1}\\\\&= \\rho ^{\\mathrm {C}_N}_\\text{reg}(g) \\mathsf {K}^{\\mathrm {C}_N}_{\\text{reg} \\rightarrow \\text{reg}}{\\rho ^{\\mathrm {C}_N}_\\text{reg}}^{-1}.$" ], [ "Minimal Implementation", "To illustrate the simplicity of our approach and better explain the tensor operation in the construction of a steerable filter, we list the minimal self-contained PyTorch implementation with only 60 lines of code in this section.", "Note that this implementation is slightly different from the final version which will be released as open source later.", "In the implementation we use $\\mathtt {grp} = (0, N)$ and $(1, N)$ to denote $\\mathrm {C}_N$ and $\\mathrm {D}_N$ respectively.", "$\\mathtt {irreps}$ , $\\mathtt {in\\_irreps}$ and $\\mathtt {out\\_irreps}$ denote irreps by a $M \\times 2$ matrix, of which each row denotes an irrep $(j, k)$ .", "The steerable convolution operators $\\mathtt {IrrepToRegular}$ , $\\mathtt {RegularToIrrep}$ and $\\mathtt {RegularToRegular}$ preserve the same interface to PyTorch $\\mathtt {nn.Conv2d}$ .", "The input and output channels are flattened from a structure of $\\mathtt {grp[0] \\times \\mathtt {grp[1]} \\times \\mathtt {feature\\_multiplicity}}$ .", "mystyle float=*t,language=Python, label=lst:example, caption=Minimal implementation of the proposed approach.", "]minimal.py" ] ]
2105.11636
[ [ "Photon Ring and Observational Appearance of a Hairy Black Hole" ], [ "Abstract Recently, the image of a Schwarzschild black hole with an accretion disk has been revisited, and it showed that the \"photon ring\", defined as highly bent light rays that intersect the disk plane more than twice, is extremely narrow and makes a negligible contribution to the total brightness.", "In this paper, we investigate the observational appearance of an optically and geometrically thin accretion disk around a hairy black hole in an Einstein-Maxwell-scalar model.", "Intriguingly, we find that in a certain parameter regime, due to an extra maximum or an \"ankle-like\" structure in the effective potential for photons, the photon ring can be remarkably wide, thus making a notable contribution to the flux of the observed image.", "In particular, there appears a wide and bright annulus, which comprises multiple concentric bright thin rings with different luminosity, in the high resolution image." ], [ "Introduction", "The announcement of the first angular resolution image of the supermassive black hole M87* by the Event Horizon Telescope (EHT) collaboration is a significant event in observing astrophysical black holes [1], [2], [3], [4], [5], [6], [7], [8], which opens a new window to test general relativity in the strong field regime.", "The image contains two prominent features, a circular dark interior, dubbed “shadow”, and a bright ring which is closely relevant to a class of circular photon orbits (i.e., photon sphere).", "The shadow and photon sphere are originated from the light deflection by the strong gravitational field near the black hole [9], [10], [11], [12].", "Thus, it is believed that the shadow image encodes valuable information of the geometry around the black hole, especially in the vicinity of the horizon.", "Modelling the M87* with the Kerr black hole geometry, the observation was found to be in good agreement with the prediction of general relativity.", "Nevertheless, due to finite resolution of the M87* image, there still exists some space for alternative models to simulate the black hole image within observational uncertainty tolerance.", "To explore these possibilities, one can parameterize the deviations from the Kerr metric and compare the corresponding shadow image with the observed image [13], [14], [15], [16], [17], [18].", "Alternatively, shadows and photon spheres of black holes are widely studied in the context of various specific theories including new physics, e.g., the nonlinear electrodynamics [19], [20], [21], [22], [23], the Gauss-Bonnet theory [24], [25], [26], [27], fuzzball [28], the Chern-Simons type theory [29], [30], $f(R)$ gravity [31], [32], and string inspired black holes [33], [34], [35], [36].", "Moreover, there could exist some exotic ultra-compact objects acting as black hole mimickers [37].", "The gravitational lensing by various horizonless objects, such as wormholes [38], [39], [40] and bosonic stars [41], [42], has been detailedly analyzed.", "Their shadow images are usually distinct from those of black holes, but it is hard to distinguish between them at the current EHT resolution.", "Interestingly, it was argued in [43], [44], [45] that naked singularities can cast a shadow in the absence of the photon sphere.", "Furthermore, the EHT observation can also be applied to impose constraints on the cosmological parameters [46], [47], [48], [49], [50], [51] and the size of extra dimensions [52], [53], [54], test the equivalence principle [55], [56], [57], and probe some fundamental physics issues including dark matter [58], [59], [60], [61], [62], [21], [63] and dark energy [64], [65], [66], [67].", "On the other hand, an astrophysical black hole is generally believed to be surrounded by a luminous accretion flow, which is an essential ingredient in obtaining the black hole image.", "In fact, the realistic image is a result of the complex interactions between the strong gravitational lensing of the black hole and the electromagnetic plasma in the accretion flow, which requires intensive numerical general relativistic magnetohydrodynamic (GRMHD) simulations [68].", "Nevertheless, simplified accretion models usually suffice to capture major features of black hole images, and hence have been widely investigated in the literature, e.g., spherical accretion flows [43], [26], [65], [66], [69], [63], thin or thick accretion disks [70], [71], [72], [73], [74], [75], [76], [77].", "In particular, the authors in [72] considered the emission from an optically and geometrically thin disk around a Schwarzschild black hole, which is divided into three classes by the number $m$ of half-orbits that an emitted photon completes around the black hole before reaching the observer: the direct emission ($m\\le 1$ ), the lensing ring ($m=2$ ) and the photon ring ($m\\ge 3$ ).", "It showed that the lensing ring superimposed upon the direct emission produces a thin ring of twice the background intensity in the black hole image, while the photon ring, which picks up larger intensity, makes negligible contributions to the total observed brightness due to its exponential narrowness.", "Nevertheless, observations of the photon ring would provide a new and powerful tool to probe a black hole spacetime [78].", "Recently, experimental methods have been proposed to detect the photon ring by measuring its interferometric signatures [79] and two-point correlation function of intensity fluctuations [80].", "No-hair theorem states that a black hole can be completely characterized by only three observable classical parameters: mass, electric charge, and spin [81].", "However, various models of hairy black holes (HBHs) have been proposed to circumvent the no-hair theorem (see [82] for a review).", "Testing the no-hair theorem with observations would provide powerful probes of alternative theories of gravity.", "The observation of the black hole shadow would enable tests of the no-hair theorem, and therefore studying shadows of HBHs with various hairs has attracted great attention, e.g., axion-like hairs [83], [84], dilaton-like hairs [85], [86], [87] and others [88], [89], [90], [91], [92], [93], [94].", "Interestingly, in [95] we found that there can exist two unstable photon spheres outside the event horizon for the HBH solutions in the Einstein-Maxwell-scalar (EMS) theory proposed in [96].", "This novel feature results in two concentric bright rings of different radii in the observed image of the HBH surrounded by an optically thin, spherical accretion flow.", "Note that the presence of more than one photon sphere has also been reported in other scenarios, e.g., horizonless ultra-compact objects [97], Morris-Thorne type wormholes [38], [98], reflection-asymmetric wormholes [39], [40], [99], [100], and a Schwarzschild black hole surrounded by a certain matter distribution [101].", "To investigate the effects of two unstable photon spheres on the photon ring, we consider the observational appearance of the aforementioned HBH surrounded by an optically and geometrically thin accretion disk.", "Remarkably, we show that, in a certain parameter regime, the existence of two photon spheres or its reminiscence can significantly extend the photon ring band, which thus makes a non-trivial contribution to the total observed intensity, comparable to that of the lensing ring.", "The rest of the paper is organized as follows.", "In Sec.", ", we briefly review the HBH solutions, the behavior of null geodesics and the observed intensity of the accretion disk.", "Sec.", "contains our main numerical results, which include effective potentials for photons, accretion disk images seen by a distant observer and the dependence of the photon ring on the HBH charge and scalar coupling.", "We conclude with a brief discussion in Sec.", ".", "We set $16\\pi G=1$ throughout the paper." ], [ "Set Up", "Consider a specific 4D EMS theory with an exponential scalar-electromagnetic coupling given by [96] $S=\\int d^{4}x\\sqrt{-g}\\left[ \\mathcal {R}-2\\partial _{\\mu }\\phi \\partial ^{\\mu }\\phi -e^{\\alpha \\phi ^{2}}F_{\\mu \\nu }F^{\\mu \\nu }\\right] , $ where $\\mathcal {R}$ is the Ricci scalar, the scalar field $\\phi $ is minimally coupled to the metric $g_{\\mu \\nu }$ and non-minimally coupled to the electromagnetic field $A_{\\mu }$ , and $F_{\\mu \\nu }=\\partial _{\\mu }A_{\\nu }-\\partial _{\\nu }A_{\\mu }$ is the electromagnetic tensor.", "Many properties of this model and its extensions have been explored in the literature, e.g., various non-minimal coupling functions [102], [103], dyons including magnetic charges [104], axionic-type couplings [105], massive and self-interacting scalar fields [106], [107], horizonless reflecting stars [108], stability analysis of the HBHs [109], [110], [111], [112], [113], higher dimensional scalar-tensor models [114], quasinormal modes of the HBHs [115], [116], two U(1) fields [117], quasi-topological electromagnetism [118], topology and spacetime structure influences [119], the Einstein-Born-Infeld-scalar theory [120] and with a negative cosmological constant [121], [122].", "Starting with the static and spherically symmetric black hole solution ansatz, $ds^{2}=-N(r)e^{-2\\delta (r)}dt^{2}+\\frac{dr^{2}}{N(r)}+r^{2}\\left( d\\theta ^{2}+\\sin ^{2}\\theta d\\varphi ^{2}\\right) ,\\qquad \\mathbf {A}=A_{t}dt=V(r)dt,$ we obtain the equations of motion $2m^{\\prime }(r)-r^{2}N(r)\\phi ^{\\prime }(r)^{2}-e^{2\\delta (r)+\\alpha \\phi (r)^{2}}r^{2}V^{\\prime }(r)^{2} & =0,\\nonumber \\\\\\delta ^{\\prime }(r)+r\\phi ^{\\prime }(r)^{2} & =0,\\nonumber \\\\\\left[ e^{-\\delta (r)}r^{2}N(r)\\phi ^{\\prime }(r)\\right] ^{\\prime }-\\alpha e^{\\delta (r)+\\alpha \\phi (r)^{2}}\\phi (r)r^{2}V^{\\prime }(r)^{2} &=0,\\\\\\left[ e^{\\delta (r)+\\alpha \\phi (r)^{2}}r^{2}V^{\\prime }(r)\\right] ^{\\prime }& =0,\\nonumber $ where $N(r)\\equiv 1-2m(r)/r$ can be expressed in terms of the Misner-Sharp mass function $m(r)$ , and primes denote derivatives with respect to $r$ .", "The last equation in Eq.", "$\\left( \\ref {eq:eom}\\right) $ yields $V^{\\prime }(r)=-e^{-\\delta (r)-\\alpha \\phi (r)^{2}}Q/r^{2}$ , where the constant $Q$ can be interpreted as the electric charge of the black hole.", "To solve the above non-linear ODEs, suitable boundary conditions at the event horizon of radius $r_{h}$ and spatial infinity shall be imposed as $m(r_{h}) & =\\frac{r_{h}}{2},\\qquad \\delta (r_{h})=\\delta _{0},\\qquad \\phi (r_{h})=\\phi _{0},\\qquad V(r_{h})=0,\\nonumber \\\\m(\\infty ) & =M,\\qquad \\delta (\\infty )=0,\\qquad \\phi (\\infty )=0,\\qquad V(\\infty )=\\Psi ,$ where $\\delta _{0}$ and $\\phi _{0}$ are two positive constants, $M$ is the ADM mass, and $\\Psi $ is the electrostatic potential.", "The scalar-free black hole solution with $\\phi =0$ corresponds to Reissner-Nordström black holes (RNBHs).", "When the dimensionless coupling $\\alpha $ is larger than $1/4$ , there exist HBH solutions with a non-trivial profile of the scalar field $\\phi (r)$ [96], [102], [120].", "In this paper, we focus on the fundamental state of the HBH solutions, which means that $\\phi (r)$ remains positive outside the event horizon.", "The behavior of a photon traveling outside the HBH can be encapsulated in the null geodesic equation, $\\frac{d^{2}x^{\\mu }}{d\\lambda ^{2}}+\\Gamma _{\\rho \\sigma }^{\\mu }\\frac{dx^{\\rho }}{d\\lambda }\\frac{dx^{\\sigma }}{d\\lambda }=0, $ where $\\lambda $ is the affine parameter, and $\\Gamma _{\\rho \\sigma }^{\\mu }$ is the Christoffel symbol.", "In the appendix, we show that light rays propagating in the HBH spacetime indeed are determined by the null geodesic equation.", "Due to the spherical symmetry, we only consider light rays moving on the equatorial plane with $\\theta =\\pi /2$ .", "Combining $ds^{2}=0$ and Eqs.", "$\\left(\\ref {eq:metric ansatz}\\right) $ and $\\left( \\ref {eq:geodesic}\\right) $ , one obtains the time, azimuthal and radial components of the null geodesic, $\\frac{dt}{d\\lambda } & =\\frac{1}{bN(r)e^{-2\\delta (r)}},\\nonumber \\\\\\frac{d\\varphi }{d\\lambda } & =\\pm \\frac{1}{r^{2}},\\\\e^{-2\\delta (r)}\\left( \\frac{dr}{d\\lambda }\\right) ^{2} & =\\frac{1}{b^{2}}-\\frac{e^{-2\\delta (r)}N(r)}{r^{2}},\\nonumber $ where $\\pm $ in the second line corresponds to the light rays moving in the counterclockwise $(+)$ or clockwise $(-)$ along $\\varphi $ -direction.", "The impact parameter $b$ is defined as $|L|/E$ , where $L$ and $E$ are the conserved angular momentum and energy of the photons, respectively.", "Note that we use a redefined affine parameter $\\lambda \\rightarrow \\lambda /|L|$ in Eq.", "$\\left( \\ref {eq:light ray eom}\\right) $ .", "From the last equation of Eq.", "$\\left( \\ref {eq:light ray eom}\\right) $ , one can define the effective potential of light rays as $V_{\\text{eff}}(r)=\\frac{e^{-2\\delta (r)}N(r)}{r^{2}}.", "$ Particularly, an unstable photon sphere (or equivalently, a circular and unstable null geodesic) is determined by $V_{\\text{eff}}(r_{ph})=\\frac{1}{b_{ph}^{2}},\\qquad V_{\\text{eff}}^{\\prime }(r_{ph})=0,\\qquad V_{\\text{eff}}^{\\prime \\prime }(r_{ph})<0,$ where $r_{ph}$ is the radius of the photon sphere, and $b_{ph}$ is the corresponding impact parameter.", "In this paper, we focus on unstable photon spheres since they play an important role in determining the accretion disk image seen by a distant observer.", "For convenience, photon spheres are referred to unstable photon spheres in the remainder of this paper unless we make an explicit statement.", "In this paper, we consider that the HBH is surrounded by a static and geometrically thin accretion disk, which is assumed to radiate isotropically in the rest frame of the matter.", "In addition, we take the disk emission to be optically thin by neglecting the absorption effect.", "Note that there exist compelling evidences indicating that an optically thin hot accretion flow may surround M87* or Sgr A* [123], [124].", "To obtain the accretion disk image perceived by a distant observer, we evolve light rays from the observer's position backwards in time.", "Furthermore, one can use $m$ , the times that a certain ray intersects with the disk plane outside the horizon, to distinguish light rays' behavior.", "Following the definitions in [72], light rays with $m\\le 1$ , $m=2$ and $m\\ge 3$ constitute the direct emission, the lensing ring and the photon ring, respectively.", "The observed total intensity $F_{o}(b)$ generated by a light ray of impact parameter $b$ is a sum of the intensities from each intersection with the disk plane outside the horizon, $F_{o}(b)=\\int _{\\nu _{o}}d\\nu _{o}I_{\\nu _{o}}(b)=\\underset{m}{\\sum }\\left.\\int _{\\nu _{e}}g(r)d\\nu _{e}g^{3}(r)I_{\\nu _{e}}(r)\\right|_{r=r_{m}(b)}=\\underset{m}{\\sum }\\left.", "g^{4}(r)F_{e}(r)\\right|_{r=r_{m}(b)},$ where $g(r)$ is the red-shift factor, $I_{\\nu _{e}}$ is the specific intensity at the emission frequency $\\nu _{e}$ , $I_{\\nu _{o}}$ is the specific intensity at the observed frequency $\\nu _{o}$ , and $F_{e}=\\int _{\\nu _{e}}I_{\\nu _{e}}d\\nu _{e}$ is the total emitted intensity.", "Here, we have applied the $I_{\\nu _{o}}=g^{3}(r)I_{\\nu _{e}}$ by Liouville's theorem in the second equality.", "The function $r_{m}(b)$ $(m=1,2,3...)$ , dubbed the transfer function, is the radial coordinate of the light ray crossing the disk plane at the $m^{th}$ time.", "Moreover, the slope of the transfer function, $dr_{m}/db$ , is the demagnification factor of the $m^{th}$ image of the accretion disk.", "For simplicity, we set the total emitted intensity $F_{e}(r)=1/r^{2}$ with $r>r_{h}$ , which suffices to illustrate the major features of the accretion disk image.", "In this case, the observed total intensity is given by $F_{o}(b)=\\underset{m}{\\sum }\\left.", "\\frac{N^{2}(r)e^{-4\\delta (r)}}{r^{2}}\\right|_{r=r_{m}(b)}, $ where we use $g(r)=\\sqrt{N(r)}e^{-\\delta (r)}$ [95].", "When viewed in a face-on orientation, the 2D image of the accretion disk is circularly symmetric.", "Taking account of the geometric interpretation of $b$ , we can depict the 2D face-on image by employing $F_{o}(b)=F_{o}(\\sqrt{X^{2}+Y^{2}})$ , where the coordinates $(X,Y)$ span the observer's celestial plane." ], [ "Numerical Results", "In this section, we present the numerical results about the optical appearance of the accretion disk surrounding the HBH black hole, viewed face-on.", "In the left panel of Fig.", "REF , we first display the domain of existence for HBH solutions to the action $\\left( \\ref {eq:action}\\right) $ , which is bounded by the sets of existence and critical solutions, denoted by $Q_{\\text{exi}}$ -line (red line) and $Q_{\\text{cr}}$ -line (blue line) in the $Q$ -$\\alpha $ plane, respectively.", "On the $Q_{\\text{cr}}$ -line, the black hole horizon radius vanishes with the black hole mass and charge remaining finite.", "Furthermore, this domain can be divided into four parameter space regions, in one of which the effective potentials $\\left( \\ref {eq:Veff}\\right) $ have distinct profiles, e.g., the number of the maxima.", "In particular, we obtain four families of the HBH solutions, Single-peak I (green region): The potential has a single maximum, which is similar to Schwarzschild and RN black holes [125], [69].", "For instance, the $\\alpha =10$ case (black line) in the middle panel of Fig.", "REF .", "Single-peak II (blue region): The potential has a single maximum and an “ankle-like” structure, e.g., the $\\alpha =0.9$ case (green line) in the middle panel.", "The ankle-like structure is characterized by a flattening of the potential, and corresponds to a transition between convexity and concavity of the effective potential.", "More precisely, its presence can be determined by the appearance of $V_{\\text{eff}}^{\\prime \\prime }(r)>0$ in the region where $V_{\\text{eff}}^{\\prime }(r)<0$ .", "Double-peak I (orange region): The potential has two maxima at two different radii, and the peak of the potential at the smaller radius is lower than that at the larger radius, e.g., the $\\alpha =0.6$ case (pink line) in the middle panel.", "Double-peak II (red region): The potential has two maxima at two different radii, and the peak of the potential at the smaller radius is higher than that at the larger radius, e.g., the $\\alpha =0.7$ case (orange line) in the middle panel.", "It is observed that the single-peak I family occupies almost the whole HBH existence regime, whereas the other three families only exist in a small $\\alpha $ region near the $Q_{\\text{cr}}$ -line.", "To illustrate how the potential profile changes among different families, we display a set of potentials along the lines with fixed $Q_{r}\\equiv (Q-Q_{\\text{exi}})/(Q_{\\text{cr}}-Q_{\\text{exi}})=0.98$ and fixed $\\alpha =0.9$ in the middle and right panels of Fig.", "REF , respectively.", "As $\\alpha $ increases with $Q_{r}=0.98$ , the middle panel (specifically, the inset therein) shows that the potential first has a single peak, then another peak at a smaller radius appears and grows, meanwhile the peak at the larger radius shrinks until disappears, indicating the single-peak I $\\rightarrow $ double-peak I $\\rightarrow $ double-peak II $\\rightarrow $ single-peak II $\\rightarrow $ single-peak I transition.", "When $Q_{r}$ increases with fixed $\\alpha =0.9$ , the right panel and the inset therein present the single-peak I $\\rightarrow $ double-peak I $\\rightarrow $ double-peak II $\\rightarrow $ single-peak II transition.", "In what follows, we display several representative cases to show the main features of each family.", "Note that the HBH mass $M$ is set to 1 without loss of generality in the rest of this section." ], [ "Single-peak potential", "For the single-peak I family, we consider a HBH with $\\alpha =0.9$ and $Q=1.03$ in Fig.", "REF .", "The potential with one maximum indicates a single photon sphere with radius $r_{ph}$ and the corresponding impact parameter $b_{ph}$ .", "To consider the accretion disk viewed from a face-on orientation, we assume that an observer is placed at the far right of the upper-right panel, corresponding to the “north pole direction”, and the disk lies in the equatorial plane with respect to the observer's orientation (dashed green line in the upper-right panel).", "Tracing a light ray backwards from the observer, the total orbit number $n$ is related to the total disk-crossing times $m$ as $n<0.5$ $\\rightarrow m=0$ , $0.5\\le n<0.75$ $\\rightarrow m=1$ , $0.75\\le n<1.25$ $\\rightarrow m=2$ and $n\\ge 1.25$ $\\rightarrow m\\ge 3$ .", "By definition, $m\\le 1$ , $m=2$ and $m\\ge 3$ correspond to the direct emission, the lensing ring and the photon ring, respectively [72].", "In the upper-middle panel, we plot $n$ as a function of $b$ and depict the direct, lensing and photon ring bands in gray, orange and red, respectively.", "To obtain the observed intensity, we first calculate the transfer functions $r_{m}(b)$ with $m=1,2,3,\\cdots $ .", "In the lower-left panel of Fig.", "REF , we depict the transfer functions for $m=1,2$ and 3, which are associated with the direct, lensing and photon ring bands, respectively.", "Since the average slope of $r_{m}(b)$ roughly reflects the demagnified level of the $m^{th}$ image of the disk plane, it shows that the secondary image is highly demagnified, and the tertiary image is extremely demagnified.", "Via Eq.", "$\\left( \\ref {eq:intensity2}\\right) $ , the observed intensity $F_{o}$ as a function of $b$ is shown in the lower-middle panel, which presents a narrow spike-like photon ring (red) and a broader bump-like lensing ring (orange) superimposed on the direct emission (gray).", "One can see that the direct emission makes the dominant contribution to the overall intensity flux, whereas the lensing$\\backslash $ photon ring makes modest$\\backslash $ little contributions.", "To present the 2D image of the accretion disk seen by a distant observer, we project $F_{o}(b)$ to the observer's celestial $(X,Y)$ -plane via $b^{2}=X^{2}+Y^{2}$ .", "In the lower-right panel, the observational appearance of the photon ring is shown to be a thin bright ring of radius $b_{ph}$ , which suggests that the photon ring makes a very small contribution to the total flux.", "Moreover, there exists a completely dark area with vanishing intensity inside the photon ring, which is determined by the $m=0$ band.", "Figure: Behavior of photons in a HBH with α=0.9\\alpha =0.9 and Q=1.074Q=1.074, whichis in the single-peak II family, and the observational appearance of anoptically and geometrically thin accretion disk around the HBH, viewed from aface-on orientation.", "The effective potential shows a flattening around logr∼1\\log r\\sim 1 (Upper Left), which leads to a finite broad peak in thenn-bb plane (Upper Middle).", "As a result, the width of the photonring is significantly increased, and comparable to that of the lensing ring.Therefore, the photon ring can make a non-trivial net contribution to theobserved flux F o (b)F_{o}(b) (Lower Middle).", "Specially, a bright and wideannulus, consisting of multiple concentric bright thin rings, appears outsidethe bright ring, corresponding to the photon sphere, in the accretion diskimage (Lower Right).", "Moreover, as indicated by the transferfunctions, the secondary and tertiary images are less demagnified than thosein the single-peak I family (Lower Left).On the other hand, the photon ring can play a non-trivial role in the observational appearance of an accretion disk around a HBH in the single-peak II family.", "A HBH with $\\alpha =0.9$ and $Q=1.074$ is considered in Fig.", "REF , the upper-left panel of which shows that the effective potential has an ankle-like structure around $\\log r\\sim 1$ .", "This distinctive feature results in more complicated behavior of null geodesics and observational appearance of emission from an accretion disk.", "Indeed, due to the flattening of the ankle-like structure, photons can revolve around the ankle-like structure multiple times, and cross the disk plane more than twice.", "So the upper-middle panel displays that, in addition to the sharp peak determined by the photon sphere at $b=b_{ph}$ , the $n(b)$ curve also possesses a much broader peak of finite height at $b>b_{ph}$ , which significantly widens the photon ring band.", "Note that a similar phenomenon has been reported in a wormhole scenario [38].", "Consequently, as shown in the lower-middle pane, the photon ring makes a noticeable contribution to the total intensity flux, comparable to that of the lensing ring.", "Therefore, the photon ring plays an important role in determining the observational appearance of the accretion disk, leading to the presence of a bright ring and a concentric bright annular region, which can be observed in the lower-right panel.", "While the bright ring at the smaller radius, corresponding to the photon sphere, is barely visible due to the sharpness of the peak of $F_{o}(b)$ at $b=b_{ph}$ , the wide bright annular region at the larger radius is quite noticeable and comprises multiple concentric thin bright rings with different luminosity.", "Interestingly, the second and third transfer functions both exist over a larger range of $b$ than those in the single-peak I family (lower-left panels of Figs.", "REF and REF ), thus resulting in less demagnification for secondary and tertiary images in the single-peak II family." ], [ "Double-peak potential", "The prominent character of a double-peak potential is the presence of two maxima at $r=r_{ph1}$ and $r=r_{ph2}$ with $r_{ph1}<r_{ph2}$ , corresponding to two photon spheres of radii $r_{ph1}$ and $r_{ph2}$ outside the horizon, respectively.", "The associated maximum values of the potential are $1/b_{ph1}^{2}$ and $1/b_{ph2}^{2}$ , where $b_{ph1}$ and $b_{ph2}$ are the impact parameters of the photon spheres (see the upper-left panels of Fig.", "REF and REF ).", "As mentioned before, there are two families of double-peak potentials according to the magnitudes of the two maximum values of the potentials.", "For the double-peak I potential, the maximum at the smaller radius is lower than that at the larger radius (or equivalently, $b_{ph1}>b_{ph2}$ ).", "So the light rays revolving around the photon sphere of smaller radius can not escape to the infinity, rendering this photon sphere invisible to a distant observer.", "As a result, accretion disk images in the double-peak I family closely resemble those in the single-peak I family.", "On the other hand, both photon spheres can be responsible for obtaining the image of an accretion disk in the double-peak II family.", "In fact, two bight rings of arbitrarily large brightness, whose radii are $b_{ph1}$ and $b_{ph2}$ , respectively, can appear in the image of the accretion disk.", "In what follows, we display two representative cases of the double-peak II family.", "In Fig.", "REF , we consider a HBH with $\\alpha =0.9$ and $Q=1.064$ , for which the difference between $b_{ph1}$ and $b_{ph2}$ is small.", "As expected, the existence of two photon spheres endows both $n(b)$ and $F_{o}(b)$ curves with two infinite peaks at radii $b_{ph1}$ and $b_{ph2}$ .", "As shown in the upper-middle panel, this two-peak structure extends the width of the photon ring band, compared to the single-peak I case.", "It is observed in the lower-middle panel that the photon ring can make a non-negligible contribution to the observed intensity, which is comparable to that of the lensing ring.", "Moreover, the photon ring leads to some internal structure between the two peaks for the observed intensity.", "However, since the photon ring is not wide enough, the lower-right panel shows that the internal structure can hardly been seen in the 2D image, and the observational appearance of the photon ring is almost indistinguishable from that of a thin bright ring.", "Figure: Behavior of photons in a HBH with α=0.9\\alpha =0.9 and Q=1.07Q=1.07, which isin the double-peak II family, and the observational appearance of an opticallyand geometrically thin accretion disk around the HBH, viewed from a face-onorientation.", "Compared to Fig.", ", the photon ring herebecomes significantly wider (Upper Middle and Lower Middle) since thetwo maximum values of the effective potential are well separated(Upper Left).", "Therefore, the photon ring can make a sizablecontribution to the total flux (Lower Middle), and the internalstructure of the photon ring, which is made up of multiple concentric brightrings, can be observed in the 2D image (Lower Right).In Fig.", "REF , we consider another HBH with $\\alpha =0.9$ and $Q=1.07$ , for which the difference between $b_{ph1}$ and $b_{ph2}$ is large.", "As shown in the upper-left panel, the two maximum values of the effective potential are quite different, thus leading to a remarkably wider photon ring.", "The lower-middle panel exhibits that the lensing and photon rings both contribute appreciably to the total observed intensity, and the photon ring is notably wider than that in Fig.", "REF .", "In the lower-right panel, the 2D observed image is shown to have a bright thin ring at $b=b_{ph1}$ and a bright annulus around $b=b_{ph2}$ , which consists of a bright ring at $b=b_{ph2}$ and multiple concentric bright rings with different luminosity.", "In other words, the inner structure of the photon ring can be seen in the observed image since the photon ring is wide enough.", "From the lower-left panels in Figs.", "REF and REF , it shows that the secondary and tertiary images in Fig.", "REF are less demagnified than those in Fig.", "REF .", "It is noteworthy that the ankle-like structure of the potential in Fig.", "REF is reminiscent of the maximum of the potential at $r=r_{ph2}$ in Fig.", "REF ." ], [ "Dependence on black hole charge and scalar coupling", "Here, we turn to investigate the dependence of the size of the photon ring and shadow on the HBH charge $Q$ and the scalar coupling $\\alpha $ .", "To study the dependence on $Q$ , we consider two cases with fixed $\\alpha =0.9$ and $\\alpha =10$ in the upper and lower rows of Fig.", "REF , respectively.", "Note that the allowed parameter regions are bounded by the $Q_{\\text{exi}}$ -line and/or the $Q_{\\text{cr}}$ -line, which are shown in Fig.", "REF .", "The upper-left panel of Fig.", "REF displays that, when $\\alpha =0.9$ , the HBHs belong to the single-peak I (region on the left of the orange region), single-peak II (region on the right of the red region), double-peak I (orange region) or double-peak II (red region) families, depending on the value of the HBH charge.", "From the upper middle panel, one can see that the photon ring (red region) of the HBHs in the single-peak II and double-peak II families are quite wide, thus indicating that the photon rings can play a relevant role in determining the observed accretion disk images.", "Moreover, as the HBH charge increases toward the $Q_{\\text{cr}}$ -line, the width of the photon ring grows until the photon ring splits into a narrow photon ring of smaller radius and a broad one of larger radius in the image plane.", "While the photon ring of smaller radius is barely visible in the observed 2D image, the one of larger radius can make a non-negligible contribution to the total intensity and the accretion disk image.", "Therefore, it is expected that the accretion disk image in the two photon rings scenario is quite similar to those shown in Figs.", "REF and REF .", "On the other hand, the HBH with $\\alpha =10$ is always in the single-peak I family, which is shown in the inset of the left panel of Fig.", "REF .", "In this case, the lower-middle panel of Fig.", "REF displays that the photon ring is always very narrow, and the lensing (orange and red regions) and photon rings decrease in size with $Q$ increasing toward $Q_{\\text{cr}}$ .", "Around $Q=Q_{\\text{cr}}$ , the lensing ring also becomes very narrow, and makes a negligible contribution to the observed intensity.", "In this paper, the term “standard shadow” is used to refer to the area inside the (smaller) photon sphere (i.e., the apparent boundary [10], [11]).", "Specifically, the radius of the standard shadow is the impact parameter of the smaller photon sphere if there exist two photon spheres [95].", "From the left column of Fig.", "REF , it is observed that the standard shadow of a HBH in the $\\alpha =0.9$ and $\\alpha =10$ cases shrinks with increasing $Q$ and vanishes at $Q=Q_{\\text{cr}}$ , where the HBH horizon becomes zero.", "For $\\alpha =0.9$ , the standard shadow radius decreases at a larger decreasing rate in the single-peak II and double-peak II families.", "In the middle column of Fig.", "REF , the black regions denote the completely dark area with vanishing intensity, and are also shown to decrease in size with increasing $Q$ and vanishes at $Q=Q_{\\text{cr}}$ .", "For comparison, we also plot the impact parameters of the photon spheres of RNBHs with $M=1$ in the right column of Fig.", "REF .", "In the coexisting $Q$ -range of the RNBH and HBH, one can see that the standard shadow of the RNBH is larger than that of the HBH in the $\\alpha =0.9$ case (upper-right panel of Fig.", "REF ), and vice verse in the $\\alpha =10$ case (lower-right panel of Fig.", "REF ), which is consistent with the result in [91].", "In Fig.", "REF , three cases with fixed $Q=0.8$ (left column), $Q=1.06$ (middle column) and $Q=2$ (right column) are presented to study the dependence on the scalar coupling $\\alpha $ .", "As shown in the upper row, the size of the standard shadow becomes larger for a stronger scalar coupling in all cases.", "For $Q=0.8$ , the ranges of the photon and lensing rings are fairly insensitive to $\\alpha $ .", "In the near $Q_{\\text{cr}}$ -line regime, HBHs with $Q=1.06$ are in the double-peak II family, and the photon ring's contribution to the observed intensity can not be neglected.", "On the other hand, the ranges of the photon and lensing rings of HBHs with $Q=2$ become zero at the $Q_{\\text{cr}}$ -line.", "Finally, it is interesting to note that the stronger $\\alpha $ , the larger the completely dark area becomes." ], [ "Blurred accretion disk images", "So far, we have considered high resolution images of HBHs surrounded by an accretion disk, which present some interesting features of the photon ring.", "To gain some insight into the effects of the photon ring on a realistic observation, we blur the images of the accretion disk in Figs.", "REF -REF with a Gaussian filter with standard derivation equal to 1/12 the field of view to mimic the EHT resolution [72], as shown in the lower row of Fig.", "REF .", "In Fig.", "REF , the white dotted circles represent critical circles, whose radii are the impact parameters of photon spheres, and the standard shadow is defined as the region inside the critical circle or that of smaller radius if there exist two photon spheres.", "In the first column of Fig.", "REF , we display the high resolution and blurred images of the HBH from Fig.", "REF , which is in the single-peak I family.", "The blurred image is primarily determined by the direct emission and the lensing ring, and has a blurred bright ring at the location of the lensing ring, which indicates that the observed shadow almost coincides with the standard shadow.", "In the second column, we exhibit the high resolution and blurred images of the HBH of the double-peak II family from Fig.", "REF , which has two critical circles of similar radii.", "Compared with the single-peak I family, the existence of two photon spheres slightly increases the brightness of the blurred bright ring.", "In addition, the observed shadow in the blurred image is almost same as the standard shadow.", "On the other hand, when the photon ring is wide enough (e.g., Figs.", "REF and REF ), its effects can become quite noticeable in the blurred images.", "Indeed, the high resolution and blurred images of the HBH of the double-peak II family from Fig.", "REF are shown in the third column of Fig.", "REF .", "In this case, the radii of the two critical circles are quite different, which significantly increases the width of the photon ring.", "Although the internal structure of the photon ring is washed out by the blurring, the wide photon ring leads to a brighter and wider blurred ring than in the single-peak I family.", "Since the inner edge of the blurred bright ring is at the smaller critical circle, the observed shadow nearly matches the standard shadow.", "In the last column, the high resolution and blurred images of the HBH of the single-peak II family from Fig.", "REF are presented.", "The ankle-like structure of the effective potential also notably increases the brightness and width of the blurred bright ring.", "Moreover, the blurred bright ring is at the location of the ankle-like structure, and hence has a large radius than the critical circle, which means that the observed shadow is larger than the standard shadow.", "In this paper, we investigated the behavior of null geodesics and observational appearance of a HBH surrounded by an optically and geometrically thin accretion disk in the EMS model with an exponential scalar coupling.", "Depending on the profile of the effective potentials for photons, we found that the HBH solutions are categorized into four different families, namely single-peak I, single-peak II, double-peak I and double-peak II.", "The single-peak I family dominates the allowed parameter space, while the other three families exist in a small region with small $\\alpha $ and large $Q$ .", "For the single-peak I solution, the photon ring is very narrow and makes little contribution to the total brightness.", "Therefore, the accretion disk images bear much similarity to these of various static black holes with a single-peak potential [72], [65], [74], [75], e.g., a bright ring due to the lensing ring shows up (Fig.", "REF ).", "On the other hand, an additional ankle or peak-like structure emerges in the single-peak II and double-peak II solutions, and results in the appearance of a new finite or infinite peak in the $n(b)$ curve.", "Consequently, the photon ring can become sufficiently broader, and its inner structure comes into play, resulting in two bright annuluses in the observed image (Figs.", "REF and REF ).", "The annulus of smaller radius is quite narrow, while the one of larger radius is much wider, comprising multiple concentric thin bight rings with different luminosity.", "To illustrate effects of the photon ring on observations, we presented the images of the HBHs at low resolutions, roughly corresponding to the EHT resolution (Fig.", "REF ).", "For the single-peak II and double-peak II solutions, the details of bright annuluses are washed out, giving a considerably bright and wide ring in the blurred image.", "We end with some comments on future studies from several perspectives.", "In this paper, we considered the concept “photon ring”, which is defined as the collection of all $m\\ge 3$ bands [72].", "Since we observed the complicate inner structure inside the wide photon ring (Figs.", "REF and REF ), it would be interesting to study this rich structure by performing detailed analysis on each $m\\ge 3$ band.", "Moreover, the ankle- and peak-like structures in the effective potentials in the aforementioned two cases behave like plateaus (upper-left panels of Figs REF and REF ), which is reminiscent of the potential profiles reported in [126], [127].", "It suggests that a marginally unstable photon sphere is likely to appear during the transition between the single-peak II and double-peak II families.", "In addition, apart from the exponential coupling of the EMS model considered in this paper, exploring other coupling functions may provide us more novel phenomena about the photon spheres and disk images.", "Finally, it would gain more insights into the modified gravity by considering a rotating HBH surrounded by an accretion disk and comparing it with black hole images released in future observations, such as the Next Generation Very Large Array [128], the Thirty Meter Telescope [129], and the BlackHoleCam [130].", "We thank Guangzhou Guo and Li Li for his helpful discussions and suggestions.", "This work is supported in part by NSFC (Grant No.", "11875196, 11375121, 11947225 and 11005016)." ], [ "Light Propagation in the EMS Model", "In this appendix, we use the geometric optics approximation [131], [132], [133] to derive the equations of geometric optics governing light propagation in the EMS model with the action $S=\\int d^{4}x\\sqrt{-g}\\left[ \\mathcal {R}-2\\partial _{\\mu }\\phi \\partial ^{\\mu }\\phi -e^{\\alpha \\phi ^{2}}F_{\\mu \\nu }F^{\\mu \\nu }\\right] .$ The equation of motion for the electromagnetic field $A^{\\mu }$ is then given by $\\partial _{\\mu }\\left[ \\sqrt{-g}e^{\\alpha \\phi ^{2}}F^{\\mu \\nu }\\right]=0.$ Assuming the wavelength of the photon is much smaller than the other physical scales of the system (i.e., the geometric-optics limit), we consider the ansatz $A^{\\mu }=\\operatorname{Re}\\left[ \\mathcal {A}^{\\mu }\\exp \\left( i\\psi \\right)\\right] ,$ where $\\mathcal {A}^{\\mu }$ is the slowly evolving amplitude, and $\\psi $ is a rapidly oscillating function of time and space.", "The wave vector $k_{\\mu }$ of light rays is identified as $k_{\\mu }\\equiv \\partial _{\\mu }\\psi .$ Putting the ansatz $\\left( \\ref {eq:ansatz}\\right) $ into Eq.", "$\\left(\\ref {eq:eomF}\\right) $ , we obtain the term of order $k^{2}$ , which is the leading term in the eikonal approximation, $k^{\\mu }k_{\\mu }=0\\text{,}$ where the Lorentz gauge $\\nabla _{\\mu }A^{\\mu }=0$ is used.", "Acting on the above equation with a covariant derivative leads to $k^{\\mu }\\nabla _{\\mu }k_{\\nu }=0,$ where we use $\\nabla _{\\mu }k_{\\nu }=\\nabla _{\\nu }k_{\\mu }$ .", "From Eq.", "$\\left(\\ref {eq:keqn}\\right) $ , it shows that light rays of the EMS model are described by null geodesics in the geometric-optics limit." ] ]
2105.11770
[ [ "Tunable Anderson Localization of Dark States" ], [ "Abstract Random scattering of photons in disordered one-dimensional solids gives rise to an exponential suppression of transmission, which is known as Anderson localization.", "Here, we experimentally study Anderson localization in a superconducting waveguide quantum electrodynamics system comprising eight individually tunable qubits coupled to a photonic continuum of a waveguide.", "Employing the qubit frequency control, we artificially introduce frequency disorder to the system and observe an exponential suppression of the transmission coefficient in the vicinity of its subradiant dark modes.", "The localization length decreases with the disorder strength, which we control in-situ by varying individual qubit frequencies.", "Employing a one-dimensional non-interacting model of coupled qubits and photons, we are able to support and complement the experimental results.", "The difference between our investigation and previous studies of localization in qubit arrays is the coupling via a common waveguide, allowing us to explore the localization of mediating photons in an intrinsically open system.", "The experiment opens the door to the study of various localization phenomena on a new platform, which offers a high degree of control and readout possibilities." ], [ "Experimental Setup", "The cryogenic microwave measurement setup is shown in Fig.", "REF .", "Figure: Sketch of the used cryogenic measurement setup." ], [ "Model Hamiltonian", "We consider a Hamiltonian $H = H_\\gamma + H_q + H_{\\gamma q}$ with terms corresponding to qubit part $H_q$ , photon part $H_\\gamma $ and coupling $H_{\\gamma q}$ between them: $H_q &= \\sum _{j=1}^N \\omega _j a_j^\\dagger a_j\\\\H_\\gamma &= \\sum _{n=1}^N \\omega _{k_n} \\tilde{b}^\\dagger _{k_n} \\tilde{b}_{k_n}\\\\H_{\\gamma q} &= g \\sum _{j, n=1}^N \\left( \\exp (\\mathrm {i} k_n d j) \\tilde{b}_{k_n}^\\dagger a_j + \\exp (-\\mathrm {i}k_n d j) a_j^\\dagger \\tilde{b}_{k_n} \\right).", "\\nonumber $ We set both qubit- and photon number to $N$ and $\\hbar \\equiv 1$ .", "$b^\\dagger $ , $b$ and $a^\\dagger $ , $a$ are photon- and qubit creation and annihilation operators.", "$g$ is the coupling strength between photons and qubits and $d$ the lattice spacing of the qubits.", "We draw the qubit frequencies $\\omega _i$ from a random uniform distribution $\\omega _i \\in \\left[-\\sqrt{3} \\sigma _{\\omega } + \\omega _{\\rm c} , \\sqrt{3} \\sigma _{\\omega } + \\omega _{\\rm c} \\right]$ characterized by its standard deviation $\\sigma _{\\omega }$ .", "With $\\omega _k =2 J \\cos (dk)$ (the photon lattice spacing is equivalent to the qubit lattice spacing, the photon band width is $4J$ ) we can equivalently write $b^\\dagger _j &:= \\sum _{n=1}^N \\exp (\\mathrm {i}k_n d j) \\tilde{b}^\\dagger _{k_n}\\\\H_\\gamma &= J\\sum _{i=1}^N \\left( b^\\dagger _i b_{i + 1} + b^\\dagger _{i + 1} b_i \\right)\\\\H_{\\gamma q} &= g \\sum _{i=1}^N \\left( b_{i}^\\dagger a_i + a_i^\\dagger b_i \\right).$ Therefore, we have the site space Hamiltonian $H &= \\sum _{j=1}^N \\omega _j a_j^\\dagger a_j + J\\sum _{i=1}^N \\delta _{\\langle i, j\\rangle } b^\\dagger _{i} b_j + g \\sum _{i=1}^N \\left( b_{i}^\\dagger a_i + a_i^\\dagger b_i \\right);\\\\\\delta _{\\langle i, j \\rangle } &:= {\\left\\lbrace \\begin{array}{ll}1 & i,\\ j \\text{ nearest neighbors}\\\\0 & \\text{else}.\\end{array}\\right.", "}$ In order to efficiently calculate transmission coefficients and localization lengths we integrate out the qubit part of the model (assuming $J, g, \\omega _j \\in \\mathbb {R}$ ): $I&:= \\int \\mathcal {D}\\lbrace a, a^*, b, b^*\\rbrace \\exp (-S(\\omega ))= \\int \\mathcal {D}\\lbrace a, a^*, b, b^*\\rbrace \\exp (-\\sum _{i=1}^N (\\omega a_i^* a_i) - \\sum _{i=1}^N(\\omega b_i^* b_i) + H)\\\\&=\\int \\mathcal {D}\\lbrace a, a^*, b, b^*\\rbrace \\exp (-\\sum _{i, j=1}^N a^*_i (\\omega - \\omega _i) \\delta _{i, j} a_j + \\sum _{i=1}^N g( a_i^* b_i + b_i^* a_i))\\exp (\\sum _{i, j=1}^N b_i^* (J\\delta _{\\langle i, j \\rangle } - \\omega \\delta _{i, j})b_j)\\\\&\\sim \\int \\mathcal {D}b \\mathcal {D}b^* \\exp (\\sum _{i, j=1}^N b_i^* \\left(J \\delta _{\\langle i, j \\rangle } + \\delta _{i, j}\\left(\\frac{g^2}{\\omega _i - \\omega } - \\omega \\right)\\right) b_j)$ $I$ is the generating integral for Hamiltonian (REF ) and $S(\\omega )$ the corresponding action.", "We read off effective action and Hamiltonian of the photons $S_\\mathrm {eff}(\\omega ) &= \\omega \\sum _{i=1}^N b_i^* b_i - \\sum _{i, j=1}^N \\left(\\delta _{i, j} \\frac{g^2}{\\omega _i - \\omega } + J b_i^\\dagger b_j \\delta _{\\langle i, j \\rangle } \\right)\\\\H_\\mathrm {eff}(\\omega ) &=\\sum _{i, j=1}^N b_i^\\dagger b_j\\left(\\delta _{i, j} \\frac{g^2}{\\omega _i - \\omega } + J\\delta _{\\langle i, j \\rangle } \\right) .", "$ The usual course of action in the context of wQED systems would be to integrate out the photons instead of the qubits, in order to include a continuum of photon modes into the calculation.", "Integrating out the photons in our model gives $I &= \\int \\mathcal {D}\\lbrace a, a^*, b, b^*\\rbrace \\exp (-\\sum _{i, j=1}^N a^*_i (\\omega - \\omega _i) \\delta _{i, j} a_j + g \\sum _{i=1}^N (a_i^* b_i + b_i^* a_i))\\exp (\\sum _{i, j=1}^N b_i^* (J\\delta _{\\langle i, j \\rangle } - \\omega \\delta _{i, j})b_j)\\\\&= \\int \\mathcal {D}\\lbrace a, a^*, b, b^*\\rbrace \\mathrm {e}^{-S_a}\\times \\nonumber \\\\&\\times g\\exp (g \\sum _{i, n=1}^N ( \\exp (\\mathrm {i} d i k_n) a_i^* \\tilde{b}_{k_n} + \\exp (-\\mathrm {i} d i k_n) \\tilde{b}_{k_n}^* a_i))\\exp (\\sum _{n, n^\\prime =1}^N \\tilde{b}_{k_n^\\prime }^* (2J\\cos (d k_n) - \\omega ) \\delta _{k_n, k_{n^\\prime }} \\tilde{b}_{k_n})\\\\&\\sim \\int \\mathcal {D}\\lbrace a, a^*\\rbrace \\mathrm {e}^{-S_a} \\exp (g^2 \\sum _{n=1}^N \\sum _{i, j=1}^N \\frac{ a_i^* a_j \\exp (\\mathrm {i}d k_n(i - j))}{\\omega - 2 J\\cos (k_n d)})$ where we denoted the qubit onsite term of the action $S_a$ .", "Effective action and Hamiltonian of the qubit system are therefore $S_\\mathrm {eff}^{q}(\\omega ) &= \\omega \\sum _{i=1}^N a_i^\\dagger a_i - \\sum _{i, j=1}^N a_i^\\dagger a_j\\left(\\delta _{i, j} \\omega _i + g^2 \\sum _{n=1}^N \\frac{ \\exp (\\mathrm {i}d k_n(i - j))}{\\omega - 2 J\\cos (k_n d)}\\right)\\\\H_\\mathrm {eff}^{ q}(\\omega ) &= \\sum _{i, j=1}^N a_i^\\dagger a_j\\left(\\delta _{i, j} \\omega _i + g^2 \\sum _{n=1}^N \\frac{ \\exp (\\mathrm {i}d k_n(i - j))}{\\omega - 2 J\\cos (k_n d)}\\right) .$ If we increased the number of photons in the model, the momentum sum in Eq.", "(REF ) could be solved in the continuum limit.", "Linearizing the dispersion relation and taking the continuum limit, we would recover the noninteracting effective qubit wQED Hamiltonian of Refs.", "[11], [29]." ], [ "Calculating transmission coefficients and localization lengths", "Due the tridiagonal form $H_{i, j} &= H_{i, j} \\delta _{i, j} + H_{i, j} \\delta _{\\langle i, j + 1\\rangle }$ of effective Hamiltonian (REF ), the corresponding Schroedinger equation can be written as $H_{i, j}\\psi _j &= \\omega \\psi _i\\\\\\Rightarrow \\omega \\psi _i &= H_{i, i} \\psi _i + \\left( H_{i, i + 1} \\psi _{i + 1} + H_{i, i - 1} \\psi _{i - 1}\\right)\\\\\\Leftrightarrow \\psi _{i + 1} &= \\frac{1}{H_{i, i + 1}} \\left[ \\left(\\omega - H_{i, i} \\right) \\psi _i - H_{i, i - 1} \\psi _{i - 1}\\right].$ This allows us to determine wave function coefficients iteratively with the matrix equation $\\begin{pmatrix}\\psi _{i + 1}\\\\\\psi _{i}\\end{pmatrix} =\\begin{pmatrix}\\frac{\\omega - H_{i, i}}{H_{i, i + 1}} & -\\frac{H_{i, i - 1}}{H_{i, i + 1}}\\\\1 & 0\\end{pmatrix}\\begin{pmatrix}\\psi _i\\\\\\psi _{i - 1}\\end{pmatrix},$ given an initial condition on sites $\\psi _i$ , $\\psi _{i - 1}$ .", "From the wave function components on each site, we can calculate an effective localization length $\\xi _i$ assuming $|\\psi _i| &\\sim \\exp (\\pm \\frac{i \\cdot d}{2\\xi _i})$ with site distance $d$ ($d:= 1$ in the following).", "It follows $\\xi &= \\lim _{i \\rightarrow \\infty } \\frac{i}{2\\log (|\\psi _i|)},$ where the limit should be chosen such that the localization length converges.", "From the localization length we can find the power transmission coefficient by $T_N := \\left| \\frac{\\psi _{N+1}}{\\psi _{0}}\\right|^2$ where we introduced the new sites $\\psi _{N+1}$ , $\\psi _0$ .", "If $\\psi $ indeed corresponds to a localized state, we can therefore determine $T_N$ from $\\xi $ as $T_N &= \\left| \\frac{\\psi _{N+1}}{\\psi _0}\\right|^2 = \\left| \\frac{\\exp (-\\frac{N}{2\\xi _N})}{1}\\right|^2,\\\\\\Rightarrow T_N &= \\exp (-\\frac{N}{\\xi _N}) \\qquad \\Leftrightarrow \\qquad \\frac{1}{\\xi _N} = -\\frac{\\log (T_N)}{N}.$ Alternatively we can directly calculate transmission coefficients and obtain effective localization lengths from $\\frac{1}{\\xi _N} = - \\frac{1}{N} \\left\\langle \\log (T_N) \\right\\rangle .$ In order to define a transmission coefficient for any incident energy we couple the system of sites $i \\in [1, N]$ to leads extending from $(-\\infty , 0]$ and $[N+1, \\infty )$ (see for example Ref.", "[34]): with lead operators $d$ , $d^\\dagger $ we write $H_\\mathrm {coupled} = &\\sum _{j=1}^N \\omega _j a_j^\\dagger a_j + J\\sum _{i, j}^N \\delta _{\\langle i, j\\rangle } b^\\dagger _i b_{j}+ g \\sum _{i=1}^N \\left( b_{i}^\\dagger a_i + a_i^\\dagger b_i \\right) \\\\&+ J_\\mathrm {lead} \\sum _{i, j\\in \\mathbb {N}\\setminus [1, N]} d^\\dagger _i \\delta _{\\langle i, j \\rangle } d_j + c_\\mathrm {L}\\left( d^\\dagger _0 b_1 + b_1^\\dagger d_0 \\right)+ c_\\mathrm {R} \\left(d_{N+1}^\\dagger b_N + b_N^\\dagger d_{N+1}\\right)$ Assuming constant couplings $c_\\mathrm {L}$ and $c_\\mathrm {R}$ between system and left / right lead.", "For our simulations $c_{\\rm L} = c_{\\rm R} = J_{\\rm lead} =1$ .", "Integrating out the qubits we find $H_\\text{eff, coupled} = &\\sum _{i, j=1}^N b_i^\\dagger \\left(\\delta _{i, j} \\frac{g^2}{\\omega _i - \\omega } + J \\delta _{\\langle i, j \\rangle } \\right)b_j \\\\&+ J_\\mathrm {lead} \\sum _{i, j\\in \\mathbb {N}\\setminus [1, N]} d^\\dagger _i \\delta _{\\langle i, j \\rangle } d_j\\\\&+ c_\\mathrm {L}\\left( d^\\dagger _0 b_1 + b_1^\\dagger d_0 \\right)+ c_\\mathrm {R} \\left(d_{N+1}^\\dagger b_N + b_N^\\dagger d_{N+1}\\right)$ Assuming a system of at least two sites we have $\\begin{pmatrix}\\psi _{N + 2}\\\\\\psi _{N + 1}\\end{pmatrix} &=\\begin{pmatrix}\\frac{\\omega }{J_\\mathrm {lead}} & - \\frac{c_\\mathrm {R}}{ J_\\mathrm {lead}}\\\\1 & 0\\end{pmatrix} \\begin{pmatrix}\\frac{\\omega -\\varepsilon _N}{c_\\mathrm {R}} & - \\frac{J}{c_\\mathrm {R}}\\\\1 & 0\\end{pmatrix}\\begin{pmatrix}\\psi _{N}\\\\\\psi _{N - 1}\\end{pmatrix}\\\\&= T_\\mathrm {right} \\underbrace{ \\left(\\prod _{i=2}^{N - 1} Q_i \\right)}_{=:T_\\mathrm {sys}}\\begin{pmatrix}\\frac{(\\omega - \\varepsilon _1)}{J} & - \\frac{c_\\mathrm {L}}{J}\\\\1 & 0\\end{pmatrix} \\begin{pmatrix}\\frac{\\omega }{c_\\mathrm {L}} & - \\frac{J_\\mathrm {lead}}{c_\\mathrm {L}}\\\\1 & 0\\end{pmatrix}\\begin{pmatrix}\\psi _0\\\\\\psi _{-1}\\end{pmatrix}\\\\&= T_\\mathrm {right} T_\\mathrm {sys} T_\\mathrm {left}\\begin{pmatrix}\\psi _0\\\\\\psi _{-1}\\end{pmatrix},\\\\Q_i &:= \\begin{pmatrix}\\frac{\\omega - \\varepsilon _i}{J} & -1\\\\1 & 0\\end{pmatrix},\\\\\\varepsilon _i &:= \\frac{g^2}{\\omega _i - \\omega }$ As an initial condition, we choose a plane wave of specified momentum $k = \\arccos (\\omega / 2J) / d$ (frequency $\\omega $ ) and convert it to coefficients: $\\psi _n &= A_n \\exp (\\mathrm {i}kdn) + B_n \\exp (-\\mathrm {i}kdn)\\\\\\begin{pmatrix}\\psi _{n + 1}\\\\\\psi _n\\end{pmatrix} &=\\underbrace{\\begin{pmatrix}\\exp (\\mathrm {i}kd(n + 1)) & \\exp (-\\mathrm {i}kd(n + 1))\\\\\\exp (\\mathrm {i}kdn) & \\exp (-\\mathrm {i}kdn)\\end{pmatrix}}_{M_{n}(k)}\\begin{pmatrix}A_n\\\\B_n\\end{pmatrix}.$ If we consider an incident wave coming from the left lead, we know that there is only a rightmoving wave to the right of the system.", "We therefore set $\\begin{pmatrix}\\psi _{N+2}\\\\\\psi _{N +1}\\end{pmatrix}(k) = M_{N + 1}(k) \\begin{pmatrix}A_{N + 1}\\\\B_{N + 1}\\end{pmatrix}= M_{N + 1}(k)\\begin{pmatrix}1\\\\0\\end{pmatrix}$ and write with $T = T_\\mathrm {right} T_\\mathrm {sys} T_\\mathrm {left}$ $\\begin{pmatrix}A_{-1}\\\\B_{-1}\\end{pmatrix}(k)&=M_{-1}^{-1}(k) \\begin{pmatrix}\\psi _{0}\\\\\\psi _{-1}\\end{pmatrix}(k)\\\\&=M_{-2}^{-1}(k) T^{-1}(k)\\begin{pmatrix}\\psi _{N+2}\\\\\\psi _{N+1}\\end{pmatrix}(k)\\\\&=M_{-2}^{-1}(k) T^{-1}(k) M_{N + 1}(k)\\begin{pmatrix}1\\\\0\\end{pmatrix}.$ Transmission and reflection coefficients $t$ and $r$ are defined as $t(k)&= \\left|\\frac{1}{A_{-1}(k)}\\right|^2\\\\r(k)&= \\frac{\\left|B_{-1}(k)\\right|^{2}}{\\left|A_{-1}(k)\\right|^2}.$ In order to find $t(k)$ , we solve for all coefficients in a single step as explained in Ref.", "[35]: We first introduce new transfer-matrices $\\tilde{Q}_i \\begin{pmatrix}\\psi _{i}\\\\\\psi _{i - 1}\\end{pmatrix} =\\begin{pmatrix}\\psi _{i + 2}\\\\\\psi _{i + 1}\\end{pmatrix} =: \\vec{\\psi }_{i + 2} \\qquad i \\in \\lbrace 2, 4, \\ldots , N - 2\\rbrace $ Furthermore, we have matrices $T_\\mathrm {left}$ , $T_\\mathrm {right}$ to match the leads to the ends of the system.", "In the right lead we specify the initial condition $\\vec{\\psi }_0 &= T_\\mathrm {left}^{-1} \\vec{\\psi }_2\\\\\\vec{\\psi }_{N + 2} &= T_\\mathrm {right} \\vec{\\psi }_N \\overset{!", "}{=} \\underbrace{M_{l + 1}(k) \\begin{pmatrix}1\\\\0\\end{pmatrix}}_{=:\\vec{i}(k)}$ and solve the matrix equation $\\begin{pmatrix}-\\mathbb {I} & T_\\mathrm {left}^{-1} & 0 &\\hdots & 0 & 0\\\\0 & -\\mathbb {I} & \\tilde{Q}_2^{-1} & \\hdots & 0 & 0\\\\\\vdots & \\vdots & \\ddots & \\ddots & \\vdots & \\vdots \\\\0 & 0 & 0 &\\hdots & - \\mathbb {I} & \\tilde{Q}_{N - 2}^{-1}\\\\0 & 0 & 0 & \\hdots & 0 & M_{l + 1}^{-1}(k) * T_\\mathrm {right}\\end{pmatrix}\\begin{pmatrix}\\vec{\\psi }_0\\\\\\vec{\\psi }_{2}\\\\\\vec{\\psi }_4\\\\\\vdots \\\\\\vec{\\psi }_{N - 2}\\\\\\vec{\\psi }_{N}\\end{pmatrix} = \\begin{pmatrix}0\\\\0\\\\0\\\\\\vdots \\\\0\\\\\\vec{i}(k)\\end{pmatrix}.$ See Fig.", "REF for an example of average transmissions coefficients $\\langle T_8\\rangle $ as well as corresponding effective localization lengths $\\xi _8$ as a function of frequency.", "Figure: Average transmission coefficients 〈T 8 〉\\langle T_8\\rangle (top panel) and corresponding effective localization lengths ξ 8 =-8 〈log(T 8 )〉\\xi _8= \\frac{-8}{\\langle \\log (T_8) \\rangle } (bottom panel) in a system of N=8N=8 sites as a function of frequency and for different strengths of disorder.", "The parameters are J=1J =1, ω c =0.164\\omega _{\\rm c} = 0.164 and g=1.1·10 -2 g =1.1\\cdot 10^{-2}." ], [ "Model localization lengths for strong disorder", "See Fig.", "REF for localization lengths $\\xi _8$ and $\\xi $ at parameters $J =1$ , $\\omega _{\\rm c} =0.164$ , $g =1.1 \\cdot 10^{-2}$ and disorder strengths up to an order of magnitude larger than what was shown in Fig.", "REF b and c; $\\sigma _{\\omega } \\cdot 10^5 \\in [1, 13]$ .", "For these disorder strengths the transmission peaks seen in Fig.", "REF c completely vanish and $\\xi _8$ is very close to $\\xi $ .", "Furthermore, the localization length does no longer behave monotonically with the disorder strength: We can see that for the considered frequencies the localization lengths for disorder strengths above $\\sigma _{\\omega } \\cdot 10^5 \\sim 8$ begin to increase with increasing disorder.", "This behaviour is expected from the form of the effective disorder in Hamiltonian (REF ) $H_{\\rm dis} = g^2 \\sum _{i=1}^N \\frac{1}{\\omega _i - \\omega } b^\\dagger _i b_i,$ from which we note, that the probability of large effective disorder values decreases with increasing disorder strength, if $\\omega \\in [\\omega _{\\rm c} - \\sqrt{3}\\sigma _{\\omega }, \\omega _{\\rm c} + \\sqrt{3} \\sigma _{\\omega }]$ .", "Figure: Localization lengths ξ\\xi (left panel) and ξ 8 \\xi _8 (right panel) for large disorder values.", "The parameters are J=1J =1, ω c =0.164\\omega _{\\rm c} =0.164 and g=1.1·10 -2 g = 1.1 \\cdot 10^{-2}." ], [ "Dissipation vs. Localization", "Since the localization length is in the experiment extracted from the power transmission coefficient $T=|S_{21}|^2$ , the effective localization length $\\xi _8$ obtained with Eq.", "(REF ) is also influenced by dissipation (i.e.", "non-radiative decoherence) of the qubits.", "This can be numerically investigated with a transfer-matrix approach based on the reflection coefficient of the individual qubits, which incorporates non-radiative decay.", "The calculation follows the methods presented in Refs.", "[15], [38].", "The left panel of Fig.", "REF shows the calculated $\\xi _8=-8/\\log (T)$ for the ordered system (all qubits at $\\omega _i/2\\pi =7.835\\,$ GHz, $\\Gamma _{10}/2\\pi =6.4\\,$ MHz) and different non-radiative decoherence rates $\\Gamma _\\text{nr}$ .", "Although the maximum value of $\\xi _8$ at the Fano-peak of the subradiant mode is strongly influenced by the amount of dissipation, it shows a distinct monotonic decrease with the disorder strength $W$ (right panel of Fig.", "REF ).", "For larger disorder strengths $\\xi _8$ converges to the value of a non-dissipative model.", "Figure: Left panel: numerically calculated ξ 8 =-8/log(T)\\xi _8=- 8/\\log (T) for the N=8N=8 ordered system obtained from a transfer-matrix calculation around the frequency of the brightest subradiant mode.", "Here, dissipation is included via the non-radiative decoherence rate Γ nr \\Gamma _\\text{nr} of the qubits, which has a comparable effect as the presence of disorder of reducing ξ 8 \\xi _8 at the position of the peak.", "Right panel: ξ 8 =-8/log(T)\\xi _8=- 8/\\left\\langle \\log (T)\\right\\rangle at the position of the peak as a function of disorder." ], [ "Normalization of Spectroscopic Measurement Data", "All spectroscopic measurement data in this work is normalized with $S^\\text{norm}_{21}(\\omega )=S^\\text{meas}_{21}(\\omega )/\\text{max}(S^\\text{meas}_{21}(\\omega ))$ ." ] ]
2105.11729
[ [ "Discovery of 22 GHz Water Masers in the Serpens South Region" ], [ "Abstract Using the Karl G. Jansky Very Large Array (VLA), we have conducted a survey for 22 GHz, 6_{1,6}-5_{2,3} H2O masers toward the Serpens South region.", "The masers were also observed with the Very Long Baseline Array (VLBA) following the VLA detections.", "We detect for the first time H2O masers in the Serpens South region that are found to be associated to three Class 0-Class I objects, including the two brightest protostars in the Serpens South cluster, known as CARMA-6 and CARMA-7.", "We also detect H2O masers associated to a source with no outflow or jet features.", "We suggest that this source is most probably a background AGB star projected in the direction of Serpens South.", "The spatial distribution of the emission spots suggest that the masers in the three Class 0-Class I objects emerge very close to the protostars and are likely excited in shocks driven by the interaction between a protostellar jet and the circumstellar material.", "Based on the comparison of the distributions of bolometric luminosity of sources hosting 22 GHz H2O masers and 162 YSOs covered by our observations, we identify a limit of L_Bol ~ 10 L_Sun for a source to host water masers.", "However, the maser emission shows strong variability in both intensity and velocity spread, and therefore masers associated to lower-luminosity sources may have been missed by our observations.", "We also report 11 new sources with radio continuum emission at 22 GHz." ], [ "Introduction", "Water masers are known to be abundant in low- and high-mass star-forming regions, where they trace collimated outflows [14], [18], [26], [37], and protoplanetary disks [13], [56], both of which are key features during the earliest phases of protostellar evolution.", "In particular, the water maser line from the $J=6_{1,6} - 5_{2,3}$ rotational transition at 22 GHz has been detected, since its discovery by [5], in hundreds of sources within both high- and low-mass star forming regions [16], [39].", "These masers are extremely bright and compact, and have become primary targets for Very Long Baseline Interferometry (VLBI), which can probe angular resolutions better than 1 mas [60], [53].", "Observations of 22 GHz water masers have been crucial primarily for the study of the dense gas and their dynamics around young stellar objects (YSOs; [38]).", "The earliest phase of low-mass protostellar evolution (the Class 0 phase in the evolutionary classes defined by [33] and [1]) is characterized by the presence of powerful outflows, which are believed to be intimately linked to the accretion process.", "These outflows can create shocked regions where the protostellar jets impact the ambient molecular cloud, which could collisionally pump H$_2$ O maser emission.", "Searches for water masers frequently target the youngest low-mass protostars, since they exhibit the most powerful collimated mass outflows.", "Several systematic surveys to search for water maser emission toward low-mass stars have been conducted in the past [58], [7], [17], [16].", "These surveys have found that the detection rate of water masers drops drastically as protostars evolve through the Class I and II phases [17], [16].", "This is explained by the dissipation of the dense gas around the central object as it evolves.", "Also, the detection rate of water masers does seem to drop significantly for very low-luminosity objects ($L\\lesssim 0.4~L_\\odot $ ; [21]).", "Only a few VLBI studies of the kinematics of water masers in low-mass stars have been conducted in the past [6], [40], [27], [11], in part because they are weaker than their counterparts associated to high-mass stars.", "These studies have shown that the masers emerge at the base of the protostellar jet, in shocks likely driven by the interaction with the disk, or in shocked gas clumps along the axis of the jet [40].", "Figure: The 48 VLA pointings used for our observations are indicated by the large circles.", "Fields where H 2 _2O masers are detected are in white.", "The circle diameters of 2. '", "72\\unknown.", ".", "{^{\\prime }}7 correspond to the field of view at 22.2 GHz.", "Cyan dots mark the positions of 90 young stars reported in , which were identified as Class 0+I objects.", "Red dots correspond to 60 YSO candidates by classified as Class 0+I objects.", "Blue dots are 67 protostars identified by from ALMA 1-mm continuum observations and IR data.", "The distribution of the VLA pointings was chosen to cover the most of these objects.", "Green dots correspond to 31 Flat-spectrum, 59 Class II and 12 Class III objects from the catalog of that fall within the observed VLA fields.", "The background is a Herschel H 2 _{2} column density map of the Serpens South star-forming region .In this paper, we focus on the Serpens South region, a well known region harboring one of nearest very young protostellar clusters.", "Serpens South was discovered by [23] from Spitzer images as an infrared dark cloud and since then it has become an interesting target to observe low-mass young stars in the earliest phases of its development.", "It is located $\\sim $ 3$^{\\rm o}$ south of the Serpens Main cloud, a region also rich in star formation activity [12].", "The W40 region, located $\\sim $ 10 arcmin to the east of Serpens South, is a more evolved star-forming region hosting a cluster of massive stars and an HII region.", "Serpens South and W40 are both projected within the broader Aquila Rift complex of molecular clouds, and often are referred to as the Aquila region [2].", "The distance to the Aquila region has been a matter of debate in the literature.", "However, recent measurements does seem to converge to $\\approx $ 440–480 pc [45], [62], [24].", "[45] obtained VLBI trigonometric parallaxes of radio continuum sources in Serpens Main and W40 and reported an average distance of $436.0\\pm 9.2$  pc.", "Later, [46] analyzed Gaia parallaxes of stars in the Aquila region (two stars are projected in the outskirts of Serpens South) and in Serpens Main, delivered as part of the 2nd data release (DR2).", "They found that the Gaia parallaxes from Aquila agree on average with those from Serpens Main, and are also consistent with the previous VLBI estimation, although their associated uncertainties are larger.", "Thus, in the present study we adopt the distance from the VLBI measurement of $436.0\\pm 9.2$  pc.", "Table: VLA observed epochsHere we use the Karl G. Jansky Very Large Array (VLA) to conduct a survey of water masers toward Serpens South covering the region with the highest density of protostellar objects.", "Follow up observations of the VLA-detected water masers were obtained with the Very Long Baseline Array (VLBA).", "The paper is organized as follows.", "Section describes the target selection, acquired observations, and data reduction.", "In Sect.", "we present our results and discuss the properties of the detected H$_2$ O emission, the spatial and velocity distribution of maser spots, and the association of the masers with outflow activity.", "This section also reports the sources detected with radio continuum emission.", "In Sect.", "we discuss the relationship between H$_2$ O maser emission and bolometric source luminosity.", "Finally, Sect.", "presents our conclusions.", "We observed the $6_{1,6} - 5_{2,3}$ H$_2$ O maser line (at rest frequency 22,235.080 MHz) with the K-band receiver at a velocity resolution of 0.1 km s$^{-1}$ (corresponding to 7.8 kHz) and a velocity coverage of $\\sim $ 100 km s$^{-1}$ .", "The observations were taken in 4 epochs in C and C$\\rightarrow $ BC$\\rightarrow $ B denotes the reconfiguration from C- to B-array.", "configurations (Table REF ) under program 18B-230.", "The epoch observed on February 2, 2019 missed 25% of the scans; therefore it was re-observed on February 8, 2019 with the C$\\rightarrow $ B configuration (Table REF ).", "The water maser line was covered by a 16-MHz wide spectral window with 2048 channels.", "Eight additional 128-MHz wide spectral windows (with 64 channels each) were observed in each baseband for the continuum, resulting in an aggregate bandwidth of 2 GHz.", "A total of 48 VLA fields (Figure REF ) were selected to cover essentially all known low-mass protostars across the region.", "Our target sample includes all Class 0+I candidatesClass 0+I refers to objects in the Class 0 or Class I phase, that cannot be separated based on IR measurements alone.", "[10] uses the IR extinction corrected spectral index, $\\alpha $ , with $\\alpha \\ge 0.3$ for Class 0+I.", "[59] uses IR colors to identify (deeply) embedded protostars as Class 0+I objects.", "reported in [59] and [10], as well as the 67 protostars identified by [49] from ALMA 1-mm continuum observations and infrared (IR) data.", "The observed area also includes 31 Flat-spectrum, 59 Class II and 12 Class III objects from the catalog of [10].", "Observing sessions consisted of series of three scans on target fields (for $\\sim $ 1.8 min each target) bracketed by phase calibrator scans of $\\sim $ 1.4min.", "The quasar 3C286 ($\\alpha $ (J2000) = 13:31:08.287984, $\\delta $ (J2000) = +30:30:32.95886), observed at the beginning of the observations, was used as the standard flux and bandpass calibrator, while J1851+0035 ($\\alpha $ (J2000) = 18:51:46.7217, $\\delta $ (J2000) = +00:35:32.414) was used as the phase calibrator.", "The total observing time in each epoch was about 2.4 hr.", "Data calibration was performed with the NRAO's Common Astronomy Software Applications (CASA, version 5.4.1) package using the VLA pipelinehttps://science.nrao.edu/facilities/vla/data-processing/pipeline/CIPL_54 provided along with the data, that has been modified to work with spectral line observations.", "The calibrated visibilities were imaged using the CASA task tclean.", "We produced maps of continuum emission for each observed field by integrating the full 2-GHz bandwidth.", "The pixel size was $0\\unknown.", ".", "{^{\\prime \\prime }}16$ in the maps from the first three epochs and $0\\unknown.", ".", "{^{\\prime \\prime }}073$ in the last epoch.", "The number of clean iterations was set to 10,000 with a threshold for cleaning of 0.066 mJy.", "We use “Briggs” weighting and applied the primary beam correction.", "For the image sizes, we used 1040$\\times $ 1040 and 2250$\\times $ 2250 pixels in C and C$\\rightarrow $ B configuration, respectively, that correspond to a field size of $2\\unknown.", ".", "{^{\\prime }}7$ .", "Maps were made for individual epochs and for the combination of the first, second and fourth epochs.", "The central frequency (wavelength) in these continuum images is 22.9 GHz (1.31 cm).", "The beam sizes and root mean square (rms) noise achieved in the continuum images are given in columns 4–6 of Table REF .", "For the images of the line data, we first fit and subtract the continuum from the uv data using the task uvcontsub, excluding the inner 900 channels for the fitting.", "Then, the task tclean was used to generate the data cubes of $2\\unknown.", ".", "{^{\\prime }}7$ in size, with 1,000 clean iterations, threshold for cleaning of 25 mJy, and the same pixel size and weighting scheme as the continuum images.", "The average rms noise in the maps not corrected by the primary beam was 16, 16, 21, and 18 mJy beam$^{-1}$ in epochs 1, 2, 3, and 4, respectively (Table REF ).", "In order to obtain the positions and fluxes of the detected spots at individual channels (c.f.", "Sect.", "REF ), we perform a 2D gaussian fit to the brightness distribution with the CASA task imfit.", "The error in the spot position is given by the astrometric uncertainty, ${\\theta _{\\rm res}}/ {(2\\times {\\rm S/N})}$ , where $\\theta _{\\rm res}$ is the FWHM size of the restoring beam, and S/N the signal-to-noise ratio of the source [55].", "The C-configuration maps of the H$_2$ O line have an average beam size of $1\\unknown.", ".", "{^{\\prime \\prime }}2$ .", "Therefore, for emission detected at S/N=5 the formal (statistical) error in position is $\\approx 0\\unknown.", ".", "{^{\\prime \\prime }}12$ .", "For the C$\\rightarrow $ B configuration, the statistical error is $\\approx 0\\unknown.", ".", "{^{\\prime \\prime }}08$ .", "Table: VLBA observed epochsccccccD,-1D,-1c Properties of the VLA detected sources with 22 GHz water emission.", "0pt Name Epoch $\\alpha $ (J2000) $\\delta $ (J2000) $V_{\\rm LSR}$ $\\Delta V_{\\rm LSR}$ Peak Flux Int.", "Flux $L_{\\rm H_2O}$ (h:m:s) ($^{\\rm o}$ :$^{\\prime }$ :$^{\\prime \\prime }$ ) (km s$^{-1}$ ) (km s$^{-1}$ ) (mJy beam$^{-1}$ ) (mJy) ($10^{-10} L_\\odot $ ) (1) (2) (3) (4) (5) (6) (7) (8) (9) 3*CARMA-7$^a$ 1 18:30:04.12 –02:03:02.56 10.46 1.2 96.70,4.30 92.83,7.70 5.6 2 18:30:04.12 –02:03:02.56 10.46 1.4 118.27,7.32 115.42,12.76 6.0 4 18:30:04.12 –02:03:02.49 10.46 0.8 132.19,6.69 172.79,14.34 5.7 4*SSTgbs J1829053-014156 1 18:29:05.32 –01:41:56.93 -6.18 0.8 104.32,6.46 139.87,14.04 4.1 2 18:29:05.33 –01:41:56.90 -6.08 0.6 119.33,6.21 89.86,10.17 2.9 3 18:29:05.34 –01:41:56.96 -6.08 0.2 126.59,7.72 140.77,15.66 1.6 4 18:29:05.33 –01:41:56.99 -5.97 0.6 166.97,6.58 231.62,14.65 4.7 4*SSTgbs J1830177-021211 1 18:30:17.72 –02:12:11.59 6.04 0.2 78.26,7.64 66.09,11.95 0.6 2 18:30:17.72 –02:12:11.64 5.30 0.6 121.87,4.36 118.22,7.55 2.2 3 18:30:17.72 –02:12:11.70 5.30 0.6 123.83,6.02 126.24,10.61 2.4 4 18:30:17.71 –02:12:11.72 5.40 0.2 86.99,7.78 71.35,12.95 1.2 Column 1 is the source name.", "Column 2 is the observed epoch; Column 3 and 4 give the coordinates of the weighted mean position of all contributing emission spots; Columns 5 and 6 are the velocity of the highest intensity channel, and velocity range of the H$_2$ O emission; Column 7 and 8 give the peak and integrated flux density, respectively, at the highest intensity channel and associated errors obtained using imfit.", "Column 9 gives the water maser luminosity.", "aDue to failures during the observations, CARMA-7 was not observed in the third epoch." ], [ "VLBA observations", "We conducted multi-epoch Very Long Baseline Array (VLBA) observations toward 4 targets, including the 3 sources that were undoubtedly detected in H$_2$ O emission with the VLA (Sect.", "REF ), and one more star with tentative detection (2MASS J18295902–0201575).", "These observations were conducted between March and November, 2020 as part of program BO061 (Table REF ).", "The data were taken at 22.2 GHz with 4 intermediate frequency (IF) bands, each of 16 MHz bandwidth.", "One IF was centered at the $6_{1,6} - 5_{2,3}$ H$_2$ O transition and correlated with a channel spacing of $\\sim $ 0.2 km s$^{-1}$ (15.625 kHz).", "We observed the quasar J1824+0119 ($\\alpha $ (J2000)=18:24:48.143436, $\\delta $ (J2000)=+01:19:34.20183) as the phase reference calibrator, which we alternated with target observations in fast switching mode, switching sources every $\\approx $ 30 seconds.", "Additional 30-min blocks of calibrators distributed over a wide range of elevations were observed at 23.7 GHz every $\\approx $ 2 hours during each $\\approx $ 9-hr observing run.", "The observations were organized in two blocks, “A” and “B”, with each block observing up to 2 targets.", "The total on-source time of the water masers was 1.5 and 3 hr for blocks observing two and one targets, respectively (see Table REF ).", "The blocks have been observed in a total of 4 epochs as of the year 2020 (labeled as 1 to 4 in Table REF ).", "Here, we only report the detections achieved so far (Sect.", "REF ).", "The full analysis of the multi-epoch VLBA data will be presented in a forthcoming paper.", "Data calibration was performed with the Astronomical Imaging System (AIPS; [22]), using the ParselTongue scripting interface [29] and following standard procedures for phase-referencing observations [52].", "Since the VLA-detected masers are relatively weak (<1 Jy; Section ), the fringe-fitting solutions were derived from the scans on the phase-reference calibrator and then applied to the target.", "Once the calibration was completed, we imaged individual channel maps of $4096\\times 4096$ pixels using a pixel size of $50~\\mu $ as.", "Spot positions and fluxes were determined by fitting a Gaussian to the brightness distribution at individual channels using the AIPS task jmfit.", "The expected statistical positional errors are of the order of 70 $\\mu $ as for emission detected at S/N=5." ], [ "VLA detected sources with H$_2$ O emission", "The cubes of the H$_2$ O line were visually inspected to search for emission at the location of the targets.", "Only 3 sources have detected H$_2$ O emission, whose properties are listed in Table REF .", "Column 1 of this table indicates the name of the source.", "Column 2 gives the epoch of detection.", "Column 3 and 4 give the mean position obtained by taking the weighted mean of the contributing maser spots, where “spot” refers to emission detected in a single channel map.", "Column 5 gives line-of-sight LSR velocity of the channel with the highest intensity.", "Column 6 gives the velocity range of H$_2$ O emission.", "Columns 7 and 8 indicate peak and integrated flux density of the highest-intensity channel, respectively.", "Column 9 gives the water maser luminosity.", "In the three detected sources, the H$_2$ O emission is weak, with fluxes below $\\sim $ 230 mJy and velocity spread $\\lesssim $ 1 km s$^{-1}$ .", "Figure REF presents extracted spectra at the position with highest intensity.", "From this figure, it is clear that the emission shows variability in both flux and velocity (the velocity range of H$_2$ O emission changes between epochs).", "This, together with the narrow widths of the lines, which are in the range from $0.7$ to $2.5$  km s$^{-1}$ , suggest that the detected H$_2$ O emission is presumably due to masers.", "We also notice that spatially distinct groups of spots contribute to the observed H$_2$ O spectra toward SSTgbs J1829053–014156 and SSTgbs J1830177–021211.", "These groups of spots may correspond to spatially separated features, where “feature” refers to emission observed in contiguous velocity channels at nearly the same position.", "Since the poor angular resolution of the VLA does not allow us to unambiguously separate the features, we average all maser spots for the positions reported in Table REF , ignoring the possibility that they may be part of distinct features.", "In the following, we discuss each detected source separately.", "CARMA-7.", "Also known as SerpS-MM18a [36], it is a Class 0 protostar [35] with strong millimeter continuum emission [47] and a highly collimated bipolar outflow extending $\\sim $ 0.16 pc [48].", "Several knots are seen along the outflow, suggesting episodic events that are attributed to variations in the accretion rate of mass onto the protostar.", "There is a nearby protostar, CARMA-6 (also known as SerpS-MM18b; [36]), located to the southwest of CARMA-7, which also has millimeter continuum emission and is classified as a Class 0+I object [28].", "The molecular outflow associated with CARMA-6 has a much wider opening angle (see Fig.", "REF ).", "The dust masses of CARMA-7 and CARMA-6 are estimated to be 1.21 and 0.43 $M_\\odot $ , respectively [47].", "They have internal luminosities of 13 and 16 $L_\\odot $ that were derived from the 70-$\\mu $ m band of Herschel assuming a distance of 350 pc [50].", "These luminosities are rescaled to 20 and 25 $L_\\odot $ for a distance of 436 pc.", "The water maser detected with the VLA toward CARMA-7 is found at the very base of the CO ($J$ =2–1) molecular outflow (see Fig.", "REF ) traced by ALMA [48].", "This position also coincides with the peak of the millimeter continuum emission (see right panel of Fig.", "REF ).", "The velocity-integrated intensity map of the CO ($J$ =3–2) line is also shown in this figure (right panel).", "In CARMA-6, the red-shifted CO ($J$ =3–2) outflow seems to correspond to the cavity walls of the CO ($J$ =2–1) outflow.", "Radio continuum sources associated with both CARMA-7 and CARMA-6 were found by [28] from observations at 4.75–7.25 GHz (their sources VLA 12 and VLA 13).", "The radio continuum emission is also detected in our observations (see Sect.", "REF and Fig.", "REF in Appendix ; sources #10 and #9).", "[28] derived radio spectral indices of $2.31\\pm 0.12$ and $0.51\\pm 0.08$ for CARMA-7 and CARMA-6, respectively, which are indicative of thermal radio emission from ionized gas, and proposed that the radio emission is tracing the base of collimated outflows.", "Figure: Large scale molecular outflows traced by CO (J=2-1J = 2-1) at 230.538 GHz from ALMA observations toward CARMA-7 and CARMA-6 .", "The integration ranges are --20 to 4 km s -1 ^{-1} for the blue-shifted component and 12 to 40 km s -1 ^{-1} for red-shifted component.", "The nnth contour is at 2 n ×S max ×p\\left(\\sqrt{2}\\right)^{n}\\times S_{\\rm max} \\times p, where S max =3.5S_{\\rm max}=3.5 and 6.3 Jy beam -1 ^{-1} km s -1 ^{-1} (for blue-shifted and red-shifted emission, respectively), nn=0, 1, 2, 3, 4 ..., and pp=10%.", "The background is an ALMA map of 1-mm continuum emission .The right panel shows a zoom-in of the central part of the mapped region.", "The contours correspond to CO (J=3-2J = 3-2) emission at 345.796 GHz from ALMA observations, integrated in the same velocity range as the CO (J=2-1J = 2-1) data, with S max =10.8S_{\\rm max}=10.8 Jy beam -1 ^{-1} km s -1 ^{-1}.In both panels, the “X”s indicate the average position of the water masers detected with the VLA (green)and VLBA (yellow; see Sect.", ").The beamsizes are shown in the bottom left corner of the images as white (molecular data) and cyan (continuum emission) ellipses.SSTgbs J1829053–014156/IRAS 18264-0143.", "This object is also a known YSO [10].", "The extinction corrected slope of the infrared spectral energy distribution (SED) is 0.96, which places the source in the Class 0+I phase [10].", "The stellar extinction corrected bolometric luminosity is $L_{\\rm Bol} = 2.9~L_\\odot $ obtained by assuming a distance of 260 pc [10].", "This value is rescaled to $8.2~L_\\odot $ for a distance of 436 pc.", "There is a 1.2 mm continuum peak close (at $\\approx 6^{\\prime \\prime }$ ) to the water maser, called Aqu-MM3, which was identified as a Class 0+I object [35].", "A dust mass of 1.7 $M_\\odot $ and a bolometric luminosity of 14.3 $L_\\odot $ (corrected for the assumed distance) was measured for the millimeter continuum source.", "We detected radio continuum emission associated to this source (see Fig.", "REF in Appendix , source #2), which may be tracing the base of the jet.", "The maser position coincides, within the position errors, with the peak of radio continuum (Fig.", "REF ).", "Figure: H 2 _2 2.12 μ\\mu m image of MHO2213, the outflow associated with SSTgbs J1829053–014156/IRAS 18264–0143 .", "The MHO features are marked with magenta ellipses and denoted with letters.The green “X” denotes the position of the maser, which coincides with the position of the outflow driving source.We searched the literature for molecular outflows that can be associated with this maser source.", "Observations of the CO ($J=3-2$ ) transition at 345.796 GHz were conducted by [41] with the ASTE 10 m telescope to study the outflow activity in Serpens South.", "In their images, there is a clear bipolar CO outflow in the vicinity of SSTgbs J1829053–014156 (the red-shifted and blue-shifted outflow components are called R6 and B11, in the nomenclature of [41]).", "The maser is very close to the base of the blue-shifted component (see Fig.", "9 of [41]).", "This outflow is also traced by H$_2$ emission at 2.12 $\\mu $ m [9], [61].", "The associated molecular hydrogen emission-line object is MHO 2213, which is thought to be driven by IRAS 18264–0143.", "The position angle of $118^{\\rm o}$ of the MHO is similar to the orientation of the CO outflow (see Fig.", "REF ).", "The maser is located at the base of the MHO feature that is associated with the blue-shifted CO lobe.", "SSTgbs J1830177–021211/IRAS 18276–0214.", "This object is a known YSO [10], [59].", "The infrared SED has a extinction corrected slope of $-2.22$ [10], which places it in the Class III phase.", "Later, based on its infrared colors, [59] classified it as a disk-bearing pre-main-sequence object (equivalent to the Class II/transition disk class of [10]).", "The extinction corrected bolometric luminosity is $L_{\\rm Bol} = 158~L_\\odot $ , which has been rescaled for a distance of 436 pc.", "Its mass has not been estimated.", "Figure: Molecular outflow lobes traced by CO (J=1-0J=1-0) at 115.27 GHz toward SSTgbs J1830177–021211.", "The integration ranges are --15 to 4 km s -1 ^{-1} for the blue-shifted component and 11 to 30 km s -1 ^{-1} for red-shifted component.", "The labels denote lobes identified by from CO (J=3-2J=3-2) observations at 345.796 GHz.", "The nnth contour is at 2 n ×S max ×p\\left(\\sqrt{2}\\right)^{n}\\times S_{\\rm max} \\times p, where S max =0.2S_{\\rm max}=0.2 K km s -1 ^{-1}, nn=0, 1, 2, 3, 4 ..., and pp=11%.", "In the background we show a Herschel 70 μ\\mu m map retrieved from the Science Archive (http://archives.esac.esa.int/hsa/whsa/).Orange circles mark the location of the Herschel protostellar cores that have been identified as the outflow driving sources .", "The source with detected H 2 _2O masers is indicated by the green circle.The beamsize is shown in white in the bottom left corner.cccccD,-1D,-1cc Properties of the maser features detected with the VLBA.", "0pt Source $\\alpha (J2000)^{a}$ $\\delta (J2000)^{a}$ Feature Epoch R.A. offset Dec. offset $V_{\\rm LSR}$$^{b}$ $L_{\\rm H_2O}$ (h:m:s) ($^{\\rm o}$ :$^{\\prime }$ :$^{\\prime \\prime }$ ) # (mas) (mas) (km s$^{-1}$ ) ($10^{-10}~L_$ ) (1) (2) (3) (4) (5) (6) (7) (8) (9) 10* SSTgbs J1830177–0212113* 18:30:17.71414943* –02:12:11.6857391B1-0.00,0.000.00,0.066.23* 46 2B10.72,0.020.98,0.044.6 3B11.48,0.040.68,0.095.3 4* 18:30:17.71414334* –02:12:11.6857341B20.00,0.040.00,0.036.24*218 2B20.79,0.010.89,0.014.6 3B21.31,0.110.81,0.045.5 4B21.75,0.071.07,0.036.6 18:30:17.7142092–02:12:11.6857891B30.00,0.050.00,0.066.2 5 18:30:17.7142152–02:12:11.6860651B40.00,0.050.00,0.046.3 17 4* CARMA–64* 18:30:03.53804* –02:03:08.3771A4-3.47,0.062.85,0.0611.34*1.2 2A4-2.81,0.052.43,0.0710.2 3A4-0.94,0.010.06,0.159.4 4A4-0.00,0.010.01,0.188.5 aReference position at the given epoch.", "bLine-of-sight velocity of the feature obtained as the intensity-weighted mean $V_{\\rm LSR}$ of the contributing spots.", "The detection of water maser emission in this source is unexpected given that earlier surveys have suggested that maser activity disappears after the main accretion and outflow (Class 0–Class I) phase [17].", "The maser has also been detected in our follow-up VLBA observations (see Sect.", "REF ).", "We did not detect radio continuum emission associated to the maser and no radio continuum has been reported in the literature either.", "There are three molecular outflow lobes (B14, B15 and R8 in the nomenclature of [41]) in the surroundings of the water maser as seen in Fig.", "REF , where we show CO ($J=1-0$ ) data at 115.27 GHz taken with the Nobeyama telescope [42].", "The outflow lobes identified by [41] are indicated in this figure, as well as the positions of the putative driving sources, which are taken from the Herschel catalog of protostellar cores [31].", "The CO ($J=1-0$ ) emission at the position of the maser is relatively weak.", "The H$_2$ 2.12 $\\mu $ m image is dominated by very strong emission from IRAS 18276–0214 (Fig.", "12.9 in [61]), so it is difficult to find an association with an H$_2$ outflow feature." ], [ "VLBA detected sources with H$_2$ O maser emission", "SSTgbs J1830177–021211/IRAS 18276–0214.", "The maser emission is seen at $V_{\\rm LSR}$ from 4.2 to $6.6$  km s$^{-1}$ (left panel in Fig.", "REF ).", "These velocities are blue-shifted with respect to the velocity of the cloud of 8 km s$^{-1}$ [30] by a few km s$^{-1}$ .", "The brightest spot has a peak flux of 0.2 Jy beam$^{-1}$ , which is higher than the highest flux detected with the VLA.", "Figure REF shows the spatial distribution of the VLBA detected maser spots in four epochs.", "We identify four main features that occupy an extent of about 2 mas ($\\approx $ 0.9 au) and are aligned roughly along the northeast-southwest direction.", "The strongest feature, labeled #1, has persisted over the four observed epochs, which cover a time baseline of $\\approx $ 7 months.", "Features #2 and #3 were detected on the first and second epochs, and feature #4 only on the second epoch.", "Table REF gives the error-weighted mean position offsets and intensity-weighted $V_{\\rm LSR}$ for each feature obtained from all contributing spots to that feature.", "These positional offsets are with respect to the position of feature #1, which we fixed at the origin in all epochs.", "Fig.", "REF shows that feature #2 (panel d) moved toward the southeast, while feature #3 (panel c) moved toward the east between two consecutive epochs separated by only 13 days.", "Since feature positions are relative to feature #1, we can investigate the internal proper motions of the two features, #2 and #3.", "In doing this, we remove the effect of the parallax, which is not well constrained by the current data.", "We obtain proper motions of $(\\mu _\\alpha \\cos \\delta ,\\mu _\\delta )=(1.9\\pm 0.8,-2.6\\pm 1.2)$  mas yr$^{-1}$ for feature #2 and $(\\mu _\\alpha \\cos \\delta ,\\mu _\\delta )=(-4.7\\pm 3.0,3.7\\pm 1.0)$  mas yr$^{-1}$ for feature #3.", "Although small, and given the fact that the positional offsets are larger than the astrometric uncertainties of about 70 $\\mu $ as (Sect.", "REF ), these motions suggest that the two features are moving toward each other.", "We attempt to estimate the absolute proper motions of feature #1 by fitting the positions of the spot detected at $V_{\\rm LSR}=6.1$  km s$^{-1}$ , where the proper motions are free parameters and the parallax is fixed to a constant value.", "We found that the resulting proper motions largely depend on the assumed value for the parallax.", "In addition, the fits yield lower residuals for parallaxes that are in the range from 0.5 to 1.0 mas.", "Further observations spanning a larger time baseline will allow us to determine if the relative motions we measured continue over time, and disentangle absolute proper motions from the parallax.", "Figure: Spatial distribution of the maser spots detected with the VLBA toward SSTgbs J1830177–021211.", "The position offsets are with respect to the error-weighted mean position of feature #1.The spots are color-coded by the LSR velocity (color bar).", "We use different symbols to distinguish between 4 epochs observed during 2020 as follows: circles – Mar 27, triangles – Apr 9, squares – Sep 29, pentagons – Nov 1.", "For each epoch and feature, the symbol with black edge indicates the error-weighted mean position of all contributing spots.", "Panels (a) to (d) show close-up views of the features plotted in panel (e).In Fig.", "REF , we see weak blue-shifted CO emission around the location of the masers that supports the presence of a molecular outflow that is too weak to be detected.", "This could happen if the star is not in Serpens South, but behind the molecular cloud, which could absorb the emission from the outflow.", "We searched the Gaia Early Data Release 3 (EDR3) catalog and found astrometric solution for the optical counterpart of SSTgbs J1830177–021211.", "The parallax reported in this catalog is $1.52\\pm 0.84$  mas [20], [19], which is still consistent (within the errors) with a distance of 436 pc, although it may suggest a larger distance.", "Additional observations of the maser spots will allow us to also fit the parallax and provide an independent measurement of the distance to the star.", "It is important to note that the classification of SSTgbs J1830177–021211 as a YSO is based on the infrared spectral index [10].", "However, Asymptotic Giant Branch (AGB) stars with infrared excesses can be misidentified as YSOs and the contamination fraction is non-negligible among Class II–Class III sources [43].", "Thus, SSTgbs J1830177–021211 could be a background AGB star with the water masers probably tracing an expanding or contracting circumstellar envelope.", "Given the small relative proper motions we measured for two maser features, and the fact that smaller parallaxes are favored from the astrometric fits and are within the $1\\sigma $ uncertainty of the Gaia based parallax measurement, we incline towards the AGB star scenario as the most plausible interpretation.", "CARMA-6.", "Although we did not detect the maser associated with CARMA-7 using the VLBA, we did find a very bright maser ($\\sim $ 12 Jy beam$^{-1}$ ) associated with CARMA-6.", "This maser was seen serendipitously in our VLBA data on September, 2020, albeit it was not detected previously with the VLA in all three observed epochs.", "Considering the rms noise level of the VLA observations (c.f.", "Table REF ), the VLBA detection of CARMA-6 implies an increase of maser flux density by more than two orders of magnitude in the highest intensity channel.", "This may correspond to a flare event, although less prominent than water maser flares seen toward massive stars [25], [57].", "Additional data correlation at the position of CARMA-6 was obtained in a subsequent epoch.", "The spectrum observed in October, 2020 is shown in Fig REF , after integrating over the area containing all maser spots.", "Figure REF shows the spatial and velocity distribution of the spots detected in the images.", "Because the maser is very bright, in this case we phase-referenced the visibility data to the maser spot at $V_{\\rm LSR}=8.5$  km s$^{-1}$ .", "We detect four groups of spots or features that are oriented in the southeast-northwest direction, covering an angular extent of about 4 mas (1.7 au).", "The groups located to the northwest (NW), hereafter the NW cluster, delineate a nearly straight filament.", "The emission is red-shifted with respect to the systemic velocity of the cloud (8 km s$^{-1}$ ), covering LSR velocities smaller than the red-shifted lobe of the CO ($J=2-1$ ) outflow traced by ALMA at larger angular scales (Fig.", "REF ).", "We see a velocity gradient through the filament with LSR velocities increasing to the north.", "The groups seen to the southeast (SE), hereafter the SE cluster, show LSR velocities close to the systemic velocity.", "Here, the maser spots are distributed along two opposite arc-like structures, displaying velocity gradients through the arcs, with LSR velocities increasing to the south.", "Similar gradients have been seen for instance in Serpens SMM1 [40].", "In Fig.", "REF , the diamonds indicate the error-weighted mean position of all contributing emission spots (indicated by the stars) to each particular feature.", "The line-of-sight velocity of each feature is obtained as the intensity-weighted mean $V_{\\rm LSR}$ of the contributing spots.", "Fig.", "REF shows that the line-of-sight velocities of the features increase to the north.", "We argue that the water masers originate in shocks between the red lobe of the molecular outflow and the surrounding material.", "As mentioned above, the NW and SE clusters draw a linear structure with the velocity gradient through this structure.", "The velocity gradient may arise from a rotating protostellar jet.", "Observationally, rotation signatures in jets have been seen as velocity gradients perpendicular to the jet axis [4], [34].", "In CARMA-6, the orientation of the protostellar jet axis is not yet very well constrained.", "In the left panel of Fig.", "REF we see that the molecular outflow is oriented close to the north-south direction, thus the jet may be oriented in the same direction.", "This seems to be supported by the orientation of the dust disk detected in the ALMA continuum map at 347 GHz shown in Fig.", "REF of Appendix .", "The deconvolved size of this disk is $0\\unknown.", ".", "{^{\\prime \\prime }}2\\times 0\\unknown.", ".", "{^{\\prime \\prime }}14$ with a position angle of 82$^{\\rm o}$ .", "If the jet is perpendicular to the disk, the jet position angle would be 172$^{\\rm o}$ , while the water maser filament has a position angle of $\\approx $ 130$^{\\rm o}$ .", "This seems to work against a rotating protostellar jet as the explanation for the observed maser velocity gradient.", "In Fig.", "REF we compare the positions of the maser spots (phase-referenced to the extragalactic calibrator) against the distribution of the ALMA continuum emission at 347 GHz.", "We see that the spots are located within the disk, but have a significatn offset of 50 mas ($\\approx 22$  au) with respect to the continuum peak; the astrometric accuracy of the ALMA observations is about 9 mashttps://help.almascience.org/kb/articles/what-is-the-astrometric-accuracy-of-alma.", "Because the water masers appear to locate at the base of the outflow (and within the protostellar disk), and the linear scale of the masers of 1.7 au is smaller than the typical size of protostellar disks ($\\lesssim 60$  au; [36]), then the velocity gradient may inherit the velocity structure of the disk.", "Therefore, the observed water maser flare and the velocity gradient may be directly linked to a disk episodic accretion burst in CARMA-6.", "The two epochs where the masers were detected are separated by only two months, covering a time baseline too short to investigate the internal kinematics of the masers.", "Additional VLBA observations will allow us to establish the kinematic structure of the water masers and further investigate the above alternative scenarios.", "Figure: Spatial distribution of the maser spots detected with the VLBA toward CARMA–6.", "The spots are color-coded by the LSR velocity (color bar).", "The stars indicate offsets measured on Oct 25, 2020, which are relative to α\\alpha (J2000)=18:30:03.538, δ\\delta (J2000)=–02:03:08.377.", "For each feature, the diamonds indicate the error-weighted mean position of all contributing spots to that feature." ], [ "Continuum sources detected with the VLA", "We performed a visual inspection of the maps that were constructed for the 48 VLA fields, first looking at the individual epochs, and then at the maps of the combination of the data from three epochs (see Sec.", "REF ).", "The visual inspection was done in the images uncorrected for the primary beam response, as this correction increases the noise toward the field edges affecting weak sources that then may appear as noise.", "However, once identified, the properties of the sources are measured in the primary beam corrected images.", "Maps of $9^{\\prime \\prime }\\times 9^{\\prime \\prime }$ in size around the location of detected sources are presented in Figures REF –REF in Appendix .", "The maps are for all available epochs, but we note that some epochs do not exhibit detection.", "Table lists the 17 sources detected with radio continuum, as well as their positions and fluxes as obtained by fitting the brightness distribution with a Gaussian model using the task imfit in CASA.", "The fluxes are listed for each epoch and for the combined image.", "Not all detected sources with radio continuum are associated with known young stars or other type of objects; there are 5 sources that have no counterparts (within a radius of 2$^{\\prime \\prime }$ ) in SIMBADhttp://simbad.u-strasbg.fr/simbad/.", "On the other hand, we found that 10 sources are associated with known or candidate YSOs [51], and other 2 are associated with known radio sources [8], [44], [28], also within a radius of 2$^{\\prime \\prime }$ .", "Table gives the names of the known sources.", "Out of the 12 objects that have an association with a known source, 6 had not been detected before in the radio according to SIMBAD.", "Therefore, we are reporting 6+5=11 new radio continuum detections.", "The newly detected radio continuum sources with no counterparts at any other wavelength are #1, #5, #14, #15 and #17.", "Source #1 is detected in the four observed epochs with fluxes of 1.7–1.9 mJy.", "The other sources (#5, #14, #15 and #17) are detected in only one epoch, with fluxes above 0.22 mJy.", "In addition, sources #3 and #12, that have reported before in the literature, do not have counterparts at any other wavelength as well.", "Following [3], we can estimate the number of expected background sources inside a field of diameter $\\theta _F$ as, $N = 1.4 \\left\\lbrace 1 -\\exp \\left[ -0.0066 \\left( \\frac{\\theta _F}{\\rm arcmin} \\right)^2 \\left( \\frac{\\nu }{\\rm 5~GHz} \\right)^2 \\right] \\right\\rbrace \\times \\left( \\frac{S_0}{\\rm mJy} \\right)^{-0.75} \\left( \\frac{\\nu }{\\rm 5~GHz} \\right)^{-2.52}$ where $S_0$ is the detectable flux density threshold and $\\nu $ the observing frequency.", "In our observations, $\\nu $ =22.2 GHz, and $S_0=3\\times {\\rm rms}\\approx 0.09$  mJy (c.f.", "Sect.", "REF ).", "Using a field size of $\\theta _F=2\\unknown.", ".", "{^{\\prime }}7$ , we obtain $\\approx 7$ expected background objects in the 48 observed fields.", "Thus, all of the unclassified sources with detected radio continuum emission are probably extragalactic objects.", "Since our targets were observed in multiple epochs, covering a timescale of about 3 weeks, we can investigate the variability of continuum emission between the epochs.", "We estimated the variability as the difference between the maximum and minimum peak flux density, normalized by the maximum flux.", "For the estimation of variability uncertainties, we adopted a flux density calibration error of 15%https://science.nrao.edu/facilities/vla/docs/manuals/oss/performance/fdscale, which was added quadratically to the statistical errors obtained from the Gaussian fits.", "We found that 9 sources show high levels of variability, with variations $\\gtrsim 50\\%$ at $3\\sigma $ .", "These sources are #4, #5, #11, #12, #13, #14, #15, #16 and #17.", "Four of these objects are YSOs; the other 5 are background candidates.", "Thus, in terms of variability, we do not see a distinction between the two groups.", "Previous works have found a similar result at shorter radio wavelengths.", "For instance, [32] showed that both YSOs and extragalactic objects show strong radio continuum variability at 7.5 GHz." ], [ "Discussion", "The four sources with H$_2$ O maser emission detected here are known to be associated with phenomena related to YSOs.", "However, while CARMA-7, CARMA-6 and SSTgbs J1829053–014156 are in the early Class 0–Class I phase, SSTgbs J1830177–021211 is probably in the more evolved Class II phase.", "Three of the sources with associated maser emission drive large-scale outflows.", "From the spatial distribution of the maser spots, we argue that in all these sources the masers originate very close to the star, and are excited by the interaction between molecular outflows with the surrounding dense material, likely of the circumstellar disk.", "Extinction corrected bolometric luminosities are available for the 162 stars of the catalog by [10] that were observed with the VLA.", "The distribution of the bolometric luminosities, which have been rescaled assuming a distance of 436 pc, are shown in the left panel of Fig.", "REF .", "Also shown in this figure are the bolometric luminosities of SSTgbs J1829053–014156, and SSTgbs J1830177–021211, and the internal luminosities of CARMA-7 and CARMA-6.", "As expected, maser emission was detected toward the objects with the highest luminosity.", "Figure REF suggests that there is a bolometric luminosity threshold of $L_{\\rm Bol}\\approx 10~L_\\odot $ to excite water maser emission.", "However, water masers have been detected in objects with lower luminosities before [16], for example, in VLA 1623 ($L_{\\rm Bol}\\approx 1~L_\\odot $ , $d$ = 138 pc; [1], [46]) and GF 9-2 ($L_{\\rm Bol}\\approx 0.3-1.7~L_\\odot $ , $d$ = 200 – 474 pc; [15], [50]).", "It is still possible that the detection of water masers associated to lower-luminosity objects in Serpens South was missed due to variability.", "For instance, in CARMA-6, the masers were not detected in the three observed epochs with the VLA, but serendipitously detected with the VLBA about 1.5 years later.", "We estimate water maser luminosities according to $L_{\\rm H_2O} = 4\\pi d^2 S_{\\rm int} \\Delta V \\nu _0/c,$ where $S_{\\rm int}$ is the maser integrated flux density, $\\Delta V$ is the velocity range of maser emission, $\\nu _0=22,235.080$  MHz is the rest frequency of the $J=6_{1,6}-5_{2,3}$ water line, $c$ the speed of light, and $d$ the distance to the source.", "The water mater luminosities are listed in Column 9 of Tables REF and REF for sources detected with the VLA and VLBA, respectively.", "Using a 3$\\sigma $ channel sensitivity of $\\approx $ 48 mJy from our VLA observations (c.f.", "Table REF ), a velocity spread of the masers of 3 channels, and $d=436$  pc, Eq.", "REF gives an H$_2$ O luminosity of $6\\times 10^{-11}~L_\\odot $ .", "Assuming the correlation between $L_{\\rm H_2O}$ and $L_{\\rm Bol}$ found by [54] for high-luminosity YSOs, according to which $L_{\\rm H_2O} = 3\\times 10^{-9}L_{\\rm Bol}^{0.94},$ this upper limit in H$_2$ O luminosity corresponds to $L_{\\rm Bol}\\approx 0.02~L_\\odot $ .", "Thus, our VLA observations were in principle sensitive enough to detect all H$_2$ O masers associated to low-luminosity protostars in Serpens South with $L_{\\rm Bol}\\gtrsim 0.02~L_\\odot $ .", "As noted by [21], the correlation between bolometric and maser luminosities may not hold for the lowest-luminosity YSOs.", "We plot in the right panel of Fig.", "REF the bolometric luminosities of the 4 objects that have water masers and the maser luminosities measured at each individual epoch observed with the VLA and the VLBA.", "Due to the strong variability in both flux and velocity spread of the maser emission, the $L_{\\rm H_2O}$ changes in all sources by more than 1 order of magnitude.", "CARMA-6 shows the highest variability, since the non-detection with the VLA implies a change in $L_{\\rm H_2O}$ by about 4 orders of magnitude.", "In Fig.", "REF , the two stars detected with the VLBA (SSTgbs J1830177–021211 and CARMA-6) fall, within one order of magnitude, close to their predicted position by the $L_{\\rm H_2O}$ versus $L_{\\rm Bol}$ empirical relationship.", "A scatter of one order of magnitude was also observed for this relationship [54]." ], [ "Conclusions", "We have conducted an interferometric survey of 22 GHz H$_2$ O masers toward the low-mass star-forming region Serpens South.", "Our observations were first carried out with the VLA covering all known protostars (Class 0–Class I objects) across the region.", "The VLA observations revealed, for the first time, three water masers in the region, which are found to be associated to CARMA-7, SSTgbs J1830177–021211 and SSTgbs J1829053–014156.", "Follow-up VLBA observations were carried out toward the VLA-detected sources to investigate the spatial distribution and kinematics of the masers.", "The VLBA observations found water maser emission associated to CARMA-6, which had not been detected with the VLA.", "Three water maser sources (CARMA-7, SSTgbs J1829053–014156 and CARMA-6) are associated with Class 0–Class I objects that drive large scale molecular outflows and also display radio continuum emission from ionized gas.", "The water masers are found at the base of the molecular outflows and we propose that in all these three objects the masers are excited in shocks driven by the interaction between a protostellar jet and the circumstellar material.", "On the other hand, the source responsible for the excitation of the water maser associated with SSTgbs J1830177–021211 is unknown.", "This source has been classified in the literature as a Class II object and has no associated molecular outflows or radio jets.", "The small relative proper motions of two maser features that persisted over two epochs, and the small parallax hinted by the astrometric fits to the brightest feature suggest that SSTgbs J1830177–021211 is most likely a background AGB star with the water masers tracing an expanding or contracting circumstellar envelope.", "Further VLBI observations will allow us to obtain the parallax and proper motions of the maser spots and to test the proposed mechanism for the water maser excitation in these objects and confirm the AGB scenario proposed for SSTgbs J1830177–021211.", "We also investigate the distributions of the bolometric luminosity of sources hosting 22 GHz H$_2$ O masers and 162 YSOs covered by our observations.", "The comparison of the two distributions suggest a luminosity threshold for the water maser emission of $L_{\\rm Bol}\\approx 10~L_\\odot $ .", "However, the water masers show strong variability, thus lower-luminosity sources may have been missed by the observations.", "Lastly, we detected 11 new sources with radio continuum emission at 22 GHz, of which 6 are known or candidate YSOs, and 5 are unknown sources without counterparts at any other wavelength.", "Based on the estimation of the number of expected background sources in the observed area, we suggest that all of these unclassified sources are probably extragalactic objects.", "The authors are grateful to the anonymous referee, whose comments helped to improve this paper.", "G.N.O.-L. acknowledges support from the von Humboldt Stiftung.", "L.L.", "acknowledges the support of DGAPA/PAPIIT grants IN112417 and IN112820, CONACyT-AEM grant 275201, and CONACyT-CF grant 263356.", "The authors acknowledge MiaoMiao Zhang for sharing his Canada–France–Hawaii Telescope near-infrared data.", "The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under a cooperative agreement by Associated Universities, Inc.", "This paper makes use of the following ALMA data: ADS/JAO.ALMA #2012.1.00769.S and #2015.1.00283.S.", "ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile.", "The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.", "Figure: Left: Bolometric luminosity distribution of 162 YSOs covered by our VLA observations in grey and the 4 objects with detected water maser emission.", "Right: Water maser luminosity versus bolometric luminosity for the objects detected with the VLA and the VLBA as described in the legend.", "The arrows indicate upper limits.", "The dashed line represents the empirical relation expressed by Eq.", ".cccccccccc Properties of the VLA detected radio continuum sources.", "700pt Source Epoch $\\alpha $ (J2000) $\\delta $ (J2000) Peak Flux Int.", "Flux Known Object$^{a}$ Ref.$^{a}$ New$^{b}$ ID (h:m:s) ($^{\\rm o}$ :$^{\\prime }$ :$^{\\prime \\prime }$ ) (mJy beam$^{-1}$ ) (mJy) Name Type Detection?", "(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 1 1 18:29:04.88 -01:30:06.2 1.75 $\\pm $ 0.02 1.78 $\\pm $ 0.03 5*Y 1 2 18:29:04.88 -01:30:06.1 1.63 $\\pm $ 0.02 1.67 $\\pm $ 0.04 1 3 18:29:04.89 -01:30:06.1 1.91 $\\pm $ 0.04 1.84 $\\pm $ 0.07 1 4 18:29:04.88 -01:30:06.2 1.89 $\\pm $ 0.05 1.93 $\\pm $ 0.10 1 all 18:29:04.88 -01:30:06.2 1.85 $\\pm $ 0.02 1.89 $\\pm $ 0.04 2 1 18:29:05.33 -01:41:57.0 0.75 $\\pm $ 0.02 0.79 $\\pm $ 0.03 5*IRAS 18264-0143 5*Y*O 5*1 5*Y, H$_2$ O 2 2 18:29:05.33 -01:41:57.0 0.73 $\\pm $ 0.02 0.78 $\\pm $ 0.03 2 3 18:29:05.32 -01:41:56.9 0.72 $\\pm $ 0.04 0.79 $\\pm $ 0.07 2 4 18:29:05.33 -01:41:57.0 0.71 $\\pm $ 0.02 0.78 $\\pm $ 0.04 2 all 18:29:05.33 -01:41:57.0 0.80 $\\pm $ 0.01 0.91 $\\pm $ 0.03 3 1 18:29:06.29 -02:07:48.1 6.94 $\\pm $ 0.29 6.78 $\\pm $ 0.51 5*PMN J1829-0207 5*Rad 5*2 3 2 18:29:06.29 -02:07:48.1 8.45 $\\pm $ 0.34 9.89 $\\pm $ 0.67 3 3 18:29:06.29 -02:07:48.2 8.51 $\\pm $ 0.28 9.05 $\\pm $ 0.52 3 4 18:29:06.29 -02:07:48.1 7.07 $\\pm $ 0.09 7.34 $\\pm $ 0.17 3 all 18:29:06.29 -02:07:48.1 8.03 $\\pm $ 0.15 8.34 $\\pm $ 0.28 4 1 – – $<$ 0.34 – 4*2MASS J18291560-0204503 4*Y*O 4*3 4*Y 4 2 18:29:15.62 -02:04:50.3 2.75 $\\pm $ 0.06 2.82 $\\pm $ 0.10 4 4 – – $<$ 0.18 – 4 all 18:29:15.62 -02:04:50.3 1.15 $\\pm $ 0.04 1.42 $\\pm $ 0.08 5 1 18:29:51.04 -01:54:24.4 0.22 $\\pm $ 0.01 0.18 $\\pm $ 0.02 4*Y 5 2 – – $<$ 0.06 – 5 4 – – $<$ 0.06 – 5 all – – $<$ 0.06 – 6 1 18:30:01.36 -02:10:25.7 0.32 $\\pm $ 0.01 0.33 $\\pm $ 0.03 4*2MASS J18300136-0210256, G028.5480+03.7663 4*Y*O 4*1 4*Y 6 2 18:30:01.36 -02:10:25.7 0.37 $\\pm $ 0.01 0.36 $\\pm $ 0.03 6 4 18:30:01.36 -02:10:25.7 0.32 $\\pm $ 0.01 0.31 $\\pm $ 0.02 6 all 18:30:01.35 -02:10:25.7 0.38 $\\pm $ 0.01 0.38 $\\pm $ 0.02 7 1 18:30:03.12 -01:36:32.9 0.16 $\\pm $ 0.02 0.16 $\\pm $ 0.03 5*G029.0540+04.0193 5*Y*?", "5*4 5*Y 7 2 18:30:03.12 -01:36:33.1 0.15 $\\pm $ 0.01 0.18 $\\pm $ 0.03 7 3 – – $<$ 0.10 – 7 4 – – $<$ 0.09 – 7 all 18:30:03.12 -01:36:32.9 0.19 $\\pm $ 0.01 0.20 $\\pm $ 0.01 8 1 – – $<$ 0.10 – 4*MHO 3247, G028.6658+03.8174, [KKT2016] VLA 11 4*Y*O, Rad 4*5 8 2 18:30:03.38 -02:02:45.8 0.15 $\\pm $ 0.01 0.13 $\\pm $ 0.02 8 4 – – $<$ 0.08 – 8 all 18:30:03.37 -02:02:45.8 0.19 $\\pm $ 0.02 0.23 $\\pm $ 0.03 9 1 18:30:03.54 -02:03:08.4 0.63 $\\pm $ 0.03 0.55 $\\pm $ 0.05 4*SSTYSV J183003.48-020308.5, CARMA-6, [KKT2016] VLA 13 4*Y*O, Rad 4*3 4* H$_2$ O 9 2 18:30:03.54 -02:03:08.3 0.73 $\\pm $ 0.02 0.71 $\\pm $ 0.04 9 4 18:30:03.54 -02:03:08.4 0.56 $\\pm $ 0.02 0.54 $\\pm $ 0.03 9 all 18:30:03.54 -02:03:08.4 0.77 $\\pm $ 0.02 0.72 $\\pm $ 0.04 10 1 18:30:04.11 -02:03:02.5 0.29 $\\pm $ 0.01 0.44 $\\pm $ 0.03 4*CARMA-7, [KKT2016] VLA 12 4*Y*O, Rad 4*1 4* H$_2$ O 10 2 18:30:04.12 -02:03:02.7 0.35 $\\pm $ 0.02 0.40 $\\pm $ 0.03 10 4 18:30:04.13 -02:03:02.6 0.28 $\\pm $ 0.01 0.37 $\\pm $ 0.03 10 all 18:30:04.12 -02:03:02.6 0.38 $\\pm $ 0.02 0.51 $\\pm $ 0.04 11 1 – – $<$ 0.09 – 5*2MASS J18300580-0201444, [KKT2016] VLA 7 5*Y*O, Rad 5*5 11 2 – – $<$ 0.11 – 11 3 18:30:05.82 -02:01:44.6 0.23 $\\pm $ 0.01 0.21 $\\pm $ 0.02 11 4 – – $<$ 0.11 – 11 all 18:30:05.82 -02:01:44.4 0.21 $\\pm $ 0.01 0.28 $\\pm $ 0.03 12 1 18:30:09.69 -02:00:32.7 0.61 $\\pm $ 0.03 0.50 $\\pm $ 0.05 5*GBS-VLA J183009.68-020032.7 5*Rad 5*6 12 2 – – $<$ 0.21 – 12 3 – – $<$ 0.27 – 12 4 – – $<$ 0.21 – 12 all 18:30:09.68 -02:00:32.7 0.48 $\\pm $ 0.03 0.53 $\\pm $ 0.05 13 1 18:30:25.88 -02:10:43.0 0.25 $\\pm $ 0.01 0.29 $\\pm $ 0.02 5*2MASS J18302593-0210420, G028.5908+03.6734 5*Y*O 5*1 5*Y 13 2 18:30:25.88 -02:10:42.8 0.39 $\\pm $ 0.03 0.34 $\\pm $ 0.06 13 3 – – $<$ 0.12 – 13 4 18:30:25.88 -02:10:42.9 0.31 $\\pm $ 0.01 0.31 $\\pm $ 0.02 13 all 18:30:25.88 -02:10:42.9 0.36 $\\pm $ 0.02 0.40 $\\pm $ 0.03 14 1 – – $<$ 0.11 – 5*Y 14 2 – – $<$ 0.09 – 14 3 – – $<$ 0.17 – 14 4 18:30:28.63 -01:53:32.6 0.23 $\\pm $ 0.01 0.21 $\\pm $ 0.02 14 all 18:30:28.63 -01:53:32.7 0.19 $\\pm $ 0.01 0.16 $\\pm $ 0.02 15 1 18:30:34.12 -01:56:37.5 0.60 $\\pm $ 0.04 0.67 $\\pm $ 0.09 4*Y 15 2 – – $<$ 0.33 – 15 4 – – $<$ 0.19 – 15 all 18:30:34.12 -01:56:37.6 0.65 $\\pm $ 0.03 0.62 $\\pm $ 0.05 16 1 – – $<$ 0.06 – 5*IRAS 18280-0210, G028.6435+03.6457 5*Y*O 5*3 5*Y 16 2 – – $<$ 0.12 – 16 3 18:30:37.61 -02:08:40.1 0.23 $\\pm $ 0.02 0.29 $\\pm $ 0.03 16 4 – – $<$ 0.12 – 16 all 18:30:37.61 -02:08:40.2 0.19 $\\pm $ 0.01 0.21 $\\pm $ 0.02 17 1 – – $<$ 0.10 – 5*Y 17 2 – – $<$ 0.08 – 17 3 – – $<$ 0.13 – 17 4 18:31:13.29 -02:05:46.2 0.65 $\\pm $ 0.02 0.63 $\\pm $ 0.03 17 all 18:31:13.29 -02:05:46.2 0.22 $\\pm $ 0.01 0.17 $\\pm $ 0.02 aClassification taken from the literature: young stellar object (Y*O), young stellar object candidate (Y*?", "), and known radio source (Rad).", "References: (1) [35]; (2) [8] (3); [59]; (4) [51]; (5) [49]; (6) [44] bThis flag indicates whether the source is a new radio continuum detection (Y) and/or has detected maser emission (H$_2$ O)." ], [ "Supplementary Figures", "In this Appendix, we show maps of radio continuum emission from the VLA data toward water maser sources (Figure REF ), YSOs with no detected water masers (Figure REF ), and candidate extragalactic sources (Figure REF ).", "Figure REF displays a 347 GHz ALMA continuum map of CARMA-6.", "Figure: Maps of radio continuum emission obtained with the VLA.", "The first 4 columns correspond to epochs 1, 2, 3 and 4,respectively (see Table ).", "The last column shows the maps of the combination of epochs 1, 2 and 4.The color scale is in Jy beam -1 ^{-1}.", "Shown are YSOs with detected water masers, whose mean positionsare indicated by the black crosses.Figure: As Fig.", "for known or candidate YSOs and no detected water masers.Figure: As Fig.", "for candidate extragalactic sources.Figure: ALMA continuum emission at 347 GHz from CARMA-6 (color scale and white contours).The nnth contour is at 2 n ×S max ×p\\left(\\sqrt{2}\\right)^{n}\\times S_{\\rm max} \\times p, where S max =0.074S_{\\rm max}=0.074 Jy beam -1 ^{-1}, nn=0, 1, 2, 3, 4 ..., and pp=10%.The green crosses indicate the positions of the water masers and the red cross correspond to the position of the ALMA continuum peak.", "The sizes of the crosses indicate 3 times the astrometric accuracy of ALMA (9 mas) at 347 GHz.The beamsize is shown in white in the bottom left corner." ] ]
2105.11747
[ [ "A Generalised Inverse Reinforcement Learning Framework" ], [ "Abstract The gloabal objective of inverse Reinforcement Learning (IRL) is to estimate the unknown cost function of some MDP base on observed trajectories generated by (approximate) optimal policies.", "The classical approach consists in tuning this cost function so that associated optimal trajectories (that minimise the cumulative discounted cost, i.e.", "the classical RL loss) are 'similar' to the observed ones.", "Prior contributions focused on penalising degenerate solutions and improving algorithmic scalability.", "Quite orthogonally to them, we question the pertinence of characterising optimality with respect to the cumulative discounted cost as it induces an implicit bias against policies with longer mixing times.", "State of the art value based RL algorithms circumvent this issue by solving for the fixed point of the Bellman optimality operator, a stronger criterion that is not well defined for the inverse problem.", "To alleviate this bias in IRL, we introduce an alternative training loss that puts more weights on future states which yields a reformulation of the (maximum entropy) IRL problem.", "The algorithms we devised exhibit enhanced performances (and similar tractability) than off-the-shelf ones in multiple OpenAI gym environments." ], [ "Introduction", "Modelling the behaviours of rational agents is a long active research topic.", "From early attempts to decompose human and animal locomotion [21] to more recent approaches to simulate human movements [16], [20], [31], the common thread is an underlying assumption that the agents are acting according to some stationary policies.", "To rationalise these behaviours, it is natural to assume that they are optimal with respect to some objective function, with evidences in the case of animal conditioned learning [28], [38], [17], [39].", "The global objective of Inverse Reinforcement Learning (IRL) is inferring such objective function given measurements of the rational agent's behaviour, its sensory inputs and a model of the environment [26].", "IRL builds upon the standard Reinforcement Learning (RL) formulation, where the goal is to find the policy that minimises discounted cumulative costs of some Markov Decision Process [23].", "It aims at finding cost functions for which the observed behaviour is “approximately optimal”.", "However this simplistic formulation admits degenerate solutions [1].", "This led to a series of innovative reformulations to lift this indeterminacy by favouring costs for which the observed behaviour is particularly better than alternative ones, namely maximum margin IRL [25] and maximum entropy IRL [42], [41].", "The latter formulation ended up as the building block of recent breakthroughs, with both tractable and highly performing algorithms [4], [11], [5].", "These improvements provided the ground for multiple practical real-life applications [42], [2], [32], [12], [18].", "We propose an orthogonal improvement to this literature.", "We question the very pertinence of characterising optimality w.r.t.", "the cumulative discounted costs as it induces a bias against policies with longer mixing times.", "We propose an extension of this criterion to alleviate this issue.", "From this novel objective, we derive reformulations for both the RL and IRL problems.", "We discuss the ability of existing RL algorithms to solve this new formulation and we generalise existing IRL algorithms to solve the problem under the new criterion.", "We back up our proposition with empirical evidence of improved performances in multiple OpenAI gym environments." ], [ "Generalised optimality criterion", "In this section, we introduce the classical settings of RL and IRL, as well as the new generalised settings we introduce to alleviate some inherent biases of current methods." ], [ "A classical RL Setting", "Consider an infinite horizon Markov Decision Process (MDP) $\\mathcal {M}=\\lbrace \\mathcal {S}, \\mathcal {A}, \\mathcal {P}, c, \\gamma , p_0 \\rbrace $ , where: – $\\mathcal {S}$ is either a finite or a compact subset of ${R}^d$ , for some dimension $d\\in {N}$ – $\\mathcal {A}$ is either a finite or a compact subset of ${R}^{d^{\\prime }}$ , for $d^{\\prime } \\in {N}$ – $\\mathcal {P}$ is the state transition kernel, i.e., a continuous mapping from $\\mathcal {S} \\times \\mathcal {A}$ to $\\Delta (\\mathcal {S})$ , where $\\Delta (\\cdot )$ denotes the set of probability measuresThe $\\sigma $ -field is always the Borel one.", "over some set, – $c:\\mathcal {S}\\times \\mathcal {A} \\rightarrow \\mathbb {R}$ is a continuous non-negative cost function, – $p_0 \\in \\Delta (\\mathcal {S})$ is the initial state distribution, and $\\gamma \\in (0,1)$ is the discount factor.", "A policy $\\pi $ is a mapping indicating, at each time step $t \\in {N}$ , the action $a_t$ to be chosen at the current state $s_t$ ; it could depend on the whole past history of states/actions/rewards but it is well known that one can focus solely, at least under mild assumptions, on stationary policies $\\pi : \\mathcal {S} \\rightarrow \\Delta (\\mathcal {A})$ .", "The choice of a policy $\\pi $ , along with a kernel $\\mathcal {P}$ and the initial probability $p_0$ , generates a unique probability distribution over the sequences of states denoted by $\\mathbb {P}_\\pi $ (the solution to the forward Chapman–Kolmogorov equation).", "The expected cumulative discounted cost of this policy, in the MDP $\\mathcal {M}$ is consequently equal to $\\mathbb {E}_{p_0,\\pi }[\\sum _t \\gamma ^t c(s_t,a_t)] = \\int _{s_0} p_0(s_0) \\sum _{t=0}^\\infty \\int _{s_t, a_t}\\hspace{-14.22636pt} \\gamma ^t \\mathbb {P}_\\pi (s_t,a_t|s_0)c(s_t,a_t).$ Optimal policies are minimisers of this quantity (existence is ensured under mild assumptions [23]).", "A standard way to compute optimal policies, is to minimise the state-action value mapping defined as: $Q^c_\\pi (s,a) = c(s,a) + \\sum _{t=1}^\\infty \\int _{s_t, a_t} \\hspace{-14.22636pt}\\gamma ^t \\mathbb {P}_\\pi (s_t,a_t|s)c(s_t,a_t)$ .", "Indeed, the expected cumulative discounted cost of a policy is the expectation of $Q$ -function against $p_0$ : $\\mathbb {E}_{p_0,\\pi }[\\sum _{t=0}^\\infty \\gamma ^t c(s_t,a_t)] = \\int _{s_0,a_0}\\hspace{-14.22636pt} p_0(s_0) \\pi (a_0|s_0)Q^c_\\pi (s_0,a_0)$" ], [ "A built-in bias in the IRL formulation", "The problem gets more complicated in Inverse Reinforcement Learning where the objective is to learn an unknown cost function $c$ whose associated optimal policy coincides with a given one $\\pi _E$ (referred to as the “expert” policy).", "This problem is unfortunately ill-posed as all policies are optimal w.r.t.", "a constant cost function [1].", "In order to lift this indeterminacy, the most used alternative formulation is called maximum entropy inverse reinforcement learning [42], [41] that aims at finding a cost function $c^*$ such that the expert policy $\\pi _E$ has a relatively small cumulative cost $\\mathbb {E}_{p_0,\\pi _E}[\\sum _{t=0}^\\infty \\gamma ^t c^*]$ while other policies incur a much higher cost.", "This implicitly boils down to learning an optimal policy (associated to some learned cost) that matches the expert's future state occupancy measure $\\rho _{\\pi _E}$ marginalised over the initial state distribution, where $\\rho _\\pi (s,a|s_0) = \\sum _t \\gamma ^t \\mathbb {P}_\\pi (s_t = s, a_t=a|s_0)$ .", "State of the art approaches [11], [5] consist, roughly speaking, in a two-step procedure.", "In the first step, given a cost function $\\hat{c}$ , an (approximately) optimal policy $\\hat{\\pi }$ of $\\hat{\\mathcal {M}}$ (the MDP $\\mathcal {M}$ with $\\hat{c}$ for cost function), is learned.", "In the second step, trajectories generated by $\\hat{\\pi }$ are compared to expert ones (in the sense of $\\rho _\\pi $ ); then $\\hat{c}$ is updated to penalise states unvisited by the expert (say, by gradient descent over some parameters).", "Obviously, those two steps can be repeated until convergence (or until the generated and the original data-sets are close enough).", "However, the presence of a discount factor in the definition of $\\rho _\\pi $ has a huge undesirable effect: the total weight of the states in the far future (say, after some stage $t^*$ ) is negligible in the global loss, as it would be of the order of $\\gamma ^{t^*}$ .", "So trying to match the future state occupancy measure will implicitly favours policies mimicking the behaviour in the short term.", "As a consequence, this would end up in penalising policies with longer mixing times even if their stationary distribution matches the experts on the long run.", "This built-in bias is a consequence of solving the reinforcement learning step with policies that optimise the cumulative discounted costs (minimises the expectation of the Q-functions against $p_0$ ) rather than policies that achieve the Bellman optimality criterion (minimises the Q-function for any state action pairs).", "Unfortunately, there is no IRL framework solving the problem under the latter assumption.", "In order to bridge this gap, we introduce a more general optimality criterion for the reinforcement learning step; it is still defined as the expectation of the $Q$ -function, yet not against $p_0$ as in traditional RL, but against both the initial and the future states distributions.", "To get some flexibility, we allow the loss to weight present and future states differently by considering a probability distribution $\\eta $ over $\\mathbb {N}$ .", "Formally, we define the $\\eta $ -weighted future state measurement distribution: $P^{\\eta }_{\\pi }(s_{+}, a_{+}|s_0):= \\sum _{n=0}^\\infty \\eta (n) \\mathbb {P}_\\pi (s_n = s_{+}, a_n=a_{+}|s_0).$ Using $P_{\\pi }^{\\eta }$ , the new criterion is defined as: ${E}^{\\eta }_{p_0,\\pi }[Q^c_{\\pi }]:=\\int _{s_0}\\hspace{-2.84544pt} p_0(s_0)\\int _{s_+,a_+}\\hspace{-14.22636pt} P^{\\eta }_{\\pi }(s_{+}, a_{+}|s_0) Q^c_\\pi (s_+,a_+)= {E}_{p_0,\\pi }\\Big [\\sum _k \\eta (k) \\sum _t \\gamma ^t c_{t+k} \\Big ]$ where $c_t$ denotes the cost at the $t^\\text{th}$ observation ($c(s_t,a_t)$ ).", "Any policy that minimises ${E}^{\\eta }_{p_0,\\pi }[Q^c_{\\pi }]$ will now be referred to as “$\\eta $ -optimal” (w.r.t.", "the cost function $c$ ).", "As mentioned before, the inverse RL problem can be decomposed in two sub-problems, learning approximate optimal strategies (given a candidate $\\hat{c}$ ) and optimizing over $\\hat{c}$ (taking into account the expert distribution $\\pi _E$ ).", "In order to avoid over-fitting when learning optimal policies, the standard way is to regularize the optimization loss [6].", "As a consequence, we consider any mapping $\\Omega : \\Delta (\\mathcal {A})^\\mathcal {S}\\rightarrow \\mathbb {R}$ that is a concave over the space of policies.", "The associated regularised loss of adopting a policy $\\pi $ given the cost function $c$ is defined as: $\\mathcal {L}_\\Omega ^\\eta (\\pi ,c) = {E}^{\\eta }_{p_0,\\pi }\\big [ Q^{c}_{\\pi } \\big ] - \\Omega (\\pi )$ The generalised RL problem is then defined as: $\\operatorname{RL}^\\eta _\\Omega (c) := \\operatornamewithlimits{argmin}_{\\pi } \\mathcal {L}_\\Omega ^\\eta (\\pi ,c)$ Similarly, in order to learn simpler cost functions [11], the optimization loss considered is in turn penalised by a convex (over the space of cost functions) regularizer $\\psi : \\mathbb {R}^{(\\mathcal {S}\\times \\mathcal {A})}\\rightarrow \\mathbb {R}$ .", "The problem of Generalised (Maximum Entropy) Inverse Reinforcement learning, whose objective is to learn an appropriate cost function $c$ , is formally defined as : $\\operatorname{IRL}^\\eta _{\\psi ,\\Omega }(\\pi _E):= \\operatornamewithlimits{argmax}_c \\min _{\\pi } \\mathcal {L}_\\Omega ^\\eta (\\pi ,c) - \\mathcal {L}_\\Omega ^\\eta (\\pi _E,c) - \\psi (c)$ We emphasise that simply choosing $\\delta _0$ (a Dirac mass at 0) for the distribution $\\eta $ induces the classical definitions of both the $\\operatorname{RL}$ and $\\operatorname{IRL}$ problems [11].", "On the other hand, choosing $\\eta =\\operatorname{Geom}(\\gamma )$ transforms the loss into the expectation of the sum of discounted Q-functions along the trajectory.", "Hypothetically, there could be other generalisations of discounted cost.", "However, preserving the compatibility of the Bellman criterion with the proposed generalisation for RL and duality properties for IRL is not trivial (for example, polynomial decay $\\frac{\\gamma }{t^n}$ would break these properties).", "In the following, we prove that the $\\eta $ -optimality framework satisfies both properties." ], [ "Generalised Reinforcement Learning ", "As in the classical setting, solving the generalised IRL problem (Equation REF ), requires solving the generalised RL problem (Equation REF ) as a sub-routine.", "Among the model free RL algorithms, value-based vs. policy gradient-based methods can be distinguished.", "In this section, we focus on the first type of methods as they can easily be used for the search of $\\eta $ -optimal policies.", "We provide a detailed discussion of the limitations of current policy gradient-based methods in Appendix , as they might be less adapted to solving $\\operatorname{RL}_\\Omega ^\\eta $ .", "Given a standard MDP $\\mathcal {M}$ and policy $\\pi $ , the Bellman operator $T_\\pi $ (from ${R}^{\\mathcal {S}}$ to ${R}^{\\mathcal {S}}$ ) is defined as [6]: $[T_{\\pi }(v)](s) = \\mathbb {E}_{a\\sim \\pi }\\Big [ c(s,a) + \\gamma \\mathbb {E}_{s^{\\prime }|s,a}[v(s^{\\prime })]\\Big ],$ and its unique fixed point is called the associated value function $v_\\pi ^c$ .", "This concept is transposed to the regularised case as follows: Given a concave regularisation function $\\Omega $ and a policy $\\pi $ , the associated regularised Bellman operator $T_{\\pi ,\\Omega }$ and the associated value function $v_{\\pi ,\\Omega }^c$ are respectively defined as: $T_{\\pi ,\\Omega } : v \\in \\mathbb {R}^\\mathcal {S} \\rightarrow T_{\\pi ,\\Omega }(v) = T_{\\pi }(v) - \\Omega (\\pi ) \\in \\mathbb {R}^\\mathcal {S},$ and as: $v_{\\pi ,\\Omega }^c= T_{\\pi ,\\Omega }(v_{\\pi ,\\Omega }^c)$ , its unique fixed point.", "As usual, the regularised Bellman optimality operator $T_{*,\\Omega }$ is in turn defined as: $T_{*,\\Omega } : v \\in \\mathbb {R}^\\mathcal {S} & \\mapsto T_{*,\\Omega }(v) \\in \\mathbb {R}^\\mathcal {S} \\\\[T_{*,\\Omega }(v)](s) & = \\min _\\pi [T_{\\pi ,\\Omega }(v)](s), \\quad \\forall s\\in \\mathcal {S}.$ Notice that given $v \\in {R}^{\\mathcal {S}}$ , the policy $\\bar{\\pi }_v(\\cdot |s)=\\delta _{\\bar{a}}$ with $\\bar{a} = \\operatornamewithlimits{argmin}_a c(s,a) + \\gamma \\mathbb {E}_{s^{\\prime }|s,a}[v(s^{\\prime })]$ achieves the minimum in the overall equation for all state $s\\in \\mathcal {S}$ .", "The policy improvement theorem [34] guarantees that if $v_{\\pi ,\\Omega }^c$ is the regularised value function of $\\pi $ , then $\\bar{\\pi }:=\\bar{\\pi }_{v_{\\pi ,\\Omega }^c}$ dominates $\\pi $ (in the sense that $v_{\\bar{\\pi }, \\Omega }^c(s) \\le v_{\\pi , \\Omega }^c(s)$ for any state $s\\in \\mathcal {S}$ ).", "If we denote by $v_{*,\\Omega }^c$ the unique fixed point of the regularised Bellman optimality operator $T_{*,\\Omega }$ , then the policy $\\pi ^*_\\Omega :=\\bar{\\pi }_{v_{*,\\Omega }^c}$ is associated to the minimum regularised value function: Proposition 1 Optimal regularised policy (Theorem 1 of [6]) : The policy $\\pi _\\Omega ^* :=\\bar{\\pi }_{v_{*,\\Omega }^c}$ is the unique optimal regularised policy in the sense that, for all policies $\\pi $ , the following holds: $\\forall s\\in \\mathcal {S}, \\quad v^{c}_{\\pi _\\Omega ^*, \\Omega }(s) = v_{*,\\Omega }^c(s) \\le v^c_{\\pi , \\Omega }(s).$ Quite interestingly, the optimal regularised policy $\\pi _\\Omega ^*$ (that minimizes the regularised cumulative discounted cost), is still minimizing the regularised $\\eta $ -weighted Q-functions: Corollary 1.1 For any given distribution $\\eta $ , the optimal regularised policy minimises $\\operatorname{RL}_\\Omega ^\\eta $ : $ \\pi _\\Omega ^* \\in \\operatornamewithlimits{argmin}_{\\pi } {E}^{\\eta }_{p_0,\\pi }\\big [ Q^c_{\\pi } \\big ] - \\Omega (\\pi ) $ This implies that such policy is not only optimal in the sense of the classical formulation (i.e.", "with $\\eta = \\delta _0$ ), but also $\\eta $ -optimal for any given distribution $\\eta $ .", "More importantly, we can directly exploit state of the art value base RL algorithms that approximates $v_{*,\\Omega }$ (such as Soft Actor Critic (SAC) [9]) to solve the generalised setting.", "Corollary REF guarantees that the Bellman optimality criterion is compatible with the proposed generalisation ($\\operatorname{RL}_\\Omega ^\\eta $ ) for any distribution $\\eta $ ." ], [ "Generalised Inverse Reinforcement Learning", "We recall that the global objective of IRL is to learn the cost function based on an expert policy $\\pi _E$ .", "In this section, we illustrate that the solution of $\\operatorname{IRL}_{\\psi ,\\Omega }^\\eta (\\pi _E)$ is a cost function $\\hat{c}$ , whose associated optimal policy $\\operatorname{RL}_\\Omega ^\\eta (\\hat{c})$ matches the expert's future state distributions $P^\\eta _{\\pi _E}$ marginalised against $\\rho _{\\pi _E}$ rather than simply matching the occupancy measure $\\rho _{\\pi _E}$ , as in usual IRL formulation.", "To alleviate notations, we denote the $\\eta $ -optimal policy $\\hat{\\pi }=\\operatorname{RL}_\\Omega ^\\eta (\\hat{c})$ as $\\operatorname{RL}_\\Omega ^\\eta \\circ \\operatorname{IRL}_{\\psi ,\\Omega }^\\eta (\\pi _E)$ .", "This policy minimises the worst-case cost weighted divergence $d_c(\\hat{\\pi }\\Vert \\pi _E): \\mathcal {S}\\mapsto \\mathbb {R}$ averaged over $p_0$ , such that: $d_c(\\hat{\\pi }\\Vert \\pi _E)(s_0):= \\hspace{-5.69046pt} \\int _{s,a,s_+,a_+} \\hspace{-31.2982pt} c(s_+,a_+) \\Big [ \\rho _{\\hat{\\pi }}(s,a|s_0)P_{\\hat{\\pi }}^{\\eta }(s_{+}, a_{+}|s,a) -\\rho _{\\pi _E}(s,a|s_0)P_{\\pi _E}^{\\eta }(s_{+}, a_{+}|s,a) \\Big ]$ This is formalised in the following proposition that requires the following notations.", "Given a policy $\\pi $ we denote by $\\mu _\\pi (s_+,a_+|s_0) = \\sum _{t,k} \\gamma ^t \\eta (k) \\mathbb {P}_\\pi (s_{t+k}=s_+,a_{t+k}=a_+|s_0)$ the frequency of $(s_{+},a_{+})$ in the $\\eta $ -weighted future steps of trajectories initialised according to $\\rho _\\pi (s,a|s_0)$ .", "Proposition 2 For any convex penalty $\\psi $ , concave regulariser $\\Omega $ (w.r.t.", "the future occupancy measure $\\mu _\\pi $ ) and any expert policy $\\pi _E$ , if $\\eta $ is geometric, then: $&\\operatorname{RL}_\\Omega ^\\eta \\circ \\operatorname{IRL}_{\\psi ,\\Omega }^\\eta (\\pi _E) = \\operatornamewithlimits{argmin}_\\pi \\max _c L(\\pi ,c) \\\\\\text{where: } \\;& L(\\pi , c) = -\\Omega (\\pi ) - \\psi (c) + \\int _{s_0} p_0(s_0) d_c(\\pi \\Vert \\pi _E)(s_0)$ As mentioned before, Proposition REF states that in its generalised formulation, solving the IRL problem can be done by matching $\\eta $ -weighted future state distributions $\\mu _\\pi $ (as opposed to matching $\\rho _\\pi $ in the classical case).", "This proves that the generalised setting preserves the duality properties of classical IRL.", "The solution of $\\operatorname{IRL}_{\\psi , \\Omega }^\\eta $ is a Nash-Equilibrium of a game between poicies and the cost functions: Corollary 2.1 Under the assumptions of Proposition REF , $(\\tilde{c},\\tilde{\\pi })= (\\operatorname{IRL}_{\\psi ,\\Omega }^\\eta (\\pi _E), \\operatorname{RL}_\\Omega ^\\eta (\\tilde{c}))$ is a Nash-Equilibrium of the following game: $\\begin{array}{ll}\\tilde{c}: \\max _c L(\\pi ,c) + \\Omega (\\pi ) \\quad ; \\quad \\tilde{\\pi }: \\min _\\pi {E}_+^\\pi \\big [ Q_{\\pi }(c) \\big ] - \\Omega (\\pi )\\end{array}$ A practical implication of Corollary REF is a template algorithm dubbed $\\operatorname{GIRL}_{(\\psi ,\\Omega ,\\eta )}$ and illustrated in Algorithm REF that can be used to solve this problem.", "$\\operatorname{GIRL}_{(\\psi ,\\Omega ,\\eta )}$ (Generalised IRL) [1] Input: Expert trajectories $\\tau _E \\sim \\pi _E$ , initial policy $\\pi _{\\theta _0}$ and initial cost function $c_{w_0}$ $e \\in [1, N]$ Sample trajectories $\\tau \\sim \\pi _{\\theta _i}$ Sample from $\\tau $ policy state action $(S^+,A^+)\\sim \\mu _{\\pi }^{\\eta }(\\tau )$ Sample from $\\tau _E$ expert state action $(S^+_E,A^+_E)\\sim \\mu _{\\pi _E}^{\\eta }(\\tau _E)$ Update the cost parameter $w_i$ to maximise $-\\psi (c_w) + \\sum _{S^+,A^+}c_{w}(s,a) - \\sum _{S^+_E,A^+_E}c_{w}(s,a)$ Update $\\theta _i$ using a value-based reinforcement learning algorithm to minimise $c_{w_{i+1}}$ Return: $(\\pi _{\\theta _N}, c_{w_N})$ We stress out now that the concavity of $\\Omega $ w.r.t.", "$\\mu _\\pi $ in Proposition REF is not too restrictive in practical settings as the $\\eta $ -weighted entropy regulariser, amongst others, satisfies it: Proposition 3 The $\\eta $ -weighted entropy regulariser $\\bar{H}^{\\eta }_{p_0}$ defined by $\\bar{H}^{\\eta }_{p_0}(\\mu _\\pi ) := H^{\\eta }_{p_0}(\\pi ) = {E}^{\\eta }_{p_0,\\pi }\\Big [\\sum _t -\\gamma ^t\\log \\big [\\pi (a_t|s_t)\\big ]\\Big ]$ is concave with respect to the occupancy measure $\\mu _\\pi $ ." ], [ "Tractability", "The tractability of $\\operatorname{GIRL}_{\\psi , \\Omega , \\eta }$ is a crucial requirement for practical implementation.", "In this section, both the regulariser term $\\Omega (\\cdot )$ and the penalty term $\\psi (\\cdot )$ are assumed to be tractably optimisable.", "For example the entropy, a widely used regulariser in the RL literature is efficiently tractable in practice.", "Indeed, Soft Actor Critic [9] uses a single sample approximation of the entropy to optimise the entropy regularised Bellman optimality operator.", "Similarly, using an indicator penalty over a subset $\\mathcal {C}$ of possible cost functions (i.e., the penalty is infinite if $c\\notin \\mathcal {C}$ and 0 otherwise) is also tractable with projected gradient updates if $\\mathcal {C}$ is convex [1], [36], [37].", "As a consequence, establishing tractability of $\\operatorname{GIRL}_{\\psi , \\Omega , \\eta }$ reduces to finding tractable sampling schemes from $\\mu _\\pi $ .", "This is equivalent to sampling sequentially from the $\\eta $ -weighted future state distribution and the occupancy measure as: $\\mu _\\pi (s_+,a_+|s_0) = \\int _{s,a} \\rho _\\pi (s,a|s_0) P_\\pi ^\\eta (s_+,a_+|s,a) = \\int _{s,a} P_\\pi ^\\eta (s,a|s_0) \\rho _\\pi (s_+,a_+|s,a)$ Given a policy $\\pi $ , the simplest approach to sample from these distributions is to sample transitions from a set of $\\pi $ -generated trajectories, denoted by $\\lbrace (s^{(i)}_t,a^{(i)}_t)_{t\\in \\lbrace 1,H\\rbrace }; i\\in \\lbrace 1,N\\rbrace \\rbrace $ , where $H$ is the horizon and $N$ is the number of trajectories.", "– For the occupancy measure $\\rho _\\pi $ : given a uniformly sampled index $i\\sim \\mathcal {U}[1,N]$ and a time sampled from truncated geometric distribution $t\\sim \\operatorname{Geom}_{[1,H]}(\\gamma )$ , the associated pair of state/action $(s_t^{(i)},a_t^{(i)})$ is an (approximate) sample from the marginal of $\\rho _\\pi (.|s_0)$ against $p_0$ .", "– For the future state distribution $P_\\pi ^\\eta $ : Given a state $s_{t}^{(i)}$ sampled as above, a time $k$ is sampled from a truncated $\\eta _{[1,H-t]}$ ; the state-action $(s_{t+k}^{(i)}, a_{t+k}^{(i)})$ is an approximate sample from $P_\\pi ^\\eta (.|s_t)$ .", "As a consequence, the above scheme shows that sampling from $\\rho _\\pi $ and $P_\\pi ^\\eta $ reduces to sampling indices from $\\operatorname{Geom}(\\gamma )$ and $\\eta $ , which is tractable from both the expert and the learned policies perspective.", "This proves that solving $\\operatorname{IRL}_{\\psi ,\\Omega }^\\eta $ does not incur any additional computational burden." ], [ "MEGAN: Maximum Entropy - Generative Adversarial Network", "This section introduces a new algorithm, called $\\operatorname{MEGAN}$ , that will improve upon state of the art IRL algorithms.", "Recent progresses in the field propose variations of GAIL [11] in order to solve a wide variety of problems.", "For example, AIRL [5] uses a particular shape for the discriminator for better transferability of the learned rewards, EAIRL [24] applies empowerment regulariser to policy updates to prevent over-fitting the expert demonstration, RAIRL [13] generalises AIRL for regularised MDPs (i.e.", "$\\Omega $ is not necessarily the entropy), s-GAIL [15] generalises the formulation for multi-task RL, etc.", "Their contributions were crucial to the progresses of IRL.", "However, we will actually focus on improving the core algorithm GAIL so that all the aforementioned approaches can be implemented with $\\operatorname{MEGAN}$ instead of GAIL with improved performances.", "We considered the rather classical penalty function [11]: $\\psi _{GAN}(c) =\\left\\lbrace \\begin{array}{cc}\\mathbb {E}_{p_0,\\pi _E}^{\\eta } [g(c(s,a))] & \\textit { if } c<0 \\\\+\\infty & \\textit { otherwise }\\end{array}\\right.\\textbf {where: } g(x) =\\left\\lbrace \\begin{array}{cc}-x - \\log (1-e^x) & \\textit {if } x<0 \\\\+\\infty & \\textit {if } x\\ge 0\\end{array}\\right.$ As in the precedent cases studied, the generalised problem boils down to using $\\mathbb {E}_\\pi ^\\eta $ : Proposition 4 Under the assumptions of Proposition REF , and for $\\psi =\\psi _{GAN}(c)$ , it holds: $\\operatorname{RL}_\\Omega ^\\eta \\circ \\operatorname{IRL}_{\\psi }^\\eta (\\pi _E) = \\operatornamewithlimits{argmin}_\\pi \\; -\\Omega (\\pi ) + \\max _{D \\in (0,1)^{\\mathcal {S}\\times \\mathcal {A}}} \\mathbb {E}_\\pi ^\\eta [\\log D]-\\mathbb {E}_{\\pi _E}^\\eta [\\log (1-D))]$ The algorithm $\\operatorname{MEGAN}$ (Maximum Entropy - Generative Adversarial Network), is then equivalent to $\\operatorname{GIRL}_{\\psi _{GAN}, H, \\operatorname{Geom}(\\gamma )}$ and a generalisation of the corner stone in state of the art IRL [11]; its pseudo-code is given in Algorithm .", "$\\operatorname{MEGAN}$ [1] Input: Expert trajectories $\\tau _E \\sim \\pi _E$ , initial policy $\\pi _{\\theta _0}$ and initial discriminator function $D_{w_0}$ $e \\in [1, N]$ Sample trajectories $\\tau \\sim \\pi _{\\theta _i}$ Sample states randomly $(S_t,A_t)\\sim \\tau $ and $(S^+,A^+) = (S_{t+k}, A_{t+k})$ where $k\\sim \\eta $ Sample states randomly $(S^{\\prime }_t,A^{\\prime }_t)\\sim \\tau _E$ and $(S^+_E,A^+_E) = (S^{\\prime }_{t+k}, A^{\\prime }_{t+k})$ where $k\\sim \\eta $ Update the cost parameter $w_i$ to maximise $\\big [ \\log D_{w}(S^+,A^+) - \\log (1-D_{w}(S^+_E,A^+_E)) \\big ]$ Update $\\theta _i$ using soft actor critic to minimise the cost $\\big [ \\log D_{w_{i+1}}\\big ]$ Return: $(\\pi _{\\theta _N}, D_{w_N})$" ], [ "Experiments", "This section is devoted to experimental evidences that $\\operatorname{MEGAN}$ achieves state of the art performances.", "It is compared to GAIL as all subsequent approaches build upon its formulation.", "The standard approach to compare IRL algorithms is to consider the best performing policies obtained during the training and evaluate their performances.", "This is an issue in practice as we do not have access to such cost function in order to implement a stopping rule once the learned policy reaches a certain performance threshold.", "A reasonable alternative criterion is to measure the divergence between generated and expert future state distributions (in the sense of $\\rho _{\\pi }$ or $\\mu _{\\pi }$ ).", "In this section, we propose to evaluate the divergence using the maximum mean discrepancy (MMD)A formal reminder on the definition of MMD divergence is provided in Appendix REF for completeness.. We will tackle the following questions empirically: -1 How does varying the parameter of a geometric $\\eta $ distribution affect performances?", "-2 How does alternative $\\eta $ distribution (e.g.", "a Poisson) compare to the use of a geometric one?", "-3 Does varying the discount factor $\\gamma $ produce similar performances?", "Due to limited space, we only analyse single-task environments in this section.", "We provide in Appendix and further investigations for the multi-task setting.", "A summary of the used hyper-parameters is also provided in Appendix REF ." ], [ "Performance improvement using a Geometric $\\eta $ distribution", "Recall that solving the IRL problem essentially boils down to finding an equilibrium between a policy that matches the expert behaviour and a cost function that discriminates generated trajectories from expert ones.", "An important property of a given algorithm is the stability of the associated equilibrium.", "In order to take into account this aspect, we propose to evaluate performances using trajectories sampled during the last 100 iterations of training.", "We will refer to these trajectories as the remaining replay buffer.", "This procedure provides an evaluation of the policies toward which the algorithm converges, while factoring in their stability.", "Notice that the goal of $\\operatorname{GAIL}$ is to match the distribution $\\rho _{\\pi _E}$ while $\\operatorname{MEGAN}$ matches the distribution $\\mu _{\\pi _E}^\\eta $ .", "In order to take into account this difference, we propose to measure performances in terms of cumulative costs, $\\operatorname{MMD}_\\rho = \\operatorname{MMD}(\\rho _\\pi , \\rho _{\\pi _E})$ and $\\operatorname{MMD}_\\mu = \\operatorname{MMD}(\\mu _\\pi ^{GEOM(0.99)}, \\mu _{\\pi _E}^{GEOM(0.99)})$ .", "We evaluate the performances of $\\operatorname{MEGAN}$ using a truncatedDue to obvious computational limitations, the trajectories are finite geometric $\\eta $ distribution with different parameters (specifically $\\lbrace 0, 0.25, 0.5, 0.75, 1\\rbrace $ ).", "Notice that using a geometric distribution with parameter 0 is equivalent to using a Dirac mass at 0 (or equivalently solving the IRL problem using $\\operatorname{GAIL}$ ).", "Similarly, using a geometric distribution with parameter 1 is equivalent to using a uniform $\\eta $ distribution.", "The remaining values can be seen as an interpolation between these extremes.", "In Figure REF , each point reports the average performances obtained using the remaining replay buffer of 3 randomly seeded instances of the algorithm.", "The blue curves report the average $\\operatorname{MMD}_\\rho $ (divergence in the sense of the classical IRL formulation), the green curves report the average $\\operatorname{MMD}_\\mu $ (divergence in the sense of the generalised IRL formulation), and the red curves report the average cumulative costs (divergence in the sense of the environment's ground truth).", "From left to right, we report the performances in three classical control settings with varying complexity from the MuJoCo based environments [22].", "In Figure REF we used the Ant environment (a state action space of dimension 118), in Figure REF we used the Half-Cheetah environment (a state action space of dimension 23) and in Figure REF we used the Hopper environment (a state action space of dimension 14).", "All the provided experiments confirmed a reduction of the average $\\operatorname{MMD}$ divergence by $25\\%$ to $60\\%$ (in the sense of both classical IRL and generalised IRL formulations) as the parameter of the $\\eta $ distribution increased to 1.", "This confirms that using the $\\eta $ -optimality objective function improves both the stability and the ability of IRL algorithms to match faithfully the expert behaviour.", "Notice that despite the fact that $\\operatorname{GAIL}$ explicitly optimises divergence in the sense of $\\rho _\\pi $ , it under-performs in the sense of $\\operatorname{MMD}_\\rho $ when compared to $\\operatorname{MEGAN}$ (the blue curves decreases as the parameter of the geometric distribution increases).", "This confirms empirically that the $\\eta $ -optimality framework proposed in this paper does indeed bridge the gap between policy-based reinforcement learning (optimising cumulative discounted costs) and value-based reinforcement learning (achieving the Bellman optimality criterion) as it even improves performances in the sense of classical IRL.", "Another important observation in Figure REF , is that for complex environments (Ant and Half-Cheetah) the decrease of the $\\operatorname{MMD}$ divergence -as we increased the parameter of the geometric distribution to 1- was correlated with a decrease of the average cumulative costs by a factor of 2 to 4.", "This was not the case of the Hopper environment, as we obtained similar cumulative costs despite the reduction of the divergence by a factor of 3.", "This is explained by the fact that the ground truth cost function of the Hopper environment produces similar cumulative costs for a wider variety of policies.", "For this reason, the IRL solution does not need to achieve a faithful expert behaviour matching in order to achieve good performances.", "This illustrates the importance of evaluating IRL algorithm with respect to $\\operatorname{MMD}_\\rho $ and $\\operatorname{MMD}_\\mu $ when the goal is to mimic behaviors.", "Figure: Performances of the policies obtained during the last 100 iteration of MEGAN\\operatorname{MEGAN}: as the parameter of η=Geom(κ)\\eta = \\operatorname{Geom}(\\kappa ) grows from 0 (equivalent to an instance of GAIL\\operatorname{GAIL}) to 1 (equivalent to an instance of MEGAN\\operatorname{MEGAN} with a uniform η\\eta ), we observe that the learned policies' generated trajectories are increasingly similar with those generated by the expert in the sense of ρ π \\rho _\\pi (classical IRL criterion), μ π \\mu _\\pi (Generalised IRL criterion) and the cumulative discounted costs (The environment's ground truth)." ], [ "Performance improvement using a Poisson $\\eta $ distribution", "Using the same experimental setting from the previous section, we evaluate $\\operatorname{MEGAN}$ 's performances with non-geometric $\\eta $ distributions.", "In Figure REF , we plot the performances of the remaining replay buffer obtained with geometric $\\eta $ distributions in blue lines, and those obtained when using a Poisson distribution in red.", "To reduce clutter, we removed the cumulative costs and only provided the divergences $\\operatorname{MMD}_\\rho $ (represented with solid lines in Figure REF ) and $\\operatorname{MMD}_\\mu $ (dashed lines in Figure REF ).", "Despite the weaker theoretical guarantees provided in our paper for the case of non-geometric $\\eta $ distributions, we observe that using a Poisson $\\eta $ can lead to comparable performances.", "Recall that the expectation of a Poisson distribution is equal to its parameter value.", "This implies that solving $\\operatorname{IRL}_{\\psi , \\Omega }^{Poisson(\\lambda )}$ searches for policies that match $\\rho _{\\pi _E}(.|s)$ for states $s$ observed around the $\\lambda ^{\\textit {th}}$ frame of the expert demonstrations.", "Now notice that the control tasks analysed in Figure REF , consist of movements cycles that are repeated perpetually.", "Quite interestingly, setting $\\lambda $ to a value around the length of an expert cycle ($\\lambda =10$ in the Ant environment, 25 in the Half-Cheetah, and 40 for the Hopper), ended up achieving the best performances.", "In a sense, the proposed $\\eta $ -optimality criterion can be seen as an inductive bias: we successfully injected qualitative knowledge (the repetitive nature of the expert behaviour) by explicitly asking the agent to focus on matching $\\rho _\\pi (.|s)$ for states $s$ observed within a single movement cycle of the expert demonstrations via careful parameterisation of the distribution $\\eta $ .", "In the case where such higher understanding/representation of the expert behaviour is unavailable, using a uniform $\\eta $ distribution (or a geometric $\\eta $ with a parameter close to 1) is a safe bet.", "Notice that in Figure REF , the both the blue and red curves have comparable minimum values.", "Figure: Performances of MEGAN\\operatorname{MEGAN} using a Poisson η\\eta distribution: Setting the parameter of η=Poisson(λ)\\eta =\\operatorname{Poisson}(\\lambda ) to a value around the length of the expert's movement cycle achieved similar/better performances than those obtained using a uniform η\\eta distribution (Geom(1)\\operatorname{Geom}(1)).", "The expert cycle is roughly 10 frames long in the Ant environment, 25 in the Half-Cheetah, and 40 in the Hopper." ], [ "Effect of varying the discount factor $\\gamma $", "Recall that in the classical problem formulation, the discount factor $\\gamma $ can be interpreted as the weight of future observations.", "In this section, we investigate whether changing this parameter can overcome the buit-in bias against policies with longer mixing times, without resorting to the $\\eta $ -optimality criterion.", "In practice, the discount factor $\\gamma $ is used separately in two building blocks of IRL algorithms (including $\\operatorname{GAIL}$ and $\\operatorname{MEGAN}$ ).", "The first instance is in the Bellman updates when solving the RL problem under a given cost function: we refer to this parameter as $\\gamma _{\\operatorname{RL}}$ .", "The second instance is in the discrimination problem when approximately sampling from $\\rho _\\pi $ : we refer to this parameter as $\\gamma _{\\operatorname{IRL}}$ .", "Due to the finite nature of the expert demonstrations, the standard approach is to approximate future state distributions by setting $\\gamma _{\\operatorname{IRL}}$ to 1 (or equivalently, sampling transitions uniformly).", "This can be seen as an asymptotic behaviour of the classical problem formulation: as the discount factor approaches 1, the associated truncated geometric distribution will converge to a uniform one.", "Reducing this parameter will only accentuate the discussed issues as it entails up-sampling the early stages of the collected demonstrations, which will inevitably favor short term imitation.", "On the other hand, it is not possible to set $\\gamma _{\\operatorname{RL}}=1$ (the Bellman operator is no longer guaranteed to admit a fixed point, and state of the art value based RL algorithms become extremely unstable).", "For this reason, most practitioners set this parameter to a value reasonably close to 1.", "In this section, we evaluated the remaining replay buffers obtained using $\\operatorname{GAIL}$ as we vary the discount factors values ($\\gamma _{RL}\\in [0.9, 0.99, 0.999]$ ).", "We emphasize that in all the reported empirical evaluations (including previous ones, i.e.", "Figures REF and REF ), we fixed $\\gamma _{\\operatorname{IRL}}$ to 1.", "In Figure REF , we report the average $\\operatorname{MMD}_\\rho $ divergence of the remaining replay buffer in solid lines, and the average $\\operatorname{MMD}_\\mu $ divergence in dashed lines.", "$\\operatorname{GAIL}$ 's performances as we vary the discount factor are reported in blue and the best performances obtained with $\\operatorname{MEGAN}$ are reported in red.", "Reducing $\\gamma _{RL}$ to $0.9$ accentuated the bias against policies with longer mixing times, and on the other hand increasing it to $0.999$ lead to a less reliable RL algorithm.", "As expected, we observe in Figure REF that both tweaks did not entail performances on par to what we obtained using $\\operatorname{MEGAN}$ .", "Figure: Performances of GAIL\\operatorname{GAIL} as we vary the discount factor γ\\gamma : Neither increasing nor decreasing the discount factor resulted in improved performances.", "Agnostic of the used parameter, GAIL\\operatorname{GAIL} was not able to match the expert behavior as well as MEGAN\\operatorname{MEGAN}." ], [ "Conclusion", "In this paper, we generalised the classical criterion of optimality in the reinforcement learning literature by putting more weights onto future observations.", "Using this novel criterion, we reformulated both the regularised RL and the maximum-entropy IRL problems.", "We reviewed existing RL algorithms and discussed their ability to search for $\\eta $ -optimal policies.", "We also generalised classical IRL solutions.", "The derived algorithm produced stable solutions with enhanced expert matching properties.", "In practice, the main difference between MEGAN and GAIL is the discriminator's sampling procedure.", "This implies that it can easily replace the latter algorithm in all subsequent contributions.", "An interesting future direction of research consists in evaluating the margin of improvement that can be gained from this modification." ], [ "Generalisability of policy gradient based RL algorithms", "Policy gradient based RL algorithms minimise the cumulative discounted cost by directly optimising the policy parameters using gradient descent.", "We discuss in this section the ability of these algorithms to search for $\\eta $ -optimal policies.", "We distinguish two types of policy gradient approaches:" ], [ "Optimising the global objective directly:", "Many methods (such as Deterministic Policy Gradient [33], and variants of the REINFORCE algorithm [40], [3]) are rooted in the policy gradient theorem [35]: $& \\nabla \\mathbb {E}_{p_0, \\pi }[\\sum _t \\gamma ^t c(s_t,a_t)] = \\int _{s_0, s} \\hspace{-11.38092pt} p_0(s_0) \\rho _\\pi (s|s_0) \\mathbb {E}_{a\\sim \\pi }[Q_\\pi ^c(s,a) \\nabla \\log \\pi (a|s)] \\\\\\text{with}\\ & \\rho _\\pi (s|s_0) = \\sum _t \\gamma ^t\\mathbb {P}_\\pi (s_t = s | s_0)$ where the policy $\\pi $ is a function of some parameter $\\theta $ and all gradients are implicitly with respect to $\\theta $ .", "In order to adapt these approaches for the search of $\\eta $ -optimal policies, the policy updates must take into account the future state distribution derivative.", "In fact, the gradient of the generalised criterion with respect to the policy induces an additional term as provided in proposition REF : Proposition 5 For any given distribution $\\eta $ : $\\nabla {E}^{\\eta }_{p_0,\\pi }[\\sum _t \\gamma ^t c(s_t,a_t)] = &\\underbrace{\\int _{s_0, s_+}\\hspace{-17.07182pt} p_0(s_0) v_{\\pi }^c(s_{+}) \\nabla P^{\\eta }_{\\pi }(s_{+}|s_0)}_\\textbf {additional term} \\\\& + \\underbrace{\\int _{s_0, s_+, s}\\hspace{-24.18501pt} p_0(s_0) P^{\\eta }_{\\pi }(s_{+}|s_0) \\rho _\\pi (s|s_{+}) \\mathbb {E}_{\\pi }[Q_{\\pi }^c(s,a) \\nabla \\log \\pi _\\theta (a|s)]}_\\textbf {modified term}$ where $P_{\\pi }^{\\eta }(s_{+}|s_0) = \\sum _{n=0}^\\infty \\eta (n) \\mathbb {P}_\\pi (s_n = s_{+}|s_0)$ .", "Notice that the modified term has the same form as the original policy gradient theorem.", "This is not an issue as it's a matter of adapting the used estimators in practice.", "However, the additional term is not taken into account.", "Furthermore, in this current form, $\\nabla P^{\\eta }_{\\pi }(s_{+}|s_0)$ is not tractable.", "This implies that current policy gradient approaches that rely on the policy gradient theorem can not search in a reliable way for $\\eta $ -optimal policies.", "On the other hand, recent policy gradient algorithms such as Trust Region Policy Optimisation (TRPO) [29] and Proximal Policy Optimisation (PPO) [30] iteratively search for a new policy $\\pi _{n}$ that improves the performances of an old policy $\\pi _{o}$ by optimising a local approximation of the right hand term in the following identity [14]: $\\mathcal {L}_0^{\\delta _0} (\\pi _{n}, c)= & \\mathcal {L}_0^{\\delta _0} (\\pi _{o}, c) + \\mathbb {E}_{p_0, \\pi _{n}}[\\sum _t \\gamma ^t A_{\\pi _{o}}^c(s_t,a_t)] \\\\= & \\mathcal {L}_0^{\\delta _0} (\\pi _{o}, c) + \\int _{s_0,s}\\hspace{-11.38092pt} p_0(s_0) \\rho _{\\pi _{n}}(s|s_0) \\int _a \\pi _{n}(a|s) A_{\\pi _{o}}^c(s,a)$ where $\\mathcal {L}_0^{\\delta _0}$ is the unregulated loss function, and $A_{\\pi }^c(s,a)$ is the advantage function: $A_{\\pi }^c(s,a) = Q_{\\pi }^c(s,a) - v_{\\pi }^c(s)$ In principle, this approach is tractable for the search of $\\eta $ -optimal policies.", "In fact, the generalised setting verifies a similar formulation of this identity: Proposition 6 For any given distribution $\\eta $ : $\\mathcal {L}_0^{\\eta } (\\pi _{n}, c) & = \\mathcal {L}_0^{\\eta } (\\pi _{o}, c) + \\mathbb {E}_{p_0, \\pi _{n}}^\\eta [\\sum _t \\gamma ^t A_{\\pi _{o}}^c(s_t,a_t)] \\\\& = \\;\\mathcal {L}_0^{\\eta } (\\pi _{o}, c) + \\int _{s_0,s}\\hspace{-11.38092pt} p_0(s_0) \\rho _{\\pi _{n}}(s|s_0) \\mathbb {E}_{\\pi _{n}}^{\\eta }[A_{\\pi _{o}}^c(s_{+},a_{+})|s] \\\\\\text{with } \\qquad &\\mathbb {E}_{\\pi _{n}}^{\\eta }[A_{\\pi _{o}}^c(s_{+},a_{+})|s] =\\hspace{-2.84544pt}\\int _{s_{+},a_{+}}\\hspace{-19.91684pt} P_{\\pi _{n}}^{\\eta }(s_{+}, a_{+}|s) A_{\\pi _{o}}^c(s_{+},a_{+})$ Propositions REF lay the ground to adapt proximal policy gradient based RL algorithms to the search of $\\eta $ -optimal policies.", "However, we do not investigate this further in this paper as we focus on the Inverse problem." ], [ "Generalisation of classical IRL algorithms", "In this section, we discuss particular classical penalisation functions $\\Omega $ and $\\Psi $ that lead to generalisation of well known $\\operatorname{IRL}$ algorithms.", "In all cases, $\\Omega $ is always chosen as the entropy regulariser, as in state of the art $\\operatorname{RL}$ algorithms.", "Let $\\mathcal {C}$ be a subset of admissible cost functions, and the penalisation function defined as: $\\psi (c) = \\imath _{\\mathcal {C}}(c) =\\left\\lbrace \\begin{array}{ll}0 & \\textit {if } c\\in \\mathcal {C} \\\\+\\infty & \\textit {if } c\\notin \\mathcal {C}\\end{array}\\right.$ Two particular subsets are studied in details in the following as they lead to generalisations of classical $\\operatorname{IRL}$ algorithms." ], [ "EMMA: Expectation Matching - Maximum Entropy", "First, consider the set of linear interpolation of some finite basis set function $\\lbrace f_i(s,a), \\ i \\in \\mathcal {I}\\rbrace $ , i.e., $\\mathcal {C}_{\\textit {linear}} = \\Big \\lbrace \\sum _{i \\in \\mathcal {I}} w_i f_i, \\ \\textit { such that } \\Vert w\\Vert _2 \\le 1 \\Big \\rbrace .$ In the classical, non-generalised IRL problem (with $\\eta =\\delta _0$ ), this problem coincides with features expectation matching $\\operatorname{IRL}$ algorithm [1], that minimises the $l_2$ expected feature vectors [11]: $L(\\pi ,c) + \\Omega (\\pi ) = \\max _{c\\in \\mathcal {C}_{\\textit {linear}}} \\mathbb {E}_{\\rho _\\pi }[c(s,a)]-\\mathbb {E}_{\\rho _{\\bar{\\pi }}}[c(s,a)] = \\big \\Vert \\mathbb {E}_{\\rho _\\pi }[f]-\\mathbb {E}_{\\rho _{\\bar{\\pi }_E}}[f] \\big \\Vert _2,$ where $f(s,a)=(f_i(s,a))_{i \\in \\mathcal {I}}$ .", "Generalising this algorithm for any geometric distribution $\\eta $ simply consists in substituting the expectation with ${E}_\\pi ^\\eta $ : Proposition 7 Under the assumptions of Proposition REF , and for $\\psi =\\imath _{\\mathcal {C}_{\\textit {linear}}}$ , it holds that: $\\operatorname{RL}_\\Omega ^\\eta \\circ \\operatorname{IRL}_{\\psi }^\\eta (\\bar{\\pi }) = \\operatornamewithlimits{argmin}_\\pi -\\Omega (\\pi ) + \\big \\Vert \\mathbb {E}_\\pi ^\\eta [f]-\\mathbb {E}_{\\pi _E}^\\eta [f] \\big \\Vert _2$ The generalised version of this algorithm (which is actually $\\operatorname{GIRL}_{\\imath {\\mathcal {C}_{\\textit {linear}}}, H, \\eta }$ ) that we called $\\operatorname{EMMA}_{\\eta }$ , is derived in the following as a generalisation of expectation matching $\\operatorname{IRL}$ algorithm [1].", "The optimal cost function $c^*_\\pi $ (given a previously learned policy $\\pi $ ) must satisfy the following equalityc.f.", "the proof of proposition REF in Appendix : $\\int _{s_0} p_0(s_0) \\big [\\mu _\\pi (s_{+},a_{+}|s_0)-\\mu _{\\pi _E}(s_{+},a_{+}|s_0)\\big ] c^*_\\pi (s_{+},a_{+}) = \\Big \\Vert \\mathbb {E}_{\\mu _\\pi }[f]-\\mathbb {E}_{\\mu _{\\pi _E}}[f] \\Big \\Vert _2$ As the objective solution is known, we propose to replace the cost optimisation step in the template procedure we provide in Algorithm REF with the following simple quadratic loss: $\\mathcal {L}_\\text{linear}(w) = \\Big ( \\mathbb {E}_{\\mu _\\pi }[wf^T]-\\mathbb {E}_{\\mu _{\\pi _E}}[wf^T] - \\Vert \\mathbb {E}_{\\mu _\\pi }[f]-\\mathbb {E}_{\\mu _{\\pi _E}}[f]\\Vert _2 \\Big )^2$ Given that the set of feasible cost function is convex, we can update the loss using projected gradient updates.", "We also use an approximation of this loss in practice: $\\bar{\\mathcal {L}}_\\text{linear}(w) = \\Big ( \\sum _{S^+,A^+} wf^T(s,a) - \\sum _{S^+_E,A^+_E} wf^T(s,a) - \\Vert \\sum _{S^+,A^+} f(s,a) - \\sum _{S^+_E,A^+_E} f(s,a)\\Vert _2 \\Big )^2$ The algorithm we propose is then defined as follows: $\\operatorname{EMMA}$ [1] Input: Expert trajectories $\\tau _E \\sim \\pi _E$ , initial policy $\\pi _{\\theta _0}$ and initial cost function $w_0$ $e \\in [1, N]$ Sample trajectories $\\tau \\sim \\pi _{\\theta _i}$ Sample states randomly $(S_t,A_t)\\sim \\tau $ and $(S^+,A^+) = (S_{t+k}, A_{t+k})$ where $k\\sim \\eta $ Sample states randomly $(S^{\\prime }_t,A^{\\prime }_t)\\sim \\tau _E$ and $(S^+_E,A^+_E) = (S^{\\prime }_{t+k}, A^{\\prime }_{t+k})$ where $k\\sim \\eta $ Update the cost weights $w_i$ to minimise $ \\bar{\\mathcal {L}}_\\text{linear}(w_i)$ Project the cost weights on the feasible set $\\mathcal {C}_\\text{linear}$ Update $\\theta _i$ using soft actor critic to minimise $w_{i+1}f^T$ Return: $(\\pi _{\\theta _N}, D_{w_N})$" ], [ "WIEM: Worst Individual cost - Entropy Maximizer:", "We now consider convex combination of basis functions: $ \\mathcal {C}_{\\textit {convex}} = \\Big \\lbrace \\sum _{i \\in \\mathcal {I}} w_i f_i, \\textit { with} \\sum _{i \\in \\mathcal {I}} w_i = 1, \\text{ and } w_i \\ge 0, \\forall i\\in \\mathcal {I}\\Big \\rbrace $ In the classical non-generalised IRL setting, this is equivalent to MWAL [37] and LPAL [36] where we minimise the worst-case excess cost among the basis functions [11]: $L(\\pi ,c) + \\Omega (\\pi ) = \\max _{c\\in \\mathcal {C}_{\\textit {convex}}} \\mathbb {E}_{\\rho _\\pi }[c(s,a)]-\\mathbb {E}_{\\rho _{\\bar{\\pi }}}[c(s,a)] = \\max _{i\\in \\mathcal {I}} \\mathbb {E}_{\\rho _\\pi }[f_i]-\\mathbb {E}_{\\rho _{\\bar{\\pi }}}[f_i]$ This setting is also simply generalised for any geometric $\\eta $ by takng the expectation w.r.t.", "${E}_\\pi ^\\eta $ : Proposition 8 Under the assumptions of Proposition REF , and for $\\psi =\\imath _{\\mathcal {\\mathcal {C}_{\\textit {convex}}}}$ , it holds that: $\\operatorname{RL}_\\Omega ^\\eta \\circ \\operatorname{IRL}_{\\psi }^\\eta (\\pi _E) = \\operatornamewithlimits{argmin}_\\pi -\\Omega (\\pi ) +\\max _i \\mathbb {E}_\\pi ^\\eta [f_i]-\\mathbb {E}_{\\pi _E}^\\eta [f_i]$ We derive $\\operatorname{WIEM}_{\\eta }$ in the following, which is equivalent to $\\operatorname{GIRL}_{\\delta _{\\mathcal {C}_{\\textit {convex}}}, H, \\eta }$ and a generalisation of worst-case excess IRL algorithms.", "The optimal cost function $c^*_\\pi $ (given a previously learned policy $\\pi $ ) must satisfy the following equalityc.f.", "the proof of proposition REF in Appendix : $\\int _{s_0} p_0(s_0) \\big [\\mu _\\pi (s_{+},a_{+}|s_0)-\\mu _{\\pi _E}(s_{+},a_{+}|s_0)\\big ] c^*_\\pi (s_{+},a_{+}) = \\max _i \\mathbb {E}_{\\mu _\\pi }[f_i]-\\mathbb {E}_{\\mu _{\\pi _E}}[f_i]$ As the objective solution is known, we propose to replace the cost optimisation step in the template procedure we provide in Algorithm REF with the following simple quadratic loss: $\\mathcal {L}_\\text{convex}(w) = \\Big ( \\mathbb {E}_{\\mu _\\pi }[wf^T]-\\mathbb {E}_{\\mu _{\\pi _E}}[wf^T] - \\max _i \\mathbb {E}_{\\mu _\\pi }[f_i]-\\mathbb {E}_{\\mu _{\\pi _E}}[f_i] \\Big )^2$ Given that the set of feasible cost function is convex, we can update the loss using projected gradient updates.", "We also use an approximation of this loss in practice: $\\bar{\\mathcal {L}}_\\text{convex}(w) = \\Big ( \\sum _{S^+,A^+} wf^T(s,a) - \\sum _{S^+_E,A^+_E} wf^T(s,a) - \\max _i \\big [\\sum _{S^+,A^+} f_i(s,a) - \\sum _{S^+_E,A^+_E} f_i(s,a)\\big ] \\Big )^2$ The algorithm we propose is then defined as follows: $\\operatorname{WIEM}$ [1] Input: Expert trajectories $\\tau _E \\sim \\pi _E$ , initial policy $\\pi _{\\theta _0}$ and initial cost function $w_0$ $e \\in [1, N]$ Sample trajectories $\\tau \\sim \\pi _{\\theta _i}$ Sample states randomly $(S_t,A_t)\\sim \\tau $ and $(S^+,A^+) = (S_{t+k}, A_{t+k})$ where $k\\sim \\eta $ Sample states randomly $(S^{\\prime }_t,A^{\\prime }_t)\\sim \\tau _E$ and $(S^+_E,A^+_E) = (S^{\\prime }_{t+k}, A^{\\prime }_{t+k})$ where $k\\sim \\eta $ Update the cost weights $w_i$ to minimise $ \\bar{\\mathcal {L}}_\\text{convex}(w_i)$ Project the cost weights on the feasible set $\\mathcal {C}_\\text{convex}$ Update $\\theta _i$ using soft actor critic to minimise $w_{i+1}f^T$ Return: $(\\pi _{\\theta _N}, D_{w_N})$" ], [ "Multi-task setting", "Classically, the multi-task setting is defined by considering a task space $\\Theta $ and for each task $\\theta \\in \\Theta $ the associated Markov decision process $\\mathcal {M}_\\theta =\\lbrace \\mathcal {S}, \\mathcal {A}, \\mathcal {P}, c_\\theta , \\gamma , p_0\\rbrace $ .", "Depending on the context, the objective is then to either solve the RL or the IRL problems for the set of MDPs $(\\mathcal {M}_\\theta )_{\\theta \\in \\Theta }$ by averaging the losses with respect to a task distribution $\\mathcal {F}$ .", "This is equivalent in principle to solving the problem for the MDP $\\bar{\\mathcal {M}}=\\lbrace \\mathcal {S}\\times \\Theta , \\bar{\\mathcal {A}}, \\bar{\\mathcal {P}}, \\bar{c}, \\gamma , \\bar{p}_0\\rbrace $ where for any states $(s,s^{\\prime })$ , tasks $(\\theta ,\\theta ^{\\prime })$ and action $a$ , the following equalities hold true: $&\\bar{\\mathcal {A}}(s,\\theta ) &= & \\; \\mathcal {A}(s) & \\\\&\\bar{\\mathcal {P}}(s^{\\prime },\\theta ^{\\prime }|s,\\theta ,a) & = & \\; \\mathcal {P}(s^{\\prime }|s,a)\\delta (\\theta ^{\\prime }=\\theta )&\\\\&\\bar{c}(s,\\theta ,a) & = & \\; c_\\theta (s,a) & \\\\&\\bar{p}_0(s,\\theta ) & = & \\; p_0(s)\\mathcal {F}(\\theta ) &$ We adapt the latter formulation here for the sake of coherence with previous sections.", "The difficulty in multi-task settings arises from ray-interference: when the cost function encourages conflicting behaviours for different tasks, the learning objective plateaus [27].", "This stagnates the progress of the policy, which in turn complicates the IRL problem as these plateaus are an opportunity for the discriminator to over-fit the replay-buffer.", "To alleviate this issue, we propose to augment the data-set as proposed in Section REF (by using $\\operatorname{MEGAN}$ coupled with the Idle subroutine." ], [ "Idle : an On-policy data augmentation routine", "If the state space $\\mathcal {S}$ is very large, or even continuous, it becomes quite unlikely to encounter the same state twice in a (finite) trajectory from a given data-set.", "In particular, this renders quite difficult the estimation of future state distribution $P_\\pi ^\\eta (.|s)$ (where $s\\sim \\rho _\\pi (.|s_0)$ ).", "To circumvent this issue, we propose to use the following on-policy data augmentation scheme, modeled as a game between a discriminator $D: (\\mathcal {S}\\times \\mathcal {A}\\times \\mathcal {S})\\rightarrow [0,1]$ and a generator $G:\\mathcal {S}\\rightarrow \\Delta (\\mathcal {S}\\times \\mathcal {A})$ .", "The objective of the generator is to produce future states similar to the gathered samples while the discriminator $D$ aims to identify true samples from generated ones, with the following score function: $V(D,G) = \\mathbb {E}\\big [ \\log (D(s_+,a_+|s)) +\\log (1-D(s_g,a_g|s)) \\big ]$ where the expectation is taken w.r.t the future state distribution $P_\\pi ^\\eta (.|s)$ for $(s_+,a_+)$ , the generator distribution $G(.|s)$ for $(s_g,a_g)$ , and the marginal over the initial state distribution of the occupancy measure $\\rho _\\pi (.|s_0)$ for $s$ .", "Solving this game approximates $P_{\\pi }^{\\eta }$ : Proposition 9 $(\\tilde{D},\\tilde{G})= (\\frac{1}{2},P_{\\pi }^{\\eta })$ is a Nash-equilibrium of the following zero-sum game: $D^*: \\ \\min _D V(D,G) \\; \\ \\text{ and } \\;G^*: \\ \\max _G V(D,G)$ From this proposition, we derive in Algorithm REF , a method to approximate future state distributions that is used as a subroutine in $\\operatorname{GIRL}$ .", "This approach is theoretically feasible given an on-policy data set (such as the expert's trajectories).", "In practice, we noticed that approximating $P_\\pi ^\\eta $ using (Idle) produces reliable generators when the variance of $\\eta $ is relatively small, so that future state samples fall within a reduced range with a high probability.", "Inversely, if $\\eta $ has a high variance, samples from $P_\\pi ^\\eta $ would be along extended horizon and the (Idle) discriminator easily picks up on the parts being learned and halts the generator’s improvement.", "This either leads to vanishing gradient updates or a mode collapse.", "Idle (an on-policy future state generator) [1] Input: On-policy trajectories $\\tau $ , initial discriminator $D_{\\phi _0}$ and initial generator $G_{\\nu _0}$ $e \\in [1, N]$ Sample states randomly $(S_t,A_t)\\sim \\tau $ Sample $(S^+,A^+) = (S_{t+k}, A_{t+k})$ where $k\\sim \\eta $ Sample $(S^+_G,A^+_G)\\sim G_{\\nu _i}(S)$ Update the discriminator parameter $\\phi _i$ to minimise: $\\sum _{\\begin{array}{c}S_t,S^+,A^+\\end{array}} \\log (D_{\\phi _i}(s_+,a_+|s)) + \\sum _{\\begin{array}{c}S_t,S^+_G,A^+_G\\end{array}} \\log (1-D_{\\phi _i}(s_+,a_+|s))$ Update the generator parameter $\\nu _i$ to minimise: $\\quad \\quad \\sum _{\\begin{array}{c}S_t\\end{array}} \\log (D(G(s)|s))$ Return: $(D_{\\phi _N}, G_{\\nu _N})$" ], [ "Fetch-Reach environment", "We consider in this section the FetchReachTo the extent of our knowledge, this is the first reported performances of IRL algorithms on a fully continuous environment task from the MuJoCo based environments [22].", "To evaluate the generalisability of the learned policies, we only generate expert trajectories for a subset of possible tasks (only target positions that are 5-10 cm away from the initial gripper's positionthe maximum range of the arm is about 25 cm to be precise).", "We evaluate the learned policies in the learned setting (same horizon and same tasks) and in a generalisability setting (twice the training horizon and the full range of tasks).", "As in the simple task setting, we asses performances in terms of normalised cumulative costs.", "Figure: Performances in the Fetch Reach settingIn Figure REF , we compare the performances of MEGAN (with and without the data augmentation) and GAIL over the training.", "We observe that both GAIL and MEGAN (without Idle) struggle in solving the problem.", "However, using the Idle generator reduces the undesirable effect of ray interference and stabilises the training.", "Performance wise, the learned policy using MEGAN outperforms the expert demonstrations in the training tasks while at the same time providing comparable performances in the remaining set of tasks (as provided in Figure REF ).", "Figure: Idly generated samples from expert trajectoryIt is arguable that the success of MEGAN in the multi-task setting is explained with the Idle procedure.", "A similar approach on GAIL might be appealing.", "However, this is not feasible in practice.", "It is true that the reasoning provided in Section REF can be developed for any distribution $\\eta $ , this entails that we can use the same approach to learn a generator that mimics $\\rho _\\pi (.|s_0)$ .", "Unfortunately, this is not feasible in practice (due to the high variance of $\\rho _\\pi $ , as explained in Section REF ).", "The issue at play here is that the latter distribution ($\\rho _\\pi )$ covers all the observations of the trajectory.", "Indeed, from our early empirical attempts, we noticed that learning such distribution is unstable: we either over-fit a sub-set of the trajectory (notably the stationary distribution, which is hurtful for our purpose) or we do not learn the distribution at all.", "On the other hand, learning $P_\\pi ^\\eta $ conditioned on some intermediate state is a lot easier when we choose $\\eta $ such that it only covers the near future transitions (for example when $\\eta =\\operatorname{Geom}(0.7)$ , the average prediction of $P_\\pi ^\\eta (|s_t)$ is 3 steps in the future).", "In Figure REF , we evaluate the learned approximation of $P_{\\pi _E}^{\\operatorname{Geom}(0.7)}$ on a sample expert trajectory.", "We plot the evolution of the (true/generated) gripper position in 3D overtime.", "For each state encountered on the trajectory, the learned generator outputs 10 samples (in blue).", "Clearly, the future state generator is reliable; this is successful because the distribution $\\eta $ is a short term prediction: the learned generator maps current states to the possible ones in the next few steps." ], [ "Multi-task 2-D navigation", "In this section we report the performances on a custom-made multi-task navigation environment.", "The goal is to navigate from an initial position to a target position while avoiding four lakes.", "The state space is constructed by concatenating the coordinates of the agent, the coordinates of the target as well as the distance from the centre of each of the four lakes.", "The action space is the norm 2 ball $\\lbrace x\\in \\mathbb {R}^2 \\textbf { s.t.\\ } \\Vert x\\Vert _2\\le 1 \\rbrace $ .", "The transition kernel is a Dirac mass at the sum of the previous position and the action vector.", "If the sum is within one of the lakes or outside the grid, then the new position is the projection of the previous position on the border according to the action direction.", "In Figure REF , we render the environment to provide an idea about the task at hand: the lakes are painted in blue, the agent is the red square, the target is the green square, and the grey pixels are the possible positions.", "These positions are the subset of $[-10,10]^2$ that excludes the points within the lakes.", "The goal is to navigate around the lakes in order to reach the target position." ], [ "Learning the Idle generator", "In this section we analyse the ability to learn $P_\\pi ^\\eta $ and $\\rho _\\pi $ using Algorithm REF .", "As discussed in Section REF , when the target distribution is a short term prediction of future states, the obtained generator is reliable.", "We consider in what follows $\\eta =\\operatorname{Geom}(0.7)$ to satisfy this condition.", "We use the same hyper-parameters to learn the generator in both cases ($P_\\pi ^\\eta $ and $\\rho _\\pi $ ).", "We use 3-layers deep, 64-neurons wide neural networks for the generator and the discriminator, a batch size of 256 and we iterate the algorithm for 10000 steps.", "In Figures REF and REF , we plot the expert trajectories with black lines, the initial position with green dots, the target position with red dots, and the sampled states with blue triangles.", "On one hand, we observe in Figure REF that the $P_\\pi ^\\eta $ learned generator provides reliable samples (in the sense that they follow the trails of expert trajectories).", "On the other hand, in Figure REF , we observe that the learned $\\rho _\\pi $ generator over-fits the stationary distribution and only samples states around the target position.", "The mode collapse is essentially explained by the fact that most of the samples from the $\\rho _\\pi $ distribution are indeed around the target position.", "Figure: P π η P_\\pi ^\\eta learned generatorFigure: ρ π \\rho _\\pi learned generator" ], [ "Learned cost function", "As discussed in Section , we only obtain expert-like performances when using the Idle generator.", "However, given that the considered state-space in this section is a 2-D plan, we can visualise the learned (state only) cost function with a heat-map.", "We consider five particular tasks that coincide with reaching the top-left, center, top-right, bottom-left and bottom-right of the map.", "Figures REF and REF coincide with such heat-maps, with darker shades for higher costs and brighter colours for lower ones.", "In Figure REF , we observe that the GAIL learned costs are particularly low in the vicinity of the target position while they are evenly spreaded elsewhere.", "This entails from the discriminator over-fitting the replay-buffer as most of the observations are drawn from the stationary distribution.", "On the other hand, in Figure REF , we observe that the MEGAN learned costs are high outside of the paths that lead to the target, and decrease exponentially as we get closer to the goal position.", "Figure: GAIL learned cost heat-map as a function of the target positionFigure: MEGAN learned cost heat-map as a function of the target position" ], [ "Maximum Mean Discrepancy evaluation", "Formally, given a reproducing kernel Hilbert space (RKHS) of real-valued functions $\\mathcal {H}$ , the MMD between two distributions $P$ and $Q$ is defined as: $\\operatorname{MMD}_{\\mathcal {H}}(P, Q) = \\sup _{f\\in \\mathcal {H}} \\mathbb {E}_{X\\sim P}[f(X)] - \\mathbb {E}_{Y\\sim Q}[f(Y)]$ .", "Recall that the reproducing property of RKHS, implies that there is a one to one correspondence between positive definite kernels $k$ and RKHSs $\\mathcal {H}$ such that every function $f\\in \\mathcal {H}$ verifies $f(x)=\\langle f,k(.,x)\\rangle _{\\mathcal {H}}$ (where $\\langle \\,,\\rangle _{\\mathcal {H}}$ denotes the RKHS inner product).", "We propose to evaluate the MMD using a kernel two-sample test with the following unbiased estimator [8]: $\\operatorname{MMD}_{\\mathcal {H}}^2(P, Q) = \\frac{1}{N(N-1)} \\sum _{i\\ne j} k(x_i, x_j) + \\frac{1}{N(N-1)} \\sum _{i\\ne j} k(y_i, y_j) - \\frac{1}{N^2} \\sum _{i, j} k(x_i, y_j)$ where $(x_i)_{i=0}^N$ are sampled according to $P$ and $(y_i)_{i=0}^N$ are sampled according to $Q$ .", "In the experimental analysis, we only consider the RKHS associated with the radial basis function $k(x,y) = \\exp (\\Vert x-y\\Vert ^2/d)$ (where $d$ is the dimension of the variables $x$ and $y$ )." ], [ "Hyper-parameters", "In this section we provide a detailed description of the used implementation as well as the selected hyper-parameters.", "Expert demonstrations of length Max-Length are stored in a demonstrator replay-buffer.", "We use two additional replay-buffers (one for the policy and one for the expert), with a maximum capacity of $10^6$ transitions that are initially empty.", "In each cycle, N trajectories from the demonstrator replay-buffer are sampled and added to the expert replay-buffer.", "The policy generates then N trajectories, that are stored in the policy replay-buffer.", "In the multi-task setting, the tasks of these trajectories are the same ones in the expert's samples.", "The policy is updated each cycle using SAC for SAC-Epoch epochs with a bath size of SAC-Batch.", "Every D-Update-Rate cycles, the discriminator is updated for D-Epoch epochs with a bath size of D-Batch.", "The algorithm runs until the policy generates Max-Transitions transitions in total.", "The policy, as well as the underlying value functions, are approximated using an N-Layer-P deep, Hidden-P wide neural networks.", "The discriminator is approximated using an N-Layer-D deep, Hidden-D wide neural network.", "Table: NO_CAPTION" ], [ "Proof of technical results", "We provide in this section proofs for all stated technical results.", "To find a particular one, please refer to the following: Section REF : Useful intermediate results as well as their proof.", "Section REF : Proofs for the theoretical claims stated in Section REF and Appendix .", "Section REF : Proofs for the theoretical claims stated in Section REF .", "Section REF : Proof for the theoretical claims stated in Section REF .", "Section REF : Proof for the theoretical claims stated in Section ." ], [ "Useful intermediate results", "For the sake of conciseness, we start by providing important intermediate results that will be used in the proofs of propositions REF , REF , and REF .", "The first one (Proposition REF ) transforms $\\eta $ -weighted $\\gamma $ discounted functional averaged over $\\pi $ -generated trajectory into expectations with respect to $\\rho _\\pi $ and $P_\\pi ^\\eta $ or, equivalently into expectations with respect to $\\mu _\\pi $ .", "The second one (Proposition REF ) guarantees a one on one mapping between occupancy measures and policies.", "Proposition 10 For any distribution $\\eta $ , and for any mapping $f:\\mathcal {S}\\times \\mathcal {A}\\rightarrow \\mathbb {R}$ , the following identity holds: $\\mathbb {E}_{p_0,\\pi }^\\eta [\\sum _t \\gamma ^t f(s_t,a_t)] & = \\int _{s_0,s,a,s_{+},a_{+}}\\hspace{-28.45274pt} p_0(s_0)\\rho _\\pi (s,a|s_0)P_\\pi ^\\eta (s_{+},a_{+}|s,a) f(s_{+},a_{+}) \\\\& = \\int _{s_0,s_{+},a_{+}}\\hspace{-14.22636pt} p_0(s_0)\\mu _\\pi (s_{+},a_{+}|s_0) f(s_{+},a_{+})$ Proposition 11 Let $(\\phi _t)_{t=0}^\\infty $ be a strictly positive real valued convergent series (i.e.", "$\\sum _t \\phi _t<\\infty $ and $\\phi _t>0$ ), and let $\\Phi _\\pi (s,a|s_0)$ be the $\\phi $ -weighted occupancy measure associated to the policy $\\pi $ : $\\Phi _\\pi (s,a|s_0) := \\sum _t \\phi _t \\mathbb {P}_\\pi (s_t=s,a_t=a|s_0)$ Then for a given $\\phi $ -weighted occupancy measure $\\Phi \\in \\lbrace \\Phi _\\pi | \\pi :\\mathcal {S}\\rightarrow \\Delta (\\mathcal {A})\\rbrace $ : 1- $\\Phi $ is the $\\phi $ -weighted occupancy measure of $\\pi _\\Phi :=\\frac{\\Phi (s,a|s_0)}{\\int _{a^{\\prime }}\\Phi (s,a^{\\prime }|s_0)}$ 2- $\\pi _\\Phi $ is the only policy whose $\\phi $ -weighted occupancy measure is $\\Phi $" ], [ "proof of proposition ", "The proof of the first equality relies on some algebraic manipulations and the law of total expectation.", "$& \\mathbb {E}_{p_0,\\pi }^\\eta [\\sum _t \\gamma ^t f(s_t,a_t)] := \\int _{s_0,s,a}\\hspace{-14.22636pt} p_0(s_0) P_\\pi ^\\eta (s,a|s_0) \\mathbb {E}_{\\pi , \\delta _{(s,a)}}[\\sum _t \\gamma ^t f(s_{t},a_{t})] \\\\& \\qquad =\\int _{s_0,s,a}\\hspace{-14.22636pt} p_0(s_0)\\sum _k \\eta (k) \\mathbb {P}_\\pi (s_k=s,a_k=a|s_0) \\mathbb {E}_{\\pi }[\\sum _t \\gamma ^t f(s_{t+k},a_{t+k})|s_k=s,a_k=a] \\\\& \\qquad = \\int _{s_0,s_k,a_k}\\hspace{-22.76228pt} p_0(s_0)\\sum _{k,t} \\gamma ^t \\eta (k) \\mathbb {E}_{\\pi }\\Big [ \\mathbb {E}_{\\pi }\\big [f(s_{t+k},a_{t+k})|s_k,a_k\\big ]\\Big |s_0\\Big ]$ where $\\mathbb {E}_{\\pi , \\delta _{(s,a)}}$ designate the expectation over trajectories initialised at the state action couple $(s,a)$ .", "Using the law of total expectation we can assert that: $\\mathbb {E}_{\\pi }\\Big [ \\mathbb {E}_{\\pi }\\big [f(s_{t+k},a_{t+k})|s_k,a_k\\big ]\\Big |s_0\\Big ] & = \\mathbb {E}_\\pi [f(s_{t+k},a_{t+k})|s_0] \\\\& = \\mathbb {E}_{\\pi }\\Big [ \\mathbb {E}_{\\pi }\\big [f(s_{t+k},a_{t+k})|s_t,a_t\\big ]\\Big |s_0\\Big ]$ From this relationship, it follows that: $&\\mathbb {E}_{p_0,\\pi }^\\eta [\\sum _t \\gamma ^t f(s_t,a_t)] = \\int _{s_0,s_t,a_t}\\hspace{-22.76228pt} p_0(s_0)\\sum _{k,t} \\gamma ^t \\eta (k) \\mathbb {E}_{\\pi }\\Big [ \\mathbb {E}_{\\pi }\\big [f(s_{t+k},a_{t+k})|s_t,a_t\\big ]\\Big |s_0\\Big ] \\\\& \\qquad = \\int _{s_0,s,a}\\hspace{-14.22636pt} p_0(s_0)\\sum _{t} \\gamma ^t \\mathbb {P}_\\pi (s_t=s,a_t=a|s_0) \\mathbb {E}_{\\pi , \\delta _{(s,a)}}[\\sum _k \\eta (k) f(s_{k},a_{k})] \\\\& \\qquad = \\int _{s_0,s,a}\\hspace{-14.22636pt} p_0(s_0) \\rho _\\pi (s,a|s_0) \\mathbb {E}_{\\pi , \\delta _{(s,a)}}[\\sum _k \\eta (k) f(s_{k},a_{k})] \\\\& \\qquad = \\int _{s_0,s,a}\\hspace{-14.22636pt} p_0(s_0) \\rho _\\pi (s,a|s_0) \\sum _k \\eta (k) \\int _{s_+,a_+}\\hspace{-14.22636pt}\\mathbb {P}_\\pi (s_{t+k}=s_{+},a_{t+k}=a_{+}|s_t=s, a_t=a) f(s_+,a_+) \\\\& \\qquad = \\int _{s_0,s,a,s_{+},a_{+}}\\hspace{-28.45274pt} p_0(s_0)\\rho _\\pi (s,a|s_0)P_\\pi ^\\eta (s_{+},a_{+}|s,a) f(s_{+},a_{+})$ This concludes the proof of the first equality in Proposition REF .", "The proof of the second equality relies on the Markov property of the environment and some algebraic manipulations.", "$&\\mathbb {E}_{p_0,\\pi }^\\eta [\\sum _t \\gamma ^t f(s_t,a_t)] = \\int _{s_0,s,a,s_{+},a_{+}}\\hspace{-28.45274pt} p_0(s_0) \\rho _\\pi (s,a|s_0)P_\\pi ^\\eta (s_{+},a_{+}|s,a) f(s_{+},a_{+}) \\\\& \\qquad = \\int _{s_0,s,a,s_{+},a_{+}}\\hspace{-28.45274pt} p_0(s_0) \\sum _{t} \\gamma ^t \\mathbb {P}_\\pi (s_t=s, a_t=a|s_0) \\sum _k \\eta (k) \\mathbb {P}_\\pi (s_{t+k}=s_+, a_{t+k}=a|s_t=s, a_t=a)f(s_{+},a_{+}) \\\\& \\qquad = \\int _{s_0,s_{+},a_{+}}\\hspace{-14.22636pt} p_0(s_0) \\sum _{t,k} \\gamma ^t \\eta (k) \\int _{s,a} \\Big ( \\mathbb {P}_\\pi (s_t=s, a_t=a|s_0) \\mathbb {P}_\\pi (s_{t+k}=s_+, a_{t+k}=a|s_t=s, a_t=a) \\Big ) f(s_{+},a_{+}) \\\\& \\qquad = \\int _{s_0,s_{+},a_{+}}\\hspace{-14.22636pt} p_0(s_0) \\sum _{t,k} \\gamma ^t \\eta (k) \\mathbb {P}_\\pi (s_{t+k}=s_+, a_{t+k}=a|s_0) f(s_{+},a_{+}) \\\\& \\qquad = \\int _{s_0,s_{+},a_{+}} \\hspace{-14.22636pt} p_0(s_0) \\mu _\\pi (s_{+},a_{+}|s_0) f(s_{+},a_{+})$ This concludes the proof of Proposition REF .", "For the first assertion of the proposition, recall that: $\\Phi _\\pi (s,a|s_0) & := \\sum _t \\phi _t \\mathbb {P}_\\pi (s_t=s,a_t=a|s_0) \\\\& = \\sum _t \\phi _t \\mathbb {P}_\\pi (s_t=s|s_0) \\pi (a|s) :=\\Phi _\\pi (s|s_0) \\pi (a|s)$ This implies that: $\\frac{\\Phi (s,a|s_0)}{\\int _{a^{\\prime }}\\Phi (s,a^{\\prime }|s_0)} &= \\frac{\\pi (a|s) \\Phi _\\pi (s|s_0)}{\\int _{a^{\\prime }}\\pi (a^{\\prime }|s) \\Phi _\\pi (s|s_0)} = \\pi (a|s)$ For the second assertion of the proposition, consider two policies $\\pi _1$ and $\\pi _2$ such that $\\Phi _{\\pi _1}=\\Phi _{\\pi _2}$ .", "Notice that: $\\forall s,s_0\\in \\mathcal {S}& & \\Phi _{\\pi _1}(s|s_0) & := \\sum _t \\phi _t \\mathbb {P}_{\\pi _1}(s_t=s|s_0) = \\int _a \\Phi _{\\pi _1}(s,a|s_0) \\\\& & & = \\int _a \\Phi _{\\pi _2}(s,a|s_0) = \\Phi _{\\pi _2}(s|s_0)$ This can further yield: $&\\forall s\\in \\mathcal {S}, a\\in \\mathcal {A} & \\Phi _{\\pi _1}(s,a|s_0) = \\Phi _{\\pi _2}(s,a|s_0) \\\\\\Rightarrow &\\forall s\\in \\mathcal {S}, a\\in \\mathcal {A} & \\Phi _{\\pi _1}(s|s_0)\\pi _1(a|s) = \\Phi _{\\pi _2}(s|s_0)\\pi _2(a|s) \\\\\\Rightarrow & \\forall s\\in \\mathcal {S}, a\\in \\mathcal {A} & \\pi _1(a|s) = \\pi _2(a|s)$ This concludes the proof of Proposition REF ." ], [ "Generalised Reinforcement Learning", "In this section we address the claims stated in section REF as well as those stated in Appendix .", "Proposition REF is recalled for the sake of comprehensiveness, a detailed proof is provided in [6]." ], [ "Proof of Corollary ", "The proof relies on the fact that optimal regularised policies are associated to the minimum regularised value function (as stated in Proposition REF ).", "By construction $\\pi ^*_\\Omega $ would minimise the Q-function for all possible state-actions.", "Recall that for any given policy $\\pi $ , the following identity hold [6]: $Q_{\\pi ,\\Omega }^c(s,a) & := Q_\\pi ^c(s,a) - \\Omega (\\pi ) = c(s,a) + \\mathbb {E}_{s^{\\prime }|s,a}[v_{\\pi ,\\Omega }^c(s^{\\prime })]$ From Proposition REF and Equation REF we can derive the following implications for any policy $\\pi $ : $& \\forall s\\in \\mathcal {S} & v_{\\pi _\\Omega ^*,\\Omega }^c(s) \\le v^c_{\\pi , \\Omega }(s) \\\\\\Rightarrow & \\forall s\\in \\mathcal {S}, a\\in \\mathcal {A} & \\mathbb {E}_{s^{\\prime }|s,a}[v_{\\pi _\\Omega ^*,\\Omega }^c(s^{\\prime })] \\le \\mathbb {E}_{s^{\\prime }|s,a}[v_{\\pi ,\\Omega }^c(s^{\\prime })] \\\\\\Rightarrow & \\forall s\\in \\mathcal {S}, a\\in \\mathcal {A} & Q_{\\pi _\\Omega ^*}^c(s,a) - \\Omega (\\pi _\\Omega ^*) \\le Q_\\pi ^c(s,a) - \\Omega (\\pi ) \\\\\\Rightarrow & \\forall \\eta \\in \\Delta (\\mathbb {N}) & {E}^{\\eta }_{p_0,\\pi _\\Omega ^*}[Q^c_{\\pi _\\Omega ^*}] - \\Omega (\\pi _\\Omega ^*) \\le {E}^{\\eta }_{p_0,\\pi }[Q^c_{\\pi }] - \\Omega (\\pi ) \\\\\\Rightarrow & \\forall \\eta \\in \\Delta (\\mathbb {N}) & \\pi _\\Omega ^* \\in \\operatornamewithlimits{argmin}_\\pi {E}^{\\eta }_{p_0,\\pi }[Q^c_{\\pi }] - \\Omega (\\pi )$ This concludes the proof of Corollary REF .", "In order to obtain the desired result, we exploit both the classical policy gradient theorem and the product derivative rule.", "Using elementary calculus, we obtain the following: $\\nabla _\\theta {E}^{\\eta }_{p_0,\\pi _\\theta }[Q_{\\pi _\\theta }^c] & = \\nabla _\\theta \\int _{s_0, s_+,a_+}\\hspace{-28.45274pt} p_0(s_0) P^{\\eta }_{\\pi _\\theta }(s_{+}, a_{+}|s_0) Q^c_{\\pi _\\theta }(s_+,a_+) \\\\& = \\nabla _\\theta \\int _{s_0, s_+,a_+}\\hspace{-28.45274pt} p_0(s_0) P^{\\eta }_{\\pi _\\theta }(s_{+}|s_0) \\pi _\\theta (a_{+}|s_{+}) Q^c_{\\pi _\\theta }(s_+,a_+)\\\\& = \\nabla _\\theta \\int _{s_0, s_+}\\hspace{-17.07182pt} p_0(s_0) P^{\\eta }_{\\pi _\\theta }(s_{+}|s_0) v_{\\pi _\\theta }^c(s_{+})\\\\& = \\int _{s_0, s_+}\\hspace{-17.07182pt} p_0(s_0) \\Big [ v_{\\pi _\\theta }^c(s_{+}) \\nabla _\\theta P^{\\eta }_{\\pi _\\theta }(s_{+}|s_0) + P^{\\eta }_{\\pi _\\theta }(s_{+}|s_0) \\nabla _\\theta v_{\\pi _\\theta }^c(s_{+}) \\Big ] \\\\& = \\int _{s_0, s_+}\\hspace{-17.07182pt} p_0(s_0) v_{\\pi _\\theta }^c(s_{+}) \\nabla _\\theta P^{\\eta }_{\\pi _\\theta }(s_{+}|s_0) + \\underbrace{\\int _{s_0, s_+}\\hspace{-17.07182pt} p_0(s_0) P^{\\eta }_{\\pi _\\theta }(s_{+}|s_0) \\nabla _\\theta v_{\\pi _\\theta }^c(s_{+})}_A$ Recall that the policy gradient theorem can be written simply as: $&\\forall s_0\\in \\mathcal {S} \\quad & \\nabla _\\theta \\mathbb {E}_{\\pi _\\theta }[\\sum _t \\gamma ^t c(s_t,a_t)|s_0] & = \\nabla _\\theta \\int _a \\pi _\\theta (a|s_0) Q_{\\pi _\\theta }^c(s_0,a) \\\\& & & = \\nabla _\\theta v_{\\pi _\\theta }^c(s_0) = \\int _{s} \\rho _\\pi (s|s_0) \\mathbb {E}_{a\\sim \\pi _\\theta }[Q_{\\pi _\\theta }^c(s,a) \\nabla _\\theta \\log \\pi _\\theta (a|s)]$ This concludes our proof as: $A = \\int _{s_0, s_+}\\hspace{-17.07182pt} p_0(s_0) P^{\\eta }_{\\pi _\\theta }(s_{+}|s_0) \\nabla _\\theta v_{\\pi _\\theta }^c(s_{+}) & = \\int _{s_0, s_+, s}\\hspace{-22.76228pt} p_0(s_0) P^{\\eta }_{\\pi _\\theta }(s_{+}|s_0) \\rho _\\pi (s|s_{+}) \\mathbb {E}_{a\\sim \\pi _\\theta }[Q_{\\pi _\\theta }^c(s,a) \\nabla _\\theta \\log \\pi _\\theta (a|s)]$ The proof relies on the definition of the advantage function to obtain the first equality and on proposition REF to obtain the second one.", "Recall that: $\\mathbb {E}_{p_0,\\pi _n}^\\eta \\Big [\\sum _t \\gamma ^t A_{\\pi _o}(s_t,a_t) \\Big ] & = \\mathbb {E}_{p_0,\\pi _n}^\\eta \\Big [\\sum _t \\gamma ^t (c(s_t,a_t) + \\gamma v_{\\pi _o}^c(s_{t+1}) - v_{\\pi _o}^c(s_t)) \\Big ] \\\\& = \\mathbb {E}_{p_0,\\pi _n}^\\eta [ - v_{\\pi _o}^c(s_0) + \\sum _t \\gamma ^t c(s_t,a_t) ] \\\\& = - \\underbrace{\\int _{s_0} p_0(s_0) v_{\\pi _o}^c(s_0)}_{\\mathcal {L}_0^\\eta (\\pi _o,c)} + \\underbrace{\\mathbb {E}_{p_0,\\pi _n}^\\eta [ \\sum _t \\gamma ^t c(s_t,a_t) ]}_{\\mathcal {L}_0^\\eta (\\pi _n,c)} \\\\\\iff \\mathcal {L}_0^\\eta (\\pi _n,c) & = \\mathcal {L}_0^\\eta (\\pi _o,c) + \\mathbb {E}_{p_0,\\pi _n}^\\eta \\Big [\\sum _t \\gamma ^t A_{\\pi _o}(s_t,a_t) \\Big ]$ This concludes the proof of the first equality.", "In addition, by observing that $A_\\pi $ is a mapping from $\\mathcal {S}\\times \\mathcal {A}$ to $\\mathbb {R}$ , we can apply proposition REF to further simplify the expectation term: $\\mathbb {E}_{p_0,\\pi _n}^\\eta \\Big [\\sum _t \\gamma ^t A_{\\pi _o}(s_t,a_t) \\Big ] = \\int _{s_0,s,a,s_{+},a_{+}}\\hspace{-28.45274pt} p_0(s_0)\\rho _\\pi (s,a|s_0)P_\\pi ^\\eta (s_{+},a_{+}|s,a) A_{\\pi _o}(s_{+},a_{+})$ This concludes the proof as we have: $\\mathcal {L}_0^\\eta (\\pi _n,c) = \\mathcal {L}_0^\\eta (\\pi _o,c) + \\int _{s_0,s,a,s_{+},a_{+}}\\hspace{-28.45274pt} p_0(s_0)\\rho _\\pi (s,a|s_0)P_\\pi ^\\eta (s_{+},a_{+}|s,a) A_{\\pi _o}(s_{+},a_{+})$" ], [ "Generalised Inverse Reinforcement Learning", "In this section, we address the claims stated in section REF ." ], [ "Proof of Proposition ", "The proof relies on the properties of saddle point [10].", "Let $\\tilde{c}$ , $\\tilde{\\pi }$ and $\\hat{\\pi }$ be respectively defined as: $\\tilde{c} & \\in \\operatorname{IRL}_{\\psi ,\\Omega }^\\eta (\\pi _E) \\\\\\tilde{\\pi } & \\in \\operatorname{RL}_\\Omega ^\\eta (\\tilde{c})=\\operatorname{RL}_\\Omega ^\\eta \\circ \\operatorname{IRL}_{\\psi ,\\Omega }^\\eta (\\pi _E) = \\operatornamewithlimits{argmin}_\\pi \\mathcal {L}_\\Omega ^\\eta (\\pi ,\\tilde{c}) \\\\\\hat{\\pi } & \\in \\operatornamewithlimits{argmin}_\\pi \\max _c L(\\pi ,c)$ Our goal is to prove that $\\tilde{\\pi }=\\hat{\\pi }$ .", "Equivalently (due to proposition REF ) this boils down to proving that $\\mu _{\\tilde{\\pi }}=\\mu _{\\hat{\\pi }}$ .", "Using Proposition REF and REF , we can re-write: $\\mathcal {L}_\\Omega ^\\eta (\\pi ,c) & := {E}^{\\eta }_{p_0,\\pi }\\big [ \\sum _t \\gamma ^t c(s_t,a_t) \\big ] - \\Omega (\\pi ) \\\\& = \\int _{s_0,s,a,s_{+},a_{+}}\\hspace{-28.45274pt} p_0(s_0)\\rho _\\pi (s,a|s_0)P_\\pi ^\\eta (s_{+},a_{+}|s,a) c(s_{+},a_{+}) - \\Omega (\\pi ) \\\\& = \\bar{\\mathcal {L}}_\\Omega ^\\eta (\\mu _\\pi ,c) = \\int _{s_0,s_{+},a_{+}}\\hspace{-14.22636pt} p_0(s_0)\\mu _\\pi (s_{+},a_{+}|s_0) c(s_{+},a_{+}) - \\Omega (\\mu _\\pi )$ This implies that : $\\mu _{\\tilde{\\pi }} & \\in \\operatornamewithlimits{argmin}_{\\mu \\in \\mathcal {D}} \\bar{\\mathcal {L}}_\\Omega ^\\eta (\\mu ,\\tilde{c}) = \\operatornamewithlimits{argmin}_{\\mu \\in \\mathcal {D}} \\bar{L}(\\mu , \\tilde{c})$ where: $\\bar{L}& : \\mathcal {D}\\times \\mathbb {R}^{\\mathcal {S}\\times \\mathcal {A}} \\rightarrow \\mathbb {R}\\\\\\mathcal {D} & = \\Big \\lbrace \\mu _\\pi : \\mu _\\pi (s,a|s_0) = \\sum _{t,k} \\gamma ^t \\eta (k) \\mathbb {P}_\\pi (s_{t+k}=s,a_{t+k}=a|s_0) \\Big | \\pi :\\mathcal {S}\\rightarrow \\Delta (\\mathcal {A})\\Big \\rbrace \\\\\\bar{L}(\\mu _\\pi , c) & = L(\\pi , c) = -\\Omega (\\pi ) - \\psi (c) + \\int _{s_0} p_0(s_0) d_c(\\pi \\Vert \\pi _E)(s_0) \\\\& = -\\Omega (\\mu _\\pi ) - \\psi (c) + \\int _{s_0,s_{+},a_{+}}\\hspace{-14.22636pt} p_0(s_0) c(s_{+},a_{+}) \\Big [ \\mu _\\pi (s_{+},a_{+}|s_0) - \\mu _{\\pi _E}(s_{+},a_{+}|s_0) \\Big ]$ In addition, for the same reasons, we have that: $\\begin{split}\\tilde{c} & \\in \\operatornamewithlimits{argmax}_c \\min _{\\pi } \\mathcal {L}_\\Omega ^\\eta (\\pi ,c) - \\mathcal {L}_\\Omega ^\\eta (\\pi _E^*,c) - \\psi (c) = \\operatornamewithlimits{argmax}_c \\min _{\\mu \\in \\mathcal {D}} \\bar{L}(\\mu , c) \\\\\\mu _{\\hat{\\pi }} & \\in \\operatornamewithlimits{argmin}_{\\mu \\in \\mathcal {D}} \\max _c \\bar{L}(\\mu , c)\\end{split}$ Notice that $\\mathbb {R}^{\\mathcal {S}\\times \\mathcal {A}}$ is convex, $\\bar{L}(\\mu , c)$ is convex w.r.t $\\mu $ and concave w.r.t $c$ (due to convexity of $\\psi $ and $-\\Omega $ ).", "Therefore, the minmax duality property holds as soon as $\\mathcal {D}$ is compact and convex." ], [ "Convexity and compacity of $\\mathcal {D}$ :", "We prove this under the assumption that $\\eta $ is a geometric distribution (i.e.", "$\\eta =\\operatorname{Geom}(\\gamma _\\eta )$ ).", "To establish the convexity, we prove that $\\mathcal {D}=\\lbrace f:\\mathcal {S}\\rightarrow \\Delta (\\mathcal {S}\\times \\mathcal {A})\\rbrace $ where $f$ is a solution of the following equations: $&\\forall s,s_0\\in \\mathcal {S}, a\\in \\mathcal {A} \\quad &\\begin{split}& \\int _{a} f(s,a|s_0) = \\int _{a} g(s,a|s_0) + \\int _{s^{\\prime },a^{\\prime }} \\gamma f(s^{\\prime },a^{\\prime }|s_0) \\mathcal {P}(s|s^{\\prime },a^{\\prime }) \\\\& \\int _{a} f(s,a|s_0) = \\int _{a} h(s,a|s_0) + \\int _{s^{\\prime },a^{\\prime }} \\gamma _\\eta f(s^{\\prime },a^{\\prime }|s_0) \\mathcal {P}(s|s^{\\prime },a^{\\prime }) \\\\& \\int _{a} g(s,a|s_0) = 1 + \\int _{s^{\\prime },a^{\\prime }} \\gamma _\\eta \\mathcal {P}(s|s^{\\prime },a^{\\prime }) g(s^{\\prime },a^{\\prime }|s_0)\\\\& \\int _{a} h(s,a|s_0) = 1 + \\int _{s^{\\prime },a^{\\prime }} \\gamma \\mathcal {P}(s|s^{\\prime },a^{\\prime }) h(s^{\\prime },a^{\\prime }|s_0)\\\\& f(s,a|s_0)\\ge 0, \\quad g(s,a|s_0)\\ge 0, \\quad h(s,a|s_0)\\ge 0\\end{split}.", "&$ To this end, notice that for any policy $\\pi $ , we can verify that $(f=\\mu _\\pi , g=P_\\pi ^\\eta , h=\\rho _\\pi )$ is a solution to Equation REF .", "We focus now on the converse statement.", "Let $(f,g,h)$ a solution to Equation REF .", "Given the third equality: $\\int _{a} g(s,a|s_0) = 1 + \\int _{s^{\\prime },a^{\\prime }} \\gamma _\\eta \\mathcal {P}(s|s^{\\prime },a^{\\prime }) g(s^{\\prime },a^{\\prime }|s_0),$ we exploit a classical result from the MDP literature [[23], Section 6.9.2], to derive the existence of a policy $\\pi _g$ such that: $\\begin{split}& g(s,a|s_0) = \\pi _g(a|s) \\Big [ 1 + \\int _{s^{\\prime },a^{\\prime }} \\gamma _\\eta \\mathcal {P}(s|s^{\\prime },a^{\\prime }) g(s^{\\prime },a^{\\prime }|s_0) \\Big ] \\\\\\textbf {where } \\forall s_0 \\in \\mathcal {S} \\quad & \\pi _g(a|s) = \\frac{g(s,a|s_0)}{\\int _{a^{\\prime }} g(s,a^{\\prime }|s_0)}\\end{split}$ From Equation REF , we conclude that $g$ is the unique fixed point of a $\\gamma _\\eta $ -contraction.", "We can verify that $g=P_{\\pi _g}^\\eta $ is the unique solution of Equation REF .", "Using a similar reasoning with respect to the fourth equality in Equation REF , we conclude that $h=\\rho _{\\pi _h}$ where for any state $s_0\\in \\mathcal {S}$ , we have $\\pi _h(a|s) = \\frac{h(s,a|s_0)}{\\int _{a^{\\prime }} h(s,a^{\\prime }|s_0)}$ .", "From this, we conclude that for any solution $(f,g,h)$ to Equation REF , there exist two policies $\\pi _h$ and $\\pi _g$ such that: $&\\forall s,s_0\\in \\mathcal {S}, a\\in \\mathcal {A} \\quad &\\begin{split}& \\int _{a} f(s,a|s_0) = P_{\\pi _g}^\\eta (s|s_0) + \\int _{s^{\\prime },a^{\\prime }} \\gamma f(s^{\\prime },a^{\\prime }|s_0) \\mathcal {P}(s|s^{\\prime },a^{\\prime }) \\\\& \\int _{a} f(s,a|s_0) = \\rho _{\\pi _h}(s|s_0) + \\int _{s^{\\prime },a^{\\prime }} \\gamma _\\eta f(s^{\\prime },a^{\\prime }|s_0) \\mathcal {P}(s|s^{\\prime },a^{\\prime }) \\\\& f(s,a|s_0)\\ge 0\\end{split}.", "&$ which is equivalent to [[23], Section 6.9.2]: $&\\forall s,s_0\\in \\mathcal {S}, a\\in \\mathcal {A} \\quad &\\begin{split}& f(s,a|s_0) = \\pi _f(a|s) \\Big [ P_{\\pi _g}^\\eta (s|s_0) + \\int _{s^{\\prime },a^{\\prime }} \\gamma f(s^{\\prime },a^{\\prime }|s_0) \\mathcal {P}(s|s^{\\prime },a^{\\prime }) \\Big ] \\\\& f(s,a|s_0) = \\pi _f(a|s) \\Big [ \\rho _{\\pi _h}(s|s_0) + \\int _{s^{\\prime },a^{\\prime }} \\gamma _\\eta f(s^{\\prime },a^{\\prime }|s_0) \\mathcal {P}(s|s^{\\prime },a^{\\prime }) \\Big ] \\\\& \\pi _f(a|s) = \\frac{f(s,a|s_0)}{\\int _{a^{\\prime }} f(s,a^{\\prime }|s_0)}, \\quad f(s,a|s_0)\\ge 0\\end{split}.", "&$ Notice that the first equality in Equation REF , implies that $f$ is the unique fixed point of a $\\gamma $ -contraction.", "We also notice that: $f_{\\pi _g, \\pi _f}^0(s_+, a_+ | s_0) & := \\sum _{k,t} \\gamma ^t \\gamma _\\eta ^k \\int _{s,a} \\mathbb {P}_{\\pi _f}^t(s_+,a_+|s) \\mathbb {P}_{\\pi _g}^k(s,a|s_0) \\\\& = \\sum _{k,t>0} \\gamma ^t \\gamma _\\eta ^k \\int _{s,a} \\mathbb {P}_{\\pi _f}^t(s_+,a_+|s) \\mathbb {P}_{\\pi _g}^k(s,a|s_0) + \\sum _k \\gamma _\\eta ^k \\int _{s,a} \\mathbb {P}_{\\pi _f}^0(s_+,a_+|s) \\mathbb {P}_{\\pi _g}^k(s,a|s_0) \\\\& = \\gamma \\pi _f(a_+|s_+)\\int _{s^{\\prime },a^{\\prime }} f_{\\pi _g, \\pi _f}^0(s^{\\prime },a^{\\prime }|s_0) \\mathcal {P}(s|s^{\\prime },a^{\\prime }) + \\sum _k \\gamma _\\eta ^k \\mathbb {P}_{\\pi _g}^k(s_+|s_0) \\pi _f(a_+|s_+) \\\\& = \\pi _f(a_+|s_+) \\Big [ \\int _{s^{\\prime },a^{\\prime }} \\gamma f_{\\pi _g, \\pi _f}^0(s^{\\prime },a^{\\prime }|s_0) \\mathcal {P}(s_+|s^{\\prime },a^{\\prime }) + P_{\\pi _g}^\\eta (s_+|s_0) \\Big ]$ where $\\mathbb {P}_\\pi ^n(s,a|s^0) = \\mathbb {P}_\\pi (s_n=s, a_n=a|s_0=s^0)$ .", "Thus, we conclude $f_{\\pi _g, \\pi _f}^0$ is the unique solution to the first equality.", "Similarly, we notice that the second equality is a $\\gamma _\\eta $ -contraction, whose unique fixed point is: $f_{\\pi _h, \\pi _f}^1(s_+, a_+ | s_0) & := \\sum _{k,t} \\gamma ^t \\gamma _\\eta ^k \\int _{s,a} \\mathbb {P}_{\\pi _f}^k(s_+,a_+|s) \\mathbb {P}_{\\pi _h}^t(s,a|s_0) \\\\& = \\sum _{k>0,t} \\gamma ^t \\gamma _\\eta ^k \\int _{s,a} \\mathbb {P}_{\\pi _f}^k(s_+,a_+|s) \\mathbb {P}_{\\pi _h}^t(s,a|s_0) + \\sum _t \\gamma ^t \\int _{s,a} \\mathbb {P}_{\\pi _f}^0(s_+,a_+|s) \\mathbb {P}_{\\pi _h}^t(s,a|s_0) \\\\& = \\pi _f(a_+|s_+) \\Big [ \\int _{s^{\\prime },a^{\\prime }} \\gamma _\\eta f_{\\pi _h, \\pi _f}^1(s^{\\prime },a^{\\prime }|s_0) \\mathcal {P}(s_+|s^{\\prime },a^{\\prime }) + \\rho _{\\pi _h}(s_+|s_0) \\Big ]$ We derive from the previously discussed statement, that if $(f,g,h)$ is a solution to Equation REF , then there exist three policies $(\\pi _f,\\pi _g, \\pi _h)$ such that: $\\begin{split}f & = f_{\\pi _g, \\pi _f}^0 = f_{\\pi _h, \\pi _f}^1 \\\\g & = P_{\\pi _g}^\\eta \\\\h & = \\rho _{\\pi _h}\\end{split}$ However, not any random choice of policies $(\\pi _f,\\pi _g, \\pi _h)$ can satisfy Equation REF .", "By varying $\\gamma $ and $\\gamma _\\eta $ , we notice that in order for the first equality to hold, the following equality must be satisfied for any integers $(k,t)$ , and for any states $(s_+,a_+,s_0)$ : $\\int _{s,a}\\mathbb {P}_{\\pi _f}^t(s_+,a_+|s) \\mathbb {P}_{\\pi _g}^k(s,a|s_0) = \\int _{s,a}\\mathbb {P}_{\\pi _f}^k(s_+,a_+|s) \\mathbb {P}_{\\pi _h}^t(s,a|s_0)$ by fixing $k$ at zero and varying $t$ and by fixing $t$ at zero and varying $k$ , we obtain the following constraints: $\\begin{split}P_{\\pi _f}^\\eta & = P_{\\pi _g}^\\eta \\\\\\rho _{\\pi _f} & = \\rho _{\\pi _h}\\end{split}$ Using Proposition REF and Equation REF , it follows that Equation REF admits a solution if and only if $\\pi _f=\\pi _g=\\pi _h$ .", "This means that if $(f,g,h)$ is a solution to Equation REF , then there exists a policy $\\pi $ such that: $\\begin{split}f & = f_{\\pi , \\pi }^0 = f_{\\pi , \\pi }^1 = \\mu _\\pi \\\\g & = P_{\\pi }^\\eta \\\\h & = \\rho _{\\pi }\\end{split}$ This concludes the converse statement, proving that $\\mathcal {D}$ is a set of occupancy measures satisfying the set of affine constraints from Equation REF .", "Consequently, $\\mathcal {D}$ is a convex set.", "In addition, the limit of any sequence of elements from $\\mathcal {D}$ will also satisfy Equation REF .", "From this we establish that $\\mathcal {D}$ is closed which implies that it is also compact.", "From this we derive that minmax duality holds and that: $\\min _{\\mu \\in \\mathcal {D}} \\max _c \\bar{L}(\\mu , c) = \\max _c \\min _{\\mu \\in \\mathcal {D}} \\bar{L}(\\mu , c)$ From Equation REF , it follows that $(\\mu _{\\hat{\\pi }}, \\tilde{c})$ is a saddle point for the function $\\bar{L}$ .", "This implies from Equation REF that: $\\mu _{\\hat{\\pi }}, \\mu _{\\tilde{\\pi }} \\in \\operatornamewithlimits{argmin}_{\\mu \\in \\mathcal {D}} \\bar{L}(\\mu , \\tilde{c})$ In addition, due to the strict convexity of $\\bar{L}$ w.r.t $\\mu $ (due to assumed strict convexity of $\\Omega $ ) we have that: $\\mu _{\\hat{\\pi }} = \\mu _{\\tilde{\\pi }}$ which concludes our proof." ], [ "Proof of Corollary ", "The proof entails directly from the duality of $\\bar{L}$ and that $(\\tilde{c}, \\mu _{\\tilde{\\pi }})$ is a saddle point of $\\bar{L}$ .", "The proof relies on re-writing the $\\eta $ -weighted entropy reguliser using Proposition REF , and then verifying its convexity with respect to $\\mu _\\pi $ using the log-sum inequality.", "In fact, notice that: $H^{\\eta }_{p_0}(\\pi ) & = {E}^{\\eta }_{p_0,\\pi }\\Big [\\sum _t -\\gamma ^t\\log \\big [\\pi (a_t|s_t)\\big ]\\Big ] \\\\& = \\int _{s_0,s_{+},a_{+}}\\hspace{-14.22636pt} p_0(s_0)\\mu _\\pi (s_{+},a_{+}|s_0) \\log \\big [\\pi (a_{+}|s_{+})\\big ] = \\bar{H}^{\\eta }_{p_0}(\\mu _\\pi )$ Consider two $\\eta $ -weighted occupancy measures $\\mu _1,\\mu _2$ , and let $\\pi _1,\\pi _2$ their respective policies.", "Let $\\lambda \\in ]0,1[$ : $\\bar{H}(\\lambda \\mu _1 + (1-\\lambda )\\mu _2) & = \\int _{s_0,s_{+},a_{+}}\\hspace{-19.91684pt} -p_0(s_0)\\big [\\lambda \\mu _1 + (1-\\lambda )\\mu _2\\big ](s_{+},a_{+}|s_0) \\log \\Big [\\frac{\\big [\\lambda \\mu _1 + (1-\\lambda )\\mu _2\\big ](s_{+},a_{+}|s_0)}{\\int _{a}\\big [\\lambda \\mu _1 + (1-\\lambda )\\mu _2\\big ](s_{+},a|s_0)}\\Big ]$ Du to the log-sum inequality we have: $& \\big [\\lambda \\mu _1 + (1-\\lambda )\\mu _2\\big ](s_{+},a_{+}|s_0) \\log \\Big [\\frac{\\big [\\lambda \\mu _1 + (1-\\lambda )\\mu _2\\big ](s_{+},a_{+}|s_0)}{\\int _{a}\\big [\\lambda \\mu _1 + (1-\\lambda )\\mu _2\\big ](s_{+},a|s_0)}\\Big ] \\\\= & \\big [\\lambda \\mu _1 + (1-\\lambda )\\mu _2\\big ](s_{+},a_{+}|s_0) \\log \\Big [\\frac{\\lambda \\mu _1(s_{+},a_{+}|s_0) + (1-\\lambda )\\mu _2(s_{+},a_{+}|s_0) }{\\lambda \\int _{a}\\mu _1(s_{+},a|s_0) + (1-\\lambda )\\int _{a}\\mu _2(s_{+},a|s_0)}\\Big ] \\\\\\le & \\lambda \\mu _1 \\log \\Big [\\frac{\\lambda \\mu _1(s_{+},a_{+}|s_0)}{\\lambda \\int _{a}\\mu _1(s_{+},a|s_0)}\\Big ] + (1-\\lambda ) \\mu _2 \\log \\Big [\\frac{(1-\\lambda )\\mu _2(s_{+},a_{+}|s_0)}{(1-\\lambda )\\int _{a}\\mu _2(s_{+},a|s_0)}\\Big ] \\\\= & \\lambda \\mu _1 \\log \\Big [\\frac{\\mu _1(s_{+},a_{+}|s_0)}{\\int _{a}\\mu _1(s_{+},a|s_0)}\\Big ] + (1-\\lambda ) \\mu _2 \\log \\Big [\\frac{\\mu _2(s_{+},a_{+}|s_0)}{\\int _{a}\\mu _2(s_{+},a|s_0)}\\Big ]$ This implies that: $\\bar{H}(\\lambda \\mu _1 + (1-\\lambda )\\mu _2) \\ge \\lambda \\bar{H}(\\mu _1) + (1-\\lambda ) \\bar{H}(\\mu _2)$ with equality if and only if $\\pi _1(a|s) := \\frac{\\mu _1(s,a|s_0)}{\\int _{a^{\\prime }}\\mu _1(s,a^{\\prime }|s_0)} = \\frac{\\mu _2(s,a|s_0)}{\\int _{a^{\\prime }}\\mu _2(s,a^{\\prime }|s_0)} := \\pi _2(a|s)$ .", "This concludes the proof of the $\\eta $ -weighted strict concavity w.r.t the set of measures $\\mu $ ." ], [ "Data augmentation", "In this section, we provide the proof of Proposition REF .", "We start by noticing that $V(D,G)$ is the loss function used by conditional generative adversarial neural networks [19], which minimum w.r.t the discriminator is achieved for the optimal Bayes classifier [7]: $D^*(s,a|s_0) = \\frac{P_\\pi ^\\eta (s,a|s_0)}{P_\\pi ^\\eta (s,a|s_0) + G(s,a|s_0)}$ where $G(s,a|s_0)$ is the probability of generating $(s,a)$ using the generator $G$ .", "From this, we can re-write the generator's loss against an infinite capacity (optimal) discriminator as: $V(D^*,G) & = D_{KL}(P_\\pi ^\\eta (s,a|s_0) \\Vert \\frac{P_\\pi ^\\eta (s,a|s_0)}{P_\\pi ^\\eta (s,a|s_0) + G(s,a|s_0)}) + D_{KL}(G(s,a|s_0) \\Vert \\frac{P_\\pi ^\\eta (s,a|s_0)}{P_\\pi ^\\eta (s,a|s_0) + G(s,a|s_0)}) - \\log (4) \\\\& = 2D_{JSC}(G(s,a|s_0) \\Vert P_\\pi ^\\eta (s,a|s_0)) - \\log (4)$ where $D_{KL}$ is the KL divergence, and $D_{JSC}$ is the Jenson-Shannon divergence.", "A global minimum is achieved when $G^*$ : $G^*(s,a|s_0)=P_\\pi ^\\eta (s,a|s_0).$ This concludes the proof as it implies that $(\\tilde{D}=\\frac{1}{2},\\tilde{G}=P_{\\pi }^{\\eta })$ is a Nash-equilibrium" ], [ "Particular settings of interest", "In this section we address the claims stated in section ." ], [ "Proof of Proposition ", "Notice that in this setting: $\\max _{c} L(\\pi , c) = & \\max _{c\\in \\mathcal {C}_\\textit {linear}} \\int _{s_0} p_0(s_0) d_c(\\pi \\Vert \\pi _E)(s_0) \\\\= & \\max _{c\\in \\mathcal {C}_\\textit {linear}} \\int _{s_0} p_0(s_0) \\big [\\mu _\\pi (s_{+},a_{+}|s_0)-\\mu _{\\pi _E}(s_{+},a_{+}|s_0)\\big ] c(s_{+},a_{+}) \\\\= & \\max _{w \\textbf { with } \\Vert w\\Vert _2\\le 1} \\int _{s_0} p_0(s_0) \\big [\\mu _\\pi (s_{+},a_{+}|s_0)-\\mu _{\\pi _E}(s_{+},a_{+}|s_0)\\big ] \\sum _i w_i f_i(s_{+},a_{+}) \\\\= & \\max _{w \\textbf { with } \\Vert w\\Vert _2\\le 1} \\sum _i w_i \\int _{s_0} p_0(s_0) \\big [\\mu _\\pi (s_{+},a_{+}|s_0)-\\mu _{\\pi _E}(s_{+},a_{+}|s_0)\\big ] f_i(s_{+},a_{+}) \\\\= & \\Big \\Vert \\int _{s_0} p_0(s_0) \\big [\\mu _\\pi (s_{+},a_{+}|s_0)-\\mu _{\\pi _E}(s_{+},a_{+}|s_0)\\big ] f_i(s_{+},a_{+}) \\Big \\Vert _2 \\\\= & \\Big \\Vert \\mathbb {E}_{\\mu _\\pi }[f]-\\mathbb {E}_{\\mu _{\\pi _E}}[f] \\Big \\Vert _2$ This concludes the proof.", "In this case, we notice the following: $\\max _{c} L(\\pi , c) = & \\max _{c\\in \\mathcal {C}_\\textit {convex}} \\int _{s_0} p_0(s_0) d_c(\\pi \\Vert \\pi _E)(s_0) \\\\= & \\max _{c\\in \\mathcal {C}_\\textit {convex}} \\int _{s_0} p_0(s_0) \\big [\\mu _\\pi (s_{+},a_{+}|s_0)-\\mu _{\\pi _E}(s_{+},a_{+}|s_0)\\big ] c(s_{+},a_{+}) \\\\= & \\max _{w_i>0 \\textbf { with } \\sum w_i = 1} \\int _{s_0} p_0(s_0) \\big [\\mu _\\pi (s_{+},a_{+}|s_0)-\\mu _{\\pi _E}(s_{+},a_{+}|s_0)\\big ] \\sum _i w_i f_i(s_{+},a_{+}) \\\\= & \\max _{w_i>0 \\textbf { with } \\sum w_i = 1} \\sum _i w_i \\int _{s_0} p_0(s_0) \\big [\\mu _\\pi (s_{+},a_{+}|s_0)-\\mu _{\\pi _E}(s_{+},a_{+}|s_0)\\big ] f_i(s_{+},a_{+}) \\\\= & \\max _i \\int _{s_0} p_0(s_0) \\big [\\mu _\\pi (s_{+},a_{+}|s_0)-\\mu _{\\pi _E}(s_{+},a_{+}|s_0)\\big ] f_i(s_{+},a_{+})\\\\= & \\max _i \\mathbb {E}_{\\mu _\\pi }[f_i]-\\mathbb {E}_{\\mu _{\\pi _E}}[f_i]$ This concludes the proof.", "We start by re-writing the cost function as an expectation with respect to the occupancy measure $\\mu _\\pi $ using Proposition REF : $\\psi _{GAN}(c) =\\left\\lbrace \\begin{array}{ll}\\int _{s,a,s_0} p_0(s_0) \\mu _{\\pi _E}(s,a|s_0) g(c(s,a)) & \\textit { if } c<0 \\\\+\\infty & \\textit { otherwise }\\end{array}\\right.$ With this, we can rewrite the objective function $L(\\pi ,c)$ in this setting as follows: $L(\\pi , c) & = -\\Omega (\\pi ) - \\psi (c) + \\int _{s_0} p_0(s_0) d_c(\\pi \\Vert \\pi _E)(s_0)\\\\& = -\\Omega (\\pi ) + \\int _{s_0} p_0(s_0) \\int _{s,a} \\Big [ \\mu _{\\pi _E}(s,a|s_0) g(c(s,a)) + \\big ( \\mu _{\\pi }(s,a|s_0) - \\mu _{\\pi _E}(s,a|s_0) \\big ) c(s,a) \\Big ]$ Notice that this is the same objective as the one used in GAIL [[11] Appendix A.2], where we compute expectations with respect to $\\mu _\\pi $ while the do it with respect to $\\rho _\\pi $ .", "Using the same change of variable, we can obtain the folowing: $\\max _c L(\\pi , c) & = \\max _{D\\in (0,1)^{\\mathcal {S}\\times \\mathcal {A}}} -\\Omega (\\pi ) + \\int _{s_0} p_0(s_0) \\int _{s,a} \\mu _{\\pi }(s,a|s_0) \\log (D(s,a) - \\mu _{\\pi _E}(s,a|s_0) \\log (1-D(s,a))$ This concludes the proof of Proposition REF , as we can state using Proposition REF : $\\operatorname{RL}_\\Omega ^\\eta \\circ \\operatorname{IRL}_{\\psi }^\\eta (\\pi _E) & = \\operatornamewithlimits{argmin}_\\pi \\max _c L(\\pi ,c) \\\\& = \\operatornamewithlimits{argmin}_\\pi -\\Omega (\\pi ) + \\max _{D \\in (0,1)^{\\mathcal {S}\\times \\mathcal {A}}} \\mathbb {E}_{p_0,\\pi }^\\eta [\\log D]-\\mathbb {E}_{p_0,\\pi _E}^\\eta [\\log (1-D))]$" ] ]
2105.11812
[ [ "Evaluation of multivariate integrals based on a duality identity for the\n Stieltjes transform" ], [ "Abstract A detailed study of a double integral representation of the Catalan's constant allows us to identify a duality identity for the Stieltjes transform on which it is based.", "This duality identity is then extended to an arbitrary dimensional integral and several special cases are deduced.", "On the way, we also highlight a relationship with some multivariate generalizations of the Riemann zeta function." ], [ "Introduction", "Catalan's constant was named after the French and Belgian mathematician Eugene Charles Catalan [2].", "Catalan's constant (henceforth denoted by $ \\mathcal {G} $ ) is defined by $\\mathcal {G} = \\beta (2) = \\sum _{n = 0}^{\\infty }\\dfrac{(-1)^n}{(2n+1)^2}$ where $ \\beta $ represents the Dirichlet Beta function.", "Till date it remains unknown whether $ \\mathcal {G} $ is irrational, let alone transcendental.", "However $ \\mathcal {G} $ has been called arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven [2].", "The numerical value of $ \\mathcal {G} $ is approximately (see sequence A006752 in the OEIS): $\\mathcal {G} = 0.91596559417721901505460351493238411077414937$ However, the number of known digits of Catalan's constant $ \\mathcal {G} $ has increased dramatically during the last decades.", "This is mostly due to increase of performance of computers.", "There is a rich and seemingly endless source of definite integrals that can be equated to or expressed in terms of Catalan's constant [2].", "One of the main goals of the present paper is to establish a magnificent triple integral representation of the Catalan's constant: Proposition 1.1 Let $ \\mathcal {G} $ represent the Catalan's constant, then we have $\\mathcal {G} = \\dfrac{1}{\\pi }\\left\\lbrace \\dfrac{7}{4}\\,\\zeta (3) + \\int _{0}^{\\pi /2}\\int _{0}^{\\pi /2}\\int _{0}^{1}\\dfrac{\\mathrm {d}\\omega \\,\\mathrm {d}x\\,\\mathrm {d}z}{2\\,(1 + (1 - \\omega )\\cos x + \\omega \\cos z)}\\right\\rbrace $ We present a quite elementary and short proof of proposition $ 1.1 $ by evaluating and finding a nice closed form for the triple integral.", "Let's first walk through the Preliminaries." ], [ "Preliminaries", "Proposition 2.1 Let $ z \\in \\mathbb {R}$ such that $ \\left|z\\right|< 1 $ , then we have $\\int _{0}^{1}\\dfrac{1}{(1+zx)\\sqrt{(1 + x)(1-x)}}\\, \\mathrm {d}x = \\dfrac{\\arccos (z)}{\\sqrt{(1 + z)(1-z)}}$ We give a real analytic proof.", "Substituting $ x \\mapsto \\sin (\\vartheta ) $ , we find that $\\int _{0}^{1}\\dfrac{1}{(1+zx)\\sqrt{(1 + x)(1-x)}}\\, \\mathrm {d}x = \\int _{0}^{\\pi /2}\\dfrac{1}{1 + z\\sin (\\vartheta )}\\,\\mathrm {d}\\vartheta $ Now we simply write $\\displaystyle 1 = \\sin ^2\\left(\\dfrac{\\vartheta }{2}\\right) + \\cos ^2\\left(\\dfrac{\\vartheta }{2}\\right)$ and $\\displaystyle z\\sin (\\vartheta ) = 2z\\sin \\left(\\dfrac{\\vartheta }{2}\\right)\\cos \\left(\\dfrac{\\vartheta }{2}\\right) $ , to get: $\\int _{0}^{\\pi /2}\\dfrac{1}{1 + z\\sin (\\vartheta )}\\,\\mathrm {d}\\vartheta &= \\int _{0}^{\\pi /2}\\dfrac{1}{\\sin ^2\\left(\\dfrac{\\vartheta }{2}\\right) + \\cos ^2\\left(\\dfrac{\\vartheta }{2}\\right) + 2z\\sin \\left(\\dfrac{\\vartheta }{2}\\right)\\cos \\left(\\dfrac{\\vartheta }{2}\\right)}\\,\\mathrm {d}\\vartheta \\\\& = \\int _{0}^{\\pi /2}\\dfrac{\\tan \\left(\\dfrac{\\vartheta }{2}\\right)^{^{\\prime }}}{1 + \\tan ^2\\left(\\dfrac{\\vartheta }{2}\\right) + 2z\\tan \\left(\\dfrac{\\vartheta }{2}\\right)}\\,\\mathrm {d}\\vartheta = 2\\int _{0}^{1}\\dfrac{1}{\\Phi ^2 + 2\\Phi z + 1}\\,\\mathrm {d}\\Phi $ where the last equality is followed simply by substituting $ \\displaystyle \\tan \\left(\\dfrac{\\vartheta }{2}\\right) \\mapsto \\Phi $ .", "Now substituting $ \\Phi + z \\mapsto \\vartheta $ and writing $ \\Phi ^2 + 2\\Phi z + 1 = (\\Phi + z)^2 + \\left(\\sqrt{1 - z^2}\\right)^2$ inside the integral we get $2\\int _{0}^{1}\\dfrac{1}{\\Phi ^2 + 2\\Phi z + 1}\\,\\mathrm {d}\\Phi = 2\\int _{0}^{1}\\dfrac{1}{(\\Phi + z)^2 + \\left(\\sqrt{1 - z^2}\\right)^2}\\,\\mathrm {d}\\Phi = 2\\int _{z}^{z+1}\\dfrac{1}{\\vartheta ^2 + \\left(\\sqrt{1 - z^2}\\right)^2}\\,\\mathrm {d}\\vartheta $ $=\\dfrac{2}{\\sqrt{1-z^2}}\\left\\lbrace \\arctan \\left(\\dfrac{z+1}{\\sqrt{1-z^2}}\\right)- \\arctan \\left(\\dfrac{z}{\\sqrt{1-z^2}}\\right)\\right\\rbrace = \\dfrac{2}{\\sqrt{(1+z)(1-z)}}\\arctan \\left(\\sqrt{\\dfrac{1-z}{1+z}}\\right)$ Now using the well-known trigonometric identity $ \\displaystyle \\arccos (z) = 2\\arctan \\left(\\sqrt{\\dfrac{1-z}{1+z}}\\right)$ we get $\\int _{0}^{1}\\dfrac{1}{(1+zx)\\sqrt{(1 + x)(1-x)}}\\, \\mathrm {d}x = \\dfrac{2}{\\sqrt{(1+z)(1-z)}}\\arctan \\left(\\sqrt{\\dfrac{1-z}{1+z}}\\right) = \\dfrac{\\arccos (z)}{\\sqrt{(1 + z)(1-z)}}$ as desired.", "This completes the proof of proposition $ \\ref {21} $ ." ], [ "Evaluating the Triple Integral", "Proposition 3.1 We have $\\dfrac{1}{2}\\int _{0}^{1}\\dfrac{\\mathrm {d}\\omega }{1 + (1 - \\omega )\\cos x + \\omega \\cos z} = \\dfrac{\\log \\left(\\cos \\left(\\dfrac{x}{2}\\right)\\right) - \\log \\left(\\cos \\left(\\dfrac{z}{2}\\right)\\right)}{\\cos (x) - \\cos (z)}$ The proof is pretty striaghtforward since $ \\cos (x) $ and $ \\cos (z) $ are constants.", "Proposition 3.2 We have $\\int _{0}^{\\pi /2}\\int _{0}^{\\pi /2}\\dfrac{\\log \\left(\\cos \\left(\\dfrac{x}{2}\\right)\\right) - \\log \\left(\\cos \\left(\\dfrac{z}{2}\\right)\\right)}{\\cos (x) - \\cos (z)}\\,\\mathrm {d}x\\,\\mathrm {d}z = \\dfrac{1}{2}\\int _{0}^{1}\\dfrac{\\arccos ^2(\\vartheta )}{(1 + \\vartheta )(1 - \\vartheta )}\\,\\mathrm {d}\\vartheta $ Using the well-known trigonometric identity $ \\displaystyle \\cos \\left(\\dfrac{B}{2}\\right) = \\sqrt{\\dfrac{1 + \\cos (B)}{2}}$ we get $& \\int _{0}^{\\pi /2}\\left\\lbrace \\int _{0}^{\\pi /2}\\dfrac{\\log \\left(\\cos \\left(\\dfrac{x}{2}\\right)\\right) - \\log \\left(\\cos \\left(\\dfrac{z}{2}\\right)\\right)}{\\cos (x) - \\cos (z)}\\,\\mathrm {d}x\\right\\rbrace \\,\\mathrm {d}z\\\\\\\\= & \\int _{0}^{\\pi /2}\\left\\lbrace \\int _{0}^{\\pi /2}\\dfrac{\\log \\left(\\sqrt{\\dfrac{1 + \\cos (x)}{2}}\\right) - \\log \\left(\\sqrt{\\dfrac{1 + \\cos (z)}{2}}\\right)}{\\cos (x) - \\cos (z)}\\,\\mathrm {d}x\\right\\rbrace \\,\\mathrm {d}z$ Changing the variable $ \\cos (x) = \\vartheta _1 $ and $ \\cos (z) = \\vartheta _2 $ , we find that $&\\int _{0}^{\\pi /2}\\left\\lbrace \\int _{0}^{\\pi /2}\\dfrac{\\log \\left(\\sqrt{\\dfrac{1 + \\cos (x)}{2}}\\right) - \\log \\left(\\sqrt{\\dfrac{1 + \\cos (z)}{2}}\\right)}{\\cos (x) - \\cos (z)}\\,\\mathrm {d}x\\right\\rbrace \\,\\mathrm {d}z\\\\\\\\= & \\dfrac{1}{2}\\int _{0}^{1}\\left\\lbrace \\int _{0}^{1}\\dfrac{\\log \\left(1 + \\vartheta _1\\right) - \\log \\left(1 + \\vartheta _2\\right)}{(\\vartheta _1 - \\vartheta _2)\\sqrt{(1 + \\vartheta _1)(1 - \\vartheta _1)}\\sqrt{(1 + \\vartheta _2)(1 - \\vartheta _2)}}\\,\\mathrm {d}\\vartheta _1\\right\\rbrace \\,\\mathrm {d}\\vartheta _2\\\\\\\\= & \\dfrac{1}{2}\\int _{0}^{1}\\left\\lbrace \\int _{0}^{1}\\left(\\int _{0}^{1}\\dfrac{1}{(1 + \\vartheta _1\\vartheta )(1+ \\vartheta _2\\vartheta )\\sqrt{(1 + \\vartheta _1)(1 - \\vartheta _1)}\\sqrt{(1 + \\vartheta _2)(1 - \\vartheta _2)}}\\,\\mathrm {d}\\vartheta _1\\right)\\,\\mathrm {d}\\vartheta _2\\right\\rbrace \\,\\mathrm {d}\\vartheta $ where the last equality is followed simply because $\\log \\left(1 + \\vartheta _1\\right) - \\log \\left(1 + \\vartheta _2\\right) = \\int _{0}^{1}\\left(\\dfrac{\\vartheta _1}{1 + \\vartheta _1\\vartheta } - \\dfrac{\\vartheta _2}{1 + \\vartheta _2\\vartheta }\\right)\\,\\mathrm {d}\\vartheta $ Now changing the order of integration, we get $& \\dfrac{1}{2}\\int _{0}^{1}\\left\\lbrace \\int _{0}^{1}\\left(\\int _{0}^{1}\\dfrac{1}{(1 + \\vartheta _1\\vartheta )(1+ \\vartheta _2\\vartheta )\\sqrt{(1 + \\vartheta _1)(1 - \\vartheta _1)}\\sqrt{(1 + \\vartheta _2)(1 - \\vartheta _2)}}\\,\\mathrm {d}\\vartheta _1\\right)\\,\\mathrm {d}\\vartheta _2\\right\\rbrace \\,\\mathrm {d}\\vartheta \\\\\\\\= & \\dfrac{1}{2}\\int _{0}^{1}\\left\\lbrace \\int _{0}^{1}\\dfrac{1}{(1+ \\vartheta _2\\vartheta )\\sqrt{(1 + \\vartheta _2)(1 - \\vartheta _2)}}\\left(\\int _{0}^{1}\\dfrac{1}{(1 + \\vartheta _1\\vartheta )\\sqrt{(1 + \\vartheta _1)(1 - \\vartheta _1)}}\\,\\mathrm {d}\\vartheta _1\\right)\\,\\mathrm {d}\\vartheta _2\\right\\rbrace \\,\\mathrm {d}\\vartheta $ Using proposition $ \\ref {21} $ , we can conclude that $\\dfrac{1}{2}\\int _{0}^{1}\\left\\lbrace \\int _{0}^{1}\\dfrac{1}{(1+ \\vartheta _2\\vartheta )\\sqrt{(1 + \\vartheta _2)(1 - \\vartheta _2)}}\\left(\\int _{0}^{1}\\dfrac{1}{(1 + \\vartheta _1\\vartheta )\\sqrt{(1 + \\vartheta _1)(1 - \\vartheta _1)}}\\,\\mathrm {d}\\vartheta _1\\right)\\,\\mathrm {d}\\vartheta _2\\right\\rbrace \\,\\mathrm {d}\\vartheta $ $= \\dfrac{1}{2}\\int _{0}^{1}\\dfrac{\\arccos ^2(\\vartheta )}{(1 + \\vartheta )(1 - \\vartheta )}\\,\\mathrm {d}\\vartheta $ as desired.", "This completes the proof of proposition $ \\ref {32} $ .", "Proposition 3.3 Let $ \\mathcal {G} $ represent the Catalan's constant, then we have $\\dfrac{1}{2}\\int _{0}^{1}\\dfrac{\\arccos ^2(\\vartheta )}{(1 + \\vartheta )(1 - \\vartheta )}\\,\\mathrm {d}\\vartheta = \\mathcal {G}\\pi - \\dfrac{7}{4}\\,\\zeta (3)$ Substituting $ \\vartheta \\mapsto \\cos (x) $ everywhere inside the integral, we find that $\\dfrac{1}{2}\\int _{0}^{1}\\dfrac{\\arccos ^2(\\vartheta )}{(1 + \\vartheta )(1 - \\vartheta )}\\,\\mathrm {d}\\vartheta = \\dfrac{1}{2}\\int _{0}^{\\pi /2}\\dfrac{x^2}{1 - \\cos ^2(x)}\\,\\mathrm {d}x = \\dfrac{1}{2}\\int _{0}^{\\pi /2}\\dfrac{x^2}{\\sin ^2(x)}\\,\\mathrm {d}x$ Now integration by parts yields $\\dfrac{1}{2}\\int _{0}^{\\pi /2}\\dfrac{x^2}{\\sin ^2(x)}\\,\\mathrm {d}x = -\\int _{0}^{\\pi /2}x\\log \\left(\\tan \\left(\\dfrac{x}{2}\\right)\\right)\\,\\mathrm {d}x = -4\\int _{0}^{\\pi /4}z\\log (\\tan (z))\\,\\mathrm {d}z$ where the last equality is simply folowed by substituting $ \\dfrac{x}{2} \\mapsto z $ .", "Now using the Fourier series expansion of $ \\log (\\tan (z)) $ 1.442 we can evaluate the above obtained integral: $-4\\int _{0}^{\\pi /4}z\\log (\\tan (z))\\,\\mathrm {d}z = 8\\int _{0}^{\\pi /4}z\\sum _{k=1}^{\\infty }\\dfrac{\\cos (2(2k-1)z)}{2k-1} \\, \\mathrm {d}z = 8\\sum _{k=1}^{\\infty }\\int _{0}^{\\pi /4}\\dfrac{z\\cos (2(2k-1)z)}{2k-1} \\, \\mathrm {d}z$ $= \\sum _{k=1}^{\\infty }\\left(\\pi \\dfrac{(-1)^{k-1}}{(2k-1)^2} - \\dfrac{2}{(2k-1)^3}\\right) = \\pi \\sum _{k=1}^{\\infty }\\dfrac{(-1)^{k-1}}{(2k-1)^2} - 2\\sum _{k=1}^{\\infty }\\dfrac{1}{(2k-1)^3} = \\mathcal {G}\\pi - \\dfrac{7}{4}\\,\\zeta (3)$ as desired.", "This completes the proof of proposition $ \\ref {33} $ .", "Combining proposition $ \\ref {31}, \\ref {32} $ and $ \\ref {33} $ , we recover the result stated in proposition $ \\ref {11} $ ." ], [ "Final Remarks", "We also recover a beautiful double integral representation of Catalan's constant: $\\mathcal {G} = \\dfrac{1}{\\pi }\\left\\lbrace \\dfrac{7}{4}\\,\\zeta (3) + \\int _{0}^{\\pi /2}\\int _{0}^{\\pi /2}\\dfrac{\\log \\left(\\cos \\left(\\dfrac{x}{2}\\right)\\right) - \\log \\left(\\cos \\left(\\dfrac{z}{2}\\right)\\right)}{\\cos (x) - \\cos (z)}\\,\\mathrm {d}x\\,\\mathrm {d}z\\right\\rbrace $" ], [ "Acknowledgements", "The authors would like to thank Haran Mouli for pointing out various typos in the draft." ] ]
2105.11771
[ [ "Least-Squares ReLU Neural Network (LSNN) Method For Linear\n Advection-Reaction Equation" ], [ "Abstract This paper studies least-squares ReLU neural network method for solving the linear advection-reaction problem with discontinuous solution.", "The method is a discretization of an equivalent least-squares formulation in the set of neural network functions with the ReLU activation function.", "The method is capable of approximating the discontinuous interface of the underlying problem automatically through the free hyper-planes of the ReLU neural network and, hence, outperforms mesh-based numerical methods in terms of the number of degrees of freedom.", "Numerical results of some benchmark test problems show that the method can not only approximate the solution with the least number of parameters, but also avoid the common Gibbs phenomena along the discontinuous interface.", "Moreover, a three-layer ReLU neural network is necessary and sufficient in order to well approximate a discontinuous solution with an interface in $\\mathbb{R}^2$ that is not a straight line." ], [ "Introduction", "During the past several decades, numerical methods for linear advection-reaction equations have been intensively studied by many researchers and many numerical schemes have been developed.", "When inflow boundary data is discontinuous, so is the solution.", "It is well-known that traditional mesh-based numerical methods often exhibit oscillations near a discontinuity (called the Gibbs phenomena).", "Such spurious oscillations are unacceptable for many applications (see, e.g, [16]).", "To eliminate or reduce the Gibbs phenomena, finite difference and finite volume methods often use numerical techniques such as limiters, filters, ENO/WENO, etc.", "[13], [15], [16], [20]; and finite element methods usually employ discontinuous finite elements [4], [9], [12] and/or adaptive mesh refinement (AMR) to generate locally refined elements along discontinuous interfaces (see, e.g., [5], [17], [18]).", "Recently, there has been increasing interests in using deep neural networks (DNNs) to solve partial differential equations (see, e.g., [7], [25], [28]).", "DNNs produce a large class of functions through compositions of linear transformations and activation functions.", "One of the striking features of DNNs is that this class of functions is not subject to a hand-crafted geometric mesh or point cloud as are the traditional, well-studied finite difference, finite volume, and finite element methods.", "The physical partition of the domain $\\Omega $ , formed by free hyper-planes, can automatically adapt to the target function.", "This is much better than the AMR generated mesh because AMR is based on a geometric mesh and subject to mesh conformity; moreover, it is not easy to remove unnecessary elements or points.", "This paper will make use of this powerful approximation property of DNNs for solving linear advection-reaction problem with discontinuous solution.", "DNN functions are nonlinear functions of the parameters.", "Hence, the advection-reaction equation will be discretized through least-squares principles.", "In the context of finite element approximations, several least-squares methods have been studied (see, e.g., [1], [2], [3], [8], [10], [11], [22]).", "Basically, there are two least-squares formulations which are equivalent to the original differential equation.", "One is a direct application of least-squares principle (see, e.g., [1], [10]) with a weighted $L^2$ norm for the inflow boundary condition, where the weight is the magnitude of the normal component of the advection velocity field.", "The other is to apply the least-squares principle to an equivalent system of the underlying problem by introducing an additional flux variable (see [11], [22]).", "Some numerical techniques such as feedback least-squares finite element method [2], adaptive local mesh refinement with proper finite elements [22], etc.", "were introduced in order to reduce the Gibbs phenomena for problems with discontinuous solutions.", "The purpose of this paper is to study the least-squares neural network (LSNN) method for solving the linear advection-reaction problem with discontinuous solution.", "The LSNN method is based on the least-squares formulation studied in ([1], [10]), i.e., a direct application of the lease-squares principle to the underlying problem, and on the ReLU neural network as the class of approximating functions.", "The class of neural network functions enables the LSNN method to automatically approximate the discontinuous solution without using a priori knowledge of the location of the discontinuities.", "Compared to various AMR methods that locate the discontinuous interface through local mesh refinement, the LSNN method is much more effective in terms of the number of the degrees of freedom (see, e.g., Fig.", "REF (c) and REF (c)).", "Theoretically, it is proved in [10] that the homogeneous least-squares functional is equivalent to a natural norm in the solution space $V_{\\beta }$ consisting of all square-integrable functions whose directional derivative along ${\\beta }$ is also square-integrable (see section 2).", "This equivalence enables us to prove Ceá's lemma for the LSNN approximation, i.e., the error of the LSNN approximation is bounded by the approximation error of the set of ReLU neural network functions.", "This result is extended to the LSNN method with numerical integration as well.", "Even though approximation theory of the ReLU neural network has been intensively studied by many researchers (see, e.g., [24] for work before 2000 and [26], [27]), we are not able to find a result which is applicable to the discontinuous solution of the advection-reaction problem.", "To explore how well the ReLU neural network approximates the discontinuous solution, we consider two-dimensional transport problem, i.e., (REF ) with $\\hat{\\gamma }=0$ .", "When the boundary data $g$ is discontinuous at point ${\\bf x}_0\\in \\Gamma _-$ , the solution of the transport problem is discontinuous across an interface: the streamline of the advection velocity field starting at ${\\bf x}_0$ .", "The solution of this problem can be decomposed as the sum of a piece-wise constant function and a continuous piece-wise smooth function (see, e.g., (REF )).", "We show that the piece-wise constant function can be approximated well without the Gibbs phenomena by either a two- or a three-layer ReLU neural network with the minimal number of neurons depending on the shape of the interface (see Lemmas 3.1 and 5.1).", "Together with the universal approximation property, this implies that a two- or three-layer ReLU neural network is sufficient to well approximate the solution of the linear transport problem without oscillation.", "These theoretical results are confirmed by numerical results.", "The procedure for determining the values of the parameters of the network is now a problem in nonlinear optimization even though the underlying PDE is linear.", "This high dimensional, nonlinear optimization problem usually has many solutions.", "In order to obtain the desired one, we need to start from a close enough first approximation, and a common way to do so is by the method of continuation.", "In this paper, we propose the method of model continuation through approximating the advection velocity field by a family of piece-wise constant vector fields.", "Numerical results for a test problem with variable velocity field show that this method is able to reduce the total number of the parameters significantly.", "The paper is organized as follows.", "Section 2 introduces the advection-reaction problem, its least-squares formulation, and preliminaries.", "The ReLU neural network and the least-squares neural network are described and analyzed in section 3.", "Initialization for the two-layer neural network and the method of model continuation for initialization are presented in sections 4 and 6, respectively.", "Finally, numerical results for various benchmark test problems are given in section 5.", "Standard notations and definitions are used for the Sobolev space $H^s(\\Omega )^d$ and $H^s(\\Gamma _{-})^d$ when $s \\ge 0$ .", "The associated norms with these two spaces are denoted by $\\Vert \\cdot \\Vert _{s,\\Omega }$ and $\\Vert \\cdot \\Vert _{s,\\Gamma _{-}}$ , and their respective inner products are denoted as $(\\cdot , \\cdot )_{s,\\Omega } $ and $(\\cdot , \\cdot )_{s,\\Gamma _{-}} $ .", "For $s=0$ case, $H^s(\\Omega )^d$ is the same as $L^2(\\Omega )^d$ , then the norm and inner product are simply denoted as $\\Vert \\cdot \\Vert $ and $(\\cdot , \\cdot )$ , respectively.", "The subscripts $\\Omega $ in the designation of norms will be suppressed when there is no ambiguity." ], [ "Problem Formulation", "Let $\\Omega $ be a bounded domain in ${\\mathbb {R}}^d$ with Lipschitz boundary, and denote the advective velocity field by ${\\beta }({\\bf x}) = (\\beta _1, \\cdots , \\beta _d)^T\\in C^1(\\bar{\\Omega })^d$ .", "Define the inflow and outflow parts of the boundary $\\Gamma =\\partial \\Omega $ by $\\Gamma _- = \\lbrace {\\bf x}\\in \\Gamma :\\, {\\beta }({\\bf x}) \\cdot {n}({\\bf x}) <0\\rbrace \\quad \\mbox{and}\\quad \\Gamma _+ = \\lbrace {\\bf x}\\in \\Gamma :\\, {\\beta }({\\bf x}) \\cdot {n}({\\bf x}) > 0\\rbrace ,$ respectively, where ${n}({\\bf x})$ is the unit outward normal vector to $\\Gamma $ at ${\\bf x}\\in \\Gamma $ .", "As a model hyperbolic boundary value problem, we consider the linear advection-reaction equation $\\left\\lbrace \\begin{array}{rccl}\\nabla \\!\\cdot \\!", "(\\beta u) + \\gamma u &= & f &\\text{ in }\\,\\, \\Omega , \\\\u&=&g &\\text{ on }\\,\\, \\Gamma _{-},\\end{array}\\right.$ where $\\gamma \\in C(\\bar{\\Omega })$ , $f \\in L^2(\\Omega )$ , and $g \\in L^2(\\Gamma _-)$ are given scalar-valued functions.", "We assume that there exist a positive constant $\\gamma _0$ such that $\\gamma ({\\bf x}) + \\frac{1}{2}\\nabla \\cdot {\\beta } ({\\bf x}) \\ge \\gamma _0 >0\\quad \\text{ for all } {\\bf x}\\in \\Omega .$ For simplicity of presentation, we also assume that $g$ is bounded so that streamline functions from $\\Gamma _{-}$ to $\\Gamma _{+}$ is not needed (see [10]).", "Denote by $v_{\\beta } = {\\beta }\\cdot \\nabla v$ the directional derivative along the advective velocity field ${\\beta }$ , then (REF ) may be rewritten as follows $\\left\\lbrace \\begin{array}{rccl}u_{\\beta } + \\hat{\\gamma }\\, u &=&f &\\text{ in }\\, \\Omega , \\\\[2mm]u&=&g &\\text{ on }\\,\\, \\Gamma _{-},\\end{array}\\right.$ where $\\hat{\\gamma } = \\gamma + \\nabla \\!", "\\cdot \\!", "{\\beta }$ .", "The solution space of (REF ) is given by $V_{\\beta } = \\lbrace v\\in L^2(\\Omega ): v_{{\\beta }}\\in L^2(\\Omega )\\rbrace ,$ which is equipped with the norm as ${\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|v\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}_{\\beta }= \\left(\\Vert v\\Vert _{0,\\Omega }^2 + \\Vert v_{\\beta }\\Vert _{0,\\Omega }^2 \\right)^{1/2}.$ Denote the weighted $L^2(\\Gamma _{-})$ norm over the inflow boundary by $\\Vert v\\Vert _{-{\\beta }}=\\left<v,v\\right>^{1/2}_{-{\\beta }}=\\left( \\int _{\\Gamma _-} |{\\beta }\\!", "\\cdot \\!", "{n}|\\, v^2\\,ds\\right)^{1/2}.$ The following trace and Poincaré inequalities are proved in [10] (see also [2]) that there exist positive constants $C_t$ and $C_p$ such that $\\Vert v\\Vert _{-{\\beta }} \\le C_t\\, {\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|v\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}_{\\beta },\\quad \\forall \\,\\,v\\in V_{\\beta }$ and $\\Vert v\\Vert _{0,\\Omega } \\le C_p\\left(\\Vert v\\Vert _{-{\\beta }} + D\\, \\Vert v_{\\beta }\\Vert _{0,\\Omega } \\right), \\quad \\forall \\,v\\in V_{\\beta },$ respectively, where $D=\\text{diam} (\\Omega )$ is the diameter of the domain $\\Omega $ .", "Let ${\\cal C}$ be the streamline of the advection velocity field ${\\beta }$ starting at ${\\bf x}_0\\in \\Gamma _-$ in two dimensions.", "Assume that the inflow boundary condition $g$ is discontinuous at ${\\bf x}_0$ .", "Then it is easy to see that the solution of (REF ) is also discontinuous across ${\\cal C}$ because the restriction of the solution on ${\\cal C}$ satisfies the same differential equation but different initial condition.", "Moreover, if $\\hat{\\gamma }=0$ , then the jump of the solution along ${\\cal C}$ is a constant $|g({\\bf x}_0^+)-g({\\bf x}_0^-)|$ , where $g({\\bf x}_0^+)$ and $g({\\bf x}_0^-)$ are the values of $g$ at ${\\bf x}_0$ from different sides.", "The streamline ${\\cal C}$ is referred to be the discontinuous interface.", "In the remainder of this section, we describe the least-squares (LS) formulation following [2], [10].", "To this end, define the LS functional $\\mathcal {L}(v;{\\bf f}) = \\Vert v_{\\beta } +\\hat{\\gamma }\\, v-f\\Vert _{0,\\Omega }^2 + \\Vert v-g\\Vert _{-\\beta }^2 $ for all $v\\in V_{\\beta }$ , where ${\\bf f} = (f,g)$ .", "Now, the corresponding least-squares formulation is to seek $u\\in V_{\\beta }$ such that $\\mathcal {L}(u;{\\bf f}) = \\min _{\\small v\\in V_{\\beta }} \\mathcal {L}(v;{\\bf f}).$ It follows from the trace, triangle, and Poincaré inequalities and assumptions on ${\\beta }$ and $\\gamma $ that the homogeneous LS functional $\\mathcal {L}(v;{\\bf 0})$ is equivalent to the norm ${\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|v\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}_{\\beta }^2$ , i.e., there exist positive constants $\\alpha $ and $M$ such that $\\alpha \\, {\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|v\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}_{\\beta }^2\\le \\mathcal {L}(v;{\\bf 0}) \\le M\\, {\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|v\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}_{\\beta }^2.$ Furthermore, problem (REF ) has a unique solution $u\\in V_{\\beta }$ satisfying the following a priori estimate ${\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|u\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}_{\\beta } \\le C\\, \\left(\\Vert f\\Vert _{0,\\Omega } + \\Vert g\\Vert _{-\\beta }\\right).$ Denote the bilinear and linear forms by $a(u,v)=(u_{\\beta } + \\hat{\\gamma }\\, u,v_{\\beta } + \\hat{\\gamma }\\, v) + \\left<u,v\\right>_{-{\\beta }}\\quad \\mbox{and}\\quad f(v)=(f,v_{\\beta } + \\hat{\\gamma }\\, v) + \\left<g,v\\right>_{-{\\beta }},$ respectively.", "Then the minimization problem in (REF ) is to find $u\\in V_{\\beta }$ such that $a(u,v)=f(v), \\quad \\forall \\,\\, v\\in V_{\\beta }.$" ], [ "Least-Squares Neural Network Method", "This section describes deep neural networks and the corresponding least-squares method for linear transport equations.", "We consider a deep neural network (DNN) with a scalar-valued output as ${\\cal N}:\\, {\\bf x}\\in \\mathbb {R}^{d}\\longrightarrow {\\cal N}({\\bf x})\\in \\mathbb {R}.$ The DNN function ${\\cal N}({\\bf x})$ is typically represented as compositions of many layers of functions: ${\\cal N}({\\bf x})=N^{(L)} \\circ \\cdots N^{(2)}\\circ N^{(1)}({\\bf x}),$ where the symbol $\\circ $ denotes the composition of functions, and $L$ is the depth of the network.", "In this case, $N^{(l)}$ is called the $l^{th}$ layer of the network.", "All layers except the last one $N^{(L)}$ are called hidden layers.", "A layer $N^{(l)}: \\mathbb {R}^{n_{l-1}} \\rightarrow \\mathbb {R}^{n_{l}}$ is defined as a composition of a linear transformation $T^{(l)}: \\mathbb {R}^{n_{l-1}} \\rightarrow \\mathbb {R}^{n_{l}}$ and an activation function $\\sigma : \\mathbb {R}\\rightarrow \\mathbb {R}$ as follows: $N^{(l)}({\\bf x}^{(l-1)})=\\sigma \\big ( T^{(l)}({\\bf x}^{(l-1)})\\big )= \\sigma (\\mbox{$\\omega $}^{(l)}{\\bf x}^{(l-1)}-{\\bf b}^{(l)})\\quad \\mbox{for } {\\bf x}^{(l-1)}\\in \\mathbb {R}^{n_{l-1}},$ where $\\mbox{$\\omega $}^{(l)}\\in \\mathbb {R}^{n_{l}\\times n_{l-1}}$ , ${\\bf b}^{(l)}\\in \\mathbb {R}^{n_{l}}$ , ${\\bf x}^{(0)}={\\bf x}$ , and application of $\\sigma $ to a vector is defined component-wise.", "There is typically no activation function in the output layer.", "Components of $\\mbox{$\\omega $}^{(l)}$ and ${\\bf b}^{(l)}$ are called weights and bias, respectively, and are parameters to be determined (trained).", "Each component of the vector-valued function $N^{(l)}$ is interpreted as a neuron and the dimension $n_{l}$ defines the width or the number of neurons of the $l^{\\text{th}}$ layer in a network.", "This paper will use the popular rectified linear unit (ReLU) activation function defined by $\\sigma (t) = \\max \\lbrace 0,\\,t\\rbrace =\\left\\lbrace \\begin{array}{rclll}0, & \\mbox{if } t\\le 0,\\\\[2mm]t, & \\mbox{if } t >0.\\end{array}\\right.$ For given integers $\\lbrace n_l\\rbrace _{l=1}^L$ , denote the set of DNN functions by ${\\cal M}({\\small \\mbox{${\\theta }$}},L)=\\big \\lbrace {\\cal N}({\\bf x})= N^{(L)} \\circ \\cdots \\circ N^{(1)}({\\bf x}) :\\, \\mbox{$\\omega $}^{(l)}\\in \\mathbb {R}^{n_{l}\\times n_{l-1}},\\,\\, {\\bf b}^{(l)}\\in \\mathbb {R}^{n_{l}} \\mbox{ for } l=1,...,L\\big \\rbrace ,$ where $N^{(l)}({\\bf x}^{(l-1)})$ is defined in (REF ) and ${\\small \\mbox{${\\theta }$}}$ denotes all parameters: $\\mbox{$\\omega $}^{(l)}$ and ${\\bf b}^{(l)}$ for $l=1,...,L$ .", "It is easy to see that ${\\cal M}({\\small \\mbox{${\\theta }$}},L)$ is a subset of $V_{\\beta }$ , but not a linear subspace.", "The least-squares approximation is to find $u_{_N}({\\bf x};{\\small \\mbox{${\\theta }$}}^*) \\in {\\cal M}({\\small \\mbox{${\\theta }$}},L)$ such that $\\mathcal {L}\\big (u_{_N}({\\bf x};{\\small \\mbox{${\\theta }$}}^*);\\,{\\bf f}\\big )= \\min \\limits _{v\\in {\\cal M}({\\scriptsize \\mbox{${\\theta }$}},L)} \\mathcal {L}\\big (v({\\bf x};{\\small \\mbox{${\\theta }$}});\\,{\\bf f}\\big )= \\min _{{\\scriptsize \\mbox{${\\theta }$}}\\in \\mathbb {R}^{N}}\\mathcal {L}\\big (v({\\bf x};{\\small \\mbox{${\\theta }$}});\\,{\\bf f}\\big ),$ where $N$ is the total number of parameters in ${\\cal M}({\\small \\mbox{${\\theta }$}},L)$ given by $N=M_d(L) =\\sum ^L_{l=1} n_{l}\\times (n_{l-1}+1).$ Let $u$ and $u_{_N}$ be the solutions of problems (REF ) and (REF ), respectively.", "Then we have ${\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|u-u_{_N}\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}_{\\beta }\\le \\left(\\dfrac{M}{\\alpha }\\right)^{1/2} \\inf _{v\\in {\\cal M}({\\scriptsize \\mbox{${\\theta }$}},L)} {\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|u-v\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}_{\\beta },$ where $\\alpha $ and $M$ are constants in (REF ).", "For any $v\\in {\\cal M}({\\small \\mbox{${\\theta }$}},L)\\subset V_{\\beta }$ , it follows from the coercivity and continuity of the homogeneous functional $ \\mathcal {L}\\big (v;\\,{\\bf 0}\\big )$ in (REF ), problem (REF ), and (REF ) that $\\alpha \\, {\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|u-u_{_N}\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}^2_{\\beta }&\\le & \\mathcal {L}\\big (u-u_{_N};\\,{\\bf 0}\\big )= \\mathcal {L}\\big (u_{_N}({\\bf x};{\\small \\mbox{${\\theta }$}}^*);\\,{\\bf f}\\big )\\\\[2mm]&\\le & \\mathcal {L}\\big (v({\\bf x};{\\small \\mbox{${\\theta }$}});\\,{\\bf f}\\big )= \\mathcal {L}\\big (u-v;\\,{\\bf 0}\\big )\\le M \\,{\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|u-v\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}^2_{\\beta },$ which implies (REF ).", "This completes the proof of the lemma.", "For a given vector $\\mbox{$\\xi $}\\in \\mathbb {R}^d$ and $c\\in \\mathbb {R}$ , assume that the hyper-plane ${\\cal P}:\\,\\mbox{$\\xi $}\\cdot {\\bf x}=c$ divides the domain $\\Omega $ into two non-empty subdomains $\\Omega _1$ and $\\Omega _2$ , i.e., $\\Omega _1=\\lbrace {\\bf x}\\in \\Omega :\\, \\mbox{$\\xi $}\\cdot {\\bf x}<c\\rbrace \\quad \\mbox{and}\\quad \\Omega _2=\\lbrace {\\bf x}\\in \\Omega :\\, \\mbox{$\\xi $}\\cdot {\\bf x}>c\\rbrace .$ Let $\\chi ({\\bf x};\\mbox{$\\xi $},c)$ be a piece-wise constant function defined by $\\chi ({\\bf x};\\mbox{$\\xi $},c)=\\left\\lbrace \\begin{array}{ll}\\alpha _1, & {\\bf x}\\in \\Omega _1,\\\\[2mm]\\alpha _2, & {\\bf x}\\in \\Omega _2.\\end{array}\\right.$ Let $p({\\bf x})$ be a two-layer neural network function given by $p({\\bf x})=\\alpha _1 +\\dfrac{\\alpha _2-\\alpha _1}{2\\varepsilon } \\Big (\\sigma (\\mbox{$\\xi $}\\cdot {\\bf x}-c + \\varepsilon ) - \\sigma (\\mbox{$\\xi $}\\cdot {\\bf x}-c-\\varepsilon )\\Big )$ for any $\\varepsilon >0$ such that intersections between the domain $\\Omega $ and the hyper-planes $\\mbox{$\\xi $}\\cdot {\\bf x}=c\\pm \\varepsilon $ are not empty.", "Then we have $\\Vert \\chi - p\\Vert _{0,\\Omega }=\\left(\\Vert \\chi - p\\Vert ^2_{0,\\Omega } + \\Vert \\chi _{\\scriptsize \\eta } - p_{\\scriptsize \\eta }\\Vert ^2_{0,\\Omega }\\right)^{1/2}\\le {\\dfrac{1}{\\sqrt{6}}}\\, D^{(d-1)/2}\\, \\big |\\alpha _1-\\alpha _2\\big |\\, \\sqrt{\\varepsilon },$ where ${\\eta }$ is a vector normal to ${\\mbox{$\\xi $}}$ and $D$ is the diameter of the domain $\\Omega $ .", "Let $\\Omega _\\varepsilon =\\Omega _{\\varepsilon ,1}\\cup \\Omega _{\\varepsilon ,2}\\equiv \\lbrace {\\bf x}\\in \\Omega :\\, c-\\varepsilon < \\mbox{$\\xi $}\\cdot {\\bf x}< c\\rbrace \\cup \\lbrace {\\bf x}\\in \\Omega :\\, c< \\mbox{$\\xi $}\\cdot {\\bf x}< c+\\varepsilon \\rbrace .$ The equality in (REF ) follows from the fact that $\\chi _{\\scriptsize \\eta } - p_{\\scriptsize \\eta }=0$ .", "To show the validity of the inequality in (REF ), first we have $\\chi -p =\\left\\lbrace \\begin{array}{cl}\\dfrac{\\alpha _1-\\alpha _2}{2\\varepsilon } \\,\\big (\\mbox{$\\xi $}\\cdot {\\bf x}-c + \\varepsilon \\big ), & {\\bf x}\\in \\Omega _{\\varepsilon ,1}, \\\\[4mm]\\dfrac{\\alpha _1-\\alpha _2}{2\\varepsilon } \\,\\big (\\mbox{$\\xi $}\\cdot {\\bf x}-c - \\varepsilon \\big ), & {\\bf x}\\in \\Omega _{\\varepsilon ,2}, \\\\[4mm]0, & {\\bf x}\\in \\Omega \\setminus \\Omega _\\varepsilon .\\end{array}\\right.$ By a rotation of the coordinates, ${\\bf x}=(s,{\\bf y})$ , it is easy to see that the domain $\\Omega _{\\varepsilon ,1}$ is bounded by the hyper-planes $s=c-\\varepsilon $ and $s=c$ and the hyper-surfaces $\\varphi _1(s)$ and $\\varphi _2(s)$ on $\\partial \\Omega $ .", "Hence, we have $ \\int _{\\Omega _{\\varepsilon ,1}} \\big (\\mbox{$\\xi $}\\cdot {\\bf x}-c + \\varepsilon \\big )^2\\,d{\\bf x}=\\int _{c-\\varepsilon }^{c}\\int ^{\\varphi _2(s)}_{\\varphi _1(s)}\\big (s -c + \\varepsilon \\big )^2\\,d{\\bf y}\\, ds\\le D^{d-1} \\int _{c-\\varepsilon }^{c}\\big (s -c + \\varepsilon \\big )^2\\,\\, ds = \\dfrac{D^{d-1}}{3}\\,\\varepsilon ^3.$ In a similar fashion, $\\int _{\\Omega _{\\varepsilon ,2}} \\big (\\mbox{$\\xi $}\\cdot {\\bf x}-c - \\varepsilon \\big )^2\\,d{\\bf x}\\le \\dfrac{D^{d-1}}{3}\\varepsilon ^3.$ The above two inequalities imply $\\Vert \\chi - p\\Vert ^2_{0,\\Omega }\\le \\left(\\dfrac{\\alpha _1-\\alpha _2}{2\\varepsilon }\\right)^2 \\dfrac{2D^{d-1}}{3}\\varepsilon ^3 = \\dfrac{D^{d-1}(\\alpha _1-\\alpha _2)^2\\varepsilon }{6}.$ This proves the inequality in (REF ) and, hence, the lemma.", "Assume that $u$ is a piece-wise smooth function with respect to the partition $\\lbrace \\Omega _1,\\Omega _2\\rbrace $ such that the jump of $u$ on the interface ${\\cal P}=\\Omega _1\\cap \\Omega _2$ is a constant $\\alpha _2-\\alpha _1$ , i.e., $[\\!", "[ u]\\!", "]_{_{\\cal P}}\\equiv u_2|_{_{\\cal P}} - u_1|_{_{\\cal P}}=\\alpha _2-\\alpha _1.$ Then $u$ has the following decomposition $u= \\chi ({\\bf x};\\mbox{$\\xi $},c) + \\hat{u},$ where $\\mbox{$\\xi $}$ is a vector normal to ${\\beta }$ .", "It is easy to see that $\\hat{u}$ is continuous in $\\Omega $ and piece-wise smooth.", "Assume that the advection velocity field ${\\beta }$ is a constant vector field and that $f\\in C(\\Omega )$ .", "Let $u$ and $u_{_N}$ be the solutions of problems (REF ) and (REF ), respectively.", "Then we have ${\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|u-u_{_N}\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}_{\\beta }\\le C\\,\\left(\\big |\\alpha _1-\\alpha _2\\big |\\, \\sqrt{\\varepsilon } + \\inf _{v\\in {\\cal M}({\\scriptsize \\mbox{${\\theta }$}},L)} {\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|\\hat{u}-v\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}_{\\beta }\\right),$ where $\\hat{u}\\in C(\\Omega )$ is given in (REF ).", "The assumptions on ${\\beta }$ and $f$ imply that the exact solution $u$ has the decomposition in (REF ).", "Now, (REF ) is a direct consequence of Lemmas 3.1 and 3.2.", "Similar to [7], we evaluate the LS functional numerically.", "To this end, let ${\\cal T}=\\lbrace K :\\, K\\mbox{ is an open subdomain of } \\Omega \\rbrace $ be a partition of the domain $\\Omega $ .", "Then ${\\cal E}_{-}=\\lbrace E=\\partial K \\cap \\Gamma _-:\\,\\, K\\in \\mathcal {T}\\rbrace $ is a partition of the inflow boundary $\\Gamma _-$ .", "Let ${\\bf x}_{_K}$ and ${\\bf x}_{_E}$ be the centroids of $K\\in {\\cal T}$ and $E\\in {\\cal E}_-$ , respectively.", "Define the discrete LS functional as follows: $\\begin{split}\\mathcal {L}_{_{\\small {\\cal T}}}\\big (v({\\bf x}; {\\small \\mbox{${\\theta }$}});{\\bf f}\\big )= \\sum _{K \\in {\\cal T}} \\big (v_{\\beta } +\\hat{\\gamma }\\, v-f \\big )^2({\\bf x}_{_K}; {\\small \\mbox{${\\theta }$}})\\,|K|+ \\sum _{E\\in {\\cal E}_-} \\big (|{\\beta } \\cdot {n}|(v-g)^2\\big )({\\bf x}_{_E}; {\\small \\mbox{${\\theta }$}})|E|,\\end{split}$ where $|K|$ and $|E|$ are the $d$ and $d-1$ dimensional measures of $K$ and $E$ , respectively.", "Then the discrete least-squares approximation is to find ${u}^N_{_{\\small {\\cal T}}}({\\bf x},{\\small \\mbox{${\\theta }$}}^*)\\in {\\cal M}({\\small \\mbox{${\\theta }$}},L)$ such that $\\mathcal {L}_{_{\\small {\\cal T}}} \\big ({u}^N_{_{\\small {\\cal T}}}({\\bf x},{\\small \\mbox{${\\theta }$}}^*);{\\bf f}\\big )= \\min \\limits _{v\\in {\\cal M}({\\scriptsize \\mbox{${\\theta }$}},L)} \\mathcal {L}_{_{\\small {\\cal T}}}\\big (v({\\bf x};{\\small \\mbox{${\\theta }$}});\\,{\\bf f}\\big )= \\min _{{\\scriptsize \\mbox{${\\theta }$}}\\in \\mathbb {R}^{N}}\\mathcal {L}_{_{\\small {\\cal T}}} \\big (v({\\bf x}; {\\small \\mbox{${\\theta }$}});{\\bf f}\\big ).$ Let $u$ , $u_{_N}$ , and $u_{_{\\cal T}}^N$ be the solutions of problems (REF ), (REF ), and (REF ), respectively.", "Then there exists a positive constant $C$ such that ${\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|u-u^{_N}_{_{\\cal T}}\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}_{\\beta }\\le C\\,\\left( \\inf _{v\\in {\\cal M}({\\scriptsize \\mbox{${\\theta }$}},L)} {\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|u-v\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}_{\\beta }+ \\big |({\\cal L}-{\\cal L}_{_{\\cal T}})(u_{_N}-u_{_{\\cal T}}^{\\small N}; {\\bf 0})\\big |+ \\big |({\\cal L}-{\\cal L}_{_{\\cal T}})(u-u_{_N}; {\\bf 0})\\big |\\right).$ By the triangle inequality, the fact that ${\\cal L}_{_{\\cal T}}(u_{_{\\cal T}}^N; {\\bf f}) \\le {\\cal L}_{_{\\cal T}}(u_{_N}; {\\bf f})$ , and the continuity of the homogeneous functional $\\mathcal {L}\\big (v;\\,{\\bf 0}\\big )$ in (REF ), we have $\\dfrac{1}{2}\\,{\\cal L}_{_{\\cal T}}(u_{_N}-u_{_{\\cal T}}^N; {\\bf 0})&\\le & {\\cal L}_{_{\\cal T}}(u_{_N}-u; {\\bf 0}) + {\\cal L}_{_{\\cal T}}(u-u_{_{\\cal T}}^N; {\\bf 0})= {\\cal L}_{_{\\cal T}}(u_{_N}; {\\bf f}) + {\\cal L}_{_{\\cal T}}(u_{_{\\cal T}}^N; {\\bf f})\\\\[2mm]&\\le & 2\\, {\\cal L}_{_{\\cal T}}(u_{_N}; {\\bf f})= 2\\,\\big (({\\cal L}_{_{\\cal T}}-{\\cal L})(u_{_N}-u; {\\bf 0})+{\\cal L}(u_{_N}-u; {\\bf 0})\\big )\\\\[2mm]&\\le & 2\\,({\\cal L}_{_{\\cal T}}-{\\cal L})(u_{_N}-u; {\\bf 0})+2M\\,{\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|u-u_{_N}\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}^2_{\\beta } ,$ which, together with the coercivity of the homogeneous functional $\\mathcal {L}\\big (v;\\,{\\bf 0}\\big )$ in (REF ), implies that $\\alpha \\, {\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|u_{_N}-u_{_{\\cal T}}^{N}\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}^2_{\\beta }&\\le & \\mathcal {L}\\big (u_{_N}-u_{_{\\cal T}}^{N};\\,{\\bf 0}\\big )= \\big (\\mathcal {L}-{\\cal L}_{_{\\cal T}}\\big )\\big (u_{_N}-u_{_{\\cal T}}^N;\\,{\\bf 0}\\big ) +\\mathcal {L}_{_{\\cal T}}\\big (u_{_N}-u_{_{\\cal T}}^N;\\,{\\bf 0}\\big )\\\\[2mm]&\\le & \\big (\\mathcal {L}-{\\cal L}_{_{\\cal T}}\\big )\\big (u_{_N}-u_{_{\\cal T}}^N;\\,{\\bf 0}\\big )+ 4\\,({\\cal L}_{_{\\cal T}}-{\\cal L})(u_{_N}-u; {\\bf 0})+4M\\,{\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|u-u_{_N}\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}^2_{\\beta }.$ Now, (REF ) is a direct consequence of the triangle inequality, the above inequality, and Lemma 3.1.", "This completes the proof of the lemma.", "Lemma 3.4 indicates that the total error of the LSNN approximation with numerical integration is bounded by the approximation error of the neural network and the error of the numerical integration." ], [ "Initialization of two-layer neural network", "The nonlinear optimization in (REF ) usually has many solutions, and the desired one is obtained only if we start from a close enough first approximation.", "In this section, we briefly describe the initialization process introduced in [21] for two-layer neural network.", "To this end, a two-layer ReLU NN with $n_1$ neurons produces the following set of functions: ${\\cal M}({\\small \\mbox{${\\theta }$}},2) = \\left\\lbrace c_0+\\sum _{i=1}^{n_1} c_i\\sigma (\\mbox{$\\omega $}_i\\cdot {\\bf x}-b_i)\\, :\\,c_i,\\, b_i\\in \\mathbb {R},\\,\\, \\mbox{$\\omega $}_i\\in {\\cal S}^{d-1} \\right\\rbrace ,$ where ${\\cal S}^{d-1}$ is the unit sphere in $\\mathbb {R}^d$ .", "Let $\\varphi _{0}({\\bf x})=1\\quad \\mbox{and}\\quad \\varphi _{i}({\\bf x})=\\sigma (\\mbox{$\\omega $}_i\\cdot {\\bf x}-b_i)\\quad \\mbox{for } i=1,...,n_1.$ For a given input weights and bias $\\mbox{$\\omega $}=(\\mbox{$\\omega $}_1, ... , \\mbox{$\\omega $}_{n_1})\\quad \\mbox{and}\\quad {\\bf b}=(b_1, ..., b_{n_1}),$ problem (REF ) may be approximated by finding $u_{n_1}=\\sum \\limits ^{n_1}_{i=0}c_i\\varphi _{i}({\\bf x})$ such that $a(u_{n_1},\\varphi _{j}) =f(\\varphi _{j})\\quad \\mbox{for } j=0,1,...,n_1.$ for $j=0,1,...,n_1$ .", "Let $A(\\mbox{$\\omega $},{\\bf b})=\\left(a(\\varphi _{j}, \\varphi _{i}) \\right)_{(n_1+1)\\times (n_1+1)}\\quad \\mbox{and}\\quad F(\\mbox{$\\omega $},{\\bf b})=\\left(f(\\varphi _{j}) \\right)_{(n_1+1)\\times 1},$ then the coefficients, ${\\bf c}=(c_0,c_1, ..., c_n)$ , of $u_{n_1}$ is the solution of the system of linear algebraic equations $A(\\mbox{$\\omega $},{\\bf b})\\, {\\bf c}= F(\\mbox{$\\omega $},{\\bf b}).$ Assume that hyper-planes $\\lbrace \\mbox{$\\omega $}_i\\cdot {\\bf x}=b_i\\rbrace _{i=1}^{n_1}$ are distinct.", "Then the coefficient matrix $A(\\mbox{$\\omega $},{\\bf b})$ is symmetric, positive definite.", "Obviously, $A(\\mbox{$\\omega $},{\\bf b})$ is symmetric.", "Positive definiteness of $A(\\mbox{$\\omega $},{\\bf b})$ follows from the lower bound in (REF ) and the linear independence of $\\lbrace \\varphi _{i}\\rbrace ^{n_1}_{i=0}$ (see Lemma 2.1 of [21]).", "As discussed in [21], the (breaking) hyper-planes ${\\cal P}_i:\\, \\mbox{$\\omega $}_i\\cdot {\\bf x}-b_i=0\\quad \\mbox{for } i=1,...,n_1$ and the boundary of the domain $\\Omega $ form a physical partition of the domain $\\Omega $ .", "It is then natural to initialize the input weights $\\mbox{$\\omega $}$ and bias ${\\bf b}$ such that the corresponding hyper-planes $\\lbrace {\\cal P}_i\\rbrace ^{n_1}_{i=1}$ form a uniform partition of the domain $\\Omega $ .", "The initial for the output weights and bias ${\\bf c}$ may be chosen to be the solution of problem (REF )." ], [ "Numerical Experiments", "In this section, we present numerical results for test problems with constant, piece-wise constant, or variable advection velocity fields.", "The solutions of these test problems are discontinuous along an interface which is a line segment, a piece-wise line segment, or a curve.", "In all experiments, the integration mesh ${\\cal T}$ is obtained by uniformly partitioning the domain $\\Omega $ into identical squares with mesh size $h=10^{-2}$ .", "The directional derivative in the least-squares functional is approximated by the backward finite difference quotient $v_{\\beta }({\\bf x}_{_K}) \\approx \\frac{v({\\bf x}_{_K})-v\\big ({\\bf x}_{_K} - \\rho \\bar{{\\beta }}({\\bf x}_{_K}\\big ))}{\\rho }$ where $\\rho \\in \\mathbb {R}$ is chosen to be smaller than the integration mesh size $h$ , and $\\bar{{\\beta }}$ is the unit vector in the ${\\beta }$ direction.", "The minimization problem in (REF ) is solved numerically by the Adam version of gradient descent [19], and variable learning rate is used during the training.", "Let $u$ be the exact solution of problem (REF ) and $\\bar{u}^N_{_{\\cal T}}$ be the LSNN approximation.", "Tables REF –REF report the numerical errors in the relative $L^2$ , $V_{\\beta }$ , and graph norms.", "In these tables, a network structure is expressed by 2-$n$ -1 for a two-layer network with $n$ neurons, by 2-$n_1$ -$n_2$ -1 for a three-layer network with $n_1$ and $n_2$ neurons in the respective first and second layers, and so on.", "Figures REF –REF depict the traces of the exact solution and the numerical approximation on a plane perpendicular to both the $x_1x_2$ -plane and the discontinuous interface, which accurately illustrate the quality of the numerical approximation." ], [ "Problems with a constant advection velocity fields", "In this section, we present numerical results for two test problems with constant advection velocity fields whose solutions are piece-wise constants (see, e.g., [22]).", "A two-layer neural network is employed and the network is initialized by the method described in section 4.", "The first test problem is the equation in (REF ) with the domain $\\Omega =(0,2) \\times (0,1)$ , the inflow boundary $\\Gamma _- = \\lbrace (x,0):\\, x\\in (0,2)\\rbrace $ , a constant advection velocity field ${\\beta } = (0,1)^T$ , $\\gamma =f=0$ , and the inflow boundary data $g(x)=0$ for $x\\in (0,\\pi /3)$ and $g(x)=1$ for $x\\in (\\pi /3,2)$ .", "Let $\\Omega _1=\\lbrace (x,y)\\in \\Omega :\\, 0<x<\\pi /3\\rbrace $ and $\\Omega _2=\\lbrace (x,y)\\in \\Omega :\\,\\pi /3<x<2\\rbrace $ , it is then easy to see that the exact solution is a piece-wise constant given by $ u(x,y)=\\left\\lbrace \\begin{array}{ll}0, & (x,y)\\in \\Omega _1, \\\\[2mm]1, & (x,y)\\in \\Omega _2.\\end{array}\\right.$ The discontinuous interface is the vertical line $x=\\pi /3$ .", "This problem was used to test various adaptive least-squares finite element methods in [22].", "In particular, the discontinuous interface $x=\\pi /3$ was chosen so that if the initial mesh does not align with the interface, so is the mesh generated by either global or local mesh refinements.", "Numerical results in [22] (see Fig.", "REF ) showed that the conforming least-squares finite element method (C-LSFEM) exhibits the Gibbs phenomena even with very fine mesh and that the newly developed flux-based LSFEM in [22] using a pair of the lowest-order elements is able to avoid overshooting on an adaptively refined mesh.", "Figure: An adaptively refined meshThe LSNN method is implemented with $\\rho =h/2$ and a fixed learning rate $0.003$ with 20000 iterations.", "Our first set of experiments are done by using networks: 2-200-1 and 2-25-15-15-1.", "These two networks have 601 and 705 parameters, respectively, and provide good approximations (similar to Fig.", "REF (a,b)) to the exact solution.", "Lemma 3.2 indicates that a two-layer network with 2 neurons is sufficient to approximate the exact solution well.", "Our second set of experiments are done by using networks: 2-2-1 and 2-4-1 with the respective 7 and 13 parameters.", "The 2-2-1 network fails to approximate the exact solution when the initial breaking lines are chosen to be the vertical line $x=1$ and the horizontal line $y=1/2$ .", "This is because the iterative solver of the nonlinear optimization is not able to move the initial horizontal breaking line to the right place.", "The initial breaking lines for the 2-4-1 network are chosen to be the vertical lines $x=2/3$ and $x=4/3$ and the horizontal lines $y=1/3$ and $y=2/3$ .", "Table: Relative errors of the problem with discontinuity along a vertical line segmentFigure: Network breaking linesErrors of numerical results are presented in Table REF .", "The second and third columns in Table REF show that the approximation of the small network is slightly more accurate than that of the large network while the values of the loss functions are reversed.", "This indicates that the large network is trapped in a local minimum.", "The numerical solution of the 4-neuron network is depicted in Fig.", "REF (a).", "The traces of the exact and numerical solutions on the plane $y=1$ are depicted in Fig.", "REF (b), which shows no oscillation.", "Fig.", "REF (c) displays breaking lines of the network with two vertical lines $x=1.02882$ and $x=1.06114$ closing to the interface $x=\\pi /3$ .", "This indicates that breaking lines of neural network are capable of automatically adapting to the discontinuous interface.", "This simple test problem shows that the LSNN method out-performs the traditional mesh-based numerical methods." ], [ "Discontinuity along the diagonal", "The second test problem is again equation (REF ) with a constant advection velocity vector and a piece-wise constant inflow boundary condition.", "Specifically, ${\\beta } = (1,1)^T/\\sqrt{2}$ , $\\Omega =(-1,1)^2$ , $\\Gamma _-=\\Gamma _-^1\\cup \\Gamma _-^2\\equiv \\lbrace (-1,y):\\, y \\in (-1,1)\\rbrace \\cup \\lbrace (x,-1):\\, x \\in (-1,1)\\rbrace $ , $\\gamma =1$ , $g$ and $f$ are piece-wise constants given by $g(x,y)=\\left\\lbrace \\begin{array}{ll}1, & (x,y)\\in \\Gamma ^1_-, \\\\[2mm]0, & (x,y)\\in \\Gamma ^2_-,\\end{array}\\right.\\quad \\mbox{and}\\quad f(x,y) = \\left\\lbrace \\begin{array}{ll}1, & (x,y)\\in \\Omega _1, \\\\[2mm]0, & (x,y)\\in \\Omega _2,\\end{array}\\right.$ where $\\Omega _1=\\lbrace (x,y)\\in \\Omega :\\, y>x\\rbrace $ and $\\Omega _2=\\lbrace (x,y)\\in \\Omega :\\, y<x\\rbrace $ .", "The exact solution of the test problem is $u(x,y) =f(x,y)$ with the discontinuous interface: $y=x$ .", "The LSNN method is implemented with $\\rho =h/2$ and a fixed learning rate 0.003 with 20000 iterations for two networks: 2-4-1 and 2-6-1.", "The numerical results are presented in Table REF which imply that the 2-4-1 network fails to accurately approximate the solution.", "Figure REF shows the NN approximation of the 2-6-1 network.", "The traces of the exact and numerical solutions on the plane $y=-x$ are depicted in Fig.", "REF (b).", "Clearly, the LSNN method with only 19 parameters approximates the exact solution accurately without the Gibbs phenomena.", "This test problem shows that the LSNN method is able to rotate and shift the initial breaking lines to approximate the discontinuous interface.", "Table: Relative errors of the problem with discontinuity along the diagonalFigure: Network breaking linesThe third test problem is a modification of the second test problem by changing the inflow boundary condition from the piece-wise constant to a discontinuous piece-wise smooth function and the domain from $\\Omega =(-1,1)^2$ to $\\Omega =(0,1)^2$ , i.e., $g(x,y)=\\left\\lbrace \\begin{array}{ll}\\sin (y), & (x,y)\\in \\Gamma ^1_-=\\lbrace (0,y):\\, y \\in (0,1)\\rbrace , \\\\[2mm]\\cos (x), & (x,y)\\in \\Gamma ^2_-=\\lbrace (x,0):\\, x \\in (0,1)\\rbrace .\\end{array}\\right.$ Set $\\gamma =f=0$ , the exact solution of this test problem is $ u(x,y) = \\left\\lbrace \\begin{array}{ll}\\sin (y-x), & (x,y)\\in \\Omega _1=\\lbrace (x,y)\\in (0,1)^2:\\, y>x\\rbrace , \\\\[2mm]\\cos (x-y), & (x,y)\\in \\Omega _2=\\lbrace (x,y)\\in (0,1)^2:\\, y<x\\rbrace .\\end{array}\\right.$ The LSNN method is employed with $\\rho =h/2$ and a fixed learning rate 0.003 for 30000 iterations.", "Numerical results of three network models are reported in Table REF and the first two models fail to approximate the solution well.", "Figure REF presents the NN approximation of the 2-40-1 network.", "The traces of the exact and numerical solutions on the plane $y=1-x$ are depicted in Fig.", "REF (b), which exhibits no oscillation.", "It is expected that the network with additional neurons is needed in order to approximate the solution well since the solution of the test problem is a piece-wise smooth function.", "Moreover, this test problem conforms Theorem 3.3 that a piece-wise smooth function having a constant jump along a line segment discontinuous interface may be approximated well by a two-layer network.", "Table: Relative errors of the problem with a piece-wise smooth solutionFigure: Network breaking lines" ], [ "Problem with two discontinuous interfaces", "The fourth test problem is again a modification of the second test problem by changing the domain to $\\Omega =(-1,1)\\times (0,1)$ , the inflow boundary condition to a combination of jumps and smooth function $g(x,y)=\\left\\lbrace \\begin{array}{ll}\\sin \\left(\\frac{\\pi (x-y+0.9)}{0.3}\\right), & (x,y)\\in \\Gamma ^1_-=\\lbrace (x,0):\\, x \\in (-0.9,-0.6)\\rbrace , \\\\[2mm]-1, & (x,y)\\in \\Gamma ^2_-=\\lbrace (x,0):\\, x \\in (-0.2,0.1)\\rbrace , \\\\[2mm]0, & (x,y)\\in \\Gamma _- \\setminus (\\Gamma ^1_-\\cup \\Gamma ^2_-)\\end{array}\\right.$ with the inflow boundary $\\Gamma _- = \\lbrace (x,0):\\, x \\in (-1,1)\\rbrace \\cup \\lbrace (-1,0)\\rbrace \\cup \\lbrace (-1,y):\\, y \\in (0,1)\\rbrace .$ Set $f$ as $f(x,y)=\\left\\lbrace \\begin{array}{ll}\\sin \\left(\\frac{\\pi (x-y+0.9)}{0.3}\\right), & (x,y)\\in \\Upsilon _1=\\lbrace (x,y)\\in \\Omega :\\, -0.9<x-y <-0.6\\rbrace , \\\\[2mm]-1, & (x,y)\\in \\Upsilon _2=\\lbrace (x,y)\\in \\Omega :\\, -0.2<x-y <0.1\\rbrace , \\\\[2mm]0, & (x,y)\\in \\Omega \\setminus (\\Upsilon _1\\cup \\Upsilon _2),\\end{array}\\right.$ then the exact solution of the test problem is $u(x,y)=f(x,y)$ with two discontinuous interfaces $y=x+0.2$ and $y=x-0.1$ , respectively.", "The LSNN method is implemented with $\\rho = h/2$ and an adaptive learning rate which starts with 0.01 and decreases by 0.002 for every 20000 iterations.", "The total number of iterations is 80000.", "We observed from the experiment that adding a weight $\\alpha $ to the inflow boundary loss in (REF ) is helpful for the training.", "Empirically, we choose $\\alpha =10$ and report the numerical results for three respective network structures in Table REF .", "The results suggest that the first 2-20-1 network model fails to approximate the solution well due to the possibility of training and/or insufficient number of neurons.", "Starting with a 2-30-1 network and applying the adaptive neuron enhancement strategy [21] once, the 2-34-1 network provides an accurate approximation (see Table REF and Figure REF ).", "The traces of the exact and numerical solutions are depicted on the plane $y=0.8$ in Figure REF (c).", "This test problem shows that the LSNN method using a small number of DoF is capable of approximating a discontinuous solution containing a smooth extrema without oscillations.", "Table: Relative errors of the problem with two discontinuous interfacesFigure: Network breaking lines" ], [ "Problem with a piece-wise constant advection velocity field", "The fifth test problem is equation (REF ) defined on $\\Omega =(0,1)^2$ with $\\gamma =f =0$ and a piece-wise constant advection velocity field.", "Specifically, the advection velocity field is given by ${\\beta } =\\left\\lbrace \\begin{array}{rclll}&(1-\\sqrt{2},1)^T, & (x,y)\\in \\Upsilon _1=\\lbrace (x,y)\\in \\Omega :\\, y<x\\rbrace , \\\\[2mm]&(-1,\\sqrt{2}-1)^T, & (x,y)\\in \\Upsilon _2=\\lbrace (x,y)\\in \\Omega :\\, y\\ge x\\rbrace .\\end{array}\\right.$ and, hence, the inflow boundary of the problem is $\\Gamma _-=\\lbrace (x,0):\\, x\\in (0,1)\\rbrace \\cup \\lbrace (1,0)\\rbrace \\cup \\lbrace (1,y):\\, y\\in (0,1)\\rbrace .$ Let $\\Gamma ^1_-=\\lbrace (x,0): x\\in (0, a)\\rbrace $ with $a=43/64$ .", "For the inflow boundary condition $g(x,y)=\\left\\lbrace \\begin{array}{rl}-1,& (x,y)\\in \\Gamma ^1_-, \\\\[2mm]1, & (x,y)\\in \\Gamma ^2_-=\\Gamma _-\\setminus \\Gamma _-^1,\\end{array}\\right.$ the exact solution is a piece-wise constant: $u=-1$ in $\\Omega _1$ and $u=1$ in $\\Omega _2$ , where $\\Omega _2=\\Omega \\setminus \\bar{\\Omega }_1$ and $\\Omega _1=\\lbrace {\\bf x}\\in \\Upsilon _1: \\mbox{$\\xi $}_1 \\cdot {\\bf x}< a\\rbrace \\cup \\lbrace {\\bf x}\\in \\Upsilon _2: \\mbox{$\\xi $}_2 \\cdot {\\bf x}< a\\rbrace .$ Here, $\\mbox{$\\xi $}_1=(1, \\sqrt{2}-1)^T$ and $\\mbox{$\\xi $}_2=(\\sqrt{2}-1, 1)^T$ are vectors normal to ${\\beta }|_{_{\\Upsilon _1}}$ and ${\\beta }|_{_{\\Upsilon _2}}$ , respectively.", "The LSNN method with $\\rho = h/2$ and a fixed learning rate 0.003 with 50000 iterations is implemented for networks: 2-30-1, 2-200-1, and 2-5-5-1.", "Initialization of the first layer is done by the approach described in section 4, and that of the subsequent layers are randomly generated.", "The numerical results are presented in Table REF and Figure REF , and the figures of the two-layer network is for the 2-200-1 model.", "The traces of the exact and numerical solutions on the plane $x=0$ and the breaking lines of these two networks are depicted in Fig.", "REF (c,d) and Fig.", "REF (e,f), respectively.", "Clearly, the two-layer network with 200 neurons (over 600 parameters) fails to approximate the solution well in average (see Table REF ) and point-wise (see Figure REF ).", "A three-layer network with less than 8% of parameters outperforms this large two-layer network in every aspects including breaking lines.", "Comparing these two networks, a three-layer network is more suitable than a two-layer network to accurately approximate the solution having a constant jump along a piece-wise line segment discontinuous interface.", "Due to the random generation of some parameters, the training of 2-5-5-1 network is replicated five times and the best result is reported.", "We observe from the training process that the network may get trapped in a local minimum and fails to accurately approximate the solution.", "To address such issue, we introduce an adaptive process in [6] for obtaining a good initialization which is crucial for nonlinear optimization problems.", "Table: Relative errors of the problem with a piece-wise constant advection velocity fieldFigure: 3-layer NN breaking linesBelow we show theoretically that a three-layer neural network is sufficient for approximating the solution well (see Lemma 5.1 below).", "To make it slightly general, let $\\chi =\\left\\lbrace \\begin{array}{ll}\\alpha _1, & {\\bf x}\\in \\Omega _1,\\\\[2mm]\\alpha _2, & {\\bf x}\\in \\Omega _2.\\end{array}\\right.$ Without loss of generality, assume that $\\alpha _1 < \\alpha _2$ .", "Let $p_1({\\bf x})$ and $p_2({\\bf x})$ be two-layer neural network functions given by $p_i({\\bf x})=\\alpha _1 +\\dfrac{\\alpha _2-\\alpha _1}{2\\varepsilon } \\Big (\\sigma (\\mbox{$\\xi $}_i\\cdot {\\bf x}-a + \\varepsilon ) - \\sigma (\\mbox{$\\xi $}_i\\cdot {\\bf x}-a-\\varepsilon )\\Big )$ for any $\\varepsilon >0$ such that intersections between the domain $\\Omega $ and the hyper-planes $\\mbox{$\\xi $}_i\\cdot {\\bf x}=a\\pm \\varepsilon $ are not empty.", "Let $p({\\bf x})=\\max \\lbrace p_1({\\bf x}),p_2({\\bf x})\\rbrace $ , then we have $\\Vert \\chi - p\\Vert _{0,\\Omega }=\\left(\\Vert \\chi - p\\Vert ^2_{0,\\Omega } + \\Vert \\chi _{\\scriptsize {\\beta }} - p_{\\scriptsize {\\beta }}\\Vert ^2_{0,\\Omega }\\right)^{1/2}\\le \\sqrt{\\dfrac{2}{3}}\\,D^{(d-1)/2} \\big |\\alpha _1-\\alpha _2\\big |\\, \\sqrt{\\varepsilon },$ where $D$ is the diameter of the domain $\\Omega $ .", "Since $p({\\bf x})=p_i({\\bf x})$ in $\\Upsilon _i$ for $i=1,\\,2$ and $\\Omega =\\Upsilon _1\\cup \\Upsilon _2$ , we have $\\Vert \\chi - p\\Vert _{0,\\Omega }^2=\\Vert \\chi - p_1\\Vert _{0,\\Upsilon _1}^2+\\Vert \\chi - p_2\\Vert _{0,\\Upsilon _2}^2.$ Combining with the fact that $\\chi _{\\scriptsize {\\beta }} - p_{\\scriptsize {\\beta }}= 0$ in $\\Omega $ , (REF ) is then a direct consequence of Lemma 3.2.", "Similar as the discussion in [14], the maximum operation can be constructed by using an additional hidden layer of the ReLU network with 4 neurons: $\\max \\lbrace a,b\\rbrace = \\dfrac{a+b}{2}+\\dfrac{|a-b|}{2} = {\\bf v}\\, \\sigma \\left(\\mbox{$\\omega $}\\left[\\!\\!\\begin{array}{l}a\\\\b\\end{array}\\!\\!\\right]\\right)$ where the row vector and the $4\\times 2$ matrix are given by $ {\\bf v}= \\dfrac{1}{2}\\,[1,-1,1,1] \\quad \\text{and} \\quad \\mbox{$\\omega $}=\\left[\\begin{array}{rr}1 & 1 \\\\-1 & -1 \\\\1 & -1 \\\\-1 & 1 \\\\\\end{array}\\right],$ respectively.", "Then this lemma indicates that a three-layer neural network is sufficient when the interface consists of two line segments.", "In a similar fashion, a three-layer network can be constructed to approximate the solution with the interface consisting of more than two line segments." ], [ "Problem with a variable advection velocity field", "The last test problem is equation (REF ) defined on the domain $\\Omega =(0,1)^2$ with a variable advection velocity field ${\\beta } = (-y,x)^T$ and $\\gamma =f=0$ (see, e.g., [2], [23]).", "With the inflow boundary condition $g$ given in (REF ), the exact solution is a piece-wise constant given by $u(x,y) = \\left\\lbrace \\begin{array}{rl}-1,& (x,y)\\in \\Omega _1, \\\\[2mm]1,& (x,y)\\in \\Omega _2,\\end{array}\\right.$ where $\\Omega _1=\\lbrace (x,y)\\in \\Omega :\\, x^2+y^2<a^2\\rbrace $ and $\\Omega _2=\\lbrace (x,y)\\in \\Omega :\\, x^2+y^2 >a^2\\rbrace $ .", "For the LSNN method, again we use a uniform integration mesh ${\\cal T}$ with the mesh size $h=10^{-2}$ ; the finite difference quotient in (REF ) is calculated with $\\rho =h/10$ to avoid using values on both sides of the interface.", "Instead of intricately choosing the $\\rho $ value, a robust approach will be developed in a forthcoming paper.", "Besides, the parameters are initialized by the method described in section 4 for the first layer and randomly for the subsequent layers.", "The learning rate starts with 0.005, and is reduced by half for every 50000 iterations.", "This learning rate decay strategy is used with 150000 iterations.", "Due to the random initialization of some parameters, numerical experiments are replicated three times and the best results for the three- and four-layer networks are reported in Table REF and Figure REF .", "The traces of the exact and numerical solutions at the plane $x=0$ are depicted in Fig.", "REF (b) and (c) for the respective three- and four-layer networks.", "As shown in Fig.", "REF (b), the LSNN approximation of the three-layer network with 40 neurons at each layer smears the discontinuity.", "A careful examination of the iterative process, it seems to us that the smear is due to the initialization (see Fig.", "REF ).", "Table: Relative errors of the problem with a variable advection velocity fieldFigure: 4-layer NN cross section on x=0x=0" ], [ "Method of model continuation", "As observed from our numerical experiments for the test problem with a curved discontinuous interface, initial of the parameters plays an important role in training neural networks.", "This is because the high dimensional nonlinear optimization usually have many solutions.", "Without a good initial, our previous simulations rely on over-parameterized neural networks to approximate the underlying problem well.", "The strategy of over-parameterization is computationally expensive.", "Based on our numerical experiments in the previous sections, to generate a good initial for the parameters, we introduce the method of continuation through models for the advection-reaction problem in (REF ) with a variable advection velocity field ${\\beta }({\\bf x})$ .", "To this end, let $\\lbrace {\\beta }_n({\\bf x})\\rbrace $ be a sequence of piece-wise constant vector fields.", "Consider the following advection-reaction problem with the advection velocity field ${\\beta }_n({\\bf x})$ : $\\left\\lbrace \\begin{array}{rccl}(u_n)_{{\\beta }_n} + \\hat{\\gamma }\\, u_n &= & f, &\\text{ in }\\,\\, \\Omega , \\\\u_n&=&g, &\\text{ on }\\,\\, \\Gamma _{-}.\\end{array}\\right.$ Let $u$ be the solution of (REF ), it is easy to see that $u-u_n$ satisfies $\\left\\lbrace \\begin{array}{rccl}(u-u_n)_{{\\beta }_n} + \\hat{\\gamma }\\, (u-u_n) &= & u_{{\\beta }_n}-u_{{\\beta }}, &\\text{ in }\\,\\, \\Omega , \\\\u-u_n&=&0, &\\text{ on }\\,\\, \\Gamma _{-},\\end{array}\\right.$ which, together with the stability estimate in (REF ), implies $\\Vert u-u_n\\Vert _{0,\\Omega }\\le {\\left|\\hspace{-1.0625pt}\\left|\\hspace{-1.0625pt}\\left|u-u_n\\right|\\hspace{-1.0625pt}\\right|\\hspace{-1.0625pt}\\right|}_{{\\beta }_n} \\le C\\, \\Vert u_{{\\beta }_n}-u_{{\\beta }}\\Vert _{0,\\Omega } =C\\,\\left(\\int _\\Omega \\big (({\\beta }_n-{\\beta })\\cdot \\nabla u\\big )\\,d{\\bf x}\\right)^{1/2}.$ Hence, if ${\\beta }_n$ is a good approximation to ${\\beta }$ , then $u_n$ is a good approximation to $u$ .", "This indicates that (REF ) provides a continuation process on the parameter $n$ for (REF ).", "For the test problem in section 5.4, since streamlines of the advection velocity field ${\\beta }=(-y,x)^T$ are quarter circles in $\\Omega =(0,1)^2$ oriented counterclockwise, it is natural to approximate the quarter-circle by $n$ line segments.", "To this end, let $t_i=\\dfrac{i\\pi }{2n}$ for $i=0,1,...,n$ and $\\Upsilon _{i+1}=\\lbrace (x,y)\\in \\Omega :\\, (\\sin t_i)x < (\\cos t_i)y\\,\\,\\mbox{ and }\\,\\, (\\sin t_{i+1})x \\ge (\\cos t_{i+1})y\\rbrace .$ Then $\\lbrace \\Upsilon _{i+1}\\rbrace _{i=0}^{n-1}$ forms a partition of $\\Omega $ (see Fig.", "REF for $n=4$ ).", "This type of approximations leads to ${\\beta }_n = (\\cos t_{i+1}-\\cos t_{i}, \\sin t_{i+1}-\\sin t_{i})^T\\quad \\mbox{in }\\Upsilon _{i+1}$ for $i=0,1,...,n-1$ .", "Note that ${\\beta }_2$ is the same vector field given in (REF ).", "Hence, (REF ) with $n=2$ and the test problem in section 5.3 are the same.", "The method of model continuation starts with a three-layer neural network (2-5-5-1) to approximate $u_2$ (see the third row of Table REF and Fig.", "REF (b,d)).", "This trained network is used as an initial for the parameters in the hidden layers of the 2-6-6-1 network to approximate $u_3$ by randomly generated the parameters of new neurons.", "The initial for the output weights and bias may be chosen as the solution of the system (REF ).", "The adaptive learning rate strategy which starts with 0.01 and decays by 20% for every 50000 iterations is implemented with the method.", "The networks for $u_n$ with $n=4,5$ and for the test problem in section 5.4 are initialized sequentially in a similar fashion.", "Numerical results for approximating $u_n$ and $u$ are reported in Table REF , and the traces of the exact and numerical solutions at the plane $x=0$ are depicted in Fig.", "REF .", "The third and fourth columns show that the difficulty of the corresponding problems increase as the number of line segments increase.", "The fifth column shows that $u_n$ approaches to $u$ monotonously.", "Comparing Table 5 with the last row of Table 6, it is clear that the method of model continuation is capable of reducing the size of the network significantly.", "Figure: Discontinuous interfaceTable: Relative errors of the problem with discontinuity along line segmentsFigure: Breaking lines of the original problem ()" ], [ "Discussions and Conclusions", "We proposed the LSNN method for solving the linear advection-reaction problem.", "The least-squares formulation, based on a direct application of the least-squares principle to the underlying problem, does not require additional smoothness of the solution if $f\\in L^2(\\Omega )$ .", "In the $V_{\\beta }$ norm, the LSNN approximation is proved to be quasi-optimal, i.e., the error of the LSNN approximation is bounded above by the approximation error of the network.", "A major challenge in numerical simulation of hyperbolic partial differential equations is the discontinuity of their solutions.", "For the linear transport problem in two dimensions, by decomposing the discontinuous solution into the sum of a piece-wise constant function and a continuous piece-wise smooth function, we are able to show theoretically and numerically that the LSNN method using a (at most) three-layer ReLU neural network is capable of approximating the discontinuous solution accurately without oscillation.", "In particular, the piece-wise constant solution can be approximated well by a ReLU network with a small number of neurons.", "Numerical results presented in section 5 show that it is important to use a proper neural network in order to accurately approximate the solution of the underlying problem with fewer parameters.", "How to automatically design such a proper network, in terms of their width and depth, is an open and fundamental question for numerically solving partial differential equations within the prescribed accuracy.", "Following our recent paper on adaptive neuron enhancement method [21], this will be addressed in the forthcoming paper.", "The procedure of training the value of the parameters is a problem in non-convex optimization which usually has many solutions and are complicated and computationally demanding.", "In order to obtain a desired solution, we introduced a method of model continuation for providing a good first approximation.", "Numerical results show that this method is effective for reducing the number of the parameters of the network.", "Moreover, a good initial is very helpful in training as well.", "Nevertheless, training is still a challenging problem since the learning rate of the methods of the gradient type is difficult to tune.", "A reasonably good learning rate can only be discovered through the method of trial and error.", "Using NNs to solve PDEs is relatively new, developing fast solvers is an open and challenging problem and requires lots of efforts from numerical analysts.", "Because of its great potential and many difficulties at the same time, machine learning is a hot research topic in scientific computing." ] ]
2105.11632
[ [ "SHAFF: Fast and consistent SHApley eFfect estimates via random Forests" ], [ "Abstract Interpretability of learning algorithms is crucial for applications involving critical decisions, and variable importance is one of the main interpretation tools.", "Shapley effects are now widely used to interpret both tree ensembles and neural networks, as they can efficiently handle dependence and interactions in the data, as opposed to most other variable importance measures.", "However, estimating Shapley effects is a challenging task, because of the computational complexity and the conditional expectation estimates.", "Accordingly, existing Shapley algorithms have flaws: a costly running time, or a bias when input variables are dependent.", "Therefore, we introduce SHAFF, SHApley eFfects via random Forests, a fast and accurate Shapley effect estimate, even when input variables are dependent.", "We show SHAFF efficiency through both a theoretical analysis of its consistency, and the practical performance improvements over competitors with extensive experiments.", "An implementation of SHAFF in C++ and R is available online." ], [ "Introduction", "State-of-the-art learning algorithms are often qualified as black-boxes because of the high number of operations required to compute predictions.", "This complexity prevents to grasp how inputs are combined to generate the output, which is a strong limitation for many applications, especially those with critical decisions at stake—healthcare is a typical example.", "For this reason, interpretability of machine learning has become a topic of strong interest in the past few years.", "One of the main tools to interpret learning algorithms is variable importance, which enables to identify and rank the influential features of the problem.", "Recently, Shapley effects have been widely accepted as a very efficient variable importance measure since they can equitably handle interactions and dependence within input variables [19], [24], [12], [15].", "Shapley values were originally defined in economics and game theory [22] to solve the problem of attributing the value produced by a joint team to its individual members.", "The main idea is to measure the difference of produced value between a subset of the team and the same subteam with an additional member.", "For a given member, this difference is averaged over all possible subteams and gives his Shapley value.", "Recently, [19] adapted Shapley values to the problem of variable importance in machine learning, where an input variable plays the role of a member of the team, and the produced value is the explained output variance.", "In this context, Shapley values are now called Shapley effects, and are extensively used to interpret both tree ensembles and neural networks.", "Next, [15] also introduced SHAP values to adapt Shapley effects to local importance measures, which break down the contribution of each variable for a given prediction.", "We focus on Shapley effects throughout the article, but our approach can be easily adpated to SHAP values as they share the same challenges.", "The objective of variable importance is essentially to perform variable selection.", "More precisely, it is possible to identify two final aims [9]: (i) find a small number of variables with a maximized accuracy, or (ii) detect and rank all influential variables to focus on for interpretation and further exploration with domain experts.", "The following example illustrates that different strategies should be used depending on the targeted objective: if two influential variables are strongly correlated, one must be discarded for objective (i), while the two must be kept in the second case.", "Indeed, if two variables convey the same statistical information, only one should be selected if the goal is to maximize the predictive accuracy with a small number of variables, i.e., objective (i).", "On the other hand, these two variables may be acquired differently and represent distinct physical quantities.", "Therefore, they may have different interpretations for domain experts, and both should be kept for objective (ii).", "Shapley effects are a relevant measure of variable importance for objective (ii), because they equitably allocate contributions due to interactions and dependence across all input variables.", "The main obstacle to estimate Shapley effects is the computational complexity.", "The first step is to use a learning algorithm to generalize the relation between the inputs and the output.", "Most existing Shapley algorithms are agnostic to the learning model.", "[16] open an interesting route by restricting their algorithm to tree ensembles, in order to develop fast greedy heuristics, specific to trees.", "Unfortunately, as mentioned by [1], the algorithm is biased when input variables are dependent.", "In the present contribution, we focus our Shapley algorithm on random forests, well known for their good behavior on high-dimensional or noisy data, and their robustness.", "Using the specific structure of random forests, we develop SHAFF, a fast and accurate Shapley effect estimate." ], [ "Shapley effects.", "To formalize Shapley effects, we introduce a standard regression setting with an input vector $\\textbf {X}= (X^{(1)}, \\hdots , X^{(p)}) \\in {R}^p$ , and an output $Y \\in {R}$ .", "We denote by $\\smash{\\textbf {X}^{(U)}}$ the subvector with only the components in $U \\subset \\lbrace 1,\\hdots ,p\\rbrace $ .", "Formally, the Shapley effect of the $j$ -th variable is defined by $Sh^{\\star }(X^{(j)}) = \\sum _{U \\subset \\lbrace 1,\\hdots ,p\\rbrace \\setminus \\lbrace j\\rbrace } \\frac{1}{p} {p - 1 \\atopwithdelims ()|U|}^{-1} \\frac{\\mathbb {E}[Y|\\textbf {X}^{(U\\cup \\lbrace j\\rbrace )}]] - \\mathbb {E}[Y|\\textbf {X}^{(U)}]]}{\\mathbb {V}[Y]}.$ In other words, the Shapley effect of $X^{(j)}$ is the additional output explained variance when $j$ is added to a subset $U \\subset \\lbrace 1,\\hdots ,p\\rbrace $ , averaged over all possible subsets.", "The variance difference is averaged for a given size of $U$ through the combinatorial weight, and then the average is taken over all $U$ sizes through the term $1/p$ .", "Observe that the sum has $2^{p-1}$ terms, and each of them requires to estimate $\\smash{\\mathbb {E}[Y|\\textbf {X}^{(U)}]]}$ , which is computationally costly.", "Overall, two obstacles arise to estimate Shapley effects: the computational complexity is exponential with the dimension $p$ ; $\\mathbb {E}[Y|\\textbf {X}^{(U)}]]$ requires a fast and accurate estimate for all variable subsets $U \\subset \\lbrace 1,\\hdots ,p\\rbrace $ .", "In the literature, efficient strategies have been developed to handle these two issues.", "They all have drawbacks: they are either fast but with a limited accuracy, or accurate but computationally costly.", "We will see how SHAFF considerably improves this trade-off." ], [ "Related work.", "The computational issue of Shapley algorithms—$1.$ above—is solved using Monte-Carlo methods in general [23], [15], [6], [29], [7].", "In the case of tree ensembles, specific heuristics based on the tree structure enable to simplify the algorithm complexity [16].", "For the second issue of conditional expectation estimates—$2.$ above, two main approaches exist: train one model for each selected subset of variables (accurate but computationally costly) [29], or train a single model once with all input variables and use greedy heuristics to derive the conditional expectations (fast but limited accuracy).", "In the latter case, existing algorithms estimate the conditional expectations with a quite strong bias when input variables are dependent.", "More precisely, [15], [6], and [7] simply replace the conditional expectations by the marginal distributions, [16] use a greedy heuristic specific to tree ensembles, and [5] leverage $k$ -nearest neighbors to approximate sampling from the conditional distributions.", "Besides, efficient algorithms exist when it is possible to draw samples from the conditional distributions of the inputs [23], [1], [5].", "However, we only have access to a finite sample in practice, and the input dimension $p$ can be large, which implies that estimating the conditional distributions of the inputs is a very difficult task.", "This last type of methods is therefore not really appropriate in our setting—see Table REF for a summary of the existing Shapley algorithms.", "Table: State-of-the-art of Shapley algorithms.As mentioned above, several of the presented methods provide local importance measures for specific prediction points, called SHAP values [15], [16], [7].", "Their final objective differs from ours, since we are interested in global estimates.", "However, SHAP values share the same challenges as Shapley effects: the computational complexity and the conditional expectation estimates, and our approach can therefore be adapted to SHAP values.", "Let us also mention that several recent articles discuss Shapley values in the causality framework [8], [11], [13], [28].", "These works have a high potential since causality is quite often the ultimate goal when one is looking for interpretations.", "However, causality methods require strong prior knowledge and assumptions about the studied system, and can therefore be difficult to apply in some applications.", "In these cases, we argue that the best way to go is to use standard Shapley effects to detect and rank influential variables, as a starting point to deepen the analysis with domain experts." ], [ "Outline.", "We leverage random forests to develop SHAFF, a fast and accurate Shapley effect estimate.", "Such remarkable performance is reached by combining two new features.", "Firstly, we improve the Monte-Carlo approach by using importance sampling to focus on the most relevant subsets of variables identified by the forest.", "Secondly, we develop a projected random forest algorithm to compute fast and accurate estimates of the conditional expectations for any variable subset.", "The algorithm details are provided in Section .", "Next, we prove the consistency of SHAFF in Section .", "To our knowledge, SHAFF is the first Shapley effect estimate, which is both computationally fast and consistent in a general setting.", "In Section , several experiments show the practical improvement of our method over state-of-the-art algorithms." ], [ "Existing approach.", "SHAFF builds on two Shapley algorithms: [15] and [29].", "From these approaches, we can deduce the following general three-step procedure to estimate Shapley effects.", "First, a set $\\mathcal {U}_{n,K}$ of $K$ variable subsets $U \\subset \\lbrace 1,\\hdots ,p\\rbrace $ is randomly drawn.", "Next, an estimate $\\hat{v}_n(U)$ of $\\smash{\\mathbb {E}[Y|\\textbf {X}^{(U)}]]}$ is computed for all selected $U$ from an available sample ${D}_{n}= \\lbrace (\\textbf {X}_1, Y_1), \\hdots , (\\textbf {X}_n, Y_n) \\rbrace $ of $n$ independent random variables distributed as $(\\textbf {X}, Y)$ .", "Finally, Shapley effects are defined as the least square solution of a weighted linear regression problem.", "If $I(U)$ is the binary vector of dimension $p$ where the $j$ -th component takes the value 1 if $j \\in U$ and 0 otherwise, Shapley effect estimates are the minimum in $\\beta $ of the following cost function: $\\ell _{n}(\\beta ) = \\frac{1}{K} \\sum _{U \\in \\mathcal {U}_{n,K}} w(U) (\\hat{v}_{n}(U) - \\beta ^T I(U))^2,$ where the weights $w(U)$ are given by $w(U) = \\frac{p - 1}{{p \\atopwithdelims ()|U|} |U| (p - |U|)},$ and the coefficient vector $\\beta $ is constrained to have its components sum to $\\hat{v}_{n}(\\lbrace 1,\\hdots ,p\\rbrace )$ ." ], [ "Algorithm overview.", "SHAFF introduces two new critical features to estimate Shapley effects efficiently, using an initial random forest model.", "Firstly, we apply importance sampling to select variable subsets $U \\subset \\lbrace 1,\\hdots , p\\rbrace $ , based on the variables frequently selected in the forest splits.", "This favors the sampling of subsets $U$ containing influential and interacting variables.", "Secondly, for each selected subset $U$ , the variance of the conditional expectation is estimated with the projected forest algorithm described below, which is both a fast and consistent approach.", "We will see that these features considerably reduce the computational cost and the estimate error.", "To summarize, once an initial random forest is fit, SHAFF proceeds in three steps: sample many subsets $U$ , typically a few hundreds, based on their occurrence frequency in the random forest (Subsection REF ); estimate $\\mathbb {E}[Y|\\textbf {X}^{(U)}]]$ with the projected forest algorithm for all selected $U$ and their complementary sets $\\lbrace 1,\\hdots , p\\rbrace \\setminus U$ (Subsection REF ); solve a weighted linear regression problem to recover Shapley effects (Subsection REF )." ], [ "Initial random forest.", "Prior to SHAFF, a random forest is fit with the training sample ${D}_{n}$ to generalize the relation between the inputs $\\textbf {X}$ and the output $Y$ .", "A large number $M$ of CART trees are averaged to form the final forest estimate $m_{M,n}(\\textbf {x}, \\Theta _{M})$ , where $\\textbf {x}$ is a new query point, and each tree is randomized by a component of $\\Theta _{M} = (\\Theta _1,\\hdots ,\\Theta _{\\ell },\\hdots ,\\Theta _M)$ .", "Each $\\smash{\\Theta _{\\ell }}$ is used to bootstrap the data prior to the $\\ell $ -th tree growing, and to randomly select mtry variables to optimize the split at each node.", "mtry is a parameter of the forest, and its efficient default value is $p/3$ .", "In the sequel, we will need the forest parameter min_node_size, which is the minimum number of observations in a terminal cell of a tree, as well as the out-of-bag (OOB) sample of the $\\ell $ -th tree: the observations which are left aside in the bootstrap sampling prior to the construction of tree $\\ell $ .", "Given this initial random forest, we can now detail the main three steps of SHAFF." ], [ "Importance Sampling", "The Shapley effect formula for a given variable $X^{(j)}$ sums terms over all subsets of variables $U \\subset \\lbrace 1,\\hdots ,p\\rbrace \\setminus \\lbrace j\\rbrace $ , which makes $2^{p-1}$ terms, an intractable problem in most cases.", "SHAFF uses importance sampling to draw a reasonable number of subsets $U$ , typically a few hundreds, while preserving a high accuracy of the Shapley estimates.", "We take advantage of the initial random forest to define an importance measure for each variable subset $U$ , used as weights for the importance sampling distribution." ], [ "Variable subset importance.", "In a tree construction, the best split is selected at each node among mtry input variables.", "Therefore, as highlighted by Proposition 1 in [21], the forest naturally splits on influential variables.", "SHAFF leverages this idea to define an importance measure for all variable subsets $U \\subset \\lbrace 1,\\hdots ,p\\rbrace $ as the probability that a given $U$ occurs in a path of a tree of the forest.", "Empirically, this means that we count the occurrence frequency of $U$ in the paths of the $M$ trees of the forest, and denote it by $\\hat{p}_{M,n}(U)$ .", "Such approach is inspired by [2] and [3].", "This principle is illustrated with the following simple example in dimension $p = 10$ .", "Let us consider a tree, where the root node splits on variable $\\smash{X^{(5)}}$ , the left child node splits on variable $\\smash{X^{(3)}}$ , and the subsequent left child node at the third tree level, on variable $\\smash{X^{(2)}}$ .", "Thus, the path that leads to the extreme left node at the fourth level uses the following index sequence of splitting variables: $\\lbrace 5, 3, 2\\rbrace $ .", "All in all, the following variable subsets are included in this tree path: $U = \\lbrace 5\\rbrace $ , $U = \\lbrace 3, 5\\rbrace $ , and $U = \\lbrace 2, 3, 5\\rbrace $ .", "Then, SHAFF runs through the forest to count the number of times each subset $U$ occurs in the forest paths, and computes the associated frequency $\\hat{p}_{M,n}(U)$ .", "If a subset $U$ does not occur in the forest, we obviously have $\\hat{p}_{M,n}(U) = 0$ .", "Notice that the computational complexity of this step is linear: $O(Mn)$ ." ], [ "Paired importance sampling.", "The occurrence frequencies $\\hat{p}_{M,n}(U)$ defined above are scaled to sum to 1, and then define a discrete distribution for the set of all subsets of variables $U \\subset \\lbrace 1,\\hdots ,p\\rbrace $ , excluding the full and empty sets.", "By construction, this distribution is skewed towards the subsets $U$ containing influential variables and interactions, and is used for the importance sampling.", "Finally, SHAFF draws a number $K$ of subsets $U$ with respect to this discrete distribution, where $K$ is a hyperparameter of the algorithm.", "We define $\\mathcal {U}_{n,K}$ the random set of the selected variable subsets $U$ .", "For all $U \\in \\mathcal {U}_{n,K}$ , SHAFF also includes the complementary set $\\lbrace 1,\\hdots , p\\rbrace \\setminus U$ in $\\mathcal {U}_{n,K}$ , as [7] show that this “paired sampling” improves the final Shapley estimate accuracy.", "Clearly, the computational complexity and the accuracy of the algorithm increase with $K$ .", "The next step of SHAFF is to efficiently estimate $\\smash{\\mathbb {E}[Y|\\textbf {X}^{(U)}]]}$ for all drawn $U \\in \\mathcal {U}_{n,K}$ ." ], [ "Projected Random Forests", "In order to estimate $\\mathbb {E}[Y|\\textbf {X}^{(U)}]]$ for the selected variable subsets $U \\in \\mathcal {U}_{n,K}$ , most existing methods use greedy algorithms.", "However, such estimates are not accurate in moderate or large dimensions when input variables are dependent [1], [25].", "Another approach is to train a new model for each subset $U$ , but this is computationally costly [29].", "To solve this issue, we design the projected random forest algorithm (PRF), to obtain a fast and accurate estimate of $\\smash{\\mathbb {E}[Y|\\textbf {X}^{(U)}]]/Y]}$ for any variable subset $U \\subset \\lbrace 1,\\hdots ,p\\rbrace $ ." ], [ "PRF principle.", "PRF takes as inputs the initial forest and a given subset $U$ .", "The general principle is to project the partition of each tree of the forest on the subspace spanned by the variables in $U$ , as illustrated in Figure REF .", "Then the training data is spread across this new tree partitions, and the cell outputs are recomputed by averaging the output $Y_i$ of the observations falling in each new cell, as in the original forest.", "The projection enables to eliminate the variables not contained in $U$ from the tree predictions, and thus to estimate $\\smash{\\mathbb {E}[Y|\\textbf {X}^{(U)}]}$ instead of $\\smash{\\mathbb {E}[Y|\\textbf {X}]}$ .", "Finally, the predictions for the out-of-bag samples are computed with the projected tree estimates, and averaged across all trees.", "The obtained predictions are used to estimate the targeted normalized variance $\\smash{\\mathbb {E}[Y|\\textbf {X}^{(U)}]]/Y]}$ , denoted by $\\hat{v}_{M,n}(U)$ .", "More formally, we let $\\smash{m_{M,n}^{(U, OOB)}(\\textbf {X}_i^{(U)}, \\Theta _M)}$ be the out-of-bag PRF estimate for observation $i$ and subset $U$ , and take $\\hat{v}_{M,n}(U) = 1 - \\frac{1}{n \\hat{\\sigma }_Y} \\sum _{i=1}^{n} \\big (Y_i - m_{M,n}^{(U, OOB)}(\\textbf {X}_i^{(U)}, \\Theta _M)\\big )^2,$ where $\\hat{\\sigma }_Y$ is the standard estimate of $Y]$ .", "Figure: Example of the partition of [0,1] 2 [0,1]^2 by a random CART tree (left side) projected on the subspace spanned by 𝐗 (U) =X (1) \\textbf {X}^{(U)} = X^{(1)} (right side).", "Here, p=2p = 2 and U={1}U = \\lbrace 1\\rbrace ." ], [ "PRF algorithm.", "The critical feature of PRF is the algorithmic trick to compute the projected partition efficiently, leaving the initial tree structures untouched.", "Indeed, a naive computation of the projected partitions from the cell edges is computationally very costly, as soon as the dimension increases.", "Instead, we simply drop observations down the initial trees, ignoring splits which use a variable outside of $U$ .", "This enables to recover the projected partitions with an efficient computational complexity.", "To explain this mechanism in details, we focus on a given tree of the initial forest.", "Thus, the training observations are dropped down the tree, and when a split involving a variable outside of $U$ is met, data points are sent both to the left and right children nodes.", "Consequently, each observation falls in multiple terminal leaves of the tree.", "We drop the new query point $\\smash{\\textbf {X}^{(U)}}$ down the tree, following the same procedure, and retrieve the set of terminal leaves where $\\smash{\\textbf {X}^{(U)}}$ falls.", "Next, we collect the training observations which belong to every terminal leaf of this collection, in other words, we intersect the collection of leaves where $\\smash{\\textbf {X}^{(U)}}$ falls.", "Finally, we average the outputs $Y_i$ of the selected training points to generate the tree prediction for $\\smash{\\textbf {X}^{(U)}}$ .", "Notice that such set of selected observations can be empty if $\\smash{\\textbf {X}^{(U)}}$ belongs to a large collection of terminal leaves.", "To avoid this issue, PRF uses the following strategy.", "Recall that a partition of the input space is associated to each tree level, and consequently, a projected tree partition can also be defined at each tree level.", "Thus, when $\\smash{\\textbf {X}^{(U)}}$ is dropped down the tree, it is stopped before reaching a tree level where it falls in an empty cell of the associated projected partition.", "Overall, this mechanism is equivalent to the projection of the tree partition on the subspace span by $\\smash{X^{(U)}}$ , because all splits on variables $\\smash{X^{(j)}}$ with $\\smash{j \\notin U}$ are ignored, and the resulting overlapping cells are intersected—see Figure REF ." ], [ "PRF computational complexity.", "An efficient implementation of the PRF algorithm is detailed in Algorithm REF in the Supplementary Material.", "The computational complexity of PRF for all $U \\in \\mathcal {U}_{n,K}$ does not depend on the dimension $p$ , is linear with $M$ , $K$ , and quasi-linear with $n$ : $O(\\smash{MKn\\log (n))}$ .", "PRF is therefore faster than growing $K$ random forests from scratch, one for each subset $U$ , which has an averaged complexity of $O(\\smash{MKpn\\log ^2(n))}$ [14].", "The computational gain of SHAFF can be considerable in high dimension, since the complexity of all competitors depends on $p$ —see the Supplementary Material for a detailed computational complexity analysis.", "Notice that the PRF algorithm is close in spirit to a component of the Sobol-MDA [4], used to measure the loss of output explained variance when an input variable $j$ is removed from a random forest.", "In particular, a naive adaptation leads to a quadratic complexity with respect to the sample size $n$ , whereas our PRF algorithm has a quasi-linear complexity, which makes it operational.", "Finally, the last step of SHAFF is to take advantage of the estimated $\\hat{v}_{M,n}(U)$ for $U \\in \\mathcal {U}_{n,K}$ to recover Shapley effects." ], [ "Shapley Effect Estimates", "The importance sampling introduces the corrective terms $\\hat{p}_{M,n}(U)$ in the final loss function.", "Thus, SHAFF estimates $\\smash{\\mathbf {\\hat{Sh}}_{M,n} = (\\hat{Sh}_{M,n}(X^{(1)}), \\hdots , \\hat{Sh}_{M,n}(X^{(p)}))}$ as the minimum in $\\beta $ of the following cost function $\\ell _{M,n}(\\beta ) = \\frac{1}{K} \\sum _{U \\in \\mathcal {U}_{n,K}} \\frac{w(U)}{\\hat{p}_{M,n}(U)} (\\hat{v}_{M,n}(U) - \\beta ^T I(U))^2,$ where the sum of the components of $\\beta $ is constrained to be the proportion of output explained variance of the initial forest, fit with all input variables.", "Finally, this can be written in the following compact form: $\\mathbf {\\hat{Sh}}_{M,n} = \\underset{\\beta \\in [0,1]^p}{\\textrm {argmin}} & \\quad \\ell _{M,n}(\\beta ) \\\\\\textrm {s.t.", "}& ||\\beta ||_1 = \\hat{v}_{M,n}(\\lbrace 1,\\hdots ,p\\rbrace ).$" ], [ "SHAFF Consistency", "We prove in this section that SHAFF is consistent, in the sense that the estimated value can be arbitrarily close to the ground truth theoretical Shapley effect, provided that the sample size is large enough.", "To our knowledge, we provide the first Shapley algorithm which requires to fit only a single initial model and is consistent in the general case.", "We insist that our result is valid even when input variables exhibit strong dependences.", "The consistency of SHAFF holds under the following mild and standard assumption on the data distribution: (A1) The response $Y \\in {R}$ follows $Y = m(\\textbf {X}) + \\varepsilon ,$ where $\\textbf {X}= (X^{(1)}, \\hdots , X^{(p)}) \\in [0,1]^p$ admits a density over $[0,1]^p$ bounded from above and below by strictly positive constants, $m$ is continuous, and the noise $\\varepsilon $ is sub-Gaussian, independent of $\\textbf {X}$ , and centered.", "To alleviate the mathematical analysis, we slightly modify the standard Breiman random forests: the bootstrap sampling is replaced by a subsampling without replacement of $a_n$ observations, as it is usually done in the theoretical analysis of random forests [21], [18].", "Additionally, we follow [27] with an additional small modification of the forest algorithm, which is sufficient to ensure its consistency.", "Firstly, a node split is constrained to generate child nodes with at least a small fraction $\\gamma > 0$ of the parent node observations.", "Secondly, the split selection is slightly randomized: at each tree node, the number mtry of candidate variables drawn to optimize the split is set to $\\texttt {mtry} = 1$ with a small probability $\\delta > 0$ .", "Otherwise, with probability $1 - \\delta $ , the default value of mtry is used.", "It is stressed that these last modifications are mild, since $\\gamma $ and $\\delta $ can be chosen arbitrarily small.", "Finally, we introduce the following two assumptions on the asymptotic regime of the algorithm parameters.", "Assumption (A2) enforces that the tree partitions are not too complex with respect to the sample size $n$ .", "On the other hand, Assumption (A3) states that the number of trees and the number of sampled variable subsets $U$ grow with $n$ .", "This ensures that all possible variable subsets have a positive probability to be drawn, which is required for the convergence of our algorithm based on importance sampling.", "(A2) The asymptotic regime of $a_n$ , the size of the subsampling without replacement, and the number of terminal leaves $t_n$ are such that $a_n \\le n-2$ , $a_n/n < 1 - \\kappa $ for a fixed $\\kappa > 0$ , $\\lim \\limits _{n \\rightarrow \\infty } a_n = \\infty $ , $\\lim \\limits _{n \\rightarrow \\infty } t_n = \\infty $ , and $\\lim \\limits _{n \\rightarrow \\infty } 2^{t_n} \\frac{(\\log (a_n))^9}{a_n} = 0$ .", "(A3) The number of Monte-Carlo sampling $K_n$ and the number of trees $M_n$ grow with $n$ , such that $M_n \\longrightarrow \\infty $ and $n.M_n/K_n \\longrightarrow 0$ .", "We also let the theoretical Shapley effect vector be $\\mathbf {Sh}^{\\star } = (Sh^{\\star }(X^{(1)}), \\hdots , Sh^{\\star }(X^{(p)}))$ to formalize our main result.", "Theorem 1 If Assumptions (A1), (A2), and (A3) are satisfied, then SHAFF is consistent, that is $\\mathbf {\\hat{Sh}}_{M_n,n} \\overset{p}{\\longrightarrow } \\mathbf {Sh}^{\\star }.$ [Sketch of proof of Theorem REF ] Firstly, we need three lemmas to prove Theorem REF , gathered in the Supplementary Material.", "Lemma 1 states that all variable subsets $U$ have a positive probability to be drawn asymptotically, which ensures that the importance sampling approach can converge.", "Lemma 2 states the consistency of the projected forest estimate, and the proof uses arguments from [10] to control both the approximation and estimation errors.", "Lemma 3 applies the two previous lemmas to state the convergence of the loss function of the weigthed regression problem solved to recover Shapley effect estimates.", "Secondly, we apply Theorem 2 from [15] to show that the minimum of the theoretical loss function are the theoretical Shapley effects.", "Finally, using Lemma 3 and Theorem $5.7$ from [26], we show that the minimum of the empirical loss function converges towards the minimum of the theoretical loss function, which gives the consistency of SHAFF." ], [ "Experiments", "We run two batches of experiments to show the improvements of SHAFF over the main competitors [5], [29], and [6].", "Experiment 1 is a simple linear case with a redundant variable, while Experiment 2 is a non-linear example with high order interactions.", "In both cases, existing Shapley algorithms exhibit a bias which significantly modifies the accurate variable ranking, as opposed to SHAFF." ], [ "Experiment settings.", "Our implementation of SHAFF is based on ranger, a fast random forest software written in C++ and R from [30].", "We implemented [29] from scratch, as it only requires to sample variable subsets $U$ , fit a random forest for each $U$ , and recover Shapley effects by solving the linear regression problem defined in Section .", "Notice that we limit tree depth to 6 when $|U| \\le 2$ to avoid overfitting.", "We implemented SAGE following Algorithm 1 from [6], and setting $m = 30$ .", "The original implementation of [5] in the R package sensitivity has an exponential complexity with $p$ .", "Even for $p=10$ , we could not have the experiments done within 24 hours when parallelized on 16 cores.", "Therefore, we do not display the results for [5], which seem to have a high bias on toy examples.", "In all procedures, the number $K$ of sampled subsets $U$ is set to 500, and we use 500 trees for the forest growing.", "Each run is repeated 30 times to estimate the standard deviations.", "For both experiments, we analytically derive the theoretical Shapley effects, and display this ground truth with red crosses in Figures REF and REF —see the Supplementary Material for the formulas." ], [ "Experiment 1.", "In the first experiment, we consider a linear model and a correlated centered Gaussian input vector of dimension 11.", "The output $Y$ follows $Y = \\beta ^T\\textbf {X}+ \\varepsilon ,$ where $\\beta \\in [0,1]^{11}$ , and the noise $\\varepsilon $ is centered, independent, and such that $\\varepsilon ] = 0.05 \\times Y]$ .", "Finally, two copies of $\\smash{X^{(2)}}$ are appended to the data as $\\smash{X^{(12)}}$ and $\\smash{X^{(13)}}$ , and two dummy Gaussian variables $\\smash{X^{(14)}}$ and $\\smash{X^{(15)}}$ are also added.", "We draw a sample ${D}_{n}$ of size $n = 3000$ .", "Figure REF shows that SHAFF is more accurate than its competitors.", "[6] has a strong bias for several variables, in particular $\\smash{X^{(4)}}$ , $\\smash{X^{(7)}}$ , $\\smash{X^{(8)}}$ , and $\\smash{X^{(10)}}$ .", "The algorithm from [29] has a lower performance since its variance is higher than for the other methods.", "Notice that [29] recommend to set $K = 2n$ ($ = 6000$ here).", "Since we use $K = 500$ to compare all algorithms, this high variance is quite expected and show the improvement due to the importance sampling of our method.", "Besides, the computational complexity of [29] is $O(n^2)$ whereas SHAFF is quasi-linear.", "Finally, in this experiment, the random forest has a proportion of explained variance of about 86%, and the noise variance is 5%, which explains the small negative bias of many estimated values.", "Figure: Shapley effects for Experiment 1.", "Red crosses are the theoretical Shapley effects." ], [ "Experiment 2.", "In the second experiment, we consider two independent blocks of 5 interacting variables.", "The input vector is Gaussian, centered, and of dimension 10.", "All variables have unit variance, and all covariances are null, except $\\textrm {Cov}(X^{(1)}, X^{(2)}) = \\textrm {Cov}(X^{(6)}, X^{(7)}) = 0.9$ , and $\\textrm {Cov}(X^{(4)}, X^{(5)}) = \\textrm {Cov}(X^{(9)}, X^{(10)}) = 0.5$ .", "The output $Y$ follows $Y = 3\\sqrt{3}& \\times X^{(1)} X^{(2)} {1}_{X^{(3)} > 0}+ \\sqrt{3} \\times X^{(4)} X^{(5)} {1}_{X^{(3)} < 0} \\\\& + 3 \\times X^{(6)} X^{(7)} {1}_{X^{(8)} > 0} + X^{(9)} X^{(10)} {1}_{X^{(8)} < 0} + \\varepsilon ,$ where the noise $\\varepsilon $ is centered, independent, and such that $\\varepsilon ] = 0.05 \\times Y]$ .", "We add 5 dummy Gaussian variables $\\smash{X^{(11)}}$ , $\\smash{X^{(12)}}$ , $\\smash{X^{(13)}}$ , $\\smash{X^{(14)}}$ , and $\\smash{X^{(15)}}$ , and draw a sample ${D}_{n}$ of size $n = 10000$ .", "In this context of strong interactions and correlations, we observe that all competitors have a strong bias for most variables, as opposed to SHAFF, which is also the only algorithm providing the accurate variable ranking given by the theoretical Shapley effects.", "In particular, SHAFF properly identifies variable $\\smash{X^{(3)}}$ as the most important one, whereas SAGE considerably overestimates the Shapley effects of variables $\\smash{X^{(1)}}$ and $\\smash{X^{(2)}}$ .", "SHAFF also clearly ranks variable $\\smash{X^{(8)}}$ as more important than $\\smash{X^{(6)}}$ and $\\smash{X^{(7)}}$ , as opposed to its competitors.", "Besides, the proportion of explained variance of the forest is about 84% in this setting, which explains the negative bias observed for several estimates.", "Figure: Shapley effects for Experiment 2.", "Red crosses are the theoretical Shapley effects." ], [ "Conclusion", "We introduced SHAFF, SHApley eFfects via random Forests, an algorithm to estimate Shapley effects based on random forests, which has an implementation in C++ and R available online.", "The challenges in Shapley estimation are the exponential computational complexity, and the estimates of conditional expectations.", "SHAFF addresses the first point by using importance sampling to favor the subsets of influential variables, which often occur along the forest paths.", "For the second point, SHAFF uses the projected forest algorithm, a fast procedure to eliminate variables from the forest prediction mechanism.", "Thanks to this approach, SHAFF only needs to fit a random forests once, as opposed to other methods which retrain many models and are computationally costly.", "Importantly, we prove that SHAFF is consistent.", "To our knowledge, we propose the first Shapley algorithm which do not retrain several models and is proved to be consistent under mild assumptions.", "Furthermore, we conducted several experiments to show the practical performance improvements over state-of-the-art Shapley algorithms.", "Notice that the adaptation of SHAFF to SHAP values is straightforward, since the projected random forests provides predictions of the output conditional on any variable subset.", "Finally, in specific settings, it is obviously possible that other learning algorithms outperform random forests.", "Then, we can use such efficient model to generate a new large sample of simulated observations, which can then feeds SHAFF and improves its accuracy." ], [ "Computational Complexity", "We provide the average computational complexity of SHAFF, as well as its competitors [5], [29], and [6].", "For these last two algorithms, random forests are used as the required black-box model.", "Only SHAFF is quasi-linear with the sample size $n$ and independent of the dimension $p$ ." ], [ "SHAFF", "We derive the computational complexity of each step of SHAFF.", "Overall, the computational complexity is $O(MKn\\log (n))$ ." ], [ "Importance sampling.", "In order to compute the variable subset importance, SHAFF counts the occurence of variable subsets $U$ in the tree paths of the forest, which has a complexity of $O(Mn)$ , since each tree has about $O(n)$ nodes.", "The sampling of $K$ subsets $U$ has a complexity of $O(K)$ ." ], [ "Projected random forests.", "An efficient implementation of the PRF algorithm is detailed in Algorithm REF .", "For the sake of clarity, we provide a version of PRF for a single variable subset $U$ and one query point $\\smash{\\textbf {X}^{(U)}}$ .", "Let us consider a given tree.", "The new observation $\\smash{\\textbf {X}^{(U)}}$ is dropped down the tree, eventually applying multiple splits at each level, because data points are sent on both sides of splits involving a variable outside of $U$ .", "At the same time, the PRF computes which training observations fall in the same projected cell as $\\smash{\\textbf {X}^{(U)}}$ , and stops going down the tree just before the size of this projected cell becomes lower than the parameter min_node_size.", "Such procedure has a complexity of $O(n)$ since we sequentially apply splits to reduce the number of training observations from about $n$ to min_node_size to reach the terminal projected cell.", "Therefore, the computational complexity to compute the PRF prediction for a given $U$ and $\\smash{\\textbf {X}^{(U)}}$ is $O(Mn)$ .", "In SHAFF, the PRF is run for all subsets $U \\in \\mathcal {U}_{n,K}$ and the full OOB sample for each tree.", "In practice, we do not naively run Algorithm REF for all $U$ and OOB observations, i.e., $O(Kn)$ times, since it would lead to a quadratic complexity with $n$ .", "Instead, for a given tree, all OOB and training observations are dropped down the tree simultaneously.", "Even if multiple splits are applied at each tree level, we are still partitioning two samples of size $O(n)$ by sequentially applying splits: splitting one time all cells of a given partition takes $O(n)$ operations, and this has to be repeated $O(\\log (n))$ times so that each cell reaches a size of min_node_size.", "Therefore, the global complexity of running PRF for the full OOB samples and the $K$ subsets $U$ is $O(MKn\\log (n))$ ." ], [ "Shapley effect estimates.", "The complexity to solve a least square problem with $p$ columns and $K$ rows is $O(p^3K)$ .", "However in practice, $K$ is always fixed to default value, and when $p > K$ , only at most $O(K)$ input variables are selected in the subsets $U$ .", "For the non-selected inputs, the Shapley effect is null, and they can be removed from the least square problem, leading to a complexity of $O(K^4)$ ." ], [ "{{cite:6ec2aedb3147f2b32e1d0c25e70d4b1a7410c4a1}}", "The conditional expectations are estimated for all $U \\in \\lbrace 1,\\hdots ,p\\rbrace $ , which makes $2^p$ estimates.", "Efficient $k$ -nearest neighbor algorithms have a complexity of $O(pn\\log (n))$ .", "Overall the complexity is $O(n\\log (n)p2^p)$ , which is exponential with respect to the dimension $p$ ." ], [ "{{cite:67c9fa6698815bc7ea5287100b930522d396e154}}", "Growing $K$ random forests from scratch, one for each subset $U$ , has an averaged complexity of $O(\\smash{MKpn\\log ^2(n))}$ [14].", "[29] recommend to use $K = O(n)$ , which makes a global complexity of $O(\\smash{Mpn^2\\log ^2(n))}$ , and is quadratic with respect to the sample size $n$ and depends on the dimension $p$ ." ], [ "{{cite:fb268d514cbaba612e2eeaa562ee8bc8eca4d5f1}}", "Running a prediction for random forests takes $O(M\\log (n))$ operations.", "Since SAGE computes $np$ predictions, the global complexity is $O(Mpn\\log (n))$ and depends on the dimension $p$ .", "Projected Random Forest [1] Inputs: A random forest fit with ${D}_{n}$ , a variable subset $U \\subset \\lbrace 1,\\hdots ,p\\rbrace $ , and a query point $\\textbf {X}^{(U)}$ .", "for all trees in the forest: # Step 1: initialize variables initialize $nodes\\_level$ as a list of nodes containing only the root node; initialize $nodes\\_child$ as an empty list of child nodes; initialize $samples$ as the list of observation indices of the full training data of the tree; for all levels in the tree:   # Step 2: drop $\\textbf {X}^{(U)}$ to the next tree level with the relevant training observations   for all nodes in $nodes\\_level$ :   if the node splits on a variable in $U$ :   compute whether $\\smash{\\textbf {X}^{(U)}}$ falls in the left or right child node;   append the child node to $nodes\\_child$ ;   set $samples\\_child$ as the observations in $samples$ which satisfy the split   else:   append both the left and right children nodes to $nodes\\_child$ ;   set $samples\\_child = samples$ ;   if the size of $samples\\_child$ is lower then $min\\_node\\_size$ :   break the loop through the tree levels;   else:   set $samples = samples\\_child$ ;   set $nodes\\_level = nodes\\_child$ ;   # Step 3: compute prediction   compute the tree prediction as the average of $Y_i$ for all $i$ in $samples$ ; average predictions of all trees; return final prediction;" ], [ "Proof of Theorem ", "We need the following three lemmas to prove Theorem REF .", "Lemma REF gives the convergence of the importance sampling, because all variable subsets $U$ have a positive probability to be drawn asymptotically.", "Lemma REF states the consistency of the projected forest estimate, and the proof follows arguments from [21].", "Lemma REF uses the two previous lemmas to state the convergence of the loss function of the weighted regression problem solved to recover Shapley effect estimates.", "Lemma 1 If Assumption (A3) is satisfied, we have ${P}\\big ( \\hat{p}_{M_n,n}(U) > 0 \\big ) \\longrightarrow 1.$ Lemma 2 If Assumptions (A1) and (A2) are satisfied, the PRF is consistent, that is, for all $M \\in \\mathbb {N}^{\\star }$ and $U \\subset \\lbrace 1,\\hdots ,p\\rbrace $ , $\\hat{v}_{M,n}(U) \\overset{p}{\\longrightarrow } \\mathbb {E}[Y|\\textbf {X}^{(U)}]]/Y] \\overset{\\rm def}{=} v^{\\star }(U).$ We let $Z$ be a discrete random variable taking values in the set of all subsets of $\\lbrace 1, \\hdots , p\\rbrace $ , excluding the full and empty sets.", "The discrete distribution of $Z$ is given by the weights $w(U)$ (the weights are scaled to sum to 1).", "Lemma 3 If Assumptions (A1), (A2), and (A3) are satisfied, we have $\\ell _{M,n}(\\beta ) \\overset{p}{\\longrightarrow } \\mathbb {E}[(v^{\\star }(Z) - \\beta ^T I(Z))^2] \\overset{\\rm def}{=} \\ell ^{\\star }(\\beta ).$ [Proof of Theorem REF ] We assume that Assumptions (A1), (A2), and (A3) are satisfied.", "Since $\\ell ^{\\star }$ is convex and $\\beta $ belongs to the compact set $[0,1]^p$ , the pointwise convergence of Lemma REF gives the uniform convergence $\\sup _{\\beta \\in [0,1]^p} |\\ell _{M,n}(\\beta ) - \\ell ^{\\star }(\\beta )| \\overset{p}{\\longrightarrow } 0.$ Additionally, since $\\ell ^{\\star }$ is a quadratic convex function and the constraint domain $[0,1]^p$ is convex, $\\ell ^{\\star }$ has a unique minimum.", "According to Theorem 2 from [15], this unique minimum is $\\mathbf {Sh}^{\\star }$ .", "Finally, since the minimum of $\\ell ^{\\star }$ is unique and $\\ell _{M,n}$ uniformly converges to $\\ell ^{\\star }$ , we apply Theorem $5.7$ from [26] to conclude that $\\mathbf {\\hat{Sh}}_{M,n} \\overset{p}{\\longrightarrow } \\mathbf {Sh}^{\\star }.$ We prove Lemmas REF , REF , and REF involved in the proof of Theorem REF .", "[Proof of Lemma REF ] We assume that Assumption (A3) is satisfied, and denote by $T_{n,\\ell }$ the random set of all variable subsets of $\\lbrace 1,\\hdots ,p\\rbrace $ belonging to a path of the $\\ell $ -th tree.", "To prove the result, we derive an upper bound for ${P}(\\hat{p}_{M,n}(U) = 0)$ .", "First, we write ${P}(\\hat{p}_{M,n}(U) = 0 | {D}_{n}) = {P}\\big (\\bigcap _{\\ell =1}^{M_n} U \\notin T_{n,\\ell } | {D}_{n}\\big ),$ and since the trees are independent conditional on ${D}_{n}$ ${P}(\\hat{p}_{M,n}(U) = 0 | {D}_{n}) = {P}(U \\notin T_{n,1} | {D}_{n})^{M_n}.$ For $n > s \\log _2(p)$ , where $s$ is the minimum number of observations in a terminal leaf, there is at least one path in each tree that has at least $p$ splits.", "Additionally, recall that the random forest algorithm is slightly modified such that mtry is randomly set to 1 with a small probability $\\delta $ .", "Thus, if we define the random event $A_n$ as mtry is set to 1 and a new variable of $U$ is selected at each node of a path of length at least $|U|$ , then $A_n$ is included in $\\lbrace U \\in T_{n,1}\\rbrace $ .", "This event $A_n$ is of probability lower bounded by $(\\delta /p)^p$ , and thus for $n > s \\log _2(p)$ ${P}(U \\in T_{n,1} | {D}_{n}) \\ge P(A_n) \\ge (\\delta /p)^p,$ and then ${P}(\\hat{p}_{M,n}(U) = 0 | {D}_{n}) \\le (1 - (\\delta /p)^{p})^{M_n}.$ Finally, Assumption (A3) gives that the number of trees increases with $n$ , and we obtain ${P}\\big ( \\hat{p}_{M,n}(U) = 0 \\big ) \\longrightarrow 0,$ which is the desired result.", "[Proof of Lemma REF ] We assume that Assumptions (A1) and (A2) are satisfied and consider $M \\in \\mathbb {N}^{\\star }$ and $U \\subset \\lbrace 1,\\hdots ,p\\rbrace $ .", "Recall that $\\hat{v}_{M,n}(U) = 1 - \\frac{1}{n \\hat{\\sigma }_Y} \\sum _{i=1}^{n} \\big (Y_i - m_{M,n}^{(U, OOB)}(\\textbf {X}_i^{(U)}, \\Theta _M)\\big )^2.$ The right hand side is expanded as follows: $\\hat{v}_{M,n}(U) = 1 - \\frac{1}{n \\hat{\\sigma }_Y} \\sum _{i=1}^{n}& \\big (m(\\textbf {X}_i) + \\varepsilon _i - m_{M,n}^{(U, OOB)}(\\textbf {X}_i^{(U)}, \\Theta _M)\\big )^2 \\\\= 1 - \\frac{1}{n \\hat{\\sigma }_Y} \\sum _{i=1}^{n}& \\big (m(\\textbf {X}_i) - \\mathbb {E}[m(\\textbf {X}_i)|\\textbf {X}_i^{(U)}] + \\varepsilon _i \\\\ & - [m_{M,n}^{(U, OOB)}(\\textbf {X}_i^{(U)}, \\Theta _M) - \\mathbb {E}[m(\\textbf {X}_i)|\\textbf {X}_i^{(U)}]] \\big )^2.", "\\\\$ Therefore, $ \\hat{v}_{M,n}(U) = 1 - &\\frac{1}{n \\hat{\\sigma }_Y} \\sum _{i=1}^{n} (m(\\textbf {X}_i) - \\mathbb {E}[m(\\textbf {X}_i)|\\textbf {X}_i^{(U)}])^2 \\nonumber \\\\ \\nonumber &+ \\varepsilon _i^2 + 2 \\varepsilon _i \\times (m(\\textbf {X}_i) - \\mathbb {E}[m(\\textbf {X}_i)|\\textbf {X}_i^{(U)}]) \\nonumber \\\\& - 2 \\varepsilon _i \\times \\big (m_{M,n}^{(U, OOB)}(\\textbf {X}_i^{(U)}, \\Theta _M) - \\mathbb {E}[m(\\textbf {X}_i)|\\textbf {X}_i^{(U)}]\\big ) \\nonumber \\\\& - 2 (m(\\textbf {X}_i) - \\mathbb {E}[m(\\textbf {X}_i)|\\textbf {X}_i^{(U)}]) \\times \\big (m_{M,n}^{(U, OOB)}(\\textbf {X}_i^{(U)}, \\Theta _M) - \\mathbb {E}[m(\\textbf {X}_i)|\\textbf {X}_i^{(U)}]\\big ) \\nonumber \\\\& + \\big (m_{M,n}^{(U, OOB)}(\\textbf {X}_i^{(U)}, \\Theta _M) - \\mathbb {E}[m(\\textbf {X}_i)|\\textbf {X}_i^{(U)}]\\big )^2.$ Now, using the law of large numbers, we obtain $\\frac{1}{n} \\sum _{i=1}^{n} (m(\\textbf {X}_i) -& \\mathbb {E}[m(\\textbf {X}_i)|\\textbf {X}_i^{(U)}])^2 + \\varepsilon _i^2 \\\\[-1em] & + 2 \\varepsilon _i \\times (m(\\textbf {X}_i) - \\mathbb {E}[m(\\textbf {X}_i)|\\textbf {X}_i^{(U)}])\\overset{p}{\\longrightarrow } \\mathbb {E}[m(\\textbf {X}) | \\textbf {X}^{(U)}]] + \\varepsilon ],$ and also $\\hat{\\sigma }_Y \\overset{p}{\\longrightarrow } Y]$ .", "Combining these two limits, we have $1 - \\frac{1}{n \\hat{\\sigma }_Y} \\sum _{i=1}^{n} (&m(\\textbf {X}_i) - \\mathbb {E}[m(\\textbf {X}_i)|\\textbf {X}_i^{(U)}])^2 + \\varepsilon _i^2 \\\\[-1em] & + 2 \\varepsilon _i \\times (m(\\textbf {X}_i) - \\mathbb {E}[m(\\textbf {X}_i)|\\textbf {X}_i^{(U)}])\\overset{p}{\\longrightarrow } 1 - (\\mathbb {E}[m(\\textbf {X}) | \\textbf {X}^{(U)}]] + \\varepsilon ])/Y].$ Rewriting this limit using the law of total variance, we are led to $1 - (\\mathbb {E}&[m(\\textbf {X}) | \\textbf {X}^{(U)}]] + \\varepsilon ])/Y] \\\\& = (Y] - \\mathbb {E}[m(\\textbf {X}) | \\textbf {X}^{(U)}]] + \\varepsilon ])/Y] \\\\& = (m(\\textbf {X})] + \\varepsilon ] - \\mathbb {E}[m(\\textbf {X}) | \\textbf {X}^{(U)}]] - \\varepsilon ])/Y] \\\\& = \\mathbb {E}[m(\\textbf {X}) | \\textbf {X}^{(U)}]]/Y] \\\\& = \\mathbb {E}[Y | \\textbf {X}^{(U)}]]/Y] \\\\& = v^{\\star }(U).$ Overall, the result of the lemma holds if the last three terms of the decomposition (REF ) converge towards 0 in probability.", "This is clearly true if the OOB PRF estimate is $\\mathbb {L}^2$ -consistent, that is for $i \\in \\lbrace 1,\\hdots ,n\\rbrace $ , $\\mathbb {E}\\big [ \\big (m_{M,n}^{(U, OOB)}(\\textbf {X}_i^{(U)}, \\Theta _M) - \\mathbb {E}[m(\\textbf {X}_i)|\\textbf {X}_i^{(U)}]\\big )^2 \\big ]\\longrightarrow 0.$ According to Lemma 2 from [4], the $\\mathbb {L}^2$ -convergence of the OOB forest estimate follows from the convergence of the standard forest estimate.", "Therefore, we only need to show the $\\mathbb {L}^2$ -convergence of the PRF estimate to get the final result.", "To do so, we adapt the proof of Theorem 1 from [21], which shows the convergence of Breiman's forests for additive models.", "The proof only differs for the approximation error.", "Indeed, we need to show that the variation of the regression function vanishes in a cell of the empirical PRF.", "[21] show that this is always true in the original forest for additive models.", "Here, the result is valid for all regression functions, using the fact that the random forest is slightly modified: splits cannot be too close from the edges of cells (at least a fraction of $\\gamma $ observations in children nodes), and $mtry$ is set to 1 at each node with a small probability $\\delta $ .", "Under these small modifications, Lemma 2 from [17] gives that the diameter of each cell of the original forest vanishes, i.e, $\\lim \\limits _{n \\rightarrow \\infty } \\textrm {diam}(A_n(\\textbf {X}, \\Theta )) = 0,$ where $A_n(\\textbf {X}, \\Theta )$ is the cell of the forest where the new query point $\\textbf {X}$ falls, and the diameter of a cell $A$ is the length of the longest line fitting in $A$ , formally $\\textrm {diam}(A) = \\sup _{\\textbf {x},\\textbf {x}^{\\prime } \\in A} ||\\textbf {x}- \\textbf {x}^{\\prime }||_2.$ By definition of the PRF algorithm, the projected cell where $\\textbf {X}^{(U)}$ falls is included in $A_n(\\textbf {X}, \\Theta )$ , and therefore the diameter of the projected cell also vanishes as $n$ increases.", "Additionally, the regression function $m$ is continuous by Assumption (A1), and consequently the approximation error converges to 0.", "Finally, the PRF estimate is $\\mathbb {L}^2$ -consistent, and we deduce the final result, $\\hat{v}_{M,n}(U) \\overset{p}{\\longrightarrow } v^{\\star }(U).$ [Proof of Lemma REF ] The loss function $\\ell _{M,n}$ contains three sources of randomness: the data ${D}_{n}$ , the forest randomization $\\Theta $ , and the importance sampling of the subsets $U$ .", "The discrete distribution used to sample the subsets $U$ is built using the occurrence frequency in the forest $\\hat{p}_{M,n}(U)$ , which depends on ${D}_{n}$ and $\\Theta $ .", "This subtle relation between the data, the forest, and the importance sampling prevent a straightforward proof for this lemma.", "We reshape the loss function and use the law of total variance to handle separately the multiple sources of randomness.", "We assume that Assumptions (A1), (A2), and (A3) are satisfied.", "First, we have $\\ell _{M,n}(\\beta ) &= \\frac{1}{K_n} \\sum _{U \\in \\mathcal {U}_{n,K}} \\frac{w(U)}{\\hat{p}_{M,n}(U)} (\\hat{v}_{M,n}(U) - \\beta ^T I(U))^2\\\\&= \\sum _{U \\subset \\lbrace 1,\\hdots ,p\\rbrace } \\frac{w(U) N_n(U)}{K_n \\hat{p}_{M,n}(U)} {1}_{\\hat{p}_{M,n}(U) > 0} (\\hat{v}_{M,n}(U) - \\beta ^T I(U))^2,$ where $N_n(U)$ is the number of times where $U$ is drawn in $\\mathcal {U}_{n,K}$ (with the convention $0/0 = 0$ ).", "Since the sum is finite, it is enough to study the convergence of the terms one by one.", "Let us consider a given variable subset $U$ .", "First, we define $\\Delta _{n,K_n} = \\frac{N_n(U) {1}_{\\hat{p}_{M,n}(U) > 0}}{K_n \\hat{p}_{M,n}(U)}.$ Next, we derive the limit of $\\Delta _{n,K_n}]$ using the law of total variance.", "We have $\\Delta _{n,K_n}] = \\mathbb {E}[\\Delta _{n,K_n}|{D}_{n},\\Theta ]] + \\mathbb {E}[\\Delta _{n,K_n}|{D}_{n},\\Theta ]].$ On one hand, since $K_n$ is a constant and $\\hat{p}_{M,n}(U)$ only depends on ${D}_{n}$ and $\\Theta $ , we have $\\Delta _{n,K_n}|{D}_{n},\\Theta ] = [\\frac{N_n(U) {1}_{\\hat{p}_{M,n}(U) > 0}}{K_n \\hat{p}_{M,n}(U)}|{D}_{n},\\Theta \\big ]= \\Big (\\frac{{1}_{\\hat{p}_{M,n}(U) > 0}}{K_n \\hat{p}_{M,n}(U)}\\Big )^2 [N_n(U) |{D}_{n},\\Theta \\big ].$ By definition, $N_n(U) = \\sum _{k = 1}^{K_n} {1}_{U_k = U}$ , where $U_1,\\hdots ,U_{K_n}$ are the variable subsets drawn at each iteration of the importance sampling.", "Since $U_1,\\hdots ,U_{K_n}$ are independent conditional on ${D}_{n}$ and $\\Theta $ , and $U$ is drawn with probability $\\hat{p}_{M,n}(U)$ , $[N_n(U) |{D}_{n},\\Theta \\big ] = K_n {1}_{U_1 = U} | {D}_{n},\\Theta ] = K_n \\hat{p}_{M,n}(U) [1 - \\hat{p}_{M,n}(U)],$ and finally $\\mathbb {E}[\\Delta _{n,K_n}|{D}_{n},\\Theta ]] = \\frac{1}{K_n} \\mathbb {E}\\big [\\frac{1 - \\hat{p}_{M,n}(U)}{\\hat{p}_{M,n}(U)} {1}_{\\hat{p}_{M,n}(U) > 0}\\big ].$ Therefore, $\\mathbb {E}[\\Delta _{n,K_n}|{D}_{n},\\Theta ]] \\le \\frac{1}{K_n} \\mathbb {E}\\big [\\frac{{1}_{\\hat{p}_{M,n}(U) > 0}}{\\hat{p}_{M,n}(U)}\\big ].$ The number of paths in the forest is upper bounded by $n \\times M_n$ , and therefore if $\\hat{p}_{M,n}(U)$ is not null, it is lower bounded by $1/(n.M_n)$ .", "Thus $\\mathbb {E}[\\Delta _{n,K_n}|{D}_{n},\\Theta ]] \\le \\frac{n.M_n}{K_n},$ which converges to 0 by Assumption (A3).", "On the other hand, $\\mathbb {E}[\\Delta _{n,K_n}|{D}_{n},\\Theta ] =\\frac{{1}_{\\hat{p}_{M,n}(U) > 0}}{K_n \\hat{p}_{M,n}(U)} \\mathbb {E}[N_n(U)|{D}_{n},\\Theta ] = {1}_{\\hat{p}_{M,n}(U) > 0},$ and then $\\mathbb {E}[\\Delta _{n,K_n}|{D}_{n},\\Theta ]] =& {P}(\\hat{p}_{M,n}(U) > 0)[1 - {P}(\\hat{p}_{M,n}(U) > 0)] \\\\=& {P}(\\hat{p}_{M,n}(U) > 0){P}(\\hat{p}_{M,n}(U) = 0).$ Lemma REF gives that ${P}\\big ( \\hat{p}_{M,n}(U) = 0 \\big ) \\longrightarrow 0$ , which implies the convergence of $\\mathbb {E}[\\Delta _{n,K_n}|{D}_{n},\\Theta ]]$ towards 0.", "Overall, the law of total variance gives that $\\Delta _{n,K_n}] \\longrightarrow 0.$ Since $\\mathbb {E}[\\Delta _{n,K_n}] = {P}\\big ( \\hat{p}_{M,n}(U) > 0 \\big ) \\longrightarrow 1$ and $\\mathbb {L}^2$ -convergence implies convergence in probability, we have $\\Delta _{n,K_n} \\overset{p}{\\longrightarrow } 1.$ Next, using Lemma REF , we obtain $\\frac{w(U) N_n(U)}{K_n \\hat{p}_{M,n}(U)} {1}_{\\hat{p}_{M,n}(U) > 0} (\\hat{v}_{M,n}(U) - \\beta ^T I(U))^2 \\overset{p}{\\longrightarrow }w(U) (v^{\\star }(U) - \\beta ^T I(U))^2.$ If $Z$ is a discrete random variable taking values in the set of all subsets of $\\lbrace 1,\\hdots ,p\\rbrace $ , excluding the full and empty sets, and distributed with the scaled weights $w(U)$ , we finally have $\\ell _{M,n}(\\beta ) \\overset{p}{\\longrightarrow } \\mathbb {E}[(v^{\\star }(Z) - \\beta ^T I(Z))^2].$" ], [ "Experiment 1.", "For a linear model with a Gaussian input vector of dimension $p$ , the theoretical Shapley effects are given by Theorem 2 in [20] as $Sh^{\\star }(X^{(j)}) = \\frac{1}{p} \\sum _{U \\subset \\lbrace 1,\\hdots ,p\\rbrace \\setminus j} {p - 1 \\atopwithdelims ()|U|}^{-1} \\frac{\\textrm {Cov}[X^{(j)}, \\textbf {X}^{(-U) T} \\beta ^{(-U)} | \\textbf {X}^{(U)}]^2}{X^{(j)}|\\textbf {X}^{(U)}]} \\Big (1 - \\frac{\\sigma _{\\varepsilon }^2}{Y]}\\Big ),$ where the conditional covariances and variances can be easily computed using standard formulas for Gaussian vectors, and $\\sigma _{\\varepsilon }^2$ is the noise variance.", "In Experiment 1, several copies of a given input $X^{(k)}$ are added to the data.", "We denote by $r$ the number of redundant variables.", "We easily deduce the updated value $Sh^{\\prime \\star }(X^{(j)})$ from the original Shapley effects $Sh^{\\star }(X^{(j)})$ for all variables.", "Then, we have $Sh^{\\prime \\star }(X^{(k)}) = \\frac{1}{p + r} \\sum _{U \\subset \\lbrace 1,\\hdots ,p\\rbrace \\setminus k} {p + r - 1 \\atopwithdelims ()|U|}^{-1} \\frac{\\textrm {Cov}[X^{(k)}, \\textbf {X}^{(-U) T} \\beta ^{(-U)} | \\textbf {X}^{(U)}]^2}{X^{(k)}|\\textbf {X}^{(U)}]} \\Big (1 - \\frac{\\sigma _{\\varepsilon }^2}{Y]}\\Big ).$ If $j \\in \\lbrace 1,\\hdots ,p\\rbrace \\setminus k$ , we have $Sh^{\\prime \\star }(X^{(j)}) = \\frac{1}{p + r} \\sum _{\\small {\\begin{array}{c}U \\subset \\lbrace 1,\\hdots ,p\\rbrace \\setminus j \\\\\\textrm {s.t. }", "k \\notin U\\end{array}}}& {p + r - 1 \\atopwithdelims ()|U|}^{-1} \\frac{\\textrm {Cov}[X^{(j)}, \\textbf {X}^{(-U) T} \\beta ^{(-U)} | \\textbf {X}^{(U)}]^2}{X^{(j)}|\\textbf {X}^{(U)}]} \\Big (1 - \\frac{\\sigma _{\\varepsilon }^2}{Y]}\\Big ) \\\\+ \\frac{1}{p + r} \\sum _{\\small {\\begin{array}{c}U \\subset \\lbrace 1,\\hdots ,p\\rbrace \\setminus j \\\\\\textrm {s.t. }", "k \\in U\\end{array}}}& \\Big [ \\sum _{\\ell =0}^{r} {r \\atopwithdelims ()\\ell }{p + r - 1 \\atopwithdelims ()|U| + \\ell }^{-1}+ \\sum _{\\ell =1}^{r} {r \\atopwithdelims ()\\ell }{p + r - 1 \\atopwithdelims ()|U| + \\ell - 1}^{-1} \\Big ] \\\\\\\\[-3em] & \\times \\frac{\\textrm {Cov}[X^{(j)}, \\textbf {X}^{(-U) T} \\beta ^{(-U)} | \\textbf {X}^{(U)}]^2}{X^{(j)}|\\textbf {X}^{(U)}]} \\Big (1 - \\frac{\\sigma _{\\varepsilon }^2}{Y]}\\Big ).$ Finally, for $j \\in \\lbrace p+1, \\hdots , p+r\\rbrace $ , clearly $Sh^{\\prime \\star }(X^{(j)}) = Sh^{\\prime \\star }(X^{(k)}),$ and dummy variables have a null Shapley effect." ], [ "Experiment 2.", "Recall that in the second experiment, we consider two independent blocks of 5 interacting variables.", "The input vector is Gaussian, centered, and of dimension 10.", "All variables have unit variance, and all covariances are null, except $\\textrm {Cov}(X^{(1)}, X^{(2)}) = \\textrm {Cov}(X^{(6)}, X^{(7)}) = \\rho _1$ , and $\\textrm {Cov}(X^{(4)}, X^{(5)}) = \\textrm {Cov}(X^{(9)}, X^{(10)}) = \\rho _2$ .", "The output $Y$ is defined as a specific case of $Y = a \\sqrt{\\alpha }& \\times X^{(1)} X^{(2)} {1}_{X^{(3)} > 0}+ b \\sqrt{\\alpha } \\times X^{(4)} X^{(5)} {1}_{X^{(3)} < 0} \\\\& + c \\sqrt{\\beta } \\times X^{(6)} X^{(7)} {1}_{X^{(8)} > 0} + d \\sqrt{\\beta } X^{(9)} X^{(10)} {1}_{X^{(8)} < 0} + \\varepsilon .$ The Shapley effects of the input variables are given by $Sh^{\\star }(X^{(1)}) = Sh^{\\star }(X^{(2)}) = \\frac{\\alpha }{\\alpha V_1 + \\beta V_2 + \\sigma ^2_{\\varepsilon }} \\Big ( \\frac{(a\\rho _1)^2}{8} + \\frac{5}{24}a^2 \\Big ),$ $Sh^{\\star }(X^{(4)}) = Sh^{\\star }(X^{(5)}) = \\frac{\\alpha }{\\alpha V_1 + \\beta V_2 + \\sigma ^2_{\\varepsilon }} \\Big ( \\frac{(b\\rho _2)^2}{8} + \\frac{5}{24}b^2 \\Big ),$ $Sh^{\\star }(X^{(3)}) = \\frac{\\alpha }{\\alpha V_1 + \\beta V_2 + \\sigma ^2_{\\varepsilon }} \\Big ( \\frac{(a\\rho _1 - b\\rho _2)^2}{4}+ \\frac{(a\\rho _1)^2}{4} + \\frac{(b\\rho _2)^2}{4} + \\frac{a^2}{12} + \\frac{b^2}{12} \\Big ),$ where $V_1 = \\Big ( \\frac{(a\\rho _1 - b\\rho _2)^2}{4}+ \\frac{(a\\rho _1)^2}{2} + \\frac{(b\\rho _2)^2}{2} + \\frac{a^2}{2} + \\frac{b^2}{2} \\Big ),$ and $V_2 = \\Big ( \\frac{(c\\rho _1 - d\\rho _2)^2}{4}+ \\frac{(c\\rho _1)^2}{2} + \\frac{(d\\rho _2)^2}{2} + \\frac{c^2}{2} + \\frac{d^2}{2} \\Big ).$ Symmetrically, we have $Sh^{\\star }(X^{(6)}) = Sh^{\\star }(X^{(7)}) = \\frac{\\beta }{\\alpha V_1 + \\beta V_2 + \\sigma ^2_{\\varepsilon }} \\Big ( \\frac{(c\\rho _1)^2}{8} + \\frac{5}{24}c^2 \\Big ),$ $Sh^{\\star }(X^{(9)}) = Sh^{\\star }(X^{(10)}) = \\frac{\\beta }{\\alpha V_1 + \\beta V_2 + \\sigma ^2_{\\varepsilon }} \\Big ( \\frac{(d\\rho _2)^2}{8} + \\frac{5}{24}d^2 \\Big ),$ $Sh^{\\star }(X^{(8)}) = \\frac{\\beta }{\\alpha V_1 + \\beta V_2 + \\sigma ^2_{\\varepsilon }} \\Big ( \\frac{(c\\rho _1 - d\\rho _2)^2}{4}+ \\frac{(c\\rho _1)^2}{4} + \\frac{(d\\rho _2)^2}{4} + \\frac{c^2}{12} + \\frac{d^2}{12} \\Big ).$ Clearly, $ Sh^{\\star }(X^{(11)}) = Sh^{\\star }(X^{(12)}) = Sh^{\\star }(X^{(13)}) = Sh^{\\star }(X^{(14)}) =Sh^{\\star }(X^{(15)}) = 0$ ." ] ]
2105.11724
[ [ "Flocculation of suspended cohesive particles in homogeneous isotropic\n turbulence" ], [ "Abstract We investigate the dynamics of cohesive particles in homogeneous isotropic turbulence, based on one-way coupled simulations that include Stokes drag, lubrication, cohesive and direct contact forces.", "We observe a transient flocculation phase characterized by a growing average floc size, followed by a statistically steady equilibrium phase.", "We analyze the temporal evolution of floc size and shape due to aggregation, breakage, and deformation.", "Larger turbulent shear and weaker cohesive forces yield elongated flocs that are smaller in size.", "Flocculation proceeds most rapidly when the fluid and particle time scales are balanced and a suitably defined Stokes number is \\textit{O}(1).", "During the transient stage, cohesive forces of intermediate strength produce flocs of the largest size, as they are strong enough to cause aggregation, but not so strong as to pull the floc into a compact shape.", "Small Stokes numbers and weak turbulence delay the onset of the equilibrium stage.", "During equilibrium, stronger cohesive forces yield flocs of larger size.", "The equilibrium floc size distribution exhibits a preferred size that depends on the cohesive number.", "We observe that flocs are generally elongated by turbulent stresses before breakage.", "Flocs of size close to the Kolmogorov length scale preferentially align themselves with the intermediate strain direction and the vorticity vector.", "Flocs of smaller size tend to align themselves with the extensional strain direction.", "More generally, flocs are aligned with the strongest Lagrangian stretching direction.", "The Kolmogorov scale is seen to limit floc growth.", "We propose a new flocculation model with a variable fractal dimension that predicts the temporal evolution of the floc size and shape." ], [ "Introduction", "Individual cohesive particles suspended in liquid or gaseous fluid flows tend to form larger aggregates, due to attractive inter-particle forces that cause the primary particles to flocculate.", "This mechanism plays a dominant role in environmental processes such as sediment erosion and transport in rivers and oceans, or soil erosion by wind [75], [25], [73], [65].", "In planetary astrophysics, corresponding processes influence the coagulation of dust during the formation of protoplanetary disks [50], [59], [49], [48].", "The emergence of large aggregates due to the flocculation of cohesive primary particles is also highly relevant in the context of a wide range of industrial processes, such as the ingestion of dust in gas turbine engines [4], [58], or the use of membrane separation technologies for wastewater treatment and the production of potable water [8], [35], [44], [29].", "Similarly, the operation of certain types of medical equipment, for example dry powder inhalers [78], [79], [66], [67], involves the formation of agglomerates or flocs.", "The flocculation process is strongly affected by the turbulent nature of the underlying fluid flow.", "Small-scale eddies modify the collision dynamics of the primary particles and hence the growth rate of the flocs, while turbulent stresses can result in the deformation and breakup of larger cohesive flocs.", "Hence the dynamic equilibrium between floc growth and breakup is governed by a complex and delicate balance of hydrodynamic and inter-particle forces.", "A host of experimental studies have provided considerable insight into key aspects of the development of flocs in turbulent shear flows, such as their growth rate [3], [81], [77], [33], the equilibrium size distribution [10], [6], [56], [34], and the transient shape of the flocs [37], [26], [23].", "Based on the early pioneering work by [36], several of these investigations have employed a population balance approach to formulate models for the temporal floc evolution [37], [60], [74], [62], [63].", "Alternative approaches based on the classical work by [61] propose statistical collision equations [28], [80], [32].", "Most of the above approaches do not incorporate detailed information on the overall floc strength, which varies with the floc size and shape, and with the strength of the bonds between the primary cohesive particles [19].", "[45], on the other hand, consider the dependence of the overall floc strength on the number and strength of the bonds within the floc.", "[46] and [24] observe that loosely structured agglomerates fragment more easily during collisions than densely packed ones.", "In recent years, highly resolved numerical simulations have begun to provide a promising new avenue for gaining insight into the interplay of hydrodynamic, inertial and inter-particle forces during the growth, deformation and breakup of aggregates [42].", "The study by [82] focuses on a conceptually simple cellular model flow in order to explore the competition between inertial, drag, and cohesive forces during the flocculation process.", "The authors find that floc growth proceeds most rapidly if the fluid and particle time scales are in equilibrium, so that a suitably defined Stokes number is of order unity.", "Based on simulations in a similar model flow, [57] suggest a criterion for the breakup of aggregates.", "[19] investigate the dynamics, collision and fragmentation of flocs in shear flows, via two-way coupled simulations that account for the modification of the flow by the particles.", "They demonstrate that the particle-fluid interaction induces vortex rings in the flow.", "[17] propose a novel stochastic vortex structure method, and proceed to show that this numerical approach produces realistic collision rates in homogeneous turbulence.", "For flocculation in turbulence, [18] show that the aggregation process influences the background turbulence only weakly.", "Quite recently, [12] and [11] conducted a detailed computational study of cohesive particle aggregation in homogeneous isotropic turbulence, based on two-way coupled direct numerical simulations combined with an adhesive discrete element method.", "The simulations presented in [12], which account for Stokes drag, lubrication and adhesive contact forces, address the early stages of flocculation before an equilibrium size distribution is reached.", "Upon the onset of flocculation, the results demonstrate a time-dependent, exponential size distribution of the flocs for all values of the cohesive force strength.", "Based on this observation, the authors develop an effective agglomeration kernel for the population balance equation that successfully reproduces the DNS results.", "In a follow-up study, [11] investigate the collision-induced breakup of agglomerates in homogeneous isotropic turbulence.", "The authors are able to quantify the fraction of collisions that result in breakage, which presents useful information for closing the population balance equation.", "However, because the simulations focus on the early stages of flocculation before the emergence of an equilibrium size distribution, and because they employ particles with diameter approximately equal to the Kolmogrov scale, they do not allow the authors to assess the role of the Kolmogorov length scale in limiting the floc size, a widely reported experimental observation [21], [15], [7], [33].", "Furthermore, the authors model the cohesive van der Waals force as a “sticky force\" that acts only on contact.", "Several previous studies, on the other hand, have indicated that this attractive force extends over a finite range even before the particles come into contact, so that it can affect the probability that two close-by particles will collide [70], [27], [76], [72].", "The present investigation aims to explore the interplay between floc aggregation, deformation and breakup from inception all the way to the dynamic equilibrium phase, with the goal of obtaining scaling laws for both of these qualitatively different stages.", "Towards this end, we will employ a simulation approach that tracks dispersed individual spherical particles of a given diameter in homogeneous isotropic turbulence.", "The simulations are one-way coupled in the sense that the particles do not modify the fluid flow, although particle-particle interactions are fully accounted for, and the grid spacing employed for calculating the fluid motion is smaller than the particle diameter.", "Sometimes this approach is referred to as “three-way coupled”.", "The simulations account for inter-particle forces based on recently developed advanced collision models for viscous flows [2], along with the cohesive force model of [72].", "The homogeneous isotropic turbulence is generated and maintained via the forcing method of [20].", "We will employ these simulations in order to investigate the floc size and shape evolution, the floc size distribution during the equilibrium stage, the orientation of the flocs with regard to the principal directions of the Eulerian strain and the Lagrangian stretching, as well as the role of the Kolmogorov length scale in limiting floc growth.", "Based on our findings, we then propose a novel flocculation model that predicts the evolution of the floc size and shape with time.", "To assess the performance of this new flocculation model, we will compare its predictions to those obtained with existing models in the literature.", "The paper is structured along the following lines.", "Section briefly reviews the governing equations for the fluid flow and the particle motion, and it describes the computational approach.", "It identifies the governing dimensionless parameters and quantifies the range over which they will be varied in the present investigation.", "The properties of the turbulent flow fields are described in Section , and their statistically stationary and isotropic nature is discussed.", "Starting from 10,000 randomly distributed individual particles, we then analyze the temporal evolution of the floc size and shape as a result of aggregation, deformation and breakage in Section .", "Here we will distinguish between the transient flocculation stage and the equilibrium stage, and we will discuss the underlying physical mechanisms.", "We will furthermore analyze the alignment of the flocs with regard to the principal strain directions of the turbulent velocity field, and we will focus on how the Kolmogorov scale affects the maximum floc size.", "Subsequently, we introduce the new flocculation model in Section , and we compare its predictions to those obtained from existing models.", "Section summarizes the main findings of the current investigation, and presents its key conclusions.", "We consider the one-way coupled motion of suspended cohesive particles in three-dimensional, incompressible homogeneous isotropic turbulence.", "The motion of the single-phase fluid with constant density $\\rho _f$ and kinematic viscosity $\\nu $ is governed by $ \\nabla \\cdot {u_f} = 0 \\ ,$ $ \\frac{\\partial {u_f} }{\\partial t} + ({u_f} \\cdot \\nabla ) {u_f} = - \\frac{1}{\\rho _f} \\nabla p + \\nu \\nabla ^2 {u_f} + F_{tur} \\ ,$ where ${u_f} = (u_f, v_f, w_f)^{\\rm T}$ denotes the fluid velocity vector and $p$ indicates the hydrodynamic pressure.", "We employ the spectral approach of [20] to obtain the forcing term $F_{tur}$ , which generates and maintains statistically stationary turbulence, as implemented in [13].", "Here, $F_{tur}$ is non-zero only in the low-wavenumber band where the wavenumber vector $|\\kappa | < \\kappa _f$ , with $\\kappa _f = 2.3\\kappa _0$ and $\\kappa _0 = 2 \\pi / L_0$ , with $L_0$ denoting the length of the physical domain.", "The origin $\\kappa = 0$ is not forced.", "In addition to the cutoff wavenumber $\\kappa _f$ , the random forcing process is governed by the dimensionless parameter $D_s = \\sigma ^2 T_0 L_0^4 / \\nu ^3$ , where $\\sigma ^2$ and $T_0$ indicate the variance and the time scale of the random process, respectively.", "Regarding the details of evaluating $F_{tur}$ from $\\kappa _f$ and $D_s$ , we refer the reader to the original work by [20].", "We approximate each primary suspended particle $i$ as a sphere moving with translational velocity ${u}_{p,i} = (u_{p,i}, v_{p,i}, w_{p,i})^{\\rm T}$ and angular velocity ${\\omega }_{p,i}$ .", "These are obtained from the linear and angular momentum equations $ m_p \\frac{\\mathrm {d}{u}_{p,i}}{\\mathrm {d}t} = {F}_{d,i} + \\underbrace{\\sum _{j=1,j \\ne i}^{N}(F_{con,ij} + F_{lub,ij} + F_{coh,ij})}_{{F}_{c,i}} \\ ,$ $ I_p \\frac{\\mathrm {d}{\\omega _{p,i}}}{\\mathrm {d}t} = \\underbrace{\\sum _{j=1,j \\ne i}^{N}(T_{con,ij} + T_{lub,ij})}_{{T}_{c,i}} \\ ,$ where the primary particle $i$ moves in response to the Stokes drag force $F_{d,i} = -3 \\pi D_p \\mu ({u}_{p,i} - {u}_{f,i})$ , and the particle-particle interaction force $F_{c,i}$ .", "Buoyancy is not considered here, so that we can investigate the effects of particle inertia in isolation.", "We only consider primary particles that are larger than 2$\\mu m$ (cf.", "table 1), so that a suitably defined Peclet number measuring the relative importance of hydrodynamic and Brownian forces is sufficiently large for their Brownian motion to be negligible [2], [12], [72], [52].", "${u}_{p,i}$ indicates the particle velocity evaluated at the particle center.", "${u}_{f,i} = \\sum _{1}^{N_i}(\\phi _{i,k} {u}_{f,k})$ represents the fluid velocity averaged over the volume of particle $i$ , where $N_{i}$ denotes the number of Eulerian grid cells covered by particle $i$ , ${u}_{f,k}$ is the fluid velocity at the center of the grid cell $k$ , and $\\phi _{i,k}$ is the volume fraction of the particle $i$ in the grid cell $k$ .", "We remark that the above implies that the diameter $D_p$ of the primary particle should be larger than the grid spacing $h$ .", "This avoids the need for interpolating the fluid velocity within one grid cell, which would be required if $D_p < h$ [12].", "$m_p$ denotes the particle mass, $\\mu $ the dynamic viscosity of the fluid, and $N$ the total number of particles in the flow.", "We assume all particles to have the same diameter $D_p$ and density $\\rho _p$ .", "$F_{c,i}$ accounts for the direct contact force $F_{con,ij}$ in both the normal and tangential direction, as well as for short-range normal and tangential forces due to lubrication $F_{lub,ij}$ and cohesion $F_{coh,ij}$ , where the subscript $ij$ indicates the interaction between particles $i$ and $j$ .", "$I_p = \\pi \\rho _p D_p^5 / 60$ denotes the moment of inertia of the particle.", "$T_{c,i}$ represents the torque due to particle-particle interactions, where we distinguish between direct contact torque $T_{con,ij}$ and lubrication torque $T_{lub,ij}$ .", "Within a large floc, we account for all of the individual binary particle interactions.", "The lubrication force $F_{lub,ij}$ is accounted for based on [16] as implemented in [82].", "We note that, although the present study is limited to monodisperse particles, polydisperse particle-particle interactions can be taken into account by an effective radius $R_{eff}=R_p R_q / (R_p + R_q)$ , where $R_p$ and $R_q$ are the radii of two interacting spheres.", "Following [2], the collision force $F_{con,ij}$ is represented by a nonlinear spring–dashpot model in the normal direction, while the tangential component is modelled by a linear spring–dashpot model capped by the Coulomb friction law to account for zero-slip rolling or sliding of particles.", "We note that the tangential component of the contact force depends on the surface roughness, a prescribed restitution coefficient $e_{dry} = 0.97$ and a friction coefficient $e_{fri} = 0.15$ are implemented to yield adaptively calibration for every collision as described by [2].", "The cohesive force $F_{coh,ij}$ , which reflects the combined influence of the attractive van der Waals force and the repulsive electrostatic force, is based on the work of [72], where additional details and validation results are provided.", "The model assumes a parabolic force profile, distributed over a thin shell surrounding each primary particle.", "Hence the cohesive force between primary particles extends over a finite range, so that it is felt by the particles even before they come into direct contact.", "We consider two primary particles to be part of the same floc when their surface distance is smaller than half the range of the cohesive force, as implemented in [82].", "We remark that, based on equations (REF ) and (REF ), the configuration of the primary particles within a floc can change with time in response to fluid forces, since the cohesive bonds are not rigid.", "Specifically, the contact points on the surface of the primary particles are not fixed, so that the primary particles can rotate individually within a floc." ], [ "Nondimensionalization", "In order to render the above governing equations dimensionless, we consider primary particles with diameter $D_p = 5 \\ \\rm {\\mu m}$ , which represents a typical value for clay or fine silt.", "The cubic computational domain has an edge length $L_0 = 125D_p = 6.25 \\times 10^{-4} \\ \\rm {m}$ .", "As time scale of the random turbulent forcing process we select $T_0 = 7.81 \\times 10^{-5} \\ \\rm {s}$ .", "By choosing $L_0$ , $T_0$ and $\\rho _f = 1,000 \\ \\rm {kg/m^{3}}$ as the characteristic length, time and density scales, we obtain the characteristic velocity scale $U_0 = L_0/T_0 = 8 \\ \\rm {m/s}$ , which is similar to values employed in previous investigations [12], [11].", "We employ $L_0$ and $U_0$ to define the turbulence Reynolds number $Re = L_0 U_0 \\rho _f/ \\mu $ .", "The dimensionless continuity and momentum conservation equations can then be expressed as $ \\tilde{\\nabla } \\cdot \\tilde{u}_f = 0 \\ ,$ $ \\frac{\\partial \\tilde{u}_f }{\\partial \\tilde{t}} + (\\tilde{u}_f \\cdot \\tilde{\\nabla }) \\tilde{u}_f = - \\tilde{\\nabla } \\tilde{p} + \\frac{1}{Re} \\tilde{\\nabla }^2 \\tilde{u}_f + \\tilde{F}_{tur} \\ ,$ while the dimensionless equations of motion for the primary cohesive particles take the form $ \\tilde{m}_p \\frac{\\mathrm {d} \\tilde{u}_{p,i}}{\\mathrm {d} \\tilde{t}} = \\underbrace{- \\frac{3 \\pi \\tilde{D}_p (\\tilde{u}_{p,i} - \\tilde{u}_{f,i})}{Re}}_{\\tilde{F}_{d,i}} + \\sum _{j=1,j \\ne i}^{N}(\\tilde{F}_{con,ij} + \\tilde{F}_{lub,ij} + \\tilde{F}_{coh,ij} ) \\ ,$ $\\tilde{I}_p \\frac{\\mathrm {d} \\tilde{\\omega }_{p,i}}{\\mathrm {d} \\tilde{t}} = \\sum _{j=1,j \\ne i}^{N}(\\tilde{T}_{con,ij} + \\tilde{T}_{lub,ij}) \\ .$ Here dimensionless quantities are denoted by a tilde.", "The dimensionless particle mass is defined as $\\tilde{m}_p = \\pi \\tilde{D}_p^3 \\tilde{\\rho }_s / 6$ , the moment of inertia $\\tilde{I}_p = \\pi \\tilde{\\rho }_s \\tilde{D}_p^5 / 60$ , and the density ratio $\\tilde{\\rho }_s = \\rho _p / \\rho _f$ .", "The dimensionless direct contact and lubrication forces, $\\tilde{F}_{con,ij}$ and $\\tilde{F}_{lub,ij}$ , are accounted for based on [82], while the dimensionless cohesive force $\\tilde{F}_{coh,ij}$ is defined as $\\tilde{F}_{coh,ij} = \\left\\lbrace \\begin{array}{ll}- 4 Co \\frac{\\tilde{\\zeta }_{n,ij}^2 - \\tilde{h}_{co} \\tilde{\\zeta }_{n,ij}}{\\tilde{h}_{co}^2} n, & \\tilde{\\zeta }_{min} < \\tilde{\\zeta }_{n,ij} \\leqslant \\tilde{h}_{co} \\ , \\\\0, & \\rm {otherwise} \\ .\\end{array}\\right.$ Here $\\tilde{\\zeta }_{min} = 0.0015 \\tilde{D}_p$ and $\\tilde{h}_{co} = 0.05 \\tilde{D}_p$ represent the surface roughness of the particles and the range of the cohesive force, respectively.", "$n$ represents the outward-pointing normal on the particle surface, while $\\tilde{\\zeta }_{n, ij}$ is the normal surface distance between particles $i$ and $j$ .", "The cohesive number $Co$ indicates the ratio of the maximum cohesive force $\\vert \\vert {F_{coh,ij}}\\vert \\vert $ at $\\tilde{\\zeta }_{n,ij} = \\tilde{h}_{co}/2$ to the characteristic inertial force $ Co = \\frac{{\\rm max} (\\vert \\vert {F_{coh,ij}}\\vert \\vert )}{U_0^2 L_0^2 \\rho _f} = \\frac{A_H D_p}{16 h_{co} \\zeta _{0} } \\frac{1}{U_0^2 L_0^2 \\rho _f } \\ ,$ where the Hamaker constant $A_H$ is a function of the particle and fluid properties, and the characteristic distance $\\zeta _0 = 0.00025D_p$ .", "[72] provide representative values of various physicochemical parameters such as $A_H$ , salt concentration and grain size of the primary particles for common natural systems.", "The present numerical approach for simulating the dynamics of cohesive sediment has been employed to predict the flocculation in simple vortical flow fields, and it was successfully validated with experimental data in our earlier work [82].", "To summarize, the simulations require as direct input parameters the turbulence Reynolds number $Re$ , the characteristic parameter of the random turbulent forcing process $D_s$ , the dimensionless particle diameter $\\tilde{D}_p$ , the total number of particles $N$ , the density ratio $\\tilde{\\rho }_s$ , and the cohesive number $Co$ .", "As we will discuss below, $Re$ and $D_s$ can equivalently be expressed by the shear rate $G$ of the turbulence, cf.", "equation (REF ), and the Stokes number $St$ defined by equation (REF ).", "A list of the relevant dimensionless parameters is provided in table REF .", "We remark that due to computational limitations the simulations consider Kolmogorov scales that are somewhat smaller than typical field values, and turbulent shear rates that are larger than field values.", "Hence the ratio of the Kolmogorov length scale to the primary particle size takes values up to 3.3 in the simulations, as compared to values up to $O(10)$ under typical field conditions.", "For convenience, the tilde symbol will be omitted henceforth.", "Table: Nondimensionalization employed in the present work: the characteristic values for length, velocity and density are L 0 =125D p =6.25×10 -4 mL_0 = 125D_p = 6.25 \\times 10^{-4} \\ \\rm m, U 0 =8m/sU_0 = 8 \\ \\rm {m/s} and ρ f =1,000 kg /m 3 \\rho _f = 1,000 \\ \\rm {kg/m^3}, respectively." ], [ "Computational set-up", "The triply periodic computational domain $\\Omega $ has a dimensionless size of $L_x \\times L_y \\times L_z = 1 \\times 1 \\times 1$ , with the number of grid cells $N_x \\times N_y \\times N_z = 128 \\times 128 \\times 128$ .", "This relatively modest number of grid points enables us to conduct the simulations over sufficiently long times for the flocculation and break-up processes to reach an equilibrium state [68], and it is in line with the earlier study of [12].", "As mentioned above, we set the diameter $D_p$ of the primary particles moderately larger than the grid size $h = L_x/N_x$ , at a constant value $D_p/h=1.024$ .", "Before introducing the particles into the flow, we simulate the single-phase turbulence until it reaches a statistically stationary state.", "Table REF gives an overview of the physical parameters for the simulations conducted within the present investigation.", "Here the Kolmogorov length scale and the root-mean-square velocity are defined as $\\eta = 1/ (Re^3 \\epsilon )^{1/4}$ and $u_{rms} = (2 k /3)^{1/2}$ , respectively, where $\\epsilon $ and $k$ denote the domain-averaged dissipation rate and kinetic energy of the fluctuations.", "The Taylor Reynolds number $Re_{\\lambda } = \\lambda u_{rms} Re$ of the turbulence is based on the Taylor microscale $\\lambda = \\sqrt{15} \\, u_{rms}/(Re \\, \\epsilon )^{1/2}$ .", "To provide a more complete quantitative description of the fluid shear, we define the vorticity fluctuation amplitude $ G = \\frac{1}{Re \\, \\eta ^2} \\ ,$ which can also be regarded as the turbulent shear rate.", "For additional details with regard to these quantities, we refer the reader to [53].", "Table: Physical parameters of the single-phase turbulence simulations.", "As input parameters we specify the fluid Reynolds number Re=L 0 U 0 /νRe = L_0 U_0 / \\nu and the characteristic parameter of the random turbulent forcing process D s =σ 2 T 0 L 0 4 Re 3 D_s = \\sigma ^2 T_0 L_0^4 Re^3.", "The simulation then yields the Taylor Reynolds number Re λ =λu rms ReRe_{\\lambda } = \\lambda u_{rms} Re, the Kolmogorov scale η\\eta , the average root-mean-square velocity u rms u_{rms}, and the shear rate G=1/(Reη 2 )G = 1/ (Re \\, \\eta ^2).", "All of these output quantities are obtained by averaging over space and time, after a statistically stationary state has evolved." ], [ "Turbulence properties for different $Re_{\\lambda }$", "One key goal of the present investigation is to study the flocculation of primary particles whose diameter $D_p$ is smaller than the Kolmogorov length scale $\\eta $ .", "Since the particle diameter needs to be larger than the grid spacing, and the number of grid points is limited, suitable values of $\\eta $ require a relatively low $Re_{\\lambda }$ .", "On the other hand, it is known that for $Re_{\\lambda } \\le O(50)$ the turbulence may not be fully developed and isotropic [41].", "Hence this section presents a more detailed discussion of the turbulence properties for $Re_{\\lambda } \\le O(50)$ .", "Figure REF shows the time-dependent evolution of the box-averaged Kolmogorov length $\\eta $ , the root-mean-square velocity $u_{rms}$ , the Taylor Reynolds number $Re_{\\lambda }$ , and the shear rate $G$ for cases Tur1 and Tur8, which have time-averaged Taylor Reynolds numbers of 9.72 and 50.34, respectively.", "Both cases are seen to reach statistically stationary states.", "We note that while case Tur8 results in $\\eta /h = 0.6656$ , [13] demonstrated the validity of the current turbulent forcing approach even when the Kolmogorov length is smaller than the grid spacing.", "Snapshots of the vorticity modulus in a slice of the computational domain are shown in Figure REF .", "They exhibit the intermittent multiscale patterns featuring eddies of different size along with thin filaments that are typical for turbulence.", "Figure: Temporal evolution of box-averaged turbulence properties for cases Tur1 and Tur8 in table : (a) Kolmogorov length scale η\\eta ; (b) Root-mean-square velocity u rms u_{rms}; (c) Taylor Reynolds number Re λ Re_{\\lambda }; (d) Shear rate GG.", "A statistically stationary state is seen to evolve for all quantities.Figure: Representative snapshots of the vorticity modulus normalized by the vorticity fluctuation amplitude GG, shown in the plane z=0.5z = 0.5.", "(a) Case Tur1; (b) Case Tur8.Figure REF shows the temporal evolution of the domain-averaged magnitude of the velocity components $\\langle |u_f| \\rangle _{\\Omega }$ , $\\langle |v_f| \\rangle _{\\Omega }$ and $\\langle |w_f| \\rangle _{\\Omega }$ .", "During the statistically stationary state the three components are seen to oscillate around similar average values for both Tur1 and Tur8, which indicates that the flow is isotropic to a good approximation.", "Figure: Temporal evolution of box-averaged magnitude of the fluid velocity components: (a) Case Tur1; (b) Case Tur8.", "The flow is seen to be isotropic to a good approximation.We define the instantaneous kinetic energy components in Fourier space, $E_{11}(\\kappa )$ , $E_{22}(\\kappa )$ and $E_{33}(\\kappa )$ , as $\\int _0^\\infty E_{11}(\\kappa )\\, d \\kappa = & \\langle \\frac{u_f \\cdot u_f}{2} \\rangle _{\\Omega } \\ , \\\\\\int _0^\\infty E_{22}(\\kappa )\\, d \\kappa = & \\langle \\frac{v_f \\cdot v_f}{2} \\rangle _{\\Omega } \\ , \\\\\\int _0^\\infty E_{33}(\\kappa )\\, d \\kappa = & \\langle \\frac{w_f \\cdot w_f}{2} \\rangle _{\\Omega } \\ ,$ where $\\kappa = |\\kappa |$ denotes the wavenumber.", "Figure REF shows the time-averaged one-dimensional energy spectra.", "Only the wavenumbers below the cutoff wavenumber ($\\kappa _f$ , shown as vertical dashed lines in Figure REF ) are forced.", "The shapes of the energy spectra are in qualitative agreement with those obtained by [13] for higher values of $Re_{\\lambda } \\approx 60$ .", "We conclude that the present forcing scheme yields statistically steady flow fields that are approximately isotropic for the current range of $Re_{\\lambda }$ -values.", "Figure: Time-averaged one-dimensional energy spectra.", "the vertical dashed lines indicate the respective cutoff wavenumber of the turbulence forcing scheme, κ f η=2.3(2π/L x )η\\kappa _f \\, \\eta = 2.3 (2\\pi /L_x) \\eta .", "(a) Case Tur1; (b) Case Tur8.", "The spectra confirm that the statistically stationary flow fields are approximately isotropic." ], [ "One-way coupling", "Once the single-phase turbulence reaches the statistically stationary regime, $N = 10,000$ identical cohesive particles with diameter $D_p=0.008$ are randomly distributed throughout the domain, resulting in a particle volume fraction $\\phi _p = 0.268\\%$ .", "Initially all particles are at rest and separated by a distance larger than the cohesive range $h_{co}$ .", "To improve the statistics, we carry out repeated simulations for different random initial conditions, as the simulation results are statistically independent of the initial particle placement.", "The simulations to be discussed in the following are one-way coupled, so that the particles do not modify the background turbulence.", "[5] find that particle loading can modify the turbulence statistics even for volume fractions as low as $10^{-5}$ , so that we expect two-way coupling effects to have an impact on the flocculation process even in moderately dilute flows.", "In addition, even for globally dilute flows the local volume fraction inside a floc will be $O(1)$ , so that the one-way coupled assumption generally will not hold inside a floc.", "However, fully two-way coupled simulations for sufficiently many particles to obtain reliable statistical information, and for sufficiently long times to explore the balance between aggregation and breakup during the equilibrium stage, are not feasible on currently available supercomputers.", "Our assumption of one-way coupling hence limits the volume and mass fractions that we can reasonably consider.", "On the other hand, the current simulations and their comparisons to experimental observations are useful in that they help address the question as to which aspects of flocculation are governed by one-way coupled dynamics, and which other aspects require fully two-way coupled dynamics.", "As we will see below, for the range of physical parameters listed in table REF , even one-way coupled simulations are able to reproduce several experimentally observed statistical features of flocculation dynamics.", "We adopt a multiscale time-stepping approach in which the fluid motion is calculated with a time step $\\Delta t$ based on the criterion that the Courant-Friedrichs-Lewy number $\\rm CFL \\le 0.5$ .", "The particle motion, on the other hand, is evaluated with a much smaller time step $\\Delta t_p = \\Delta t / 15$ .", "Since the computational approach maintains a contact duration of $T_c = 10 \\Delta t = 150 \\Delta t_p$ [2], each particle collision is effectively resolved by 150 substeps, at the price of a marginal increase in the computational cost.", "The dynamics of the primary particles are characterized by the Kolmogorov-scale Stokes number $ St = \\frac{\\rho _s}{18} \\frac{D_p^2}{\\eta ^2} Re_{\\eta } \\ ,$ where the Kolmogorov Reynolds number $Re_{\\eta } = \\eta \\, u_{rms} \\, Re$ .", "Since the particle diameter $D_p$ is constant throughout the present investigation, $St$ depends on the density ratio $\\rho _s$ and the fluid properties.", "A particle with a small Stokes number tends to follow the fluid motion, while the dynamics of a particle with a large Stokes number is dominated by its inertia, so that it tends to continue along its initial direction of motion.", "Table REF summarizes the physical parameters of the simulations that we conducted.", "Following our analysis from §REF and the examples given in Appendix A of [72], these values correspond to primary silica particles with a grain size of fine to medium silt in ocean water.", "In the following, we will investigate how the flocculation dynamics are influenced by the cohesive number $Co$ , the Stokes number $St$ and the shear rate $G$ .", "We remark that the density ratio $\\rho _s$ and the size ratio $\\eta /D_p$ are implicitly accounted for by $St$ and $G$ .", "Table: Physical parameters of the flocculation simulations.", "We separately investigate the influence of the cohesive number CoCo (based on Flo1-5), the shear rate GG (Flo6-9), and the Stokes number StSt (Flo10-13).", "The effects of ρ s \\rho _s and η/D p \\eta /D_p are implicitly accounted for by StSt and GG." ], [ "Flocculation and equilibrium stages", "When the surface distance between two particles is smaller than half the range of the cohesive force, $h_{co}/2$ , we consider these particles to be part of the same floc.", "Hence, in terms of a physical force balance breakage occurs when the net force pulling the particles apart is sufficiently strong to overcome the maximum of the cohesive force holding the particles together.", "An individual particle is considered to be the smallest possible floc.", "Figure REF shows the evolution of the number of flocs $N_f(t)$ with time for the representative case Flo9, with $Co = 1.2 \\times 10^{-7}$ , $St = 0.06$ , $G = 0.62$ , $\\rho _s = 2.65$ and $\\eta /D_p = 2.25$ .", "As a result of flocculation, $N_f$ decreases rapidly with time from its initial value of 10,000, before levelling off around a constant value $N_{f,eq}$ that reflects a stable equilibrium between aggregation and breakage.", "This tendency of $N_f$ is consistent with our previous observation of flocculation in steady cellular flow fields [82].", "Consequently, we can identify two pronounced stages of the flow, viz.", "an initial flocculation stage and a subsequent equilibrium stage.", "We define the end of the flocculation stage, i.e., the onset of the equilibrium stage, as the time $t_{eq}$ when $N_f$ first equals $N_{f,eq}$ .", "Figure REF shows separately the number of flocs with $N_{p} = 1, 2, 3$ and more than three primary particles.", "While the number of flocs with two or three particles initially grows quickly, they soon reach a peak and subsequently decline, as more flocs of larger sizes form.", "Toward the end of the flocculation stage, a stable equilibrium of the different floc sizes begins to emerge, although the distribution of flocs with different numbers of primary particles is still changing slowly.", "In order to gain insight into the dynamics of floc growth and breakage, we keep track of the evolution of three different types of flocs over a suitably specified time interval $\\Delta T$ : a) those flocs that maintain their identity, i.e., they consist of the same primary particles at the start and the end of the time interval; b) those that add additional primary particles while keeping all of their original ones; and c) all others, i.e., all those who have undergone a breakage event during the time interval.", "We denote the fractions of these respective groups as $\\theta _{id} = N_{f,id} / N_f$ , $\\theta _{ad} = N_{f,ad} / N_f$ , and $\\theta _{br} = N_{f,br} / N_f$ .", "It follows that $ \\theta _{id} + \\theta _{ad} + \\theta _{br} = 1 \\ .$ We found that a value of $\\Delta T = 3$ is suitable for obtaining insight into the dynamics of the flocculation process, as it allows most of the flocs to maintain their identity during the time interval, while smaller but still significant numbers undergo primary particle addition or breakage.", "Figure REF shows the evolution of $\\theta _{id}$ , $\\theta _{ad}$ and $\\theta _{br}$ for case Flo9.", "After an initial transient stage, all three fractions reach statistically steady states.", "Interestingly, even during the equilibrium stage when $N_f \\approx const.$ , we observe that $\\theta _{ad} \\ne \\theta _{br}$ .", "This reflects events such as when one floc breaks into three smaller parts, two of which then merge with other flocs.", "Here the total number of flocs remains unchanged at three, in spite of only one break-up but two particle addition events." ], [ "Evolution of floc size and shape", "While the number of primary particles in a floc, $N_{p}$ , provides a rough measure of its size, flocs with identical values of $N_p$ can have very different shapes.", "In order to capture this effect, we define the characteristic diameter $D_f$ of the floc, also known as the Feret diameter, as $D_f = 2 {\\rm max}(\\Vert x_{p,i} - x_c \\Vert ) + D_p, \\ \\ \\ 1 \\leqslant i \\leqslant N_{p} \\ ,$ as well as its gyration diameter $D_g$ [12], $D_g = \\left\\lbrace \\begin{array}{lll}2\\sqrt{\\frac{1}{N_{p}} \\sum _{i=1}^{N_{p}} \\Vert x_{p,i} - x_c \\Vert ^2}, & N_{p} > 2 \\ , \\\\\\\\\\sqrt{1.6}D_p, & N_{p} = 2 \\ , \\\\\\\\D_p, & N_{p} = 1 \\ .", "\\\\\\end{array}\\right.$ Here $x_{p,i}$ denotes the position of the center of primary particle $i$ , and the floc's center of mass is evaluated as $x_c = \\sum _{i=1}^{N_{p}} x_{p,i} / N_{p}$ .", "While the characteristic diameter $D_f$ more closely represents the true spatial extent of the floc, the gyration diameter $D_g$ also accounts for the irregularity of the floc shape.", "Following [30], [31], we then calculate the fractal dimension $n_f$ of the floc $ n_f = \\frac{\\log N_{p}}{\\log \\frac{D_f}{D_p}} \\ ,$ as a measure of its compactness.", "A dense, nearly spherical floc has $n_f \\approx 3$ , while for a linear floc $n_f \\approx 1$ .", "When $N_{p} = 1$ and $D_f/D_p = 1$ , the above definition of the fractal dimension does not yield a finite value, and we set $n_f = 1$ .", "In this way, the definition of the fractal dimension is continuous between $N_p=1$ and $N_p=2$ .", "It is important to note that this differs from previous studies, which usually set $n_f = 3$ for this case [30], [31], [37], [63].", "For a typical floc with $N_p = 7$ that maintains its identity, figure REF shows the evolution of $D_f$ and $n_f$ over time.", "During the time interval $200 \\leqslant t \\leqslant 210$ , hydrodynamic forces deform the floc so that it becomes more compact, which reduces $D_f$ and increases $n_f$ .", "Later on, near $t \\approx 240$ , the floc is being stretched, which modifies $D_f$ and $n_f$ in the opposite directions.", "Figure: Temporal evolution of the characteristic diameter D f D_f and the fractal dimension n f n_f of a typical floc that maintains its identity over the time interval considered.", "Three instants are marked by vertical dashed lines, and the corresponding floc shapes are shown.", "In response to the fluid forces acting on it, the floc first changes from a slightly elongated to a more compact shape, and subsequently to a more strongly elongated one.", "The floc with seven primary particles is taken from case Flo10 with governing parameters Co=1.2×10 -7 Co = 1.2 \\times 10^{-7}, St=0.1St = 0.1, G=0.91G = 0.91.Figure REF shows the evolution with time of the various floc size measures, for cases Flo6-9 in table REF with different turbulent shear rates $G$ .", "The other parameters are kept approximately constant at $Co = 1.2 \\times 10^{-7}$ , $St = 0.06$ , $\\rho _s = 2.65$ , and $2.24 \\leqslant \\eta /D_p \\leqslant 2.28$ .", "As can be seen from figure REF , a smaller shear rate results in a longer transient phase before the average number of primary particles per floc $\\overline{N}_{p} = N / N_f$ reaches an equilibrium.", "A smaller value of $G$ furthermore gives rise to an equilibrium stage characterized by fewer flocs with more primary particles, since the weaker hydrodynamic stresses cannot break up the flocs as easily.", "Figures REF and REF indicate that both the average characteristic diameter $\\overline{D}_f$ and the average gyration diameter $\\overline{D}_g$ increase for smaller $G$ .", "This is consistent with previous observations by other authors in both laboratory experiments [26], [23] and river estuaries [39], [40].", "Both $\\overline{D}_g$ and $\\overline{D}_f$ remain smaller than the Kolmogorov length scale $0.0179 \\leqslant \\eta \\leqslant 0.0183$ for all cases.", "Since flocs with one or two primary particles have a constant fractal dimension $n_f = 1$ , we evaluate the average fractal dimension $\\overline{n}_{f,lar}$ from only those flocs with three or more particles.", "Figure REF shows that $\\overline{n}_{f,lar}$ increases for smaller shear rates, which demonstrates that for weaker turbulence the floc shape tends to be more compact.", "This finding is consistent with experimental observations by [26], whereas previous numerical work by [12] reports a constant value $\\overline{n}_{f,lar} = 1.64$ .", "Figure: Temporal evolution of various floc size measures for different turbulent shear rates GG, with Co=1.2×10 -7 Co = 1.2 \\times 10^{-7}, St=0.06St = 0.06, ρ s =2.65\\rho _s = 2.65, and 2.24⩽η/D p ⩽2.282.24 \\leqslant \\eta /D_p \\leqslant 2.28 (cases Flo6-9).", "(a) Average number of primary particles per floc N ¯ p \\overline{N}_p; (b) Average characteristic floc diameter D ¯ f \\overline{D}_f; (c) Average floc gyration diameter D ¯ g \\overline{D}_g; (d) Average fractal dimension n ¯ f,lar \\overline{n}_{f,lar} of flocs with three or more primary particles.", "Larger turbulent shear results in smaller flocs, with fewer primary particles and more elongated shapes.Figure REF discusses the floc growth during the very early flow stages, as a function of the turbulent shear rate $G$ .", "As seen in Figure REF , the evolution of $\\overline{D}_f(t)$ can be closely approximated by an exponential function of the form $ \\overline{D}_f = b_1 (e^{b_2 t} - 1) + D_p \\ ,$ where $b_1$ and $b_2$ represent fitting coefficients.", "Based on corresponding fits for different values of $G$ , Figure REF displays the time-dependent floc growth rate $d\\overline{D}_f /dt$ for different $G$ .", "Consistent with the experimental observations by [26], we find stronger shear to cause more rapid flocculation for $t < 5$ .", "After this early stage the trends reverse, which reflects the fact that the equilibrium stage is reached faster for stronger turbulence.", "This agrees with the experimental findings by [7], who also reported the equilibrium stage to emerge more quickly for stronger turbulence, due to more frequent floc collisions.", "We remark that the evolution of $\\overline{N}_{p}$ (not shown) exhibits corresponding trends.", "Figure: Early-stage flocculation rate for different turbulent shear rates GG, with Co=1.2×10 -7 Co = 1.2 \\times 10^{-7}, St=0.06St = 0.06, ρ s =2.65\\rho _s = 2.65, and η/D p ≈2.26\\eta /D_p \\approx 2.26 (cases Flo6-9).", "(a) The early-stage simulation results for D ¯ f (t)\\overline{D}_f(t) can be accurately fitted by an exponential relation, as shown for the representative case Flo6 with G=1.49G = 1.49; (b) The flocculation rate d(D ¯ f )/dtd(\\overline{D}_f)/dt obtained from the exponential fits of D ¯ f (t)\\overline{D}_f(t).", "Initially flocs grow fastest in strong turbulence.", "Subsequently their growth rate decays, as the equilibrium stage is reached more rapidly for strong turbulence.Figure: Temporal evolution of various floc size measures, for different Stokes number values StSt, with Co=1.2×10 -7 Co = 1.2 \\times 10^{-7}, G=0.91G = 0.91 and η/D p =1.85\\eta /D_p = 1.85 (cases Flo10-13).", "(a) Average number of primary particles per floc N ¯ p \\overline{N}_p; (b) Average characteristic floc diameter D ¯ f \\overline{D}_f; (c) Early-stage flocculation rate d(D ¯ f )/dtd(\\overline{D}_f)/dt obtained from exponential fits of D ¯ f (t)\\overline{D}_f(t); (d) Average fractal dimension n ¯ f,lar \\overline{n}_{f,lar} of flocs with three or more primary particles.", "During the equilibrium stage, the number of primary particles per floc, the characteristic floc diameter, and the fractal dimension all increase for smaller Stokes numbers.", "Initially, flocs with St≈O(1)St \\approx O(1) exhibit the fastest growth.Figure REF presents corresponding floc size results for different Stokes numbers, obtained from cases Flo10-13 in table REF .", "These simulations all employ the same turbulent flow Tur6, so that they have constant parameter values $Co = 1.2 \\times 10^{-7}$ , $G = 0.91$ and $\\eta /D_p = 1.85$ .", "$St$ is varied by changing the density ratio $\\rho _s$ .", "Figures REF and REF indicate that the equilibrium values of both $\\overline{N}_{p}$ and $\\overline{D}_f$ increase for smaller $St$ .", "This reflects the fact that cohesive forces become more dominant for smaller $St$ , due to the lower drag force and the shorter particle response time.", "By again employing exponential fits for the early stages, we obtain the floc growth rate $d\\overline{D}_f / dt$ for different $St$ -values, as shown in Figure REF .", "Initially flocs with intermediate Stokes numbers of O(1) are seen to grow most rapidly, consistent with our earlier findings for two-dimensional cellular flows [82].", "This trend changes for $t > 20$ , due to the later onset of the equilibrium stage for small Stokes numbers.", "The time evolution of the average fractal dimension $\\overline{n}_{f,lar}$ of flocs with three or more primary particles is shown in Figure REF .", "It demonstrates that smaller Stokes numbers result in more compact flocs.", "Figure REF analyzes the influence of the cohesive number $Co$ by comparing cases Flo1-5 in table REF .", "The other parameters are held constant at $St = 0.02$ , $G = 0.29$ $\\rho _s = 2.65$ and $\\eta /D_p = 3.30$ .", "We note that due to the small values of $St$ and $G$ , the emergence of an equilibrium stage takes longer in these simulations.", "In fact, for case Flo5 with $Co = 1.2 \\times 10^{-7}$ , an equilibrium had not yet formed by $t=20,000$ , when the simulation terminated.", "Nevertheless, the simulations demonstrate the tendency of higher $Co$ to result in larger values of $\\overline{N}_{p}$ during all phases of the flow, cf.", "Figure REF .", "Interestingly, however, we observe that during the transient flow stages the flocs for $Co = 6 \\times 10^{-8}$ have larger average diameters $\\overline{D}_f$ and $\\overline{D}_g$ than those for $Co = 1.2 \\times 10^{-7}$ , even though they contain fewer primary particles, cf.", "Figures REF and REF .", "The explanation for this finding is given by Figure REF , which indicates that for $Co = 1.2 \\times 10^{-7}$ the flocs have a higher average fractal dimension $\\overline{n}_f$ and are more compact than those for $Co = 6 \\times 10^{-8}$ , which can be deformed more easily by turbulent stresses.", "In summary, as a general trend we observe that during the equilibrium stages weaker turbulence, lower Stokes numbers and higher cohesive numbers result in larger and more compact flocs.", "Figure: Temporal evolution of various floc size measures for different values of the cohesive number CoCo, with St=0.02St = 0.02, G=0.29G = 0.29, ρ s =2.65\\rho _s = 2.65 and η/D p =3.30\\eta /D_p = 3.30 (cases Flo1-5).", "(a) Average number of primary particles per floc N ¯ p \\overline{N}_p; (b) Average characteristic floc diameter D ¯ f \\overline{D}_f; (c) Average floc gyration diameter D ¯ g \\overline{D}_g; (d) Average fractal dimension of flocs n ¯ f \\overline{n}_f.", "Note that case Flo5 with Co=1.2×10 -7 Co = 1.2 \\times 10^{-7} has not yet reached the equilibrium stage by the end of the simulation.", "For higher CoCo-values, the equilibrium stage is characterized by larger flocs with more primary particles.", "During the transient stages, however, intermediate CoCo-values can give rise to flocs that are more elongated and hence larger than those at higher CoCo-values, in spite of having fewer primary particles." ], [ "Floc size distribution during the equilibrium stage", "In order to discuss the floc size distribution during the equilibrium stage, we sort the flocs into bins of width $\\Delta (D_f/D_p) = 0.7$ .", "Figure REF shows that for all values of the turbulent shear $G$ the size distribution peaks at the smallest flocs and then decreases exponentially with the floc size.", "The decay rate is largest for the strongest turbulence, confirming our earlier observation that strong turbulence breaks up large flocs and reduces the average floc size, cf.", "Figure REF .", "This finding is consistent with the experimental observations by [7] in an energetic tidal channel.", "Corresponding results for different $St$ -values display a similar trend (not shown).", "Figure REF shows the size distributions for different values of the cohesive number.", "For larger values of $Co$ , we find that the peak of the distribution decreases and shifts to larger flocs, while the exponential decay rate with increasing floc size is reduced.", "Figure: Floc size distribution during the equilibrium stage, obtained by sorting all flocs into bins of constant width Δ(D f /D p )=0.7\\Delta (D_f/D_p) = 0.7.", "(a) Results for different shear rates GG, with Co=1.2×10 -7 Co = 1.2 \\times 10^{-7} and St=0.06St = 0.06, during the time interval 1,000⩽t⩽4,0001,000 \\leqslant t \\leqslant 4,000 (cases Flo6-9); (b) Results for different cohesive numbers CoCo, with St=0.02St = 0.02 and G=0.29G = 0.29, for the time interval 15,000⩽t⩽19,00015,000 \\leqslant t \\leqslant 19,000 (cases Flo1-4)." ], [ "In the following, we analyze the deformation in time of those flocs that maintain their identity over the time interval $\\Delta T$ , by keeping track of their characteristic diameter $D_f$ .", "Accordingly, we distinguish between those flocs within the fraction $\\theta _{id}$ whose value of $D_f$ increases or stays constant during $\\Delta T$ , $\\theta _{id,gro}$ , and those whose diameter decreases, $\\theta _{id,shr}$ $ \\theta _{id} = \\theta _{id,gro} + \\theta _{id,shr} \\ .$ Equations (REF ) and (REF ) thus yield $ \\theta _{br} + \\theta _{id,gro} + \\theta _{id,shr} + \\theta _{ad} = 1 \\ .$ For the choice of $\\Delta T = 3$ , Figure REF displays the evolution of these fractions for the representative case Flo6.", "Interestingly, we find that $\\theta _{id,gro}$ is consistently much larger than $\\theta _{id,shr}$ , which indicates that of those flocs who maintain their identity during $\\Delta T$ , many more see their value of $D_f$ increase than decrease.", "Hence, it is much more common for these flocs to deform from a compact shape to an elongated one than vice versa.", "This consistent difference between $\\theta _{id,gro}$ and $\\theta _{id,shr}$ can be maintained only if the elongated flocs eventually break.", "As a general trend, turbulent stresses thus stretch cohesive flocs before eventually breaking them.", "This confirms earlier numerical results by [46] and [24], who employed conceptually simpler models with `sticky' cohesive particles and observed that compact flocs have greater strength than elongated ones.", "Figure: Evolution of the floc number fractions displaying different behaviors.", "(a) Of those flocs that maintain their identity during ΔT\\Delta T, many more are being stretched than shrink, resulting in θ id,gro ≫θ id,shr \\theta _{id,gro} \\gg \\theta _{id,shr} (case Flo6 with G=1.49G = 1.49); (b) The fraction θ id,gro \\theta _{id,gro} that is being stretched increases for more intense turbulence; (c) The fraction θ id,shr \\theta _{id,shr} that shrinks decreases for stronger turbulence.", "For (b) and (c) the color coding of the curves is identical, and the other parameter values are Co=1.2×10 -7 Co = 1.2 \\times 10^{-7} and St=0.06St = 0.06 (cases Flo6-9).The influence of the shear rate $G$ on the fractions $\\theta _{id,gro}$ and $\\theta _{id,shr}$ during equilibrium is displayed in Figures REF and REF , respectively.", "For larger values of $G$ , the fraction $\\theta _{id,gro}$ grows, while $\\theta _{id,shr}$ is reduced, which reflects the fact that more intense turbulence tends to elongate the cohesive flocs more strongly.", "Figures REF and REF indicate that larger $St$ -values also promote the stretching of those flocs that maintain their integrity, as they increase $\\theta _{id,gro}$ and reduce $\\theta _{id,shr}$ .", "Figures REF and REF show that smaller $Co$ -values result in the elongation of those flocs that maintain their identity, whereas stronger cohesive forces prompt the flocs to assume a more compact shape.", "Figure: Evolution of floc number fractions for different values of StSt.", "(a) Of those flocs that maintain their identity during ΔT\\Delta T, the fraction θ id,gro \\theta _{id,gro} that is stretched increases with StSt; (b) The fraction θ id,shr \\theta _{id,shr} whose diameter D f D_f decreases is reduced for larger StSt.", "The other parameter values are Co=1.2×10 -7 Co = 1.2 \\times 10^{-7} and G=0.91G = 0.91 (cases Flo10-13).Figure: Evolution of floc number fractions for different values of CoCo.", "(a) Of those flocs that maintain their identity during ΔT\\Delta T, the fraction θ id,gro \\theta _{id,gro} that is stretched increases for weaker cohesive forces; (b) The fraction θ id,shr \\theta _{id,shr} whose diameter D f D_f decreases is reduced for weaker cohesive forces.", "The other parameter values are St=0.02St = 0.02 and G=0.29G = 0.29 (cases Flo1-5)." ], [ "Orientation of elongated flocs", "We now investigate the alignment of the elongated flocs with the principal strain directions of the turbulent velocity field.", "Towards this end, we define an Eulerian fluid velocity difference tensor $A$ for each floc at time $t$ as $ {A}(m,n) = \\frac{u_{f,c}(n) - u_{f,j}(n)}{x_{c}(m) - x_{p,j}(m)} \\ ,$ where $m,n = 1, 2, 3$ represent the $x$ -, $y$ - and $z$ -components, respectively, of the tensor and vectors.", "${x}_c = (x_{c}, y_{c}, z_{c})^{\\rm T}$ denotes the location of the floc's center of mass, and the fluid velocity averaged over the volume of the floc is written as ${u}_{f,c} = \\sum _{1}^{N_p}({u}_{f,i}) / N_p$ .", "The location and fluid velocity at the center of the primary particle $j$ that is located the farthest away from the floc's center of mass are denoted as ${x}_{p,j} = (x_{p,j}, y_{p,j}, z_{p,j})^{\\rm T}$ and ${u}_{f,j} = (u_{f,j}, v_{f,j} w_{f,j})^{\\rm T}$ .", "The orientation of the floc is defined as ${x}_{f} = {x}_{p,j} - {x}_{c}$ .", "Especially for large flocs, ${x}_c$ and ${x}_{p,j}$ can be multiple grid spacings apart from each other.", "We remark that $A$ is defined by sampling the velocity difference at points separated along a line, and it thus represents a simplified approach for considering the influence of the fluid velocity gradients on the whole floc, compared with employing the full coarse-grained velocity gradient tensor [54].", "Hence $A$ differs from the standard, locally evaluated fluid velocity gradient tensor [1], [55], [71].", "We decompose this Eulerian velocity difference tensor $A = S + Q$ into the symmetric velocity difference tensor $S = {S}^{\\rm T}$ , which is similar but not identical to the strain rate tensor, and the anti-symmetric tensor $Q = -{Q}^{\\rm T}$ .", "The three eigenvalues $r_m$ of the velocity difference tensor $S$ are ordered as $r_1 > r_2 > r_3$ .", "We remark that the intermediate eigenvalue $r_2$ is automatically zero, by nature of the definition of $S$ .", "With the three eigenvalues we associate three corresponding orthonormal eigenvectors ${e}_m$ $ S {e}_m = r_m {e}_m \\ .$ We define a modified vorticity vector $\\omega = \\omega {e}_{\\omega }$ based on the anti-symmetric tensor $Q$ , with magnitude $\\omega $ and unit direction vector ${e}_{\\omega }$ [55].", "We furthermore define a modified deformation gradient tensor $B$ that characterizes the Lagrangian deformation experienced by a fluid element extending from the floc's center of mass to its primary particle $j$ , over the time interval from $t$ to $(t + \\Delta t)$ , as $ {B}(m,n) = \\frac{x_{c}(m) - x_{p,j}(m)}{[x_{c}(n)+ \\Delta t u_{f,c}(n)] - [x_{p,j}(n) + \\Delta t u_{f,j}(n)]} \\ .$ This modified deformation gradient tensor $B$ provides a Lagrangian description of the fluid stretching [51], [47].", "It differs from the standard locally evaluated deformation gradient tensor, for the same reasons mentioned earlier for the Eulerian velocity difference tensor $A$ .", "The Lagrangian stretching tensor $C = B B^T$ , obtained from the two symmetric inner products of $B$ with itself, is similar but not identical to the left Cauchy–Green strain tensor commonly used to define stretching in a Lagrangian basis [9].", "The three eigenvalues of the Lagrangian stretching tensor $C$ are ordered as $r_{L1} > r_{L2} > r_{L3}$ , and the three corresponding orthonormal eigenvectors are ${e}_{Lm}$ $ C {e}_{Lm} = r_{Lm} {e}_{Lm} \\ .$ In the following, we investigate the alignment of ${x}_{f}$ and ${e}_{\\omega }$ with ${e}_{m}$ and ${e}_{Lm}$ , respectively.", "We focus on those elongated flocs with $n_f \\leqslant 1.2$ and $N_{p} \\geqslant 2$ , and firstly analyze their alignment with the eigendirections ${e}_m$ of the Eulerian velocity difference tensor and the vorticity vector ${e}_{\\omega }$ in terms of the magnitude of the angle $\\alpha $ between them.", "We divide the elongated flocs into three different groups, according to the ratio of their characteristic diameter $D_f$ and the Kolmogorov length scale $\\eta $ .", "The alignment of small flocs with $D_f/\\eta < 0.8$ and medium-size flocs with $0.8 \\leqslant D_f/\\eta \\leqslant 1.2$ is indicated in Figures REF and REF , respectively.", "The alignment of large flocs with $D_f/\\eta > 1.2$ is not shown.", "The results indicate that the modified vorticity vector ${e}_{\\omega }$ is always aligned with the intermediate eigenvector ${e}_2$ , which is consistent with the previous finding by [1].", "We observe that medium-size flocs are strongly aligned with the intermediate eigenvector ${e}_2$ and the vorticity vector ${e}_{\\omega }$ , as shown in Figure REF .", "This result is consistent with previous findings for microscopic axisymmetric rod-like particles in turbulence by [55], who noticed that the vortex stretching term ${A} {\\omega }$ promotes, and the viscous term $\\nabla ^2 {\\omega } / Re$ opposes, the alignment of ${x_f}$ with ${e}_{\\omega }$ .", "In contrast, Figure REF shows that small flocs tend to align themselves with the extensional strain direction ${e}_1$ .", "For large flocs, we did not observe preferential alignment of the flocs with any of the three eigendirections of the Eulerian velocity difference tensor (not shown).", "The alignment of the elongated flocs and the modified vorticity vector with the eigendirections ${e}_{Lm}$ of the Lagrangian stretching tensor is shown in Figures REF and REF , respectively.", "The results indicate that the elongated flocs are perfectly aligned, and the modified vorticity vector is strongly aligned with the direction corresponding to the largest eigenvalue ${e}_{L1}$ of the Lagrangian stretching tensor $C$ .", "This alignment is consistent with, but even more pronounced than the corresponding previous findings by [51] and [47], due to our definition of the modified deformation gradient tensor $B$ .", "The perfect alignment of ${x}_{f}$ with ${e}_{L1}$ suggests that the present Lagrangian stretching tensor $C$ is well suited for analyzing the instantaneous alignment of flocs in turbulent flows.", "Figure: Floc alignment with the principal directions of the symmetric Eulerian velocity difference tensor for the representative case Flo 9.", "Results include both the flocculation and the equilibrium stages, for all elongated flocs with n f ⩽1.2n_f \\leqslant 1.2 and N p,local ⩾2N_{p,local} \\geqslant 2.", "The upper two frames show the alignment of the floc orientation x f {x}_f with the eigendirections e m {e}_m of the symmetric Eulerian velocity difference tensor, and with the vorticity vector e ω {e}_{\\omega }: (a) Small flocs with D f /η<0.8D_f / \\eta < 0.8; (b) Medium-size flocs with 0.8⩽D f /η⩽1.20.8 \\leqslant D_f / \\eta \\leqslant 1.2.", "Small flocs are preferentially aligned with the extensional strain direction, while medium-size flocs tend to align themselves with the intermediate strain direction.", "The lower two frames show the alignment with the eigendirections e Lm {e}_{Lm} of the Lagrangian deformation tensor: (c) The floc orientation x f {x}_f; (d) The vorticity vector e ω {e}_{\\omega }.", "Both the flocs and the vorticity vector tend to be aligned with the strongest Lagrangian stretching direction." ], [ "Floc size vs. Kolmogorov length scale", "Several authors have hypothesized that for sufficiently strong turbulence the median floc size should be of the same order as the smallest turbulent eddies [43], [21], [15], [33].", "Others have suggested that even the largest flocs cannot exceed the Kolmogorov length scale [69].", "In the following, we discuss data from the present simulations in order to explore this issue.", "Figure REF discusses case Flo9, with $\\eta /D_p = 2.25$ , $G = 0.62$ , $St = 0.06$ , and $Co = 1.2 \\times 10^{-7}$ .", "Figure REF compares both the average and the maximum floc size to the Kolmogorov scale.", "It demonstrates that for all times the average floc diameter $\\overline{D}_f$ is smaller than the Kolmogorov length scale $\\eta $ .", "However, at any given time the largest floc diameter $D_{f,max}$ is several times larger than $\\eta $ .", "We now define \"big\" flocs as those whose diameter $D_f$ is larger than $\\eta $ , and we indicate their fraction as $\\theta _{big} = \\frac{N_{f,big}}{N_f} \\ ,$ where $N_{f,big}$ is the number of big flocs at a given moment.", "Analogous to equation (REF ), we also define the fractions of big flocs that grow, break or maintain their identity, so that we have $\\theta _{big,br} + \\theta _{big,id,gro} + \\theta _{big,id,shr} + \\theta _{big,ad} = \\theta _{big} \\ .$ Here the subscripts $br$ , $ad$ , $id$ , $gro$ and $shr$ have the same meanings as in eqn.", "(REF ).", "Figure REF demonstrates that $\\theta _{big}$ plateaus around a value of 0.2, so that at any given time approximately 20% of all flocs are larger than the Kolmogorov scale.", "$\\theta _{big,id,shr}$ levels off around 0.1, which indicates that a substantial fraction of these big flocs deform towards a more compact shape while maintaining their identity over $\\Delta T = 3$ .", "Figure REF shows that the ratio $\\theta _{big,br}/\\theta _{br}$ is stable around 0.6, so that about 60% of those flocs that break are larger than the Kolmogorov scale $\\eta $ .", "The ratio $\\theta _{big,id,gro}/\\theta _{id,gro}$ levels off around 0.2, meaning that of those flocs that become elongated while maintaining their identity, only about 20% are big.", "Hence we can conclude that most of the big flocs tend to either become more compact or to break, but that some continue to grow.", "This finding is consistent with previous experimental observations by [64], who found that the breakage of big flocs is not instantaneous and depends on the floc strength.", "Figure REF addresses the time scale over which big flocs grow.", "The duration of the continuous growth of the big flocs is denoted by $\\Delta t_{big,gro}$ .", "We remark that $\\Delta t_{big,gro}$ is measured for all big flocs until their $D_f$ is smaller than $\\eta $ .", "The results indicate that, on average, flocs larger than the Kolmogorov scale keep growing only for the relatively short time period of $\\Delta t_{big,gro} \\approx 4.8$ .", "This is consistent with previous observations for controls on floc growth in tidal cycle experiments by [7], who found that big flocs cannot resist the turbulent stresses for long, and that they are torn apart quickly.", "This relatively quick breakage of large flocs in the simulations also agrees with our findings in Section REF , which showed that flocs are being continually stretched until they break.", "Figure: Constraint on the floc size by the Kolmogorov length scale, for case Flo9 with η/D p =2.25\\eta /D_p = 2.25, G=0.62G = 0.62, St=0.06St = 0.06, and Co=1.2×10 -7 Co = 1.2 \\times 10^{-7}.", "(a) Temporal evolution of the average and maximum floc diameters, D ¯ f \\overline{D}_f and D f,max D_{f,max}.", "The dashed horizontal line indicates the Kolmogorov length scale η\\eta ; (b) The fraction θ big \\theta _{big} of flocs that are larger than η\\eta , and the fraction θ big,id,shr \\theta _{big,id,shr} of big flocs maintaining their identity that become more compact; (c) The ratios θ big,br /θ br \\theta _{big,br}/\\theta _{br}, θ big,id,gro /θ id,gro \\theta _{big,id,gro}/\\theta _{id,gro} and θ big,ad /θ ad \\theta _{big,ad}/\\theta _{ad}; (d) Average time interval Δt big,gro \\Delta t_{big,gro} over which big flocs exhibit continuous growth.To summarize, while the size of an individual floc can be larger than the Kolmogorov length for a brief period of time, once $D_f$ becomes bigger than $\\eta $ , the floc tends to break relatively soon.", "Given that the physical parameter ranges listed in table REF represent common fluid-particle systems in nature, our simulation data suggest that the average floc size $\\overline{D}_f$ is effectively limited by the Kolmogorov length scale $\\eta $ in such systems.", "We remark, however, that for other classes of primary particles with potentially much stronger bonds it may be possible, in principle, to form flocs that are significantly larger than the Kolmogorov scale.", "For cases Flo14 and Flo15, Figure REF discusses corresponding results regarding the time scale over which big flocs grow.", "Flo14 employs an increased shear rate $G = 2.7$ along with $\\eta /D_p = 1.08$ , while Flo15 has $G = 7.4$ and $\\eta /D_p = 0.65$ .", "We remark that the ratio $\\eta /D_p$ is widely used to classify the primary particles as either `small' if $\\eta /D_p > 1$ , or as `finite-size' if $\\eta /D_p \\leqslant 1$ [22], [14], [13].", "Hence Flo14 addresses the small particle scenario, while Flo15 considers finite-size particles.", "Interestingly, Figure REF shows that the time interval $\\Delta t_{big,gro}$ over which big flocs grow for Flo14 is smaller than the corresponding value for Flo9 in Figure REF .", "This observation indicates that the constraint on floc growth by the turbulent eddies becomes stronger for increasing shear rate $G$ , which is consistent with experimental findings for small particles by [7].", "Those authors had found that the time lag before big flocs break becomes shorter for larger $G$ .", "However, a further increase of the shear rate to $G = 7.4$ in case Flo15, which means that the primary particles now fall into the finite-size category, yields a longer time lag $\\Delta t_{big,gro} \\approx 20.5$ , as shown in Figure REF .", "While the detailed reasons for this observation will require further investigation, we can conclude that the enhanced control on floc growth by the Kolmogorov length scale for stronger turbulent shear is seen to hold for small primary particles with $\\eta /D_p > 1$ , although it does not necessarily apply for finite-size primary particles with $\\eta /D_p \\leqslant 1$ .", "Figure: Average time interval Δt big,gro \\Delta t_{big,gro} over which big flocs exhibit continuous growth.", "(a) η/D p =1.08\\eta /D_p = 1.08, Co=1.2×10 -7 Co = 1.2 \\times 10^{-7}, St=0.38St = 0.38, and G=2.7G = 2.7 (case Flo14); (b) η/D p =0.65\\eta /D_p = 0.65, Co=1.2×10 -7 Co = 1.2 \\times 10^{-7}, St=1.25St = 1.25, and G=7.4G = 7.4 (case Flo15)." ], [ "A new flocculation model with variable fractal dimension", "As indicated by Figures REF - REF , the average characteristic floc diameter $\\overline{D}_f$ and the average fractal dimension $\\overline{n}_f$ both increase during the flocculation stage, and then remain constant during the equilibrium stage.", "This indicates that flocs of larger size generally have a more compact shape, and that it is difficult for elongated flocs to keep growing in turbulent shear without breaking.", "Closer inspection indicates that for all of the cases listed in table REF the relationship between these two quantities can be approximated well by a power law of the form $ \\overline{n}_f = k_1 \\left( \\frac{\\overline{D}_f}{D_p} \\right)^{k_2} \\ .$ The condition that $\\overline{n}_f = 1$ for an individual primary particle requires that $k_1 = 1$ , while the value of $k_2$ varies as a function of $St$ , $Co$ and $G$ .", "Typical fitting results are shown in Figure REF for cases Flo4 and Flo5.", "This power law relationship allows us to obtain the average fractal dimension $\\overline{n}_f$ during flocculation as a function of the average floc diameter $\\overline{D}_f$ , rather than assuming a constant fractal dimension, as was done in earlier investigations [74], [33], [82].", "Figure: (a) The relationship between the average fractal dimension n ¯ f \\overline{n}_f and the average value D ¯ f /D p \\overline{D}_f/D_p, during the flocculation and equilibrium stages.", "Simulation data and power law fits according to eqn.", "() are shown for Flo4 with Co=6.0×10 -8 Co = 6.0 \\times 10^{-8}, St=0.02St = 0.02, and G=0.29G = 0.29; and for Flo5 with Co=1.2×10 -7 Co = 1.2 \\times 10^{-7}, St=0.02St = 0.02, and G=0.29G = 0.29; (b) Comparisons between the experimental data of , predictions by the relation of , , and the new relation ().", "The experimental parameters are D p =5μmD_p = 5 \\ \\rm {\\mu m}, ρ p =2,650 kg m -3 \\rho _p = 2,650 \\ \\rm {kg \\ m^{-3}}, c=0.5gL -1 c = 0.5 \\ \\rm {g \\ L^{-1}}, ρ f =1,000 kg m -3 \\rho _f = 1,000 \\ \\rm {kg \\ m^{-3}}, μ=0.001 Pa s\\mu = 0.001 \\ \\rm {Pa \\ s} and G=5∼40s -1 G = 5 \\sim 40 \\ \\rm { s^{-1}}.", "Khelifa's relation (-) has constant coefficient values n ¯ f,char =2\\overline{n}_{f,char} = 2, D ¯ f,char =2,000μm\\overline{D}_{f,char} = 2,000 \\rm {\\mu m} and updated k 1 =1k_1 = 1.", "The calibration of the empirical coefficient for the new relation () yields a 3 =4×10 -6 a_3 = 4 \\times 10^{-6} for G=5s -1 G = 5 \\ \\rm { s^{-1}}, and a 3 =4×10 -5 a_3 = 4 \\times 10^{-5} for G=40s -1 G = 40 \\ \\rm { s^{-1}}.The power law (REF ) is closely related to the earlier study by [30], [31].", "However, those authors assumed that an individual primary particle has $n_f = 3$ , and consequently they set $k_1 = 3$ .", "For the exponent $k_2$ they proposed an empirical correlation of the form $ k_2 = \\frac{\\log (\\overline{n}_{f,char} / k_1)}{\\log (\\overline{D}_{f,char}/D_p)} \\ ,$ where $\\overline{D}_{f,char}$ denotes the characteristic floc size that exhibits the characteristic fractal dimension $\\overline{n}_{f,char}$ .", "As a general rule, $\\overline{D}_{f,char}$ and $\\overline{n}_{f,char}$ should be evaluated from experiments before one can then determine $k_2$ from (REF ).", "Should that not be feasible, [30], [31] suggested assuming constant values of $\\overline{n}_{f,char} = 2$ and $\\overline{D}_{f,char} = 2,000 \\rm {\\mu m}$ , which yields a constant value for $k_2$ that depends only on the primary particle size $D_p$ .", "As Figure REF indicates, however, $k_2$ should be a function of $G$ , $St$ and $Co$ even for a constant $D_p$ , since $\\overline{n}_{f,char} = 2$ is associated with different average floc sizes $\\overline{D}_{f,char}/D_p$ in cases Flo4 and Flo5.", "Hence, even though equation (REF ) has been widely used to describe the fractal dimension of flocs [37], [63], [32], we will now try to refine this scaling law by accounting for the dependence of $k_2$ on $St$ , $Co$ and $G$ .", "By fitting the simulation results for all of the cases Flo1 - 15, we obtain a relationship for $k_2$ of the form $ k_2 = 0.44St^{-0.018}Co^{0.096}G^{-1.5} \\ ,$ with an R-squared value of 0.97.", "We remark that in a laboratory experiment or field investigation it may be challenging to evaluate the Stokes number $St$ as defined in equation (REF ), if the $rms$ -velocity $u_{rms}$ is unknown.", "To overcome this difficulty, we follow the approach taken in our earlier work [82], where we defined the characteristic Stokes number $St_{char}$ and cohesive number $Co_{char}$ by employing the characteristic fluid velocity $u_{char} = 0.25(G/Re)^{0.5}$ instead of $u_{rms}$ , so that $ St_{char} = \\frac{St \\, u_{char}}{u_{rms}} = \\frac{\\rho _s D_p^2 \\, u_{char} Re}{18 \\eta } \\ ,$ $ Co_{char} = \\frac{Co}{\\eta ^2 \\, u_{char}^2} \\ .$ Here $Re$ and $Co$ are of the form defined in (REF ) and (REF ), respectively.", "Note that $u_{char}$ and $\\eta $ in equations (REF ) and (REF ) are dimensionless.", "Based on $St_{char}$ and $Co_{char}$ , a fit of the simulation data yields the relationship for $k_2$ $ k_2 = \\frac{St_{char}^{-1.9}Co_{char}^{0.1}}{1.3 \\times 10^5} \\ ,$ which has an R-squared value of 0.86.", "Here $St_{char}$ captures the strongly inverse influence of the shear rate $G$ on $k_2$ .", "By substituting (REF ) into (REF ), we obtain a new model for the average fractal dimension $\\overline{n}_f$ of the form $ \\overline{n}_f = \\left(\\frac{\\overline{D}_f}{D_p}\\right)^{a_3 \\, St_{char}^{-1.9}Co_{char}^{0.1}/(1.3 \\times 10^5)}\\ .$ For the specific range of physical parameters listed in table REF , $a_3=1$ yields optimal agreement with a maximum deviation of $\\pm 30\\%$ from the simulation data.", "As we will see below, this value of $a_3$ is not universally optimal, so that $a_3$ will have to be recalibrated for other parameter ranges.", "In the following, we will compare predictions for the fractal dimension by the new relation (REF ) with corresponding ones by the earlier relation of Khelifa & Hill (REF -REF ).", "Employing the approach of [38], [37] estimate the time evolution of the average fractal dimension $\\overline{n}_f$ and floc size $\\overline{D}_f$ in experiments with constant turbulent shear rates $G = 5, 10, 20$ and 40 $\\rm {s^{-1}}$ , respectively.", "The suspended cohesive sediment in the experiments has a primary particle diameter $D_p = 5 \\ \\rm {\\mu m}$ , density $\\rho _p = 2,650 \\ \\rm {kg \\ m^{-3}}$ , and concentration $c = 0.5 \\ \\rm {g \\ L^{-1}}$ .", "Since the authors assume $\\overline{n}_f = 3$ for flocs with one particle while we set $\\overline{n}_f = 1$ for that situation, we have to convert their original experimental data before we can compare them with the present simulation results.", "The details of the conversion are discussed in appendix , and the converted data are presented in Figure REF .", "In addition, we set the dimensional Kolmogorov length $\\eta = [\\mu / (\\rho _f G)]^{0.5} \\rm {m}$ and the Hamaker constant $A_H = 1.0 \\times 10^{-20} \\ \\rm {J}$ to obtain the characteristic values $St_{char}$ and $Co_{char}$ according to (REF )-(REF ).", "Since the experimental shear rates $G = 5 \\sim 40 \\ \\rm { s^{-1}}$ are much smaller than the simulation values $G = 3.7 \\times 10^3 \\sim 9.5 \\times 10^4 \\ \\rm { s^{-1}}$ , we have to recalibrate the constant $a_3$ required for our model (REF ) from the experimental data.", "Based on the fact that the exponent $k_2$ should decrease for increasing $G$ , we obtain $a_3 = 4 \\times 10^{-6}$ for the minimum experimental shear rate $G = 5 \\rm { s^{-1}}$ , and $a_3 = 4 \\times 10^{-5}$ for the maximum experimental shear rate $G = 40 \\rm { s^{-1}}$ , respectively.", "Figure REF demonstrates that the present relation successfully reproduces the range of experimental data for different $G$ -values, whereas Khelifa & Hill's relation does not account for variations in $G$ .", "At the same time, we do need to keep in mind that the present model does require a recalibration of $a_3$ for different experimental parameter ranges.", "In order to develop a variable fractal dimension model for the transient stages, we build on the approach taken in our recent investigation [82].", "There we conducted cohesive sediment simulations for a steady, two-dimensional cellular flow model.", "Based on the simulation data, we proposed an analytical flocculation model of the form $\\overline{D}_f & = & (\\overline{N}_{p}) ^ {\\frac{1}{\\overline{n}_f}} D_p \\ , \\\\\\overline{N}_p & = & \\frac{1}{(1 / \\overline{N}_{p,in} - 1 / \\overline{N}_{p,eq}) e^{b t} + 1 / \\overline{N}_{p,eq}} \\ ,\\\\\\overline{N}_{p,eq} & = & \\left\\lbrace \\begin{array}{ll}N, \\quad if \\ \\overline{N}_{p,eq} \\geqslant N \\ , \\\\\\\\8.5 a_1 St_{char}^{0.65} Co_{char}^{0.58} D_{p,char}^{-2.9} \\phi _p^{0.39} \\rho _s^{-0.49} (W+1)^{-0.38} \\ , \\quad otherwise \\ ,\\end{array}\\right.\\\\b & = & \\left\\lbrace \\begin{array}{ll}-0.7 a_2 St_{char}^{0.36} Co_{char}^{-0.017} D_{p,char}^{-0.36} \\phi _p^{0.75} \\rho _s^{-0.11} (W+1)^{-1.4}, \\quad St_{char} \\leqslant 0.7 \\ , \\\\\\\\-0.3 a_2 St_{char}^{-0.38} Co_{char}^{0.0022} D_{p,char}^{-0.61} \\phi _p^{0.67} \\rho _s^{0.033} (W+1)^{-0.46}, \\quad St_{char} > 0.7 \\ .\\end{array}\\right.$ Here $\\overline{N}_{p,in}$ and $\\overline{N}_{p,eq} = N/N_{f,eq}$ indicate the average number of primary particles per floc at the initial time and during the equilibrium stage, respectively.", "$|b|$ denotes the rate of change in the number of flocs, where a bigger $|b|$ indicates a faster increase of the mean number of primary particles per floc $\\overline{N}_{p}$ during flocculation.", "$D_{p,char} = D_p/\\eta $ is the characteristic primary particle diameter, and $W$ represents the Stokes settling velocity.", "$a_1$ and $a_2$ are empirical coefficients that need to be calibrated via comparison with experiments or simulations.", "Under the assumption of a constant average fractal dimension $\\overline{n}_f = 2$ , and for given values of $N$ , $\\overline{N}_{p,in}$ , $St_{char}$ , $Co_{char}$ , $D_{p,char}$ , $\\phi _p$ , $\\rho _s$ and $W$ , this model predicts the transient floc size $\\overline{D}_f$ and the average number of particles per floc $\\overline{N}_{p}$ as functions of time.", "Model results were presented in [82].", "As the present simulations show, however, assuming a constant average fractal dimension represents a serious limitation, cf.", "Figures REF , REF and REF , which we aim to overcome in the following.", "Towards this end, we combine equations (REF ) and () to obtain a new flocculation model (termed the `present model') that allows for a variable fractal dimension.", "This model yields predictions of the floc size $\\overline{D}_f$ , the number of particles per floc $\\overline{N}_{p}$ , and the fractal dimension $\\overline{n}_f$ as functions of time.", "Since equations (REF ) and (REF ) need to be solved concurrently, the model cannot be written in closed form.", "However, due to the narrow range of the average fractal dimension $1 \\leqslant \\overline{n}_f \\leqslant 3$ , an iterative solution can easily be obtained.", "In analogous fashion, we can link the variable fractal dimension relation (REF ) - (REF ) by [30], [31] to our previous flocculation model (), to obtain the `combined model.'", "A list of all models discussed here is provided in table REF for convenience.", "We now proceed to assess their performance.", "Table: Typical models cited, proposed and implemented in the present work.By calibrating with the average floc size data for simulation Flo4, we determine the empirical coefficients for the `present model' as $a_1 = 8$ , $a_2 = 0.5$ and $a_3 = 1$ , shown as solid red line in Figure REF .", "We then employ the present model to predict the average fractal dimension for Flo4 as function of time.", "Figure REF indicates good agreement between the predictions and the simulation data.", "In complete analogy, we determine the empirical coefficients for the `combined model' as $a_1=2$ , $a_2 = 0.5$ and $k_1 = 1$ , which yields the solid blue line in Figure REF .", "The average fractal dimension $\\overline{n}_f$ predicted by the combined model is very close to that of the present model and to the simulation data, which suggests that both models are able to predict the average fractal dimension quite accurately.", "In applications, it may be difficult to obtain precise calibration values for $a_1$ , so that it is important to establish the robustness of the present model with regard to uncertainties in the value of $a_1$ .", "In order to assess this robustness, we ran the present model for $a_1=2$ and $a_1=32$ , instead of the optimal value $a_1=8$ that we had obtained earlier from the calibration.", "The results, shown in Figure REF as dashed and dotted red lines, indicate that the model predictions are reasonably robust with regard to uncertainties in the value of $a_1$ .", "To summarize, our new fractal relation (REF ) no longer has the limitation associated with assuming a constant value for $k_2$ in equations (REF ) and (REF ) when predicting the variable fractal dimension $\\overline{n}_f$ .", "In addition, we observe that predictions of the floc size $\\overline{D}_f$ and the fractal dimension $\\overline{n}_f$ as functions of time by the present flocculation model (REF ) and () are fairly robust with respect to uncertainties that arise when calibrating the empirical coefficients by means of experimental data.", "Figure: Comparisons between the numerical data and predictions by the present model and the combined model listed in table , simulation data of case Flo4 with Co=6.0×10 -8 Co = 6.0 \\times 10^{-8}, St=0.02St = 0.02 and G=0.29G = 0.29 is selected.", "(a) Calibration predictions for the temporal evolution of the average floc izes D ¯ f \\overline{D}_f, the calibrated coefficients in the models are a 2 =0.5a_2 = 0.5 and a 3 =1a_3 = 1, the constant k 1 =1k_1 = 1; (b) Comparisons for the temporal evolution of the average fractal dimension n ¯ f \\overline{n}_f." ], [ "Conclusions", "In the present investigation we have employed one-way coupled simulations to explore the dynamics of cohesive particles in homogeneous isotropic turbulence.", "The simulations account for the Stokes drag, as well as lubrication, cohesive and direct contact forces.", "They demonstrate the existence of a transient flocculation phase which is characterized by the growth of the average floc size.", "This flocculation phase is followed by a statistically steady equilibrium phase governed by a balance between floc growth and breakup.", "The simulations provide information about the temporal evolution of the floc size and shape, as a result of aggregation, breakage and deformation, and as function of the governing parameters.", "In general, we find that larger turbulent shear and weaker cohesive forces limit the floc size and result in elongated floc shapes.", "Flocculation proceeds most rapidly during the transient stage when the Stokes number of the primary particles based on the Kolmogorov scales is of order unity.", "During the transient stage cohesive forces of intermediate strength yield the largest flocs.", "On one hand, these intermediate cohesive forces are strong enough to result in the rapid aggregation of primary particles, but on the other hand they are not so strong as to pull them into a compact shape.", "During the equilibrium stage, stronger cohesive forces produce larger flocs.", "Small Stokes numbers and weak turbulence typically lead to a later onset of the equilibrium stage.", "The equilibrium floc size distribution exhibits a preferred size as function of the cohesive number.", "This distribution decays exponentially for larger floc sizes.", "The simulation results indicate that flocs are generally elongated by turbulent stress before they eventually break.", "We observe that flocs close to the Kolmogorov scale in size preferentially align themselves with the intermediate strain direction and the vorticity vector.", "Flocs that are smaller than the Kolmogorov scale, on the other hand, tend to align themselves with the direction of extensional strain.", "The simulation results furthermore demonstrate that flocs generally align themselves with the strongest Lagrangian stretching direction.", "The simulations show that the average floc size is effectively limited by the Kolmogorov scale, and can at most exceed it marginally.", "However, individual flocs can grow larger than the Kolmogorov scale for a limited amount of time.", "Based on the simulation data we propose a novel flocculation model that allows for a variable fractal dimension, which enables us to predict the temporal evolution of the floc size and shape, as a function of the governing dimensionless parameters, after some limited calibration.", "Predictions by the new model are fairly robust and cover a broad range of parameters.", "Acknowledgements EM gratefully acknowledges support through NSF grants CBET-1803380 and OCE-1924655, as well as by the Army Research Office through grant W911NF-18-1-0379.", "TJH received support through NSF grant OCE-1924532.", "KZ is supported by the National Natural Science Foundation of China through the Basic Science Center Program for Ordered Energy Conversion through grant 51888103, as well as by the China Scholarship Council.", "BV gratefully acknowledges support through German Research Foundation (DFG) grant VO2413/2-1.", "Computational resources for this work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by NSF grant TG-CTS150053.", "Declaration of Interests The authors report no conflict of interest." ], [ "Conversion of the experimental data", "[37] measured the floc size and evaluated the fractal dimension $\\overline{n}_{f,ori}$ in experiments by setting the fractal dimension of an individual primary particle to three.", "Taking their experimentally measured floc size as the characteristic floc diameter $D_f$ in (REF ), the original experimental data are shown in Figure REF .", "For each pair of $\\overline{n}_{f,ori}$ and $\\overline{D}_f/D_p$ , we can obtain the average floc size $\\overline{D}_{f,char}$ as $ \\overline{D}_{f,char} = D_p10^{[\\log (\\overline{n}_{f,char} / k_{1,ori})]/k_{2,ori}} \\ ,$ where the characteristic fractal dimension $\\overline{n}_{f,char} = 2$ , the diameter of primary particles $D_p = 5 \\ \\rm {\\mu m}$ , $k_{1,ori} = 3$ , and $ k_{2,ori} = \\frac{\\log (\\overline{n}_{f,ori}/k_{1,ori})}{\\log (\\overline{D}_{f}/D_p)} \\ .$ The converted fractal dimension $\\overline{n}_{f}$ for the corresponding $\\overline{D}_f/D_p$ in the experiments can then be obtained from $ \\overline{n}_{f} = k_{1}(\\frac{\\overline{D}_{f}}{D_p})^{k_2} \\ ,$ where $k_1 = 1$ and $ k_{2} = \\frac{\\log (\\overline{n}_{f,char}/k_{1})}{\\log (\\overline{D}_{f,char}/D_p)} \\ .$ The converted experimental data are shown in Figure REF .", "Figure: Original experimental data of for the experimental parameter values D p =5μmD_p = 5 \\ \\rm {\\mu m}, ρ p =2650 kg m -3 \\rho _p = 2650 \\ \\rm {kg \\ m^{-3}}, c=0.5gL -1 c = 0.5 \\ \\rm {g \\ L^{-1}}, ρ f =1,000 kg m -3 \\rho _f = 1,000 \\ \\rm {kg \\ m^{-3}}, μ=0.001 Pa s\\mu = 0.001 \\ \\rm {Pa \\ s} and G=5∼40s -1 G = 5 \\sim 40 \\ \\rm { s^{-1}}." ] ]
2105.11735
[ [ "Room temperature ferroic orders in Zr and (Zr, Ni) doped SrTiO$_3$" ], [ "Abstract We synthesized strontium titanate SrTiO$_3$ (STO), Zr doped $\\text{Sr}_\\text{1-x}\\text{Zr}_\\text{x}\\text{Ti}\\text{O}_3$ and (Zr, Ni) co-doped $\\text{Sr}_\\text{1-x}\\text{Zr}_\\text{x}\\text{Ti}_\\text{1-y}\\text{Ni}_\\text{y}\\text{O}_3$ samples using solid state reaction technique to report their structural, electrical and magnetic properties.", "The cubic $Pm$-$3m$ phase of the synthesized samples has been confirmed using Rietveld analysis of the powder X-ray diffraction pattern.", "The grain size of the synthesized materials was reduced significantly due to Zr doping as well as (Zr, Ni) co-doping in STO.", "The chemical species of the samples were identified using energy-dispersive X-ray spectroscopy.", "We observed forbidden first order Raman scattering at 148, 547 and 797 cm$^{-1}$ which may indicate nominal loss of inversion symmetry in cubic STO.", "The absence of absorption at 500 cm$^{-1}$ and within 600-700 cm$^{-1}$ band in Fourier Transform Infrared spectra corroborates Zr and Ni as substitutional dopants in our samples.", "Due to 4% Zr doping in $\\text{Sr}_\\text{0.96}\\text{Zr}_\\text{0.04}\\text{Ti}\\text{O}_3$ sample dielectric constant, remnant electric polarization, remnant magnetization and coercivity were increased.", "Notably, in the case of 4% Zr and 10% Ni co-doping we have observed clearly the existence of both FE and FM hysteresis loops in $\\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}_{0.90}\\text{Ni}_{0.10}\\text{O}_3$ sample.", "In this co-doped sample, the remnant magnetization and coercivity were increased by $\\sim$1 and $\\sim$2 orders of magnitude respectively as compared to those of undoped STO.", "The coexistence of FE and FM orders in (Zr, Ni) co-doped STO might have the potential for interesting multiferroic applications." ], [ "Introduction", "Strontium titanate SrTiO3 (STO) is perhaps one of the most prominent prototypical ABO3 perovskite oxide materials having potentials in diverse fields of applications such as high-k dielectrics [1], substrate materials [2], tunable microwave devices [3], [4], non-volatile memory applications [5], O2 and H2S sensing [6], [7] and photocatalytic water splitting [8], to name just a few.", "At room temperature, STO exists in cubic structure with space group $Pm$ -$3m$ where the $\\text{Sr}^{2+}$ occupies A site which is surrounded by four TiO6 octahedra and $\\text{Ti}^{4+}$ occupies B site which is located at the octahedral void formed by six $\\text{O}^{2-}$ ions situated at the faces of the cube [9].", "Pure stoichiometric STO is a band-insulator whose quantum paraelectric behaviour excludes the emergence of the ferroic phases such as ferroelectric (FE) and ferromagnetic (FM) orders [10], [11], [12], [13], [14].", "Over the years numerous attempts have been made to induce FE and FM orders in STO by means of cation doping for achieving multiferroic behaviour.", "The FM ordering is induced in STO by doping transition metals (Mn, Fe, Co, Cd) and rare-earth elements (Yb, Mn)[15], [16], [17], [18], [19], [20].", "The FE behavior has been observed at room temperature in Pr doped STO [21] whereas Mn doping introduced FE relaxor type behaviour at low temperature [22], [14], [23], [24], [25].", "Moreover, (Nd, Dy) and (La, Mn) co-doped STO showed promising results for multiferroic applications [21], [22], [14], [23], [24], [25].", "Here we doped STO with two different transition metal elements which are Zr and Ni.", "In the stable Zr$^{4+}$ oxidation state, the empty $4d$ orbital has the possibility of inducing the ferroelectric order in STO and thereby may enhance dielectric properties in Zr doped STO [10], [26], [27].", "The rationale behind co-doping of Zr$^{4+}$ and Ni$^{2+}$ is that the magnetic moments of the unpaired electrons in the partially filled $3d$ orbitals in Ni$^{2+}$ oxidation state might have the potential to tune the magnetic properties of STO.", "We assume substitutional doping following the general rule where ions with smaller radius Ni$^{2+}$ ($r_o=0.69$ Å) tend to substitute Ti$^{4+}$ in B site and ions with larger radius Zr$^{4+}$ ($r_o=0.80$ Å) prefer to go in A site occupied by Sr$^{2+}$ [28].", "The objective of this investigation is to synthesize a number of Zr and Ni co-doped STO materials and extensively characterize their structural, electrical and magnetic properties [29], [30], [31], [32].", "To this aim, undoped STO, Zr-doped $\\text{Sr}_\\text{1-x}\\text{Zr}_\\text{x}\\text{Ti}\\text{O}_3$ and (Zr, Ni) co-doped $\\text{Sr}_\\text{1-x}\\text{Zr}_\\text{x}\\text{Ti}_\\text{1-y}\\text{Ni}_\\text{y}\\text{O}_3$ samples were synthesized using solid state reaction technique.", "Based on the findings from the pertinent materials characterization techniques, we report a particular composition of this (Zr, Ni) co-doped samples with improved multiferroic properties which might have potential for multifunctional applications [30].", "A number of samples for undoped SrTiO3, Zr-doped SrTiO3 and (Zr, Ni) co-doped SrTiO3 have been synthesized using the standard solid state reaction method [33], [34].", "For the starting materials, proper combinations of analytical grade SrCO3 (98% pure), TiO2 (99.9% pure), ZrO2 (99.9% pure) and NiO (99.9% pure) were used with desired stoichiometric ratio.", "For the Zr-doped samples, $x = 0.02, 0.04, 0.06$ were prepared with the chemical formula $\\text{Sr}_\\text{1-x}\\text{Zr}_\\text{x}\\text{Ti}\\text{O}_3$ .", "In case of (Zr, Ni) co-doped samples with chemical formula $\\text{Sr}_\\text{1-x}\\text{Zr}_\\text{x}\\text{Ti}_\\text{1-y}\\text{Ni}_\\text{y}\\text{O}_3$ , a series of combinations with $\\text{y} = 0.05, 0.10, 0.15~\\text{and}~0.20$ for fixed $\\text{x}=0.04$ have been synthesized.", "The powder mixtures were hand milled for 6 hours using mortar and pestle to produce a homogeneous solid mixture with fine constituent particles in proximity with each other.", "Several droplets of polyvinyl alcohol were mixed with the samples in a steel die to facilitate binding before being subjected to uniaxial force of 20 kN in a hydraulic press to form circular disk-shaped pellets.", "These pellets were calcined at $800^{\\circ }$ C for 4 hours to promote reaction among the mixture constituents.", "The pre-sintered disk-shaped pellets were smashed into fine powders by hand milling in a ceramic mortar and pestle for 3 hours to expedite solid state reactions probabilities in the subsequent sintering of the samples.", "The crystallization temperatures of the udoped, Zr doped and (Zr, Ni) co-doped samples were measured to be $\\sim $ 1080$^{\\circ }$ , $\\sim $$1085^{\\circ }$ and $\\sim $$942^{\\circ }$ C respectively using a differential scanning calorimeter.", "The pre-sintered powder materials were pressed into circular disk shaped pellets and toroid rings by using a hydraulic press and sintered at $1250^{\\circ }$ C for 4 hours." ], [ "Characterization Techniques", "To estimate the crystallization temperature of our synthesized samples, the differential scanning calorimetry was performed using a NETZSCH STA 449 F3 Jupiter simultaneous thermal analyzer.", "The high temperature sintering of the samples was done in a Nabertherm Muffle Furnace LT 5/14.", "The X-ray Diffraction (XRD) patterns for the synthesized samples were obtained from $10^{\\circ }$ to $80^{\\circ }$ at 35 kV accelerating voltage with an emission current of 20 mA using a Rigaku SmartLab SE multipurpose XRD system with Cu K$\\alpha $ radiation ($\\lambda =0.15418$ nm).", "The surface morphology and chemical species identification were performed with Scanning Electron Microscopy (SEM) and Energy-dispersive X-ray spectroscopy (EDX) respectively using a AVO 18 Research Scanning Electron Microscope from ZEISS.", "The room temperature Raman scattering spectroscopy was performed with a Confocal Raman Microscope MonoVista CRS+ using a 532.090 nm laser.", "To characterize the chemical bond vibrations inside our samples, we used the KBr pellet technique in Fourier Transform Infrared (FTIR) PerkinElmer Spectrum spectrometer.", "The dielectric constant and resistivity of the samples were measured from complex impedance spectroscopy performed by Wayne Kerr 6500B Impedance Analyzer.", "For the electric hysteresis measurements, the electric polarization $P$ vs electric field $E$ loops were recorded using a Precision Multiferroic II Ferroelectric Test System from Marine India.", "The magnetization $M$ vs. magnetic field $H$ hysteresis loops of the samples were obtained using a vibrating sample magnetometer VSM from Quantum Design PPMS DynaCool.", "Table: Lattice parameters a, b, c, α\\alpha , β\\beta and γ\\gamma extracted from Rietveld refinement of XRD patterns of Sr 1-x Zr x Ti 1-y Ni y O 3 \\text{Sr}_\\text{1-x}\\text{Zr}_\\text{x}\\text{Ti}_\\text{1-y}\\text{Ni}_\\text{y}\\text{O}_3 for (x, y) = (0.00 ,0.00), (0.02, 0.00), (0.04, 0.00), (0.06, 0.00), (0.04, 0.05), (0.04, 0.10), (0.04, 0.15) and (0.04, 0.20).", "The bulk density, X-ray density, porosity, crytallite size were derived from standard formulas.", "The values for FWHM were for the most intense XRD peak corresponding to (110) plane." ], [ "X-ray Diffraction Analysis", "We have investigated the crystallographic structure, phase and purity of the synthesized samples using the XRD patterns, see Fig.", "REF .", "The pure STO exhibit cubic $Pm$ -$3m$ phase (space group no.", "221) according to the standard JCPDS data (01-084-0443) [35], [17].", "The phase purity and crystalline nature of the samples were evident from the sharp and intense diffraction peaks.", "In case of Zr doped $\\text{Sr}_\\text{1-x}\\text{Zr}_\\text{x}\\text{Ti}\\text{O}_3$ samples, extraneous peaks started to appear for $\\text{x}=0.06$ indicating the presence of additional phases on top of the cubic phase.", "Hence, to avoid the influence of these secondary phases in (Zr, Ni) co-doped samples, we fixed Zr concentration x to be at 0.04 as $\\text{Sr}_{0.96}\\text{Zr}_\\text{0.04}\\text{Ti}_\\text{1-y}\\text{Ni}_\\text{y}\\text{O}_3$ .", "For y$\\ge 0.10$ , the small extraneous peaks at 37.18$^{\\circ }$ , 43.20$^{\\circ }$ , 62.75$^{\\circ }$ and 75.25$^{\\circ }$ were observed due excess NiO [36].", "The lattice parameters were extracted by Rietveld refinement of the XRD profile using FullProf software.", "We estimated bulk density, X-ray density, porosity, crystallite size and full-width-half-maximum using standard techniques [37], see Table REF .", "The bulk density remained unchanged for undoped and doped STO.", "For Zr doped and (Zr, Ni) co-doped samples, an increasing trend in the X-ray-density was observed with Zr and Ni concentrations.", "These increments may have originated from the change in molecular weights due to the incorporation of the Zr and Ni dopants in the sample.", "A nominal variation in porosity has been observed across the synthesized samples.", "As for the crystallite size, monotonic decrements have been detected with increasing Zr concentration in STO.", "The change in the full-width-at-half-maximum (FWHM) for (Zr, Ni) co-doped samples indicates slight distortion and disorder due to size differences of dopants and interstitial dopants respectively [38].", "From the peak shift analysis (as shown in Fig.", "S1 of the Supplementary Information), it is evident that the peak shift between pure STO and $\\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}\\text{O}_3$ is very nominal almost conforming with pure cubic phase.", "We superimposed the Rietveld refined pattern on experimentally measured XRD data for three selected samples; SrTiO3, Sr0.96Zr0.04TiO3 (due to superior dielectric properties in Section REF ) and Sr0.96Zr0.04Ti0.90Ni0.10O3 (manifested superior magnetic properties as shall be seen later in Section REF ) in Fig.", "REF .", "The goodness of fitting parameter $\\chi ^2$ for the three samples were found to be 3.784, 3.590 and 3.806 which indicate simulated patterns are in good agreement with experimental observations conforming cubic structure.", "The auxiliary crystallographic parameters such as atomic positions in Wyckoff coordinates, relevant bond lengths and bond angles have also been extracted from the refinement (see Table S1 in the Supplementary Information).", "Figure: X-ray diffraction patterns of Sr1-xZrxTi1-yNiyO3 for (x, y) = (0.00, 0.00), (0.02, 0.00), (0.04, 0.00), (0.06, 0.00), (0.04, 0.05), (0.04, 0.10), (0.04, 0.15) and (0.04, 0.20).", "The unknown impurity phase is marked with black diamonds in case of (x, y)= (0.06, 0.00).Figure: Simulated Rietveldleast square minimized XRD patterns superimposed on experimentally observed data of Sr1-xZrxTi1-yNiyO3 for (x,y) = (0.00, 0.00), (0.04, 0.00) and (0.04, 0.10).", "The red solid circles are the experimental data points (Y obs _\\text{obs}), the dark solid line represents calculated refined pattern Y cal _\\text{cal}, the bottom blue curve Y diff _\\text{diff} shows difference between the experimental Y obs _\\text{obs} and calculated Y cal _\\text{cal} values and pink vertical lines mark the positions of Bragg peaks for cubic SrTiO3 with PmPm-3m3m space group.", "The green vertical lines represent NiO phase.Figure: SEM micrographs (a) SrTiO3, (b) Sr0.96Zr0.04TiO3 and (c) Sr0.96Zr0.04Ti0.90Ni0.10O3 samples." ], [ "Morphological and EDX Analysis", "To understand the effect of doping on the microstructure and to perform chemical species identification of our samples, SEM micrographs and EDX spectra of the three selected samples SrTiO3, Sr0.96Zr0.04TiO3 and Sr0.96Zr0.04Ti0.90Ni0.10O3 have been obtained.", "The average grain size was estimated to be 2 $\\mu \\text{m}$ in case of undoped SrTiO3, see SEM micrograph in Fig.", "REF (a).", "The estimated average grain size is comparable with the previous studies on STO in Refs.", "[28], [39].", "The detection of Sr, Ti and O peaks in the corresponding EDX spectra for undoped STO excludes the presence of unwanted chemical species in the sample (see Fig.", "S2(a) in the Supplementary Information).", "The average grain size was reduced to 0.343 $\\mu \\text{m}$ for Sr0.96Zr0.04TiO3 sample as showed in Fig.", "REF (b).", "This shrinkage of the gain size can be due to the presence of Zr or Ti at the grain boundaries [28].", "In addition, Zr substitution prompted grain irregularity and inhomogeneity.", "The presence of Zr peak in addition with Sr, Ti and O peaks in the corresponding EDX spectra for Sr0.96Zr0.04TiO3 corroborates its incorporation as a dopant in the sample (see Fig.", "S2(b) in the Supplementary Information).", "For (Zr, Ni) co-doped Sr0.96Zr0.04Ti0.90Ni0.10O3 sample, the estimated grain size was found to be 0.341 $\\mu \\text{m}$ according to the SEM micrograph in Fig.", "REF (c).", "The corresponding EDX spectra containing Zr and Ni peaks elucidates the co-doping of the STO (see Fig.", "S2(c) in the Supplementary Information).", "Moreover the atomic weights (%) of chemical species in all aforementioned samples were compared with the corresponding theoretical values in Table S2 of the Supplementary Information." ], [ "Raman Analysis", "We performed room temperature Raman scattering spectroscopy to characterize the vibrational phonon in terms of transverse acoustic (TA), longitudinal acoustic (LA), transverse optical (TO) and longitudinal optical (LO) modes of our samples.", "For a three dimensional ($d=3$ ) unit cell with cubic symmetry, the STO contains $n=5$ atoms (one Sr, one Ti and three O) that generates $3n=15$ vibrational phonon modes; out of which the 3 low frequency acoustic modes (F$_{1u}$ ) are degenerate, 3 degenerate optical modes (F$_{1u}$ ) are Raman and infrared (IR) inactive and the rest of 9 optical modes (F$_{2u}$ ) are IR-active [40], [41], [42], [43], [44].", "As cubic symmetry forbids first order Raman scattering, the Raman modes in cubic STO originate from second order scattering processes [40].", "The two photon momentum conservation processes made the second order Raman scattering peaks to appear broad and continuous, see Fig.", "REF .", "The broad intensity peaks of the Raman scattered radiation appeared in 250-500 cm$^{-1}$ and 600-800 cm$^{-1}$ wave number ranges.", "These Raman peaks were identified with corresponding vibrational phonon modes of the samples according to Refs.", "[43], [42] in Table S3 of the Supplementary Information.", "The 2TA mode appeared around 208 cm$^{-1}$ , whereas the combined acoustic and optical modes such as TO$_1+$ LA, LO$_1+$ TA and LA+LO$_3$ contributed at 275, 383 and 702 cm$^{-1}$ respectively.", "The modes corresponding to two optical phonons such as TO$_1$ +TO$_4$ (641 cm$^{-1}$ ) and LO$_3$ +TO$_2$ (762 cm$^{-1}$ ) were there for both Zr and (Zr, Ni) co-doped STO.", "The presence of peaks at 148 cm$^{-1}$ (TO$_1$ ), 547 cm$^{-1}$ (TO$_4$ ) and 797 cm$^{-1}$ (LO$_4$ ) for undoped STO can be attributed to forbidden first order Raman scattering.", "This may indicate a nominal loss of inversion symmetry due to surface frozen dipoles and oxygen vacancies [45], [46].", "In case of doped samples, this nominal symmetry breaking can happen due to incorporation of dopants in the host STO [18].", "Figure: FTIR spectra of (a) SrTiO 3 _3, (b) Sr 0.96 Zr 0.04 TiO 3 \\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}\\text{O}_3 and (c) Sr 0.96 Zr 0.04 Ti 0.90 Ni 0.10 O 3 \\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}_{0.90}\\text{Ni}_\\text{0.10}\\text{O}_3 samples.Figure: (a) The real ϵ real \\epsilon _{\\text{real}} vs frequency ff and (b) the imaginary part ϵ imag \\epsilon _{\\text{imag}} vs frequency ff of the complex dielectric constant ϵ=ϵ real +iϵ imag \\epsilon = \\epsilon _{\\text{real}}+i\\epsilon _{\\text{imag}} of Sr1-xZrxTi1-yNiyO3 for (x, y)=(0.00, 0.00), (0.02, 0.00), (0.04, 0.00), (0.06, 0.00), (0.04, 0.05), (0.04, 0.10), (0.04, 0.15) and (0.04, 0.20)." ], [ "Fourier Transform Infrared Spectroscopy", "We also measured FTIR spectra for the undoped STO, 4% Zr doped $\\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}\\text{O}_3$ and (4% Zi, 10% Ni) co-doped $\\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}_{0.90}\\text{Ni}_{0.10}\\text{O}_3$ samples at room temperature from 350 to 4000 cm$^{-1}$ and displayed in Fig.", "REF .", "The measured FTIR absorption peaks were identified to corresponding chemical bond vibrations following Refs.", "[47], [48], [49], [50] in Table S4 of the Supplementary Information.", "The absorption peaks at 1652 cm$^{-1}$ and 3465 cm$^{-1}$ can be designated to hydroxyl –OH stretching vibration, see Fig.", "REF (a)&(b).", "The stemming of H2O or -OH in our sample may imply adsorption of water molecules from air surrounding the sample.", "The FTIR peaks at 1462, 1460 and 1485 cm$^{-1}$ for the three samples hinted deformed -OH in C-OH bond [47].", "The trace of C may have its origin in SrCO3 even after the calcination process.", "The FTIR bands in 300-600 cm$^{-1}$ represent characteristic IR absorptions due to Ti-O in STO.", "The peaks around $\\sim $ 366 cm$^{-1}$ for different samples can appear from TiO2 bending vibrations.", "Moreover the one near $\\sim $ 598 cm$^{-1}$ can be ascribed to TiO6 stretching vibration connected to Sr. We have not observed any sharp absorption peak at 500 cm$^{-1}$ in $\\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}\\text{O}_3$ which corresponds to Zr-O stretching vibrations in ZrO2, see Fig.", "REF (b).", "This corroborates the fact that Zr$^{4+}$ ions have been incorporated in STO lattice.", "But Zr$^{4+}$ ions, as it replaces Sr$^{2+}$ , shorten different Ti-O bond lengths to different degrees; effectively generate several very closely spaced IR absorption peaks.", "The combined effect of these peaks is to widen the Sr–Ti–O absorption peak at 594 cm$^{-1}$ .", "In case of (Zr, Ni) co-doped sample, as Ni$^{2+}$ ions replace the Ti$^{4+}$ , they affect Ti-O bonds and shift the Sr–Ti–O absorption peak to 536 cm$^{-1}$ in Fig.", "REF (c)." ], [ "Dielectric Measurements", "The circular disk-shaped pellets were used to form parallel plate capacitors with a geometric capacitance $C_0$ giving rise to a frequency $f$ and complex dielectric constant $\\epsilon = \\epsilon _{\\text{real}}+i\\epsilon _{\\text{imag}}$ ( $i=\\sqrt{-1}$ ) dependent impedance $Z(f)=1/i2\\pi fC_0 \\epsilon $ which was measured with the impedance analyzer [51], [52].", "The real part $\\epsilon _\\text{real}$ of the complex dielectric constant was plotted for $\\text{Sr}_\\text{1-x}\\text{Zr}_\\text{x}\\text{Ti}_\\text{1-y}\\text{Ni}_\\text{y}\\text{O}_3$ samples as a function of excitation frequency $f$ in Fig.", "REF (a).", "The $\\epsilon _\\text{real}$ gradually decreases with increasing $f$ for undoped, Zr doped and (Zr, Ni) co-doped STO samples.", "This indicates electric dipoles inside the material struggle to synchronize and fall out of steps with high frequency electric field.", "The dielectric behaviour is controlled by different constituents polarizations arising from interfacial charge, space charge, oriental dipolar, ionic and electronic contributions [53].", "Owing to the fact that the $\\epsilon _\\text{real}$ decays rapidly beyond 1 kHz, we attribute this to interfacial and space charge polarization effects [35], [54], [55].", "This polarization may arise due to charges trapped at the interface between the sample and the electrodes, space charges at the grain boundaries, interstitial and voids.", "These trapped charges are sluggish in responding to the applied field in the high frequency regime beyond 1 kHz and usually modelled within the general framework of Maxwell-Wagner relaxation processes occurring inside the sample [56].", "Figure: The ϵ real \\epsilon _{\\text{real}} of Sr1-xZrxTiO3 sample for x = 0.00, 0.02, 0.04 and 0.06.", "Inset: The ϵ real \\epsilon _{\\text{real}} of Sr0.96Zr0.04Ti1-yNiyO3 sample for y= 0.05, 0.10, 0.15 and 0.20.", "All estimations were done at ff = 1 kHz.For the dielectric loss analysis, the imaginary part $\\epsilon _\\text{imag}$ of the complex $\\epsilon $ was plotted as a function of frequency $f$ in Fig.", "REF (b).", "This $\\epsilon _\\text{imag}$ quantifies the energy dissipation of the electric dipoles due to random collisions or phase lag during their orientation change in response to the oscillating field.", "The frequency response of $\\epsilon _\\text{imag}$ has similar trend as compared to $\\epsilon _\\text{real}$ , i.e.", "it diminishes with increasing $f$ for undoped, Zr doped and (Zr, Ni) co-doped samples.", "This is expected as higher losses occur at low frequencies due to interfacial and space charge polarizations.", "The effect of doping on $\\epsilon _\\text{real}$ was analyzed in Fig.", "REF for a fixed frequency of 1 kHz.", "For Zr doped $\\text{Sr}_\\text{1-x}\\text{Zr}_\\text{x}\\text{Ti}\\text{O}_3$ samples, the $\\epsilon _\\text{real}$ increases monotonically with increasing composition x=$0.02,~04~\\text{and}~0.06$ .", "This enhancement of dielectric constant can be attributed to more space charge accumulation due to higher oxidation state of $\\text{Zr}^{4+}$ as compared to $\\text{Sr}^{2+}$ in A site of STO.", "But in case of $\\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}_\\text{1-y}\\text{Ni}_\\text{y}\\text{O}_3$ samples, $\\epsilon _\\text{real}$ decreases steadily with increasing Ni concentrations as shown in the inset of Fig.", "REF .", "This $\\epsilon _\\text{real}$ reduction may imply depletion of space charges as $\\text{Ni}^{2+}$ substitutes $\\text{Ti}^{4+}$ ion at the B-site of STO.", "Moreover, the substitutional $\\text{Ni}^{2+}$ can restrain the rattling of $\\text{Ti}^{4+}$ ions in TiO6 octahedra causing the reduction of dielectric constant [28], [57], [58].", "Figure: The resistivity ρ\\rho of Sr1-xZrxTi1-yNiyO3 as a function of frequency ff for (x, y)= (0.00, 0.00), (0.02, 0.00), (0.04, 0.00), (0.06, 0.00), (0.04, 0.05), (0.04, 0.10), (0.04, 0.15) and (0.04, 0.20).Figure: The resistivity ρ\\rho at a fixed frequency ff = 1 kHz of Sr1-xZrxTiO3 samples for x = 0.00, 0.02, 0.04 and 0.06.", "Inset: The ρ\\rho of Sr0.96Zr0.04Ti1-yNiyO3 samples for y= 0.05, 0.10, 0.15 and 0.20.Figure: PP-EE hysteresis loops of (a) Sr 1-x Zr x TiO 3 \\text{Sr}_\\text{1-x}\\text{Zr}_\\text{x}\\text{Ti}\\text{O}_3 for x=0.0,0.02,0.04and0.06\\text{x}=0.0,0.02,0.04~\\text{and}~0.06, (b) Sr 0.96 Zr 0.04 Ti 1-y Ni y O 3 \\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}_\\text{1-y}\\text{Ni}_\\text{y}\\text{O}_3 with y=0.00,0.05,0.10,0.15and0.20\\text{y}=0.00, ~0.05,~0.10,~0.15~\\text{and}~0.20." ], [ "Resistivity Measurements", "The frequency dependant ac resistivity $\\rho $ of the samples was measured from the complex impedance $Z = Z_\\text{real}+iZ_\\text{imag}$ by using the relation $\\rho =A_\\text{s}Z_\\text{real}/d_\\text{s}$ , where $A_\\text{s}$ and $d_\\text{s}$ represent area and thickness of the circular disc shape pellets respectively [59].", "The resistivity $\\rho $ decays with $f$ in Fig.", "REF .", "This indicates enhanced mobile charge hopping in the grain boundaries and sample-electrode interfaces [60], [35].", "The interfacial charges can produce a thin conductive layer at the sample surface effectively reducing the resistivity at high frequencies.", "Moreover the carrier transport in high frequency is dominated by bulk of the grains whereas grain boundary dictates the low frequency transport [61].", "The carrier activation energy at the grain boundaries may fall at high frequencies resulting in enhanced charge conduction [62].", "To analyze the effect of doping, we plot $\\rho $ for undoped, Zr doped $\\text{Sr}_\\text{1-x}\\text{Zr}_\\text{x}\\text{Ti}\\text{O}_3$ and (Zr, Ni) co-doped $\\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}_\\text{1-y}\\text{Ni}_\\text{y}\\text{O}_3$ samples at a fixed $f$ of 1 kHz in Fig.", "REF .", "The insulating characteristics of undoped STO is well captured in its high resitivity value.", "As we dope STO with Zr, the resistivity $\\rho $ of the $\\text{Sr}_{1-x}\\text{Zr}_x\\text{Ti}\\text{O}_3$ samples decreases with increasing concentrations $x = ~0.02,~0.04$ .", "This can be attributed to the fact that $\\text{Zr}^{4+}$ acts as an n-type dopant for $\\text{Sr}^{2+}$ and raises the conductivity of the samples.", "The Ni, as a second dopant in $\\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}_\\text{1-y}\\text{Ni}_\\text{y}\\text{O}_3$ samples, seems to gradually increase the $\\rho $ for $y=0.10,~0.15.~\\text{and}~0.20$ as shown in the inset of Fig.", "REF .", "A number of reasons can be ascribed to this conductivity depletion.", "The $\\text{Ni}^{2+}$ ion can act as an acceptor dopant for $\\text{Ti}^{4+}$ .", "This suppresses the n-type conductivity between the grain bulks as acceptor dopants can act as a non-conductive layer for mobile charge carriers across the grain boundaries [63], [64].", "Moreover, any oxygen vacancies induced by $\\text{Ni}^{2+}$ can act as a charge trapping center reducing the carrier mobility." ], [ "Electric Hysteresis Measurements", "The room temperature electric polarization ($P$ ) vs electric field ($E$ ) hysteresis $P$ -$E$ loops were measured with external triangular ac field excitation up to $\\pm $ 3 kV$\\text{cm}^{-1}$ at 50 Hz for undoped, Zr doped and (Zr, Ni) co-doped samples.", "The standard $P$ -$E$ loop parameters such as coercive electric field $E_\\text{c}$ , remnant polarization $P_\\text{r}$ , maximum polarization $P_\\text{max}$ and leakage current $I_\\text{d}$ at $P_\\text{max}$ were extracted and displayed in Table REF .", "The undoped STO exhibits a small $P$ -$E$ hysteresis loop where the $P$ is almost linear with $E$ and does not reach any saturation, see Fig.", "REF (a).", "At $E=$ 3 kVcm$^{-1}$ the sample showed a $P_\\text{max}=0.103$ $\\mu \\text{Ccm}^{-2}$ , a tiny $P_\\text{r}=0.017$ $\\mu \\text{Ccm}^{-2}$ and $E_\\text{c}=0.154$ $\\text{kVcm}^{-1}$ .", "The incorporation of Zr seems to enhance FE behaviour in STO.", "For Zr doped $\\text{Sr}_\\text{1-x}\\text{Zr}_\\text{x}\\text{Ti}\\text{O}_3$ samples, the $P_\\text{r}$ and $E_\\text{c}$ values monotonically increase with the Zr concentration, see Table REF .", "For (Zr, Ni) co-doped $\\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}_\\text{1-y}\\text{Ni}_\\text{y}\\text{O}_3$ sample with a composition of y=0.05, the Ni dopants diminish the ferroelectric characteristics by narrowing down the $P$ -$E$ loop ( $E_\\text{c}=$ 0.791 $\\text{kVcm}^{-1}$ , $P_\\text{r}=$ 0.039 $\\mu \\text{Ccm}^{-2}$ and $P_\\text{max}=$ 0.112 $\\mu \\text{Ccm}^{-2}$ ) as compared to $\\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}\\text{O}_3$ ( $E_\\text{c}=$ 1.216 $\\text{kVcm}^{-1}$ , $P_\\text{r}=$ 0.083 $\\mu \\text{Ccm}^{-2}$ and $P_\\text{max}=$ 0.162 $\\mu \\text{Ccm}^{-2}$ ), see Fig.", "REF (b).", "The FE hysteresis existed for all (Zr, Ni) co-doped samples even for higher Ni concentrations; for example in case of $\\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}_{0.90}\\text{Ni}_\\text{0.10}\\text{O}_3$ sample $E_\\text{c}=$ 1.420 $\\text{kVcm}^{-1}$ , $P_\\text{r}=$ 0.079 $\\mu \\text{Ccm}^{-2}$ and $P_\\text{max}=$ 0.125 $\\mu \\text{Ccm}^{-2}$ .", "For higher doping concentration in Zr and (Zr, Ni) co-doped samples, the area of the PE loop was enlarged which indicates increased dielectric losses were present inside the sample.", "This is consistent with the increased leakage current in the samples as displayed in Table REF .", "The origin of the leakage current can be due to carriers originated from the oxygen vacancies [65].", "The dominance of the leakage current prevented complete saturation in polarization to occur in our samples [66].", "Table: The coercive field (E c E_\\text{c}), the remnant polarization (P r P_\\text{r}), the maximum polarization (P max P_\\text{max}) and leakage current I d I_\\text{d} at P max P_\\text{max} of Sr 1-x Zr x Ti 1-y Ni y O 3 \\text{Sr}_\\text{1-x}\\text{Zr}_\\text{x}\\text{Ti}_\\text{1-y}\\text{Ni}_\\text{y}\\text{O}_3 for different values of x and y.Figure: Magnetic hysteresis loop of Sr1-xZrxTi1-yNiyO3 samples using vibrating sample magnetometer for (x, y) = (0.00, 0.00), (0.04, 0.00), (0.04, 0.05), (0.04, 0.10), (0.04, 0.15), (0.04, 0.20).", "Inset: The enlarged view of the hysteresis loop for 4% Zr and 10% Ni co-doped sample." ], [ "Magnetization Measurements", "We have recorded magnetization ($M$ ) vs. magnetic field ($H$ ) hysteresis loops of the as-prepared samples using a vibrating sample magnetometer applying a maximum magnetic field of $\\pm $ 20 kOe and displayed in Fig.", "REF .", "The inset of Fig.", "REF shows an enlarged view of the hysteresis loop for the 4 % Zr and 10 % Ni co-doped $\\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}_{0.90}\\text{Ni}_\\text{0.10}\\text{O}_3$ sample.", "We extracted different $M$ -$H$ loop parameters such as remnant magnetization ($M_\\text{r}$ ), the coercive field ($H_\\text{c}$ ) and the saturation magnetization ($M_\\text{s}$ ); and the values were inserted in Table REF .", "The $H_\\text{c}$ was quantified following the formula $H_\\text{c}=(H_\\text{c1}-H_\\text{c2})/2$ where $H_\\text{c1}$ and $H_\\text{c2}$ are the left and right coercive fields respectively [67].", "Table: The remnant magnetization (M r M_\\text{r}), the coercive field (H c H_\\text{c}) and the saturation magnetization (M s M_\\text{s}) of Sr 1-x Zr x Ti 1-y Ni y O 3 \\text{Sr}_\\text{1-x}\\text{Zr}_\\text{x}\\text{Ti}_\\text{1-y}\\text{Ni}_\\text{y}\\text{O}_3 for different values of x and y.The saturated hysteresis loop with a saturation magnetization $M_\\text{s}$ of 2.80 emug$^{-1}$ and coercive field of 4 Oe revealed soft FM nature of the as-synthesized STO sample.", "The origin of FM behaviour can be attributed to the presence oxygen vacancies $\\text{V}_\\text{O}^{2-}$ induced as a result of charge imbalance due to loss of Sr$^{2+}$ at high sintering temperature [68], [69], [70], [71], [72], [73].", "Moreover the variable oxidation state of Ti ($\\text{Ti}^{4+}\\Leftrightarrow \\text{Ti}^{3+}$ ) in STO can also facilitate oxygen vacancy to maintain charge equilibrium.", "The reduction in $M_\\text{s}$ to 0.50 emug$^{-1}$ in case of $4\\%$ Zr doped $\\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}\\text{O}_3$ sample can be ascribed to diamagnetic Zr$^{4+}$ ion with its empty $4d$ orbitals [74].", "In addition, the presence of Zr$^{4+}$ in the grain boundaries may reduce the defects at the grain surfaces that can cause suppression in $M_\\text{s}$ [70].", "Moreover we observed increments in $M_\\text{r}$ and $H_\\text{c}$ as compared to those of undoped STO which indicates reduction in softness of FM order due to Zr doping.", "The incorporation of Ni dopants notably increased $H_\\text{c}$ and $M_\\text{r}$ in (Zr, Ni) co-doped STO.", "In particular, for 4% Zi and 10 % Ni co-doped sample, the values of $H_\\text{c}$ , $M_\\text{r}$ and $M_\\text{s}$ are mentionable.", "For a further increment of the amount of Ni to 15 % and 20 %, $H_\\text{c}$ and $M_\\text{r}$ do not change significantly, however, $M_\\text{s}$ reduced notably.", "Due to Zr and Ni co-doping in STO, the coercivity enhancement was very high compared to that of undoped STO which might have originated form the inflated exchange coupling between Ni$^{2+}$ ions mediated by trapped electron in the oxygen vacancy [75], [76].", "Note also the variation in $H_\\text{c}$ with doping concentration is not surprising as it depends on large number of factors defining the microstructure of the sample such as grain homogeneity, grain size distribution and domain wall pinning [77].", "For the case of remnant magnetization $M_\\text{r}$ , we observed an order of magnitude enhancement in $\\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}_\\text{0.90}\\text{Ni}_\\text{0.10}\\text{O}_3$ sample ($322\\times 10^{-3}$ emug$^{-1}$ ) as compared to that of undoped STO ($35\\times 10^{-3}$ emug$^{-1}$ ).", "Overall the presence of hysteresis loop and the remnant magnetization corroborates long range FM order in our (Zr, Ni) co-doped samples." ], [ "Conclusion", "Undoped, Zr doped and (Zr, Ni) co-doped SrTiO3 samples were synthesized with varying degrees of doping concentrations and were characterized comprehensively using the appropriate techniques.", "We confirmed the cubic phase up to 4% Zr doping in STO from Rietveld analysis of the powder X-ray diffraction pattern.", "The substitution of 4% Zr instead of Sr in STO improved the morphological, electrical and magnetic properties.", "Therefore, Zr and Ni co-doped samples were prepared for this fixed % of Zr to improve further the electrical and magnetic properties of STO.", "Interestingly, 4% Zr and 10% Ni co-doped $\\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}_{0.90}\\text{Ni}_\\text{0.10}\\text{O}_3$ sample demonstrated $\\sim $ 1 and $\\sim $ 2 orders of magnitude enhancement in remnant magnetization and coercivity respectively at room temperature.", "Along with a clear ferroelectric hysteresis loop we observed also a ferromagnetic hysteresis loop for this co-doped sample.", "We may anticipate that simultaneous existence of ferromagnetic and ferroelectric phases in this as-synthesized (Zr, Ni) co-doped $\\text{Sr}_{0.96}\\text{Zr}_{0.04}\\text{Ti}_{0.90}\\text{Ni}_\\text{0.10}\\text{O}_3$ sample may open up the potentials as a multiferroic material for use in multifunctional applications." ], [ "Acknowledgments", "We gratefully acknowledge the support from Dr. Ishtiaque M. Syed for providing access to high temperature sintering facility in Semiconductor Technology Research Center, University of Dhaka.", "S.A. and A. K. M. S. H. F. contributed equally.", "Competing interests: The authors declare no competing interests.", "Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTION" ] ]
2105.11740
[ [ "Electroweak effects in $e^+ e^- \\to Z H$ process" ], [ "Abstract Electroweak radiative corrections to the cross section of the process $e^+ e^- \\to Z H$ are considered.", "The complete one-loop electroweak radiative corrections are evaluated with the help of the SANC system.", "Higher-order contributions of the initial state radiation are computed in the QED structure function formalism.", "Numerical results are produced by a new version of the ReneSANCe event generator and MCSANCee integrator for the conditions of future electron-positron colliders.", "The resulting theoretical uncertainty in the description of this process is estimated." ], [ "Introduction", "The Standard Model (SM) is extremely successful in describing various phenomena in particle physics.", "In spite of this fact, there are many reasons to consider the SM as an effective model, i.e., a low-energy approximation of a more general theory.", "Looking for the limits of the SM applicability domain is one of the most valuable problems in modern fundamental physics.", "On the other hand, a deep investigation of the SM properties at the quantum level is still an important task since this model is relevant for many applications in high-energy physics as well as astrophysics and cosmology.", "In this context, exploring the Higgs boson sector of the SM is crucial for checking the mechanism of spontaneous symmetry breaking and finalizing the verification of the model within the energy range achieved at modern accelerators.", "To perform an in-depth verification of the Standard Model and define the energy region of its applicability, we certainly need a new high-energy accelerator.", "An electron-positron collider with energies of a few hundred GeV looks now to be the best option.", "Several projects of this kind of machine are being under consideration, e.g., ILC [1], CLIC [2], CEPC [3], and FCC-ee [4].", "Programs of all these colliders necessarily include the option to run as Higgs factories with center-of-mass system (c.m.s.)", "energies of about 240 GeV.", "At this energy, the maximal count rate of events of the $e^+e^-\\rightarrow ZH$ processes can be reached.", "Collecting several million such events will substantially increase the precision of the Higgs boson mass and the determination of the partial decay widths [5], [6].", "The expected high statistics of events with Higgs bosons challenges the theory to provide very accurate SM predictions for the corresponding processes with uncertainty at the permille level.", "So we need to take into account radiative corrections in the first and higher orders of perturbation theoryNon-perturbative effects due to strong interactions are also relevant in running EW couplings and in producing extra meson pairs.", "The status of high-precision calculations for FCC-ee (and other future $e^+e^-$ colliders in general) is described in [7].", "In this work, we analyse QED and electroweak (EW) radiative corrections to the higgsstrahlung process $e^+\\ +\\ e^-\\ \\rightarrow Z\\ + H.$ This process is the most promising one in studying the Higgs boson properties.", "So the accuracy of its theoretical description should be higher than the experimental precision so that the combined uncertainty in the results of data analysis would not be spoiled by the theory.", "The uncertainty estimate should be as complete as possible.", "In this paper, we evaluate the complete one-loop corrections supplemented by higher-order (HO) QED contributions in the leading logarithmic approximation (LLA) [8].", "Our aim is to analyze the size of different HO contributions, estimate the resulting theoretical uncertainty, and verify the necessity to include other HO corrections.", "Note that in this work we do not consider decays of $Z$ and $H$ , which are left for further study.", "The complete one-loop electroweak radiative corrections to the process under consideration were computed with the help of the SANC computer system and reported in [9].", "Here we will concentrate on the analysis of the HO QED effects.", "Two-loop QED corrections due to the initial state radiation (ISR) for a general process of high-energy electron-positron annihilation through a virtual photon or $Z$ boson were calculated in [10] and recently corrected in [11].", "Higher-order QED ISR contributions in the leading and next-to-leading logarithmic approximations up to the order $\\mathcal {O}(\\alpha ^6L^5)$ were given in [12].", "These results are performed in an inclusive set-up where only the distribution in the invariant mass of the final state particles is available.", "So they provide a benchmark for comparisons while for practical applications one needs a Monte Carlo simulation with complete kinematics.", "The paper is organized as follows.", "In Section , we outline the contributions due to the higher-order QED initial state radiation order by order.", "In Section , we present the numerical results for the cross section of associated $ZH$ production in the energy region $\\sqrt{s} = 200-500$ GeV.", "Our conclusions are given in Section ." ], [ "General notes", "Let us consider ISR corrections to processes of high-energy electron-positron annihilation within the LLA.", "They can be evaluated with the help of the QED structure function formalism [8].", "For ISR corrections in the annihilation channel the large logarithm is $L=\\ln ({s}/{m_e^2})$ where the total c.m.s.", "energy $\\sqrt{s}$ is chosen as factorization scale.", "The master formula reads $ \\sigma ^{\\mathrm {LLA}} =\\int \\limits ^{1}_{0}\\mathrm {d}x_1 \\int \\limits ^{1}_{0}\\mathrm {d}x_2\\ \\mathcal {D}_{ee}(x_1) \\mathcal {D}_{ee}(x_2)\\sigma _0(x_1,x_2,s) \\Theta (\\mathrm {cuts}),$ where $\\sigma _0(x_1,x_2,s)$ is the Born level cross section of the annihilation process with reduced energies of the incoming particles.", "Here we do not take into account “photon induced” contributions, since the corresponding kernel cross sections $\\sigma (\\gamma e\\rightarrow e ZH)$ and $\\sigma (\\gamma \\gamma \\rightarrow ZH)$ are very much suppressed by extra powers of the fine structure constant $\\alpha $ .", "The electron structure functions $\\mathcal {D}_{ee}$ describe the density of probability to find an electron with an energy fraction $x$ in the initial electron [8], [13], [14].", "In the LLA approximation we can separate the pure photonic corrections (marked “$\\gamma $ ”) and the rest ones which include the pure pair and mixed photon-pair effects (marked as “pair”) as follows: $ && \\mathcal {D}_{ee}(x)= \\mathcal {D}_{ee}^{\\gamma }(x)+ \\mathcal {D}_{ee}^{\\mathrm {pair}}(x),\\\\&& \\mathcal {D}_{ee}^{\\gamma }(x)= \\delta (1-x)+ \\frac{\\alpha }{2\\pi }(L-1)P^{(1)}(x)+ \\left(\\frac{\\alpha }{2\\pi }(L-1)\\right)^2\\frac{1}{2!", "}P^{(2)}(x)\\nonumber \\\\ && \\qquad + \\left(\\frac{\\alpha }{2\\pi }(L-1)\\right)^3\\frac{1}{3!", "}P^{(3)}(x)+ \\left(\\frac{\\alpha }{2\\pi }(L-1)\\right)^4\\frac{1}{4!", "}P^{(4)}(x)+ {\\mathcal {O}}\\left(\\alpha ^5L^5\\right),\\\\&& \\mathcal {D}_{ee}^{\\mathrm {pair}}(x)=\\left(\\frac{\\alpha }{2\\pi }L\\right)^2\\biggl [\\frac{1}{3}P^{(1)}(x)+ \\frac{1}{2}R_s(x) \\biggr ]\\nonumber \\\\ && \\qquad + \\left(\\frac{\\alpha }{2\\pi }L\\right)^3\\biggl [\\frac{1}{3}P^{(2)}(x)+ \\frac{4}{27}P^{(1)}(x)+ \\frac{1}{3}R_p(x) - \\frac{1}{9}R_s(x) \\biggr ]+ {\\mathcal {O}}\\left(\\alpha ^4L^4\\right),$ Pair corrections can be separated into singlet $(\\sim R_{s,p})$ and non-singlet ones $(\\sim P^{(n)})$ .", "We take into account both by default.", "Non-singlet splitting functions can be represented in the form $ P^{(n)}(x) = \\lim _{\\Delta \\rightarrow 0}\\biggl \\lbrace \\delta (1-x)P^{(n)}_\\Delta + P^{(n)}_\\Theta (x)\\Theta (1-\\Delta -x) \\biggr \\rbrace $ with $\\Delta \\ll 1$ being the soft-hard separator.", "For example, $ P^{(1)}_\\Delta = 2\\ln \\Delta + \\frac{3}{2}, \\qquad P^{(1)}_\\Theta (x) = \\frac{1+x^2}{1-x}\\, .$ Higher-order non-singlet pure photonic splitting functions are obtained by iterations of convolution $ P^{(n+1)}(x) = \\int \\limits _0^1\\mathrm {d}x_1 \\int \\limits _0^1\\mathrm {d}x_2\\ \\delta (x-x_1x_2)P^{(n)}(x_1)P^{(1)}(x_2),$ see the details in Ref. [14].", "The singlet splitting functions $R_s$ and $R_p$ are not singular at $x\\rightarrow 1$ , so they do not contain $\\Delta $ parts.", "Explicit expressions for all relevant splitting functions are given in Ref.", "[14], [15].", "The Born level partonic cross section $\\sigma _0(x_1,x_2,s)$ is known in the partonic c.m.s.", "as $\\sigma _{\\mathrm {Born}}(\\hat{s})$ , where $\\hat{s}=x_1 x_2 s$ .", "The transition from the partonic c.m.s.", "into the laboratory reference frame is required if $x_1 x_2 \\ne 1$ .", "Let us classify contributions with different kinematics: I.", "$(SV)_1\\times (SV)_2$    The Born kinematics: additional contributions to Born+Soft+Virt.", "II.", "$H_1\\times (SV)_2$    One hard photon collinear to the first initial particle with possible soft and/or virtual (Soft+Virt) corrections to the second one.", "Hereafter “One hard photon” means “at least one hard photon in the same direction”.", "III.", "$(SV)_1\\times H_2$    Soft+Virt to the second initial particle and one hard photon along the first initial particle.", "IV.", "$H_1\\times H_2$    One hard photon along the first initial particle and one along the second one.", "Separation of hard and soft photon emission is provided by the dimensionless parameter $\\Delta \\ll 1$ with typical values $10^{-3}$ , $10^{-4}$ .", "Under all integrals relevant (process dependent) cuts on the lower values of $x_1$ and $x_2$ values should be applied.", "Application of representation (REF ) in structure functions (REF ) and their substitution into the master equation (REF ) gives the corrected cross section in the LLA approximation.", "We expand the result in $\\alpha $ and look at the second, third, and fourth order contributions.", "A few general comments are in order: $\\bullet $ The first lower index below denotes the order in $\\alpha L$ .", "$\\bullet $ Factorials and coefficients are given explicitly in order to see their structure.", "$\\bullet $ For pure photonic LLA corrections the traditional shift $L\\longrightarrow (L-1)$ is carried out, it takes into account part of the next-to-leading (NLO) corrections.", "However, for pair corrections such a shift is not well justified and we keep the large log unchanged." ], [ "First order LLA contributions", "There are only photonic corrections in ${\\mathcal {O}}\\left(\\alpha \\right)$ .", "Below we list the contributions of different kinematics.", "I.", "Born kinematics $&&\\!\\!", "\\delta \\sigma _{1,\\gamma }^{(I)} =\\biggl (\\frac{\\alpha }{2\\pi }(L-1)\\biggr )\\sigma _0(1,1,s)\\biggl \\lbrace 2 P_\\Delta ^{(1)} \\biggr \\rbrace .$ II.", "Emission only along the first particle $&&\\!\\!\\delta \\sigma _{1,\\gamma }^{(II)} =\\biggl (\\frac{\\alpha }{2\\pi }(L-1)\\biggr )\\!\\!\\int \\limits _{0}^{1-\\Delta }\\mathrm {d}x_1 \\sigma _0(x_1,1,s)\\biggl \\lbrace P_\\Theta ^{(1)}(x_1) \\biggr \\rbrace .$ III.", "Emission only along the second particle $&&\\!\\!\\delta \\sigma _{1,\\gamma }^{(III)} =\\biggl (\\frac{\\alpha }{2\\pi }(L-1)\\biggr )\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_2 \\sigma _0(1,x_2,s)\\biggl \\lbrace P_\\Theta ^{(1)}(x_2) \\biggr \\rbrace .$ IV.", "Emission along both initial particles $\\delta \\sigma _{1,\\gamma }^{(IV)} = 0.$" ], [ "Second order LLA contributions", "I.", "Born kinematics $&&\\!\\!", "\\delta \\sigma _{2,\\gamma }^{(I)} =\\biggl (\\frac{\\alpha }{2\\pi }(L-1)\\biggr )^2\\sigma _0(1,1,s)\\biggl \\lbrace 2\\frac{1}{2!", "}P_\\Delta ^{(2)} + P_\\Delta ^{(1)}P_\\Delta ^{(1)} \\biggr \\rbrace ,\\\\&&\\!\\!\\delta \\sigma _{2,\\mathrm {pair}}^{(I)} =\\biggl (\\frac{\\alpha }{2\\pi }L\\biggr )^2\\sigma _0(1,1,s)\\biggl \\lbrace 2\\frac{1}{3}P_\\Delta ^{(1)} \\biggr \\rbrace .$ II.", "Emission only along the first particle $&&\\!\\!\\delta \\sigma _{2,\\gamma }^{(II)} =\\biggl (\\frac{\\alpha }{2\\pi }(L-1)\\biggr )^2\\!\\!\\int \\limits _{0}^{1-\\Delta }\\mathrm {d}x_1 \\sigma _0(x_1,1,s)\\biggl \\lbrace \\frac{1}{2!", "}P_\\Theta ^{(2)}(x_1) + P_\\Theta ^{(1)}(x_1)P_\\Delta ^{(1)} \\biggr \\rbrace ,\\\\&&\\!\\!\\delta \\sigma _{2,\\mathrm {pair}}^{(II)} =\\biggl (\\frac{\\alpha }{2\\pi }L\\biggr )^2\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_1 \\sigma _0(x_1,1,s)\\biggl \\lbrace \\frac{1}{3}P_\\Theta ^{(1)}(x_1) + \\frac{1}{2}R_s(x_1) \\biggr \\rbrace .$ III.", "Emission only along the second particle $&&\\!\\!\\delta \\sigma _{2,\\gamma }^{(III)} =\\biggl (\\frac{\\alpha }{2\\pi }(L-1)\\biggr )^2\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_2 \\sigma _0(1,x_2,s)\\biggl \\lbrace \\frac{1}{2!", "}P_\\Theta ^{(2)}(x_2) + P_\\Theta ^{(1)}(x_2)P_\\Delta ^{(1)} \\biggr \\rbrace ,\\\\&&\\!\\!\\delta \\sigma _{2,\\mathrm {pair}}^{(III)} =\\biggl (\\frac{\\alpha }{2\\pi }L\\biggr )^2\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_2 \\sigma _0(1,x_2,s)\\biggl \\lbrace \\frac{1}{3}P_\\Theta ^{(1)}(x_2) + \\frac{1}{2}R_s(x_2) \\biggr \\rbrace .$ IV.", "Emission along both initial particles $&&\\!\\!\\delta \\sigma _{2,\\gamma }^{(IV)} =\\biggl (\\frac{\\alpha }{2\\pi }(L-1)\\biggr )^2\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_1 \\!\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_2 \\sigma _0(x_1,x_2,s)\\biggl \\lbrace P_\\Theta ^{(1)}(x_1)P_\\Theta ^{(1)}(x_2) \\biggr \\rbrace ,\\\\&&\\!\\!\\delta \\sigma _{2,\\mathrm {pair}}^{(IV)} = 0.$" ], [ "Third order LLA contributions", "I.", "Born kinematics $&&\\!\\!", "\\delta \\sigma _{3,\\gamma }^{(I)} =\\biggl (\\frac{\\alpha }{2\\pi }(L-1)\\biggr )^3\\sigma _0(1,1,s)\\biggl \\lbrace 2\\frac{1}{3!", "}P_\\Delta ^{(3)} + 2P_\\Delta ^{(1)}\\frac{1}{2!", "}P_\\Delta ^{(2)} \\biggr \\rbrace ,\\\\&&\\!\\!\\delta \\sigma _{3,\\mathrm {pair}}^{(I)} =\\biggl (\\frac{\\alpha }{2\\pi }L\\biggr )^3\\sigma _0(1,1,s)\\biggl \\lbrace 2\\frac{1}{3}P_\\Delta ^{(2)}+ 2\\frac{4}{27}P_\\Delta ^{(1)}+ 2\\frac{1}{3}P_\\Delta ^{(1)} P_\\Delta ^{(1)}\\biggr \\rbrace .$ II.", "Emission only along the first particle $&&\\!\\!\\delta \\sigma _{3,\\gamma }^{(II)} =\\biggl (\\frac{\\alpha }{2\\pi }(L-1)\\biggr )^3\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_1 \\sigma _0(x_1,1,s)\\biggl \\lbrace \\frac{1}{3!", "}P_\\Theta ^{(3)}(x_1)+ \\frac{1}{2!", "}P_\\Theta ^{(2)}(x_1)P_\\Delta ^{(1)}\\nonumber \\\\ && \\qquad + P_\\Theta ^{(1)}(x_1)\\frac{1}{2!", "}P_\\Delta ^{(2)} \\biggr \\rbrace ,\\\\&&\\!\\!\\delta \\sigma _{3,\\mathrm {pair}}^{(II)} =\\biggl (\\frac{\\alpha }{2\\pi }L\\biggr )^3\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_1 \\sigma _0(x_1,1,s)\\biggl \\lbrace \\frac{1}{3}P_\\Theta ^{(2)}(x_1)+ \\frac{4}{27}P_\\Theta ^{(1)}(x_1) + \\frac{1}{3}R_p(x_1)\\nonumber \\\\ && \\qquad - \\frac{1}{9}R_s(x_1)+ 2\\frac{1}{3}P_\\Theta ^{(1)}(x_1) P_\\Delta ^{(1)}+ \\frac{1}{2}R_s(x_1) P_\\Delta ^{(1)}\\biggr \\rbrace .$ III.", "Emission only along the second particle $&&\\!\\!\\delta \\sigma _{3,\\gamma }^{(II)} =\\biggl (\\frac{\\alpha }{2\\pi }(L-1)\\biggr )^3\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_2 \\sigma _0(1,x_2,s)\\biggl \\lbrace \\frac{1}{3!", "}P_\\Theta ^{(3)}(x_2)+ \\frac{1}{2!", "}P_\\Theta ^{(2)}(x_2)P_\\Delta ^{(1)}\\nonumber \\\\ && \\qquad + P_\\Theta ^{(1)}(x_2)\\frac{1}{2!", "}P_\\Delta ^{(2)} \\biggr \\rbrace ,\\\\&&\\!\\!\\delta \\sigma _{3,\\mathrm {pair}}^{(II)} =\\biggl (\\frac{\\alpha }{2\\pi }L\\biggr )^3\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_2 \\sigma _0(1,x_2,s)\\biggl \\lbrace \\frac{1}{3}P_\\Theta ^{(2)}(x_2)+ \\frac{4}{27}P_\\Theta ^{(1)}(x_2) + \\frac{1}{3}R_p(x_2)\\nonumber \\\\ && \\qquad - \\frac{1}{9}R_s(x_2)+ 2\\frac{1}{3}P_\\Theta ^{(1)}(x_2) P_\\Delta ^{(1)}+ \\frac{1}{2}R_s(x_2) P_\\Delta ^{(1)}\\biggr \\rbrace .$ IV.", "Emission along both initial particles $&&\\!\\!\\delta \\sigma _{3,\\gamma }^{(IV)} =\\biggl (\\frac{\\alpha }{2\\pi }(L-1)\\biggr )^3\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_1 \\!\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_2 \\sigma _0(x_1,x_2,s)\\biggl \\lbrace \\frac{1}{2!", "}P_\\Theta ^{(2)}(x_1)P_\\Theta ^{(1)}(x_2)\\nonumber \\\\ && \\qquad + P_\\Theta ^{(1)}(x_1)\\frac{1}{2!", "}P_\\Theta ^{(2)}(x_2)\\biggr \\rbrace ,\\\\&&\\!\\!\\delta \\sigma _{3,\\mathrm {pair}}^{(IV)} =\\biggl (\\frac{\\alpha }{2\\pi }L\\biggr )^3\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_1 \\!\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_2 \\sigma _0(x_1,x_2,s)\\biggl \\lbrace \\frac{1}{3}P_\\Theta ^{(1)}(x_1)P_\\Theta ^{(1)}(x_2)+ \\frac{1}{2}R_s(x_1)P_\\Theta ^{(1)}(x_2)\\nonumber \\\\ && \\qquad + P_\\Theta ^{(1)}(x_1)\\frac{1}{3}P_\\Theta ^{(1)}(x_2)+ P_\\Theta ^{(1)}(x_1)\\frac{1}{2}R_s(x_2)\\biggr \\rbrace .$" ], [ "Fourth order LLA contributions", "Here we list only pure photonic contributions due to the smallness of pair corrections in the fourth order.", "I.", "Born kinematics $&&\\!\\!", "\\delta \\sigma _{4,\\gamma }^{(I)} =\\biggl (\\frac{\\alpha }{2\\pi }(L-1)\\biggr )^4\\sigma _0(1,1,s)\\biggl \\lbrace 2\\frac{1}{4!", "}P_\\Delta ^{(4)} + 2P_\\Delta ^{(1)}\\frac{1}{3!", "}P_\\Delta ^{(3)}+ \\left(\\frac{1}{2!", "}P_\\Delta ^{(2)}\\right)^2 \\biggr \\rbrace .$ II.", "Emission only along the first particle $&&\\!\\!\\delta \\sigma _{4,\\gamma }^{(II)} =\\biggl (\\frac{\\alpha }{2\\pi }(L-1)\\biggr )^4\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_1 \\sigma _0(x_1,1,s)\\biggl \\lbrace \\frac{1}{4!", "}P_\\Theta ^{(4)}(x_1)+ \\frac{1}{3!", "}P_\\Theta ^{(3)}(x_1)P_\\Delta ^{(1)}\\nonumber \\\\ && \\qquad + \\frac{1}{2!", "}P_\\Theta ^{(2)}(x_1)\\frac{1}{2!", "}P_\\Delta ^{(2)}+ P_\\Theta ^{(1)}(x_1)\\frac{1}{3!", "}P_\\Delta ^{(3)} \\biggr \\rbrace .$ III.", "Emission only along the second particle $&&\\!\\!\\delta \\sigma _{4,\\gamma }^{(II)} =\\biggl (\\frac{\\alpha }{2\\pi }(L-1)\\biggr )^3\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_2 \\sigma _0(1,x_2,s)\\biggl \\lbrace \\frac{1}{4!", "}P_\\Theta ^{(4)}(x_2)+ \\frac{1}{3!", "}P_\\Theta ^{(3)}(x_2)P_\\Delta ^{(1)}\\nonumber \\\\ && \\qquad + \\frac{1}{2!", "}P_\\Theta ^{(2)}(x_2)\\frac{1}{2!", "}P_\\Delta ^{(2)}+ P_\\Theta ^{(1)}(x_2)\\frac{1}{3!", "}P_\\Delta ^{(3)} \\biggr \\rbrace .$ IV.", "Emission along both initial particles $&&\\!\\!\\delta \\sigma _{4,\\gamma }^{(IV)} =\\biggl (\\frac{\\alpha }{2\\pi }(L-1)\\biggr )^3\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_1 \\!\\!\\!\\int \\limits _{0}^{1-\\Delta }\\!\\!\\mathrm {d}x_2 \\sigma _0(x_1,x_2,s)\\biggl \\lbrace \\frac{1}{2!", "}P_\\Theta ^{(2)}(x_1)\\frac{1}{2!", "}P_\\Theta ^{(2)}(x_2)\\nonumber \\\\ && \\qquad + P_\\Theta ^{(1)}(x_1)\\frac{1}{3!", "}P_\\Theta ^{(3)}(x_2)+ \\frac{1}{3!", "}P_\\Theta ^{(3)}(x_1)P_\\Theta ^{(1)}(x_2)\\biggr \\rbrace .$" ], [ "LLA contributions for helicity states", "The leading order (LO) splitting function $P_{ee}(x)$ given by Eq.", "(REF ) preserves helicity [16], i.e., $P^{(n)}_{e_+e_+}(x)=P^{(n)}_{e_-e_-}(x)=P^{(n)}_{ee}(x),\\qquad P^{(n)}_{e_+e_-}(x)=P^{(n)}_{e_-e_+}(x)=0.$ However, singlet contributions of pair corrections can be separated for different helicities: $&& R_{s}(x)=R_{s,e_+e_+}(x)+R_{s,e_-e_+}(x),\\qquad R_{p}(x)=R_{p,e_+e_+}(x)+R_{p,e_-e_+}(x),\\\\&& R_{s,e_+e_+}(x) = 3(1-x) + 2(1+x)\\ln {x} + \\frac{2(1-x^3)}{3x},\\\\&& R_{s,e_-e_+}(x) = - 2(1-x) + \\frac{2(1-x^3)}{3x},\\\\&& R_{p,e_+e_+}(x) =- \\frac{31}{6}(1-x)+ 6(1-x)\\ln (1-x)- 2\\ln {x}+ 4x\\ln {x}+ \\frac{4(1-x^3)}{3x}\\ln (1-x)\\nonumber \\\\ && \\qquad + \\frac{4}{3}x^2\\ln {x}+ 4(1+x)\\ln {x}\\ln (1-x)- (1+x)\\ln ^2{x}+ 4(1+x){\\mathrm {Li}}_{2}\\left(1-x\\right) ,\\\\&& R_{p,e_-e_+}(x) =\\frac{4}{3}(1-x)+ \\frac{4(1-x^3)}{3x}\\ln (1-x)- 4(1-x)\\ln (1-x)\\nonumber \\\\ && \\qquad + 2(1-x)\\ln {x}+ \\frac{4}{3}x^2\\ln {x}.$ Since QED preserves parity, $R_{i,e_-e_+}(x) = R_{i,e_+e_-}(x), \\qquad R_{i,e_-e_-}(x) = R_{i,e_+e_+}(x).$" ], [ "Scheme with exponentiation", "In the master equation (REF ), the electron structure functions can be taken in the exponentiated form [17], [18].", "That would mean continuous integration over $x_{1,2}$ without the auxiliary parameter $\\Delta $ .", "The result of [18] contains only the pure photonic corrections and corresponds to the inclusion of exact leading logs up to the order ${\\mathcal {O}}\\left(\\alpha ^5L^5\\right)$ together with approximate (incomplete) HO LLA effects.", "Note that the HO exponentiated effects become exact (in LLA) in the soft photon limit.", "$\\bullet $ The pair LLA corrections can be added to the result of [18] as, e.g., in Ref.", "[19] with a possible update for higher-order pair corrections listed above.", "$\\bullet $ The exponentiated structure functions include the LLA part of one-loop QED radiative corrections.", "To avoid double counting with complete one-loop corrections, we need to subtract the first order leading logarithmic terms from one-loop corrections.", "So the final result reads $ && \\sigma ^{\\mathrm {corr.}}", "=\\int \\limits ^{1}_{0}\\mathrm {d}x_1 \\int \\limits ^{1}_{0}\\mathrm {d}x_2\\mathcal {D}_{ee}^{\\mathrm {exp}}(x_1) \\mathcal {D}_{ee}^{\\mathrm {exp}}(x_2)\\sigma _0(x_1,x_2,s) \\Theta (\\mathrm {cuts})\\nonumber \\\\ && \\qquad + \\biggl [ \\sigma ^{\\mathrm {Soft+Virt}} - \\sigma _{\\mathrm {LLA}}^{\\mathrm {Soft+Virt}}\\biggr ]+ \\biggl [ \\sigma ^{\\mathrm {Hard}} - \\sigma _{\\mathrm {LLA}}^{\\mathrm {Hard}}\\biggr ].$ The “Soft+Virt” part has the Born-like kinematics: $ \\sigma _{\\mathrm {LLA}}^{\\mathrm {Soft+Virt}} = \\sigma ^{\\mathrm {Born}}2\\frac{\\alpha }{2\\pi }(L-1)\\left[2\\ln \\omega + \\frac{3}{2}\\right],$ where $\\omega $ is the dimensionless soft-hard separator from the original complete one-loop formulae implemented in the Monte-Carlo generator.", "The “Hard LLA” term is rewritten to match \"Hard\" kinematics: $ \\sigma _{\\mathrm {LLA}}^{\\mathrm {Hard}} = \\frac{\\alpha }{2\\pi }\\int \\frac{\\mathrm {d}^3k}{k_0^2 2\\pi }\\biggl [\\frac{E^2}{kp_1}\\sigma ^{\\mathrm {Born}}(x_1,1,s)+ \\frac{E^2}{kp_2}\\sigma ^{\\mathrm {Born}}(1,x_2,s) \\biggr ]\\biggl (2-2\\frac{k_0}{E}+\\frac{k_0^2}{E^2}\\biggr ).$" ], [ "Numerical results\n", "In this section, we show numerical results for one-loop EW and HO QED radiative corrections to the $e^+e^- \\rightarrow ZH $ process.", "The input parameters can be found in [9].", "The results are obtained without any angular cuts.", "The relative correction $\\delta $ (in %) is defined as $\\delta = \\frac{\\sigma _{\\tt LLA}}{\\sigma ^{\\text{Born}}} - 1.$ Figure: Relative corrections (in %)(a) 𝒪(α 2 L 2 )γ\\mathcal {O}({\\alpha }^2L^2) {\\gamma },(b) 𝒪(α 2 L 2 )e + e - \\mathcal {O}({\\alpha }^2L^2) {e^+e^-},(c) 𝒪(α 2 L 2 )μ + μ - \\mathcal {O}({\\alpha ^2}L^2) {\\mu ^+\\mu ^-},(d) 10 ×\\times 𝒪(α 3 L 3 )γ\\mathcal {O}({\\alpha }^3L^3) {\\gamma },(e) 10 ×\\times 𝒪(α 3 L 3 )e + e - \\mathcal {O}({\\alpha }^3L^3) {e^+e^-},(f) 10 ×\\times 𝒪(α 3 L 3 )μ + μ - \\mathcal {O}({\\alpha }^3L^3) {\\mu ^+\\mu ^-},(g) 100 ×\\times 𝒪(α 4 L 4 )γ\\mathcal {O}({\\alpha }^4L^4) {\\gamma }and (h) their sum vs. c.m.s.", "energy.To illustrate the trends of the ISR contribution behaviour, we present separate distributions for each $\\mathcal {O}({\\alpha }^nL^n)$ , $n=2-4$ term and their sum as a function of the c.m.s.", "energy.", "Figure REF shows the values of the contribution of each relative correction (in %) : (a) $\\mathcal {O}({\\alpha }^2L^2) {\\gamma }$ , (b) $\\mathcal {O}({\\alpha }^2L^2) {e^+e^-}$ , (c) $\\mathcal {O}({\\alpha ^2}L^2) {\\mu ^+\\mu ^-}$ , (d) 10 $\\times $ $\\mathcal {O}({\\alpha }^3L^3) {\\gamma }$ , (e) 10 $\\times $ $\\mathcal {O}({\\alpha }^3L^3) {e^+e^-}$ , (f) 10 $\\times $ $\\mathcal {O}({\\alpha }^3L^3) {\\mu ^+\\mu ^-}$ , (g) 100 $\\times $ $\\mathcal {O}({\\alpha }^4L^4) {\\gamma }$ , and (h) their sum vs. c.m.s.", "energy.", "The dominant contribution is $\\mathcal {O}({\\alpha }^2L^2) {\\gamma }$ which is about 3% at the c.m.s.", "energy $\\sqrt{s}=220$ GeV, then it crosses the zero line approximately at $\\sqrt{s}=250$ GeV and goes to $-0.5\\%$ at $\\sqrt{s}=500$ GeV.", "The second order contributions due to light pair emission are smaller and have the opposite sign.", "The third and fourth order corrections are approximately 10 (100) times smaller, respectively.", "The sum is mainly determined by the $\\mathcal {O}({\\alpha }^2L^2) {\\gamma }$ term in the region of c.m.s energies $\\sqrt{s}=220-290$ GeV and becomes close to zero at $\\sqrt{s}=500$ GeV.", "One can see that in the threshold energy region there are several competing contributions with different behavior.", "This confirms the necessity to take into account HO QED ISR contributions in the studies of the higgsstrahlung process at future colliders.", "It is natural to consider the contributions of the corrections in some sets, that is, to compare the contributions of the same order of magnitude: (a)-(c), (d)-(f).", "The most significant value comes from the photonic contributions (a) and (d).", "The contributions from a pair is much less, but there is a kinematical region where their contribution must be taken into account.", "The suppression of pair corrections with respect to photonic ones in the same order is typical for high-energy annihilation processes [20].", "Figure: Relative corrections (in %) for the(a) 𝒪(αL)\\mathcal {O}({\\alpha }L),(b) 10 ×\\times 𝒪(α 2 L 2 )γ\\mathcal {O}({\\alpha }^2L^2) {\\gamma },(c) 10 ×\\times 𝒪(α 2 L 2 )e + e - \\mathcal {O}({\\alpha }^2L^2) {e^+e^-},(d) 10 ×\\times 𝒪(α 2 L 2 )μ + μ - \\mathcal {O}({\\alpha }^2L^2) {\\mu ^+\\mu ^-},(e) 100 ×\\times 𝒪(α 3 L 3 )γ\\mathcal {O}({\\alpha }^3L^3) {\\gamma },(f) 100 ×\\times 𝒪(α 3 L 3 )e + e - \\mathcal {O}({\\alpha }^3L^3) {e^+e^-},(g) 100 ×\\times 𝒪(α 3 L 3 )μ + μ - \\mathcal {O}({\\alpha }^3L^3) {\\mu ^+\\mu ^-}and(h) 1000 ×\\times 𝒪(α 4 L 4 )γ\\mathcal {O}({\\alpha }^4L^4) {\\gamma }vs. c.m.s.", "energy.Figure REF presents the contributions of the orders $\\mathcal {O}({\\alpha }^nL^n), n=1-4$ to the QED ISR.", "The largest effect corresponds to the first order $\\mathcal {O}({\\alpha }L)$ term which varies from about $-25\\%$ at the c.m.s.", "energy $\\sqrt{s}=220$ GeV to $+14\\%$ at $\\sqrt{s}=500$ GeV.", "This contribution is approximately 10 times larger than the second order contributions and 100 (1000) than the third (fourth) order terms.", "Figure: Cross-sections (in fb) vs. c.m.s.", "energy:(a) the Born, (b) the one with 𝒪(α)\\mathcal {O}({\\alpha }) QED corrections,(c) the one with the complete one-loop EW contributions.Figure REF illustrates the behaviour of the cross-sections with respect to the c.m.s.", "energy.", "It is seen that at the peak in the threshold region, the one-loop QED corrections change not only the height of the peak but also its form and position.", "Figure REF complements Fig.", "REF .", "It shows the size of the relative RC in different approximations.", "One can see that the HO ISR LLA contributions provide a small but visible shift (the difference between lines (b) and (c)) from the complete one-loop EW correction.", "Moreover, this shift changes its sign.", "Figure: Relative corrections (in %) (a) for the QED 𝒪(α)\\mathcal {O}({\\alpha }),(b) for the complete one-loopand (c) for the sum of (b) and∑ n=2 4 𝒪(α n L n )\\sum \\limits _{n=2}^{4}\\mathcal {O}(\\alpha ^nL^n) ISRcontributions vs. c.m.s.", "energy.In Tables REF and REF we show the ISR corrections of different order of ${\\cal O}(\\alpha ^nL^n), n=2-4$ in the LLA approximation for the c.m.s.", "energies $\\sqrt{s}=240$  GeV and 250 GeV in the $\\alpha (0)$ EW scheme.", "Table: ISR corrections in the LLA approximation for the e + e - →ZHe^+e^- \\rightarrow ZH process at s=240\\sqrt{s} = 240 GeV.", "No cuts are imposed.", "Here δ ISRLLA ≡δσ ISRLLA /σ 0 \\delta _{\\text{\\tiny {ISR LLA}}} \\equiv \\delta \\sigma _{\\text{\\tiny {ISR LLA}}}/\\sigma _{0}.", "The Born cross section is σ 0 =225.74(1)\\sigma _0 = 225.74(1) fb.Table: ISR corrections in the LLA approximation for the e + e - →ZHe^+e^- \\rightarrow ZH process at s=250\\sqrt{s} = 250 GeV.No cuts are imposed.", "Here δ ISRLLA ≡δσ ISRLLA /σ 0 \\delta _{\\text{\\tiny {ISR LLA}}} \\equiv \\delta \\sigma _{\\text{\\tiny {ISR LLA}}}/\\sigma _{0}.The Born cross section is σ 0 =225.59(1)\\sigma _0 = 225.59(1) fb.It is seen in Tables REF and REF that the corrections for the sum of all considered orders of the ISR terms $\\sum _{n=2}^4{\\cal O}(\\alpha ^nL^n)$ are about 0.322% for the c.m.s.", "energy $\\sqrt{s}=240$  GeV and about -0.207% for the c.m.s.", "energy $\\sqrt{s}=250$  GeV.", "For the c.m.s.", "energy $\\sqrt{s}=240$  GeV the most significant HO contribution is of course the photonic one of the order ${\\cal O}(\\alpha L)^2$ .", "It composes about half a percent while from pairs we get about $-(0.1-0.2) \\%$ .", "For the c.m.s.", "energy $\\sqrt{s}=250$  GeV the dominant contributions of the second order are about $-0.099\\%$ for $\\gamma $ and $-0.119\\%$ for $e^+e^-$ -pairs ($-0.070\\%$ for $\\mu ^+\\mu ^-$ -pairs), respectively.", "When considering HO corrections, we see that it is certainly sufficient to take into account corrections up to the fourth order.", "Variation of the factorization scale in the argument of the large logarithm can simulate the next-to-leading corrections, e.g., $\\mathcal {O}(\\alpha ^2L)$ .", "In the same manner as in estimates of scale variation uncertainties in QCD, we apply factors 2 and 1/2 in the argument of the large logarithm.", "This leads to the following values of the HO LLA corrections at $\\sqrt{s}=240$  GeV: $\\delta _{\\text{\\tiny {LLA}}}(2\\sqrt{s})=0.361(1)\\%$ and $\\delta _{\\text{\\tiny {LLA}}}(\\sqrt{s}/2)=0.286(1)\\%$ , respectively.", "And for $\\sqrt{s}=250$  GeV we get $\\delta _{\\text{\\tiny {LLA}}}(2\\sqrt{s})=-0.228(1)\\%$ and $\\delta _{\\text{\\tiny {LLA}}}(\\sqrt{s}/2)=-0.187(1)\\%$ .", "Table: The Born and pure weak corrections in different EW schemes at the c.m.s.", "energy s=240\\sqrt{s} = 240 GeV.Table: The Born and pure weak corrections in different EW schemes at the c.m.s.", "energy s=250\\sqrt{s} = 250 GeV.In Tables REF and REF , the results of the Born cross sections, the sum of the Born and pure weak (PW) contributions as well as the relative corrections $\\delta $ (%) for the c.m.s.", "energies $\\sqrt{s}=240$ and 250 GeV in the $\\alpha (0)$ , $G_\\mu $ and $\\alpha (M_Z^2)$ EW schemes are presented.", "The $\\alpha (M_{\\rm {Z}}^2)$ = 1/129.02 value was used in the calculations.", "As it is seen the corrections in the $\\alpha (0)$ scheme are positive and equal to 2.72% for the c.m.s.", "energy $\\sqrt{s}=240$  GeV and 2.47% for $\\sqrt{s}=250$  GeV.", "The calculations in the $G_{\\mu }$ scheme reduce RC to about 5-6 %, they become negative and equal to -2.99% for the c.m.s.", "energy $\\sqrt{s}=240$  GeV and -3.24% for the c.m.s energy $\\sqrt{s}=250$  GeV.", "In the case of the $\\alpha (M_Z^2)$ scheme, RCs get even more negative and achieve the value -8.97% and -9.22% for the c.m.s.", "energies $\\sqrt{s}=240$  GeV and $\\sqrt{s}=250$  GeV, respectively.", "These results show that there is no most suitable EW scheme of calculations for minimizing the value of the pure weak corrections for the $e^+ e^- \\rightarrow ZH$ reaction.", "However, the sensitivity to the choice of input EW scheme is reduced for the Born+PW cross-sections compared to the Born one.", "In [21] and [22], the mixed QCD and EW NNLO corrections were considered and a further reduction of the EW scheme dependence was observed.", "Table: Comparison between results with order-by-order and exponentiated structure functions.", "Only pure photonic corrections are taken in account.", "HereR i =σ i /σ ExpMul (3) R_{i} = \\sigma _{i}/\\sigma ^{(3)}_{\\text{{Exp Mul}}},i=(LLA,ExpAdd,ExpMul)i=(\\text{{LLA}},\\text{{Exp Add}},\\text{{Exp Mul}}),σ ExpAdd \\sigma _{\\text{{Exp Add}}} calculated withthe electron structure functions taken in the additive exponentiated form and σ ExpMul \\sigma _{\\text{{Exp Mul}}} in the multiplicative exponentiated form .In Table REF , we verified the difference between order-by-order and exponentiated (\"additive\" according to the prescription of Kuraev and Fadin [8] and \"multiplicative\" proposed by Jadach and Ward [23]) realizations of the electron structure function.", "Results are shown up to $\\mathcal {O}(\\alpha ^3L^3)$ finite terms for exponentiated forms and up to $\\mathcal {O}(\\alpha ^4L^4)$ for order-by-order calculation.", "It can be seen that result using multiplicative exponentiated form converges faster.", "But taking into account four orders in the order-by-order calculation is enough to reach the $10^{-4}$ accuracy." ], [ "Conclusions ", "So we considered the contributions due to the QED initial state radiation (photons and pairs) to the higgsstrahlung process.", "Their impact has been analyzed order by order.", "The complete one-loop electroweak one-loop corrections were presented.", "Higher-order ISR QED contributions were calculated within the leading logarithmic approximation.", "The known expressions for contributions of the collinear electron structure function of the orders ${\\cal O}(\\alpha ^n L^n), n=2-4$ for photons and pairs were used.", "These corrections are known to be very important in the case of resonances, e.g., at the $Z$ -boson peak studied at LEP.", "We would like to emphasize that higher-order QED ISR corrections can be large not only at resonances but also near the reaction thresholds.", "Note that the cross section of this process has a peak at the threshold.", "By looking at the magnitude of the complete one-loop electroweak and higher-order LLA QED corrections, we can estimate the theoretical uncertainty and define what other contributions should be taken into account.", "Namely, a safe estimate of the theoretical uncertainty in EW and LLA RC can be derived by variation of the EW scheme and the factorization scale, respectively.", "One can see that to meet high precision of future experiments, we need to go beyond the approximation explored here.", "At least the next-to-leading QED ISR logarithmic corrections should also be taken into account.", "One needs to improve the uncertainty in pure weak contributions.", "That can be done by taking into account higher-order EW and mixed QCD and EW effects in the $Z$ boson propagator and vertices.", "Note also that corrections for the whole processes with different decay modes of $Z$ and Higgs bosons should be evaluated.", "For the one permille precision tag relevant for future studies of the higgsstrahlung process, we see that there is a good agreement between the order-by-order results and the known exponentiated QED LLA corrections [17], [18].", "So either approach can be used.", "Presumably, the exponentiated one is more suitable for Monte Carlo simulations, while the order-by-order one can be used for benchmarks and cross-checks.", "The numerical results presented here were obtained by means of the Monte Carlo generator ReneSANCe [24] and MCSANCee integrator [25] which allow evaluation of arbitrary differential cross sections.", "These computer codes can be downloaded from the SANC project homepage http://sanc.jinr.ru and the ReneSANCe HEPForge page https://renesance.hepforge.org." ], [ "Acknowledgments", "This research was funded by RFBR grant 20-02-00441.", "The authors are grateful to Ya.", "Dydyshka for fruitful discussions." ] ]
2105.11708
[ [ "Bridging Few-Shot Learning and Adaptation: New Challenges of\n Support-Query Shift" ], [ "Abstract Few-Shot Learning (FSL) algorithms have made substantial progress in learning novel concepts with just a handful of labelled data.", "To classify query instances from novel classes encountered at test-time, they only require a support set composed of a few labelled samples.", "FSL benchmarks commonly assume that those queries come from the same distribution as instances in the support set.", "However, in a realistic set-ting, data distribution is plausibly subject to change, a situation referred to as Distribution Shift (DS).", "The present work addresses the new and challenging problem of Few-Shot Learning under Support/Query Shift (FSQS) i.e., when support and query instances are sampled from related but different distributions.", "Our contributions are the following.", "First, we release a testbed for FSQS, including datasets, relevant baselines and a protocol for a rigorous and reproducible evaluation.", "Second, we observe that well-established FSL algorithms unsurprisingly suffer from a considerable drop in accuracy when facing FSQS, stressing the significance of our study.", "Finally, we show that transductive algorithms can limit the inopportune effect of DS.", "In particular, we study both the role of Batch-Normalization and Optimal Transport (OT) in aligning distributions, bridging Unsupervised Domain Adaptation with FSL.", "This results in a new method that efficiently combines OT with the celebrated Prototypical Networks.", "We bring compelling experiments demonstrating the advantage of our method.", "Our work opens an exciting line of research by providing a testbed and strong baselines.", "Our code is available at https://github.com/ebennequin/meta-domain-shift." ], [ "Introduction", "In the last few years, we have witnessed outstanding progress in supervised deep learning [34], [27].", "As the abundance of labelled data during training is rarely encountered in practice, ground-breaking works in Few-Shot Learning (FSL) have emerged [57], [53], [21], particularly for image classification.", "This paradigm relies on a straightforward setting.", "At test-time, given a set of unseen classes during training and few (typically 1 to 5) labelled examples for each one of those classes, the task is to classify query samples among them.", "We usually call the set of labelled samples the support set, and the set of query samples the query set.", "Well-adopted FSL benchmarks [57], [50], [56] commonly sample the support and query sets from the same distribution.", "We stress that this assumption does not hold in most use cases.", "When deployed in the real-world, we expect an algorithm to infer on data that may shift, resulting in an acquisition system that deteriorates, lighting conditions that vary, or real world objects evolving [1].", "Figure: FSL under Support / Query ShiftThe situation of Distribution Shift (DS) i.e., when training and testing distributions differ, is ubiquitous and has dramatic effects on deep models [28], motivating works in Unsupervised Domain Adaptation [46], Domain Generalization [25] or Test-Time Adaptation [58].", "However, the state of the art brings insufficient knowledge on few-shot learners' behaviours when facing distribution shift.", "Some pioneering works demonstrate that advanced FSL algorithms do not handle cross-domain generalization better than more naive approaches [11].", "Despite its great practical interest, FSL under distribution shift between the support and query sets is an under-investigated problem and attracts a very recent attention [20].", "We refer to it as Few-Shot Learning under Support/Query Shift (FSQS) and provide an illustration in Figure REF .", "It reflects a more realistic situation where the algorithm is fed with a support set at the time of deployment and infers continuously on data subject to shift.", "The first solution is to re-acquire a support set that follows the data's evolution.", "Nevertheless, it implies human intervention to select and annotate data to update an already deployed model, reacting to a potential drop in performances.", "The second solution consists in designing an algorithm that is robust to the distribution shift encountered during inference.", "This is the subject of the present work.", "Our contributions are: FewShiftBed: a testbed for FSQS available at https://github.com/ebennequin/meta-domain-shift.", "The testbed includes 3 challenging benchmarks along with a protocol for fair and rigorous comparison across methods as well as an implementation of relevant baselines, and an interface to facilitate the implementation of new methods.", "We conduct extensive experimentation of a representative set of few-shot algorithms.", "We empirically show that Transductive Batch-Normalization [8] mitigates an important part of the inopportune effect of FSQS.", "We bridge Unsupervised Domain Adaptation (UDA) with FSL to address FSQS.", "We introduce Transported Prototypes, an efficient transductive algorithm that couples Optimal Transport (OT) [47] with the celebrated Prototypical Networks [53].", "The use of OT follows a long-standing history in UDA for aligning representations between distributions [4], [22].", "Our experiments demonstrate that OT shows a remarkable ability to perform this alignment even with only a few samples to compare distributions and provide a simple but strong baseline.", "In Section  we provide a formal statement of FSQS, and we position this new problem among existing learning paradigms.", "In Section , we present FewShiftBed.", "We detail the datasets, the chosen baselines, and a protocol that guarantees a rigorous and reproducible evaluation.", "In Section , we present a method that couples Optimal Transport with Prototypical Networks [53].", "In Section , we conduct an extensive evaluation of baselines and our proposed method using the testbed.", "Finally, we present in Section  the related works, while in Section we draw perspectives of improvement and interesting research directions.", "We consider an input space $\\mathcal {X}$ , a representation space $\\mathcal {Z} \\subset \\mathbb {R}^d$ ($d>0$ ) and a set of classes $\\mathsf {C}$ .", "A representation is a learnable function from $\\mathcal {X}$ to $\\mathcal {Z}$ and is noted $\\varphi (\\cdot ; \\theta )$ with $\\theta \\in \\Theta $ for $\\Theta $ a set of parameters.", "A dataset is a set $\\Delta (\\mathsf {C}, \\mathsf {D})$ defined by a set of classes $\\mathsf {C}$ and a set of domains $\\mathsf {D}$ i.e., a domain $\\mathcal {D} \\in \\mathsf {D}$ is a set of IID realizations from a distribution noted $p_{\\mathcal {D}}$ .", "For two domains $\\mathcal {D},\\mathcal {D}^{\\prime } \\in \\mathsf {D}$ , the distribution shift is characterized by $p_{\\mathcal {D}} \\ne p_{\\mathcal {D}^{\\prime }}$ .", "For instance, if the data consists of images of letters handwritten by several users, $\\mathcal {D}$ can consist of samples from a specific user.", "Referring to the well known UDA terminology of source / target [46], we define a couple of source-target domains as a couple $(\\mathcal {D}_s, \\mathcal {D}_t)$ with $p_{\\mathcal {D}_s} \\ne p_{\\mathcal {D}_t}$ , thus presenting a distribution shift.", "Additionally, given $\\mathcal {C} \\subset \\mathsf {C}$ and $\\mathcal {D} \\in \\mathsf {D}$ , the restriction of a domain $\\mathcal {D}$ to images with a label that belongs to $\\mathcal {C}$ is noted $\\mathcal {D}^{\\mathcal {C}}$ .", "Figure: During meta-learning (Train-Time), each episode contains a support and a query set sampled from different distributions (for instance, illustrated by noise and contrasts as in Figure ) from a set of training domains (𝖣 train \\mathsf {D}_{\\mathrm {train}}), reflecting a situation that may potentially occurs at test-time.", "When deployed, the FSL algorithm using a trained backbone is fed with a support set sampled from new classes.", "As the algorithm is subject to infer continuously on data subject to shift (Test-Time), we evaluate the algorithm on data with an unknown shift (𝖣 test \\mathsf {D}_{\\mathrm {test}}).", "Importantly, both classes (𝖢 train ∩𝖢 test =∅{\\mathsf {C}}_{\\mathrm {train}} \\cap {\\mathsf {C}}_{\\mathrm {test}} = \\emptyset ) and shifts (𝖣 train ∩𝖣 test =∅\\mathsf {D}_{\\mathrm {train}} \\cap \\mathsf {D}_{\\mathrm {test}} = \\emptyset ) are not seen during training, making the FSQS a challenging problem of generalization." ], [ "Dataset splits.", "We build a split of $\\Delta (\\mathsf {C}, \\mathsf {D})$ , by splitting $\\mathsf {D}$ (respectively $\\mathsf {C}$ ) into $\\mathsf {D}_{\\text{train}}$ and $\\mathsf {D}_{\\text{test}}$ (respectively $\\mathsf {C}_{\\text{train}}$ and $\\mathsf {C}_{\\text{test}}$ ) such that $\\mathsf {D}_{\\text{train}} \\cap \\mathsf {D}_{\\text{test}} = \\emptyset $ and $\\mathsf {D}_{\\text{train}} \\cup \\mathsf {D}_{\\text{test}} = \\mathsf {D}$ (respectively $\\mathsf {C}_{\\text{train}} \\cap \\mathsf {C}_{\\text{test}} = \\emptyset $ and $\\mathsf {C}_{\\text{train}} \\cup \\mathsf {C}_{\\text{test}} = \\mathsf {C}$ ).", "This gives us a train/test split with the datasets $\\Delta _{\\text{train}} = \\Delta (\\mathsf {C}_{\\text{train}}, \\mathsf {D}_{\\text{train}})$ and $\\Delta _{\\text{test}} = \\Delta (\\mathsf {C}_{\\text{test}}, \\mathsf {D}_{\\text{test}})$ .", "By extension, we build a validation set following this protocol." ], [ "Few-Shot Learning under Support-Query Shift (FSQS).", "Given: $\\mathsf {D}^{\\prime } \\in \\lbrace \\mathsf {D}_{\\mathrm {train}}, \\mathsf {D}_{\\mathrm {test}}\\rbrace $ and $\\mathsf {C}^{\\prime } \\in \\lbrace \\mathsf {C}_{\\mathrm {train}}, \\mathsf {C}_{\\mathrm {test}}\\rbrace $ , a couple of source-target domains $(\\mathcal {D}_s, \\mathcal {D}_t)$ from $\\mathsf {D}^{\\prime }$ , a set of classes $\\mathcal {C} \\subset \\mathsf {C}^{\\prime }$ ; a small labelled support set $\\mathcal {S} = {(x_i, y_i)}_{i=1, \\dots , |\\mathcal {S}|}$ (named source support set) such that for all $i$ , $y_i \\in \\mathcal {C}$ , and $x_i \\in \\mathcal {X}$ such that $(x_i,y_i)$ is an instance of $\\mathcal {D}_s^{\\mathcal {C}}$ i.e., $\\mathcal {S} \\subset \\mathcal {D}_s^{\\mathcal {C}}$ ; an unlabelled query set $\\mathcal {Q} = {(x_i)}_{i=1, \\dots , |\\mathcal {Q}|}$ (named target query set) such that for all $i$ , $x_i$ is an instance of $\\mathcal {D}_t^{\\mathcal {C}}$ i.e., $\\mathcal {Q} \\subset \\mathcal {D}_t^{\\mathcal {C}}$ .", "The task is to predict the labels of query set, instances of $\\mathcal {Q}$ , in $\\mathcal {C}$ .", "When $|\\mathcal {C}| = n$ and the support set contains $k$ labelled instances for each class, this is called an $n$ -way $k$ -shot FSQS classification task.", "Note that this paradigm provides an additional challenge compared to classical Few-shot classification tasks, since at test time, the model is expected to generalize to both new classes and new domains while support set and query set are sampled from different distributions.", "As presented above, in order to evaluate the model's capacity to generalize to both new classes and new domains, we split the dataset into train, validation and test sets (respectively named $\\Delta _{\\mathrm {train}}$ , $\\Delta _{\\mathrm {val}}$ and $\\Delta _{\\mathrm {test}}$ ) controlling that there is no overlap of both classes and domains of these sets.", "Model's parameters are trained on $\\Delta _{\\mathrm {train}}$ and selected on $\\Delta _{\\mathrm {val}}$ .", "Finally, the model is tested on $\\Delta _{\\text{test}}$ .", "This paradigm is illustrated in Figure REF ." ], [ "Episodic training.", "We build an episode by sampling some classes $\\mathcal {C} \\subset \\mathsf {C}_{\\text{train}}$ , and a source and target domain $\\mathcal {D}_s, \\mathcal {D}_t$ from $\\mathsf {D}_{\\text{train}}$ .", "We build a support set $\\mathcal {S} = {(x_i, y_i)}_{i=1 \\dots |\\mathcal {S}|}$ of instances from source domain $\\mathcal {D}_s^{\\mathcal {C}}$ , and a query set $\\mathcal {Q} = {(x_i, y_i)}_{i=|\\mathcal {S}|+1 , \\dots , |\\mathcal {S}|+|\\mathcal {Q}|}$ of instances from target domain $\\mathcal {D}_t^{\\mathcal {C}}$ , such that $\\forall i \\in \\left[ 1, |\\mathcal {S}|+|\\mathcal {Q}| \\right]$ , $y_i \\in \\mathcal {C}$ .", "Using the labelled examples from $\\mathcal {S}$ and unlabelled instances from $\\mathcal {Q}$ , the model is expected to predict the labels of $\\mathcal {Q}$ .", "The parameters of the model are then trained using a cross-entropy loss between the predicted labels and ground truth labels of the query set." ], [ "Positioning", "To highlight FSQS's novelty, our discussion revolves around the problem of inferring on a given Query Set provided with the knowledge of a Support Set.", "We refer to this class of problems as SQ problems.", "Intrinsically, FSL falls into the category of SQ problems.", "Interestingly, Unsupervised Domain Adaptation [46] (UDA), defined as labelling a dataset sampled from a target domain based on labelled data sampled from a source domain, is also a SQ problem.", "Indeed, in this case, the source domain plays the role of support, while the target domain plays the query's role.", "Notably, an essential line of study in UDA leverages the target data distribution for aligning source and target domains, reflecting the importance of transduction in a context of adaptation [4], [22] i.e., performing prediction by considering all target samples together.", "Transductive algorithms also have a special place in FSL [19], [39], [50] and show that leveraging a query set as a whole brings a significant boost in performances.", "Nevertheless, UDA and FSL exhibit fundamental differences.", "UDA addresses the problem of distribution shift using important source data and target data (typically thousands of instances) to align distributions.", "In contrast, FSL focuses on the difficulty of learning from few samples.", "To this purpose, we frame UDA as both SQ problem with large transductivity and Support / Query Shift, while Few-Shot Learning is a SQ problem, eventually with small transductivity for transductive FSL.", "Thus, FSQS combines both challenges: distribution shift and small transductivity.", "This new perspective allows us to establish fruitful connections with related learning paradigms, presented in Table REF , that we review in the following.", "A thorough review is available in Appendix A." ], [ "Adaptation.", "We review the UDA works that are the more related to our problem.", "Inspired from the principle of invariant representations [4], [22], the seminal work [16] brings Optimal Transport [47] as an efficient framework for aligning data distributions.", "UDA requires a whole target dataset for inference, limiting its applications.", "Recent pioneering work, referred to as Test-Time Adaptation (TTA), adapts at test-time a model provided with a batch of samples from the target distribution.", "The proposed methodologies are test-time training by self-supervision [54], updating batch-normalization statistics [52] or parameters [58], or meta-learning to condition predictions on the whole batch of test samples for an Adaptative Risk Minimization (ARM) [63]." ], [ "Few-Shot Classification.", "We usually frame Few-Shot Classification methods [11] as either metric-based methods [57], [53], or optimization-based methods that learn to fine-tune by adapting with few gradient steps [21], or hallucination-based methods that augment the rare labelled data [26].", "A promising line of study leverages transductivity (using the query set as unlabelled data while inductive methods predict individually on each query sample) and brings Semi-Supervised principles to FSL.", "Transductive Propagation Network [39] meta-learns label propagation from the support to query set concurrently with the feature extractor.", "Transductive Fine-Tuning [19] minimizes the prediction entropy of all query instances during fine-tuning.", "At last, Ren et al.", "[50] use the query set to improve the estimation of prototypes.", "Evaluating cross-domain generalization of FSL (FSCD), i.e., a distributional shift between meta-training and meta-testing, attracts the attention of a few recent works [11].", "In particular, naive approaches perform equally, even better than advanced FSL as shown in [11].", "Zhao et al.", "propose a Domain-Adversarial Prototypical Network [65] in order to both align source and target domains in the feature space while maintaining discriminativeness between classes.", "Sahoo et al.", "combine Prototypical Networks with adversarial domain adaptation at the task level [51].", "Notably, Cross-Domain Few-Shot Learning [11] (CDFSL) addresses the distributional shift between meta-training and meta-testing assuming that the support set and query set are drawn from the same distribution, not making it a SQ problem with support-query shift." ], [ "Datasets", "We designed three new image classification datasets adapted to the FSQS problem.", "These datasets have two specificities.", "They are dividable into groups of images, assuming that each group corresponds to a distinct domain.", "A key challenge is that each group must contain enough images with a sufficient variety of class labels, so that it is possible to sample FSQS episodes.", "They are delivered with a train/val/test split, along both the class and the domain axis.", "This means that $\\Delta _{\\text{train}}$ (resp.", "$\\Delta _{\\text{val}}$ , $\\Delta _{\\text{test}}$ ) contains images from any domain $\\mathsf {D}_{\\text{train}}$ (resp.", "$\\mathsf {D}_{\\text{val}}$ , $\\mathsf {D}_{\\text{test}}$ ), that have their labels in $\\mathsf {C}_{\\text{train}}$ (resp.", "$\\mathsf {C}_{\\text{val}}$ , $\\mathsf {C}_{\\text{test}}$ ), with $\\forall a,b \\in \\lbrace \\text{train, val, test} \\rbrace $ , $\\mathsf {D}_{\\text{a}} \\cap \\mathsf {D}_{\\text{b}} = \\emptyset $ and $\\mathsf {C}_{\\text{a}} \\cap \\mathsf {C}_{\\text{b}} = \\emptyset $ .", "Therefore, these datasets provide true few-shot tasks at test time, in the sense that the model will not have seen any instances of test classes and domains during training.", "Furthermore, we split the classes with respect to intuitive semantic similarity, e.g.", "the training classes are intuitively closer to each other than they are from classes in the testing set.", "Note that since we split along two axes, some data may be discarded (for instance images from a domain in $\\mathsf {D}_{\\text{train}}$ with a label in $\\mathsf {C}_{\\text{test}}$ ).", "Therefore it is crucial to find a split that minimizes this loss of data." ], [ "Meta-CIFAR100-Corrupted (MC100-C).", "CIFAR-100 [33] is a dataset of 60,000 three-channel square images of size $32 \\times 32$ , evenly distributed in 100 classes.", "Classes are evenly distributed in 20 superclasses.", "We use the same method used to build CIFAR-10-C [28], which makes use of 19 image perturbations, each one being applied with 5 different levels of intensity, to evaluate the robustness of a model to domain shift.", "We modify their protocol to adapt it to the FSQS problem: (i) we split the classes with respect to the superclass structure, and assign 13 superclasses (65 classes) to the training set, 2 superclasses (10 classes) to the validation set, and 5 superclasses (25 classes) to the testing set; (ii) we also split image perturbations (acting as domains), following the split of [63].", "As a result, when using a model on this benchmark, the model will be trained on images from domains and with labels that are disjoint from both the validation domains and labels, and the testing domains and labels.", "We obtain 2,184k transformed images for training, 114k for validation and 330k for testing.", "The detailed split is available in the documentation of our code repository." ], [ "miniImageNet [57] is a popular benchmark for few-shot image classification.", "It contains 60k images from 100 classes from the ImageNet dataset.", "64 classes are assigned to the training set, 16 to the validation set and 20 to the test set.", "Like MC100-C, we build mIN-C using the image perturbations proposed by [28] to simulate different domains.", "We use the original split from [57] for classes, and use the same domain split as for MC100-C.", "Although the original miniImageNet uses $84 \\times 84$ images, we use $224 \\times 224$ images.", "This allows us to re-use the perturbation parameters calibrated in [28] for ImageNet.", "Finally, we discard the 5 most time-consuming perturbations.", "We obtain a total of 1.2M transformed images for training, 182k for validation and 228k for testing.", "The detailed split in the documentation of our code repository." ], [ "FEMNIST-FewShot (FEMNIST-FS).", "EMNIST [13] is a dataset of images of handwritten digits and uppercase and lowercase characters.", "Federated-EMNIST [9] is a version of EMNIST where images are sorted by writer (or user).", "FEMNIST-FS consists in a split of the FEMNIST dataset adapted to few-shot classification.", "We separate both users and classes between training, validation and test sets.", "We build each group as the set of images written by one user.", "The detailed split is available in the code.", "Note that in FEMNIST, many users provide several instances for each digits, but less than two instance for most letters.", "Therefore it is hard to find enough samples from a user to build a support set or a query set.", "As a result, our experiments are limited to classification tasks with only one sample per class in both the support and query sets." ], [ "Algorithms", "We implement in FewShiftBed two representative methods of the vast literature of FSL, that are commonly considered as strong baselines: Prototypical Networks (ProtoNet) [53] and Matching Networks (MatchingNet) [57].", "Besides, for transductive FSL, we also implement with Transductive Propagation Network (TransPropNet) [39] and Transductive Fine-Tuning (FTNet) [19].", "We also implement our novel algorithm Transported Prototypes (TP) which is detailed in Section .", "FewShiftBed is designed for favoring a straightforward implementation of a new algorithm for FSQS.", "To add a new algorithm, we only need to implement the blueset_forward method of the class blueAbstractMetaLearner.", "We provide an example with our implementation of the Prototypical Network [53] that only requires few line of codes: [firstnumber=last]python class ProtoNet(AbstractMetaLearner): def setforward(self, supportimages, supportlabels, queryimages): zsupport, zquery = self.extractfeatures(supportimages, queryimages) zproto = self.getprototypes(zsupport, supportlabels) return - euclideandist(zquery, zproto)" ], [ "Protocol", "To prevent the pitfall of misinterpreting a performance boost, we draw three recommendations to isolate the causes of improvement rigorously." ], [ "How important is episodic training?", "Episodic training has been, notably through its elegance, responsible for the success and the wide adoption of Meta-Learning for FSL.", "Nevertheless, in some situation, episodic training does not perform better than more naive approaches [11], including using a backbone trained by standard Empirical Risk Minimization (ERM).", "Therefore we recommend to report both the result obtained using episodic training and standard ERM (see the documentation of our code repository)." ], [ "How does the algorithm behave in the absence of Support-Query Shift?", "An algorithm that specifically addresses the distribution shift problem should not provide degraded performance in an ordinary context.", "To certify this unpleasant outcome does not occur, we also recommend reporting the model's performance when we do not observe, at test-time, a support-query shift.", "It also provides a top-performing baseline by measuring how far the method is from an ideal case.", "Note that it is equivalent to evaluate the performance in cross-domain generalization, as firstly described in [11]." ], [ "Is the algorithm transductive?", "The assumption of transductivity has been responsible of several improvements in FSL [3], [50], [8] while it has been demonstrated in [8] that MAML [21] benefits strongly from the Transductive Batch-Normalization (TBN).", "Thus, we recommend specifying if the method is transductive and adapting the choice of the batch-normalization accordingly (Conventional Batch-Normalization [30] and Transductive Batch-Normalization for inductive and transductive methods, respectively) since transductive batch normalization brings a significant boost in performance [8]." ], [ "Overall idea", "We present a novel method that brings UDA to FSQS.", "As aforementioned, FSQS presents new challenges since we no longer assume that we sample the support set and the query set from the same distribution.", "As a result, it is unlikely that the support set and query sets share the same representation space region (non-overlap).", "In particular, the $L^2$ distance, adopted in the celebrated Prototypical Network [53], may not be relevant for measuring similarity between query and support instances, as presented in Figure REF .", "We emphasize the discovery of new source and target domain couples at test-time may accentuate the phenomenon of non-overlap significantly.", "To overcome this issue, we develop a two-phase approach that combines Optimal Transport (Transportation Phase) and the celebrated Prototypical Network (Prototype Phase).", "We give some background about Optimal Transport (OT) in Section REF and the whole procedure is presented in Algorithm REF ." ], [ "Definition.", "We provide some basics about Optimal Transport (OT).", "A thorough presentation of OT is available at [47].", "Let $p_s$ and $p_t$ be two distributions on $\\mathcal {X}$ , we note $\\Pi (p_s, p_t)$ the set of joint probability with marginal $p_s$ and $p_t$ i.e., $\\forall \\pi \\in \\Pi (p_s, p_t), \\forall x \\in \\mathcal {X}, \\pi (\\cdot , x) = p_s, \\pi (x, \\cdot ) = p_t$ .", "The Optimal Transport, associated to cost $c$ , between $p_s$ and $p_t$ is defined as: $W_c(p_s, p_t) := \\min _{\\pi \\in \\Pi (p_s, p_t)} \\mathbb {E}_{(x_s,x_t) \\sim \\pi } \\left[ c(x_s,x_t)\\right]$ with $c(\\cdot ,\\cdot )$ any metric.", "We note $\\pi ^\\star (p_s, p_t)$ the joint distribution that achieves the minimum in equation REF .", "It is named the transportation plan from $p_s$ to $p_t$ .", "When there is no confusion, we simply note $\\pi ^\\star $ .", "For our applications, we will use as metric the euclidean distance in the representation space obtained from a representation $\\varphi (\\cdot ; \\theta )$ i.e., $c_\\theta (x_s,x_t) := ||\\varphi (x_s; \\theta )-\\varphi (x_t; \\theta )||_2$ ." ], [ "Discrete OT.", "When $p_s$ and $p_t$ are only accessible through a finite set of samples, respectively $(x_{s,1},..., x_{s,n_s})$ and $(x_{t, 1},...,x_{t, n_t})$ we introduce the empirical distributions $\\hat{p}_s := \\sum _{i=1}^{n_s} w_{s,i} \\delta _{x_{s,i}}, ~~ \\hat{p}_t := \\sum _{j=1}^{n_t} w_{t,j} \\delta _{x_{t,j}}$ , where $w_{s,i}$ ($w_{t,j}$ ) is the mass probability put in sample $x_{s,i}$ ($x_{t,j}$ ) i.e., $\\sum _{i=1}^{n_s} w_{s,i}=1$ ($\\sum _{j=1}^{n_t} w_{t,j}=1$ ) and $\\delta _x$ is the Dirac distribution in $x$ .", "The discrete version of the OT is derived by introducing the set of couplings $\\mathbf {\\Pi }(p_s,p_t) := \\left\\lbrace \\mathbf {\\pi }\\in \\mathbb {R}^{n_s\\times n_t}, \\mathbf {\\pi }\\mathbf {1}_{n_s} = \\mathbf {p}_s, \\mathbf {\\pi }^\\top \\mathbf {1}_{n_t} = \\mathbf {p}_t \\right\\rbrace $ where $\\mathbf {p}_s := (w_{s,1}, \\cdots , w_{s, n_s})$ , $\\mathbf {p}_t := (w_{t,1}, \\cdots , w_{1, n_t})$ , and $\\mathbf {1}_{n_s}$ (respectively $\\mathbf {1}_{n_t}$ ) is the unit vector with dim $n_s$ (respectively $n_t$ ).", "The discrete transportation plan $\\mathbf {\\pi }^\\star _\\theta $ is then defined as: $\\mathbf {\\pi }_\\theta ^\\star := \\underset{\\mathbf {\\pi }\\in \\mathbf {\\Pi }(p_s, p_t)}{\\mathrm {argmin}} \\langle \\mathbf {\\pi }, \\mathbf {C}_\\theta \\rangle _F$ where $\\mathbf {C}_\\theta (i,j) := c_\\theta (x_{s,i}, x_{t,j})$ and $\\langle \\cdot , \\cdot \\rangle _F$ is the Frobenius dot product.", "Note that $\\mathbf {\\pi }^\\star _\\theta $ depends on both $p_s$ and $p_t$ , and $\\theta $ since $\\mathbf {C}_\\theta $ depends on $\\theta $ .", "In practice, we use Entropic regularization [17] that makes OT easier to solve by promoting smoother transportation plan with a computationally efficient algorithm, based on Sinkhorn-Knopp’s scaling matrix approach [31] (see the Appendix C)." ], [ "Method", "[t] Transported Prototypes.", "blueBlue lines highlight the OT's contribution in the computational graph of an episode compared to the standard Prototypical Network [53].", "Input: Support set $\\mathcal {S} := (x_{s,i}, y_{s,i})_{1\\le i \\le n_s}$ , query set $\\mathcal {Q} := (x_{q,j}, y_{q,j})_{1 \\le j \\le n_q}$ , classes $\\mathcal {C}$ , backbone $\\varphi _\\theta $ .", "Output: Loss $\\mathcal {L}(\\theta )$ for a randomly sampled episode.", "[1] $z_{s,i}, z_{q,j} \\leftarrow \\varphi (x_{s,i}; \\theta ), \\varphi (x_{q,j}; \\theta )$ , for $i,j$ Get representations.", "blue $\\mathbf {C}_\\theta (i,j) \\leftarrow ||z_{s,i} - z_{q,j}||^2$ , for $i,j$ Cost-matrix.", "$\\mathbf {\\pi }_\\theta ^\\star \\leftarrow $ Solve Equation REF Transportation plan.", "$\\hat{\\mathbf {\\pi }}_\\theta ^\\star (i,j) \\leftarrow \\mathbf {\\pi }^\\star _\\theta (i,j) / \\sum _j \\mathbf {\\pi }_\\theta ^\\star (i,j) $ , for $i,j$ Normalization.", "$\\hat{\\mathbf {S}} = (\\hat{z}_{s,i})_i \\leftarrow $ Given by Equation REF Get transported support set.", "$\\hat{\\mathbf {c}}_k \\leftarrow \\frac{1}{|\\hat{\\mathbf {S}}_k|} \\sum _{\\hat{z}_s \\in \\hat{\\mathbf {S}}_k} \\hat{z}_s$ , for $k \\in \\mathcal {C}$ .", "Get bluetransported prototypes.", "$p_\\theta (y| x_{q,j}) \\leftarrow $ From Equation REF , for $j$ Return: $\\mathcal {L}(\\theta ) := \\frac{1}{n_q} \\sum _{j=1}^{n_q}- \\log p_\\theta (y_{q,j}| x_{q,j})$ ." ], [ "Transportation Phase.", "At each episode, we are provided with a source support set $\\mathcal {S}$ and a target query set $\\mathcal {Q}$ .", "We note respectively $\\mathbf {S}$ and $\\mathbf {Q}$ their representations from a deep network $\\varphi (\\cdot ; \\theta )$ i.e., $z_s \\in \\mathbf {S}$ is defined as $z_s := \\varphi (x_s;\\theta )$ for $x_s \\in \\mathcal {S}$ , respectively $z_q \\in \\mathbf {Q}$ is defined as $z_q := \\varphi (x_q;\\theta )$ for $x_q \\in \\mathcal {Q}$ .", "As these two sets are sampled from different distributions, $\\mathbf {S}$ and $\\mathbf {Q}$ are likely to lie in different regions of the representation space.", "In order to adapt the source support set $\\mathcal {S}$ to the target domain, which is only represented by the target query set $\\mathcal {Q}$ , we follow [16] to compute $\\hat{\\mathbf {S}}$ the barycenter mapping of $\\mathcal {S}$ , that we refer to as the transported support set, defined as follows: $\\hat{\\mathbf {S}} := \\hat{\\mathbf {\\pi }}_{\\theta }^\\star \\mathbf {Q}$ where $\\mathbf {\\pi }_{\\theta }^\\star $ is the transportation plan from $\\mathbf {S}$ to $\\mathbf {Q}$ and $\\hat{\\mathbf {\\pi }}^\\star _\\theta := \\mathbf {\\pi }^\\star _\\theta (i,j) / \\sum _{j=1}^{n_t} \\mathbf {\\pi }^\\star _\\theta (i,j)$ .", "The transported support set $\\hat{\\mathbf {S}}$ is an estimation of labelled examples in the target domain using labelled examples in the source domain.", "The success relies on the fact that transportation conserves labels, i.e., a query instance close to $\\hat{z_ s} \\in \\hat{\\mathbf {S}}$ should share the same label with $x_s$ , where $\\hat{z_s}$ is the barycenter mapping of $z_s \\in \\mathbf {S}$ .", "See step (3) of Figure REF for a visualization of the transportation phase." ], [ "Prototype Phase.", "We compute the mean, called prototype, of the features vectors for each class $k\\in \\mathcal {C}$ for the transported support set $\\hat{\\mathbf {S}}$ to obtain the transported prototypes $\\hat{\\mathbf {c}}_k:= \\frac{1}{|\\hat{\\mathbf {S}}_k|} \\sum _{\\hat{z}_s \\in \\hat{\\mathbf {S}}_k} \\hat{z}_s$ (with $\\hat{\\mathbf {S}}_k$ the transported support set with class $k$ where $\\mathcal {C}$ are classes of current episode).", "We classify each query $x_q$ with representation $z_q= \\varphi (x_q; \\theta )$ using its euclidean distance to each transported prototypes; $p_\\theta (y = k|x_q) := \\frac{\\exp \\left( - || z_q - \\hat{\\mathbf {c}}_k ||^2 \\right)}{\\sum _{k^{\\prime } \\in \\mathcal {C}} \\exp \\left( - || z_q - \\hat{\\mathbf {c}}_{k^{\\prime }} ||^2 \\right)}$ Crucially, the standard Prototypical Networks [53] computes euclidean distance to each prototypes while we compute the euclidean to each transported prototypes, as presented in step (4) of Figure REF .", "Note that our formulation involves the query set in the computation of $(\\hat{\\mathbf {c}}_k)_{k\\in \\mathcal {C}}$ ." ], [ "Genericity of OT.", "FewShiftBed implements OT as a stand-alone module that can be easily plugged into any FSL algorithm (bluetransportation_module).", "We report additional baselines in Appendix B where the FSL algorithm is equipped with OT.", "This technical choice reflects our insight that OT may be ubiquitous for addressing FSQS and makes its usage in the testbed straightforward." ], [ "Experiments", "We compare the performance of baseline algorithms with Transported Prototypes on various datasets and settings.", "We also offer an ablation study in order to isolate the source to the success of Transported Prototypes.", "Extensive results are detailed in Appendix B.", "Instructions to reproduce these results can be found in the code's documentation." ], [ "Setting and details.", "We conduct experiments on all methods and datasets implemented in FewShiftBed.", "Hyperparameters and specific implementation choices are availale in Appendix B.", "We use a standard 4-layer convolutional network for our experiments on Meta-CIFAR100-C and FEMNIST-FewShot, and a ResNet18 for our experiments on miniImageNet.", "Transductive methods are equipped with a Transductive Batch-Normalization.", "All episodic training runs contain 40k episodes, after which we retrieve the “best\" state of the model, based on the best validation accuracy.", "We run each individual experiment on three different random seeds.", "All results presented in this paper are the average accuracies obtained with these random seeds." ], [ "Analysis.", "Table REF reveals that Transported Prototypes (TP) outperform all baselines by a strong margin on all datasets and settings.", "Importantly, baselines perform poorly on FSQS, demonstrating they are not equipped to address this challenging problem, stressing our study's significance.", "It is also interesting to note that transductive approaches, which significantly improve performance in a standard FSL setting [39], [19], perform similarly than simpler methods (Prototypes and Matching Networks)Notably, TransPropNet [39] fails loudly without Transductive Batch-Normalization showing that propagating label with non-overlapping support/query can have a dramatic impact, see Appendix B..", "Thus, FSQS deserves a fresher look to be solved.", "Transported Prototypes mitigate a significant part of the performance drop caused by support-query shift while benefiting from the simplicity of combining a popular FSL method with a time-tested UDA method.", "This gives us strong hopes for future works in this direction.", "Table: Top-1 accuracy of our method with various ablations with 8 instances perclass in the query set (except for FEMNIST-FS: 1 instance per class in the query set).TP stands for Transported Prototypes, OT denotes Optimal Transport, TBN is Transductive Batch-Normalization, OT-TT refers to the setting where Optimal Transport isapplied at test time but not during episodic training, and ET means episodic trainingi.e., w/o ET refers to the setting where training is performed through standard Empirical Risk Minimization.", "TP w/o SQS reports model’s performance in the absence ofsupport-query shift." ], [ "Ablation study.", "Transported Prototypes (TP) combines three components: Optimal Transport (OT), Transductive Batch-Normalization (TBN) and episode training (ET).", "Which of these components are responsible for the observed gain?", "Following recommendations from Section REF , we ablate those components in Table REF .", "We observe that both OT and TBN individually improve the performance of ProtoNet for FSQS, and that the best results are obtained when the two of them are combined.", "Importantly, OT without TBN performs better than TBN without OT (except for 1-shot mIN-C), demonstrating the superiority of OT compared to TBN for aligning distributions in the few samples regime.", "Note that, the use of TaskNorm [8] is beyond the scope of the paperThese normalizations are implemented in FewShiftBed for future works.", "; we encourage future work to dig into that direction and we refer the reader to the very recent work [20].", "However, results are mixed.", "There is no clear evidence that using OT at train-time is better than simply applying it at test-time on a ProtoNet's backbone i.e., without OT, trained with episodic training (except for 5-shot MC100-C).", "Additionnally, the value of episodic training compared to standard Empirical Risk Minimization (ERM) is not obvious.", "For instance, simply using ERM and using Transported Prototypes is better than adding ET on 1-shot MC100-C, 1-shot mIN-C and FEMNIST-FS, making it an another element to add to the study [35] who put into question the value of ET.", "Note that Table REF reports TP results when all components are used, even if some dataset-specific choices can boost performances significantly.", "Understanding why and when we should use ET or only OT at test-time is interesting for future works.", "Additionally, we compare Transported Prototypes with MAP [29] which implements an OT-based approach for transductive FSL.", "Their approach includes a power transform to reduce the skew in the distribution, so for fair comparaison we also implemented it into Transported Prototypes for these experiments.", "Interestingly, our experiments in Table REF show that MAP is able to handle SQS.", "For fair comparison, we compare TP with MAP by using the OT module only at test-time for PT and two backbones: ProtoNet [53] and a backbone obtained by ERM.", "Note that we added in TP the Power-Transform described in [29] explaining differences with results presented in Table REF .", "Table: Top-1 accuracy with 8 instances per class in the query set when applying Transported Prototypes and MAP on two different backbones: ☆\\star is standard ERM (i.e., without Episodic Training) and †\\dag is ProtoNet .", "Transported Prototypes performs equally or better than MAP .", "Here TP includes power transform in the feature space." ], [ "Behaviour in absence of Support-Query Shift.", "In order to evaluate the performance drop related to Support-Query Shift compared to a setting with support and query instances sampled from the same distribution, we test Transported Prototypes on few-shot classification tasks without SQS (TP w/o SQS in Table REF ), making a setup equivalent to CDFSL.", "Note that in both cases, the model is trained in an episodic fashion on tasks presenting a Support-Query Shift.", "These results show that SQS presents a significantly harder challenge than CDFSL, while there is considerable room for improvements." ], [ "Related Works", "This paper makes several contributions.", "We first define a new problem and we propose algorithms to solve it.", "We bring to the community the formal statement of FSQS and a testbed including datasets, a protocol and several baselines.", "Releasing benchmark has always been an important factor for progress in the Machine Learning field, the most outstanding example being ImageNet [18] for the Computer Vision community.", "Recently, DomainBed [25] aims to settle Domain Generalization research into a more rigorous process, where FewShiftBed takes inspiration from it.", "Meta-Dataset [56] is an other example, this time specific to FSL.", "Concerning the novelty of FSQS, we acknowledge the very recent contribution of Du et al.", "[20] which studies the role of learnable normalization for domain generalization, in particular when support and query sets are sampled from different domains.", "Note that our statement is more ambitious: we evaluate algorithms on both source and target domains that were unseen during training, while in their setting the source domain has already been seen during training.", "We also discuss deeply in Section REF the positioning of FSQS with respect to existing learning paradigms.", "Following [8], we study the role of Batch-Normalization for SQS, that points out the role of transductivity.", "Our conviction was that the batch-normalization is the first lever for aligning distributions [52], [58].", "Besides, we bridge the gap between UDA and FSL using Optimal Transport (OT) [47].", "OT has a long standing story in UDA [16] and has been recently applied in a context of transductive FSL [29] while our proposal (TP) is to provide a simple and strong baseline following principle of OT as it is applied in UDA." ], [ "Conclusion", "We release FewShiftBed, a testbed for the under-investigated and crucial problem of Few-Shot Learning when the support and query sets are sampled from related but different distributions, named FSQS.", "FewShiftBed includes three datasets, relevant baselines and a protocol for reproducible research.", "Inspired from recent progress of Optimal Transport (OT) to address Unsupervised Domain Adaptation, we propose a method that efficiently combines OT with the celebrated Prototypical Network [53].", "Following the protocol of FewShiftBed, we bring compelling experiments demonstrating the advantage of our proposal compared to transductive counterparts.", "We also isolate factors responsible for improvements.", "Our findings suggest that Batch-Normalization is ubiquitous, as described in related works [8], [20], while episodic training, even if promising on paper, is questionable.", "Moving beyond the transductive algorithm, as well as understanding when meta-learning brings a clear advantage, to address FSQS remains an open and exciting problem, where FewShiftBed brings the first step for its progress." ], [ "Acknowledgements", "Etienne Bennequin is funded by Sicara and ANRT (France), and Victor Bouvier is funded by Sidetrade and ANRT (France), both through a CIFRE collaboration with CentraleSupélec.", "This work was performed using HPC resources from the “Mésocentre” computing center of CentraleSupélec and École Normale Supérieure Paris-Saclay supported by CNRS and Région Île-de-France (http://mesocentre.centralesupelec.fr/)." ], [ "Few-Shot Classification.", "Methods to solve the Few-Shot Classification problem [36] are usually put into one of these three categories [11]: metric-based, optimization-based, and hallucination-based.", "Most metric-learning methods are built on the principle of Siamese Networks [32], while also exploiting the meta-learning paradigm: they learn a feature extractor across training tasks [57].", "Prototypical Networks [53] classify queries from their euclidean distances to one prototype embedding per class.", "Relation Networks [55] add an other deep network on top of Prototype Networks to replace the euclidean distance.", "Optimization-based methods use an other approach: learning to fine-tune.", "MAML [21] and Reptile [44] learn a good model initialization, i.e.", "model parameters that can adapt to a new task (with novel classes) in a small number of gradient steps.", "Other methods such as Meta-LSTM [49] and Meta-Networks [42] replace standard gradient descent by a meta-learned optimizer.", "Hallucination-based methods aim at augmenting the scarce labeled data, by hallucinating feature vectors [26], using Generative Adversarial Networks [2], or meta-learning [59].", "Recent works also suggest that competitive results in Few-Shot Classification can be achieved with more simple methods based on fine-tuning [11], [23]." ], [ "Transductive Few-Shot Classification.", "Some methods aim at solving few-shot classification tasks by using the query set as unlabeled data.", "Transductive Propagation Network [39] meta-learns label propagation from the support to query set concurrently with the feature extractor.", "Antoniou & Storkey [3] proposed to use a meta-learned critic network to further adapt a classifier on the query set in an unsupervised setting.", "Ren et al.", "[50] extend Prototypical Networks in order to use the query set in the prototype computation.", "Transductive Information Maximization [6] aims at maximizing the mutual information between the features extracted from the query set and their predicted labels.", "Finally, Transductive Fine-Tuning [19] augments standard fine-tuning using the classification entropy of all query instances." ], [ "Unsupervised Domain Adaptation.", "UDA has a long standing story [46], [48].", "The analysis of the role of representations from [4] has led to wide literature based on domain invariant representations [22], [40].", "Outstanding progress have been towards learning more domain transferable representations by looking for domain invariance.", "The tensorial product between representations and prediction promotes conditional domain invariance [41], the use of weights [10], [62], [7], [14] has dramatically improved the problem of label shift theoretically described in [64], hallucinating consistent target samples [38], penalizing high singular values of batch of representations [12] or by enforcing the favorable inductive bias of consistence through various data augmentation in the target domain [45].", "Recent works address the problem of adaptation without source data [37], [61].", "The seminal work [16], followed by [15], [5], brings Optimal Transport (OT) to UDA by transporting source samples in the target domain." ], [ "Test-Time Adaptation.", "Test-time Adaptation (TTA) is the subject of recent pioneering works.", "In [54], adaptation is performed by test-time training of representations through a self-supervision task which consists in predicting the rotation of an image.", "This leads to a successful adaptation when the gradient of fine-tuning procedure is correlated with the gradient of the cross-entropy between the prediction and the label of the target sample, which is not available.", "Inspired from UDA methods based on domain invariance of representations, a line of works [43], [52] aims to align the mean and the variance of train and test distribution of representations.", "This is simply done by updating statistics of the batch-normalization layer.", "In a similar spirit of leveraging the batch-normalization layer for adaptation, [58] suggests to minimize prediction entropy on a batch of test samples, as suggested in semi-supervised learning [24].", "As pointed by authors of [58], updating the whole network is inefficient and exposes to a risk of test batch overfit.", "To adress this problem, authors suggest to only update batch-normalization parameters for minimizing prediction's entropy.", "The paradigm of Adaptative Risk Minimization (ARM) is introduced in [63].", "ARM aims to adapt a classifier at test-time by conditioning its prediction on the whole batch of test samples (not only one sample).", "Authors demonstrate that such classifier is meta-trainable as long as the training data exposes a structure of group.", "Consequently, [63] is closer work to ours, while we have more ambitious perspectives as we address the problem of few-shot learning i.e., few-shot are available per class while new classes are discovered at test-time." ], [ "Few-Shot Classification under Distributional Shift.", "Recent works on few-shot classification tackle the problem of distributional shift between the meta-training set and the meta-testing set.", "Chen et al.", "[11] compare the performance of state-of-the-art solutions to few-shot classification on a cross-domain setting (meta-training on miniImageNet [57] and meta-testing on Caltech-UCSD Birds 200 [60]).", "Zhao et al.", "propose a Domain-Adversarial Prototypical Network [65] in order to both align source and target domains in the feature space and maintain discriminativeness between classes.", "Considering the problem as a shift in the distribution of tasks (i.e.", "training and testing tasks are drawn from two distinct distributions), Sahoo et al.", "combine Prototypical Networks with adversarial domain adaptation at task level [51].", "While these works address the key issue of distributional shift between meta-training and meta-testing, they assume that for each task, the support set and query set are always drawn from the same distribution.", "We find that this assumption rarely holds in practice.", "In this work we consider a distributional shift both between meta-training and meta-testing and between support and query set." ], [ "All experimental results", "In this section we present the extended results of our experiments.", "Prototypical Networks, Matching Networks and Transductive Propagation Networks have been declined in 10 distinct versions: Original algorithms: episodic training, with Conventional Batch-Normalization (CBN) and not Optimal Transport (Vanilla); Episodic training and CBN, with Optimal Transport applied at test time (OT-TT); Episodic training and CBN, with Optimal Transport integrated into the algorithm both during training and testing (OT); Episodic training, with Transductive Batch-Normalization (TBN) and not Optimal Transport (Vanilla); Episodic training and TBN, with OT-TT; Episodic training and TBN, with OT; Standard Empirical Risk Minimization (ERM) instead of episodic training, with CBN and not Optimal Transport (Vanilla); ERM with CBN and OT; ERM with TBN and no Optimal Transport (Vanilla); ERM with TBN and OT.", "Transductive Fine-Tuning (FTNet) is not compatible with episodic training.", "Also the integration of Optimal Transport into this algorithm is non trivial.", "Therefore we only applied FTNet with ERM and without OT.", "Every result presented in the following tables is the average over three runs with three random seeds (1, 2 and 3).", "For clarity, we do not report the 95% confidence interval for each result.", "Keep in mind that this interval is different for each result, but we found that it is always greater than $\\pm $ 0.2% and smaller than $\\pm $ 0.8%.", "Details of the experiments and instructions to reproduce them are available in the code.", "Table: Ablation for Meta-CIFAR100-C 1-shot 8-target.Table: Ablation for Meta-CIFAR100-C 1-shot 16-target.Table: Ablation of Meta-CIFAR100-C 5-shot 8-targetTable: Ablation of Meta-CIFAR100-C 5-shot 16-targetTable: Ablation of FEMNIST-FewShot 1-shot 1-target.Table: Top-1 accuracy of MAP compared to Transported Prototypes (ours).", "Both methods incorporate Optimal Transport into Few-Shot Learning.", "MAP is originally designed for standard transductive FSL.", "Interestingly, MAP and TP perform quite similarly demonstrating that OT is a powerful tool for addressing FSQS.", "Note that MAP leverages a Power-Transform that we also plug in TP for comparison, resulting in a boost of performance.", "Understanding which learners operate best with Optimal Transport is an exciting question.", "In particular, by proposing TP, we have shown that we result in a strong, interpretable and theoretically motivated method by following principles when applying OT in UDA." ], [ "Training details", "Entropic regularization for Optimal Transport was proposed in [17] and makes OT easier to solve.", "It is defined as $\\mathbf {\\pi }^\\star _\\theta (\\hat{p}_s,\\hat{p}_t) := \\arg \\min _{\\mathbf {\\pi }\\in \\mathbf {\\Pi }} \\langle \\mathbf {\\pi }, \\mathbf {C}_\\theta \\rangle _F + \\varepsilon \\Omega (\\mathbf {\\pi })$ with $\\varepsilon >0$ and $\\Omega (\\mathbf {\\pi }) = \\sum _{i,j=1}^{n_s, n_t} \\mathbf {\\pi }(i,j)\\log \\mathbf {\\pi }(i,j)$ is the negative entropy.", "It promotes smoother transportation plan while allowing to derive a computationally efficient algorithm, based on Sinkhorn-Knopp’s scaling matrix approach [31].", "In our experiment, we set $\\varepsilon =0.05$ , but it is possible to tune it, eventually meta-learning it." ] ]
2105.11804
[ [ "Deep Neural Networks and End-to-End Learning for Audio Compression" ], [ "Abstract Recent achievements in end-to-end deep learning have encouraged the exploration of tasks dealing with highly structured data with unified deep network models.", "Having such models for compressing audio signals has been challenging since it requires discrete representations that are not easy to train with end-to-end backpropagation.", "In this paper, we present an end-to-end deep learning approach that combines recurrent neural networks (RNNs) within the training strategy of variational autoencoders (VAEs) with a binary representation of the latent space.", "We apply a reparametrization trick for the Bernoulli distribution for the discrete representations, which allows smooth backpropagation.", "In addition, our approach allows the separation of the encoder and decoder, which is necessary for compression tasks.", "To our best knowledge, this is the first end-to-end learning for a single audio compression model with RNNs, and our model achieves a Signal to Distortion Ratio (SDR) of 20.54." ], [ "Introduction", "The recent improvements of deep neural network (DNN) models and learning methods have open new horizons for highly structured sequential data-related tasks, including automatic speech recognition (ASR) ([1]) as well as audio compression ([2], [3], [4]).", "There have been particular interests in developing end-to-end DNN models that can handle raw input audio signals with little-to-none biased intervention.", "For audio compression, for example, [2] proposed a complete single DNN pipeline consisting mainly of residual networks and autoencoders.", "Several variants of this network aim to reduce its computational cost, e.g., CMRL ([3]) or LPCNet ([4]).", "The audio compression task is particularly challenging due to several issues related to the type of data and the required complexity of the model’s architecture.", "First, we need robust and flexible models to capture the complex relations between the data variables through time.", "Essentially, an end-to-end compression architecture aims to map an audio signal into a more concise portable representation.", "This characteristic is shared with deep generative models, targeting capturing critical aspects of data to generate new data instances ([5], [6], [7], [8]).", "A notable example of this approach consists of a combination of Variational Autoencoders (VAEs) ([9]) and Recurrent Neural Networks (RNNs) ([10], [11]).", "RNNs can handle extensive time-dependent data samples, but their deterministic hidden variables cannot capture the variability present in raw audio.", "On the other hand, VAEs provide high-level latent random variables and flexible mapping to the model’s output.", "It has been shown that applying a VAE to the input sequence and using an RNN to model the output distribution succeeded in the modeling and recognition of speech signals ([12], [13], [14], [15]).", "However, such method has not been studied in detail for audio compression tasks.", "Second, when raw audio signals are encoded into discrete representation, it makes training by backpropagation unfeasible because discrete representation blocks the flow of gradient information.", "Some compression models circumvent this issue using vector quantization (VQ) in the coding layer ([2]), which does not train the discrete representation by backpropagation.", "Third, for real-world implementations, the encoder and decoder of the compressor should be decoupled and placed in different devices.", "This small detail should be taken into account when designing the architecture.", "In this work, we propose a Variational Recurrent Encoder-Decoder (VRED), an end-to-end DNN for audio compression.", "In addition, we can use trainable features with convolutional and deconvolutional layers before and after VRED, respectively.", "Thus, the proposed network consists of a convolution layer to extract features, VRED, and a deconvolution layer to reconstruct audio signals.", "To make training more efficient, we train the audio compressor in three stages: (1) we pre-train the feature extractor (convolutional and deconvolutional layers), (2) we freeze these layers and train VRED, and (3) we fine-tune the whole architecture together.", "The model presents the following benefits: Simple objective function compared to other end-to-end models ([2]).", "Possible for compression compared to other VAE+RNN models like Variational Recurrent Neural Networks (VRNN) ([13]).", "More natural architecture for the discrete variables compared to VQ-VAE ([16]).", "In experiment results, the proposed network achieves a signal-to-distortion ratio (SDR) of 20.53, with a compression ratio of 1/11.", "To our knowledge, this is the first approach that uses an entirely separable audio compression model using an end-to-end VAE-RNN model." ], [ "Recurrent Neural Networks", "Given a sequential input $\\mathbf {x}$ of length $T$ , a simple RNN recursively updates its intermediate hidden state layer by $h_t = f_\\theta (x_t, h_{t-1}),$ for each $x_t \\in \\mathbf {x}$ using a $\\theta -$ parameterized function $f$ .", "This deterministic non-linear transition function $f$ can be implemented using gated mechanisms such as GRUs, LSTMs, and so on ([17], [18], [19]).", "RNNs model the sequences of probability distribution by the joint distribution: $ p(x_1,x_2,...,x_T) = \\prod _{t=1}^{T} p(x_t|x_{<t}).$ Each conditional probability $p(x_t|x_{<t})$ in Eq.", "REF is represented by the output of an RNN at a single step.", "We can interpret them as a parameterized function $d_{\\tau }$ that maps the RNN hidden state $h_{t-1}$ to a probability distribution over possible outputs: $p(x_t|x_{<t}) = d_{\\tau }(h_{t-1})$ .", "This function $d_{\\tau }$ is responsible for the representational power of an RNN, as it contains information of previous sequences." ], [ "Variational Autoencoder", "VAEs ([9]) are responsible for the expressivity in the model’s latent space, as they aim to capture the variations in the input variables $\\mathbf {x}$ through a set of random variables $\\mathbf {z}$ which aim to be disentangled ([20], [11]).", "Their goal is to train the generative model following the joint distribution $p(\\mathbf {x},\\mathbf {z}) = p(\\mathbf {x}|\\mathbf {z})p(\\mathbf {z})$ .", "Here $p(\\mathbf {x}|\\mathbf {z})$ is the likelihood (decoder) that generates $\\mathbf {x}$ data from $\\mathbf {z}$ , and $p(\\mathbf {z})$ is a prior over latent variables $\\mathbf {z}$ .", "For inference, input is encoded to a distribution $p(\\mathbf {z}|\\mathbf {x}) = p(\\mathbf {x}|\\mathbf {z}) p(\\mathbf {z}) / p(\\mathbf {x})$ .", "The true posterior $p(\\mathbf {z}|\\mathbf {x})$ is intractable, so the model parameterizes an approximate posterior distribution (encoder) $q(\\mathbf {z}|\\mathbf {x})$ instead.", "In most applications, the parameters of the encoder, decoder, and prior are computed using neural networks.", "Usually, the prior is chosen to be normally distributed with diagonal covariance, which allows for the Gaussian reparameterization trick to be used to compute the gradient of the objective function ([9], [20]).", "The use of the approximate posterior $q(\\mathbf {z}|\\mathbf {x})$ allows VAEs the use of the Evidence Lower Bound (ELBO): $ \\log p(x) \\ge -KL(q(\\mathbf {z}|\\mathbf {x})||p(\\mathbf {x})) + \\mathbb {E}_{q(\\mathbf {z}|\\mathbf {x})} [\\log p(\\mathbf {x}|\\mathbf {z})],$ where $KL$ is the Kullback-Leibler divergence between two distributions." ], [ "Variational RNN", "Out of all the previous VAE-RNN models, VRNN ([13]) has the most similar structure to ours.", "The prior on the latent variable $z_t$ used in VRNN follows a normal distribution parametrized by a function of the previous hidden state $h_{t-1}$ of the RNN.", "During the generation, the decoder is influenced by both $h_{t-1}$ and $z_t$ .", "Similarly, during inference, the approximate posterior depends on the input $x_t$ and on $h_{t-1}$ .", "The general structure of the model is shown in Figure REF (a).", "The novelty this model introduces is that the prior distribution of the latent random variable at timestep $t$ is dependent on all the preceding inputs via the RNN hidden state $h_{t-1}$ .", "Thus, the prior distribution also includes temporal dependency.", "If we consider $\\mathbf {z}$ as a sequence $\\mathbf {z}=(z_1,z_2,...,z_T)$ , the ELBO of Eq.", "REF becomes: $ \\mathbb {E}_{q(\\mathbf {z}_{\\le T}|\\mathbf {x}_{\\le T})} \\bigg [ \\sum _{t=1}^{T}(-KL(q(\\mathbf {z}_t|\\mathbf {x}_{\\le t}, \\mathbf {z}_{<t})||p(\\mathbf {z}_t|\\mathbf {x}_{<t}, \\mathbf {z}_{<t})) + \\log p(\\mathbf {x}_t|\\mathbf {z}_{\\le t}, \\mathbf {x}_{<t}))\\bigg ].$ The training of this architecture is in core the same as the VAE.", "Figure: Difference between the training of VRNN (a) and our VRED model (b).", "The dashed square represents the encoder and the pointed square, the decoder." ], [ "VQ-VAE", "The main difference between our VRED model and the previous works is discrete (instead of continuous) latent representations in the model.", "For audio compression, discrete representations are better suited but challenging to train the model with.", "Vector Quantised Variational Autoencoder (VQ-VAE) ([16]) was introduced to address this issue.", "The model successfully includes discretization in VAEs using VQ (vector quantization), reaching a similar performance to the continuous counterparts.", "To backpropagate through the discrete variables, however, the model gives up the benefit of the sampling method of VAEs.", "In addition, the model has to learn a suitable uniform categorical prior.", "Due to these, the training objective changes and is no longer the ELBO." ], [ "Variational Recurrent Encoder-Decoder", "Even though VRNN with RNN encoder/decoder seems to be a good architecture for audio compression, it presents two out of three issues mentioned above.", "It does not have a discrete latent variable with a feasible backpropagation, and it has a non-separable encoder and decoder during inference.", "Separating the encoding and decoding of the inputs may not seem necessary for the model to generate sequences, but in real-world applications of the audio compression task, the encoder and decoder would typically be in different devices.", "We propose a novel model, Variational Recurrent Encoder-Decoder (VRED) for audio compression.", "We redesign its encoder and decoder completely separable.", "Furthermore, we propose the use of discrete latent variables based on the Bernoulli distribution, with a reparameterization trick that allows a smooth backpropagation.", "The variational inference on VRED is the same as on VAE ([9].)", "Since we have feature extraction and reconstruction layers in addition to VRED, for more efficient training, the learning process can be decomposed into three stages.", "First, we pre-train signal extractor-constructor layers to learn the most significant features of the raw audio.", "Then, we freeze this feature codec and train the VRED model with the resulting feature samples.", "Finally, we fine-tune the whole model.", "Figure: Decoding structure of the VRED during testing.", "The encoder and decoder can be completely separated via introducing φ(dec t )\\phi (dec_t).", "The encoding process is given by Figure (b)." ], [ "Feature Learning", "In this work, we aim for an end-to-end DNN audio codec that can work without the use of hand-crafted or biased feature extractors such as LPC (Linear Predictive Coding) coefficients, LPS (Log Power Spectrum) and mel-spectrogram.", "Instead, we use a simple CNN layer to extract/construct from the input/output of the codec.", "It has been argued in previous works that the use of feature extractors on top of the model can significantly enhance the performance of the primary model ([13]).", "The feature extraction can be understood as a simple pre-processing step, in where the layer aims to capture prominent features of the signal by the use of filters/kernels.", "It is worth mentioning that maxpooling cannot be used for the compression task, as it downsamples the data and thus harms the reconstruction.", "The parameters and implementation details are specified in the next section." ], [ "VRED training", "Like the VRNN, the VRED contains a VAE conditioned on the state variable $h_{t-1}$ of an RNN during the generation step (decoder).", "However, the prior on the latent variable is discrete and follows a Bernoulli distribution: $ z_t \\sim Bern(p_t), \\;\\;\\;\\; \\mathrm {where} \\;\\; p_t=\\varphi _{prior}(h_{t-1}).$ Then, the generating distribution will not only be conditioned on $z_t$ but also on $h_{t-1}$ such that: $x_t|z_t \\sim \\mathcal {N}\\bigg (p_{x,t}, diag(\\sigma _{x,t}^{2})\\bigg ),$ where $p_{x,t}=dec_t(\\phi _z(z_t),h_{t-1})$ and $\\sigma _{x,t}^{2} = p_{x,t}(1-p_{x,t})$ , $p_{x,t}$ denotes the parameters of the generating distribution, $\\varphi _{prior}$ and $dec_t$ can be any highly flexible function such as neural networks.", "The RNN updates the hidden state using the recurrent equation: $h_t = f_{\\theta }(\\phi (dec_t),\\phi _z(z_t), h_{t-1}),$ where $f$ is the original transition function from Eq.", "REF .", "$\\phi _z$ and $\\phi _{x}$ are NNs that extract features from $z_t$ and $x_t$ , respectively.", "The introduction of $\\phi (dec_t)$ allows to decouple the encoder from the decoder, as there is no link between the encoding step and the latent variable during inference (Figure REF ).", "During inference, the approximate posterior will be a function of $x_t$ and $h_{t-1}$ following the equation: $z_t|x_t \\sim Bern(p_{z,t}), \\;\\;\\;\\; \\mathrm {where} \\;\\; p_{z,t}=enc_t(\\phi _x(x_t), h_{t-1}),$ where $p_{z,t}$ denotes the parameter of the approximate posterior.", "We note that the encoding of the approximate posterior and the decoding of generation are tied through the RNN hidden state $h_{t-1}$ .", "The objective function remains the same timestep-wise variational lower bound as in Eq.", "REF , but in this case, the Kullback-Lieber divergence is given by: $ KL(q||p)_{Ber} = q \\log \\frac{q}{p} + (1-q) \\log (\\frac{1-q}{1-p}),$ where $q$ and $p$ follow a Bernoulli distribution.", "To facilitate the comparison, the notations in the equations of this section are similar to [13]." ], [ "Reparameterization trick", "The main issue with using a discretized prior is that backpropagation is hard to flow through the discretized nodes.", "To solve this, we use the same parametrization trick presented in [21].", "The main idea is that given a sample $t\\sim Bern(p)$ as in Eq.", "REF , we define a quantity $c=t(1-p) - (1-t)p$ and detach it from the computation graph.", "Instead of backpropagating through $t$ , we use $p+c$ equivalently and avoid the sampling block this way." ], [ "Fine-tuning", "After training the VRED, we de-freeze the CNN codec layer and fine-tune the whole model with a single objective function which is given by Eq.", "REF and Eq.", "REF , $\\mathbb {E}_{q(\\mathbf {z}_{\\le T}|\\mathbf {x}_{\\le T})} \\bigg [ \\sum _{t=1}^{T}(-q(z_t|x_{\\le t}, z_{<t}) \\log \\frac{q(z_t|x_{\\le t}, z_{<t})}{p(z_t|x_{<t}, z_{<t})} + (1-q(z_t|x_{\\le t}, z_{<t})) \\log (\\frac{1-q(z_t|x_{\\le t}, z_{<t})}{1-p(z_t|x_{<t}, z_{<t})}) + \\log p(\\mathbf {x}_t|\\mathbf {z}_{\\le t}, \\mathbf {x}_{<t}))\\bigg ].$ The end-to-end architecture is shown in Figure REF .", "Figure: End-to-end model used for training and testing.", "Note that during testing the encoder and decoder are separable." ], [ "Experiments", "To train the VRED model, we used the BBC sound effects dataset BBC sound effects library, http://www.sound-ideas.com/sound-effects/bbc-sound-effects.html./, 2015 sampled at 44.1kHz, which consists of 15,000 audio files of different lengths (cars passing, plane engines, metal objects falling).", "For consistent processing and batching, we split the audio files into chunks of 0.02s.", "For the testing dataset, we did not use a partition of the training dataset, as we wanted to try the compression of more complex audio files.", "We used 15 audio files between 8s and 17s, including music, speech over noise, single instruments, and others Evaluation Guidelines for Unified Speech and Audio Proposals, ISO/IEC SC29 WG11 N9638, MPEG, Jan. 2008.. For both sets, the format of the audio files is “.wav\", and the sampling rate is 44.1 kHz.", "The signals are converted from stereo (2 channels) to mono.", "To determine the quality of the reconstructed signal, we used the Signal-to-Distortion Ratio score ([22]).", "The score is in decibels (dB), and reflects how similar the estimated signal $\\hat{s}$ is to the clean signal $s_{target}$ .", "It is defined as $SDR = 10 \\log _{10} \\frac{||s_{target}||^2}{||\\hat{s}-s_{target}||^2}.$ The higher the SDR score means the higher the recovery rate between the estimated and target signals).", "For example, an SDR 60dB is considered a nearly perfect reconstruction, and a reconstruction with an SDR of around 30dB is considered successful.", "For the feature extraction of the raw signals, we used a one-dimensional convolution layer.", "As the signal constructor we used the deconvolution version of the CNN feature extractor.", "We experimented with several number of kernels and kernel dimensions, which will be detailed in next section.", "To learn the parameters of the prior, approximate posterior and generative models we used feedforward networks.", "The encoder and decoder of the VRED consisted in three feedforward layers with hidden dimension 128, which was also the dimension of latent variable $\\mathbf {z}$ .", "We applied recurrences via LSTM.", "The Bernoulli prior consisted in two linear layers.", "For fine-tuning we used Adam optimization with learning rate re-scheduling." ], [ "Feature extractor", "The experiment results of the CNN features without VRED are shown in Table REF .", "Several combinations of stride and kernel sizes were tried.", "We also varied the number of filters.", "Some results achieved an SDR of almost 48 for our test dataset, although the training SDR was not very high in comparison.", "What we want is a configuration that would have a high SDR for both the training and testing sets while using a low kernel number to achieve a lower compression rate.", "Table: SDR for training and testing datasets when training only the CNN feature extraction of VRED (Section ) for 500 epochs.", "The results come from different configurations of the CNN structure." ], [ "VRED activations", "The VRED is trained using the pre-trained feature extractor with the best train/testing SDR value configuration.", "If we focus on the VRED training part only (Section 3.2), we analyze the input to the model and output of the RNN decoder.", "From Figure REF , we observe how the location of bright and dark pixels representing the extracted features values are somewhat correctly located.", "Even though the reconstruction may not seem perceptually different, the SDR score is directly proportional to the difference in the target and reconstructed features.", "Figure: Examples of VRED input and output for different encoded signals, after 3000 epochs." ], [ "Fine-tuned VRED results", "We fine-tuned VRED with a CNN feature extractor of 32 kernels with size 88 and a stride of 44.", "The dimension of the latent space was set to 128.", "The total compression rate of the audio signals resulted in $CR = CR_{CNN} \\times CR_{VRED} = \\frac{32}{44} \\times \\frac{128}{(32\\times 32)} = \\frac{1}{11},$ where $CR_{CNN}$ is the compression of the feature extractor and $CR_{VRED}$ corresponds to the compression achieved by the VRED.", "The SDR after the fine-tuning reached 20.53 dB.", "An example of the reconstruction is shown in Figure REF .", "Figure: Example of a test audio reconstruction after compression.", "We present the spectrogram of the signals (top) and their amplitudes through time (bottom).", "The reconstruction signal is noisier than the original signal.", "Furthermore, the non-smooth bands in the reconstruction spectrogram suggest that the model does not capture the original frequency spectrum in high detail." ], [ "Conclusion", "In the present work, we showed a novel, completely separable end-to-end model using deterministic latent variables.", "Our model is simpler and more straightforward to train than previous models.", "Though the obtained results are not commercially acceptable, the proposed approach opens a new door for researching audio compression using unified deep learning models with end-to-end learning." ], [ "Acknowledgments", "This work was supported by Electronics and Telecommunications Research Institute (ETRI) grant funded by the Korean government.", "[21ZH1200, The research of the basic media・contents technologies]" ] ]
2105.11681
[ [ "Structured Convolutional Kernel Networks for Airline Crew Scheduling" ], [ "Abstract Motivated by the needs from an airline crew scheduling application, we introduce structured convolutional kernel networks (Struct-CKN), which combine CKNs from Mairal et al.", "(2014) in a structured prediction framework that supports constraints on the outputs.", "CKNs are a particular kind of convolutional neural networks that approximate a kernel feature map on training data, thus combining properties of deep learning with the non-parametric flexibility of kernel methods.", "Extending CKNs to structured outputs allows us to obtain useful initial solutions on a flight-connection dataset that can be further refined by an airline crew scheduling solver.", "More specifically, we use a flight-based network modeled as a general conditional random field capable of incorporating local constraints in the learning process.", "Our experiments demonstrate that this approach yields significant improvements for the large-scale crew pairing problem (50,000 flights per month) over standard approaches, reducing the solution cost by 17% (a gain of millions of dollars) and the cost of global constraints by 97%." ], [ "Introduction", "Since crew costs are the second-highest spending source for air passenger carriers, crew scheduling is of crucial importance for airlines.", "The crew pairing problem (CPP) searches for a minimum-cost set of anonymous feasible pairings (rotations) from the scheduled flights, such that all flights are covered exactly once, and all airline regulations and collective agreements are respected.", "The complexity of this problem lies in the large number of possible pairings, as the selection of pairings at minimal cost—a large integer programming problem—cannot be performed using standard solvers.", "Seeking to obtain an efficient algorithm for large-scale monthly CPPs (up to 50,000 flights) and building on the column generation-based solver by [10], [41] proposed Commercial-GENCOL-DCA, an improved solver starting with an aggregation, in clusters, of flights.", "The initial aggregation partition permits replacing all flight-covering constraints of flights in a cluster by a single constraint, thus allowing the solver to cope with larger instances.", "Initial clusters can either be extracted from the initial solution [10] or given separately, as in [41], where authors used convolutional neural networks (CNN) to solve the flight-connection problem (a supervised multi-class classification problem).", "The objective of this problem is to predict the next flight that a crew follows in its schedule given the previous flight.", "They used CNN to harness the spatial locality (localized spatial features) and used a similarity-based input, where neighboring factors have similar features.", "By passing initial clusters of flights to the CPP solver, the reported reduction of solution cost averages between 6.8% and 8.52%, mainly due to the reduction in the cost of global constraints between 69.79% and 78.11%.", "The cost of global constraints refers to the penalties incurred when the workload is not fairly distributed among the bases in proportion to the available personnel at each base.", "However, a major weakness of their approach is that they can only produce initial clusters and not an initial solution, since they use a greedy predictor making one prediction at a time (predicting sequentially the next flight given only the previous flight).", "This prevents the predictor from incorporating constraints on the output and the produced solutions cannot be used as initial solutions for the solver as they are not sufficiently close to being feasible.", "A pairing is deemed feasible if it satisfies safety rules and collective agreement rules [16]; examples include minimum connection time between two flights, minimum rest time, and maximum number of duties in a pairing.", "By providing an initial solution, we not only accelerate the optimization process and calculate the feasibility of proposed pairings, but we also propose clusters similar to the initial solution, thus reducing the degree of incompatibility between current solution and proposed pairings [11].", "In this paper, we address this lack of constraint modeling, while still enabling the use of a convolutional architecture.", "For this purpose, we investigate the convolutional kernel network (CKN), an approximation scheme similar to CNN proposed by [25].", "To bypass the major limitations in [41], we incorporate local constraints on the outputs (imposing that each flight has to be preceded by at most one flight).", "We harness the spatiotemporal structure of the CPP problem by combining kernel methods and structured prediction.", "The outputs start the optimizer and solve a large-scale CPP, where small savings of a mere 1% translate into an increase of annual revenue for a large airline by dozens of millions of dollars.", "Note that, to the best of our knowledge, we are not aware of any ML approach that can directly solve the CPP (which has complex airline-dependent costs and constraints that are not necessarily available to the ML system at train time).", "We thus consider instead to use the ML system to propose good initial clusters and an initial solution for the CPP solver.", "The results of training on the flight-connection dataset [40], a flight-based network structure modeled as a general conditional random field (CRF) graph, demonstrate that the proposed predictor is more suitable than other methods.", "Specifically, it is more stable than CNN-based predictors and extensive tuning is not required, in that no Bayesian optimization (to find a suitable configuration) is needed, as we observe in our experiments.", "This is crucial to integrate ML into a solver for the CPP or any real-world scheduling problem.", "Note that unlike recurrent neural networks (RNNs) or neural networks by [41] which cannot produce initial solutions that are sufficiently close to being feasible, the proposed predictor incorporates local constraints in the learning process.", "Furthermore, note that while previous studies focused on using ML (either through imitation learning or reinforcement learning) to solve small-scale CO problems such as vehicle routing ($\\le $ 100 customers) and airline crew scheduling ( $\\le $ 714 flights) problems, we use the proposed predictor to warm-start a monthly CPP solver (up to 50,000 flights).", "For an extensive literature review on using machine learning for combinatorial optimization, see [2]." ], [ "Contributions.", "Bridging the gap between kernel methods and neural networks, we propose the structured convolutional kernel network (Struct-CKN)The code is available at the following link: https://github.com/Yaakoubi/Struct-CKN.", "We first sanity check the approach on the OCR dataset [37] yielding a test accuracy comparable to state of the art.", "Then, to warm-start an airline crew scheduling solver, we apply the proposed method on a flight-connection dataset, modeled as a general CRF capable of incorporating local constraints in the learning process.", "We show that the constructed solution outperforms other approaches in terms of test error and feasibility, an important metric to initialize the solver.", "The predicted solution is fed to the solver as an initial solution and initial clusters, to solve a large-scale CPP (50,000 flights).", "Our experiments demonstrate that this approach yields significant improvements, reducing the solution cost by 17% (a gain of millions of dollars) and the cost of global constraints by 97%, compared to baselines." ], [ "Outline.", "The remainder of this paper is structured as follows.", "Section  describes related methods.", "Section  presents CKNs.", "CRFs are outlined in Section .", "Section  presents Struct-CKN.", "Section  reports Computational results on OCR dataset, flight-connection dataset, and CPP." ], [ "Related Work", "Upon a succinct review of previous work to compare available approaches in the literature to Struct-CKN, we argue for the use of the latter on the flight-connection dataset to solve CPPs.", "Combining networks and energy-based models is a well-known approach since the 1990s.", "For instance, [4] introduced graph transformer networks trained end-to-end using weighted acyclic directed graphs to represent a sequence of digits in handwritten character recognition.", "Furthermore, inspired by Q-learning, [15] used an oracle value function as the objective for energy-based deep networks, and [1] introduced structured prediction energy networks (SPENs) to address the inductive bias and to learn discriminative features of the structured output automatically.", "By assigning a score to an entire prediction, SPENs take into consideration high-order interactions between predictors using minimal structural assumptions.", "Nevertheless, due to the non-convexity, optimizing remains challenging, which may cause the learning model to get stuck in local optima.", "Another approach is to move step by step and predict one output variable at a time by applying the information gathered from previous steps.", "The linking between the steps is learned using a predefined order of input variable where the conditional is modeled with RNNs [44].", "Although this method has achieved impressive results in machine translation [23], its success ultimately depends on the neural network's ability to model the conditional distribution and it is often sensitive to the order in which input data is processed, particularly in large-size graphs, as in CPPs (50,000 nodes).", "In contrast to these approaches, instead of using continuous relaxation of output space variables [1], Struct-CKN uses supervised end-to-end learning of CKNs and CRF-based models.", "Accordingly, any of the existing inference mechanisms—from belief propagation to LP relaxations—can be applied.", "This allows us to naturally handle general problems that go beyond multi-label classification, and to apply standard structured loss functions (instead of extending them to continuous variables, as in the case of SPENs).", "More importantly, Struct-CKN allows us to apply our method to a large-scale CRF graph containing up to 50,000 nodes.", "Furthermore, in contrast to methods in the literature (e.g., CNN-CRF [8], CRF-RNN [43], and deep structured models [7]), it has far fewer parameters, thus bypassing the need for extensive tuning.", "Finally, note that in recent papers, convolutional graph neural networks (ConvGNNs) [18] are used either (1) to warm-start a solver (trained under the imitation learning framework) [29], (2) to solve the optimization problem end-to-end [17], or (3) to guide an optimization process (variable selection in branch-and-bound [12]), and ConvGNNs might appear to be a good candidate-solution for CPPs when coupled with a CRF layer to impose constraints on the output.", "However, in the case of graphs with up to 50,000 nodes, the number of parameters used by ConvGNN and the computational limitations prevents us from considering it.", "In fact, we are not aware of any prior work where ConvGNNs were used at this scale.", "Furthermore, the implicit motivation for our proposed approach is an end-to-end solution method that can (1) harness the predictive capabilities of the ML predictor and the decomposition capacity of the solver [41], and (2) can be used on a standard machine with no specific resource requirements, to replace existing solvers in the industry.", "Future research will look into the possibility of integrating a distributed version of ConvGNNs into the proposed framework." ], [ "Convolutional Kernel Networks", "CKN is a particular type of CNN that differs from the latter in the cost function to be optimized to learn filters and in the choice of non-linearities.", "We review CKNs, with the same notation as in [24], [3].", "For further detail, see Appendix ." ], [ "Unsupervised Convolutional Kernel Networks {{cite:ae23995115648ff669512ebe0a42e2797d4fc1ca}}", "Unsupervised Convolutional Kernel Networks We consider an image $I_0: \\Omega _0 \\rightarrow {\\mathbb {R}}^{p_0}$ , where $p_0$ is the number of channels, e.g., $p_0=3$ for RGB, and $\\Omega _0 \\subset [0,1]^2$ is a discrete set of pixel locations.", "Given two image patches $x$ , $x^{\\prime }$ of size $e_0 \\times e_0$ , represented as vectors in ${\\mathbb {R}}^{p_0 e_0^2}$ , we define a kernel $K_1(x,x^{\\prime }) = \\Vert \\ x \\Vert \\Vert \\ x^{\\prime } \\Vert \\cdot \\kappa _1( \\langle \\tfrac{x}{\\Vert \\ x \\Vert } , \\tfrac{x^{\\prime }}{\\Vert \\ x^{\\prime } \\Vert } \\rangle )$ if $x$ , $x^{\\prime }$ $\\ne $ 0 and 0 otherwise, where $\\Vert .\\Vert $ and $\\langle , \\rangle $ denote the Euclidian norm and inner-product, respectively, and $\\kappa _1(\\langle \\cdot ,\\cdot \\rangle )$ is a dot-product kernel on the sphere.", "We have implicitly defined the reproducing kernel Hibert space (RKHS) $\\mathcal {H}_1$ associated to $K_1$ and a mapping $\\varphi _1:{\\mathbb {R}}^{p_0 e_0^2} \\rightarrow \\mathcal {H}_1$ .", "First, we build a database of $n$ patches ${\\mathbf {x}}_1, \\ldots ,{\\mathbf {x}}_n$ randomly extracted from various images and normalized to have unit $\\ell _2$ -norm.", "Then, we perform a spherical $K$ -means algorithm [5], acting as a Nyström approximation, to obtain $p_1$ centroids ${\\mathbf {z}}_1,\\ldots ,{\\mathbf {z}}_{p_1}$ with unit $\\ell _2$ -norm.", "Given a patch ${\\mathbf {x}}$ of $I_0$ , the projection of $\\varphi _1({\\mathbf {x}})$ onto $\\mathcal {F}_1 :=\\text{Span}(\\varphi _1({\\mathbf {z}}_1),\\ldots , \\varphi _1({\\mathbf {z}}_{p_1}))$ admits a natural parametrization given in (REF ) where ${\\mathbf {Z}}=[{\\mathbf {z}}_1,\\ldots ,{\\mathbf {z}}_{p_1}]$ , and $\\kappa _1$ is applied pointwise to its arguments.", "$\\begin{split}\\Gamma _1({\\mathbf {x}}) & :=\\Vert {\\mathbf {x}}\\Vert \\kappa _1({\\mathbf {Z}}^{\\top } {\\mathbf {Z}})^{-1/2} \\kappa _1\\left({\\mathbf {Z}}^\\top \\frac{{\\mathbf {x}}}{\\Vert \\ x \\Vert }\\right) \\\\&~\\text{if}~~{\\mathbf {x}}\\ne 0~~\\text{and}~~0~~\\text{o.w.", "}\\end{split}$ Consider all overlapping patches of $I_0$ .", "We set $M_1(z) = \\Gamma _1({\\mathbf {x}}_z), \\quad z \\in \\Omega _0$ where ${\\mathbf {x}}_z$ is the patch from $I_0$ centered at pixel location $z$ .", "The spatial map $M_1: \\Omega _0 \\rightarrow {\\mathbb {R}}^{p_1}$ thus computes the quantities ${\\mathbf {Z}}^\\top {\\mathbf {x}}$ for all patches ${\\mathbf {x}}$ of image $I$ (spatial convolution after mirroring the filters ${\\mathbf {z}}_j$ ), then applies the pointwise non-linear function $\\kappa _1$ .", "The previous steps transform the image $I_0: \\Omega _0 \\rightarrow {\\mathbb {R}}^{p_0}$ into a map $M_1: \\Omega _0 \\rightarrow {\\mathbb {R}}^{p_1}$ .", "Then, the CKNs involve a pooling step to gain invariance to small shifts, leading to another finite-dimensional map $I_1:\\Omega _1 \\rightarrow {\\mathbb {R}}^{p_1}$ with a smaller resolution: $I_1(z) = \\sum _{z^{\\prime } \\in \\Omega _0} M_1(z^{\\prime }) e^{-\\beta _1 \\Vert z^{\\prime }-z\\Vert _2^2}, \\quad z \\in \\Omega _1$ , where $\\beta _1$ is a subsampling factor.", "We build a multilayer image representation by stacking and composing kernels.", "Similarly to the first CKN layer transforming $I_0: \\Omega _0 \\rightarrow {\\mathbb {R}}^{p_0}$ to the map $I_1: \\Omega _1 \\rightarrow {\\mathbb {R}}^{p_1}$ , we apply the same procedure to obtain $I_2: \\Omega _2 \\rightarrow {\\mathbb {R}}^{p_2}$ , where $p_2$ is the number of centroids in the second layer, then $I_3: \\Omega _3 \\rightarrow {\\mathbb {R}}^{p_3}$ , etc." ], [ "Supervised Convolutional Kernel Networks {{cite:d56d829baa3a0256466be4f06fdc5ab30f2d80e4}}", "Supervised Convolutional Kernel Networks Let $I_0^1$ , $I_0^2$ , ..., $I_0^n$ be the training images with respective labels $y_1$ , ..., $y_n$ in {-1 ; +1} for binary classification.", "We also have $L$ : $\\mathbb {R} \\times \\mathbb {R} \\rightarrow \\mathbb {R} $ , a convex smooth loss function.", "Given a positive definite kernel $K$ on images, the classical empirical risk minimization formulation consists of finding a prediction function in the RKHS $\\mathcal {H}$ associated to $K$ by minimizing the objective $\\min _{f \\in \\mathcal {H}} \\frac{1}{n} \\sum _{i=1}^{n} L(y_i , f(I_0^i)) + \\frac{\\lambda }{2} \\left\\Vert f \\right\\Vert _\\mathcal {H}^2$ , where the parameter $\\lambda $ controls the smoothness of the prediction function $f$ with respect to the geometry induced by the kernel, hence regularizing and reducing overfitting.", "After training a CKN with $k$ layers, such a positive definite kernel $K_\\mathcal {Z}$ may be defined as in (REF ) where $I_k$ , $I_k^{\\prime }$ are the $k$ -th finite-dimensional feature maps of $I_0$ and $I_0^{\\prime }$ , respectively, and $f_k$ , $f_k^{\\prime }$ are the corresponding maps in $\\Omega _k \\rightarrow \\mathcal {H}_k $ , which have been defined in Section REF .", "$K_\\mathcal {Z} (I_0 , I_0^{\\prime }) = \\sum _{z \\in \\Omega _k} \\langle f_k(z), f_k^{\\prime }(z) \\rangle _{\\mathcal {H}_k} = \\sum _{z \\in \\Omega _k} \\langle I_k(z), I_k^{\\prime }(z) \\rangle $ The kernel $K_\\mathcal {Z}$ is also indexed by $\\mathcal {Z}$ , representing network parameters (subspaces $\\mathcal {F}_1$ , ..., $\\mathcal {F}_k$ , or equivalently the set of filters $Z_1$ , ..., $Z_k$ ).", "Then, the formulation becomes as in (REF ) where $\\left\\Vert \\cdot \\right\\Vert _F$ is the Frobenius norm extending the Euclidean norm to matrices and, with an abuse of notation, the maps $I_k^i$ are seen as matrices in $\\mathbb {R}^{p_k \\times \\vert \\Omega _k \\vert }$ .", "Then, the supervised CKN formulation consists of jointly minimizing (REF ) w.r.t.", "$W$ in $\\mathbb {R}^{p_k \\times \\vert \\Omega _k \\vert }$ and with respect to the set of filters $Z_1$ , ..., $Z_k$ , whose columns are constrained to be on the Euclidean sphere.", "$\\min _{W \\in \\mathbb {R}^{p_k \\times \\vert \\Omega _k \\vert } } \\frac{1}{n} \\sum _{i=1}^{n} L(y_i , \\langle W , I_k^i \\rangle ) + \\frac{\\lambda }{2} \\left\\Vert W \\right\\Vert _F^2$" ], [ "Graph-Based Learning", "In structured prediction, models are typically estimated with surrogate structured loss minimization, such as with structured SVM (SSVM) or CRFs.", "We used CRFs for the structured prediction that we briefly review below.", "We also tested SSVM integration instead of CRFs, but it yielded slightly worse results, see Appendices REF and  for details and experimental results using SSVM.", "A CRF models the conditional probability of a structured output $y \\in \\mathcal {Y}$ given an input $x \\in \\mathcal {X}$ where the probability to observe $y$ when $x$ is observed is $ p(y | x; w) \\propto \\exp (\\langle w, F(x,y)\\rangle ) $ , $F$ is the feature mapping, and $w$ is the vector of weights (to be learned).", "The CRF predictor of $y$ when $x$ is observed is: $ h_w(x)=\\operatornamewithlimits{arg\\,max}_{y \\in \\mathcal {Y}} \\langle w, F(x,y) \\rangle $ .", "The CRF primal problem formulation is shown in (REF ), where $\\mathcal {L}^{CRF}$ denotes the negative log likelihood loss.", "$\\min _{\\omega }\\lambda \\Vert \\omega \\Vert ^2+\\frac{1}{n}\\sum _{i=1}^{N}\\mathcal {L}^{CRF}(x_i, y_i; \\omega )$ To optimize CRFs, we use stochastic dual coordinate ascent (SDCA) [35], [36] as [22] showed it yielded state-of-the-art results for CRFs.", "Although one can also use the stochastic average gradient (SAG) algorithm [34] or the online exponentiated gradient (OEG) algorithm, an advantage of SDCA over OEG (and SAG) is that it enables performing an “exact” line search with only one call to the marginalization oracle.", "We now rewrite (REF ) using the notation for the SDCA setup for multi-class classification [36].", "Denote $M_i = | \\mathcal {Y}_i |$ the number of labelings for sequence $i$ .", "Denote $A_i$ the matrix whose columns are the corrected features $\\lbrace \\psi _i(y):=F(x_i,y_i)-F(x_i,y)\\rbrace _{y \\in \\mathcal {Y}_i}$ .", "Denote also $\\Phi _i(s) :=\\log ( \\sum _{y \\in \\mathcal {Y}_i} \\exp (s_y))$ the log-partition function for the scores $s \\in \\mathbb {R}^{M_i}$ .", "Since the negative log-likelihood can be written as $-\\log (p(y_i|x_i;w)) = \\Phi _i(-A_i^\\top w)$  [28], the primal objective function to minimize over $w \\in R^d$ becomes $P(w) :=\\frac{\\lambda }{2} \\left\\Vert w \\right\\Vert _2^2 + \\frac{1}{n} \\sum _{i=1}^{n} \\Phi _i(-A_i^\\top w).$ This minimization problem has an equivalent Fenchel convex dual problem.", "Denote $\\Delta _M$ the probability simplex over $M$ elements.", "Denote $\\alpha _i \\in \\Delta _{M_i}$ the set of dual variables for a given $x_i$ , we define the conjugate weight function $\\hat{w}$ as follows: $\\hat{w}(\\alpha ) = \\tfrac{1}{\\lambda n} \\sum _i A_i \\alpha _i = \\tfrac{1}{\\lambda n} \\sum _{i=1}^{n} \\mathbb {E}_{y \\sim \\alpha _i} (\\psi (y))$ .", "We can show that $\\hat{w}(\\alpha ^*) = w^*$ where $w^*$ and $\\alpha ^*$ are respectively the optimal primal parameters and the optimal dual parameters.", "As such, we can also define the primal sub-optimality as: $P(w) - P(w^*)$ .", "[22] adapted SDCA to CRF by considering marginal probabilities over cliques of the graphical model.", "Because the dual variable $\\alpha _i$ is exponentially large in input size $x_i$ , $\\alpha $ is replaced by $\\mu $ = ($\\mu _1$ , ..., $\\mu _n$ ), where $\\mu _i \\in \\Pi _C^{\\Delta _C}$ is the concatenation of all the clique marginal vectors for sample $i$ .", "Given its state-of-the-art performance, we decided to use it, although we will compare it with other optimizers in Section  (see Appendix REF for further details)." ], [ "The Struct-CKN Framework", "As in Figure REF , the Struct-CKN model consists of two components intended to train CRFs: (1) the CKN and (2) the structured predictor, using CRF loss, and SDCA.", "Upon initializing the CKN layers and the structured predictor, for each iteration, we pass the input image through the CKN multilayers.", "The last map of CKN is passed through to the structured predictor to infer probabilities, which are employed to train CKN weights by backpropagating using rules in [24].", "Figure: The Architecture Diagram for the Struct-CKN PredictorTraining the Struct-CKN Model [1] Initialize CKN parameters in unsupervised manner as described in Sec.", "REF Initialize structured predictor and CRF model as in Alg.", "REF (see Appendix REF , steps 1-2) $t=0\\dots $ For each input, construct an unary feature map, as described in Sec.", "REF (Optional) Center and rescale these representations to have unit $\\ell _2$ -norm on average Infer probabilities by providing image map as input to structured predictor Train the structured predictor using the feature map as an input for $n_{Ep}$ epochs Use the inferred probabilities to compute the gradient by using the chain rule (backpropagation) and update the CKN weights [24] To do inference for CRF models with a small number of nodes (as in Section REF ), we use max-product belief propagation, since chains can be solved exactly and efficiently.", "For the flight-connection dataset (as in Section REF ), since CRF models contain 50,000 nodes, we use AD3 (alternating directions dual decomposition) [26] for approximate maximum a posteriori (MAP) inference.", "It allies the modularity of dual decomposition with the effectiveness of augmented Lagrangian optimization via the alternating directions method of multipliers and has some very interesting features in comparison to other message-passing algorithms.", "Indeed, AD3 has been empirically shown to reach consensus faster than other algorithms [26] and outperforms state-of-the-art message-passing algorithms on large-scale problems.", "Besides, AD3 provides a library of computationally-efficient factors that allow handling declarative constraints within an optimization problem.", "This is particularly interesting for the CPP use case since we add a constraint to impose that each flight is preceded by one flight at most.", "To use SDCA (requiring a marginalization oracle) with AD3 (used to do approximate MAP), we propose a simple approximation, using MAP label estimates.", "See Appendix REF for details on integrating SDCA and AD3.", "Note that an embedding layer may be used before passing the input through to the CKN layers.", "Indeed, for categorical variables with a large number of categories, the input matrix is sparse, making the learning process difficult, as the extracted patches have mostly null values.", "Furthermore, as in [6], using deep structured predictors may require using scaling.", "Specifically, the last map of the “deep” layer needs to be rescaled before being passed to the “structured layer”.", "We propose to use one of the following scalers within the Scikit-learn library: Min-Max scaler, Normalizer scaler, Standard scaler, and Robust scaler.", "When using the SDCA optimizer, line search requires computing the entropy of the marginals.", "Since this is costly, in order to minimize the number of iterations, we used the Newton-Raphson algorithm.", "This requires storing the logarithm of the dual variable, which may be expensive, so a decent amount of memory should be allocated.", "Finally, note that we use a batch-version of the Struct-CKN predictor on the flight-connection dataset [40], where one batch corresponds to one CRF model (CPP instance): (1) We initialize the CKN weights (in an unsupervised manner) and the CRF model sequentially by considering each one of the six instances separately.", "(2) We ”flatten” the input by considering each one of the six instances separately.", "(3) We center and rescale the representations for all the instances at once.", "(4) We pass small batches of image maps as inputs to the structured predictor (e.g., 128 image maps) to infer the probabilities and train the structured predictor.", "Then, we use the inferred probabilities of the batch to update the CKN weights." ], [ "Experiments", "In this section, we report the results of experiments using Struct-CKN.", "First, we sanity-check Struct-CKN on the standard OCR dataset in Section REF , showing that it's comparable to the state of the art.", "Then, in Section REF , we use the proposed predictor on the flight-connection dataset to warm-start Commercial-GENCOL-DCA.We use Pytorch [30] to declare said model and perform operations on a 40-core machine with 384 GB of memory, and use K80 (12 GB) GPUs.", "The CRF model is implemented using PyStruct [27], while the SDCA optimizer is implemented using SDCA4CRF.", "Scalers are implemented using Scikit-learn [31]." ], [ "OCR - Chain CRF", "Each example in the OCR dataset [37] consists of a handwritten word pre-segmented into characters, with each character represented as a 16$\\times $ 8 binary image.", "The task is to classify the image into one of the 26 characters (a$-$ z).", "It comes with pre-specified folds; one fold is considered the test set, while the rest as the training set, as in max-margin Markov networks [37].", "Since the CRF optimizers (SAG-NUS, SAG-NUS* [34], SDCA, SDCA-GAP [22], and OEG [34]) yield similar test errors (11.8-12%), we only report SDCA (with linear features) in Table REF .", "LSTM (standard two-layer) [14] and CNN-CRF (standard two-layer) [8] yield comparable results (4.4-4.6%).", "Sequence Classification Restricted Boltzmann Machine (SCRBM) [38] and NLStruct [13] provide better results (4.0%, and 3.6%).", "However, Struct-CKN outperforms all aforementioned methods, reducing test errors to 3.40%.", "Note that some structured predictors can lower test error to $1-3\\%$  [32], such as SeaRNN [23], which adapts RNN to the learning-to-search approach.", "However, such models approximate the cost-to-go for each token by computing the task loss for as many roll-outs as the vocabulary size at each time step, and are thus difficult to scale to real-world datasets (with long sequences or large vocabulary), such as the flight-connection dataset (see Section REF ).", "Figure REF reports primal sub-optimality w.r.t parameter updates (see Appendix ).", "Struct-CKN outperforms other methods for the first 50 epochs and is comparable to other methods for subsequent epochs.", "We usually train predictors only for several epochs; thus, the precision of primal sub-optimality (below $10^{-5}$ ) is negligible." ], [ "Airline Crew Scheduling Dataset - Graph CRF", "This section reports results of using Struct-CKN to warm-start Commercial-GENCOL-DCA [41] and solve large-scale CPP.", "Section REF presents CPP.", "Section REF outlines the flight-connection prediction problem and CRF models.", "Section REF reports results of predictions.", "Section REF describes the optimization process and analyzes the feasibility of proposed monthly solutions.", "Section REF reports results of solving CPPs using the solver." ], [ "Crew Pairing Problem", "The CPP aims to find a set of pairings at minimal cost for each category of the crew and each type of aircraft fleet [9].", "A flight sequence operated by a single crew forms a duty and a consecutive sequence of duty periods is named a pairing.", "A pairing is deemed feasible if it satisfies safety rules and collective agreement rules [16], such as: minimum connection time between two consecutive flights and minimum rest-time between two duties; maximum number of flights per duty and maximum span of a duty; maximum number of landings per pairing and maximum flying time in a pairing; maximum number of days and maximum number of duties in a pairing.", "In addition, CPPs use base constraints (referred to as global constraints) to distribute the workload fairly amongst the bases proportionally to the personnel available at each base.", "Penalties when the workload is not fairly distributed is called the cost of global constraints.", "In our approach, the predictor does not consider these very complex airline-dependent constraints as well as the solution cost and the cost of global constraints, as doing so would require more data than are currently available.", "Whenever a flight appears in more than one pairing, we use deadheads: one crew operates the flight, while the others are transferred between two stations for repositioning.", "The CPP has been traditionally modelled as a set partitioning problem, with a covering constraint for each flight and a variable for each feasible pairing [9], [16].", "Formally, we consider $F$ to be a set of flights that must be operated during a given period and $\\Omega $ to be the set of all feasible pairings that can be used to cover these flights.", "It is computationally infeasible to list all pairings in $\\Omega $ when solving CPPs with more than hundreds of flights.", "Therefore, it is not tractable to do so in this context (CPPs with 50,000 flights).", "For each pairing $p \\in \\Omega $ , let $c_p$ be its cost and $a_{fp}$ , $f \\in F$ , be a constant equal to 1 if it contains leg $f$ and 0 otherwise.", "Moreover, let $x_p$ be a binary variable that takes value 1 if pairing $p$ is selected, and 0 otherwise.", "Using a set-partitioning formulation, the CPP can be modelled as follows: $& \\underset{x}{\\text{minimize}}& \\sum _{p \\in \\Omega }{c_{p} x_p} \\\\& \\text{subject to}& \\sum _{p \\in \\Omega }{a_{fp} x_p} = 1& & \\forall f \\in F \\\\&& x_p \\in \\lbrace 0 , 1 \\rbrace & & \\forall p \\in \\Omega $ The objective function (REF ) minimizes the total pairing costs.", "Constraints () ensure each leg is covered exactly once, and constraints () enforce binary requirements on the pairing variables.", "The methodology to solve CPPs depends on the size of the airline's network, rules, collective agreements, and cost structure [42].", "Since the 1990s, the most prevalent method has been column generation inserted in branch-&-bound [9].", "This algorithm was combined with multiple methods in [10] to solve large-scale CPPs.", "[41] proposed Commercial-GENCOL-DCA with a dynamic control strategy and used CNNs to develop initial monthly crew pairings.", "Because the constructed solution contained too many infeasible pairings, it was passed to the solver only as initial clusters and a generic standard initial solution was used.", "In this work, we first use Struct-CKN to construct initial monthly crew pairings passed to the solver both as initial clusters and an initial solution.", "First, we use the GENCOL solver (used to assign crews to undercovered flights),http://www.ad-opt.com/optimization/why-optimization/column-generation/ then we run Commercial-GENCOL-DCA.", "Because solvers can only handle a few thousand flights, the windowing approach is used: the month is divided into multiple windows, where each window is solved (sequentially) while flights in pairings from previous windows are frozen.", "GENCOL requires using two-day windows and one-day overlap period, while Commercial-GENCOL-DCA permits to use one-week windows and two-day overlap period.", "The latter starts with an aggregation, in clusters, of flights.", "The initial aggregation partition permits replacing all flight-covering constraints of the flights in a cluster by a single constraint, allowing to cope with larger instances." ], [ "Prediction Problem Formulation", "We aim to provide the CPP solver with an initial solution and initial clusters.", "Using the flight-connection dataset built in [40], the flight-connection prediction problem is a multi-class classification problem, formulated as follows: “Given the information about an incoming flight in a specific connecting city, choose among all possible departing flights from this city the one that the crew should follow” (see Appendix ).", "Thus, each input contains information on the previous flight performed by the crew as well as information on all possible next flights (up to 20 candidates), sorted by departure time, and the output (class) is the rank of the flight performed by the crew in past solutions.", "In [40], the authors propose a similarity-based input where neighboring factors have similar features, allowing the use of CNNs.", "We extend their approach with our Struct-CKN model, thus starting the CPP solver with an initial solution, and not only the initial clusters.", "The training set in the flight-connection dataset consists of six monthly crew pairing solutions (50,000 flights per month) and the test set is a benchmark that airlines use to decide on the commercial solver to use.", "Each flight is characterized by the cities of origin and destination, the aircraft type, the flight duration, and the departure and arrival time.", "For each incoming flight, the embedded representation of the candidate next flights is concatenated to construct a similarity-based input, where neighboring factors have similar features.", "The intuition is that each next flight is considered a different time step, enabling the use of convolutional architecture across time.", "[40] compare multiple predictors on the flight-connection dataset and empirically confirm this intuition (see Appendix ).", "The dataset is used to define a pairwise CRF on a general graph where each example consists of a flight-based network structure with nodes corresponding to flights and arcs representing the feasibility of two flights being successive.", "Each flight corresponds to a node connected to nodes that are possible successors/predecessors; the true label is the rank of the next flight in the set of sorted possible successors.", "We impose local constraints to the output by imposing that each flight has to be preceded by at most one flight (using a XOR constraint, which contributes $-\\infty $ to the potential).", "Unlike for the OCR dataset where “max-product” is used for belief propagation, in this section, we use AD3 (alternating directions dual decomposition) [26] for approximate maximum a posteriori (MAP) inference." ], [ "Results on the Flight-connection Dataset", "Results on the Flight-Connection Dataset As Struct-CKN has fewer hyperparameters than CNNs, we observed that Struct-CKN is more stable than CNNs in our experiments, in that it does not requires Bayesian optimization to find a good architecture, thus justifying our use of CKN (see Appendix ).", "Table REF reports the number of parameters and the test error on the flight-connection dataset using (1) CNNs and Bayesian optimization to search for the best configuration of hyperparameters [41]; (2) standard CNN-CRF (with non-exhaustive hyperparameter tuning) [8]; and (3) Struct-CKN.", "Struct-CKN outperforms both CNN and CNN-CRF while having far fewer parameters (97% fewer parameters).", "While Bayesian optimization can be used to fine-tune hyperparameters, this is not feasible in a real-case usage scenario, as practitioners cannot perform it each time new data become available and the need for extensive fine-tuning makes a predictor impossible to integrate into any scheduling solver.", "Table: Test Error on the Flight-connection Dataset" ], [ "Construction and Feasibility of a Monthly Solution", "As in Figure REF , we compare four approaches to construct monthly pairings.", "The first approach is a standard monthly solution, called “Standard - initial”, a “cyclic” weekly solution (from running the optimizer on a weekly CPP) rolled to cover the whole month [10].", "A cyclic solution is where the number of crews in each city is the same at the beginning and end of the horizon.", "In the second approach, CNNs predict the flight-connection probabilities.", "Then, using the same heuristics as in [41] (see Appendix REF ), we build a monthly crew pairing called “CNN - initial”.", "In the third and fourth approaches, CNN-CRF [8] and Struct-CKN are used to build monthly crew pairings called “CNN-CRF - initial”, and “Struct-CKN - initial”.", "Then, we break all illegal pairings and freeze the legal sub-part in initial crew pairings, resulting in “Standard - feasible”, “CNN - feasible”, “CNN-CRF - feasible” and “Struct-CKN - feasible”.", "We generate deadheads on all flights and pass it to the solver.", "Figure: The Optimization Process for the CPPNote that to reproduce the following results, we obtain the commercial dataset from [40], which cannot be distributed because it contains too much flight-data information sensitive to airlines' operations.", "Since the first step of the CPP solver can solve up to several thousand flights, we are constrained to use two-day windows.", "Since pairings extend to over two days, it cannot “fix” the mispredicted and illegal pairings.", "Therefore, the more illegal pairings the constructed solution contains, the longer it will take the solver to find a suitable final solution.", "Furthermore, when the constructed solution is highly infeasible, it can only be proposed as initial clusters as in all past research (e.g., [40], [41]) and not as an initial solution.", "In this case, a generic standard initial solution is used, and since the initial solution and the initial clusters are different, an adaptation strategy is required to adapt the proposed clusters of the current window to the solution of the previous window.", "This is a major limitation of past research since the explored neighborhood needed to be large enough to reach a good LP (linear programming) solution but small enough to maintain a small number of fractional variables permitting to have an efficient heuristic branch-&-bound.", "Thus, not only can we conclude that the primary metric of interest in our case is the feasibility of the constructed monthly solution.", "But, it also becomes crucial to propose a feasible pairing solution that can be proposed both as initial clusters and as an initial solution, therefore bypassing the adaptation strategy and permitting to reduce the resolution time (by reducing the neighborhood in which to explore in order to find a suitable solution).", "Table REF summarizes the computational results on the feasibility and characteristics of constructed monthly pairings.", "First, breaking all illegal pairings in “Standard - initial solution” removes 50.56% of the pairings, while that in “CNN - initial solution” led to removing 21.05%.", "Note that even though the test error is low for CNNs, due to the large number of infeasible pairings, running the optimization using this initial solution is problematic.", "Breaking all illegal pairings in “CNN-CRF - initial solution” removes 12.12% of the pairings, while only 11.07% are removed from “Struct-CKN - initial solution”.", "Clearly, although Struct-CKN has 97% fewer parameters than other methods and its hyperparameters are not exhaustively fine-tuned, it outperforms other methods in terms of test error and feasibility.", "In what follows, we provide the constructed monthly pairings to the solver and compare the resulting solutions when using the baseline solution [10], CNN [41] and the proposed approach (Struct-CKN).", "Table: Characteristics of Monthly Solutions" ], [ "Results on the Crew Pairing Problem", "As in Figure REF , the baseline solution (“Baseline”) for the monthly CPP is obtained by feeding the solver initial clusters from “Standard - feasible” [10].", "The previous state-of-the-art was obtained by feeding the solver initial clusters from “CNN - feasible”, yielding a solution called “CNN”.", "Instead of CNNs, we can use Struct-CKN to propose both initial clusters and an initial solution from “Struct-CKN - feasible” to the solver, yielding a monthly solution called “Struct-CKN”.", "To work on finding the best monthly solution possible and overcome the limitations of using the windowing approach, we feed the solution obtained from “Struct-CKN” to the solver (again) as initial clusters and initial solution, yielding “Struct-CKN+”.", "Because we cannot use the constructed monthly pairing as an initial solution, when using CNNs, we claim that re-running the optimization using the solution “CNN” as initial clusters does not improve the monthly solution much.", "To support our claim, we feed solution “CNN” to the solver as initial clusters, yielding “CNN+” (see Figure REF ).", "See Appendix REF for detailed statistics of the optimization process.", "Table REF reports computational results for the final monthly solution.", "Note that we do not report variances since the CPP solver is deterministic.", "Struct-CKN outperforms both Baseline and CNN reducing the solution cost (in millions of dollars) and cost of global constraints by 9.51% and 80.25%, respectively, while also being 33% faster than CNN.", "By re-running the optimization, on the one hand, CNN+ does not improve the solution much.", "On the other hand, the Struct-CKN+ solution yields the best statistics, reducing solution cost and cost of global constraints by 16.93% and 97.24%, respectively.", "More interestingly, the number of deadheads (see Section REF ) is reduced by 41.23%, compared to Baseline.", "Therefore, we can conclude that proposing a feasible monthly solution both as initial clusters and as an initial solution allowed us to achieve better results in less time and to provide the possibility to re-optimize the solution and improve it further.", "Thus, this permits to update the training solutions, suggesting that Struct-CKN can be further optimized and that further research to avoid the windowing approach and use a one-month window can present better results than the current version of solver.", "Table: Computational Results for Monthly Solutions" ], [ "Conclusion", "Seeking an initial solution to a crew pairing solver, this study proposes Struct-CKN, a new deep structured predictor.", "Its supervised use outperforms state-of-the-art methods in terms of primal sub-optimality of the structured prediction “layer” and test accuracy on the OCR dataset.", "The proposed method is then applied on a flight-connection dataset, modeled as a general CRF capable of incorporating local constraints in the learning process.", "To warm-start the solver, we use Struct-CKN to propose initial clusters and an initial solution to the solver, reducing the solution cost by 17% (a gain of millions of dollars) and the cost of global constraints by 97%, compared to baselines.", "Future research will look into combining deep structured methods with various operations research methods and designing new reactive/learning metaheuristics that learn to guide the search for better solutions in real-time." ], [ "Acknowledgements", "We are thankful to the anonymous reviewers and the meta-reviewer for their valuable comments that improved the quality of this work.", "We also would like to thank Andjela Mladenovic, Gina Arena, Joey Bose, Mehdi Abbana Bennani, and Stephanie Cairns for their constructive comments regarding this work.", "We thank Iban Harlouchet for his involvement during the first phase of the project.", "This work was supported by IVADO, a Collaborative Research and Development Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC), by the Canada CIFAR AI Chair Program, and AD OPT, a division of IBS Software.", "We would like to thank these organizations for their support and confidence.", "Simon Lacoste-Julien is a CIFAR Associate Fellow of the Learning in Machines & Brains program.", "The appendix is organized as follows: we first provide details about the convolutional kernel networks (CKNs) and the multilayer convolutional kernel (MCK) in Appendix .", "Next, we provide details on the Structured SVM approach in Appendix REF .", "Then, we present additional results on the OCR dataset in Appendix .", "Then, we present the prediction problem formulation in Appendix REF and describe the flight-connection dataset in Appendix REF .", "Finally, we provide details about the optimization process and instances used in our paper in Appendix REF and present additional results for the Crew Pairing Problem (CPP) optimization in Appendix REF .", "Note that in the appendix, we will refer to Struct-CKN using CRF models as Struct-CKN-SDCA.", "Instead of CRFs, we can also use SSVM predictors, in particular the block-coordinate Frank-Wolfe (BCFW) algorithm, which we will refer to as Struct-CKN-BCFW." ], [ "Convolutional Kernel Networks", "This section contains additional details on CKNs [25].", "We first present the Kernel trick, the reproducing kernel Hilbert space, and the Nyström low-rank approximation in Sections REF , REF , and REF , respectively.", "Then, Section REF presents MCKs." ], [ "Kernel Trick", "The kernel trick consists of embedding a “raw input space” $\\mathcal {X}$ into a high dimensional Hilbert “feature space” $\\mathcal {H}$ , through a possibly non-linear mapping $\\varphi : \\mathcal {X} \\rightarrow \\mathcal {H}$ , where the inner product in $\\mathcal {H}$ admits the computation formula as in (REF ), with $K: \\mathcal {X} \\times \\mathcal {X} \\mapsto \\mathbb {R}$ .", "$\\langle \\varphi (x), \\varphi (x^{\\prime })\\rangle _\\mathcal {H}=K(x,x^{\\prime }), \\quad x, x^{\\prime } \\in \\mathcal {X}$ The “kernel function” $K$ is thought of as specifying similarity between elements of $\\mathcal {X}$ .", "In contrast, the similarities in $\\mathcal {H}$ are expressed as simple inner products.", "The choice of an appropriate $\\varphi $ can result in making the image of the data subset of $\\mathcal {X}$ quasi-linearly separable, in the sense that the image of $\\mathcal {X}$ and the image of another subset disjoint to $\\mathcal {X}$ could (almost) be separated with a linear predictor.", "When a linear predictor on $\\mathcal {H}$ only uses inner products of $\\mathcal {H}$ elements, then the formula (REF ) makes the training of this linear predictor computationally feasible.", "The next section reviews a specific class of feature spaces that are built from positive definite kernels." ], [ "Reproducing Kernel Hilbert Space", "Definition 1 (Positive definite kernels) A positive definite (p.d.)", "kernel on a set $\\mathcal {X}$ is a function $K : \\mathcal {X}\\times \\mathcal {X} \\rightarrow \\mathbb {R}$ that is symmetric (i.e.", "$K(x,x^{\\prime })=K(x^{\\prime },x), \\forall x,x^{\\prime }\\in \\mathcal {X}$ ), and for which, for any $N \\in \\mathbb {N}$ and any $(x_1, \\ldots , x_N)\\in \\mathcal {X}^N$ , the Gram matrix (or similarity matrix) $[K]_{ij}=K(x_i,x_j)$ is positive semi-definite (which means that for any $(a_1, \\ldots , a_N) \\in \\mathbb {R}^n$ , $\\sum _{i=1}^{N}\\sum _{j=1}^{N}a_ia_jK(x_i, x_j) \\ge 0$ ).", "Theorem 1 (RKHS [3]) Let $K$ be a p.d.", "kernel on a set $\\mathcal {X}$ .", "There exists a unique Hilbert space $\\mathcal {H} \\subset \\mathcal {X}^{\\mathbb {R}}$ such that $\\lbrace K: y \\mapsto K(x,y), \\mathcal {X} \\mapsto \\mathcal {R} \\rbrace _{x \\in \\mathcal {X}} \\subset \\mathcal {H}$ $\\forall f \\in \\mathcal {H}, \\forall x \\in \\mathcal {X}, f(x)=\\langle f, K \\rangle _\\mathcal {H}$ In particular, the application $\\varphi :\\mathcal {X} &\\mapsto \\mathcal {H} \\\\x &\\mapsto K$ which maps $\\mathcal {X}$ in the “feature space” $\\mathcal {H}$ satisfies $\\langle \\varphi (x), \\varphi (x^{\\prime }) \\rangle _\\mathcal {H} = K(x, x^{\\prime }) \\quad x,x^{\\prime } \\in \\mathcal {X}$ We give now two classical examples of RHKSs associated to the euclidean space $\\mathcal {X}=\\mathbb {R}^d$ : the linear kernel and the Gaussian kernel.", "The linear kernel on $\\mathcal {X}$ is defined to be : $K_\\text{lin}(x,x^{\\prime })=\\langle x, x^{\\prime } \\rangle _{\\mathbb {R}^d}$ The RKHS associated to $(\\mathcal {X}, K_{lin})$ is the Hilbert space $\\mathcal {H}=\\left\\lbrace f_w = \\langle \\bullet , w \\rangle _{\\mathbb {R}^d} \\right\\rbrace _{w \\in \\mathbb {R}^d}$ endowed with the inner product $ \\langle f_w, f_v \\rangle _{\\mathcal {H}} = \\langle w, v \\rangle _{\\mathbb {R}^d}$ .", "This RKHS is isometric to $\\mathbb {R}^d$ .", "The Gaussian Kernel with bandwidth $\\sigma $ on $\\mathcal {X}$ is defined to be $K_\\text{Gauss}(x,y)=e^{-\\frac{\\Vert x-y\\Vert ^2}{2\\sigma ^2}}$ The RKHS $\\mathcal {H}$ associated to $(\\mathcal {X}, K_{Gauss})$ is an infinite-dimensional feature space, and all points of $\\mathcal {X}$ are mapped to the unit sphere of $\\mathcal {H}$ ($\\Vert \\varphi (x) \\Vert _\\mathcal {H}^2=K(x,x)=1$ ).", "Theorem 2 (Representer theorem) Let $\\mathcal {X}$ be a set endowed with a p.d.", "kernel $K$ , let $\\mathcal {H}$ be its RKHS and $S=\\lbrace x_1, \\ldots , x_n\\rbrace \\subset \\mathcal {X}$ a training data set.", "If $\\Psi : \\mathbb {R}^{n+1} \\mapsto \\mathbb {R}$ is strictly increasing with respect to the last variable, then $f^* = & \\operatornamewithlimits{arg\\,min}_{f\\in \\mathcal {H}}\\Psi \\left(f(x_1), \\ldots , f(x_n), \\Vert f \\Vert _\\mathcal {H} \\right)\\\\\\Rightarrow & f^* \\in \\text{Span}\\left(K_{x_1}, \\ldots , K_{x_n} \\right)$ Remark: $f^*$ lives in a subspace of dimension $n$ , although $\\mathcal {H}$ can be of infinite dimension." ], [ "Nyström Low-rank Approximation", "Consider a p.d.", "kernel $K:\\mathcal {X} \\times \\mathcal {X} \\mapsto \\mathbb {R}$ and its RKHS $\\mathcal {H}$ , with the mapping $\\varphi : \\mathcal {X} \\mapsto \\mathcal {H}$ such that $K(x,x^{\\prime })=\\langle \\varphi (x), \\varphi (x^{\\prime })\\rangle _\\mathcal {H}$ .", "The Nyström approximation consists in replacing any point $\\varphi (x)$ in $\\mathcal {H}$ by its orthogonal projection $\\Pi _{\\mathcal {F}}(x)$ onto a finite-dimensional subspace $\\mathcal {F} :=\\text{Span}(f_1, \\ldots , f_p) \\text{ with } p \\ll n$ where the $f_i^{\\prime }$ are anchor points in $\\mathcal {H}$ (defined below).", "This projetion is equivalent to $\\Pi _{\\mathcal {F}}(x) :=\\sum _{j=1}^{p} \\beta _j^*(x)f_j$ with $\\beta ^*(x) = \\operatornamewithlimits{arg\\,min}_{\\beta \\in \\mathbb {R}^p} \\left\\Vert \\varphi (x)-\\sum _{j=1}^{p}\\beta _j f_j \\right\\Vert _\\mathcal {H}^2$ Noting $[K_f]_{jl}=\\langle f_j, f_l \\rangle _\\mathcal {H}$ and $f(x)=(f_1(x), \\ldots , f_p(x))\\in \\mathbb {R}^p$ , it is quickly checked that $\\beta ^*(x)=K_f^{-1}f(x)$ , where $K_f^{-1}$ is the inverse of the kernel matrix $K_f$ (or its pseudo-inverse, when the matrix $K_f$ is not full rank).", "Thus, $\\varphi (x) \\approx \\sum _{j=1}^p \\beta _j^*(x)f_j$ and $\\langle \\varphi (x), \\varphi (x^{\\prime }) \\rangle _\\mathcal {H} \\approx \\beta ^*(x)^\\top K_f\\beta ^*(x^{\\prime })$ Defining the mapping $\\Gamma : \\mathcal {X} & \\mapsto \\mathbb {R}^p \\\\x & \\mapsto \\Gamma (x)=K_f^{-1/2}f(x)$ as $\\Gamma (x)=K_f^{1/2}\\beta ^*(x)$ , we see that $K(x,x^{\\prime }) \\approx \\langle \\Gamma (x), \\Gamma (x^{\\prime }) \\rangle _{\\mathbb {R}^d}$ Consequently, the mapping $\\varphi : \\mathcal {X} \\mapsto \\mathcal {H}$ is approximated by a mapping $\\Gamma : \\mathcal {X} \\mapsto \\mathbb {R}^p$ such that $\\langle \\varphi (x), \\varphi (x^{\\prime })\\rangle _\\mathcal {H} \\approx \\langle \\Gamma (x), \\Gamma (x^{\\prime }) \\rangle _{\\mathbb {R}^d}$ There exists various methods for choosing the anchor points $f_j$ 's.", "When $\\mathcal {X} = \\mathbb {R}^d$ , one such method consists in performing a $K$ -means algorithm on a training data set $S=\\lbrace x_1, \\ldots , x_n \\rbrace \\subset \\mathcal {X}$ , to obtain $p$ centroids ${\\mathbf {z}}_1, \\ldots , {\\mathbf {z}}_{p_1}\\in \\mathbb {R}^d$ , and define anchor points as $f_j=\\varphi ({\\mathbf {z}}_j)$ , for $j=1, \\ldots , p$ ." ], [ "Multilayer Convolutional Kernel {{cite:d56d829baa3a0256466be4f06fdc5ab30f2d80e4}}", "Multilayer Convolutional Kernel Two principal advantages of a CNN over a fully-connected neural network are: (i) sparsity - each nonlinear convolutional filter acts only on a local patch of the input, (ii) parameter sharing - the same filter is applied to each patch.", "Definition 2 (Image feature map and patch feature map) An image feature map $\\varphi $ is a function $\\varphi : \\Omega \\rightarrow \\mathcal {H}$ , where $\\Omega $ is a discrete subset of $[0,1]^2$ representing a set of pixel locations, and $\\mathcal {H}$ is a Hilbert space representing a feature space.", "A patch feature map is the same as an image feature map with $\\Omega $ replaced by a patch of pixels $\\mathcal {P}$ centered at 0.", "Definition 3 (Convolutional kernel) Let us consider two images represented by two image feature maps, respectively $\\varphi $ and $\\varphi ^{\\prime }:\\Omega \\rightarrow \\mathcal {H}$ .", "The one-layer convolutional kernel between $\\varphi $ and $\\varphi ^{\\prime }$ is defined as $K(\\varphi , \\varphi ^{\\prime })= \\sum _{z \\in \\Omega }\\sum _{z^{\\prime } \\in \\Omega } \\left\\lbrace \\Vert \\varphi (z) \\Vert _{\\mathcal {H}}\\Vert \\varphi ^{\\prime }(z^{\\prime }) \\Vert _{\\mathcal {H}}e^{-\\frac{1}{2\\beta ^2}\\Vert z-z^{\\prime }\\Vert ^2_2} \\right.", "\\nonumber \\left.", "e^{-\\frac{1}{2\\sigma ^2}\\Vert \\tilde{\\varphi }(z)-\\tilde{\\varphi ^{\\prime }}(z^{\\prime })\\Vert ^2_2} \\right\\rbrace $ where $\\tilde{\\varphi }(z)=\\left( 1/\\Vert \\varphi (z) \\Vert _{\\mathcal {H}}\\right)\\varphi (z)$ if $\\varphi (z) \\ne 0$ and $\\tilde{\\varphi }(z)=0$ otherwise.", "Idem for $\\tilde{\\varphi ^{\\prime }}(z)$ .", "Remark 1: The role of $\\beta $ is to control how much the kernel is locally shift-invariant.", "Remark 2: $K$ is built on a kernel $k$ on $\\mathcal {H}$ , of the form $k(h,h^{\\prime })=\\Vert h \\Vert _\\mathcal {H}\\Vert h^{\\prime } \\Vert _\\mathcal {H} \\kappa \\left( \\frac{\\langle h, h^{\\prime } \\rangle }{\\Vert h \\Vert _\\mathcal {H}\\Vert h^{\\prime } \\Vert _\\mathcal {H}}\\right), \\quad h,h^{\\prime }\\in \\mathcal {H}$ According to a classical result (Schoenberg, 1942), when $\\kappa $ is smooth with non-negative Taylor expansion coefficients, $k$ is a p.d.", "kernel.", "Consequently, $K$ is also a p.d.", "kernel.", "Definition 4 (Multilayer convolutional kernel) Let us consider a discrete set of pixel locations $\\Omega _{k\\text{--}1} \\subseteq [0,1]^2$ and a Hilbert space $\\mathcal {H}_{k\\text{--}1}$ .", "We build a new set $\\Omega _k$ and a new Hilbert space $\\mathcal {H}_k$ as follows: (i) choose a patch shape ${\\mathcal {P}}_k$ defined as a discrete symmetric subset of $[-1,1]^2$ , and a set of pixel locations $\\Omega _k$ such that for all location ${\\mathbf {z}}_k$ in $\\Omega _k$ , the patch $\\lbrace {\\mathbf {z}}_k\\rbrace + {\\mathcal {P}}_k$ is a subset of $\\Omega _{k\\text{--}1}$ ; In other words, each pixel location ${\\mathbf {z}}_k$ in $\\Omega _k$ corresponds to a valid patch of pixel locations in $\\Omega _{k\\text{--}1}$ centered at ${\\mathbf {z}}_k$ .", "(ii) define the convolutional kernel $K_k$ on the “patch” feature maps ${\\mathcal {P}}_k \\rightarrow \\mathcal {H}_{k\\text{--}1}$ , by replacing in (REF ): $\\Omega $ by ${\\mathcal {P}}_k$ , $\\mathcal {H}$ by $\\mathcal {H}_{k\\text{--}1}$ , and $\\sigma ,\\beta $ by appropriate smoothing parameters $\\sigma _k,\\beta _k$ .", "We denote by $\\mathcal {H}_k$ the Hilbert space for which the positive definite kernel $K_k$ is reproducing.", "An image represented by a feature map $\\varphi _{k\\text{--}1}: \\Omega _{k\\text{--}1} \\rightarrow \\mathcal {H}_{k\\text{--}1}$ at layer $k\\text{--}1$ is now encoded in the $k$ -th layer as $\\varphi _k:\\Omega _k \\rightarrow \\mathcal {H}_k$ , where for all ${\\mathbf {z}}_k$ in $\\Omega _k$ , $\\varphi _{k}({\\mathbf {z}}_k)$ is the representation in $\\mathcal {H}_k$ of the patch feature map ${\\mathbf {z}}\\mapsto \\varphi _{k\\text{--}1}({\\mathbf {z}}_k + {\\mathbf {z}})$ for ${\\mathbf {z}}$ in ${\\mathcal {P}}_k$ .", "In other words, given $\\varphi _{k\\text{--}1}: \\Omega _{k\\text{--}1} \\rightarrow \\mathcal {H}_{k\\text{--}1}$ , we first define $\\Omega _{k}$ and ${\\mathcal {P}}_k$ such that for any ${\\mathbf {z}}_k \\in \\Omega _{k}$ , ${\\mathbf {z}}_k + {\\mathcal {P}}_k \\subset \\Omega _{k\\text{--}1}$ .", "Second, we define $\\mathcal {H}_{k}$ as being the RKHS of the set of patch feature maps $\\lbrace {\\mathbf {z}}\\mapsto \\varphi ({\\mathbf {z}}_k + {\\mathbf {z}}), {\\mathcal {P}}_k \\mapsto \\mathcal {H}_{k\\text{--}1}: \\varphi :\\Omega _{k\\text{--}1} \\rightarrow \\mathcal {H}_{k\\text{--}1}, {\\mathbf {z}}_k \\in \\Omega _k \\rbrace $ endowed with the positive definite kernel $K_k$ adapted from (REF ), and we set $\\varphi _{k}({\\mathbf {z}}_k)$ as being the representation in $\\mathcal {H}_k$ of the patch feature map ${\\mathbf {z}}\\mapsto \\varphi _{k\\text{--}1}({\\mathbf {z}}_k + {\\mathbf {z}})$ .", "Thus we obtain $\\varphi _{k}: \\Omega _{k} \\rightarrow \\mathcal {H}_{k}$ , and for any $\\varphi _{k\\text{--}1}, \\varphi ^{\\prime }_{k\\text{--}1}: \\Omega _{k\\text{--}1} \\rightarrow \\mathcal {H}_{k\\text{--}1}$ , we have $K_k \\left(\\varphi _{k\\text{--}1}, \\varphi ^{\\prime }_{k\\text{--}1} \\right)= \\langle \\varphi _{k}, \\varphi ^{\\prime }_{k} \\rangle _{\\mathcal {H}_k}$ ." ], [ "Conditional Random Fields", "Algorithm REF describes this variant of SDCA, biasing the sampling towards examples with large duality gaps $g$ .", "If we assume that the graph has a junction tree structure $\\mathbb {T} = (\\mathbb {C}, \\mathbb {S})$ , where $\\mathbb {C}$ is the set of maximal cliques and $\\mathbb {S}$ the set of separators, we can run message passing on the junction tree to infer new marginals given weights $w$ : $ \\hat{\\mu }_i(w) = p (y_C = \\cdot | x_i ; w) $ , and recover joint probability $\\alpha _i(y)$ as function of marginals $\\mu _{i,C}$ : $\\alpha _i(y) = \\tfrac{ \\prod _{C \\in \\mathbb {C}}^{} \\mu _{i,C}(y_C) }{ \\prod _{S \\in \\mathbb {S}}^{} \\mu _{i,S}(y_S)}$ .", "This allows computing the entropy as in (REF ) and the Kullback-Leibler divergence of joints as in (REF ), using only the marginals $\\mu _i$ and $\\nu _i$ of $\\alpha _i$ and $\\beta _i$ , respectively.", "With (REF ), we compute the dual objective value, and perform the line search, as in (REF ).", "$\\widetilde{H}(\\mu _i) = H(\\alpha _i) = \\sum _{C}{} H ( \\mu _{i,C} ) - \\sum _{S}{} H ( \\mu _{i,S} )$ $\\widetilde{D}(\\mu _i || \\nu _i) = \\sum _{C}D_{KL}(\\mu _{i,C} || \\nu _{i,C}) -\\sum _{S}D_{KL}(\\mu _{i,S} || \\nu _{i,S})$ $\\gamma ^{*} = {\\arg \\!\\max }_{\\gamma \\in [ 0,1 ]} \\widetilde{H}( \\mu _i^{(t)} + \\gamma \\delta _i ) - \\frac{\\lambda n}{2} \\Vert w^{(t)} + \\gamma v_i \\Vert ^{2}$ [H] SDCA for CRF [22] [1] Initialize $\\mu _i^{(0)} \\in \\Pi _C^{\\Delta _C}, \\forall i$ ; $w^{(0)} = \\hat{w}(\\mu ^{(0)}) = \\tfrac{1}{\\lambda n}\\sum _{i}B_i \\mu _i^{(0)}$ , where $B_i $ is the horizontal concatenation of the feature vectors over cliques (Optional) Initialize the duality gaps $g_i = 100, \\forall i$ t=1 : $n_{Ep\\_SCDA}$ Sample $i$ uniformly in {1, ..., n} (Alternatively) Sample $i$ with $P(i) \\propto g_i$ Let $\\nu _{i,C}(y_C) = p( y_C | x_i ; w^{(t)} ) , \\forall C \\in \\mathbb {C}$ (Optional) Let $g_i = \\tilde{D}(\\mu _i || \\nu _i)$  (REF ) Let ascent direction be : $ \\delta _i = \\nu _i - \\mu _i^{t}$ , and the primal direction be : $v_i = \\tfrac{1}{\\lambda n} \\hat{w}(\\delta _i)$ Use line search to get the optimal step size (REF ) Update $\\mu _i^{(t+1)} = \\mu _i^{(t)} + \\gamma ^{*} \\delta _i$ , and $w^{(t+1)} = \\hat{w}(\\mu ^{(t+1)}) = w^{(t)} + \\gamma ^{*} v_i$ Note that as in Section , to do inference for CRF models, we use AD3 [26] for approximate maximum a posteriori (MAP) inference.", "SDCA needs a marginalization oracle, whereas AD3 is used to do an approximate MAP.", "In order to use SDCA with AD3, we propose a simple approximation, using MAP label estimates.", "We first run an approximate MAP inference algorithm and then imagine that the marginals put a unit probability at the approximate MAP solution and zero elsewhere [19], [45], [33].", "This is a heuristic method, but it can be expected to work well when the estimated MAP solution is close to the true MAP, and the conditional distribution is strongly “peaked”." ], [ "Structural SVM - Block-Coordinate Frank-Wolfe", "To train CRFs, we either use a Structured SVM solver using BCFW algorithm [21], or a stochastic optimization algorithm using SDCA [22].", "In this Section, we briefly review SSVM.", "Given a training set of input-output structure pairs $\\lbrace (x_1,y_1), \\ldots , (x_n,y_n) \\rbrace $ $\\in $ $\\mathcal {X} \\times \\mathcal {Y}$ , a loss function $\\Delta ~:~\\mathcal {Y} ~\\times ~\\mathcal {Y}~\\rightarrow \\mathbb {R}$ and a joint feature map $F: \\mathcal {X} \\times \\mathcal {Y} \\rightarrow \\mathbb {R} $ that encodes the input/output information necessary for prediction, we aim to learn $w$ so that the prediction function $f(x) = \\operatornamewithlimits{arg\\,max}_{y} \\langle w, F(x, y) \\rangle $ minimizes the expected loss on future data.", "To do so, we use SSVM predictors derived by maximum margin framework, and enforce correct output to be better than others by a margin: $ \\langle w, F(x_n, y_n) \\rangle \\ge \\Delta (y_n, y) + \\langle w, F(x_n, y) \\rangle ; ~ \\forall ~ y \\in \\mathcal {Y}$ .", "This is a convex optimization problem, but non-differentiable, with many equivalent formulations, resulting in different training algorithms.", "The Frank-Wolfe algorithm, one of such training algorithms, considers the convex minimization problem $\\min _{\\alpha \\in M}f(\\alpha )$ , where $M$ is compact, and $f$ is continuously differentiable.", "It only requires optimizing linear functions over $M$ .", "Since we experiment on large datasets, we use block-coordinate Frank-Wolfe (BCFW) algorithm [20] showing better convergence than the original or the batch version." ], [ "Additional results for the Struct-CKN predictor on the OCR dataset", "The hyperparameters for CKN are the number of layers, the number of filters and the size of patches.", "We use one layer and perform non-exhaustive grid search to set the number of filters (200) and the size of patches (5x5).", "We use cross-validation on the training set to measure the quality of the configuration and use different months for different samples to simulate the more realistic scenario in which we make a forecast over a new period.", "To run SDCA, we use $\\lambda = \\tfrac{1}{n}$ , where $n$ is the number of data points (sequences).", "As described in Algorithm REF , we use the gap sampling strategy: we sample proportionally to our current estimate of the duality gaps, with 80% of uniform sampling.", "Finally, we use the Newton-Raphson algorithm on the derivative of the line search function.", "We train the Struct-CKN-SDCA predictor using different values for $n_{Ep\\_SCDA}$ .", "As shown in Figure REF , we obtain similar results if we use values for $n_{Ep\\_SCDA}$ between 5 and 25.", "Performing a single update of the SDCA parameter before updating the CKN weights makes the learning process difficult and using a value of 50 leads to the optimizer overfitting, making the learning process more difficult afterwards.", "Figure: Struct-ckn With Gap Sampling With Various n Ep_SCDA n_{Ep\\_SCDA} Values, as Indicated by the Number in the LegendFigure: Comparison of Primal Sub-optimality According to the Number of Oracle Calls (Left) or Parameter Updates (Right).", "SDCA Refers to Uniform Sampling.", "SDCA-GAP Refers to Sampling Gap Sampling 80% of the Time.", "SAG-NUS Performs a Line Search at Every Iteration.", "SAG-NUS* Implements a Line-search Skipping Strategy.", "Struct-CKN-BCFW and Struct-CKN-SDCA Use BCFW and SDCA Respectively as Structured Predictors.", "Struct-CKN-SDCA-5 and Struct-CKN-SDCA-10 Use 5 and 10 Epochs Respectively to Train the SDCA Weights for Each Iteration." ], [ "Oracle Calls.", "[34] compared the algorithms based on the number of oracle calls.", "This metric was suitable for the methods they compared.", "Both OEG and SAG-NUS use a line search where they call an oracle on each step.", "SDCA, SDCA-GAP and Struct-CKN-SDCA do not require the oracle to perform their line search.", "However, the oracle is message passing on a junction tree.", "It has a cost proportional to the size of the marginals.", "Each iteration of the line search requires computing the entropy of these marginals, or their derivatives.", "These costs are roughly the same." ], [ "Parameter Updates.", "To give a different perspective, we also report the log of the sub-optimality against the number of parameter updates.", "This removes the additional cost of the line search for all methods." ], [ "Test Error on the OCR Dataset.", "We report in Table REF the test errors on the OCR dataset.", "Table: Test Error on the OCR Dataset" ], [ "Crew Pairing Problem", "This section contains additional details on the Crew Pairing Problem (CPP).", "We first describe the prediction problem formulation and the flight-connection dataset in Sections REF and REF , respectively.", "We describe the construction of the monthly crew pairing, once the ML predictor is trained, in Section REF .", "Then, we present in Section REF in full the optimization process used to solve the CPP, once we obtain the initial solution and initial clusters using the ML predictor.", "Finally, we report additional computational results for the CPP optimization per window and for monthly solution." ], [ "Prediction Problem Formulation", "The classification problem is the following: given the information about an incoming flight in a specific connecting city, choose among all the possible departing flights from this city (which can be restricted to the next 48 hours), the one that the crew should follow.", "These departing flights will be identified by their flight code (approximately 2,000 possible flight codes in the flight-connection dataset).", "Different flights may share the same flight codes in some airline companies, as flights performed multiple times a week usually use the same flight code.", "Nevertheless, a flight is uniquely identified by its flight code, departing city, and day, information that can be derived from the information on the incoming flight and the 48-hour window.", "Each flight gives the following 5 features that we can use in our classification algorithm: city of origin and city of destination ($\\sim $ 200 choices); aircraft type (5 types); duration of flight (in minutes) arrival time (for an incoming flight)or departure time (for a departing flight).", "Given that the aircrew arrived at a specific airport at a given time, we can use a priori knowledge to define which flights are possible.", "For example, it is not possible to make a flight that starts ten minutes after the arrival, nor is it possible five days later.", "Furthermore, it is rare that the type of aircraft changes between flights since each aircrew is formed to use one or two types of aircraft at most.", "The reader is referred to [16] for further details on the likelihood of these scenarios.", "Note that even though we use the aircraft routing in the pre-processing steps, the classifier cannot know which next flight uses the same aircraft, as the prediction performance of the ML predictor is similar whether we use this information as a feature or not.", "The prediction problem is, therefore, not sensitive to this information and can do without it.", "To construct the flight-connection dataset, the following conditions are used, which must always be met for the next flight performed by the crew: The departure time of the next flight should follow the arrival time of the previous flight to the connecting city; The departure time of the next flight should not exceed 48 hours following the arrival time of the previous flight to the connecting city; The departure city of the next flight should be identical to the connecting city in the previous flight; The aircraft type should be the same.", "Indeed, crew scheduling is separable by crew category and aircraft type or family [16].", "For each incoming flight, all the departing flights in the next 48 hours are considered as a set.", "Then, only those that are feasible according to the masking constraints introduced in [40], [39] are considered to filter this set.", "The maximum number of possible flights is limited to 20, as it is sufficient in the airline industry, then the task is to predict the rank of the true label among this set of departing flights." ], [ "Flight-connection Dataset", "Note that the instances in the flight-connection dataset [40] originate from an anonymous major airline and consist of six monthly crew pairing solutions for approximately 200 cities and approximately 50,000 flights per month.", "Each instance contains a one-month flight schedule, engine rotations, and crew pairings, as well as a list of airports and bases.", "This information is used to create input for the learning phase.", "The dataset consists of approximately 1,000 different source-destination pairs, 2,000 different flight codes, and pairings start from 7 different bases.", "As shown in Figure REF , this entry describes a pairing that takes place every Tuesday, Thursday, Friday, and Saturday between August 9, 2018 and September 3, 2018, except for the dates: 08/10, 08/20, and 08/29.", "The pairing contains flights between Buffalo Niagara International Airport (BUF) and O'Hare International Airport (ORD).", "Apart from this, we know the distance between the two airports is 761 Km, from which we can extrapolate the duration of the flight, based on an average aircraft flight speed, which is equivalent to roughly 1 hour and 26 minutes.", "Figure: Data Format for Aircrew PairingsNote that the flight-connection dataset [40] contains an embedded representation of the flights' features and not the original features.", "An advantage of this approach is to anonymize further the dataset, an important condition to release the dataset in public domain.", "The dataset contains the output of the embedding layer, therefore an embedded representation of the input.", "For each incoming flight, the embedding layer (dimension $n_d$ ) is used to construct a feature representation for each of the 20 possible next flights.", "The concatenated matrix ($2D$ matrix) is a $n_d \\times 20$ input, enabling the use of convolutional architecture across time.", "These representations were obtained during the last training epoch with a small tuned learning rate.", "This way, the representation of a city is not unique throughout the whole dataset.", "However, two representations of the same city are still similar.", "Furthermore, note that to use Struct-CKN on the flight-connection dataset, we had to contact the authors of [40] in order to obtain the arcs (the connections between flights).", "This information is not part of the flight-connection dataset accessible in the public domain since disclosing it would easily permit the user to reconstruct the aeronautical map and identify the anonymous airline.", "The reader interested in reproducing our results on the flight-connection dataset is encouraged to contact authors of the paper or us.", "Finally, note that in contrast to the paper where the authors tackle a weekly crew pairing problem (approximately 10,000), and thus where the test set consists of the flights within a period of one week, our manuscript tackles a monthly crew pairing (50,000 flights/month).", "Therefore, although our test set is part of the flight-connection dataset, it is different from the test set considered in [40], [39]." ], [ "From CNN Flight-connection Probabilities to Monthly Crew Pairings {{cite:bbe4ea92692df90bd90b5e677ffa9fed92c36a71}}", "From CNN Flight-connection Probabilities to Monthly Crew Pairings In what follows, we describe the construction of the monthly crew pairing once the CNN predictor is trained.", "As stated in Section REF , we use the same methods and heuristics as in [41] in order to obtain “CNN - initial”.", "Upon the finalization of the flights-connection prediction CNN model training, we can use the same architecture to solve two other prediction problems on the test set (50,000 flights): (i) predict if each of the scheduled flights is the beginning of a pairing or not; and (ii) predict whether each flight is performed after a layover or not.", "In reality, the three predictors share the same representation.", "To solve these independent classification problems, we sum the three prediction problems' cross-entropy losses when learning, therefore performing a multi-output classification." ], [ "Start in a Base.", "While training the CNN predictor to recognize whether a flight is the beginning of a pairing or not, it is possible that it misclassifies that a flight departing from a non-base city is indeed the beginning of a pairing.", "It is imperative to correct such false predictions in order to avoid pairings starting away from the base.", "Even though it is possible to construct a predictor to which only flights departing from the bases are given, it is more efficient and robust to use all flights in the training step.", "That way, the predictor learns a better representation of the input." ], [ "No Layover Below a Threshold.", "While training the CNN predictor to recognize whether there is a layover or not between two flights, and since this decision is independent of finding the next flight, we can use a threshold on the number of hours between the previous and the next flight, below which it does not make sense to make a layover.", "This threshold should be defined considering previous solutions or by a practitioner.", "To build the crew pairing, we use a greedy heuristic to build a crew pairing.", "Specifically, we consider each flight that the model predicts at the beginning of a pairing as a first flight.", "Given this incoming flight, we predict whether the crew is making a layover or not.", "In both cases, we consider the incoming flight and predict the next one.", "The pairing ends when the maximal number of days permitted per pairing is approached.", "We can use the above heuristic to construct a solution for the testing data, obtaining a monthly crew pairing that can be fed as initial clusters to the CPP solver.", "Unfortunately, if one flight in the pairing is poorly predicted, as the flights are predefined, the crew can finish its pairing away from the base.", "To correct the pairings, we define a heuristic in which all pairings where the crews finish away from the base are deleted.", "Because the monthly problem is solved using a windowing approach with one-week windows and two days of overlap period, constructing an initial partition for the entire month and using the subset in each window to feed the solver can be a major flaw.", "Such initial partition will have many inconsistencies with the solution of the previous window, particularly during the overlap period, such as flights belonging to two different clusters.", "We propose to adapt the proposed clusters to the solution of the previous window using the heuristics described above to construct the clusters of the current window in accordance with the solution found for the previous window and any inconsistency with the previous window is avoided, so the proposed partition is adapted to the current resolution.", "In addition, instead of only considering the flights that the model predicts at the beginning of a pairing as a first flight, incomplete clusters from the solution of the previous window starting during the overlap period are completed." ], [ "Optimization Process", "As shown in Figure REF , upon the finalization of training the Struct-CKN predictor weights, we build a monthly solution that we can provide to the optimization process as an initial solution and as initial clusters.", "Indeed, the optimization process requires a standard initial solution that is usually a “cyclic” weekly solution rolled to cover the whole month.", "In this monthly solution, pairings containing flights that have disappeared or have changed hours are broken; new pairings are then built to cover these flights as well, along with new flights that have appeared.", "Instead of using a standard initial solution called “Standard - initial”, we use the Struct-CKN predictor to construct the monthly solution.", "One can use a greedy heuristic to build the crew pairing.", "Specifically, we consider each flight that the model predicts at the beginning of a pairing as a first flight.", "Given this incoming flight, we predict the next one.", "The pairing ends when the maximal number of days permitted per pairing is approached.", "We can use the above heuristic to construct a solution for test data, obtaining a monthly solution that can be fed both as an initial solution and as an initial partition to the solver.", "Unfortunately, if one flight in the pairing is poorly predicted, the crew can finish its pairing away from the base.", "Therefore, for all pairings where crews finish away from the base, we delete all flights performed after the last time that the crew arrives at the base.", "Note that pairings may begin before the bidding period to cover all active flights.", "Therefore, using the Struct-CKN predictor, we propose pairings on an extended period to cover the maximal number of flights.", "This leads to a monthly crew pairing schedule that we consider as an initial solution called “Struct-CKN - initial - extended period”.", "We can also restrict the pairings to be in the bid period.", "This leads to another monthly crew pairing that can also considered as an initial solution called “Struct-CKN - initial - bid period”.", "Note that we only consider “Struct-CKN - initial” in the main paper.", "Then, we freeze the legal sub-part of pairings contained in the initial crew pairings by breaking all illegal pairings.", "Note that pairings starting before the bidding period are considered illegal since we would like to prevent the solver from generating pairings before the bid period starts.", "By breaking all illegal pairings from “Standard - initial”, we obtain a monthly solution called “Standard - feasible”.", "Similarly, by breaking all illegal pairings either from “Struct-CKN - initial - extended period” or “Struct-CKN - initial - bid period”, we obtain the same monthly solution called “Struct-CKN - feasible”.", "To cover the flights that just became uncovered, we generate deadheads on all flights and use the GENCOL solver to optimize with legal sub-blocks imposed covering open (active) flights.", "In this approach, the problem is solved by a “rolling time horizon” approach.", "Because the GENCOL solver is able to solve up to a few thousand flights per window, we are constrained to use two-days windows.", "This means that the month is divided into overlapping time slices of equal length.", "Then, a solution is constructed greedily in chronological order by solving the problem restricted to each time slice sequentially, taking into account the solution of the previous slice through additional constraints.", "The monthly solution obtained is then passed through to the CPP solver.", "The clusters can either be extracted from this solution (as is usually done) or given separately (as is the case in our approach).", "Therefore, by providing an initial solution, we can not only expedite the optimization process, calculate the number of pairings considered legal, the number of flights covered but also propose clusters that are similar to the initial solution, thus reducing the degree of incompatibility between the current solution and the proposed pairings.", "Using the standard approach, we propose “Standard - feasible” as an initial solution to GENCOL and obtain a solution called “GENCOL init.”.", "Similarly, using the Struct-CKN predictor, we propose “Struct-CKN - feasible” as an initial solution to GENCOL and obtain a solution called “Struct-CKN + GENCOL”." ], [ "Additional Results for The Crew Pairing Problem Optimization", "In what follows, we include additional results on the flight-connections dataset, and the characteristics of the monthly solutions constructed using probabilities yielded by different predictors in Table REF and Table REF , respectively.", "By using the monthly crew pairings to start the CPP solver, we report in Table REF and REF the computational results per window for the optimization process, and computation results for the final monthly solution, respectively.", "We report in Table REF the test error on the flight-connection dataset using (1) CNNs and Gaussian Process (GP) to search for the best configuration of hyperparameters, as in [41], (2) a standard CNN-CRF (with non-exhaustive hyperparameter tuning), and (3) our proposed predictor Struct-CKN.", "We obtain a test error of 0.32%, 0.38%, 0.35% and 0.28% using CNN, CNN-CRF, Struct-CKN-BCFW and Struct-CKN-SDCA, respectively.", "Therefore, our predictor outperforms both CNN (using hundreds of iterations of GP) and CNN-CRF.", "Note that although one can use GP to fine-tune hyperparameters, this is not feasible in a real-case usage scenario as practitioners cannot perform GP, each time new data is available.", "Computational results on the feasibility and characteristics of initial solutions are summarized in Table REF .", "We report the number of covered flights, pairings, and the cost of the monthly solution (i.e.", "cost of undercovered flights + cost of overcovered flights + solution cost of the covered flights).", "Note that the cost of the monthly solution is inflated and does not represent the real cost of the solution because the cost of undercovered and overcovered flights is expressed in large values.", "First, note that once we break all illegal pairings in “Standard - initial” , 51% of the pairings and 45% of the covered flights are removed, while we only remove 11% of the pairings and 6% of the covered flights for “Struct-CKN - initial” .", "Table REF reports the mean of computational results per window for all algorithms, namely, GENCOL-DCA, CNN, Struct-CKN and Struct-CKN+.", "For each window and each algorithm, we provide the LP value at the root node of the search tree N0 (LP-N0), computational time at N0 (N0 time), the number of fractional variables (# VF-N0) in the current MP solution at N0, number of branching nodes resolved (# Nodes), best LP value found (Best-LP), pairing cost of the best feasible solution (INT) and total computational time (T time); all times are in seconds.", "While CNN gives better LP-N0, Best-LP, and INT values with an average reduction factor of 14.37%, 14.17% and 11.48% respectively, compared to GENCOL-DCA, Struct-CKN yields better results with a reduction factor of 32.05%, 32% and 30.94% respectively.", "On the other hand, note that whilst CNN has a computational time on average 108% larger than that of GENCOL-DCA, Struct-CKN has a computational time only 40% larger on average.", "This time increase is due to the number of fractional variables at N0 with an increase factor of 81% for CNN and 18% for Struct-CKN, compared to GENCOL-DCA.", "When base constraints are restrictive, the root node solutions contain a more significant number of fractional-valued pairing variables in order to split the worked time between the bases evenly.", "This causes an increase in the number of branching nodes required to obtain a good integer solution.", "While the time increase in CNN is due to the large number of fractional variables at N0, caused by inconsistencies between the initial solution and initial clusters, Struct-CKN has a lower number of nodes and computational time than CNN for two reasons.", "First, the prediction process is not a greedy process, in the sense that the links (connections) between flights are predicted jointly.", "This makes the solution more feasible, as shown in Section REF .", "Second, and unlike NN-based heuristics, we use the proposed pairings not only as initial clusters, but also as an initial solution.", "Next, we compare GENCOL-DCA and Struct-CKN+.", "Struct-CKN+ gives better LP-N0 and Best-LP values for all windows providing an average reduction factor of 56.31% and 55.73% respectively.", "Likewise, Struct-CKN+ gives better feasible solutions with a reduction factor of 56.26%.", "This is explained by the ability of Struct-CKN+ to propose a better initial solution, better than those used as training set for the Struct-CKN predictor.", "Computational results for the monthly solution obtained at the end of the optimization process are reported in Table REF .", "We report the total solution cost, cost of global constraints, and number of deadheads obtained with Commercial-GENCOL-DCA fed by GENCOL init (baseline), CNN, CNN+, Struct-CKN, and Struct-CKN+.", "While CNN reduces the solution cost and the cost of global constraints by 8.52% and 78.11%, Struct-CKN outperforms CNN reducing the solution cost and the cost of global constraints by 9.51% and 80.25%.", "Furthermore, while Struct-CKN reduces the number of deadheads in the solution by 7.76%, note that for CNN, the number of deadheads in the solution is slightly larger with an increase factor of 2.17%, compared to GENCOL-DCA.", "The cost of deadheads is accounted for in the solution cost.", "Because the solution cost is a multi-objective function, We believe that using slightly more deadheads permitted to get better solutions, enhancing both the solution cost and the cost of global constraints.", "The solutions found by Struct-CKN+ present better statistics than GENCOL-DCA, CNN and Struct-CKN, with a reduction factor in solution cost and cost of global constraints of 16.93% and 97.24%, respectively.", "Even more interesting, the number of deadheads used is reduced by 41.23%, compared to GENCOL-DCA, which shows that the Struct-CKN monthly solution can further be optimized and that further research to avoid the windowing approach and use a one-month window can present better results than the current version of Commercial-GENCOL-DCA.", "Table: Test Error on the Flight-connection DatasetTable: Characteristics of Monthly SolutionsTable: Computational Results per WindowTable: Computational Results for Monthly Solutions" ] ]
2105.11646
[ [ "Phase Transitions in Ehrenfest Urns Model with Interactions: Coexistence\n of uniform and non-uniform states" ], [ "Abstract A model based on the classic non-interacting Ehrenfest urn model with two-urns is generalized to $M$ urns with the introduction of interactions for particles within the same urn.", "As the inter-particle interaction strength is varied, phases of different levels of non-uniformity emerge and their stabilities are calculated analytically.", "In particular, coexistence of locally stable uniform and non-uniform phases connected by first-order transition occurs.", "The phase transition threshold and energy barrier can be derived exactly together with the phase diagram obtained analytically.", "These analytic results are further confirmed by Monte Carlo simulations." ], [ "Introduction", "In 1872, when Boltzmann formulated the H-theorem[1] to explain how a system approaches equilibrium from non-equilibrium and the irreversibility associated with the second-law of thermodynamics, it also lead to the microscopic time-reversal and the Poincaré recurrence paradoxes[2], which were not fully understood at that time.", "Decades later, the Ehrenfest two-urn model[3] was proposed in 1907 to resolve the paradoxes and clarify the relationship between reversible microscopic dynamics and irreversible thermodynamics.", "The classic Ehrenfest model[3] considered a total of $N$ particles distributed in two urns with each particle in an urn to be chosen randomly and put into the other with equal probability.", "The Ehrenfest urn model is a simple and tractable model to understand or illustrate the conceptual foundation of statistical mechanics and the relaxation to equilibrium.", "This model was solved exactly by Kac[4] and has been often used to demonstrate the second law of thermodynamics and the approach to equilibrium.", "Later on, the Ehrenfest model was generalized to the case of unbalanced jumping rates between the two urns [5], [6].", "The two-urn Ehrenfest model was subsequently extended to multi-urn systems[7], [8], [9], [10] to investigate the associated non-equilibrium steady-states.", "Its various generalizations have been applied to investigate a variety of non-equilibrium phenomena.", "The continuous time limit of the evolution of the population probability state lead to a linear Fokker- Planck equation[4], [11] which was further modified to incorporate the nonlinear contribution [12], [13], [14], which is motivated by the processes associated with anomalous-diffusion phenomena[15], [16], [17].", "The associated generalized H-theorem for the nonlinear Fokker-Planck equation was also studied[18], [19], [20], [21], [22].", "However, most of such generalization is non-interacting, or the inclusion of interaction is phenomenological and not explicit.", "Until recently, the two-urn Ehrenfest model was extended to included particle interactions inside an urn[23].", "In the two-urn Ehrenfest model with interaction, particles can interact with all other particles inside the same urns, but particles belonging to different urns do not interact.", "In addition, a jumping rate (asymmetric in general) from one urn to another is introduced, which is independent of the particle interaction.", "The system can exhibit interesting phase transitions and the Poincaré cycle and relaxation times can be calculated[23].", "In this paper, we extend the interacting Ehrenfest model to $M$ urns ($M>2$ ).", "In particular, we focus on the equilibrium case when detailed balance can be achieved.", "A possible application for the present equilibrium model and its generalization is the optimization in partitioning problem[24], [25], such as distributing a fixed amount of total resource to $M$ locations with a certain cost to be minimized.", "The equilibrium phase behavior of the model is rather rich and can be investigated in detail.", "Analytic and exact results are derived for the conditions of the emergence of coexistence of uniform or non-uniform phases and the associated first-order phase transition and energy barrier.", "Monte Carlo simulations are also performed to verified our theoretical findings." ], [ "The M-urns model with Interactions", "The two-urn interacting model in [23] is extended to the case of $M$ -urns.", "Similar to the two-urn case[23], $N$ particles are distributed into the $M$ urns ($M\\geqslant 3$ is considered in this paper).", "Pairwise all-to-all interaction is introduced only for particles in the same urn and particles in different urns do not interact.", "Besides particle interactions, direct jumping rates is further introduced between a pair of urns.", "In general these jump rates can be asymmetric (unbalanced) and the system is non-equilibrium with non-zero net particle fluxes.", "On the other hand, if the particles in any urn are free to make transitions back and forth with another urn with balanced jump rates such that detailed balance is obeyed, the system can achieve an equilibrium state.", "In this paper, we will focus on such an equilibrium situation and the associated phase transition.", "The energy or Hamiltonian of the interacting particles in the urns are given by $\\beta {\\cal H}=\\frac{1}{2N}\\sum _{i=1}^M g_i n_i(n_i-1),$ where $\\beta \\equiv 1/(k_B T)$ is the inverse temperature and $g_i$ is the pair-wise interaction (in unit of $k_B T$ ) of the particles inside the $i^{th}$ urn.", "The urns can be thought of as arranged in some periodic lattice, such as a one-dimensional ring, a completely connected network, or in any undirected network such that the jump rates between neighboring urns are balanced.", "Under such conditions, with suitable choice of transition dynamics, such as the Metropolis rule, detailed balance is obeyed and the system can achieve thermal equilibrium with the equilibrium population distribution in the urns being Boltzmann, given by $\\rho _{eq}({\\vec{n}})\\propto \\frac{N!", "}{\\prod _{i=1}^M n_i !}", "e^{-\\beta {\\cal H}}\\propto \\frac{N!", "}{\\prod _{i=1}^M n_i !}", "e^{-\\frac{1}{2N}\\sum _{i=1}^M g_i n_i(n_i-1)},$ where ${\\vec{n}}\\equiv (n_1,\\cdots , n_M)^\\intercal $ .", "The fraction of particles in the $i^{th}$ urn is denoted by $x_i$ , with the constraint $\\sum _{i=1}^Mx_i=1$ .", "In the large $N\\rightarrow \\infty $ limit, using Stirling approximation and with the fraction $x_i\\equiv \\frac{n_i}{N}$ , ${\\vec{x}}\\equiv (x_1,\\cdots , x_{M-1})^\\intercal $ , and $x_M=1-x_1-x_2-\\cdots - x_{M-1}$ , one has $\\rho _{eq}({\\vec{x}})&=&{\\cal N}\\frac{e^{Nf({\\vec{x}})}}{\\sqrt{\\prod _{i=1}^M x_i}}, \\quad \\text{where}\\\\f({\\vec{x}})&=&-\\sum _{i=1}^{M-1} \\left(x_i\\ln x_i+{g_i\\over 2}x_i^2\\right)-\\left(1-\\sum _{i=1}^{M-1} x_i\\right)\\ln \\left( 1-\\sum _{i=1}^{M-1} x_i\\right)-{g_M\\over 2}\\left( 1-\\sum _{i=1}^{M-1} x_i\\right)^2\\\\\\hbox{ and }& &\\quad {\\cal N}^{-1}\\equiv \\int _{\\sum _{i=1}^{M-1}x_i\\leqslant 1} \\prod _{i=1}^{M-1}dx_i \\frac{e^{Nf({\\vec{x}})}}{\\sqrt{\\prod _{i=1}^M x_i}}.$ The saddle-point, ${\\vec{x}}^*$ , is obtained from $\\partial f/\\partial x_\\alpha \\vert _{{\\vec{x}}^*}=0$ , $\\alpha =1,2,\\cdots , M-1$ , which leads to the saddle-point equations $x_i^*e^{g_ix_i^*}&=&\\text{the same constant}, \\qquad i=1,2,\\cdots , M\\\\\\sum _{i=1}^M x_i^*&=&1, \\qquad 0<x_i^*<1.$ Hereafter, unless otherwise stated, we shall consider the case of identical pairwise interactions for all the urns, i.e.", "$g_i=g$ for $i=1,2,\\cdots , M$ ." ], [ "Uniform and Non-uniform Equilibrium states", "Since $g_i=g$ for every urn, the uniform solution of ${\\vec{x}^{(0)}}\\equiv ({1\\over M},\\cdots ,{1\\over M})^\\intercal $ is always a saddle-point solution of (REF ).", "In addition $M$ non-uniform saddle-points (related by symmetry) with different values for $x_i^*$ 's can exist.", "Notice that the saddle-points are also the fixed points in the corresponding dynamical system which describes the general non-equilibrium physics of the system.", "Since the function $xe^{gx}$ is monotonic increasing in the domain $0\\leqslant x\\leqslant 1$ for $g\\geqslant -1$ , all $x_i^*$ satisfying (REF ) can take one possible value and hence only the uniform state is possible.", "On the other hand, the function has one peak in $0\\leqslant x\\leqslant 1$ for $g<-1$ , thus each $x_i^*$ (satisfying (REF ) with $g_i=g$ ) can take one of the two possible values, t allowing the possibility of non-uniform solution in (REF ).", "Therefore, if $n$ urns have the fraction being one of the roots, say $x$ , the other $M-n$ urns will take the fraction $(1-nx)/(M-n)$ .", "Hence one can derive an equation for the saddle-point(s) $x e^{g x}=\\frac{(1-n x)}{M-n}e^{\\frac{g (1-n x)}{M-n}}, \\qquad n=0,1,\\cdots , M-1,$ which can also be written as ${1\\over {x}}=n+(M-n)e^{g\\frac{Mx-1}{M-n}}.$ $n=0$ represents uniform distribution (${\\vec{x}^{(0)}}$ ) of particles in which all $M$ urns have the same fraction of $1/M$ .", "$n$ corresponds to number of urns having the same fraction (say $x$ ) and the other $M-n$ urns having the same fraction of a different value ($\\frac{1-nx}{M-n}$ ).", "Notice that $x=1/M$ is always a solution in (REF ).", "It is also easy to see that if $x$ is root of (REF ) for $n=k\\geqslant 1$ , then $\\frac{1-kx}{M-k}$ is also a root for $n=M-k$ .", "Hence $n$ and $M-n$ have the same saddle-points and it is suffice to consider $k=0,1,\\cdots , \\lfloor {M\\over 2}\\rfloor $ different states, where $k=0$ is the uniform state and the others $k=1,\\cdots , \\lfloor {M\\over 2}\\rfloor $ are non-uniform states with different level of non-uniformity." ], [ "Saddle-node Bifurcations for the non-uniform saddle-points", "Now consider first the simpler case of $M=3$ , take for example $n=2$ in (REF ) with the saddle-point $(x_1,x_2)=(x,x)$ , where $x$ is given by the roots of $xe^{gx}=(1-2x)e^{g(1-2x)}.$ The stability of the saddle-point is determined by the $2\\times 2$ Hessian matrix of $f$ in (REF ) ${\\bf f^{\\prime \\prime }}= -\\begin{pmatrix} 2g+\\frac{1}{x}+\\frac{1}{1-2x} & g+\\frac{1}{1-2x} \\\\ g+\\frac{1}{1-2x} & 2g+\\frac{1}{x}+\\frac{1}{1-2x} \\end{pmatrix}.$ The saddle-point is stable if $\\text{Tr} {\\bf f^{\\prime \\prime }}<0$ and $\\det {\\bf f^{\\prime \\prime }}>0$ , i.e.", "the real part of the two eigenvalues of $ {\\bf f^{\\prime \\prime }}$ are both negative.", "Using (REF ), one can show that the uniform $(x_1,x_2)=(1/3,1/3)$ saddle-point is stable for $g>-3$ .", "On the other hand, careful examination reveals that $x={1\\over 3}$ is always a root in (REF ) and two smaller roots emerges in a pair (one stable and one unstable) for some negative values of $g<g_c$ , characteristics of a saddle-node bifurcation.", "At the bifurcation point $g_c$ can be determined by the condition of emergence of the pair of (stable and unstable) fixed point together with the condition $(xe^{g x})^{\\prime }=((1-2x)e^{g(1-2x)})^{\\prime }.$ $x$ can be eliminated from (REF ) and (REF ), then $g_c$ is simply given by the root of the following transcendental equation: $1-\\sqrt{1+{8\\over {3g}}}=2\\left(1+\\sqrt{1+{8\\over {3g}}}\\right)\\exp [{g\\over 4}(1+3\\sqrt{1+{8\\over {3g}}})],$ which has only a single root of $g_c=-2.74564...$ .", "In fact, for $g<g_c$ three other stable saddle-points related by symmetry emerge in the $x_1$ -$x_2$ phase plane.", "See Fig.", "REF for the Monte Carlo simulation results displaying $\\rho _{eq}(x_1,x_2)$ in the co-existing regime.", "Thus stable non-uniform equilibrium state exists for $g<g_c$ , stable uniform equilibrium state exists for $g>-3$ , and bi-stable coexisting equilibrium states of uniform and non-uniform populations occurs for $-3<g<g_c $ .", "For $M$ urns, the condition of saddle-node bifurcation is obtained by equating the slopes of lhs and rhs of (REF ), and using (REF ) one can derive $1+g x+ n x \\left(\\frac{g}{M-n}+\\frac{1}{1-n x}\\right)=0.$ (REF ) and (REF ) will determine the critical value $g_c(n)$ for the new fixed points to emerge via saddle-node bifurcations.", "For $n=0$ , $x=-1/g$ is the solution of (REF ) and $x={1\\over {2n}}\\left[ 1\\pm \\sqrt{1+\\frac{4n(M-n)}{gM}} \\right] \\quad \\text{ for } n=1,2,\\cdots , M-1.$ The threshold values $g_c$ at which new fixed point solutions emerge can be obtained by substituting the solution for $x$ in (REF ) back to (REF ) to give $g_c(n=0)= -M$ and for $n>0$ , $g_c(n)$ is given by the root of the following transcendental equation $1&+&\\text{sgn}(\\frac{M}{2}-n)\\sqrt{1+\\frac{4 n (M-n)}{g M}}=\\frac{n }{M-n}\\left(1-\\text{sgn}(\\frac{M}{2}-n)\\right)\\sqrt{1+\\frac{4 n (M-n)}{g M}} \\nonumber \\\\&\\times & \\exp \\left[\\frac{g}{M-n} \\left(1-\\frac{M}{2n}\\left(1+\\text{sgn}(\\frac{M}{2}-n)\\sqrt{1+\\frac{4 n (M-n)}{g M}}\\right)\\right)\\right]\\\\& &\\text{ where } \\text{sgn}(x)\\equiv 1 \\text{ for } x\\geqslant 0 \\text{ and } -1 \\text{ for } x< 0.$ Notice that $n=k$ and $n=M-k$ ($k \\geqslant 1$ ) have the same $g_c$ and hence the (non-uniform) fixed points emerge together via saddle-node bifurcation.", "Furthermore, for $n=M/2$ , (i.e.", "even $M$ ), the only solution to (REF ) and (REF ) is $g_c=-M$ and $x=1/M$ , and hence there is no non-uniform fixed point emerge due to saddle-node bifurcation.", "This special non-uniform fixed point emerges at $g_c=-M$ is via pitchfork bifurcation for even $M$ , but careful examination of the Hessian matrix (REF ) reveals that this saddle-point is unstable.", "Thus the number of distinct $g_c$ 's is $\\left\\lfloor {{M}\\over 2}\\right\\rfloor $ , and the number of distinct non-uniform phases (only 1 stable and the rest is unstable as shown below) is $M-1$ ." ], [ "Stability for the saddle-points", "The stability condition of the saddle-point ${{\\vec{x}}^*}$ is determined by the $(M-1)\\times (M-1)$ Hessian matrix $({\\bf f^{\\prime \\prime }})_{\\alpha \\beta }\\equiv \\frac{ \\partial ^2f}{\\partial x_\\alpha \\partial x_\\beta }\\vert _{{\\vec{x}}^*}$ .", "Direct calculations gives $\\frac{ \\partial ^2f}{\\partial x_\\alpha \\partial x_\\beta }\\bigg \\vert _{{\\vec{x}}^*}=-(\\frac{1}{x_M}+g)-(\\frac{1}{x_\\alpha }+g)\\delta _{\\alpha \\beta },\\qquad x_M\\equiv 1-x_1-x_2-\\cdots , x_{M-1}.$ For the uniform saddle-point ${\\vec{x}^{(0)}}\\equiv ({1\\over M},\\cdots ,{1\\over M})^\\intercal $ $\\frac{ \\partial ^2f}{\\partial x_\\alpha \\partial x_\\beta }\\bigg \\vert _{{\\vec{x}^{(0)}}}=-(M+g)(1+\\delta _{\\alpha \\beta })$ whose eigenvalues are $-M(g+M)$ and $-(g+M)$ (with $(M-2)$ degeneracy).", "Thus the uniform phase becomes unstable for $g<-M$ , i.e.", "when the inter-particle attraction is strong enough, the uniform phase becomes unstable.", "For the first non-uniform saddle-point ${\\vec{x}^{(1)}}\\equiv (y,\\cdots ,y)^\\intercal $ , $y\\ne {1\\over M}$ and $y$ is the root of (REF ) with $n=M-1$ or $n=1$ , we have from (REF ) $\\frac{ \\partial ^2f}{\\partial x_\\alpha \\partial x_\\beta }\\bigg \\vert _{{\\vec{x}^{(1)}}}=-(\\frac{1}{1-(M-1)y}+g)-({1\\over y}+g)\\delta _{\\alpha \\beta }$ whose eigenvalues are $-(Mg+\\frac{1}{y[1-(M-1)y]})$ and $-(g+{1\\over y})$ (with $(M-2)$ degeneracy).", "In the case of even $M$ , the non-uniform saddle-point ${\\vec{x}^{({M\\over 2})}}\\equiv (y,\\cdots ,y,\\frac{2}{M}-y,\\cdots , \\frac{2}{M}-y)$ exists, where $y$ is the root in (REF ) with $n={M\\over 2}$ , the Hessian matrix is $\\frac{ \\partial ^2f}{\\partial x_\\alpha \\partial x_\\beta }\\bigg \\vert _{{\\vec{x}^{({M\\over 2})}}}={\\left\\lbrace \\begin{array}{ll}-(\\frac{M}{2-My}+g)-({1\\over y}+g)\\delta _{\\alpha \\beta }, & \\text{if } \\alpha , \\beta \\leqslant {M\\over 2}\\\\-(\\frac{M}{2-My}+g)(1+\\delta _{\\alpha \\beta }) , & \\text{otherwise}.\\end{array}\\right.", "}$ The eigenvalues of (REF ) are $-(\\frac{M}{2-My}+g)$ (with $({M\\over 2}-2)$ degeneracy), $-(g+{1\\over y})$ (with $({M\\over 2}-1)$ degeneracy), and $-{1\\over 2} [M(\\frac{M}{2-My}+g)+ g+{1\\over y} \\pm \\sqrt{M(M-2)(\\frac{M}{2-My}+g)^2+(\\frac{1}{y}+g)^2}]$ .", "The $k^{th}$ $(k\\geqslant 1)$ non-uniform saddle point can be obtained by putting $n=M-k$ in the saddle-point equation (REF ).", "Apart from the uniform saddle-point, there are in general two non-uniform root from (REF ), except for $k={M\\over 2}$ (even $M$ ) in which there is only 1 non-uniform root.", "The eigenvalues of the non-uniform saddle-points can be evaluated as a function of $g$ to reveal the stability of the non-uniform phases (see Appendix for detail calculations).", "Careful examination of the eigenvalues indicated that only one of the first non-uniform phases is stable and all other non-uniform ($k>1$ ) phases always have at least one eigenvalue with a positive real part.", "Fig.", "REF illustrates the results of eigenvalues for the first two non-uniform phases for the case of $M=5$ .", "Only one of the first non-uniform phases has all its eigenvalues negative for all range of $g$ , as depicted in Fig.", "REF a for the case of $M=5$ .", "For the second non-uniform phase, there is always a positive eigenvalue for both saddle-points in the relevant range of $g$ and hence is an unstable non-uniform phase (see Fig.", "REF b).", "Figure: The eigenvalues as a function of gg in a M=5M=5 system for: (a) the first non-uniform state (solid curve with degeneracy M-2M-2 and a non-degenerate one denoted by the dashed curve).", "(b) The second (k=2k=2) non-uniform state (solid curve with degeneracy M-3M-3 and two non-degenerate ones with dashed and dotted curves).", "The two different colors (brown with symbols and blue without symbol) denote the two non-uniform saddle points x + x_+ and x - x_- respectively.", "The value of g c g_c is marked by a vertical dot-dashed line, and the horizontal dotted line marks the zero value.It should be noted that one can also employ a dynamical model of the form $\\frac{d{\\vec{x}}}{dt}={\\vec{A}}({\\vec{x}})$ whose fixed points are identical with the saddle-point of $f({\\vec{x}})$ .", "And the stability of the fixed points deduced from the Jacobian matrix $\\frac{\\partial {\\vec{A}}}{\\partial {\\vec{x}}}\\vert _{{\\vec{x}}^*}$ is the same as obtained from the Hessian matrix $\\frac{ \\partial ^2f}{\\partial x_\\alpha \\partial x_\\beta }\\vert _{{\\vec{x}}^*}$ .", "For equilibrium transition between the coexisting uniform and first non-uniform states as $g$ varies, it is convenient to project onto some line in the phase space and consider the projected equilibrium distribution function $\\hat{\\rho }_{eq}(x)$ parametrized by a single variable $x$ .", "For instance with $M=3$ , one can define $\\hat{\\rho }_{eq}(x_1) =\\int \\rho _{eq}(x_1,x_2)\\delta (x_2-x_1)dx_2\\propto \\frac{e^{Nf( x_1,x_1)}}{\\sqrt{x_1(1-2x_1)}}$ which has two maxima at $1/3$ and ${\\tilde{x}}<1/3$ .", "First-order transition occurs at $g=g_t$ , which is given by $\\frac{\\partial }{\\partial x}\\left( \\frac{e^{Nf( x,x)}}{\\sqrt{x(1-2x)}} \\right)\\bigg \\vert _{\\tilde{x}} &=&0\\\\\\frac{e^{Nf({1\\over 3},{1\\over 3})}}{{1\\over 3}\\sqrt{{1\\over 3}}} &=& \\frac{e^{Nf( {\\tilde{x}},{\\tilde{x}})}}{\\sqrt{{\\tilde{x}}(1-2{\\tilde{x}})}}.$ For $N\\rightarrow \\infty $ , one can solve to get ${\\tilde{x}}={1\\over 6}$ and $g_t=-4\\ln 2=-2.77259...$ At $g=g_t$ , $\\hat{\\rho }_{eq}(x)$ has a local minima at $x={1\\over 4}$ which in turn gives the energy barrier at the transition, $ \\frac{E_b}{N}=\\ln 3-\\frac{19}{12}\\ln 2= 0.00112925...$ In general for $M$ urns, first-order transition occurs at $g=g_t$ which is given by $\\frac{e^{Nf({\\vec{\\tilde{x}}})}}{\\sqrt{{\\tilde{x}}^{M-1}[1-(M-1){\\tilde{x}}]}}&=&\\frac{e^{Nf({\\vec{x}^{(0)}})}}{\\sqrt{\\frac{1}{M^M}}} \\\\\\left.", "\\frac{\\partial }{\\partial x}\\left( \\frac{e^{Nf({\\vec{x}})}}{\\sqrt{x^{M-1}[1-(M-1)x]}} \\right)\\right|_{\\tilde{x}} &=&0 \\qquad {\\tilde{x}}\\ne {1\\over M}$ For $N\\rightarrow \\infty $ , one can solve the above equations to get ${\\tilde{x}}&=& \\frac{1}{M(M-1)}\\\\g_t&=&-\\frac{2 (M-1)}{M-2} \\ln (M-1).$ At $g=g_t$ , one can define $\\hat{\\rho }_{eq}(x)\\equiv \\rho _{eq}(x,\\cdots ,x)$ to characterize the energy barrier.$\\hat{\\rho }_{eq}(x)$ has a local minima at $x={1\\over {2(M-1)}}$ which in turn gives the energy barrier at the transition, $\\frac{E^b}{N}=\\ln {M\\over 2} - \\frac{3 M - 2}{4 M}\\ln (M - 1).$ Fig.", "REF a plots the first-order transition threshold as a function of $M$ , together with $g_c$ at which the first non-uniform phase emerges.", "The characteristic energy barrier at the first-order transition as a function of $M$ is shown in Fig.", "REF b.", "Figure: (a) The threshold g t g_t for first-order transitions between 0↔10\\leftrightarrow 1 (solid curve) and the critical value of gg at which the locally stable first non-uniform phase emerges, g c (n=1)g_c(n=1) (dashed curve), plotted as a function of MM.", "The dotted line denotes g=-Mg=-M at which the uniform phase becomes unstable.", "(b) Energy barrier E b /NE_b/N vs MM for the first-order transitions in (a)." ], [ "Equilibrium Phase Diagram", "As the inter-particle attraction becomes stronger ($g$ becomes more negative), the system undergoes a first-order transition from the uniform phase with the emergence of coexisting a locally stable non-uniform phase at $g=g_c(n=1)$ .", "As $g$ becomes more negative, various other non-uniform phases emerge, albeit not locally stable.", "As $g$ decreases to $g=-M$ , the uniform phase becomes unstable and only the stable first non-uniform phase remains.", "Fig.", "REF displays the phase diagrams for odd ($M=7$ ) and even ($M=8$ ) values of $M$ .", "The values of $g_c$ 's at which various non-uniform phase emerge are calculated analytically.", "The first-order transition point $g_t$ as given by (REF ) is also shown.", "Figure: Phase diagrams for (a) M=7M=7 and (b) M=8M=8 showing various phases.Uniform state is denoted by 0 and various non-uniform states of different degree of non-uniformity are denoted by 1, 2, ..., with decreasing non-uniformity.", "State with a locally stable phase is labeled with a bold font.", "The most non-uniform (k=1k=1) state always has a stable phase.", "The thermal first-order phase transition that occurs at g t g_t is marked by an arrow." ], [ "Monte Carlo Simulations", "To explicitly verify the theoretical results in previous sections, we carry out Monte Carlo simulations for the $M$ urns system.", "In the simulation, a total of $N$ ($N$ is an integer multiple of $M$ ) particles are in the system consisting of $M$ urns and the population of the $i$ urn is denoted by $n_i$ .", "The transition probability that a particle from the $i$ th urn jumps to the $j$ th urn is $T_{i\\rightarrow j}=\\frac{1}{1+e^{-\\frac{g}{N}(n_i-n_j-1)}}.$ It is easy to see that detailed balance is obeyed with the above transition probability and equilibrium will be achieved after sufficient Monte Carlo steps.", "In principle, since we are interested in the equilibrium properties, the urns can be placed on any bidirectional network with balanced jump rates between all connected pair of urns and particle transition rules made to satisfy the detailed balance condition such that there is vanishing net particle flux between every connected pair of urns.", "A particle is chosen in random out of all the particles in the $M$ urns (say the $i^{th}$ urn is chosen) and a transition jump is made according to the probability given in (REF ).", "In practice, for the purpose of investigating equilibrium properties, we put the $M$ urns on a one-dimensional ring for simplicity.", "For urns on a one-dimensional ring, the possible transitions are $j=i\\pm 1$ with equal jump rate to the left and right urns.", "After some long transient time for equilibration, the populations in each urn or the fraction $x_i(t)$ is recorded for a long sampling time.", "Time is in Monte Carlo Steps per particle (MCS/N).", "One MCS/N means that on average every particle has attempted a jump.", "To quantify how non-uniform the state is, we define $\\psi =\\sqrt{\\frac{1}{M(M-1)}\\sum _{i\\ne j}(x_i-x_j)^2}$ as the non-uniformity of the state.", "$\\psi $ can also serve as an order-parameter for the phase transition: $\\psi \\simeq 0$ for the uniform (disordered) state and $\\psi >0$ for the non-uniform (order) state.", "$\\psi $ can be calculated for states of different degree of non-uniformity (labeled by $k$ ) as given by (REF ).", "One can see that from (REF ) that $\\psi $ decreases monotonically with $k$ and thus $k=1$ is the most non-uniform phase.", "Monte Carlo simulations for the 3-urn and 4-urn systems as a function of $g$ were carried out results are shown in Fig.", "REF .", "Fig.", "REF a shows the mean population fraction ($x_1$ ) of one of the 3 urns drops from the uniform value of ${1\\over 3}$ to a smaller value as the inter-particle attraction increases.", "The fluctuation of the population fraction, measured by the variance of $x_1$ also shows a peak across the expected first-order transition point.", "The mean non-uniformity of the system $\\langle \\psi \\rangle $ also increases as $g$ decreases across the transition.", "The analytical non-uniformity of the first non-uniform state $ \\psi ^{(1)}$ is also shown (see Fig.", "REF b).", "For inter-particle attraction stronger than $|g_c|$ (marked by vertical dashed line), the first non-uniform phase emerge coexisting with the uniform state.", "Fig.", "REF c shows the mean population fractions of all the urns as a function of $g$ for the 4-urn system at equilibrium.", "For low attractive strengths, the urns are equally populated with $\\langle x_i\\rangle \\simeq {1\\over 4}$ .", "As the inter-particle attraction increases across the predicted first-order transition point ($g_t=-3\\ln 3=-3.29584$ from (REF ))the populations become inhomogeneous with one urn is more populated and the other three are less but equally populated.", "The mean non-uniformity of the system $\\langle \\psi \\rangle $ also shows a sharp rise as shown in Fig.", "REF d. Figure: Monte Carlo simulation results of the urns model for at equilibrium.", "(a) 3-urns system with N=3000N=3000.", "The mean population and its fluctuation of one of the urns vs. gg.", "The urn with lowest population in the non-uniform state is chosen.", "Solid curve is the theoretical value of the mean population which is obtained from the smallest root of the saddle-point equation ().", "The theoretical first-order transition point is marked by a vertical dotted line.", "The uniform state with population fraction of 1 3{1\\over 3} is marked by a horizontal dot-dashed line.", "10 5 10^5 MCS/N are used in the sampling.", "(b) The mean non-uniformity as a function of gg in (a).", "The theoretical non-uniformity of the first non-uniform state given by () is also shown (solid curve).", "The vertical dashed line marked the theoretical value at which the non-uniform (meta-stable) state emerges.", "(c) 4-urns system with N=1000N=1000.", "The mean populations of the urns vs. gg.", "The theoretical first-order transition point is marked by a vertical dotted line.", "The uniform state with population fraction of 1 4{1\\over 4} is marked by a horizontal dot-dashed line.", "2×10 5 2\\times 10^5 MCS/N are used in the sampling.", "(d) The mean non-uniformity as a function of gg in (c).", "The theoretical non-uniformity of the first non-uniform state given by () is also shown (solid curve).", "The vertical dashed line marked the theoretical value at which the non-uniform (meta-stable) state emerges.For $M=3$ , there are only two independent variables $x_1$ and $x_2$ and the population distribution can be visualized in the two-dimensional density maps shown in Fig.", "REF .", "For $g>g_c$ the population map has a single peak at the uniform state (Fig.", "REF a), and the non-uniform state emerges and coexist as $g\\lesssim g_c$ (Fig.", "REF b).", "As the inter-particle attraction becomes stronger ($g_t<g<g_c$ ) the non-uniform population become more significant (Fig.", "REF c).", "Finally at $g<g_t$ , the uniform state vanishes and only the non-uniform phase remains (Fig.", "REF d).", "Figure: Monte Carlo simulation results for the population distribution map of the 3-urns model with N=3000N=3000 at equilibrium.", "(a) g=-2.7g=-2.7 in the uniform state.", "(b) g=-2.75g=-2.75 and (c) g=-2.8g=-2.8 in the co-existing regime.", "(d) g=-3.1g=-3.1 in the non-uniform state.The uniform phase of x i =1 3x_i={1\\over 3} is denoted by the yellow filled circle, and the non-uniform phase is denoted by filled triangles.The time courses of the population fractions of the 3-urn system above and below the first-order transitions are shown in Fig.", "REF a and REF b respectively.", "For $g\\gtrsim g_t$ the system spends most of the time around the uniform state with occasion hopping to the non-uniform meta-stable phases (Fig.", "REF a).", "On the other hand for $-3<g<g_t$ , the system is predominantly in the non-uniform phase but can hop between the degenerate permutation non-uniform phases in long time scales (Fig.", "REF b).", "The coexistence of the uniform and non-uniform phases is explicitly spelt out in the distribution functions of ach urns.", "As shown in Fig.", "REF c, the system is dominated by the uniform phase with a prominent peak at $x_i={1\\over 3}$ , but the two local peaks from the non-uniform phases are clearly seen.", "For $g<g_t$ , the two peaks of the non-uniform phases grow at the expense of the uniform peak, as shown in Fig.", "REF d. Figure: Monte Carlo simulation results for the time course of the populations in the 3-urns model at equilibrium.", "N=3000N=3000.", "The horizontal dashed line is the uniform state of x i =1 3x_i={1\\over 3}.", "Black (darker) curve shows x 1 x_1 and (grey) curve shows x 2 x_2.", "(a) g=-2.75g=-2.75 (b) g=-2.8g=-2.8 in the co-existing regime.", "Time in Monte Carlo Steps per particle (MCS/N).", "(c) P(x i )P(x_i) for the case in (a).", "10 6 10^6 MCS/N are used.", "(d) P(x i )P(x_i) for the case in (b).", "10 8 10^8 MCS/N are used in order to obtain good statistics." ], [ "Summary and Outlook", "In this paper, the equilibrium properties of the Ehrenfest $M$ -urn model with inter-particle attractions within the same urn is investigated.", "It is shown that phases of different levels of population non-uniformity can exist, but only the uniform and the most non-uniform phases are local stable.", "In addition, these two phases can coexist in a range of attraction strengths whose values can be calculated analytically.", "These two phases are also connected by a first-order transition whose transition interaction strength (Eq.", "(REF )) and energy barrier (Eq.", "(REF )) can be derived explicitly for arbitrary values of $M$ .", "For weak $|g|$ , the system is in the symmetric (uniform) phase with the same mean population $x_i=1/M$ , and for strong $|g|$ , the system is the asymmetric phase, and the only stable asymmetric phase is the ($k=1$ ) most non-uniform state.", "This first-order phase transition is associated with the breaking of $Z_M$ symmetry as $|g|$ is increased.", "The theoretical findings are further verified by Monte Carlo simulations and the agreement is excellent.", "It is remarkable that as the inter-particle attraction increases, the population changes from the entirely uniform state (in which entropy effects dominates) to the case with the emergence of the locally stable most non-uniform $k=1$ state (in which energy dominates), rather than emerging with a less (or least) non-uniform state.", "And when the attraction is increased further, less non-uniform states ($k>1$ ) can emerge, but they are all proved to be unstable.", "As a result, the most non-uniform state persists and remains stable for $g<g_c(n=1)$ due to the domination of the all-to-all inter-particle attractions within the urn over the entropy effects.", "These analytical results and physical picture can enhance our fundamental understanding of equilibrium phase transitions with multi-phase coexistence.", "The present model can be extended to the case in which the particles can possess internal energy levels.", "For instance, suppose that the energy spacing of the energy levels at each urn are the same, and the lowest one being zero.", "Now consider the coupling constant to be negative so that the particles interaction is attractive.", "When the temperature is lowered to zero, $g$ approaches to $-\\infty $ .", "In this case, inside a urn, the occupation will be dominated by its lowest energy level state.", "Because of mutual attraction between particles in the same urn, the total number of particles will be located at the lowest energy level of a specific urn.", "Hence if one generalizes the classical particles to Bosons, also assuming the weak coupling regime and the transition between different urns is classical (no coherence between different urns), then it could possibly lead to Bose condensation in a specific urn.", "Here we focused on the equilibrium behavior in which detailed balance is obeyed.", "But by allowing the jump rates between a pair of urns to be unbalanced, for instance in a one-dimensional ring, the clockwise and anti-clockwise jump rates are $p$ and $q$ respectively with $p>q$ , then a non-equilibrium state with a net clockwise flux results.", "With the particle interaction explicitly imposed in the model, the interplay of energy and entropy can lead to interesting equilibrium and non-equilibrium phase transitions.", "For example, although the less non-uniform states are found to be unstable, it may be plausible to stabilize them if the inter-urn interactions are introduced in a proper way.", "On the other hand, our model can also be extended to other non-equilibrium cases: such as by allowing the particles in the urns be active particles modeled by noise with non-trivial correlations; or the particles are subjected to noises with non-trivial spectrum, then it may lead to additional contributions that could affect the breaking of the ergodicity[26], [27] in the broken symmetry non-uniform states.", "These systems are intrinsically non-equilibrium in nature which is beyond the scope of the present study, but can be investigated in future.", "Finally, we emphasize that the $M$ -urn with interaction model can serve as a new paradigm model to study various non-trivial equilibrium and non-equilibrium statistical mechanics in a more analytically tractable way, including non-equilibrium steady states or even far from equilibrium situations such as oscillations and even complex spatial-temporal patterns.", "These are under our current investigations and the results will be presented in future publications." ], [ "Appendix: Stability calculations for the non-uniform phases", "In this Appendix, we give more details on the definitions of the non-uniform phases and derive their stability conditions.", "The possible phases are given by the roots of $x$ in the saddle-point equation (REF ).", "As discussed in Sec.", "II.A, the function $xe^{gx}$ can have at most two distinct values for $0\\leqslant x \\leqslant 1$ , thus at equilibrium the population fractions can only take at most two possible values for a given value of $g$ .", "Hence we define the $k^{th}$ phase as the particle distributions such that there are $k$ urns with the same occupation fraction, say $y$ , and the rest ($M-k$ ) of the urns having the same population fraction, say $x$ .", "Thus it follows that the $k^{th}$ and phases are the same and it suffices to consider $k=0,1,\\cdots , \\lfloor {M\\over 2}\\rfloor $ possible phases.", "In general $x\\ne y$ and $k\\ne 0$ for the non-uniform phases, otherwise a uniform phase results.", "It would be more intuitive to rewrite (REF ) as $xe^{gx}=ye^{gy}\\\\x=\\frac{1-(M-k)y}{k},$ where the relation between $x$ and $y$ in () simply follows from the requirement that the sum of all population fractions must be unity.", "Since the system possesses permutation symmetry of the $M$ identical urns, one has the freedom to choose the independent coordinates $x_1,x_2,\\cdots ,x_{M-1}$ , i.e.", "freedom to label the urns using distinct labels.", "For actual calculations, we need to choose a convenient labelling.", "For instance, one can choose the $k^{th}$ phase as given by the $M-1$ component vector ${\\vec{x}^{(k)}}= (y,\\cdots ,y,x \\cdots ,x)^\\intercal \\qquad 1\\leqslant k \\leqslant \\left\\lfloor {{M}\\over 2}\\right\\rfloor $ whose first $M-k$ components have the same value $y$ (but $y\\ne {1\\over M}$ ) and the rest $k-1$ components having the same value of $x=\\frac{1-(M-k)y}{k}$ .", "The value of $y$ can be solved by substituting () into (REF ) to give $k y=[1-(M-k)y]e^{{g\\over k}(1-My)}.$ The non-uniformity of the $k^{th}$ phase can be computed from (REF ) to be $\\psi ^{(k)}=\\sqrt{\\frac{2}{M(M-1)}(\\frac{M}{k}-1)} |1-My|.$ In the strong attraction limit, $g\\rightarrow -\\infty $ , (REF ) gives $y\\simeq \\frac{e^{-|g|/k}}{k}\\rightarrow 0$ and $ \\psi ^{(k)}(g\\rightarrow -\\infty )\\simeq \\sqrt{\\frac{2}{M(M-1)}(\\frac{M}{k}-1)}\\left(1-{M\\over k}e^{-|g|/k}\\right)\\rightarrow \\sqrt{\\frac{2}{M(M-1)}(\\frac{M}{k}-1)}$ , which is a decreasing function in $k$ .", "Thus the first non-uniform phase ($k=1$ ) is the most non-uniform state.", "Apart from the uniform saddle-point ${1\\over M}$ , there are in general two non-uniform roots of $y$ from (REF ).", "More insight can be gained by examining on the $x$ -$y$ plane unit square (see Fig.", "REF ) in which the intersection of the curve (REF ) and the line () gives the roots for the saddle-points.", "Consider the case of $x\\ne y$ and $k\\ne 0$ (non-uniform phases) and $g<-1$ , it can be shown[28] that the curve (REF ) always lies outside the square boxes $[0,-{1\\over g}] \\times [0,-{1\\over g}] $ and $[-{1\\over g},1] \\times [-{1\\over g},1] $ , and hence the saddle point must satisfy the condition that one of the $x$ or $y$ is $>-{1\\over g}$ (but not both), and the other one is $<-{1\\over g}$ .", "Figure: Plots of the curve () (for x≠yx\\ne y) and the line ().The two intersections at x=x + x=x_+ and x=x - x=x_- are indicated by filled and open circles respectively.", "The x=-1 gx=-{1\\over g} and y=-1 gy=-{1\\over g} are indicated by dot-dashed lines.The x=yx=y line is indicated by the dotted line.Here we compute the eigenvalues of Hessian matrix at the $k^{th}$ non-uniform saddle-point which is given by the root of the saddle-point equation (REF ).", "The stability condition of the $k^{th}$ phase is determined by the $(M-1)\\times (M-1)$ Hessian matrix from(REF ) and can be computed by choosing saddle-point ${{\\vec{x}}^{(k)}}$ as in (REF ) to give $\\frac{ \\partial ^2f}{\\partial x_\\alpha \\partial x_\\beta }\\bigg \\vert _{{\\vec{x}}^{(k)}}=-\\left[\\frac{1}{x}+g\\right]-\\left[\\frac{1}{y}+g\\right]\\delta _{\\alpha \\beta },$ whose eigenvalues can be solved[29] to give (for $2\\leqslant k\\leqslant \\lfloor {M\\over 2} \\rfloor $ ) $\\lambda _B&\\equiv & -\\left({1\\over y}+g\\right)\\qquad \\text{(with $(M-k-1)$ degeneracy)},\\\\\\lambda _A&\\equiv & -\\left(\\frac{1}{x}+g\\right) \\qquad \\text{(with $(k-2)$ degeneracy)}\\\\& & {1\\over 2}\\left\\lbrace M\\lambda _A +\\lambda _B \\pm \\sqrt{( M\\lambda _A +\\lambda _B)^2-4\\lambda _A[(M-k)\\lambda _A+k\\lambda _B]}\\right\\rbrace .$ These eigenvalues depends on the roots $x$ and $y$ which in turns depend on $g$ .", "Now it is easy to see for $k\\geqslant 3$ , since one of the $x$ or $y$ is $>-{1\\over g}$ and hence either $\\lambda _A$ or $\\lambda _B$ is positive, thus rendering these phases to be always unstable.", "$\\lambda _A$ is absent for $k=2$ , but we can choose another convenient coordinate such as ${\\vec{x}^{(k)}}= (x,x,y,\\cdots ,y)^\\intercal $ and one can compute directly to see that both $\\lambda _A$ and $\\lambda _B$ are eigenvalues and hence the $k=2$ phases are also unstable.", "For $k=1$ , it is convenient to choose the coordinate such that ${\\vec{x}^{(1)}}= (y,\\cdots ,y)^\\intercal $ and $x=1-(M-1)y$ .", "One can compute directly to find the eigenvalues to be $\\lambda _B$ (with $(M-2)$ degeneracy) and $\\Lambda \\equiv (M-1)\\lambda _A+\\lambda _B=-Mg-\\frac{M-1}{x(1-x)}$ , where $\\lambda _A$ and $\\lambda _B$ are given as in () and (REF ).", "For $g<g_c(n=1)\\equiv g_c$ two $k=1$ phases emerges with the corresponding roots $x_+$ and $x_-$ via saddle-node bifurcation, which occurs at $x=x_c$ .", "As $g$ is further decreased, $x_+$ keeps increasing while $x_-$ keeps decreasing.", "As discussed in previous subsection, $\\lambda _B >0 $ if the root $x < -{1\\over g}$ and $\\lambda _B<0$ if $x>-{1\\over g}$ .", "Since the stability also depends on the sign of $\\Lambda $ , we first find out the conditions that $\\Lambda =0$ .", "Vanishing $\\Lambda $ occurs for $x$ satisfying $x=\\frac{1-x}{M-1}e^{-\\frac{1-Mx}{Mx(1-x)}}$ .", "Careful examination of the roots of this equation reveals that there are two roots at $x=x_c$ (the saddle-node bifurcation point at $g=g_c$ ) and at $x=x_-={1\\over M}$ (which occurs at $g=-M$ ).", "The eigenvalues $\\lambda _B$ and $\\Lambda $ evaluated at $x_+$ and $x_-$ determine the stability of these two phases, which are considered for the following two regimes in $g$ ." ], [ "$-M\\leqslant g < g_c$", "We first consider the case of weaker inter-particle attraction $-M\\leqslant g < g_c$ .", "The condition for saddle-node bifurcation give the relation between $g_c$ and $x_c$ : $M-1=-g_c Mx_c(1-x_c)$ , which in turn shows that the eigenvalue $\\Lambda |_{x_c} =0$ at the saddle-node bifurcation point.", "For $g<g_c$ , two roots $x_+ >x_c$ and $x_-<x_c$ emerges, and we will show that the corresponding eigenvalues $\\Lambda |_{x_+} <0$ $\\Lambda |_{x_-} >0$ in this regime of $g$ .", "As $g$ becomes more and more negative, $x_-$ decreases and at $g=-M$ , $x_-={1\\over M}$ and the corresponding eigenvalue $\\Lambda =0$ .", "Since $\\Lambda |_{x_-}=0$ occurs only at $g=g_c$ and $g=-M$ , thus $\\Lambda |_{x_-}$ does not change sign in the $-M\\leqslant g < g_c$ region.", "Similarly, $\\Lambda |_{x_+} $ will not change sign in the $g<g_c$ region.", "We now use perturbation to show that for $g\\lesssim g_c$ , $\\Lambda |_{x_+} <0$ and $\\Lambda |_{x_-} >0$ .", "With $g=g_c-\\epsilon $ and writing $x\\simeq x_c+\\delta $ , expanding the saddle-point equation to leading order in $\\epsilon $ gives $\\delta ^2=\\frac{2x_c^2(1-x_c)^2(Mx_c-1)}{(M-1)(2x_c-1)}{\\epsilon }$ .", "Thus we have $\\Lambda |_{x_\\pm }=\\mp \\frac{2x_c-1}{x_c^2(1-x_c)^2}(x_\\pm -x_c),$ and hence $\\Lambda |_{x_+} <0$ and $\\Lambda |_{x_-} >0$ once the saddle-node bifurcation occurs.", "Since $\\Lambda |_{x_-}$ does not change sign in the regime of $g$ , $x_-$ is unstable.", "For $x_+$ , $\\Lambda |_{x_+}$ also does not change sign and remains $<0$ , also the other eigenvalue $\\lambda _B<0$ (since $x_+>-{1\\over g}$ and $y_+ <{1\\over g}$ ), thus it is stable." ], [ "$ g < -M$", "In this case, $x_-<-{1\\over g}$ and its eigenvalue $\\lambda _B>0$ and this phase is unstable.", "On the other hand, $x_+$ remains $>-{1\\over g}$ and both of its eigenvalues $\\lambda _B<0$ and $\\Lambda |_{x_+} <0$ ensuring that this is a stable phase.", "This work has been supported by Ministry of Science and Technology of Taiwan under the grant no.", "107-2112-M-008-003-MY3, and NCTS of Taiwan." ] ]
2105.11658
[ [ "Towards Understanding the Condensation of Neural Networks at Initial\n Training" ], [ "Abstract Empirical works show that for ReLU neural networks (NNs) with small initialization, input weights of hidden neurons (the input weight of a hidden neuron consists of the weight from its input layer to the hidden neuron and its bias term) condense on isolated orientations.", "The condensation dynamics implies that the training implicitly regularizes a NN towards one with a much smaller effective size.", "In this work, we illustrate the formation of the condensation in multi-layer fully connected NNs and show that the maximal number of condensed orientations in the initial training stage is twice the multiplicity of the activation function, where \"multiplicity\" indicates the multiple roots of activation function at origin.", "Our theoretical analysis confirms experiments for two cases, one is for the activation function of multiplicity one with arbitrary dimension input, which contains many common activation functions, and the other is for the layer with one-dimensional input and arbitrary multiplicity.", "This work makes a step towards understanding how small initialization leads NNs to condensation at the initial training stage." ], [ "Introduction", "Mildly overparameterized neural networks often show good generalization performance on real-world problems by minimizing a loss function without explicit regularizations [1], [2].", "From the perspective of approximation, for over-parameterized NNs, there are infinite possible sets of training parameters that can reach a satisfying training loss.", "However, their generalization performances can be very different.", "It is important to study what implicit regularization is imposed aside to the loss function during the training that leads the NN to a specific type of solutions.", "Recent studies on the training behavior have rendered significant understanding on NNs.", "Empirical works suggest that NNs may learn the data from simple to complex patterns [3], [4], [5], [6], [7], [8].", "For example, an implicit bias of frequency principle (or spectral bias) is widely observed that NNs often learn the target function from low to high frequency [4], [5], [6].", "However, frequency is a coarse-grained characterization of a function without much detail.", "At the initial training stage, the low-frequency function learned by the NNs can be various types, such as low-frequency sinusoidal functions or low-order polynomials.", "[8] find that the initial learning of ReLU NNs may be similar to a linear model (linear w.r.t.", "the input but not parameters) by the measurement of mutual information.", "However, it is still not clear how simple the NN output can be at the initial stage of learning.", "The NN output, either simple or complex, is a collective result of all neurons.", "The study of how neuron weights evolve during the training is central to understanding the collective behavior, including the complexity, of NN output.", "An infinite-width NN in a the well-studied neural tangent kernel (NTK) regime or lazy training regime resembles a random feature model [9], [10], [11], [12], [13], which uses the features given by the initialization to learn training data.", "Thus, NTK analysis cannot show the superiority of NNs over random feature models.", "Both empirical and theoretical studies show that an infinite-width NN with the initialization in the mean-field regime (or rich regime) exhibits highly nonlinear learning dynamics [14], [15], [16], [17].", "[18] establish a phase diagram to study the effect of initialization for two-layer ReLU NN at the infinite-width limit and find that NTK initialization and mean-field initialization are special cases of linear regime and critical regime, respectively.", "Rest to the linear and the critical regime in the phase diagram is a largely unexplored non-linear regime.", "This non-linear regime, where parameters of a NN are initialized towards infinitesimal, is named condensed regime because the features of hidden neurons (orientations of the input weight) condense in several isolated orientations during the training [18].", "The condensation is a feature learning process, which is important to learning of DNNs.", "Note that in the following, condensation is accompanied by a default assumption of small initialization.", "The condensation transforms a large network to a network of only a few effective neurons throughout the training, leading to a simple output function.", "Therefore, the study of condensation could provide insight into how NNs are implicitly regularized to achieve good generalization performance in practice.", "For two-layer ReLU, [19] prove that, as the initialization of parameters goes to zero, the features of hidden neurons condensed at finite number of orientations depending on the input data; when performing a linearly separable classification task with infinite data, [20] show that at mean-field limit, a two-layer infinite-width ReLU NN is effectively a NN of one hidden neuron, i.e., condensation on a single orientation.", "Both work [19], [20] study the condensation behavior for ReLU-NNs at a initial training stage in which the magnitudes of NN parameters are far smaller from well-fitting an $O(1)$ target function.", "However, it still remains unclear that for NNs of more general activation functions, how the condensation emerges at the initial training stage.", "In this work, we show that the condensation at the initial stage is closely related to the multiplicity $p$ at $x=0$ , which means derivative of activation at $x=0$ is zero up to the $(p-1)th$ -order and is non-zero for the $p$ -th order.", "For finite-width two-layer NNs with small initialization at the initial training stage, each hidden neuron's output in a finite domain around 0 can be approximated by a $p$ -th order polynomial and so is the NN output function.", "Based on the $p$ -th order approximation, we prove that the NN at most condenses at $2p$ orientations by experiments and a preliminary theory.", "Therefore, small initialization imposes an implicit regularization that restricts the hypothesis function space to be low-complexity at the initial learning stage.", "This implicit regularization holds regardless of how overparameterized a NN is.", "Thus, the generalization gap of NN at initial stage of training should be small, conforming with previous empirical results [1], [2].", "Our study of initial training behavior lays a solid ground for the further study of the dynamics and implicit regularization of NNs throughout the training." ], [ "Related works", "[3] measure the number of critical training samples which is close to the classification boundary and find that such critical number increases as the training, thus, intuitively suggesting that the classification function may be increasingly complex during the training.", "[4], [5], [6] utilize frequency to quantitatively find that NNs often learn the target function from low to high frequency, which is called frequency principle (or spectral bias).", "Frequency principle shows an implicit bias in the frequency domain during the training that qualitatively explains why NNs work well for the low-frequency dominant problems but badly for the high-frequency dominant problems [6], and further inspired a series of algorithms for fast learning high frequency [6], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34].", "Another research line studies how initialization affects the weight evolution of NNs with a sufficient large or infinite width.", "For example, with an initialization in neural tangent kernel (NTK) regime or lazy training regime (weights change slightly during the training), the gradient flow of infinite-width NN, can be approximated by a linear dynamics of random feature model  [9], [10], [11], [12], [13], whereas for the initialization in the mean-field regime (weights change significantly during the training), the gradient flow of infinite-width NN exhibits highly nonlinear dynamics [35], [15], [16], [17].", "[20] analyze how the dynamics of each parameter transforms from a lazy regime (NTK initialization) to a rich regime (mean-field initialization) for an two-layer infinite-width ReLU NN to perform a linearly separable classification task with infinite data.", "[18] systematically study the effect of initialization for two-layer ReLU NN with infinite width by establishing a phase diagram, which shows three distinct regimes, i.e., linear regime (similar to the lazy regime), critical regime and condensed regime (similar to the rich regime), based on the relative change of input weights as the width approaches infinity, which tends to 0, $O(1)$ and $+\\infty $ , respectively.", "NTK initialization is a specific example of the linear regime, while the mean-field initialization is a specific example of the critical regime, which serves as the boundary between the other two regimes.", "[18] also empirically finds that, in the condensed regime, the features of hidden neurons (orientation of the input weight) condense in several isolated orientations, which is a strong feature learning behavior, an important characteristic of deep learning." ], [ "Preliminary: Two-layer neural networks", "In this work, we study the following two-layer NN $f_{}() = \\sum _{j=1}^{m}a_j \\sigma (_j \\cdot ), $ where $\\sigma (\\cdot )$ is the activation function, $_j=(\\bar{_j},_j)\\in ^{d+1}$ is the neuron feature including the input weight and bias terms, and $= (\\bar{},1)\\in ^{d+1}$ is combination of the input sample and scalar 1, $$ is the set of all parameters.", "For simplicity, we call $_j$ weight and $$ input sample.", "The target function is denoted as $f^{*}()$ .", "The training loss function is mean squared error $R_{S}() = \\frac{1}{2n} \\sum _{i=1}^{n} (f_{}(_i)-f^{*}(_i))^2.$ We consider the gradient flow training, by which dynamics of each neuron $j$ is governed by $\\dot{a_j} =- \\frac{1}{n} g_{}(_j), \\quad \\dot{_j} = - \\frac{a}{n} \\nabla g_{}(_j),$ where $g_{}() := \\sum _{i=1}^{n} (f_{}(_i)-f^{*}(_i))\\sigma (\\cdot _i).$ From the particle point of view, $g_{}()$ serves as potential energy directing the evolution of each feature vector $_j$ .", "To understand the impact of $g_{}()$ given $$ , our analysis focuses on the following dynamics for a test neuron at an arbitrary location $(a,)$ $\\dot{a} =- \\frac{1}{n} g_{}(), \\quad \\dot{} = - \\frac{a}{n} \\nabla g_{}().$ For analysis, we further define $r:=\\Vert \\Vert _{2},\\quad :=/\\Vert \\Vert _{2},$ whose evolution follows $\\dot{r} = \\cdot \\dot{}, \\quad \\dot{} = \\frac{\\dot{} - (\\dot{}\\cdot ) }{r}.$ For convenience, we characterize the activation function by the following definition.", "Definition 1 (multiplicity $p$ ) Suppose that $\\sigma (x)$ satisfies the following condition, there exists a $p\\in $ and $p\\ge 1$ , such that the $k$ -th order derivative $\\sigma ^{(k)}(0)=0$ for $k=1,2,\\cdots ,p-1$ , and $\\sigma ^{(p)}(0)\\ne 0$ , then we say $\\sigma $ has multiplicity $p$ ." ], [ "Initial condensation of input weights", "It is intuitively believed that NNs are powerful at learning data features, which should be an important reason behind the success of deep learning.", "A simple way to define a learned feature of a neuron is by the orientation of its input weights.", "Previous work in [18] show that there is a condensed regime, where the neuron features condense on isolated orientations during the training for two-layer ReLU NNs.", "The condensation implicates that although there are many more neurons than samples, the number of effective neurons, i.e., the number of different used features in fitting, is often much less than samples.", "Therefore, the condensation provides a potential mechanism that helps overparameterized NNs avoid overfitting [1], [2].", "However, it is still unclear how the condensation, for general NNs with small initialization, emerges during the training.", "In this section, we would empirically show how the condensation differs among NNs with activation functions of different multiplicity, followed by theoretical analysis in the next section.", "We consider the evolution of the features in two-layer NN in (REF ), i.e., the director of the input weight, defined by $=/\\Vert \\Vert _2$ .", "Here, we use “director” instead of “orientation” in order not to distinguish the two directions of a line, i.e.", "$$ and $-$ are equivalent.", "In (REF ) the direction of $$ depends on the sign of $a$ .", "Without distinguishing the two opposite directions, we can ignore the effect of $a$ but focus on the evolution of $$ ." ], [ "High-dimensional data", "We first show the condensation at initial training stage in fitting high-dimensional data set.", "Since the input is a high-dimensional vector, the director is also high-dimensional.", "To characterize the condensation, we define the distance (denoted by $D(,)$ ) between the two directors (unit vectors of input weights, denoted by $$ , $$ ) by their inner product, $D(,) = ^.$ The 80 training data in the experiment in Fig.", "REF are sampled from a 5-dimensional function $\\sum _{k=1}^{5} 3.5 x_k \\sin (5x_k+1)$ , where $= (x_1, x_2,\\cdots ,x_5)^^{5}$ .", "As shown in Fig.", "REF (a), the NN with activation function $\\tanh (x)$ condensed at two opposite directions, i.e., one line.", "As the multiplicity increasing, NNs with $x\\tanh (x)$ (Fig.", "REF (b)) and $x^{2}\\tanh {x}$ (Fig.", "REF (c)) condensed at two and three different lines, respectively.", "For $$ , for which the multiplicity definition cannot apply, in Fig.", "REF (d), the NN condenses at three directions, in which two are opposite.", "Therefore, the multiplicity of the activation function may underlie the condensation complexity at initial training.", "Note that if the target function is simpler, directors may condense at fewer directions.", "For example, as shown in Fig.", "REF (a), compared with the high frequency function in Fig.", "REF , we only change the target function to be a simpler function, i.e., $\\sum _{k=1}^{5} 3.5 x_k \\sin (x_k+1)$ , the NN with $x^2 \\tanh (x)$ only condenses at two lines.", "For MNIST data http://yann.lecun.com/exdb/mnist/ in Fig.", "REF (b), we find that, the NN with $x^2 \\tanh (x)$ condenses at one line, which may suggest that MNIST dataset is a simple dataset.", "To understand the mechanism of the initial condensation, we turn to 1-dimensional experiments, which can be clearly visualized in the next subsection.", "Figure: (x)(x)Figure: MNIST" ], [ "1-dimensional data", "For 1-dimensional data, we can visualize the evolution of the NN output and each weight, which would further confirm the connection between the condensation and the multiplicity of the activation function.", "The experiments are set up as follows.", "The training data is 40 points sampled from $\\sin (3x)+\\sin (6x)/2$ , illustrated by green dots in Fig.", "REF .", "We train a two-layer NN of 100 hidden neurons with Adam optimizer, full batch, learning rate $0.0005$ , and mean squared error loss.", "All parameters are initialized by samples following Gaussian distribution $N(0,0.005)$ .", "We display the output at initial training, epoch 1000, for NNs with activation function $\\tanh (x)$ , $x\\tanh (x)$ , $x^{2}\\tanh (x)$ , and $(x)$ in Fig.", "REF .", "Due to the small values of parameters, an activation function with multiplicity $p$ can be well approximated by a $p$ -th order polynominal, thus, the NN output can also be approximated by a $p$ -th order polynominal.", "As shown in Fig.", "REF (a-c), the NN output with activation function $\\tanh (x)$ , $x\\tanh (x)$ and $x^2\\tanh (x)$ overlaps well with the auxiliary of linear, quadratic and cubic polynominal curve, respectively.", "In Fig.", "REF (d), the NN output with ReLU activation function deviates from a linear function (red auxiliary line).", "Particularly, the NN output has several sharp turning points.", "Figure: (x)(x)This experiment, although simple, but convincingly shows that NN does not always learn a linear function at the initial training stage and the complexity of such learning depends on the activation function.", "We then visualize the direction field for input weight $:=(w,b)$ of the dynamics in (REF ) in Fig.", "REF with the fixed $$ at different training steps.", "Note that, in (REF ), $$ and $a$ are independent with $$ , therefore, the dynamics can be understood as the direction field of a test neuron, and by implementing an interested neuron's value to the test neuron, we can obtain the interested neuron's velocity.", "Since the direction of $$ depends on the sign of $a$ , we set $a\\equiv 1$ .", "Therefore, we are only interested in the line to which the director of $$ is parallel.", "Around the original point, the field has one, two, three stables lines, on which a neuron would keep its director, for $\\tanh (x)$ , $x\\tanh (x)$ , and $x^2\\tanh (x)$ , respectively.", "We also display the weight of each neuron at the corresponding training step on the field by the green dots and their velocity direction by the orange arrows.", "Similarly to the high-dimensional case, NN with multiplicity $p$ activation function condenses at $p$ different lines for $p=1,2,3$ .", "As shown in the fourth row in Fig.", "REF , the field and the condensation for NN with $(x)$ is much more complex.", "Taken together, we have empirically show that the multiplicity of the activation function is a key factor that determines the complexity of the initial output and initial condensation.", "Figure: (x)(x), step 400" ], [ "Analysis of the initial condensation of input weights", "In this section, we would present a preliminary analysis to understand how the multiplicity of the activation function affects the initial condensation.", "Suppose the activation function satisfies the multiplicity $p$ , i.e., $\\sigma ^{(k)}(0) = 0$ for $k=1,2,\\cdots , p-1$ , and $\\sigma ^{(p)}(0) \\ne 0$ .", "For convenience, we define $e_i:=f_{}(_i)-f^{*}(_i)$ , an operator $$ satisfying $:=\\dot{} - (\\dot{} \\cdot ),$ where $:=/\\Vert \\Vert _2$ .", "Condensation refers to that the weight evolves following dynamics (REF ) towards a direction that will not change in the direction field and is defined as follows, $\\dot{}=0 \\ \\Leftrightarrow \\ :=\\dot{} - (\\dot{} \\cdot ) = 0.$ Since $\\dot{} \\cdot $ is a scalar, $\\dot{}$ is parallel with $$ .", "$$ is a unit vector, therefore, we have $=\\dot{}/\\Vert \\dot{}\\Vert _2 $ In this work, we consider NNs with sufficiently small parameters.", "For small $r=\\Vert \\Vert _2$ , dynamics (REF ) shows that $$ would moves quickly to its stable direction.", "In the following, we study the case of (i) $p=1$ and (ii) $d=1$ .", "We left other situations for future study." ], [ "Case 1: $p=1$", "Consider $\\sigma ^{\\prime }(0) \\ne 0$ , $e_i=f_{}(_i)-f^{*}(_i)$ fixed and $= o(1)$ .", "By Taylor expansion, $\\overset{\\text{leading order}}{\\approx } := -\\frac{a}{ n}\\sum _{i=1}^{n}\\sigma ^{\\prime }(0)e_i _i + (\\frac{a}{ n}\\sum _{i=1}^{n}\\sigma ^{\\prime }(0)e_i _i \\cdot ) = 0.$ WLOG, we assume $a \\ne 0$ , then $Q= 0 \\ \\Leftrightarrow \\ \\sum _{i=1}^{n}e_i _i = \\sum _{i=1}^{n} e_i (_i \\cdot ) .", "$ We have $= \\frac{\\sum _{i=1}^{n}e_i _i}{{ \\sum _{i=1}^{n}e_i _i}_2}.", "$ This indicates that there is only one solution for $Q= 0$ , which only depends on the training data.", "Therefore, when parameters are sufficiently small, all input weights would converge to the same direction or the opposite direction, i.e., condensation on a line." ], [ "Case 2: $p>1$ , {{formula:95a52ef0-6a51-4277-abdb-162a4d013624}}", "By the definition of the multiplicity $p$ , we have $\\sigma ^{\\prime }(\\cdot _i) = \\frac{\\sigma ^{(p)}(0)}{(p-1)!", "}(\\cdot _i )^{p-1} + o((\\cdot _i )^{p-1}).$ Then up to the leading order in terms of the size of $\\theta $ , we have $\\overset{\\text{leading order}}{\\approx } := -\\frac{a}{ n}\\sum _{i=1}^{n}\\frac{\\sigma ^{(p)}(0)}{(p-1)!", "}(\\cdot _i )^{p-1}e_i _i + (\\frac{a}{ n}\\sum _{i=1}^{n}\\frac{\\sigma ^{(p)}(0)}{(p-1)!", "}(\\cdot _i )^{p-1}e_i _i \\cdot ) .$ Without loss of generality, we also assume $a \\ne 0$ , then, we have $= 0 \\ \\Leftrightarrow \\ = \\frac{\\sum _{i=1}^{n}(\\cdot _i )^{p-1}e_i _i}{\\Vert \\sum _{i=1}^{n}(\\cdot _i )^{p-1}e_i _i\\Vert _{2}}.$ Inner product with $$ for both hand sides, we have $ \\sum _{i=1}^{n}(\\cdot _i )^{p}e_i = { }_2^2 ~ { \\sum _{i=1}^{n}(\\cdot _i )^{p-1}e_i _i}_{2}.$ Since $d+1 =2$ , we denote $= (u_1,u_2)^^2$ and $_i = ((_{i})_{1},(_{i})_{2})^^2$ , then, $\\frac{\\sum _{i=1}^{n}(u_1 (_i)_{1} + u_2 (_i)_{2} )^{p-1}e_i (_i)_{1}}{\\sum _{i=1}^{n}(u_1 (_i)_{1} + u_2 (_i)_{2} )^{p-1}e_i (_i)_{2}} = \\frac{u_1}{u_2} \\triangleq \\hat{u} .$ We obtain the equation for $\\hat{u}$ , $ \\sum _{i=1}^{n}(\\hat{u} (_i)_{1} + (_i)_{2} )^{p-1}e_i (_i)_{1} = \\hat{u} \\sum _{i=1}^{n}(\\hat{u} (_i)_{1} + (_i)_{2} )^{p-1}e_i (_i)_{2}.$ Since it is an univariate $p$ -th order equation, $\\hat{u} = \\frac{u_{1}}{u_{2}}$ has at most $p$ complex roots.", "Because $$ is a unit vector, $$ at most has $p$ pairs of values, in which each pair are opposite.", "Taken together, our preliminary theoretical analysis is consistent with our experiments, that is, the multiplicity of activation function at $x=0$ may underlie the number of directors the NN would condense on when parameters are infinitesimal." ], [ "Discussion", "In this work, we have shown that the characteristic of the activation function, i.e., multiplicity, is a key factor to understanding the complexity of NN output and the weight condensation at initial training.", "The condensation restricts the NN to be effectively low-capacity at the initial training stage, even for finite-width NNs.", "During the training, the NN increases its capacity to better fit the data, leading to a potential explanation for their good generalization in practical problems.", "This work also serves as a starting point for further studying the condensation for multiple-layer neural networks throughout the training process.", "How small the initialization should be in order to see a clear condensation is studied in [18] for two-layer ReLU NNs with infinite width.", "For general activation functions, the regime of the initialization for condensation depends on the NN width.", "A further study of the phase diagram for finite width NNs would be important.", "For general multiplicity with high-dimensional input data, the theoretical analysis for the initial condensation is a very difficult problem, which is equivalent to count the number of the roots of a high-order high-dimensional polynomial with a special structure originated from NNs.", "This work is sponsored by the National Key R&D Program of China Grant No.", "2019YFA0709503 (Z.", "X.", "), the Shanghai Sailing Program, the Natural Science Foundation of Shanghai Grant No.", "20ZR1429000 (Z.", "X.", "), the National Natural Science Foundation of China Grant No.", "62002221 (Z.", "X.", "), Shanghai Municipal of Science and Technology Project Grant No.", "20JC1419500 (Y.Z.", "), Shanghai Municipal of Science and Technology Major Project No.", "2021SHZDZX0102, and the HPC of School of Mathematical Sciences and the Student Innovation Center at Shanghai Jiao Tong University." ], [ "Checklist", " For all authors... Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?", "Did you describe the limitations of your work?", "See Section .", "Did you discuss any potential negative societal impacts of your work?", "Have you read the ethics review guidelines and ensured that your paper conforms to them?", "If you are including theoretical results... Did you state the full set of assumptions of all theoretical results?", "See Section .", "Did you include complete proofs of all theoretical results?", "See Section .", "If you ran experiments... Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?", "In the material.", "Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?", "See Section Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?", "Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?", "the provider information Will be shown in Acknowledgement.", "If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...", "If your work uses existing assets, did you cite the creators?", "See Section REF for MNIST dataset.", "Did you mention the license of the assets?", "The MNIST datatset is well known.", "Did you include any new assets either in the supplemental material or as a URL?", "Did you discuss whether and how consent was obtained from people whose data you're using/curating?", "Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?", "If you used crowdsourcing or conducted research with human subjects... Did you include the full text of instructions given to participants and screenshots, if applicable?", "Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?", "Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?" ] ]
2105.11686
[ [ "The distribution of roots of Ehrhart polynomials for the dual of root\n polytopes of type C" ], [ "Abstract In this paper, we study the Ehrhart polynomial of the dual of the root polytope of type C of dimension $d$, denoted by $C_d^*$.", "We prove that the roots of the Ehrhart polynomial of $C_d^*$ have the same real part $-1/2$, and we also prove that the Ehrhart polynomials of $C_d^*$ for $d=1,2,\\cdots$ has the interlacing property." ], [ "Introduction", "A lattice polytope is a convex polytope in $\\mathbb {R}^d$ all of whose vertices are on the lattice $\\mathbb {Z}^d$ .", "For a $d$ -dimensional lattice polytope $P\\subset \\mathbb {R}^d$ there exists a polynomial in $k$ of degree $d$ , called the Ehrhart polynomial of $P$ , that counts up the total lattice points in the $k$ th dilated polytope of $P$ .", "Note that we write $E_P(k)$ or $|kP\\cap \\mathbb {Z}^d|$ for the Ehrhart polynomial of $P$ .", "A lattice polytope $P$ whose dual polytope $P^{*}:=\\lbrace u\\in \\mathbb {Z}^{d}\\;|\\;\\langle u,v\\rangle \\le 1\\text{ for all } v\\in P\\rbrace $ is also a lattice polytope is called reflexive, where $\\langle \\cdot , \\cdot \\rangle $ denotes the usual inner product.", "It is known that what a lattice polytope $P$ is reflexive is equivalent to the property that the roots of the Ehrhart polynomial of $P$ distribute symmetrically with respect to the line of $\\mathrm {Re}(z)=-1/2$ (see [2]).", "The vertical line $\\mathrm {Re}(z)=-1/2$ is called the canonical line.", "Reflexive polytopes where all roots of their Ehrhart polynomials lie on the canonical line are called a CL-polytope ([4]).", "Such polytopes have been intensively studied in, e.g., [3], [4], [7], [8].", "For two polynomials $f$ of degree $d$ and polynomial $g$ of degree $d-1$ , assume all the roots of $f$ and $g$ lie on the same line $L=\\beta +\\mathbb {R}\\gamma $ on the complex plane $\\mathbb {C}$ with $\\beta ,\\gamma \\in \\mathbb {C}$ .", "Let $\\beta +t_{1}\\gamma ,\\cdots ,\\beta +t_{d}\\gamma $ and $\\beta +s_{1}\\gamma ,\\cdots ,\\beta +s_{d-1}\\gamma $ be the roots of $f$ and $g$ .", "We say $f$ is $L$ -interlaced by $g$ if we have $t_{1}\\le s_{1}\\le t_{2}\\le \\cdots \\le t_{d-1}\\le s_{d-1}\\le t_{d}.$ For a series of polynomials $\\lbrace f_{i}\\rbrace _i$ of degree $i$ , where all the roots of each $f_i$ lie on the same line $L$ , we say the series of polynomials satisfies the interlacing property if $f_{i+1}$ is $L$ -interlaced by $f_{i}$ for all $i$ .", "This interlacing property is the key in [6] for the proof of the property that all the roots have the real part equal to $-1/2$ for the Ehrhart polynomials of several polytopes including $A_{d}$ and $C_{d}$ .", "A root system is a finite set of vectors in $\\mathbb {R}^d$ satisfying certain conditions.", "As is well-known, irreducible root systems are classified as four infinite families of classical root systems of type A, B, C, and D, and five exceptional root systems.", "A root polytope is the convex hull of a root system.", "The root polytope of type A, denoted by $A_{d}$ , is the lattice polytope given as the convex hull of the classical root system of type A: $\\lbrace \\pm (\\mathbf {e}_i+\\cdots +\\mathbf {e}_j)\\; :\\;1\\le i \\le j\\le d\\rbrace ,$ where $\\mathbf {e}_i$ denotes the $i$ th unit vector of $\\mathbb {R}^d$ .", "The Ehrhart polynomial of the root polytope $A_{d}$ is calculated by Bacher, Harpe, and Venkov [1] as follows: $|kA_{d}\\cap \\mathbb {Z}^d|=\\sum _{i=0}^{d}\\binom{d}{i}^2\\binom{k+d-i}{d}.$ For the dual polytope $A_{d}^{*}$ of the root polytope $A_{d}$ , its Ehrhart polynomial is calculated by Higashitani, Kummer, and Michałek [6] as follows: $|kA_{d}^{*}\\cap \\mathbb {Z}^d|=\\sum _{i=0}^{d}\\binom{d+1}{i}k^i.$ Furthermore, they proved that both $A_{d}$ and $A_{d}^{*}$ have the property that all the roots of the Ehrhart polynomial have their real parts equal to $-1/2$ , i.e., those are CL-polytopes.", "See [6], respectively.", "The classical root system C is given by the following set of vectors: $\\lbrace \\pm (\\mathbf {e}_i+\\cdots +\\mathbf {e}_{j-1})\\; :\\;1\\le i<j\\le d\\rbrace \\cup \\lbrace \\pm (2\\mathbf {e}_i+\\cdots +2\\mathbf {e}_{d-1}+\\mathbf {e}_{d})\\; :\\;1\\le i\\le d-1\\rbrace .$ The Ehrhart polynomial of the corresponding root polytope, denoted by $C_{d}$ , is also calculated in [1] as follows: $|kC_{d}\\cap \\mathbb {Z}^d|=\\sum _{i=0}^{d}\\binom{2d}{2i}\\binom{k+d-i}{d},$ and in [6], it is proved that $C_d$ is a CL-polytope.", "In this paper, we consider the dual polytope $C_{d}^{*}$ of the root polytope $C_{d}$ .", "We prove the following theorems.", "Theorem 1.1 For the dual polytope $C_{d}^{*}$ of the root polytope $C_{d}$ , we have $|k C_{d}^{*}\\cap \\mathbb {Z}^d|=(k+1)^{d}+k^d.$ Theorem 1.2 All the roots of the Ehrhart polynomial $|kC_{d}^{*}\\cap \\mathbb {Z}^d|$ have the real part equal to $-1/2$ .", "Namely, $C_d^*$ is a CL-polytope.", "Moreover, the Ehrhart polynomial of $C_d^*$ satisfies the interlacing property.", "We prove Theorem REF in Section , and we prove Theorem REF in Section .", "Remark 1.3 (cf.", "[6]) For root polytopes of other types, as remarked in [6], the root polytope $B_d$ is not reflexive, hence the roots of the Ehrhart polynomial do not distribute symmetrically with respect to the line $\\mathrm {Re}(z)=-1/2$ .", "The root polytope $D_d$ is reflexive and the roots of the Ehrhart polynomial distribute symmetrically with respect to the line $\\mathrm {Re}(z)=-1/2$ , however, some of the roots are not on the line." ], [ "Acknowledgements", "The authors would like to thank Masahiro Hachimori for a lot of his helpful comments on the results.", "The first named author is partially supported by JSPS Grant-in-Aid for Scientists Research (C) 20K03513." ], [ "The proof of Theorem ", "In this section, we give the proof of Theorem $1.1$ .", "To prove the equation in the statement, we use the Ehrhart polynomial of $A_{d}^{*}$ .", "First, deform the Ehrhart polynomial of $A_{d}^{*}$ as follows: $|k A_{d}^{*}\\cap \\mathbb {Z}^d|=\\sum _{i=0}^{d}\\binom{d+1}{i}k^i = \\sum _{i=0}^{d+1}\\binom{d+1}{i}k^i-k^{d+1} =(k+1)^{d+1}-k^{d+1}.\\nonumber $ Here we focus on the boundary of $A_{d}^{*}$ .", "Recall that a lattice polytope $P$ is reflexive if and only if there is no lattice point between $kP$ and $(k-1)P$ ([5]).", "Thus, for $A_{d}^{*}$ , the following equation holds: $|k \\partial A_{d}^{*}\\cap \\mathbb {Z}^d|=|k A_{d}^{*}\\cap \\mathbb {Z}^d|-|(k-1) A_{d}^{*}\\cap \\mathbb {Z}^d|,$ where $\\partial X$ denotes the boundary of $X \\subset \\mathbb {R}^d$ .", "Moreover, $|k \\partial A_{d-1}^{*}\\cap \\mathbb {Z}^d|+2|(k-1) A_{d-1}^{*}\\cap \\mathbb {Z}^d|&=|k A_{d-1}^{*}\\cap \\mathbb {Z}^d|+|(k-1) A_{d-1}^{*}\\cap \\mathbb {Z}^d| \\\\&=(k+1)^d-k^d+k^d-(k-1)^d \\\\&=(k+1)^d-(k-1)^d.$ To prove the equation of the theorem, it is enough to show the following equation: $|k \\partial C_{d}^{*}\\cap \\mathbb {Z}^d|=|k \\partial A_{d-1}^{*}\\cap \\mathbb {Z}^d|+2|(k-1) A_{d-1}^{*}\\cap \\mathbb {Z}^d|\\nonumber .$ In fact, $|k \\partial C_{d}^{*}\\cap \\mathbb {Z}^d|&=(k+1)^{d}-{(k-1)^{d}} \\\\|(k-1) \\partial C_{d}^{*}\\cap \\mathbb {Z}^d|&=k^{d}-{(k-2)^{d}} \\\\|(k-2) \\partial C_{d}^{*}\\cap \\mathbb {Z}^d|&={(k-1)^d}-{(k-3)^d} \\\\&\\vdots \\\\|2 \\partial C_{d}^{*}\\cap \\mathbb {Z}^d|&={3^d}-{1^d} \\\\|1 \\partial C_{d}^{*}\\cap \\mathbb {Z}^d|&={2^d} \\\\|\\lbrace 0\\rbrace \\cap \\mathbb {Z}^d|&={1}.$ Because $C_{d}^{*}$ is also reflexive, the sum of the left-hand sides of the above equations is $|k \\partial C_{d}^{*}\\cap \\mathbb {Z}^d|+|(k-1) \\partial C_{d}^{*}\\cap \\mathbb {Z}^d|+\\cdots +|\\lbrace 0\\rbrace \\cap \\mathbb {Z}^d|=|k C_{d}^{*}\\cap \\mathbb {Z}^d|.$ On the other hand, the sum of the right-hand sides equals to $(k+1)^d+k^d$ .", "Let $f$ be a map $f : k\\partial C_{d}^{*} &\\longrightarrow k\\partial A_{d-1}^{*}\\sqcup (k-1)A_{d-1}^{*}\\sqcup (k-1)A_{d-1}^{*}, \\\\(\\alpha _{1},\\cdots ,\\alpha _{d})&\\mapsto (\\alpha _{1},\\cdots ,\\alpha _{d-1}),$ and let $g$ be a map $g : k\\partial A_{d-1}^{*}\\sqcup (k-1)A_{d-1}^{*}\\sqcup (k-1)A_{d-1}^{*}&\\longrightarrow k\\partial C_{d}^{*}, \\\\(\\alpha _{1},\\cdots ,\\alpha _{d-1})&\\mapsto (\\alpha _{1},\\cdots ,\\alpha _{d}),$ where $\\alpha _{d}$ will be determined suitably later.", "Here, two copies of $(k-1)A_{d-1}^{*}$ are in the range of $f$ .", "(In other view, $f$ maps singly on $k\\partial A_{d-1}^{*}$ and doubly on $(k-1)A_{d-1}^{*}$ .)", "Let $\\mathcal {A}=k\\partial C_{d}^{*}$ and $\\mathcal {B}=k\\partial A_{d-1}^{*}\\sqcup (k-1)A_{d-1}^{*}\\sqcup (k-1)A_{d-1}^{*}$ .", "We show that $g\\circ f=id_\\mathcal {A}$ and $f\\circ g=id_\\mathcal {B}$ , where $\\circ $ is a composition map of the two maps and $id_{X}$ is an identity map on the set $X$ ." ], [ "The definition of $g$ and the proof of {{formula:57517f31-d314-43f5-8a4b-59772c3c42c6}}", "Before proving $f\\circ g=id_\\mathcal {B}$ , we consider how to configure $\\alpha _{d}$ in the map $g$ : $g : k\\partial A_{d-1}^{*}\\sqcup (k-1)A_{d-1}^{*}\\sqcup (k-1)A_{d-1}^{*}&\\longrightarrow k\\partial C_{d}^{*},\\\\g(\\alpha _{1},\\cdots ,\\alpha _{d-1})&\\mapsto (\\alpha _{1},\\cdots ,\\alpha _{d}).$ First we define $\\alpha _{d}$ of $g(\\alpha _{1},\\cdots ,\\alpha _{d-1})$ for $(\\alpha _{1},\\cdots ,\\alpha _{d-1})\\in k\\partial A_{d-1}^{*}$ so that $g(\\alpha _{1},\\cdots ,\\alpha _{d-1})$ $\\in k\\partial C_{d}^{*}$ .", "By definition of $A_d$ , the polytope $A_d^*$ is the set of $(\\alpha _{1},\\cdots ,\\alpha _{d})\\in \\mathbb {R}^d$ satisfying the following inequalities: $|\\alpha _{i}+\\cdots +\\alpha _{j}|\\le 1\\;\\;(1\\le i\\le j \\le d).\\nonumber $ Since the point $(\\alpha _{1},\\cdots ,\\alpha _{d-1})$ lies on the boundary of $k A_{d-1}^{*}$ , $(\\alpha _{1},\\alpha _{2},\\cdots ,\\alpha _{d-1})$ satisfies the following equation: $|\\alpha _{i}+\\cdots +\\alpha _{j}| = k ~~\\mbox{for some}~ i,~j ~\\mbox{with}~ 1\\le i\\le j\\le d-1.$ For $(\\alpha _{1},\\cdots ,\\alpha _{d-1})\\in k \\partial A_{d-1}^{*}$ , we define $\\alpha _{d}$ of $g(\\alpha _{1},\\cdots ,\\alpha _{d-1})$ as follows: (i) If there exist $i$ and $j$ such that $\\alpha _{i}+\\cdots +\\alpha _{j}=k$ , then let $\\alpha _{d}=-k-2(\\alpha _{j+1}+\\cdots +\\alpha _{d-1}).$ (ii) If there exist $i$ and $j$ such that $\\alpha _{i}+\\cdots +\\alpha _{j}=-k$ , then let $\\alpha _{d}=k-2(\\alpha _{j+1}+\\cdots +\\alpha _{d-1}).$ By definition of $C_d$ , the polytope $C_{d}^{*}$ is the set of $(\\alpha _{1},\\cdots ,\\alpha _{d})\\in \\mathbb {R}^{d}$ satisfying the following inequalities: $|\\alpha _{i}|&\\le 1\\;\\; (1\\le i\\le d),\\\\|\\alpha _{i}+\\cdots +\\alpha _{j}|&\\le 1\\;\\; (1\\le i<j\\le d-1), \\mbox{ and} \\\\|2(\\alpha _{i}+\\cdots +\\alpha _{d-1})+\\alpha _{d})|&\\le 1\\;\\; (1\\le i\\le d-1).$ Now we verify that for each case $(\\alpha _{1},\\cdots ,\\alpha _{d})$ lies in $k\\partial C_{d}^{*}$ .", "By definition of $C_d^*$ , we see that $(\\alpha _1,\\cdots ,\\alpha _d) \\in kC_d^*$ if and only if it satisfies the following inequalities: $|\\alpha _{i}|&\\le k\\;\\; (1\\le i\\le d), \\\\|\\alpha _{i}+\\cdots +\\alpha _{j}|&\\le k\\;\\; (1\\le i<j\\le d-1), \\mbox{ and} \\\\|2(\\alpha _{i}+\\cdots +\\alpha _{d-1})+\\alpha _{d})|&\\le k\\;\\; (1\\le i\\le d-1).", "$ Note that the equality of (REF ) holds for some $i,j$ by (REF ).", "Since $(\\alpha _1,\\cdots ,\\alpha _{d-1}) \\in kA_{d-1}^*$ , we see that () holds and (REF ) also holds for $1\\le i\\le d-1$ .", "Hence, what we have to check is the following: $|\\alpha _{d}| &\\le k, \\text{ and} \\\\|2(\\alpha _i+\\cdots +\\alpha _{d-1})+\\alpha _{d}| &\\le k \\;\\;(1\\le i\\le d-1).", "$ We assume the case (i).", "Then we can see that the points on $k\\partial C_{d}^{*}$ satisfy the following three inequalities: ${\\left\\lbrace \\begin{array}{ll}\\alpha _{l}+\\cdots +\\alpha _{i-1}\\le 0 & \\;\\;(l\\le i-1),\\\\\\alpha _{i}+\\cdots +\\alpha _{l}\\ge 0 & \\;\\;(i\\le l\\le j),\\\\\\alpha _{j+1}+\\cdots +\\alpha _{l}\\le 0 & \\;\\;(j+1\\le l).\\end{array}\\right.", "}$ In fact, if there exists $l$ with $\\alpha _{l}+\\cdots +\\alpha _{i-1}>0$ , then $\\alpha _l+\\cdots +\\alpha _{i-1}+\\alpha _i+\\cdots +\\alpha _j>0+k$ , a contradiction to $(\\alpha _1,\\cdots ,\\alpha _{d-1}) \\in k\\partial A_{d-1}^{*}$ .", "We can verify the other equalities in the same way.", "Here, from () and (REF ), we have $-k\\le \\alpha _{j+1}+\\cdots +\\alpha _{d-1}\\le 0$ , which implies that $-k\\le -k-2(\\alpha _{j+1}+\\cdots +\\alpha _{d-1})\\le k$ .", "Therefore, we have $-k\\le \\alpha _{d}\\le k,$ and (REF ) is verified.", "On the other hand, for $l\\le i-1$ , we have $2(&\\alpha _{l}+\\cdots +\\alpha _{d-1})+\\alpha _{d} \\\\&=2(\\alpha _{l}+\\cdots +\\alpha _{i-1})+2k+2(\\alpha _{j+1}+\\cdots +\\alpha _{d-1})-k-2(\\alpha _{j+1}+\\cdots +\\alpha _{d-1}) \\\\&=k+2(\\alpha _{l}+\\cdots +\\alpha _{i-1}),$ and from () and (REF ), we have $-k\\le \\alpha _{l}+\\cdots +\\alpha _{i-1}\\le 0$ , which implies that $-2k\\le 2(\\alpha _{l}+\\cdots +\\alpha _{i-1})\\le 0$ .", "Therefore, we have $-k\\le k+2(\\alpha _{j+1}+\\cdots +\\alpha _{d-1})\\le k,$ and () is verified.", "For the cases that $i\\le l\\le j$ and $j+1\\le l$ , () is similarly verified.", "Hence, we conclude $(\\alpha _{1},\\cdots ,\\alpha _{d})\\in k\\partial C_{d}^{*}$ .", "The discussion for the case (ii) is similar.", "In this case, we use that the points on $k\\partial C_{d}^{*}$ satisfy the following three inequalities: ${\\left\\lbrace \\begin{array}{ll}\\alpha _{l}+\\cdots +\\alpha _{i-1}\\ge 0 & (l\\le i-1),\\\\\\alpha _{i}+\\cdots +\\alpha _{l}\\le 0 & (i\\le l\\le j),\\\\\\alpha _{j+1}+\\cdots +\\alpha _{l}\\ge 0 & (j+1\\le l).\\end{array}\\right.", "}$ We can verify (REF ) and () from these inequalities, and we conclude $(\\alpha _{1},\\cdots ,\\alpha _{d})\\in k\\partial C_{d}^{*}$ .", "Next, we define $\\alpha _{d}$ of $g(\\alpha _{1},\\cdots ,\\alpha _{d-1})$ for $(\\alpha _{1},\\cdots ,\\alpha _{d-1}) \\in (k-1)A_{d-1}^{*}\\sqcup (k-1)A_{d-1}^{*}$ so that $g(\\alpha _{1},\\cdots ,\\alpha _{d-1}) \\in k\\partial C_{d}^{*}$ .", "From the definition of $A_{d}^{*}$ , all $(\\alpha _{1},\\cdots ,\\alpha _{d-1})\\in (k-1)A_{d-1}^{*}$ satisfy the following inequalities: $|\\alpha _{i}+\\cdots +\\alpha _{j}| \\le k-1 ~~(1\\le i\\le j\\le d-1).$ Here, we set $p:=\\max \\lbrace \\alpha _{i}+\\cdots +\\alpha _{d-1}\\;|\\;1\\le i\\le d-1\\rbrace $ and $q:=\\min \\lbrace \\alpha _{i}+\\cdots +\\alpha _{d-1}\\;|\\;1\\le i\\le d-1\\rbrace $ .", "Note that $-k+1\\le p\\le k-1$ , $-k+1\\le q\\le k-1$ , and $p-q\\le k-1$ .", "We set two types of $\\alpha _{d}$ as follows: $\\alpha _{d}^{(1)}={\\left\\lbrace \\begin{array}{ll}k,& \\text{if }p\\le 0,\\\\k-2p,& \\text{if }0<p,\\\\\\end{array}\\right.", "}~~~~~~~~{\\rm and}~~~~~~~\\alpha _{d}^{(2)}={\\left\\lbrace \\begin{array}{ll}-k-2q, & \\text{if }q<0,\\\\-k, & \\text{if }0\\le q.", "\\end{array}\\right.", "}$ Note that each $(\\alpha _{1},\\cdots ,\\alpha _{d-1})\\in (k-1)A_{d-1}^{*}$ has two different $\\alpha _{d}$ 's ($\\alpha _{d}^{(1)}$ and $\\alpha _{d}^{(2)}$ ).", "Now, we verify that each $(\\alpha _{1},\\cdots ,\\alpha _{d})$ satisfies the definitions of $k\\partial C_{d}^{*}$ .", "Note that one of the three inequalities (REF ),() and () must hold with equality since $(\\alpha _{1},\\cdots ,\\alpha _{d})$ lies on the boundary of $kC_{d}^{*}$ .", "In particular, from (REF ) the inequality () does not hold with equality and the inequality (REF ) does not hold with equality for $1\\le i\\le d-1$ , either.", "Thus, we check each $(\\alpha _{1},\\cdots ,\\alpha _{d})$ satisfies either $|\\alpha _{d}|&= k \\text{ or } \\\\|2(\\alpha _i+\\cdots +\\alpha _{d-1})+\\alpha _{d}|&= k. $ When $\\alpha _{d}^{(1)}=k~(p\\le 0)$ , (REF ) clearly holds.", "On the other hand, we have $-k<2(-k+1)+k \\le 2(\\alpha _{i}+\\cdots +\\alpha _{d-1})+\\alpha _{d}\\le 2p+k\\le k.$ Hence, () is verified.", "When $\\alpha _{d}^{(1)}=k-2p~(0<p)$ , () clearly holds by choosing $\\max \\lbrace \\alpha _{i}+\\cdots +\\alpha _{d-1}\\rbrace $ .", "On the other hand, we have $-k<k-2(k-1) \\le k-2p=\\alpha _d <k.$ Hence, (REF ) holds.", "When $\\alpha _{d}^{(2)}=-k-2q~(q<0)$ , () clearly holds by choosing $\\min \\lbrace \\alpha _{i}+\\cdots +\\alpha _{d-1}\\rbrace $ .", "On the other hand, we have $-k<-k-2q=\\alpha _d \\le -k-2(-k+1)<k.$ Hence, (REF ) holds.", "When $\\alpha _{d}^{(2)}=-k~(0\\le q)$ , (REF ) clearly holds.", "On the other hand, we have $-k\\le 2q - k \\le 2(\\alpha _{i}+\\cdots +\\alpha _{d-1})+\\alpha _{d}\\le 2(k-1)-k<k.$ Hence, () is also verified.", "Therefore, we conclude $(\\alpha _{1},\\cdots ,\\alpha _{d})\\in k\\partial C_{d}^{*}$ .", "Note that $\\alpha _{d}^{(1)}$ and $\\alpha _{d}^{(2)}$ do not coincide.", "For example, we consider the situations of $0<p$ and $q<0$ , and assume that there exists only one $\\alpha _{d}$ i.e.", "$\\alpha _{d}^{(1)}=\\alpha _{d}^{(2)}$ .", "Then we have $k-2p=-k-2q$ , so we have $k=p-q\\le k-1$ , a contradiction.", "For the other cases, we can check in the same way.", "Therefore, $\\alpha _{d}^{(1)}$ and $\\alpha _{d}^{(2)}$ are different.", "Finally, by definition of $g$ , we clearly have $f\\circ g=id_\\mathcal {B}$ ." ], [ "The proof of $g\\circ f=id_\\mathcal {A}$", "In this subsection, we show that $f$ and $g$ satisfy the equality $g\\circ f=id_\\mathcal {A}$ .", "Consider $(\\alpha _{1},\\cdots ,\\alpha _{d})\\in k\\partial C_{d}^{*}$ .", "Since it lies on the boundary, one of the inequalities (REF ), () and () holds with equality.", "The map $f$ just removes $\\alpha _{d}$ , so in what follows, we show that the initial $\\alpha _{d}$ equals to the $\\alpha _{d}$ added by the map $g$ .", "First, we consider the case of $(\\alpha _{1},\\cdots ,\\alpha _{d-1})\\in k\\partial A_{d-1}^{*}$ .", "From the definition of $k\\partial A_{d-1}^{*}$ , there exist $i$ and $j$ $(1\\le i\\le j\\le d-1)$ such that $|\\alpha _{i}+\\cdots +\\alpha _{j}|= k.$ Note that from the definition of $k\\partial C_{d}^{*}$ , the inequalities (REF ) and () hold at the same time.", "Assume that $i$ and $j$ satisfy $\\alpha _{i}+\\cdots +\\alpha _{j}=k$ in (REF ).", "From (), we see the following: $2(\\alpha _{i}+\\cdots +\\alpha _{d-1})+\\alpha _{d}&\\le k, \\\\\\alpha _{d}&\\le k-2(\\alpha _{i}+\\cdots +\\alpha _{j})-2(\\alpha _{j+1}+\\cdots +\\alpha _{d-1}), \\\\\\alpha _{d}&\\le -k-2(\\alpha _{j+1}+\\cdots +\\alpha _{d-1}).\\nonumber $ On the other hand, we also have $-k&\\le 2(\\alpha _{j+1}+\\cdots +\\alpha _{d-1})+\\alpha _{d},\\\\-k-2(\\alpha _{j+1}+\\cdots +\\alpha _{d-1})&\\le \\alpha _{d}.$ Thus, $\\alpha _{d}=-k-2(\\alpha _{j+1}+\\cdots +\\alpha _{d-1}).$ Since this equals to $\\alpha _{d}$ of $g(\\alpha _{1},\\cdots ,\\alpha _{d-1})$ , the original $\\alpha _{d}$ is restored by $g$ and we have that $g\\circ f(\\alpha _{1},\\cdots ,\\alpha _{d})=(\\alpha _{1},\\cdots ,\\alpha _{d})$ .", "See (i) in Subsection REF .", "In the case $\\alpha _{i}+\\cdots +\\alpha _{j}=-k$ in (REF ), from (), we have $-k &\\le 2(\\alpha _{i}+\\cdots +\\alpha _{d-1})+\\alpha _{d}, \\\\-k-2(\\alpha _{i}+\\cdots +\\alpha _{j})-2(\\alpha _{j+1}+\\cdots +\\alpha _{d-1}) &\\le \\alpha _{d}, \\\\k-2(\\alpha _{j+1}+\\cdots +\\alpha _{d-1})&\\le \\alpha _{d}.$ On the other hand, we also have $2(\\alpha _{j+1}+\\cdots +\\alpha _{d-1})+\\alpha _{d}&\\le k, \\\\\\alpha _{d}&\\le k-2(\\alpha _{j+1}+\\cdots +\\alpha _{d-1}).$ Thus, $\\alpha _{d}=k-2(\\alpha _{j+1}+\\cdots +\\alpha _{d-1}).$ Hence, we have $g\\circ f(\\alpha _{1},\\cdots ,\\alpha _{d})=(\\alpha _{1},\\cdots ,\\alpha _{d})$ also in this case.", "See (ii) in Subsection REF .", "Next, we consider the case of $(\\alpha _{1},\\cdots ,\\alpha _{d-1})\\in (k-1)A_{d-1}^{*}\\sqcup (k-1)A_{d-1}^{*}$ .", "From the definition of $(k-1)A_{d-1}^{*}$ , there exist $i$ and $j$ such that $|\\alpha _{i}+\\cdots +\\alpha _{j}|\\le k-1~~~(1\\le i\\le j\\le d-1).$ Thus, (REF ) and () are satisfied but not satisfied with equality for $1\\le i\\le d-1$ .", "Hence, $\\text{either }|\\alpha _{d}|=k \\text{ or }|2(\\alpha _{l}+\\cdots +\\alpha _{d-1})+\\alpha _{d}|=k \\;\\text{(for some $1 \\le l \\le d-1$)}$ holds.", "In what follows, we use the same notation on $p$ and $q$ as in Subsection REF .", "Let $\\alpha _{d}=k$ .", "From (), we have $2p+k \\le k$ , i.e., $p \\le 0$ .", "Hence, we have $\\max \\lbrace \\alpha _{i}+\\cdots +\\alpha _{d-1}\\;|\\;1 \\le i \\le d-1\\rbrace \\le 0.$ Therefore, this $\\alpha _{d}=k$ satisfies the condition of $\\alpha _{d}^{(1)}$ in (REF ).", "Let $\\alpha _{d}=-k$ .", "We can show $q \\ge 0$ in the same way.", "Thus, $\\alpha _d=-k$ satisfies the condition of $\\alpha _d^{(2)}$ in (REF ).", "Let $2(\\alpha _{l}+\\cdots +\\alpha _{d-1})+\\alpha _{d}=k$ .", "If there exists $l^{\\prime }$ with $\\alpha _{l}+\\cdots +\\alpha _{d-1}<\\alpha _{l^{\\prime }}+\\cdots +\\alpha _{d-1}$ , since $2(\\alpha _{l^{\\prime }}+\\cdots +\\alpha _{d-1})+\\alpha _{d}\\le k$ from (), we have $2(\\alpha _{l^{\\prime }}+\\cdots +\\alpha _{d-1})+\\alpha _{d} \\le 2(\\alpha _{l}+\\cdots +\\alpha _{d-1})+\\alpha _{d}$ , a contradiction.", "Thus, $\\alpha _{l}+\\cdots +\\alpha _{d-1}=\\max \\lbrace \\alpha _{i}+\\cdots +\\alpha _{d-1}\\;|\\;1 \\le i \\le d-1\\rbrace =p.$ Moreover, from (REF ), we have $k-2p \\le k$ , i.e., $p \\ge 0$ .", "In particular, since $\\alpha _{d}=k$ when $p=0$ , we only need to assume $p>0$ .", "Hence, this $\\alpha _d=k-2(\\alpha _l+\\cdots +\\alpha _{d-1})+\\alpha _d=k-2p$ satisfies the condition of $\\alpha _d^{(1)}$ in (REF ).", "The case where $2(\\alpha _{l}+\\cdots +\\alpha _{d-1})+\\alpha _{d}=-k$ is similar to the above.", "Therefore, each $\\alpha _{d}$ satisfies the condition in (REF ).", "This completes the proof of Theorem  REF" ], [ "The proof of Theorem ", "Let $\\alpha \\in \\mathbb {C}$ be a root of the Ehrhart polynomial $|kC_{d}^{*}\\cap \\mathbb {Z}^d|=(k+1)^d+k^d$ .", "Then we have $\\alpha ^d=-(\\alpha +1)^d \\;\\Longleftrightarrow \\; \\left(\\frac{\\alpha }{\\alpha +1}\\right)^d=-1\\;\\Longleftrightarrow \\;\\frac{\\alpha }{\\alpha +1}=e^{i\\frac{2k-1}{d}\\pi }~~~(k=1,\\cdots ,d~).$ By setting $r=e^{i\\frac{2k-1}{d}\\pi }$ , we have $\\alpha =\\frac{r}{1-r}.$ On the other hand, we have $r=\\cos \\theta _{(d,k)}+i\\sin \\theta _{(d,k)}$ , where we let $\\theta _{(d,k)}=\\frac{2k-1}{d}\\pi $ .", "Hence, $\\alpha &=\\frac{r}{1-r}=\\frac{\\cos \\theta _{(d,k)}+i\\sin \\theta _{(d,k)}}{1-\\cos \\theta _{(d,k)}-i\\sin \\theta _{(d,k)}}=\\frac{\\cos \\theta _{(d,k)}-1+i\\sin \\theta _{(d,k)}}{2(1-\\cos \\theta _{(d,k)})} \\\\&=-\\frac{1}{2}+\\frac{\\sin \\theta _{(d,k)}}{2(1-\\cos \\theta _{(d,k)})}i.$ This completes the proof of the first argument from Theorem REF .", "Moreover, since $\\frac{d}{d\\theta }\\frac{\\sin \\theta }{2(1-\\cos \\theta )}=\\frac{\\cos \\theta -1}{2(1-\\cos \\theta )^2}<0 \\;\\; (0<\\theta <2\\pi ),$ we see that the imaginary part of the roots is a monotonic decreasing function.", "Hence, the interlacing property directly follows from the following relations: $\\frac{1}{d+1}\\pi <\\frac{1}{d}\\pi <\\frac{3}{d+1}\\pi <\\cdots <\\frac{2d-1}{d+1}\\pi <\\frac{2d-1}{d}\\pi <\\frac{2d+1}{d+1}\\pi .$ This completes the proof of the second argument from Theorem REF ." ] ]
2105.11677
[ [ "FNAS: Uncertainty-Aware Fast Neural Architecture Search" ], [ "Abstract Reinforcement learning (RL)-based neural architecture search (NAS) generally guarantees better convergence yet suffers from the requirement of huge computational resources compared with gradient-based approaches, due to the rollout bottleneck -- exhaustive training for each sampled generation on proxy tasks.", "In this paper, we propose a general pipeline to accelerate the convergence of the rollout process as well as the RL process in NAS.", "It is motivated by the interesting observation that both the architecture and the parameter knowledge can be transferred between different experiments and even different tasks.", "We first introduce an uncertainty-aware critic (value function) in Proximal Policy Optimization (PPO) to utilize the architecture knowledge in previous experiments, which stabilizes the training process and reduces the searching time by 4 times.", "Further, an architecture knowledge pool together with a block similarity function is proposed to utilize parameter knowledge and reduces the searching time by 2 times.", "It is the first to introduce block-level weight sharing in RLbased NAS.", "The block similarity function guarantees a 100% hitting ratio with strict fairness.", "Besides, we show that a simply designed off-policy correction factor used in \"replay buffer\" in RL optimization can further reduce half of the searching time.", "Experiments on the Mobile Neural Architecture Search (MNAS) search space show the proposed Fast Neural Architecture Search (FNAS) accelerates standard RL-based NAS process by ~10x (e.g.", "~256 2x2 TPUv2 x days / 20,000 GPU x hour -> 2,000 GPU x hour for MNAS), and guarantees better performance on various vision tasks." ], [ "Introduction", "The architecture of a convolutional neural network (CNN) is crucial for many deep learning tasks such as image classification [31] and object detection [32].", "The widespread use of neural architecture search (NAS) methods such as differentiable, one-shot, evolutional, and RL-based approaches have effectively dealt with architecture design problems.", "Despite having high performance due to its sampling-based mechanism [30], [41], [31], RL-based NAS tends to require unbearable computing resources which discourages the research community from exploring it further.", "The main obstacles to the propagation of RL-based NAS algorithm come from the following two aspects: a) it's necessary to sample a large number of architectures from the search space to ensure the convergence of the RL agent, b) the inevitable training and evaluation cost of these architecture samples on proxy tasks.", "For example, the seminal RL-based NAS [40] approach requires 12,800 generations of architectures.", "The state-of-the-art MNAS [30] and MobileNet-V3 [12] require 8000 or more generations to find the optimal architecture.", "Coupled with $\\sim $ 5 epochs training for each generation, the whole search process costs nearly 64 TPUv2 devices for 96 hours or 20,000 GPU hours on V100 for just one single searching process.", "With no access to reduce the unbearable computational cost, RL-based NAS is hard to make more widespread influence than differential [21], [4], and one-shot based [1], [9] methods.", "On the contrary, the high efficiency of one-shot NAS family brings it continuous research attention.", "Instead of sampling a huge number of sub-networks, one-shot NAS assembles them into a single super-network.", "The parameters are shared between different sub-networks during the training of the super-network.", "In this way, the training process is condensed from training thousands of sub-networks into training a super-network.", "However, this weight sharing strategy may cause problems of inaccurate performance estimation of sub-networks.", "For example, two sub-networks may propagate conflicting gradients to their shared components, which may converge to favor one of the sub-networks and repel the other one randomly.", "This conflicting phenomenon may result in instability of the search process and inferior final architectures, compared with RL-based methods.", "In this work, we aim at combining advantages of both RL-based methods and one-shot methods.", "The proposed method is based on two important key observations: First, the optimal architectures for different tasks have certain common architecture knowledge (similar sub-architectures in different search processes' optimal architectures).", "Second, the parameter knowledge (weights at samples' training checkpoints) can also be transferred across different searching settings and even tasks.", "Based on the two observations, to transfer architecture knowledge, we develop Uncertainty-Aware Critic (UAC) to learn the architecture-performance joint distribution from previous search processes in an unbiased manner, utilizing the transferability of the architecture knowledge, which reduces the needed samples in RL optimization process by 50%.", "For the transferable parameter knowledge, we propose an Architecture Knowledge Pool (AKP) to restore the block-level [30] parameters and fairly share them as new sample architectures' initialization, which speed up each sample's convergence for $\\sim $ 2 times.", "Finally, we also develop an Architecture Experience Buffer (AEB) with an off-policy correctness factor to store the previously trained models for reusing in RL optimization, with half of the search time saved.", "Under the same environment as MNAS [30] with MobileNet-v3 [12], FNAS speeds up the search process by 10$\\times $ and the searched architecture performs even better.", "To summarize, our main contributions are as follows: We propose FNAS, which introduces three acceleration modules, uncertainty-aware critic, architecture knowledge pool, and architecture experience buffer, to speed up reinforcement-learning-based neural architecture search by $\\sim $ 10$\\times $ .", "We show that the knowledge of neural architecture search processes can be transferred, which is utilized to improve sample efficiency of reinforcement learning agent process and training efficiency of each sampled architecture.", "We demonstrate new state-of-the-art accuracy on ImageNet classification, face recognition, and COCO object detection with comparable computational constraints.", "Figure: The pipeline of FNAS.", "The proposed modules are highlighted in orange.", "Architectures are sampled by the RL agent and then passed to Uncertainty-Aware Critic (UAC) for predicting performance and the corresponding uncertainty.", "Then a decision module will determine whether the sample needs to be trained by Trainer.", "The Architecture Knowledge Pool (AKP) helps to initialize new samples for training.", "Half of the samples in one batch come from Architecture Experience Buffer (AEB), the other half come from Trainer or UAC's Value Network." ], [ "Related Works", "From the perspective of how to estimation the performance of architectures, NAS methods can be classified into two categories, sampling-based and weight-sharing-based methods.", "Sampling-based methods generally sample a large number of architectures from the architecture search space and train them independently.", "Based on the evaluated performance of the well-trained sampled architectures, multiple approaches can be utilized to identify the best-performing one, including Bayesian optimization [14], evolutionary algorithm [25], and optimization of an RL agent [40].", "The main drawback of this type of methods is their tremendous time and computational consumption on training the sampled architectures.", "To alleviate this issue, a common practice is to shorten the training epochs and use proxy networks with fewer filters and cells [41], [30].", "Besides, Liu et al.", "[20] proposed to train a network to predict the final performance.", "We also aim to reduce the training cost, by leveraging the accumulated architecture knowledge and parameter knowledge to accelerate the searching process.", "Instead of training many architectures independently, the second type of methods resort to training a super-network and estimate the performance of architectures with shared weights from the super-network [1], [34], [21], [4], [36], [3], [29], [9].", "With the easy access to performance estimation of each sub-architecture, DARTS [21] introduced a gradient-based method to search for the best architecture in an end-to-end manner.", "However, as pointed in [17], the estimated architecture performances based on weight-sharing networks might be unreliable.", "Chen et al.", "[4] proposed to progressively shrink the search space so that the estimation can be gradually more accurate.", "Cai et al.", "[3] introduced a shrinking based method to train the super-network so as to generate networks of different scales without re-training.", "Besides, some existing works have tried to combine these two types of methods and reserve both of their advantages [28], [39].", "BONAS, introduced in  [28], is a sampling-based algorithm that utilizes weight-sharing to evaluate a batch of architectures simultaneously, which reduces the training cost significantly.", "Although weight-sharing in a batch can make the training more fair, the sub-networks in a batch are selected based on Bayesian Optimization method and can interfere each other in the training process, making the estimate of performance unreliable.", "Zhao et al.", "[39] propose to use multiple super-networks to alleviate the undesired co-adaption, which is highly sensitive to the splitting strategy of the search space.", "Cai et al.", "[2] propose to transform the architecture repeatedly in the search process, where the weight of network can be reused to save computational cost.", "In our pipeline, we also propose to share weights between architectures but in a different way.", "We construct a general weight pool with many trained architectures.", "Whenever a new architecture is trained, we initialize the architecture by the trained architectures in the pool.", "In this way, the number of training epochs for the new architecture can be reduced without harming the reliability of performance estimation Figure REF ." ], [ "Preliminary Observation", "RL-based NAS generally consumes quite expensive computing resources.", "MNAS [30] needs to train 8,000 models for training its RL agent until convergence, which costs 20,000 GPU hours on V100.", "Each architecture sample trained for one NAS process would not be used again.", "However, state-of-the-art differentiable-based NAS [21], [35], [4], [34] demonstrated that, with various weight-sharing techniques, the NAS algorithms can be significantly accelerated.", "In this section, we will show that the knowledge of previous searched processes can be reused, which can accelerate the NAS processes." ], [ "Architecture knowledge can be transferred", "Optimal architectures for different tasks have common architecture knowledge.", "It can be observed in many applications that a good network architecture in one task tends to generalize to work well on other tasks.", "An illustrative example of the observation is adopting the pre-trained ImageNet [6] models as the backbone networks for object detection [19], semantic segmentation [18], face recognition [7], etc.", "In NAS, however, this assumption needs to be carefully verified as there exist a huge search space of network architectures Here, We statistically verify whether this observation also holds for NAS.", "Figure: Expectation of each operator of optimal models of face and ImageNet architecture search processes.", "Calculated by the 100 optimal models of face and ImageNet architecture search processes and sorted by the significance of the difference.Figure: On the left, the value function pretrained on face recognition tasks converges much faster.", "On the right, Spearman rank-order correlation along the training process of random initialization and block-level initialization.In Figure REF , we sample 100 optimal architectures of one face recognition search process and one ImageNet classification search process, respectively.", "For each architecture, we firstly expand its tokens to one-hot representation following [22].", "After that, we can compare statistical divergence between the architecture family of face and ImageNet search processes.", "The results are shown in Figure REF .", "Similar conclusions can be obtained that the operators can be divided into two categories, one with large differences and the other with small differences.", "Many previous works [20], [16], [24], [33], [23] use a predictor to predict an architecture's performance to speed up the NAS process.", "However, as the predictor requires thousands of samples to train, they usually evolves in a progressive [20] or semi-supervised manner [23].", "Inspired by the interesting observation above, we implement it in a unified way where different search processes' samples are used together to train a unified value network to map each architecture's one-hot representation [22] to its performance.", "When running a new search experiment, we just use directly the unified network trained by the old data and keep updating it in the new task during the search process, which speeds up the convergence of the value network.", "As shown in Figure REF , when transferring a value network trained on ImageNet to face recognition task, the network converges much faster." ], [ "Parameter knowledge can be transferred", "Initializing the network by ImageNet pre-trained models and training the model on other tasks has generally been a standard way as it can speed up the convergence process.", "However, pretraining has been ignored in NAS as it may break the rank orders of different models.", "In our experiments, we observe that the parameter knowledge can help us to obtain the accurate rank correlations faster than training from scratch.", "Besides, this property holds regardless of the data distribution.", "We randomly sample 50 models and train them on ImageNet in two ways: from scratch or by initializing with parameter knowledge from face recognition models.", "Then, we compare the rank order of validation set performance with the actual rank (i.e.", "fully trained rank) along the training process.", "As shown in Figures REF and REF , with parameter knowledge from face recognition models, we can obtain more accurate rank in fewer epochs.", "Figure: Reward along sample generation between FNAS and MNAS.", "Blue dots are the searching result of MNAS, while green dots are the results of FNAS." ], [ "Uncertainty-Aware Neural Architecture Search", "In this section, we introduce how we utilize the observations above to design three core modules to inherit common architecture knowledge and parameter knowledge from other tasks." ], [ "Architecture search with Reinforcement Learning", "Following [30], [41], we use Proximal Policy Optimization (PPO) [27] to find our Pareto optimal solutions for our multi-objective search problem.", "Concretely, we follow the same idea as [30], [41] and map each sample architecture in the search space to a list of tokens.", "These tokens are determined by a sequence of actions $a_{1:T}$ from the RL agent with policy $\\pi _\\theta $ .", "Our goal is to maximize the expected reward $ \\mathbb {J} = \\mathbb {E}_{P_{(a_{1:T};m)}}R(m)$ where $m$ is a sampled model determined by action $a_{1:T}$ , and $R(m)$ is the reward of $m$ .", "We use the same definition of $R(m)$ of [30] for fair comparison $ R(m) = {{ACC(m)} \\times {\\left[{\\frac{LAT(m)}{T}}\\right]^\\alpha }},$ where $ACC(m)$ is the accuracy on the proxy task, $LAT(m)$ is the latency on target hardware, $T$ is the target latency, and $\\alpha $ is the weight factor.", "Following [30], we use a well known sample-eval-update loop to update the policy $\\pi _\\theta $ .", "At each iteration, $\\pi _\\theta $ firstly generates a batch of samples by predicting a sequence of tokens with its LSTM.", "For each sample $m$ , we train it on the proxy task to obtain $ACC(m)$ and run it on target hardware to obtain $LAT(m)$ .", "$R(m)$ is calculated with Eq.", "(REF ).", "We then update the policy $\\pi _\\theta $ to maximize the expected reward (Eq.", "(REF )) using PPO." ], [ "Uncertainty-Aware Critic in Proximal Policy Optimization (PPO)", "The value function is a common module and is widely used in RL algorithms such as PPO but rarely used in traditional NAS.", "Usually, training a value function requires a large number of samples (e.g., million-level steps in Atari environment trained by rayhttps://github.com/ray-project/rl-experiments), which is unbearable for NAS as it is equivalent to training thousands of models needed to be trained and it's expensive.", "In our algorithm, we propose the Uncertainty-Aware Critic (UAC) to deal with this issue, which is inspired by our observations as mentioned in Section .", "Given an architecture $m$ sampled from the search space, a value network $V$ is utilized to predict the reward $V(m)$ of this sample, while $R(m)$ is the actual reward of it.", "The loss function to update $V$ is formulated as $L_V = |V(m)-R(m)|$ Besides, an uncertainty network $U$ is utilized to predict the uncertainty $U(m)$ of this sampled architecture $m$ , which is used to learn discriminately whether a sample is in the distribution of learned samples.", "The loss function to supervise $U$ is formulated as $L_U = |U(m)-L_V|$ If $U(m)$ is greater than a threshold, the sample may locate in an untrusted region, which indicates that the sampled architecture $m$ has not been effectively learned by the value network.", "As a result, it would be trained on a proxy task from scratch to get its reward $R(m)$ .", "Otherwise, it can be assumed that the prediction $V(m)$ is accurate, and $V(m)$ would be regarded as $R(m)$ to update the RL agent.", "The threshold is set to ensure 2 times speedup while avoiding the risk of over-fitting.", "The whole process is illustrated in Figure REF .", "Samples that need to be trained from scratch are named as untrusted samples, while the remaining ones whose reward comes from $V$ are named as trusted samples.", "With more untrusted samples obtaining their rewards along the search process, the value network becomes more accurate, thus the uncertainty predicted by $U$ gradually decreases.", "Considering an extreme case, where each sample in a batch obtains a reward with low uncertainty and is classified as trusted sample, the agent trained with these samples is likely to overfit, which is not conducive to the exploration of the RL agent and would lead to inferior performance.", "In our implementation, we use the following constraint to balance the exploration and the exploitation of the RL agent to speed up its convergence without over-fitting.", "[leftmargin=2em] Constraint: In each batch, when the number of trusted samples is greater than 50% of the batch size, the extra trusted samples would be thrown away and the architectures would be resampled until enough untrusted samples are obtained to fill the batch.", "With the above constraint, the algorithm can get a decent performance with accelerated search, and the result is shown in Figure REF (with UAC).", "Figure: Different tasks share the same global knowledge pool." ], [ "Uncertainty-Aware Architecture Knowledge Pool", "Parameter knowledge can be transferred among different tasks to speed up the convergence of the training process of the sampled architectures as shown in Section REF .", "However, traditional pretrain is not feasible in NAS, as there are thousands of different architectures in the search space and we can not afford to pretrain each architecture on a different task.", "To address this problem, we propose to initialize each architecture in a factorized way and use a fuzzy matching algorithm to guarantee the hit ratio, which is defined as the division between the number of matched blocks and total queried blocks.", "Following [30], we define an architecture as a combination of $n$ blocks $\\lbrace b_1, b_2, \\dots , b_n\\rbrace $ .", "For any two architectures $m_i$ and $m_j$ , although generally, their structures might be quite different, some of their blocks are similar to each other, (e.g., $b_2$ of $m_i$ == $b_2$ of $m_j$ ), thus the weights of these parts could be shared.", "So we build a Architecture Knowledge Pool (AKP) to store all the previously trained models' blocks in a key-value table, where the key is the expand embedding [22] of each block and the value is the Parameter knowledge of the block.", "Recent research has found that fairness in weight sharing has great influence on the final performance of searched architecture [5].", "So we apply the following two strategies to solve the problem of fairness.", "[leftmargin=2em] The checkpoints stored in the AKP are trained with equal iterations.", "For each block query, the proposed uncertainty function is used to ensure that the match ratio reaches more than 99%, which means less than 1% blocks have been unfairly initialized.", "Given a query block $b_i$ , we calculate the cosine similarity of the expanded embedding as in Section REF between $b_i$ and each element in AKP.", "The block with the highest similarity would be retrieved to initialize $b_i$ .", "We show the overall process in Figure REF .", "Our experiments shows that using AKP created from multiple tasks can speed up the search process by 2$\\times $ (Figure REF ), as shown in Figure REF (with AKP)." ], [ "Architecture Experience Buffer (3AEB)", "In a general RL task, there are a lot of discussions about sample reuses.", "However, in RL-based NAS, improving sampling efficiency is rarely investigated.", "In our algorithm, we propose an architecture experience buffer to store the sampled architectures in the form of architecture-performance pairs, and for each iteration in the future, the stored samples may be used again to update the RL agent to speed up its convergence.", "We call the samples stored in the experience buffer as exploited sample and the newly generated samples as exploring samples.", "Different from the traditional RL works, the proposed experience buffer has the following features: [leftmargin=2em] The buffer size is relatively small (usually 10 in our experiments).", "As the convergence of the RL agent is much faster than RL tasks, if the buffer size is set too large, the agent will focus on exploited samples and the convergence speed would be slow.", "In each batch, both exploited samples and exploring samples would be selected.", "To prevent the RL updating from biasing to the exploited samples, the percentage of the exploited samples in one batch is constrained to no more than 50%.", "Some recent works [26] suggested that the samples of different properties should be selected in the buffer.", "We follow the strategy by choosing samples with different reward values.", "Then, we sample from the buffer and reweight the samples according to their priorities as defined below.", "For each sample $\\lbrace s_1, s_2, \\dots , s_n\\rbrace $ with their rewards $R \\lbrace r_1, r_2 \\dots , r_n\\rbrace $ in AEB, their priority scores are defined as $P_i = \\frac{exp(r_i)}{\\sum _{j}^{ }exp(r_j)}$ Following [26], each sample would also be reweighted by their importance sampling weights.", "The reweighted priority scores $S$ can be written as $S_i = (N \\cdot P_i)^{-\\beta }$ , where $N$ is the buffer size and $\\beta $ is the annealing coefficient and would be increased from 0 to 1 as the search proceeds.", "And the result is shown in Figure REF (with AEB).", "Table: Performance Results on ImageNet Classification.", "FNAS-Image×\\times 1.3 means scale up FNAS-Image for 1.3×\\times along width." ], [ "Fast Neural Architecture Search (FNAS) on vision tasks", "In this section, we conduct different experiments on both ImageNet and million-level face recognition tasks to verify the effectiveness of FNAS.", "The details and results are as follows:" ], [ "Implementation details", "Following the standard searching algorithm as NASNet [41], MNAS [30] and AKD [22], we use an RNN-based agent optimized by PPO algorithm [27].", "The RL agent is implemented with a one-layer LSTM [11] with 100 hidden units at each layer.", "The $V$ and $U$ of UAC are implemented with four-layer MLP with 200 hidden units at each layer and PReLU [10] nonlinearity.", "For ImageNet experiments, we sample 50K images from the training set to form the mini-val set and use the rest as the mini-training set.", "In each experiment, 8K models are sampled to update the RL agent.", "Note that when equipped with UAC or AEB, not all samples need to be activated, as many samples' rewards are directly returned from these two modules.", "For face experiments, we use MS1M [8] as the mini-training set, LFW [13] as the mini-val set.", "The final performance is evaluated on MegaFace [15].", "Table: Performance on COCO." ], [ "Proxyless FNAS on ImageNet", "Just as MNAS [30] has done, we also use a multi-objective reward to directly search on ImageNet.", "After the search process, we retrain the top 10 models with the largest reward near the target Mult-Adds from scratch to verify the search results.", "In Table REF , we get a relatively higher result than the current SOTA network MBv3 [12].", "Note that the model we search does not go through the pruning operation NetAdapt [37], which can reduce 10%$\\sim $ 15% computation and keep performance nearly unchanged.", "Compared with EfficientNetB0 [31], FNAS improves top 1 accuracy by 1 point under comparable computation budget.", "And still, there is nearly 10$\\times $ of acceleration in the entire search process compared to MNAS [30] or MBv3 [12]." ], [ "Proxyless FNAS on fine-grained facial recognition", "Besides verifying the performance of FNAS on ImageNet, we also test it on the fine-grained facial recognition task.", "As can be seen in Table REF , compared with MBv3, verification accuracy improves 2 points in comparable Mult-Adds under 1e6 distractors.", "When compared with MBv2, FNAS improves verification accuracy for nearly 4 points with 24% Mult-Adds reduction.", "The result shows: 1) FNAS has an obvious acceleration effect on different tasks and 2) the importance of searching directly on the target task." ], [ "Transferability on object detection", "We combine the model found on ImageNet in Table REF with the latest pipeline of detection to verify its generalization.", "Table REF shows the performance of the model on COCO [19].", "It can be seen that compared to MBv3, there is a significant improvement with our searched model.", "Table: The effectiveness of the three proposed modules, MBv2×\\times 0.38 means scale up MBv2 for 0.38×\\times along widthTable: Transferability of UAC and AKP" ], [ "The effectiveness of the three proposed modules.", "In this section, the effectiveness of Uncertainty-Aware Critic (UAC), Architecture Knowledge Pool (AKP), Architecture Experience Buffer (AEB) is verified when they are used alone or combined.", "Details are shown in Table REF .", "Three conclusions can be observed: 1.", "Sampling with AKP initialization gets real rank faster; 2.", "Fewer samples are required when NAS is equipped with UAC and AEB; and 3.", "10$\\times $ speedup can be achieved when NAS is equipped with AKP, UAC, and AEB." ], [ "The transferability of the proposed modules.", "In Section , we mentioned that knowledge between NAS processes is transferable, which is also verified in the experiment.", "We use the UAC trained on the face as a pre-trained model and then transfer it to the ImageNet architecture search process.", "In the absence of 3/4 of activated samples, the optimal model surpasses baseline by 0.67% with fewer Mult-Adds, showing in Table REF .", "In addition, we use AKP with the checkpoints from face architecture search process and then search on ImageNet.", "In the absence of 1/2 activated samples, performance increases by 0.6%." ], [ "Conclusion", "This paper proposes three modules (UAC, AKP, AEB) to speed up the entire running process of RL-based NAS, which consumes large amounts of computing power before.", "With these modules, fewer samples and less training computing resources are needed, making the overall search process 10$\\times $ faster.", "We also show the effectiveness of applying those modules on different tasks such as ImageNet, face recognition, and object detection.", "More importantly, the transferability of UAC and AKP is being tested by our observation and experiments, which will guide us in tapping the knowledge of the NAS process." ] ]
2105.11694
[ [ "orvara: An Efficient Code to Fit Orbits using Radial Velocity, Absolute,\n and/or Relative Astrometry" ], [ "Abstract We present an open-source Python package, Orbits from Radial Velocity, Absolute, and/or Relative Astrometry (orvara), to fit Keplerian orbits to any combination of radial velocity, relative astrometry, and absolute astrometry data from the Hipparcos-Gaia Catalog of Accelerations.", "By combining these three data types, one can measure precise masses and sometimes orbital parameters even when the observations cover a small fraction of an orbit.", "orvara achieves its computational performance with an eccentric anomaly solver five to ten times faster than commonly used approaches, low-level memory management to avoid python overheads, and by analytically marginalizing out parallax, barycenter proper motion, and the instrument-specific radial velocity zero points.", "Through its integration with the Hipparcos and Gaia intermediate astrometry package htof, orvara can properly account for the epoch astrometry measurements of Hipparcos and the measurement times and scan angles of individual Gaia epochs.", "We configure orvara with modifiable .ini configuration files tailored to any specific stellar or planetary system.", "We demonstrate orvara with a case study application to a recently discovered white dwarf/main sequence (WD/MS) system, HD 159062.", "By adding absolute astrometry to literature RV and relative astrometry data, our comprehensive MCMC analysis improves the precision of HD 159062B's mass by more than an order of magnitude to $0.6083^{+0.0083}_{-0.0073}\\,M_\\odot$.", "We also derive a low eccentricity and large semimajor axis, establishing HD 159062AB as a system that did not experience Roche lobe overflow." ], [ "Introduction", "The history of orbit fitting extends back to ancient times, when it culminated in the Ptolemataic model of the Solar system.", "During the Age of Enlightenment astronomers fit the first Keplerian orbits to visual binary stars.", "More recently, masses derived from orbital fits, known as dynamical masses, anchor models of stellar evolution [95].", "Precise dynamical masses are used to infer the dynamical evolution of objects that reside off of the main sequence and rapidly evolve in the Hertzsprung-Russell diagram, such as young stars, brown dwarfs, giant planets and white dwarfs [18], [49], [77].", "Orbits have assumed a central role in exoplanet research, since exoplanets are often detected only through their effects on their host stars.", "For a planet detected by radial velocity monitoring, a Keplerian orbital fit may be the only observational result.", "Exoplanet demographics show that the mass distribution of companions has a gap between 10 $\\mbox{$M_{\\rm Jup}$}$ and 100 $\\mbox{$M_{\\rm Jup}$}$ [108], [89], which suggests that exoplanets form very differently from low-mass stellar companions.", "Precise masses, along with luminosities and ages, provide a powerful way to test different planet formation mechanisms, substellar evolutionary models, and white dwarf cooling and atmospheric models [7], [68].", "Brown dwarfs and planets cool and fade with time: evolutionary models predict their luminosities as a function of age and mass [14], [3], [87].", "Systems with independent, dynamical masses provide the strongest tests and calibrations of these models [20], [25], [13], [65].", "For many years, orbital analyses focused on stars and used techniques appropriate for the computing power available.", "Some approaches used a grid search over a small number of parameters combined with linear least-squares fitting of the remaining parameters to map out $\\chi ^2$ surfaces [46], [88].", "Other common approaches used non-linear least-squares fitting to all parameters at once [34], [44] until Markov Chain Monte Carlo methods became more widespread [74], [53], [61].", "In the meantime, orbital analysis has become central to the study of exoplanets, inspiring the development of many orbit-fitting tools over the last several years, including $\\tt ExoFast$ [26], $\\tt PyAstrOFit$ [104], BATMAN [57], $\\tt ExoSOFT$ [70], $\\tt RadVel$ [37] and $\\tt orbitize!$ [8].", "In many cases, only one type of data (astrometry or radial velocity) is available to fit an orbit.", "Some of these tools, like BATMAN and RadVel, are designed with a particular data type in mind, while others including ExoSOFT and orbitize!", "are designed to deal with multiple data types.", "Multiple types of data can combine to offer much stronger constraints than any one type alone.", "Absolute astrometry from Hipparcos and Gaia is now capable of offering meaningful constraints [78], especially when combined with relative astrometry and radial velocity measurements.", "The Gaia spacecraft [40], [38], [39] has been surveying the nearby stellar or exoplanetary systems by taking photometric, spectroscopic and astrometric measurements across the sky with on-board instruments since 2014.", "The conceptually similar Hipparcos satellite [28], [99] obtained astrometric measurements from 1989 to 1993.", "Hipparcos and Gaia measured the positions and motions of stars in an inertial reference frame called the International Celestial Reference System (ICRS) defined by distant quasars [63], [30].", "The difference in their separate measurements of proper motions indicates acceleration in an inertial frame, which can be used to refine the orbital parameters of the accelerators.", "However, neither Hipparcos nor Gaia achieved a perfect realization of the ICRS, and both have low-level systematics and uncertainties that are sometimes underestimated.", "[12] has cross-calibrated Hipparcos and Gaia DR2 to account for systematics as a function of position on the sky, putting them in the same reference frame.", "Brandt (accepted) has performed a similar cross-calibration for Gaia EDR3.", "Absolute astrometry can enable precise constraints on systems [5], [6] even when only a small fraction of the orbit is observed [13].", "The combination of Gaia and Hipparcos proper motions provides a measurement of acceleration in the plane of the sky; a radial velocity trend adds a third dimension to the accelerations.", "Finally, direct imaging provides projected separations and position angles of the companions, allowing orbital constraints even without observing a substantial fraction of an orbit.", "Long-baseline precision RV surveys can identify stellar, substellar or planetary companions exhibiting accelerations, while a growing number of exoplanets are being discovered and characterized via high-contrast imaging [67], [59], [83], [58], [64], [10].", "Many authors have recently combined absolute astrometry with other data types to measure masses and orbits.", "These applications include radial velocity-detected exoplanets [29], [56], [21], [107], [22], directly imaged planets and brown dwarfs [16], [91], [24], [13], [72], [65], and stars [94], [96], [78].", "Our goal is to introduce a generalized, optimized, open source and flexible piece of software that can easily incorporate astrometric acceleration measurements.", "In this paper, we present orvara, a Python Package designed for fast orbit-fitting and plotting of companions.", "We adopt a similar approach as [13] to jointly fit absolute astrometry, relative astrometry, and/or radial velocities of a given star and companion.", "Furthermore, we reduce the computational cost of our approach by introducing a more efficient eccentric anomaly solver, marginalizing out four or more nuisance parameters in the likelihood function, and using low-level memory management to avoid python overheads.", "orvara requires (and is distributed with) the Hipparcos-Gaia Catalog of Accelerations.", "The paper is structured as follows.", "Section reviews the basics of Keplerian orbits including equations that govern the measured positions and velocities as a function of time and corresponding orbital parameters.", "This is followed by a discussion of the implementation of these equations in orvara in Section .", "Section describes the marginalized likelihood function over four or more parameters used in MCMC fitting.", "The computational performance of our implementation is examined in Section .", "The configuration and use of orvara, post-processing of the output, and plotting of a suite of eight plots relevant to astrometry and radial velocity are discussed in Section .", "We present a case study application of orvara to a recently discovered WD/MS system HD 159062 in Section where new constraints on the companion mass and the eccentricity are presented.", "We conclude with our results and findings in Section ." ], [ "Keplerian Orbits", "A Keplerian orbit is a solution of the two-body problem for Newtonian gravity.", "It is fully described by six orbital elements plus the masses of the two components.", "We wish to use measured positions and velocities to derive posterior probability distributions on these eight parameters, marginalizing over several nuisance parameters (e.g., the position, parallax, and velocity of the system's barycenter and astrophysical jitter in the measured radial velocities).", "In this section we briefly review the equations that give the measured positions and velocities as a function of time and the orbital parameters.", "We will describe our implementation of these equations and of the likelihood in subsequent sections.", "In a Keplerian orbit, the mean anomaly $M$ varies linearly with time as $M = \\frac{2\\pi }{P} \\left(t - t_p \\right)$ where $P$ is the system period and $t_p$ is the epoch of periastron passage.", "The position and velocity may be computed using the eccentric anomaly, which is given implicitly by $M = E - \\varepsilon \\sin E$ where $\\varepsilon $ is the eccentricity.", "The radial velocity RV is given through the true anomaly $\\nu $ by $\\nu &= 2\\, {\\rm atan2} \\left[ \\sqrt{1 + \\varepsilon } \\sin \\frac{E}{2}, \\sqrt{1 - \\varepsilon } \\cos \\frac{E}{2} \\right] \\\\{\\rm RV} &= k \\left( \\cos \\left[ \\nu + \\omega \\right] + \\varepsilon \\cos \\omega \\right), $ where $\\omega $ is the argument of periastron, $k$ is the radial velocity amplitude, and ${\\rm atan2}$ is the two-argument arctangent.", "We adopt a convention that the orbital parameters all refer to the companion(s).", "The orbital parameters for the primary are the same except that $\\omega _{\\rm pri} = \\omega + \\pi $ .", "The projected offset between the two bodies may be computed through the elliptical rectangular coordinates $X &= \\cos E - \\varepsilon \\\\Y &= \\left(\\sin E \\right) \\sqrt{1 - \\varepsilon ^2}, $ and the Thiele-Innes constants $A &= \\cos \\Omega \\cos \\omega - \\sin \\Omega \\sin \\omega \\cos i \\\\B &= \\sin \\Omega \\cos \\omega + \\cos \\Omega \\sin \\omega \\cos i \\\\F &= - \\cos \\Omega \\sin \\omega - \\sin \\Omega \\cos \\omega \\cos i \\\\G &= - \\sin \\Omega \\sin \\omega + \\cos \\Omega \\cos \\omega \\cos i.", "$ In Equations (REF )–(), $i$ is the inclination, and $\\Omega $ is the longitude of the ascending node.", "The projected offsets of the secondary with respect to the primary star in declination $\\Delta \\delta $ and right ascension $\\Delta \\alpha * = \\Delta (\\alpha \\cos \\delta )$ are then given by $\\Delta \\delta &= a(AX + FY) \\\\\\Delta \\alpha * &= a(BX + GY) $ where $a$ is the semimajor axis in angular units.", "Constraints on a system's orbit may come from measurements of the primary star's radial velocity over time, the projected angular offset of the two bodies, and/or the projected motion of either component relative to the system's barycenter in an inertial reference frame.", "orvara is designed to account for all of these within the context of Gaussian uncertainties.", "While it is straightforward to apply Equations (), (REF ), and () to compute predicted positions and velocities, implementing them as shown is typically inefficient and may require many trigonometric function calls per observational epoch.", "In the following sections we describe our computational implementation of the equations given in this section and our calculation of the likelihood." ], [ "The Eccentric Anomaly Solver", "Equation (REF ) for the eccentric anomaly is known as Kepler's Equation.", "Its importance, combined with its lack of an analytic solution, has inspired centuries worth of work on efficient computational approaches [19].", "A first-order Newton-Raphson approach requires an evaluation of sine and cosine at every iteration.", "Depending on the quality of the initial guess, this could result in anywhere from a few to $\\approx $ ten trigonometric evaluations per epoch.", "We have developed a more efficient approach for orvara: an eccentric anomaly solver based closely on that of [84], hereafter RPP, with several modifications.", "RPP's basic approach is to obtain a very good initial guess for the eccentric anomaly through fitting formulae, followed by a single step of a modified Newton-Raphson method.", "We begin by reducing the range of the mean anomaly $M$ to $(-\\pi , \\pi ]$ ; we further reduce the range to $[0, \\pi ]$ and save the sign of $M$ for later use.", "We then adopt the piecewise quintic fitting function for eccentric anomaly $E$ as a function of mean anomaly given in RPP, with the function values and the first two derivatives fixed where the eccentric anomaly is an integer multiple of $\\pi /12$ (for twelve polynomials).", "Each polynomial $i$ is then defined over a range of mean anomalies $M_i$ , $M_{i + 1}$ given in Table 3 of RPP.", "To facilitate the calculation of the coefficients, we define the $i^{\\rm th}$ quintic polynomial as $E \\approx \\sum _{k=0}^5 a_{i,k} \\left(M - M_i \\right)^k$ where $M_i$ is the minimum mean anomaly in the domain of the $i$ -th quintic polynomial.", "With this definition, $a_{i,0} &= E(M_i), \\\\a_{i,1} &= E^\\prime (M_i),~~{\\rm and} \\\\a_{i,2} &= \\frac{1}{2} E^{\\prime \\prime }(M_i), $ where the values and derivatives are tabulated in Table 3 of RPP.", "The other three coefficients may be derived by matching the values at $M_{i + 1}$ , $E(M_{i + 1}) &= \\sum _{k=0}^5 a_{i,k} \\left(M_{i + 1} - M_i \\right)^k, \\\\E^\\prime (M_{i + 1}) &= \\sum _{k=1}^5 a_{i,k} k \\left(M_{i + 1} - M_i \\right)^{k - 1},~~{\\rm and} \\\\E^{\\prime \\prime }(M_{i + 1}) &= \\sum _{k=2}^5 a_{i,k} k(k - 1)\\left(M_{i + 1} - M_i \\right)^{k - 2}.", "$ Since $a_{i,0}$ , $a_{i,1}$ , and $a_{i,2}$ are known from Equations (REF )–(), Equations (REF )–() form a linear system of three equations for the three remaining coefficients.", "We solve this linear system by hand.", "With a good initial guess for the eccentric anomaly $E$ , we deviate very slightly from RPP and use Halley's method to refine the solution for eccentricities $\\varepsilon < 0.78$ .", "This requires the sine and cosine of the eccentric anomaly.", "We use a series expansion for one and a square root call for the other; these are significantly faster than trigonometric calls and remain within a factor of a few of double precision floating point accuracy.", "If $E < \\pi /4$ , we use the Taylor series for sine up to 15th order for a cost of seven additions and nine multiplications.", "We then compute $\\cos E = \\sqrt{1 - \\sin ^2 E}$ , with no ambiguity in the sign given the range reduction.", "If $E > 3\\pi /4$ , we use $\\pi - E$ in the same Taylor expansion for sine.", "If $\\pi /4 < E < 3\\pi /4$ , we use the identity $\\cos E = \\sin (\\pi /2-E)$ and the Taylor expansion of sine to compute $\\cos E$ ; we then use a square root call to evaluate $\\sin E$ .", "For $\\varepsilon < 0.78$ , a single iteration of Halley's method brings the accuracy of our computed eccentric anomaly to nearly double precision.", "We then update $\\sin E$ and $\\cos E$ , which we initially computed using series and square roots, with the summation trigonometric identities for $\\sin (E + \\delta E)$ and $\\cos (E + \\delta E)$ .", "We evaluate $\\sin \\delta E$ and $\\cos \\delta E$ to second order in keeping with the second order accuracy of Halley's method.", "Finally, we multiply $E$ and $\\sin E$ by the sign of the mean anomaly.", "This brings $E$ , $\\sin E$ , and $\\cos E$ all close to double precision, with worst case errors below $10^{-15}$ for $\\varepsilon < 0.78$ .", "Figure: Left: residuals of the eccentric anomaly in radians, its sine and cosine, as a function of eccentricity for a uniform distribution of mean anomalies.", "The increasing errors in EE and sinE\\sin E as ε\\varepsilon approaches unity are due to their mutual cancellation in Kepler's equation: the residuals in E-εsinEE - \\varepsilon \\sin E remain below 10 -15 10^{-15}.", "Right: time per epoch for the eccentric anomaly solvers in orvara, radvel , the solver of exoplanet , the goatherd/contour integral approach of , and batman using a single core of an Intel Xeon E5-2630, 2.2 GHz processor, and for 25, 100, or 500 epochs per eccentricity (dot-dashed, dashed, and solid lines, respectively).", "Our implementation of the algorithm includes the calculation of sine and cosine; we add a calculation of sine and cosine to the other methods.The dashed black curve shows the computational cost of a single call each to sine and cosine for each epoch.", "Our RPP implementation also has an additional overhead of ∼\\sim 400 ns/orbit for the allocation and computation of the polynomials for the initial guess.", "This, combined with small Python overheads, accounts for the difference in cost between 25, 100, and 500 epochs.At higher eccentricities, the initial polynomial guess behaves poorly when $M \\ll 1$ .", "If $2M + (1 - \\varepsilon ) < 0.2$ , we adopt the second-order series expansion derived in RPP (their Equation (19)) for our starting point, and revert to the quintic polynomial fits otherwise.", "We find that at higher eccentricities and low mean anomalies, we sometimes require a third-order step to reach double precision accuracy in a single iteration.", "We therefore use Householder's third-order method if both $\\varepsilon > 0.78$ and $M < 0.4$ , and Halley's second-order method otherwise.", "As before, we refine $\\sin E$ and $\\cos E$ using the summation identities.", "Where $\\varepsilon > 0.78$ and $M < 0.4$ , we use series expansions up to third order to match the overall accuracy of Householder's (third order) method.", "This approach is slightly slower than at lower eccentricities due to the higher-order variant of Newton's method and to the need to occasionally use a series expansion for the initial guess.", "Up to eccentricities of 0.9999, however, the absolute errors in $E$ and $\\sin E$ remain below $2 \\times 10^{-14}$ at all mean anomalies, and up to eccentricities of 0.99, they remain below $3 \\times 10^{-15}$ for all mean anomalies.", "Figure REF summarizes the performance of the eccentric anomaly solver.", "The left panel shows the overall accuracy, both the average and maximum errors as a function of eccentricity for a uniform distribution of eccentric anomalies.", "Even at an eccentricity of 0.9999 the algorithm is accurate to $2 \\times 10^{-14}$ in $E$ and $\\sin E$ , and to $10^{-15}$ in $\\cos E$ .", "The increasing error at high eccentricity is a result of a cancellation between $E$ and $\\sin E$ in Kepler's equation; it could be overcome by rewriting the equation in terms of $1 - \\varepsilon $ and expanding $\\sin E$ as a series for small values of $E$ .", "However, this would not solve the more fundamental problem that we can only compute the mean anomaly from the observation epoch with limited precision, and that $dE/dM$ can be large for $\\varepsilon \\approx 1$ .", "The right panel of Figure REF shows the computational cost of our implementation of RPP compared to some other widely-used eccentric anomaly solvers, all evaluated on a single core of an Intel Xeon E5-2630, 2.2 GHz processor.", "Our implementation gives $\\sin E$ and $\\cos E$ as byproducts of the calculation.", "We add a call to sine and cosine when assessing other methods, as $\\sin E$ and $\\cos E$ are required to calculate radial velocities and/or positional offsets.", "There is an overhead of $\\approx $ 170 ns associated with calculating the 72 polynomial coefficients at a given eccentricity, making the performance of our RPP implementation only slightly faster than other solvers for a single epoch.", "We have also implemented a version of this solver tailored to single epoch calculations.", "The single epoch version eliminates most of the overhead by constructing only one of the quintic polynomials (six coefficients).", "All methods shown in the right panel of Figure REF also incur an overhead of a few hundred ns for the Cython function call.", "For 25 epochs at the same eccentricity, our implementation of RPP is around five times faster than these other approaches.", "With $\\gtrsim $ 100 epochs at a given eccentricity, it can be faster than one call per epoch to both sine and cosine.", "Our implementation is $\\sim $ 5–8 times faster than the SDG code [84] because of its avoidance of explicit trigonometric calls and its approach to computing the polynomial coefficients." ], [ "Radial Velocities", "Given the eccentric anomaly $E$ , eccentricity $\\varepsilon $ , and radial velocity amplitude $k$ , the radial velocity can be calculated by Equations (REF ) and ().", "Written this way, the radial velocity evaluation requires five trigonometric calls (four if range reduction is used to enable the use of the single-argument arctangent).", "Only one of these, $\\cos \\omega $ , can be computed once for all epochs in an orbit.", "However, Equations (REF ) and () are equivalent by trigonometric identities to $g &= \\left(\\sqrt{\\frac{1 + \\varepsilon }{1 - \\varepsilon }}\\right) \\left(\\frac{1 - \\cos E}{\\sin E}\\right) \\\\{\\rm RV} &= k \\left( \\left(\\frac{1 - g^2}{1 + g^2} + \\varepsilon \\right) \\cos \\omega - \\left(\\frac{2g}{1 + g^2} \\right) \\sin \\omega \\right) .$ The quantities $\\sin \\omega $ and $\\cos \\omega $ appearing in Equation () only need to be computed once per orbit, not once per epoch.", "The quantities $\\sin E$ and $\\cos E$ , needed for Equation (REF ), were already evaluated in the course of solving for the eccentric anomaly.", "Once $\\sin \\omega $ and $\\cos \\omega $ are computed for an orbit, computing the radial velocities using Equations (REF ) and () does not require a single trigonometric evaluation.", "In comparison tests, we found the use of Equations (REF ) and () to be $\\sim $ 10–20 times faster than Equations (REF ) and ().", "Equation (REF ) can be problematic when $E \\approx 0$ .", "If $|E| < 0.015$ , we use the series expansion up to fifth order (there is no sixth order term); this gives a value of $g$ accurate to better than $10^{-15}$ .", "If $E \\approx \\pm \\pi $ , Equation () presents no numerical problems unless $\\sin E$ is precisely zero (in which case we simply set ${\\rm RV} = k (\\varepsilon - 1) \\cos \\omega $ )." ], [ "Absolute and Relative Projected Positions", "The relative offsets of the secondary from the primary star in right ascension $\\alpha *$ and declination $\\delta $ (where $\\alpha * = \\alpha \\cos \\delta $ ) are given by Equations (REF ) and ().", "These equations present no computational difficulties.", "With $\\sin E$ and $\\cos E$ already computed from the eccentric anomaly solver, Equations (REF )–() require five trigonometric evaluations plus one square root per orbit; zero per epoch.", "We typically fit orbits using both absolute and relative astrometry.", "The displacement of the primary star from the system's barycenter is related to the relative separations of Equations (REF ) and () by $\\Delta \\delta _{\\star } &= \\left( \\frac{-M_{\\rm B}}{M_{\\rm A} + M_{\\rm B}} \\right) \\Delta \\delta \\\\\\Delta \\alpha *_{\\star } &= \\left( \\frac{-M_{\\rm B}}{M_{\\rm A} + M_{\\rm B}} \\right) \\Delta \\alpha * $ where $M_{\\rm A}$ is the mass of the primary star and $M_{\\rm B}$ is the mass of its companion.", "Astrometric missions like Hipparcos and Gaia measure the position of a star many times and fit an astrometric sky path.", "We use the Hundred Thousand Orbit Fitter (htof) (Brandt et al.", "submitted) to compute synthetic Hipparcos and Gaia catalog positions and proper motions from the offsets given in Equations (REF ) and ().", "We then compare the htof synthetic catalog values to the cross-calibrated absolute astrometry of HGCA [12].", "We refer the reader to the source code at https://github.com/gmbrandt/htof and the publication (when available) for further details." ], [ "Multiple Companions", "Keplerian orbits only describe two-body systems.", "To fit more than one companion in orvara, we use a set of simplifying approximations that we describe here.", "We approximate the star's motion by a superposition of Keplerian orbits, one due to each companion.", "We use the orbital elements of each companion (shifting the argument of periastron by $\\pi $ to give the stellar orbit), but we modify the total mass.", "For the mutual orbit between the star and a given companion indexed by $i$ , we add the mass of all other companions that orbit closer to the star than companion $i$ itself.", "Effectively, we approximate the star and inner companions as a single body and solve for its motion about its barycenter due to the $i$ -th companion.", "We then solve for the orbital motion within the inner system when we account for those companions.", "We add the contributions to the host star's motion from all companions for both absolute astrometry and radial velocity.", "We treat relative astrometry slightly differently.", "In this case, we ignore the influence of all companions orbiting beyond the companion of interest.", "We first compute the relative astrometry between this companion and the barycenter of the star and all inner companions.", "We then add the offset of the star relative to this barycenter due to the inner companions.", "This calculation is the same as the shift in absolute astrometry due to these inner companions.", "These approximations do not fully capture interactions between companions.", "While incorrect in detail, our approach does capture some of the effects of a multi-body system.", "It can be used to fit extremely accurate relative astrometry (like that from GRAVITY, [43]), though not with the fidelity of a full (and expensive) suite of $N$ -body simulations.", "We compute the likelihood ${\\cal L}$ of an orbit as $-2 \\ln {\\cal L} = \\chi ^2 = \\chi ^2_{\\rm RV} + \\chi ^2_{\\rm rel\\,ast} + \\chi ^2_{\\rm abs\\,ast} .$ We treat $\\chi ^2_{\\rm RV}$ , the radial velocity component, first.", "We then take the latter two terms together as we marginalize out the barycenter's proper motion and the system's parallax." ], [ "Radial Velocity", "For the radial velocity, we take $\\chi ^2_{\\rm RV} = \\sum _{j=1}^{N_{\\rm inst}}\\sum _{k=1}^{N_{\\rm RV}} \\bigg (& \\frac{\\left({\\rm RV}_{k}+{\\rm ZP}_j-{\\rm RV}\\left[t_k\\right] \\right)^2}{\\sigma ^2[{\\rm RV}_k] + \\sigma ^2_{\\rm jit}} \\nonumber \\\\&\\quad + \\ln \\left[ \\sigma ^2[{\\rm RV}_k] + \\sigma ^2_{\\rm jit} \\right] \\bigg ),$ where ${\\rm ZP}_j$ is the instrument-specific radial velocity zero point and $\\sigma ^2_{\\rm jit}$ is a jitter term.", "We use ${\\rm RV}_k$ to denote the measured RV at epoch $t_k$ , ${\\rm RV}[t_k]$ for the model-predicted RV and $\\sigma ^2[{\\rm RV}_k]$ for its variance.", "Our approach differs from [13] in that we adopt a single jitter for all instruments, attributing it to stellar activity.", "This has the disadvantage of assuming that instrumental uncertainties are properly estimated and that instruments sample similar regimes of stellar activity (the latter assumption could be problematic if combining visible and near-infrared data).", "However, it makes the fit robust to the inclusion of instruments with just a few data points, and it reduces the number of parameters to fit.", "orvara does include the option of fitting a different jitter to every RV instrument.", "Equation (REF ) is the only place in the likelihood where the radial velocity zero point appears.", "We therefore marginalize it out, assuming a flat prior, by integrating ${\\cal L} \\propto \\int _{-\\infty }^\\infty d {\\rm ZP} \\exp \\left[ -\\frac{\\chi ^2_{\\rm RV}}{2} \\right].$ We perform the integral separately for each instrument.", "This requires only one pass through the radial velocity data set, and adds a negligible amount of computation.", "Performing the integral results in replacing $\\chi ^2_{\\rm RV}$ with $\\chi ^2_{\\rm RV} = \\sum _{j=1}^{N_{\\rm inst}} &\\left( -\\frac{B_j^2}{4A_j} + C_j + \\ln A_j \\right) \\nonumber \\\\&+ \\sum _{j=1}^{N_{\\rm inst}}\\sum _{k=1}^{N_{\\rm RV}} \\ln \\left[ \\sigma ^2[{\\rm RV}_k] + \\sigma ^2_{\\rm jit} \\right] $ where $A_j &= \\sum _{k=1}^{N_{\\rm RV}} \\frac{1}{\\sigma ^2[{\\rm RV}_k] + \\sigma ^2_{\\rm jit}}, \\\\B_j &= \\sum _{k=1}^{N_{\\rm RV}} \\frac{2 \\left({\\rm RV}_{k} - {\\rm RV}\\left[t_k\\right] \\right)}{\\sigma ^2[{\\rm RV}_k] + \\sigma ^2_{\\rm jit}}, \\\\C_j &= \\sum _{k=1}^{N_{\\rm RV}} \\frac{\\left({\\rm RV}_{k} - {\\rm RV}\\left[t_k\\right] \\right)^2}{\\sigma ^2[{\\rm RV}_k] + \\sigma ^2_{\\rm jit}} ,$ and the sums over $k$ are restricted to radial velocity data points from instrument $j$ .", "orvara uses Equations (REF )–(REF ) as shown." ], [ "Astrometry", "The $\\chi ^2$ for relative astrometry consists of two components: the relative separation $\\rho $ and the position angle $\\theta $ measured east of north.", "The model orbit's relative separation is the product of the projected relative separation $\\rho $ in AU and the system's parallax $\\varpi $ .", "In general, the measurements of separation and position angle may be covariant: we take $c_{\\rho \\theta ,k} \\in (-1, 1)$ to be the correlation coefficient between the two measurements at epoch $k$ .", "The contribution to $\\chi ^2$ is then $\\chi ^2_{\\rm rel\\,ast} &= \\sum _{k=1}^{N_{\\rm ast}} \\frac{\\lfloor \\theta _k - \\theta [t_k]\\rfloor ^2}{(1 - c^2_{\\rho \\theta ,k}) \\sigma ^2[\\theta _k]} + \\sum _{k=1}^{N_{\\rm ast}} \\frac{\\left( \\rho _k-\\varpi \\rho \\left[t_k\\right] \\right)^2}{(1 - c^2_{\\rho \\theta ,k})\\sigma ^2 [\\rho _k]} \\nonumber \\\\&\\qquad -2 \\sum _{k=1}^{N_{\\rm ast}}\\frac{c_{\\rho \\theta ,k} \\lfloor \\theta _k - \\theta [t_k]\\rfloor \\left( \\rho _k-\\varpi \\rho \\left[t_k\\right] \\right)}{(1 - c^2_{\\rho \\theta ,k})\\sigma [\\theta _k]\\sigma [\\rho _k]}.$ where $\\rho _k$ and $\\theta _k$ are the observed separation and position angle at time $t_k$ , $\\rho [t_k]$ and $\\theta [t_k]$ are the model-predicted values, and $\\lfloor \\theta _k - \\theta [t_k] \\rfloor $ is the difference between the measured and the predicted position angles, reduced to the range $(-\\pi ,\\pi ]$ .", "The absolute astrometry in angular units is similarly proportional to parallax, and also has a velocity zero point.", "This is the proper motion of the system barycenter in the plane of the sky $\\overline{\\mathbf {\\mu }}$ , an almost perfect analog of the radial velocity zero point.", "This component of the likelihood then reads $\\chi ^2_{HG} = &\\left( {\\mathbf {\\mu }_{H, \\rm o}} - \\overline{\\mathbf {\\mu }} - \\varpi \\mathbf {\\mu }_{H} \\right)^T {\\bf C}_H^{-1} \\left( {\\mathbf {\\mu }_{H, \\rm o}} - \\overline{\\mathbf {\\mu }} - \\varpi \\mathbf {\\mu }_{H} \\right) \\nonumber \\\\&+\\left( {\\mathbf {\\mu }_{HG, \\rm o}} - \\overline{\\mathbf {\\mu }} - \\varpi \\mathbf {\\mu }_{HG} \\right)^T {\\bf C}_{HG}^{-1} \\left( {\\mathbf {\\mu }_{HG, \\rm o}} - \\overline{\\mathbf {\\mu }} - \\varpi \\mathbf {\\mu }_{HG} \\right) \\nonumber \\\\&+\\left( {\\mathbf {\\mu }_{G, \\rm o}} - \\overline{\\mathbf {\\mu }} - \\varpi \\mathbf {\\mu }_{G} \\right)^T {\\bf C}_G^{-1} \\left( {\\mathbf {\\mu }_{G, \\rm o}} - \\overline{\\mathbf {\\mu }} - \\varpi \\mathbf {\\mu }_{G} \\right)$ where, e.g., $\\mathbf {\\mu }_{H,\\rm o}$ is the observed Hipparcos proper motion and $\\mathbf {\\mu }_{H}$ is the model orbit's predicted Hipparcos proper motion in AU yr$^{-1}$ (which must then be multiplied by the parallax to obtain a proper motion in angular units).", "In some cases, Gaia obtains an astrometric solution for both stars in a binary.", "In this case, the observed epochs and scan angles will generally be the same for both stars, and the position and proper motion of the secondary may be easily computed from those of the primary.", "In the frame of the system barycenter (in which the orbit code operates), we have $\\mathbf {\\mu }_{\\rm B} = -\\mathbf {\\mu } \\left( \\frac{M_{\\rm A}}{M_{\\rm B}} \\right)$ where $\\mathbf {\\mu }$ is the proper motion of the primary star and $\\mathbf {\\mu }_{\\rm B}$ is the proper motion of its companion.", "Denoting the Gaia proper motion of the secondary by ${\\mathbf {\\mu }_{G, \\rm o, B}}$ , this measurement contributes an extra term to the astrometric $\\chi ^2$ , $\\Delta \\chi ^2_{HG} = \\left( {\\mathbf {\\mu }_{G, \\rm o, B}} - \\overline{\\mathbf {\\mu }} - \\varpi \\mathbf {\\mu }_{G, \\rm B} \\right)^T {\\bf C}_{G, \\rm B}^{-1} \\left( {\\mathbf {\\mu }_{G, \\rm o, B}} - \\overline{\\mathbf {\\mu }} - \\varpi \\mathbf {\\mu }_{G, \\rm B} \\right)$ where ${\\bf C}_{G, \\rm B}^{-1}$ is the inverse of the Gaia covariance matrix for the secondary and $\\overline{\\mathbf {\\mu }}$ remains the center-of-mass motion of the system's barycenter.", "Including this additional term assumes that both stars have their proper motion in the same reference frame and have well-calibrated errors.", "Either assumption may break down in practice: Gaia DR2 has magnitude-dependent systematics in the reference frame of $\\sim $ 0.2 mas yr$^{-1}$ [60], while the HGCA required spatially variable error inflation by a typical factor $\\sim $ 1.7 [12].", "Gaia EDR3 retains magnitude-dependent systematics of up to $\\approx $ 100 $\\mu $ as yr$^{-1}$ [17].", "We also neglect covariance between the secondary's measured proper motion and the projected separation of the two bodies.", "Equations (REF ) and (REF ) (with or without the additional term from Equation (REF )) are quadratic equations in parallax and the two components of the proper motion of the system's barycenter.", "We adopt a uniform prior on $\\mathbf {\\overline{\\mu }}$ , but use a Gaussian prior in parallax to incorporate the measured Gaia value.", "This is equivalent to adding one additional component to the log likelihood, $\\chi ^2_{\\varpi } = \\frac{\\left( \\varpi - \\varpi _{\\it Gaia} \\right)^2}{\\sigma _{\\varpi ,\\it Gaia}^2}.", "$ With the addition of Equation (REF ), we can write the design matrix of the system and solve for the maximum likelihood values of the parameters and their covariance matrix.", "Integrating the likelihood over parallax and barycentric proper motion is equivalent to substituting these maximum likelihood values into the expressions for $\\chi ^2$ and multiplying the likelihood by the square root of the determinant of the covariance matrix.", "This requires solving a $3 \\times 3$ linear system and computing the determinant of a $3 \\times 3$ matrix, neither of which incurs a significant computational cost.", "We write out this linear system in the Appendix.", "We solve the $3 \\times 3$ linear system using the singular value decomposition, but with a C routine to avoid the overheads associated with python function calls and memory management." ], [ "The Marginalized Likelihood", "The integrations discussed in this section remove four parameters from the fit for a system when all radial velocities are from a single instrument, and more parameters otherwise.", "The computational cost of doing so is small relative to the cost of computing the orbit.", "The maximum posterior probability values of the parallax and barycenter proper motion may be of interest and are difficult to reconstruct from an output chain.", "We therefore save these quantities at each step for later use, along with the values for the Hipparcos, Hipparcos–Gaia, Gaia, relative separation, position angle components of $\\chi ^2$ , and maximum likelihood radial velocity zero points of all instruments.", "It is much more difficult to analytically integrate out other parameters.", "[106] show that the log likelihood can be linear in additional parameters if the parallax is accurately known.", "However, the priors on the combinations of parameters that enter the problem linearly can be complex even for simple choices of the underlying Keplerian parameters; the product of the likelihood and prior is difficult to integrate.", "Introducing relative astrometry (as we do for many of our targets) breaks the linearity entirely.", "If we consider parallax to be known, we may still integrate out the secondary mass if we adopt the total system mass as the other mass parameter in the problem, as long as we use a flat prior.", "Given these restrictions, we retain parallax as a linear parameter and do not analytically integrate out any others.", "Our likelihood, after marginalizing four parameters, is a function of the six Keplerian orbital elements, the masses of the two components, and a radial velocity jitter, for a total of nine parameters.", "If there is more than one companion in the system, the likelihood becomes a function of two parameters (mass of the primary and radial velocity jitter) plus seven per companion (the six Keplerian orbital elements and the companion's mass).", "We sample from this marginalized likelihood using the parallel-tempering MCMC sampler ptemcee [102], a fork of emcee [33]." ], [ "Performance", "The computational cost of a single step includes contributions from the computation of the eccentric anomaly, the radial velocity and positional offsets, and the likelihood calculation.", "A typical data set consists of a few hundred measurements.", "Of these, $\\sim $ 150–200 might be individual epochs from Hipparcos and Gaia.", "For a sample data set of Gl 758 [13], there are 652 radial velocity measurements, 181 epochs for the absolute astrometry, and 4 epochs for relative astrometry.", "This data set requires 85 $\\mu $ s per iteration on a single core of a 2.2 GHz Intel Xeon E5-2630.", "Of this time, 44 $\\mu $ s is spent in the calculation of the eccentric anomalies, 5 $\\mu $ s on radial velocities, 2 $\\mu $ s on absolute astrometry for individual epochs, 9 $\\mu $ s on fitting the Hipparcos and Gaia models to the epoch astrometry, and 10 $\\mu $ s on the likelihood calculation.", "The remaining 15 $\\mu $ s are in overheads associated with emcee and the main python functions.", "For a sample data set of HD 4747, with 49 radial velocity measurements, 159 epochs for the absolute astrometry, and 8 epochs for relative astrometry, the total cost is $\\sim $ 40 $\\mu $ s per iteration.", "Due to the smaller number of data points, the times spent calculating the eccentric anomaly, radial velocity, and likelihood fall to 11 $\\mu $ s, 1 $\\mu $ s, and 5 $\\mu $ s, respectively.", "The cost of fitting the Hipparcos and Gaia models to the epoch astrometry remains $\\sim $ 9 $\\mu $ s. By comparison, a single-planet fit to 216 epochs of radial velocity data using radvel takes 1.0 ms per iteration with one core of the same CPU, or about 25 times slower than a fit to the same number of epochs (but with a diversity of data types) using orvara.", "Overheads associated with allocating numpy arrays could take the majority of the computational time if we operated exclusively on arrays.", "We therefore explicitly handle memory allocation in Cython, and use C structures with pointers rather than numpy arrays.", "Replacing each pointer allocation with an ndarray and creating a memoryview would more than double the run time for HD 4747.", "For similar reasons, we use a C implementation of the singular value decomposition rather than a python call to the linear algebra routine in numpy.", "For a typical system with a few hundred observational epochs, the total computational cost to compute an orbit and evaluate its likelihood (marginalizing out parallax, radial velocity zero point, and barycenter proper motion) is equivalent to about three to seven trigonometric evaluations per epoch.", "orvara uses the parallelization built into the packages emcee and ptemcee.", "The scaling under typical use is poor when using more than a handful of processors.", "Using five processors results in a speedup by just a factor of two on our machines, while adding further processors gives almost no additional speedup at all.", "Parallelization also works differently on different operating systems, and may not be supported on all machines.", "Regardless, multiple instances of the program may be run simultaneously for significantly better performance." ], [ "Configuration, Use and Plotting", "orvara is an open-source Python package that performs parameter fits for multi-planetary or binary star systems using a combination of the Hipparcos-Gaia Catalog of Accelerations, literature radial velocities, and/or relative astrometry.", "orvara includes Python plotting routines to produce a comprehensive orbit-fitting and plotting package.", "It is available on GitHubhttps://github.com/t-brandt/orvara.", "The detailed installation procedure of orvara and its updated documentation can be found in the $\\tt README.rst$ file.", "Most applications of orvara use radial velocity and/or relative astrometry data.", "Our example in Section uses both.", "Each type of data is given in an input ASCII text file with fields separated by spaces or tabs; lines beginning with # are ignored.", "We provide examples in the repository and summarize their required structure in tab:fileformats.", "The radial velocity file must include BJD, RV, and RV error (both in units of m/s) as its first three columns.", "An optional fourth column may include an integer $\\ge $ 0 to distinguish instruments from one another; each instrument will have its own unique RV zero point.", "If the fourth column is not supplied orvara will assume that all radial velocities share the same zero point.", "The relative astrometry file must include date (either BJD or decimal year, with values $<$ 3000 interpreted as years), separation and its error (both in arcseconds), and position angle and its error (east of north, both in degrees) as its first five columns.", "An optional sixth column gives the correlation coefficient ($\\in (-1, 1)$ ) between separation and position angle.", "An optional seventh column gives the ID of the measured companion.", "The first companion is indexed as 0, the second as 1, etc.", "If the sixth and seventh columns are not supplied, orvara will use default values of 0 for each.", "lcccr Format of the input data files 0pt Column Number Description Units Required?", "Default Value 5cRadial Velocity Data File 1 Observation epoch BJD yes ... 2 Radial velocity m/s yes ... 3 Radial velocity error m/s yes ... 4 RV instrument IDa ... no 0 5cRelative Astrometry Data File 1 Observation epoch Decimal year or BJD yes ... 2 Angular Separation arcsec yes ... 3 Separation error arcsec yes ... 4 Position angle (E of N) degrees yes ... 5 Position angle error degrees yes ... 6 Sep/PA correlation coefficienta ... no 0 7 Companion IDa ... nob 0 aValid values: sequential integers from 0 to $n_{\\rm inst} - 1$ for RV instrument ID, integers from 0 to $n_{\\rm companions} - 1$ for companion ID, real numbers between $-1$ and 1 for sep/PA correlation coefficient.", "bColumn 6 must be included (even if all zeros) when using column 7 for companion ID.", "Orbit fitting and plotting can be accessed with the $\\tt fit\\_orbit$ and $\\tt plot\\_orbit$ commands from the command line, respectively.", "The $\\tt fit\\_orbit$ output is a single Flexible Image Transport System (FITS) file [103] containing the MCMC chains and described in more detail below.", "The header of this FITS file includes the configuration parameters used to produce the chain.", "The $\\tt plot\\_orbit$ output is a suite of up to eight plots relevant to RV and relative astrometry and also discussed individually below.", "A star-specific .ini configuration file containing the appropriate file directories and settings serves as the input for both commands.", "This configuration file is comprised of four main sections denoted in square brackets: the data paths to all related file directories ($\\tt [data\\_paths]$ ), settings for MCMC fitting ($[\\tt mcmc\\_settings]$ ), settings for the primary mass prior and uncertainty ($\\tt [priors\\_settings]$ ), settings for plotting ($\\tt [plotting]$ ), and for saving a table of results ($\\tt save\\_results$ ).", "A complete list of file directories and customizable input parameters including their descriptions and functionalities is provided in tab:configfile.", "Sample configuration files for Gl 758 and HD 4747, along with their sample data sets of radial velocity and relative astrometry, are included in the package.", "A user can omit data (for example, if there is no relative astrometry) by providing a blank or invalid file path.", "A user can also omit absolute astrometry by providing an invalid Hipparcos ID and supplying an explicit prior on the system parallax.", "The first few lines in the .ini file specify the locations of data files and the Hipparcos ID of the primary star.", "The initial guesses for the nine MCMC parameters (plus seven for each companion beyond the first) may also be set in a file referenced in the .ini configuration file.", "orvara draws starting values for each walker from a normal or lognormal distribution with the means and variances specified.", "We assume log-flat priors for semimajor axis $a$ , primary mass $\\mathrm {M_{\\rm pri}}$ , companion mass $\\mathrm {M_{\\rm sec}}$ , and radial velocity jitter $\\mathrm {\\sigma _{jit}}$ ; a prior of sin$\\textit {i}$ for inclination; and uniform priors for all other fitted parameters.", "Alternatively, the prior on the primary mass and its uncertainty can be set in $\\tt [priors\\_settings]$ to incorporate other knowledge of the star.", "In addition to radial velocity fitting, we typically fit orbits using both absolute and relative astrometry (though only absolute astrometry is strictly required).", "Relative astrometry can be utilized by providing the data path to the relative astrometry file containing the astrometric epochs from different high-resolution imaging instruments.", "For epoch astrometry, the Hundred Thousand Orbit Fitter (htof, Brandt et al.", "submitted) package mentioned in Section REF is employed in orvara to provide parameter fits to the astrometric intermediate data from either Hipparcos data reduction or from Gaia.", "In order to use epoch astrometry, the file path to observational epochs and scan angles for Gaia, or to the intermediate data of Hipparcos (original or re-reduction) must be provided.", "The Gaia predicted scan epochs and angle can be queried using the Gaia Observation Forecast Tool, publicly available at $\\mathrm {https://gaia.esac.esa.int/gost/index.jsp}$ .", "htof parses the given intermediate data and extracts (inverse-)covariance matrices and epochs of observations to compute synthetic Hipparcos and Gaia catalog positions and proper motions described in Section REF .", "The output of orvara is a single FITS file with the chain and other calculated parameters in a FITS table in header-data unit (HDU) 1.", "The header of the first extension (HDU 0) contains the configuration parameters read from the .ini file and used to construct the chain.", "The FITS table in HDU 1 contains columns with names, arrays, and units (where appropriate) for each fitted parameter, for the natural logarithm of the likelihood, and for other quantities computed at each step of the chain.", "Each array has dimensions ${\\tt nwalkers} \\times {\\tt nstep//thin}$ .", "lccr 0pt Description of the configuration file contents.", "Parameter Name Data Type Default Description [data_paths] ${\\tt HipID}$ Integer 0 Hipparcos number.", "If valid, load HGCA absolute astrometry ${\\tt HGCAFile}$ Data File [required] The Hipparcos-Gaia Catalog (either the DR2 or EDR3 edition) ${\\tt RVFile }$ Data File ” File containing the radial velocity time series for the star ${\\tt AstrometryFile}$ Data File ” File containing the relative astrometry for the companion(s) ${\\tt GaiaDataDir}$ Directory ” Path to the Gaia scans as output by GOST in .csv format ${\\tt Hip1DataDir}$ Directory ” Path to the Hipparcos (original reduction) intermediate data ${\\tt Hip2DataDir}$ Directory ” Path to the Hipparcos (re-reduction) intermediate data ${\\tt start\\_file}$ Data File 'none' File with the initial parameter guesses.", "If 'none', use default guesses [mcmc_settings] ${\\tt ntemps}$ Integer 10 Number of temperatures to use in the parallel tempering chain ${\\tt nwalkers}$ Integer 100 Number of walkers; each with ntemps number of chains ${\\tt nplanets}$ Integer [required] Number of companions to fit ${\\tt nstep}$ Integer [required] Number of steps contained in each chain ${\\tt thin}$ Integer 50 Thinning of the chain (keep every ${\\tt thin}$ step) ${\\tt nthreads}$ Integer 1 Number of threads to use (the built-in parallelization is poor) ${\\tt use\\_epoch\\_astrometry}$ Boolean False Use the epoch astrometry in GaiaDataDir, Hip1DataDir, etc.?", "jit_per_inst Boolean False Fit a separate RV jitter for each instrument?", "[priors_settings] ${\\tt mpri}$ Float 1 Mean of a Gaussian prior on primary mass (in $M_\\odot $ ) ${\\tt mpri\\_sig}$ Float inf Uncertainty in the stellar priors.", "If inf, use the default $1/M$ prior minjitter Float 1e-5 Minimum allowable RV jitter (m/s).", "Should be $>0$ maxjitter Float 1e3 Maximum allowable RV jitter (m/s).", "Must be $> {\\tt minjitter}$ m_secondary0 Float 1 Mean (in $M_\\odot $ ) of the (Gaussian) mass prior for companion 0 m_secondary0_sig Float 1 Standard deviation of the (Gaussian) mass prior for companion 0 parallax Float None Parallax prior: required if star is not in the HGCA, ignored otherwise parallax_error Float None Parallax prior if star is not in the HGCA [secondary_gaia] Gaia data of the secondary (set companion_ID = -1 if undetected) companion_ID Integer -1 ID of the Gaia companion, should match entry in AstrometryFile pmra Float 0 Gaia proper motion (RA) of companion pmdec Float 0 Gaia proper motion (Dec) of companion epmra Float 1 Gaia proper motion uncertainty (RA) of companion epmdec Float 1 Gaia proper motion uncertainty (Dec) of companion corr_pmra_pmdec Float 0 Correlation ($\\in (-1, 1)$ ) between pmra and pmdec [plotting] ${\\tt McmcDataFile}$ Data File [required] Path to MCMC chain produced from the $\\tt fit\\_orbit$ command ${\\tt burnin}$ Integer 0 Burnin length for thinned chains ${\\tt check\\_convergence}$ Boolean False Make diagnostic plots to help check for convergence?", "iplanet Integer 0 ID of the companion to plot ($0 \\le {\\tt iplanet} < {\\tt nplanets}$ ) ${\\tt target}$ String ” Name of the target, used for file naming ${\\tt start\\_epoch}$ Integer 1950 Customized range of dates (fractional years) ${\\tt end\\_epoch}$ Integer 2030 Customized range of dates (fractional years) ${\\tt num\\_orbits}$ Integer 50 Number of random orbits drawn from the posterior distribution ${\\tt num\\_steps}$ Integer 1000 Points per plotted orbit (aliasing can occur if num_steps is too small) ${\\tt predicted\\_years}$ Float(s) 2010,2020 Labeled year(s) on the best-fit orbit in the Astrometry plot ${\\tt position\\_predict}$ Float 2020 Epoch (fractional year) for the Astrometric_prediction_plot ${\\tt Astrometry\\_orbits\\_plot}$ Boolean True Plot the astrometic orbits?", "${\\tt Astrometric\\_prediction\\_plot}$ Boolean True Plot the density plot for the predicted epoch?", "${\\tt RV\\_orbits\\_plot}$ Boolean True Plot the full RV orbits?", "${\\tt RV\\_plot}$ Boolean True Plot RV orbits vs. epoch and O-C over the baseline of RV data?", "${\\tt RV\\_Instrument}$ Integer/String All ${\\tt All}$ or Instrument number ${\\tt Relative\\_separation\\_plot }$ Boolean True Plot relative separation vs. epoch and O-C?", "${\\tt Position\\_angle\\_plot}$ Boolean True Plot position angle vs. epoch and O-C?", "${\\tt Proper\\_motion\\_plot}$ Boolean True Plot the proper motions in RA and Dec and O-C?", "${\\tt Proper\\_motion\\_separate\\_plots}$ Boolean False True if two separate plots for the proper motions are desired ${\\tt Corner\\_plot }$ Boolean True Plot a two dimensional corner plot from MCMC chain?", "${\\tt set\\_limit}$ Boolean False Use user-specified axis limits?", "${\\tt xlim }$ Floats None If ${\\tt set\\_limit}$ is True, set x limits with two comma-separated values ${\\tt ylim}$ Floats None If ${\\tt set\\_limit}$ is True, set y limits with two comma-separated values ${\\tt marker\\_color}$ String blue Matplotlib color of the marker for the observed data points ${\\tt use\\_colorbar}$ Boolean True Turn on/off colorbars.", "${\\tt colormap}$ String viridis Colormap name from the Matplotlib colormap library ${\\tt reference}$ String msec_jup Colormap reference, ${\\tt msec\\_jup}$ , ${\\tt msec\\_solar}$ or ${\\tt ecc}$ .", "${\\tt show\\_title}$ Boolean False Turn on/off the title of the plot ${\\tt add\\_text}$ Boolean False True if adding text_name somewhere on the plot ${\\tt text\\_name}$ String None If ${\\tt show\\_title}$ is True, specify the text ${\\tt x\\_text}$ Float None If ${\\tt add\\_text}$ is True, enter the x coordinate of the text ${\\tt y\\_text}$ Float None If ${\\tt add\\_text}$ is True, enter the y coordinate of the text [save_results] ${\\tt save\\_params}$ Boolean True Save the posterior parameters to a .txt file?", "${\\tt err\\_margin}$ Float(s) 0.16,0.5,0.84 Quantiles for posterior parameters and uncertainties We have implemented a plotting routine in Python to visualize the results obtained from the FITS file containing the MCMC chains produced by a joint orbit fit.", "In the $\\tt [plotting]$ section of the .ini configuration file, the file path to the FITS file and a $\\tt burnin$ length for the (thinned) MCMC chains must be specified.", "Also, users have the option to select which plots to generate using $\\tt True$ and $\\tt False$ values for the plot keywords.", "Here, we briefly describe the plots that orvara is currently configured to produce.", "Section will provide an example fit and sample plots." ], [ "Astrometric orbits", "The relative astrometric orbits for companions of stellar or planetary systems can be generated by setting $\\tt Astrometry\\_orbits\\_plot$ to $\\tt True$ .", "Several features are worth mentioning in this plot.", "The thick black line indicates the highest likelihood orbits; thin lines are orbits randomly drawn from the posterior distributions and colored according to either the companion mass or eccentricity.", "This is the case for all the plots described below, except for the astrometric prediction plot which is a 2D contour.", "The black dashed line inside the most-likely orbit is the line of nodes joining the ascending node and the descending node of the most-likely orbit.", "This line of nodes indicates the position of the orbital plane of the system with respect to the sky plane.", "The unfilled circles plotted along the most-likely orbit are the predicted positions of the companion at specific user-defined epochs from $\\tt predicted\\_years$ .", "If an astrometric file is provided from $\\tt AstrometryFile$ , the observations will be plotted as filled circles.", "If there is more than one companion included in the fit, the user may specify which one is plotted using the keyword iplanet (with $0 \\le {\\tt iplanet} < {\\tt nplanets}$ )." ], [ "Astrometic prediction", "orvara can predict the location of a companion relative to its host star at a specified epoch.", "The resulting density plot can be obtained by setting $\\tt position\\_predict$ to the desired epoch, and setting $\\tt Astrometric\\_prediction\\_plot$ to $\\tt True$ .", "This contour plot shows the posterior probability density of the predicted positions of the companion in terms of relative offsets from the primary star in right ascension and declination, with the inner contour being the most likely position of the companion at that future epoch.", "The 1-$\\sigma $ , 2-$\\sigma $ , and 3-$\\sigma $ contours enclose 68.3%, 95.4%, and 99.7% of the posterior probabilities for the future location." ], [ "Radial velocity orbits", "orvara can make two plots of the stellar radial velocity: one restricted to the observational time frame, and one spanning a longer, user-specified range of dates to show longer-term behavior.", "To make the latter plot the user may set $\\tt RV\\_orbits\\_plot$ to $\\tt True$ .", "The former plot, restricted to the observational baseline, requires the user to give the path for $\\tt RVFile$ .", "The $\\tt RVFile$ containing the observed radial velocity time series must have the format specified by tab:fileformats.", "The filled circles on both plots represent RV data from all the radial velocity instruments.", "The observed RV data are shifted by an offset according to the maximum likelihood zero point for each RV instrument; the maximum likelihood offsets for each instrument are given by fields named, e.g., `RV_ZP_0_ML' for instrument 0. orvara can also make a plot of the RVs restricted to the time range sampled by the RV instruments; $\\tt RV\\_plot=True$ enables this plot.", "This is essentially a zoomed-in view of the part of the RV orbits plot described above.", "It contains RV data from single- or multi-instrument RV observations, plus a corresponding Observed$-$ Calculated (hereafter O$-$ C) residual shown underneath.", "To choose to plot the RVs from a specific or from all the RV instruments, users may set $\\tt RV\\_Instrument$ to the instrument number or to `All'.", "The observed data are represented by the solid circles with error bars.", "The O$-$ C residual indicates both the deviation of the observed value from the most-likely orbit and the variation in RV across the orbits randomly drawn from the posterior.", "The error bars include the RV jitter of the best-fit orbit." ], [ "Relative separation and position angle", "For relative astrometry, orvara offers two more plots in addition to the astrometric orbits plot: the relative separation (in arcseconds) and the position angle (in degrees) of the imaged companion relative to the primary star in a time range sampled by the relative astrometric instruments, and their corresponding O$-$ C residuals.", "These two plots can be generated if the user sets $\\tt Relative\\_separation\\_plot$ or $\\tt Position\\_angle\\_plot$ to True.", "In each case, the path to $\\tt AstrometryFile$ should be specified.", "The orbit's relative separation in arcseconds is the product of the projected relative separation $\\rho $ in AU and the parallax $\\varpi $ ; we use the best-fit parallax for each set of orbital parameters.", "We refer the reader to Section REF for a detailed description of the equations, and to the Appendix for the calculation of the best-fit parallax.", "The solid circles with error bars indicate the relative astrometric data from direct imaging instruments.", "Due to each instrument's having different uncertainties, data reduction methods, and variations in the field rotation and plate scale, users may choose to adjust the imaging data in $\\tt AstrometryFile$ to keep or discard a set of imaging data.", "Only one companion will be plotted.", "If there is more than one companion fit, the user may specify which one to plot using iplanet." ], [ "Proper motion", "Variations in proper motion induced by the companion(s) on the primary star as measured from absolute astrometry from Hipparcos and Gaia can be plotted with $\\tt Proper\\_motion\\_plot=True$ .", "orvara computes these proper motions as instantaneous values using the time derivative of the eccentric anomaly.", "In contrast, both Hipparcos and Gaia fit sky paths to the star's position as measured over several years.", "As a result, the observed data points and error bars do not correspond to the plotted lines in the same way that they do for, e.g., radial velocity.", "orvara does not plot the Hipparcos-Gaia mean proper motion, as this measurement does not represent an instantaneous proper motion at the mean epoch.", "It is, rather, a measurement of the integral of the proper motion.", "The O$-$ C curves should be taken with caution.", "The actual $\\chi ^2$ values for each set of measurements (which are meaningful and also include measurement covariance) are available in the FITS file written by orvara.", "To plot the two components of the proper motion as two separate, individual plots (one for right ascension and one for declination), users may set $\\tt Proper\\_motion\\_separate\\_plots$ to True.", "The data with error bars are from the cross-calibrated absolute astrometry of the HGCA.", "The HGCA provides three proper motions: one near 1991.25, another near 2015.5, and the positional difference between the Hipparcos re-reduction and the Gaia catalog (either the DR2 or EDR3 edition) scaled by the time between them.", "orvara plots data points for only the Hipparcos and Gaia proper motions.", "orvara adds the best-fit barycenter proper motion, calculated as described in the appendix, to each orbit." ], [ "Corner plot", "A two dimensional corner plot of the MCMC chain can be obtained by setting $\\tt Corner\\_plot$ to True.", "A $\\tt burnin$ phase for the thinned MCMC chain can be specified for plotting.", "We have modified $\\tt corner.py$ from [31] to format the titles displayed on top of the histograms in this plot.", "We chose to keep two significant figures in the uncertainties with a matching number of decimal places in the values.", "The corner plot shows astrophysically meaningful Keplerian orbital elements including semi-major axis $a$ , eccentricity $\\varepsilon $ , and inclination $i$ , and the masses of the primary star in $\\mbox{$M_{}$}$ and its companion in $\\mbox{$M_{\\rm Jup}$}$ from the joint RV and astrometric MCMC analysis.", "If more than one companion is fit, the corner plot will use the parameters from the companion designated by iplanet." ], [ "Case study: application to HD 159062B", "llllllll HGCA proper motions 0pt Source $\\mu _{\\alpha *}$  (mas yr$^{-1}$ ) $\\mu _{\\delta }$  (mas yr$^{-1}$ ) corr $t_{\\alpha *}$ $t_{\\delta }$ $\\varpi $ (mas) RV (km s$^{-1}$ ) Hipparcos $174.316 \\pm 0.666$ $75.598 \\pm 0.612$ 0.27 1991.20 1991.12 HG $172.499 \\pm 0.019$ $75.776 \\pm 0.020$ 0.11 Gaia EDR3 $169.814 \\pm 0.026$ $77.133 \\pm 0.029$ 0.22 2016.07 2016.27 $46.118 \\pm 0.024$ $-84.11 \\pm 0.18$ In this section, we provide a case study application of orvara to a nearby white dwarf/main sequence (WD/MS) binary system, HD 159062.", "The white dwarf was discovered by [50], hereafter H19, who fit for its orbit and mass.", "We first review the system, a widely-separated binary with a degenerate companion to a main sequence star, before summarizing the results of H19.", "We then discuss our own orbital fit and its implications for the binary system's past evolution." ], [ "Background on HD 159062", "Main sequence stars with masses $\\lesssim $ 8 $\\mbox{$M_{}$}$ will evolve off the main sequence through the asymptotic giant branch (AGB) phase at the stellar evolutionary end-point and cool over billions of years until they eventually become dense white dwarfs [23].", "More than 97% of stars are expected to evolve into white dwarfs [85].", "White dwarfs are used as powerful tools to trace the evolution of the Galaxy, and to provide constraints on global stellar populations.", "White dwarfs in binaries can also be potential progenitors of Type 1a supernovae [48].", "Spectroscopic and photometric surveys such as SDSS, Pan-STARRS and Gaia can reveal fundamental properties of white dwarfs including mass, cooling rate and age, and atmospheric and internal composition.", "The recent Gaia DR2 catalog has provided updated precise astrometric and photometric data that can be used to infer the local white dwarf population [42].", "Since most stars end their lives as WDs and most stars reside in binaries [71], it is estimated that around one quarter of the more than two hundred known white dwarfs within 25 pc of the Sun reside in WD/MS or WD/WD binary systems [51].", "Population synthesis modeling shows this number is reasonable but the observed rate of WD/MS binaries likely suffers from selection effects due to low detection sensitivity to faint white dwarfs near bright main sequence stars [93].", "The host star of the system we study here, HD 159062A, is revealed spectroscopically as an old main sequence G-K dwarf star with low metallicity based on its $\\log g$ and $T_{\\rm eff}$ (H19).", "The observed Ca II HK emission $R^{^{\\prime }}_{\\rm HK}$ and rotation period led to an age diagnostic of $\\sim $ 7 Gyr.", "Literature reports on HD 159062A's age as derived from high-resolution spectroscopy range from approximately 5 Gyrs (e.g.", "[97]; [81]) to 8 Gyrs (e.g.", "[76]) to 14 Gyrs (e.g.", "[62]; [2]).", "Before the discovery of the white dwarf HD 159062B, [36] predicted the existence of a degenerate companion around HD 159062A due to a barium overabundance of $[{\\rm Ba/Fe}] = +0.4$ dex in HD 159062A.", "This is based on an empirical relation of the barium-to-iron enrichment as a function of age derived from an all-sky local population study of ancient Population II and intermediate-disk stars.", "Deviants from this Ba/Fe ratio versus age relation included five known stars with degenerate companions; HD 159062A was the most extreme outlier with an overabundance of barium.", "These five known systems almost certainly underwent mass transfer.", "[36] characterize them as field blue stragglers due to high rotation, excessive chromospheric activity, lithium depletion, low orbital eccentricity and noticeable age discrepancies with activity-based age indicators [35].", "Solar-type Population II stars are old; they rotate at low rotational velocities and show low levels of chromospheric and coronal activity [4], [66].", "Additional observed chromospheric activity could be explained by mass and angular momentum transfer from binary companions.", "Many of the stars with degenerate companions studied by [36] had slight barium depletions and $\\lesssim $ 1000-day orbital periods, suggestive of mass transfer from first-ascent red giant progenitors.", "HD 159062, however, had an anomalously high Ba/Fe ratio unlikely to have arisen from a field anomaly at its formation epoch; it must have been instead due to the s-process nucleosynthesis product of a binary companion.", "HD 159062's overabundance of barium suggests wind accretion of s-process elements from a more distant AGB progenitor rather than Roche lobe overflow from first-ascent red giant progenitor [9], [45], [54], so HD 159062B must have a longer period than $\\sim $ 1000 days.", "Indeed, [36] argue that barium enrichment via wind accretion only works for orbital periods from about 10 to 1000 years.", "This claim is backed up by an example of HD 114174, a cool white dwarf companion separated by $59.8 \\pm 0.4 $ AU from the G-type host star, whose orbital period is estimated to be between 154 and 881 years for eccentricities in the range 0$\\le $ e $\\le $ 0.5 [69].", "A barium overabundance of $[{\\rm Ba/Fe}] = +0.24$ dex is observed of the G star.", "The projected separation excludes the possibility of any substantial past mass exchange via Roche lobe overflow, however, [69] speculated that its orbit may have been pushed significantly outward during dynamical events such as the ejection of a third body.", "The case of HD 114174 suggests a similar scenario for HD 159062, and the presence of a wide, white dwarf companion.", "HD 159062B could be located at a wider separation of tens of AUs away from HD 159062A whose observed barium overabundance can be explained by wind accretion of s-process materials from a former AGB primary that survived as a white dwarf companion, as well as any other dynamical events that may have widened its orbit.", "Apart from HD 114174 and HD 159062, [36] also identified two systems, HR 3578 and 104 Tau, from their Ba/Fe ratio studies, both of which are without known companions but with slight barium anomalies.", "However, both stars' lack of strong astrometric accelerations are not suggestive of any orbiting white dwarf companions at orbital periods of 10 - 1000 years.", "HR 3578 does show a $\\sim $ 3$\\sigma $ acceleration in declination in the HGCA.", "104 Tau has been reported to be a visual binary [27], but more recent interferometric non-detections [82], and stable radial velocities with a shallow trend [15] strongly suggest otherwise.", "HR 3578 and 104 Tau both represent good targets for follow-up imaging." ], [ "Data and Previous Fit", "HD 159062B was discovered by H19 in 14 years of precise radial velocity (RV) data from the HIRES spectrograph on Keck [101], [52], and imaged using multi-epoch multi-band imaging observations from the ShaneAO system on the Lick Observatory 3-meter Shane telescope [41], [92], the PHARO AO system on the Palomar Observatory 5-meter telescope [47], and the NIRC2 AO system at the Keck II 10-meter telescope [105].", "H19 performed a joint MCMC analysis using 45 radial velocities and three NIRC2 astrometric measurements to derive the best orbital parameters.", "They assumed Gaussian priors on the mass of the primary star, the difference between the pre-upgrade velocity zero point and the post-upgrade velocity zero point and parallax, and uniform priors on all other fitted parameters.", "H19 derived a new spectroscopic mass of $0.76 \\pm 0.03 \\mbox{$M_{}$}$ , but adopted a prior of $0.80 \\pm 0.05 \\mbox{$M_{}$}$ based on literature values.", "H19 used 10 temperatures and 300 walkers in their MCMC in a total of $10^5$ steps.", "After the initial MCMC analysis, an additional prior on the white dwarf cooling age was added based on white dwarf cooling models and photometric constraints from ShaneAO, PHARO, and NIRC2s' photometry measurements in $J$ , $K_{s}$ and $L^{\\prime }$ .", "H19 concluded that the companion is an old $\\mathrm {M_{B} = 0.65^{+0.12}_{-0.04} \\mbox{$M_{}$}}$ with an orbital period of $P = 250^{+130}_{-76}$ years, and a cooling age of $\\mathrm {\\tau = 8.2^{+0.3}_{-0.5}}$ Gyr.", "More recently, [11] performed an orbital fit using similar data and orvara with the DR2 version of the HGCA.", "Our results are similar to theirs, but are slightly more precise with the updated absolute astrometry from Gaia EDR3.", "We omit the additional RVs from [11] (which do not extend the observational baseline) and the additional relative astrometry.", "This enables a direct comparison of our results with those of H19 and shows the power of absolute astrometry to constrain the orbit of HD 159062AB." ], [ "A New Fit", "We use orvara to perform a comprehensive joint MCMC analysis of HD 159062B using the same RV and imaging data as H19, but adding the cross-calibrated absolute astrometry of the HGCA.", "tab:HGCApm summarizes the HGCA proper motions for HD 159062B, including the correlation coefficients between proper motions in right ascension and declination, and the central epoch for each measurement.", "The relative astrometry data are from three direct imaging instruments: ShaneAO, PHARO and NIRC2.", "Similarly to H19, we restrict our analysis to NIRC2 with its well-measured distortion correction and track record of precision astrometry [90], [86].", "ll 0pt Basic Parameters and Default Priors Parameter Prior RV Jitter $\\sigma _{\\rm jit}$ $1/\\sigma $ (log-flat) Primary Mass $M_{\\rm pri}$ $1/M$ (log-flat) Secondary Mass $M_{\\rm sec}$ $1/M$ (log-flat) Semimajor axis $a$ $1/a$ (log-flat) $\\sqrt{\\varepsilon } \\sin \\omega $ uniform $\\sqrt{\\varepsilon } \\cos \\omega $ uniform Inclination $i$ $\\sin (i)$ , 0$^{\\circ }<i<180 ^{\\circ }$ Mean longitude at 2010.0 $\\lambda _{\\rm ref}$ uniform Ascending node $\\Omega $ uniform Parallax $\\varpi $ $\\exp [-\\frac{1}{2} (\\varpi - \\varpi _{\\it Gaia})^2/\\sigma _{\\varpi ,\\it Gaia}^2 ]$ Using orvara, we ran 10 temperatures and 100 walkers over $10^{5}$ steps to fit for nine parameters, keeping every 50th step.", "Our results are based on the `coldest' of 10 chains, with the `hottest' chain being the chain that effectively samples all of the allowed parameter space.", "The total time taken using two cores was 2177 s $\\sim $ 0.59 hr, and the mean acceptance fraction (cold chain) was 0.074. tab:priors provides a complete list of the parameters that we fit and the priors we used.", "We have marginalized out four parameters: the parallax $\\varpi $ , proper motion of the system barycenter, and RV offset as described in Section and the Appendix.", "lll 0pt MCMC Results Parameter Median$\\pm 1\\sigma $ 95.4% C.I.", "3cFitted parameters RV Jitter $\\sigma _{\\rm jit}$ $({\\rm m\\,s}^{-1})$ ${1.26}_{-0.30}^{+0.32}$ (0.65, 1.917) Primary Mass $M_{\\rm pri}$ $(M_{\\odot })$ $0.80 \\pm 0.05$ (0.70, 0.90) Secondary Mass $M_{\\rm comp}$ $(M_{\\odot })$ ${0.608}_{-0.0073}^{+0.0083}$ (0.594, 0.625) Semimajor axis $a$ (AU) ${62.0}_{-7.1}^{+7.0}$ (46.9, 83.3) $\\mathrm {\\sqrt{e}\\, sin\\, \\omega }$ ${-0.17}_{-0.11}^{+0.15}$ ($-$ 0.35, 0.13) $\\mathrm {\\sqrt{e}\\, cos\\, \\omega }$ ${0.00 \\pm 0.32}$ ($-$ 0.52, 0.55) $\\mathrm {Inclination}~i~(^\\circ )$ ${63.0}_{-2.3}^{+1.8}$ (56.1, 66.6) Mean longitude at $\\mathrm {\\lambda _{ref}~(^{\\circ })}$ a ${151.5}_{-6.5}^{+8.0}$ (131.845, 167.314) Ascending node $\\mathrm {\\Omega ~(^{\\circ })}$ ${133.4}_{-1.3}^{+1.7}$ (130.97, 138.92) Parallax $\\varpi $ (mas) $46.1856 \\pm 0.0042$ (46.177, 46.194) 3cDerived parameters Period (years) ${411}_{-70}^{+71}$ (270, 641) Argument of periastron $\\omega \\, (^{\\circ })$ ${260}_{-76}^{+70}$ (40, 348) Eccentricity $\\varepsilon $ ${0.102}_{-0.065}^{+0.11}$ (0.006, 0.37) Semimajor axis $a$ (mas) ${2862}_{-330}^{+320}$ (2170, 3850) Time of periastron $T_{0} = \\mathrm {t_{ref}} - P\\frac{\\lambda - \\omega }{360^{\\circ }}$ (JD) ${2506737}_{-31268}^{+16428}$ (2463393.772, 2583267.863) Mass ratio ${0.761}_{-0.045}^{+0.050}$ (0.677, 0.866) aThe reference epoch is $\\mathrm {t_{ref} = 2455197.5\\, JD\\, (\\lambda _{ref}; 2010\\, Jan\\, 1\\, 00:00\\, UT)}$ .", "We examine the chains of each parameter to make sure that the burn-in phase was complete and all walkers had stabilized in the mean and standard deviation of the posterior.We discard the first 500 recorded steps (the first 25000 overall, as we save every 50th) as the burn-in phase so that the chains are ready to be used for inference.", "The derived posterior probabilities from our joint orbit fit are given in tab:posteriors.", "With this case study, we provide an example of the eight plots produced by orvara, described in Section .", "Two types of reference schemes are available for coloring the curves randomly drawn from the posterior distributions, either based on the mass of the secondary companion $M_{\\rm sec}$ or eccentricity $\\varepsilon $ .", "We demonstrate both reference schemes for HD 159062B.", "Figure: Relative astrometric orbits and predicted future position for HD 159062B.", "Left: relative astrometric orbits colored according to the secondary mass.", "Middle: relative astrometric orbits colored by eccentricity.", "Right: astrometric prediction of the location of HD 159062B in 2040.", "𝐋𝐞𝐟𝐭𝐚𝐧𝐝𝐌𝐢𝐝𝐝𝐥𝐞:\\bf Left\\, and\\, Middle: the thick black lines indicate the highest likelihood orbit and the colorful thin lines from purple to yellow are 50 orbits drawn randomly from the posterior distributions colored according to the companion mass (left panel) or eccentricity (middle panel).", "The dotted lines connect the host star to the periastron passages.", "The black empty circles along the maximum likelihood deprojected orbit indicate epochs spaced by 10 years from 1990 to 2030.", "The dashed lines are the line of nodes.", "The filled orange circles are plotted along the maximum likelihood at the epochs corresponding to the relative astrometry used in our analysis.", "The arrows mark the direction of motion of the secondary companion.", "𝐑𝐢𝐠𝐡𝐭:\\bf Right: The contour lines demonstrate the likelihood of the location of HD 159062B in 2040, ranging from light yellow (least likely) to red to black (most likely).Figure REF shows the astrometric orbit of HD 159062B.", "The RVs, both over a full orbit and restricted to the time frame of observations, are displayed in Figure REF .", "The relative separation and position angles of the companion HD 159062B with respect to the host star HD 159062A, and the absolute astrometry of the host star from HGCA are illustrated in Figure REF .", "Finally, to evaluate the full behavior of the joint posterior distributions, we show a corner plot of the derived parameters of HD 159062B in Figure REF .", "Our joint orbit fit yields a companion mass of ${0.608}_{-0.0073}^{+0.0083} \\mbox{$M_{}$}$ , an orbiting period of $P = {411}_{-70}^{+71}$ years and an eccentricity of ${0.102}_{-0.065}^{+0.11}$ for the white dwarf HD 159062B.", "With our use of HGCA astrometry, we improve H19's constraint on HD 159062B's mass by an order of magnitude.", "Our tight constraints on the companion mass and eccentricity help to refine our picture of what the stellar system looks like.", "This can be used to infer the mass of the progenitor via the initial-final mass relation [55], which could serve as an input on the Barium enrichment of the primary star.", "We firmly place the system on the long period and low eccentricity tail of the distributions shown in Figure 6 of H19.", "This favors barium enrichment via wind accretion and rules out mass transfer by Roche lobe overflow.", "If the abundance measurement of $[{\\rm Ba/Fe}] = 0.4$ dex is accurate, HD 159062 would add to the few known cases of barium enrichment from wind accretion.", "This new information can be used to guide studies on the local stellar populations and Ba stars.", "Figure: Top: RV orbits induced by the directly imaged companion over a significant fraction of its very long period, ranging from 1990 to 2280.", "Bottom: RVs and the observed-calculated residuals, restricted to the observational time frame.", "In all of the panels, the thick black line is the highest likelihood orbit and the colorful lines are 50 orbits randomly drawn from the posterior probability distribution for HD 159062B.", "They are colored according to secondary mass (left panels) or eccentricity (right panels).", "The red solid points with error bars are RV data from Keck/HIRES with the best-fit RV zero point added.", "Error bars are too small to be visible except for some points on the plot of residuals.An RV of zero represents the system's barycentric velocity for the maximum likelihood orbit.The small bottom panel in each plot shows residuals after subtracting the RV orbit from the measurements.", "Due to the long orbital period of HD 159062B, more measurements are required to better constrain the orbital parameters.", "Moreover, we observe that with eccentricity as the reference coloring instead of the secondary mass, the RVs and residuals show more distinguishable structures.", "Lower eccentricity (more purple) orbits agree better with the highest likelihood orbit.Figure: Observed and fitted relative and absolute astrometry for the HD 159062AB system.", "From top to bottom, the panels show the relative separation and position angle of HD 159062B, and the proper motion of HD 159062A in right ascension and declination.", "The thicker black lines represent the best-fit orbit in the MCMC chain while the other 50 lines represent random draws from the chain, color-coded by either companion mass (left panels) or eccentricity (right panels).", "The lower insets should the difference between observed and calculated astrometry.", "The mean proper motion is used to compute the MCMC chain, but is not shown in the proper motion plots.", "This constraint is an integral over the proper motion between the Hipparcos and Gaia epochs.Figure: Corner plot of the derived parameters including mass of the primary star, mass of the secondary companion, semi-major axis, eccentricity, and inclination.", "The semimajor axis and eccentricity are highly covariant with one another and, to a lesser degree, with inclination." ], [ "Conclusions", "In this paper, we have presented the orbit fitting package orvara, designed to fit one or more Keplerian orbits to any combination of radial velocity, relative astrometry, and absolute astrometry.", "orvara achieves high performance by using a combination of a fast eccentric anomaly solver (faster than a standalone call to sine and cosine for a large number of data points), analytic marginalization of parallax and barycenter proper motion, and low-level memory management.", "orvara is free and open-source.", "It depends on other free packages including emcee [33], ptemcee [102], htof (Brandt et al.", "submitted), astropy [1], [80], along with numpy [75], [98] and scipy [100].", "orvara may be downloaded from github and installed by pip.", "We have tested its installation and operation on Windows, Mac and Linux machines.", "We have demonstrated orvara on the system HD 159062, a white dwarf-main sequence binary discovered by [50].", "Those authors found a white dwarf mass of $0.65_{-0.04}^{+0.12}~M_\\odot $ , and a semimajor axis and eccentricity that were compatible with a close pericenter passage.", "By combining absolute astrometry with the radial velocity and relative astrometry used by [50], we improve the precision of the mass by an order of magnitude, to $0.617_{-0.012}^{+0.013}~M_\\odot $ .", "We exclude orbits with a close pericenter passage, establishing that the binary was a case of wind accretion system in its past.", "orvara supports a variety of applications beyond what we have demonstrated here with HD 159062.", "It can account for multiple companions, it appropriately treats epoch astrometry from Hipparcos and/or Gaia, and it can account for a companion proper motion measured by Gaia.", "orvara is also designed to produce publication-ready plots.", "orvara may be configured with a single .ini file.", "We plan to keep orvara current, particularly when new Gaia data releases enable substantial improvements in astrometric precision.", "We thank an anonymous referee for many helpful comments, suggestions that improved the paper and the structure and usability of the code.", "G. M. B. is supported by the National Science Foundation (NSF) Graduate Research Fellowship under grant no.", "1650114.", "Here we derive the multivariate Gaussian that describes the parallax and barycenter proper motion at fixed values of the other orbital parameters.", "Marginalizing over this multivariate Gaussian is equivalent to replacing the parallax and proper motion with their mean (i.e.", "best-fit) values and multiplying by the square root of the determinant of the covariance matrix.", "We first write out the components of $\\chi ^2$ that involve the parallax $\\varpi $ and the proper motion of the system barycenter $\\mathbf {\\mu }$ .", "We assume here that we have a measurement of the secondary star's proper motion from Gaia; this measurement has a proper motion $\\mathbf {\\mu }_{G, \\rm o, B}$ and an inverse covariance matrix ${\\bf C}_{G, \\rm B}^{-1}$ .", "Measurements of the primary star use a suffix A, and proper motion model values refer to the proper motion of the primary star unless they carry the subscript B.", "If the secondary lacks a proper motion measurement in Gaia (as is the case for the companions in [13] and [24]), then we set ${\\bf C}_{G, \\rm B}^{-1} = 0$ .", "We also take model proper motions to be in physical units (e.g.", "AU yr$^{-1}$ ), so that multiplying by parallax gives proper motions in angular units.", "$\\chi ^2 = & \\frac{\\left(\\varpi - \\varpi _{\\it Gaia}\\right)^2}{\\sigma ^2_{\\varpi ,\\it Gaia}} + \\sum _{k=1}^{N_{\\rm ast}} \\frac{\\left( \\rho _k-\\varpi \\rho \\left[t_k\\right] \\right)^2}{(1 - c^2_{\\rho \\theta ,k}) \\sigma ^2 [\\rho _k]} - 2 \\sum _{k=1}^{N_{\\rm ast}} \\frac{c_{\\rho \\theta ,k}\\lfloor \\theta _k - \\theta [t_k]\\rfloor \\left( \\rho _k-\\varpi \\rho \\left[t_k\\right] \\right)}{(1 - c^2_{\\rho \\theta ,k})\\sigma [\\rho _k]\\sigma [\\theta _k]} \\nonumber \\\\&+\\left( {\\mathbf {\\mu }_{H, \\rm o, A}} - \\overline{\\mathbf {\\mu }} - \\varpi \\mathbf {\\mu }_{H} \\right)^T {\\bf C}_{H, \\rm A}^{-1} \\left( {\\mathbf {\\mu }_{H, \\rm o, A}} - \\overline{\\mathbf {\\mu }} - \\varpi \\mathbf {\\mu }_{H} \\right) \\nonumber \\\\&+\\left( {\\mathbf {\\mu }_{HG, \\rm o, A}} - \\overline{\\mathbf {\\mu }} - \\varpi \\mathbf {\\mu }_{HG} \\right)^T {\\bf C}_{HG, \\rm A}^{-1} \\left( {\\mathbf {\\mu }_{HG, \\rm o, A}} - \\overline{\\mathbf {\\mu }} - \\varpi \\mathbf {\\mu }_{HG} \\right) \\nonumber \\\\&+\\left( {\\mathbf {\\mu }_{G,\\rm o, A}} - \\overline{\\mathbf {\\mu }} - \\varpi \\mathbf {\\mu }_{G} \\right)^T {\\bf C}_{G, \\rm A}^{-1} \\left( {\\mathbf {\\mu }_{G, \\rm o, A}} - \\overline{\\mathbf {\\mu }} - \\varpi \\mathbf {\\mu }_{G} \\right) \\nonumber \\\\&+ \\left( {\\mathbf {\\mu }_{G, \\rm o, B}} - \\overline{\\mathbf {\\mu }} - \\varpi \\mathbf {\\mu }_{G,\\rm B} \\right)^T {\\bf C}_{G, \\rm B}^{-1} \\left( {\\mathbf {\\mu }_{G, \\rm o, B}} - \\overline{\\mathbf {\\mu }} - \\varpi \\mathbf {\\mu }_{G,\\rm B} \\right)$ We can minimize $\\chi ^2$ with respect to parallax and proper motion of the barycenter by solving ${\\bf M}\\begin{bmatrix}\\varpi \\\\\\overline{\\mu }_{\\alpha *} \\\\\\overline{\\mu }_{\\delta }\\end{bmatrix}={\\bf b}$ where $M$ is the symmetric matrix with elements $M_{\\varpi \\varpi } &= \\mathbf {\\mu }_{H}^T {\\bf C}_{H, \\rm A}^{-1} \\mathbf {\\mu }_{H}+ \\mathbf {\\mu }_{HG}^T {\\bf C}_{HG, \\rm A}^{-1} \\mathbf {\\mu }_{HG}+ \\mathbf {\\mu }_{G}^T {\\bf C}_{G, \\rm A}^{-1} \\mathbf {\\mu }_{G}+ \\mathbf {\\mu }_{G,\\rm B}^T {\\bf C}_{G, \\rm B}^{-1} \\mathbf {\\mu }_{G,\\rm B}+ \\frac{1}{\\sigma ^2_{\\varpi ,\\it Gaia}} + \\sum _{k=1}^{N_{\\rm ast}} \\frac{\\rho ^2\\left[t_k\\right]}{(1 - c^2_{\\rho \\theta ,k})\\sigma ^2 [\\rho _k]} \\\\M_{\\varpi \\alpha } &= \\mathbf {\\mu }_{H} \\cdot \\begin{bmatrix} C^{-1}_{H,{\\rm A},\\alpha \\alpha } \\\\ C^{-1}_{H,{\\rm A},\\alpha \\delta } \\end{bmatrix}+ \\mathbf {\\mu }_{HG} \\cdot \\begin{bmatrix} C^{-1}_{HG,{\\rm A},\\alpha \\alpha } \\\\ C^{-1}_{HG,{\\rm A},\\alpha \\delta } \\end{bmatrix}+ \\mathbf {\\mu }_{G} \\cdot \\begin{bmatrix} C^{-1}_{G,{\\rm A},\\alpha \\alpha } \\\\ C^{-1}_{G,{\\rm A},\\alpha \\delta } \\end{bmatrix}+ \\mathbf {\\mu }_{G,\\rm B} \\cdot \\begin{bmatrix} C^{-1}_{G,{\\rm B},\\alpha \\alpha } \\\\ C^{-1}_{G,{\\rm B},\\alpha \\delta } \\end{bmatrix} \\\\M_{\\varpi \\delta } &= \\mathbf {\\mu }_{H} \\cdot \\begin{bmatrix} C^{-1}_{H,{\\rm A},\\alpha \\delta } \\\\ C^{-1}_{H,{\\rm A},\\delta \\delta } \\end{bmatrix}+ \\mathbf {\\mu }_{HG} \\cdot \\begin{bmatrix} C^{-1}_{HG,{\\rm A},\\alpha \\delta } \\\\ C^{-1}_{HG,{\\rm A},\\delta \\delta } \\end{bmatrix}+ \\mathbf {\\mu }_{G} \\cdot \\begin{bmatrix} C^{-1}_{G,{\\rm A},\\alpha \\delta } \\\\ C^{-1}_{G,{\\rm A},\\delta \\delta } \\end{bmatrix}+ \\mathbf {\\mu }_{G,\\rm B} \\cdot \\begin{bmatrix} C^{-1}_{G,{\\rm B},\\alpha \\delta } \\\\ C^{-1}_{G,{\\rm B},\\delta \\delta } \\end{bmatrix} \\\\M_{\\alpha \\alpha } &= C^{-1}_{H,{\\rm A},\\alpha \\alpha } + C^{-1}_{HG,{\\rm A},\\alpha \\alpha } + C^{-1}_{G,{\\rm A},\\alpha \\alpha } + C^{-1}_{G,{\\rm B},\\alpha \\alpha } \\\\M_{\\delta \\delta } &= C^{-1}_{H,{\\rm A},\\delta \\delta } + C^{-1}_{HG,{\\rm A},\\delta \\delta } + C^{-1}_{G,{\\rm A},\\delta \\delta } + C^{-1}_{G,{\\rm B},\\delta \\delta } \\\\M_{\\alpha \\delta } &= C^{-1}_{H,{\\rm A},\\alpha \\delta } + C^{-1}_{HG,{\\rm A},\\alpha \\delta } + C^{-1}_{G,{\\rm A},\\alpha \\delta } + C^{-1}_{G,{\\rm B},\\alpha \\delta }$ and ${\\bf b}$ is given by $b_{\\varpi } &= \\mathbf {\\mu }_{H,\\rm o}^T {\\bf C}_{H, \\rm A}^{-1} \\mathbf {\\mu }_{H}+ \\mathbf {\\mu }_{HG,\\rm o}^T {\\bf C}_{HG, \\rm A}^{-1} \\mathbf {\\mu }_{HG}+ \\mathbf {\\mu }_{G,\\rm o}^T {\\bf C}_{G, \\rm A}^{-1} \\mathbf {\\mu }_{G}+ \\mathbf {\\mu }_{G,\\rm o, B}^T {\\bf C}_{G, \\rm B}^{-1} \\mathbf {\\mu }_{G,\\rm B} + \\frac{\\varpi _{\\it Gaia}}{\\sigma ^2_{\\varpi ,\\it Gaia}}\\nonumber \\\\ &\\qquad \\qquad + \\sum _{k=1}^{N_{\\rm ast}} \\frac{\\rho _k \\rho [t_k]}{(1 - c^2_{\\rho \\theta ,k})\\sigma ^2[\\rho _k]} - \\sum _{k=1}^{N_{\\rm ast}} \\frac{c_{\\rho \\theta ,k}\\rho [t_k]\\lfloor \\theta _k - \\theta [t_k]\\rfloor }{(1 - c^2_{\\rho \\theta ,k})\\sigma [\\rho _k]\\sigma [\\theta _k]}\\\\b_{\\alpha } &= \\mathbf {\\mu }_{H,\\rm o} \\cdot \\begin{bmatrix} C^{-1}_{H,{\\rm A},\\alpha \\alpha } \\\\ C^{-1}_{H,{\\rm A},\\alpha \\delta } \\end{bmatrix}+ \\mathbf {\\mu }_{HG,\\rm o} \\cdot \\begin{bmatrix} C^{-1}_{HG,{\\rm A},\\alpha \\alpha } \\\\ C^{-1}_{HG,{\\rm A},\\alpha \\delta } \\end{bmatrix}+ \\mathbf {\\mu }_{G,\\rm o} \\cdot \\begin{bmatrix} C^{-1}_{G,{\\rm A},\\alpha \\alpha } \\\\ C^{-1}_{G,{\\rm A},\\alpha \\delta } \\end{bmatrix}+\\mathbf {\\mu }_{G,\\rm o, B} \\cdot \\begin{bmatrix} C^{-1}_{G,{\\rm B},\\alpha \\alpha } \\\\ C^{-1}_{G,{\\rm B},\\alpha \\delta } \\end{bmatrix} \\\\b_{\\delta } &= \\mathbf {\\mu }_{H,\\rm o} \\cdot \\begin{bmatrix} C^{-1}_{H,{\\rm A},\\alpha \\delta } \\\\ C^{-1}_{H,{\\rm A},\\delta \\delta } \\end{bmatrix}+ \\mathbf {\\mu }_{HG,\\rm o} \\cdot \\begin{bmatrix} C^{-1}_{HG,{\\rm A},\\alpha \\delta } \\\\ C^{-1}_{HG,{\\rm A},\\delta \\delta } \\end{bmatrix}+ \\mathbf {\\mu }_{G,\\rm o} \\cdot \\begin{bmatrix} C^{-1}_{G,{\\rm A},\\alpha \\delta } \\\\ C^{-1}_{G,{\\rm A},\\delta \\delta } \\end{bmatrix}+\\mathbf {\\mu }_{G,\\rm o, B} \\cdot \\begin{bmatrix} C^{-1}_{G,{\\rm B},\\alpha \\delta } \\\\ C^{-1}_{G,{\\rm B},\\delta \\delta } \\end{bmatrix}.$ The covariance matrix of $\\varpi $ and the two components of $\\overline{\\mathbf {\\mu }}$ is the inverse of ${\\bf M}$ .", "Integrating, or marginalizing, over $\\overline{\\mathbf {\\mu }}$ and $\\varpi $ is equivalent to replacing $\\overline{\\mathbf {\\mu }}$ and $\\varpi $ in Equation (REF ) with the solutions of Equation (REF ) and dividing by the square root of the determinant of ${\\bf M}$ (multiplying by the square root of the determinant of the covariance matrix ${\\bf M}^{-1}$ )." ] ]
2105.11671
[ [ "Benzene Radical Anion Microsolvated in Ammonia Clusters: Modelling the\n Transition from an Unbound Resonance to a Bound Species" ], [ "Abstract The benzene radical anion, well-known in organic chemistry as the first intermediate in the Birch reduction of benzene in liquid ammonia, exhibits intriguing properties from the point of view of quantum chemistry.", "Notably, it has the character of a metastable shape resonance in the gas phase, while measurements in solution find it to be experimentally detectable and stable.", "In this light, our previous calculations performed in bulk liquid ammonia explicitly reveal that solvation leads to stabilization.", "Here, we focus on the transition of the benzene radical anion from an unstable gas-phase ion to a fully solvated bound species by explicit ionization calculations of the radical anion solvated in molecular clusters of increasing size.", "The computational cost of the largest systems is mitigated by combining density functional theory with auxiliary methods including effective fragment potentials or approximating the bulk by polarizable continuum models.", "Using this methodology, we obtain the cluster size dependence of the vertical binding energy of the benzene radical anion converging to the value of $-$2.3 eV at a modest computational cost." ], [ "Basis set benchmark", "The disperssion-corrected revPBE0-D3 [1], [2], [3], [4], [5], [6] density functional was used in line with our previous work on the benzene radical anion where we have documented the need for a hybrid functional to overcome the unphysical delocalization of the excess electron that is observed at the generalized-gradient-approximation level [7].", "Using this functional, the effect of the basis set size on the calculated VBEs was benchmarked.", "The VBEs were calculated for a randomly chosen cluster consisting of the benzene radical anion surrounded by 13 ammonia molecules.", "At this size, the system is already electronically bound and the corresponding VBE values are thus quantitative.", "As shown in Figure REF , Dunning type [8] basis sets provided roughly similar results to those obtained using Karlshure basis sets [9].", "The use of triple-$\\zeta $ basis sets lead to a significant improvement in comparison to the smaller double-$\\zeta $ ones.", "A further augmentation by diffuse (def2-TZVPD and aug-cc-pVTZ) or polarization functions (def2-QZVP and cc-pVQZ) resulted in an non-significant VBE decrease of less than 0.1 eV but in an order-of-magnitude elevation of the computational cost.", "Therefore, we conclude that the def2-TZVP basis set represents the best compromise between the accuracy of the method and its computational requirements.", "Figure: A benchmark of Karlsruhe (blue) and Dunning (orange) basis sets performance in combination with the revPBE0-D3 density functional.The binding energies are plotted as curves and their values are shown at the left-hand side yy-axis.The corresponding CPU times are depicted as bars with the same color coding.Moreover, for each basis set employed, the number of basis set functions is given in brackets under the basis set names on the top and bottom xx-axes.Note that the augmentation with diffuse functions (def2-TZVPD and aug-cc-pVTZ) is applied only on carbon atoms." ], [ "Force-Field Molecular Dynamics Details", "All auxiliary MD simulations were realized in the Gromacs 2020.4 [10], [11] software.", "A 100 ps long NVT equilibration run was performed at 223 K employing a 0.5 fs integration time step and the stochastic velocity rescaling thermostat [12].", "Energy and forces were evaluated using the rigid force field for liquid ammonia [13] and the generalized Amber force field describing the benzene radical anion [14].", "The method of resolvation relies on several steps that are summarized below.", "Carve a spherical cluster out of the original periodic AIMD trajectory frame.", "Its size is defined by distance between the center of mass of the benzene radical anion and the furthermost ammonia nitrogen atom within the cluster.", "Determine the Mulliken partial charges [15] at all atoms of a specific benzene structure at the revPBE0-D3/def2-TZVP level.", "Modify the topology of benzene radical anion with the obtained Mulliken charges.", "Center the cluster from the point 1 into a cubic unit cell with 73.85 Å side length.", "Fill this box with ammonia molecules randomly placed around the core cluster such that the minimal distance between atoms is not less than 2 Å. Equilibrate the system using NVT molecular dynamics with the original core constrained.", "Carve a resolvated cluster out of the last NVT equilibration trajectory frame with respect to an overall cutoff radius defining the cluster size including the inner core.", "Scaling of the computational cost for methods of QM-MM and QM-EFP is shown in Figure REF as a function of the overall cutoff radius of the resolvated cluster.", "Figure: The dependence of the CPU time needed to perform either a QM-EFP (top) or a QM-MM (bottom) calculation of the VBE on the cluster size.Different QM subsystem sizes are distinguished by the color scheme used consistently in the main text and explained in the legend." ], [ "Resolvation Employing the flexible Ammonia Force Field", "The VBE curves obtained from resolvation by the flexible liquid ammonia force field [16] are presented in Figure REF .", "Note the pronounced difference between the QM-MM curves here and those presented in Figure 5 of the main text.", "Figure: The mean VBEs of the clusters resolvated using the flexible ammonia force field  as a function of the overall cutoff radius (bottom xx-axis) as well as the inner core cutoff which is defined by the color coding.The QM-EFP curves are shown in the top panel and the QM-MM curves in the bottom panel.Representative numbers of average number of ammonia molecules in the clusters are given in the upper xx-axis." ], [ "PCM details", "The non-equlibrium PCM was based on the Marcus partition scheme [17].", "All PCM cavities were discretized into 194 surface points for the heavy atoms while for hydrogens we used 194 points per atom in the SES case and 110 for the vdW and SAS surfaces.", "The liquid ammonia environment was characterized for the non-equilibrium PCM purposes by a pair of dielectric constants splitting the solvent response into a slow (nuclear) and fast (electronic) parts: the low-frequency one of 22.66 and the high-frequency one of 1.9444 [18]." ] ]
2105.11757
[ [ "Robust Principal Component Analysis Using a Novel Kernel Related with\n the L1-Norm" ], [ "Abstract We consider a family of vector dot products that can be implemented using sign changes and addition operations only.", "The dot products are energy-efficient as they avoid the multiplication operation entirely.", "Moreover, the dot products induce the $\\ell_1$-norm, thus providing robustness to impulsive noise.", "First, we analytically prove that the dot products yield symmetric, positive semi-definite generalized covariance matrices, thus enabling principal component analysis (PCA).", "Moreover, the generalized covariance matrices can be constructed in an Energy Efficient (EEF) manner due to the multiplication-free property of the underlying vector products.", "We present image reconstruction examples in which our EEF PCA method result in the highest peak signal-to-noise ratios compared to the ordinary $\\ell_2$-PCA and the recursive $\\ell_1$-PCA." ], [ "Introduction", "In data analysis problems with a large number of input variables, dimension reduction methods are very useful to reduce the size of the input by decreasing the complexity of the problem while sacrificing negligible accuracy.", "Principal Component Analysis (PCA) and related methods are widely used in data analysis field as dimension reduction techniques [1], [2], [3], [4].", "In most problems, the lower dimensional subspaces that are obtained using the eigenvectors effectively capture the nature of the input data structure.", "As a result, PCA can be also used in a variety of applications including novelty detection [5], [6], data clustering [7], [8], [9], [10], [11], [12], [13], denoising [14], [15], [16], [17] and outlier detection [18], [19], [20], [21].", "Although the conventional PCA based on the regular dot-product and the $\\ell _2$ -norm has successfully solved many problems, it is sensitive to outliers in data because the effects of the outliers are not suppressed by the $\\ell _2$ -norm.", "It turns out that $\\ell _1$ -PCA is more robust to outliers and it can be iteratively solved in $ {O}(N^{rK-K+1})$ for $D$ dimensional vectors, where $N$ is the number of data vectors, $ 1\\le K<r=$ (rank of the $N\\times D$ data matrix) [22].", "Therefore, researchers proposed iterative methods to compute $\\ell _1$ -PCA to achieve robustness against outliers in data [22], [23].", "The recursive $\\ell _1$ -PCA method requires some parameters to be properly adjusted.", "On the other hand, the proposed kernel based approach does not need any hyperparameters to be adjusted.", "This is because we construct a sample covariance matrix using the kernel and obtain the eigenvalues and eigenvectors to define the orthogonal linear transformation instead of solving an optimization problem.", "We recently introduced a family of operators related with $\\ell _1$ -norm to extract features from image regions and to design Additive neural Networks (AddNet) in a wide range of computer vision applications [24], [25], [26], [27].", "We call the new family of operators Energy-Efficient (EEF) operators because they do not require any multiplications which consume more energy compared to additions and binary operations in most processors.", "Instead of a multiplication, the operators use the sign of multiplication and either sum the absolute values of operands, or calculate the minimum or maximum of operands.", "When we construct dot-product like operations from the EEF operators they induce the $\\ell _1$ -norm.", "Details of the EEF-operator are provided in Section .", "In this paper, we define three multiplication-free dot products and construct the corresponding multiplication-free covariance matrices.", "The fact that the underlying dot product is not an ordinary Euclidean inner product implies that the covariance matrix is not necessarily symmetric and positive semi-definitive.", "Nevertheless, we analytically prove that two of our vector products yield symmetric and positive semi-definite covariances.", "Correspondingly, we find the eigenvalues and eigenvectors of the matrices as in regular $\\ell _2$ -PCA.", "The resulting eigenvectors are orthogonal to each other and one can perform orthogonal projection onto the subspace formed by the eigenvectors to reduce the dimension, perform denoising and other similar PCA applications used in data analysis.", "In addition, the dot products defined by the operators can be computed without performing any multiplications.", "Consequently, the matrices of the new kernels can be computed in an energy efficient manner because the new kernels are based on sign operations, binary operations and additions." ], [ "Energy-Efficient (EEF) Vector Products", "In this section, we motivate and introduce the family of multiplication-free dot products and establish their relationship to the $\\ell _1$ -norm." ], [ "Motivation", "Let $\\mathbf {w} = [w_1 \\cdots w_n]^T \\in \\mathbb {R}^{D\\times 1}$ and $\\mathbf {x} = [x_1 \\cdots x_n]^T \\in \\mathbb {R}^{D\\times 1}$ be two $D$ -dimensional column vectors.", "The standard Euclidean inner product is defined as $\\langle \\mathbf {w}, \\mathbf {x} \\rangle = \\mathbf {w}^T \\mathbf {x} \\triangleq \\sum _{i=1}^D w_i x_i$ Note that because the product $\\langle \\cdot , \\cdot \\rangle $ induces the $\\ell _2$ -norm in the sense that for any $\\mathbf {x}$ , we have $\\langle \\mathbf {x}, \\mathbf {x}\\rangle = \\Vert \\mathbf {x}\\Vert ^2 = \\sum _{i=1}^D |x_i|^2$ .", "The $D$ multiplication operations that appear in the inner product Eq.", "(REF ) may be costly in terms of energy consumption and time.", "The existence of multiplications are also undesirable in the presence of outliers: For example, if a component is an outlier with a relatively large magnitude, multiplication will further amplify its effect, making the result of the inner product unreliable.", "In this context, it has been recently observed that in many applications, $\\ell _1$ -based methods outperform $\\ell _2$ -based methods thanks to their better resilience against outliers or impulse-type noise.", "These observations motivate us to define the new dot products that induce the $\\ell _1$ -norm.", "The new dot products should avoid multiplications both for the sake of computational and energy efficiency as well as robustness." ], [ "Multiplication-Free (MF) Dot Products", "In this work, we will evaluate the performance of three different MF operators, described in what follows.", "Given a real number $a\\in \\mathbb {R}$ , let $\\mathrm {sign}(a) ={\\left\\lbrace \\begin{array}{ll}-1,& a<0,\\\\0,& a=0,\\\\1,& a>0,\\end{array}\\right.", "}$ denote the sign of $a$ .", "Unlike [26] where we define $\\text{sign}(0)=1$ or $\\text{sign}(0)=-1$ to take advantage of bit-wise operations, we utilize the standard signum function for better precision here.", "First, we introduce our original MF dot product [24], [25].", "It is defined as $\\mathbf {w}^T\\oplus _{mf} \\mathbf {x} = \\sum _{i=1}^D \\text{sign}(w_i x_i)(|w_i| + |x_i|)$ Note that the only multiplication operations that appears in Eq.", "(REF ) correspond to sign changes and can be implemented with very low complexity.", "For this reason, we do not count the sign changes towards multiplication operations and thus call Eq.", "(REF ) an MF dot product.", "It can easily be verified that the product in Eq.", "(REF ) induces a scaled version of $\\ell _1$ -norm as $\\mathbf {x}^T \\oplus _{mf} \\mathbf {x} = \\sum _{i=1}^n |x_i|+|x_i| = 2\\Vert \\mathbf {x}\\Vert _1$ Notice that the original MF dot product conducts scale of 2, we are seeking another $\\ell _1$ -norm based method without any scaling.", "We then define a min-based MF dot product: $\\mathbf {w}^T \\odot \\mathbf {x} & \\triangleq \\sum _{i=1}^D \\text{sign}(w_i x_i) \\min (|w_i|, |x_i|).$ and its variation: $\\mathbf {w}^T\\odot _m \\mathbf {x} \\triangleq \\sum _{i=1}^D \\mathbf {1}\\left(\\text{sign}(w_i) = \\text{sign}(x_i)\\right) \\min (|w_i|, |x_i|)$ Here, $\\mathbf {1}(\\cdot )$ is the indicator function.", "The variant is related to the XX similarity measure [28].", "In Eq.", "(REF ), components of opposite sign $\\text{sign}(w_i) \\ne \\text{sign}(x_i)$ have no contribution towards the dot product, while in Eq.", "(REF ), they contribute as a subtractive term.", "Both of them induce $\\ell _1$ -norm as $\\mathbf {x}^T \\odot \\mathbf {x} = \\sum _{i=1}^n \\min (|x_i|, |x_i|) = \\Vert \\mathbf {x}\\Vert _1$ $\\mathbf {x}^T \\odot _m \\mathbf {x} = \\sum _{i=1}^n \\min (|x_i|, |x_i|) = \\Vert \\mathbf {x}\\Vert _1$ Vector dot products described above can be extended to matrix multiplications as follows: Let $\\mathbf {W} \\in \\mathbb {R}^{n\\times m}$ and $\\mathbf {X} \\in \\mathbb {R}^{n\\times p}$ be arbitrary matrices.", "We then define $\\mathbf {W^T} \\!\\oplus \\!", "\\mathbf {X} \\triangleq \\begin{bmatrix}\\mathbf {w}_1^T\\oplus \\mathbf {x}_1&\\mathbf {w}_1^T\\oplus \\mathbf {x}_2&\\dots &\\mathbf {w}_1^T\\oplus \\mathbf {x}_p\\!\\!\\!\\!\\!\\!\\\\\\mathbf {w}_2^T\\oplus \\mathbf {x}_1&\\mathbf {w}_2^T\\oplus \\mathbf {x}_2&\\dots &\\mathbf {w}_2^T\\oplus \\mathbf {x}_p\\!\\!\\!\\!\\!\\!\\\\\\vdots &\\vdots &\\ddots &\\vdots &\\\\\\mathbf {w}_m^T\\oplus \\mathbf {x}_1&\\mathbf {w}_m^T\\oplus \\mathbf {x}_2&\\dots &\\mathbf {w}_m^T\\oplus \\mathbf {x}_p\\!\\!\\!\\!\\!\\!\\end{bmatrix}$ where $\\oplus \\in \\lbrace \\oplus _{mf}, \\odot , \\odot _m\\rbrace $ , $\\mathbf {w}_i$ is the $i$ th column of $\\mathbf {W}$ for $ i = 1,\\ 2,\\ \\dots ,\\ m$ and $\\mathbf {x}_j$ is the $j$ th column of $\\mathbf {X}$ for $j = 1,\\ 2,\\ \\dots ,\\ p$ .", "In brief, the definition is similar to the matrix production $\\mathbf {W}^T\\mathbf {X}$ by only changing the element-wise product to element-wise MF-operation or element-wise min-operation." ], [ "Robust Principal Component Analysis", "Suppose that we collect members of a $D$ -dimensional dataset $\\lbrace \\mathbf {x}_1,\\ldots ,\\mathbf {x}_N\\rbrace $ to a $D\\times N$ matrix $\\mathbf {X}=[\\mathbf {x}_1 \\ \\mathbf {x}_2 \\ ... \\ \\mathbf {x}_N]\\in \\mathbb {R}^{D\\times N}$ .", "The well-known $\\ell _2$ -PCA method relies on investigating the eigendecomposition of the sample covariance matrix $\\mathbf {C} = \\mathbf {X}\\mathbf {X}^T.$ We have omitted normalization by the number of elements $N$ of the dataset as it will not change the final eigenvectors and the order of eigenvalues.", "Elementary linear algebra guarantees that $\\mathbf {C}$ has non-negative eigenvalues (i.e.", "$\\mathbf {C}$ is positive semi-definite) and thus the eigenvector corresponding to the $i$ th largest eigenvalue becomes the $i$ th principal vector.", "In this work, we propose to investigate the analogue of Eq.", "(REF ) for MF operators.", "In other words, we consider the eigendecomposition of $\\mathbf {A} = \\mathbf {X} \\oplus \\mathbf {X}^T,$ where $\\oplus \\in \\lbrace \\oplus _{mf},\\odot ,\\odot _m\\rbrace $ .", "Matrix $\\mathbf {A}$ is called as MF-covariance matrix.", "Note that the ordinary matrix product in Eq.", "(REF ) is replaced by the MF product in Eq.", "(REF ).", "On the other hand, since $\\mathbf {A}$ is no longer constructed using $\\ell _2$ -products, it is not guaranteed to be symmetric or positive semi-definite.", "Still, we have the following result.", "Theorem 1 Let $\\oplus \\in \\lbrace \\odot ,\\odot _m\\rbrace $ .", "Then, $\\mathbf {A} = \\mathbf {X} \\oplus \\mathbf {X}^T$ is symmetric and positive semi-definite for any $\\mathbf {X}$ .", "The proof can be found in the appendix.", "In particular, the theorem shows that $\\odot $ and $\\odot _m$ describe Mercer-type kernels.", "Theorem REF paves the way for extending PCA to multiplication-free operators $\\odot $ and $\\odot _m$ , as shown via Algorithm .", "[h] Algorithm for $L_1$ PCA using MF operators [1] $\\mathbf {X}=[\\mathbf {x}_1 \\ \\mathbf {x}_2 \\ ... \\ \\mathbf {x}_N]\\in \\mathbb {R}^{D\\times N}$ $\\mathbf {W} \\in \\mathbb {R}^{D\\times K}$ Construct the MF covariance matrix $\\mathbf {A}$ of $\\mathbf {X}$ based on Eq.", "(REF ).", "$[\\mathbf {W}, \\mathbf {D}] = \\text{eigs}(\\mathbf {A}, K)$ $\\mathbf {W}$ .", "Comment: Step 2 represents eigendecomposion of A and returns a subset of diagonal matrix $\\mathbf {D}$ of $K$ largest eigenvalues and matrix $\\mathbf {W}$ whose columns are the corresponding right eigenvectors, so that $\\mathbf {AW} = \\mathbf {WD}$ .", "Compared with the conventional $L_2$ -PCA Algorithm, we can see that the only difference is at Step 1.", "We replace the standard covariance matrix by the multiplication-free covariance matrix.", "The conclusions of Theorem REF does not hold for the $\\oplus _{mf}$ operator.", "A counterexample is provided by the dataset $\\mathbf {x}_1 = [1\\,\\,2]^T,\\,\\mathbf {x}_2 = [-1\\,-2]^T$ , which yields a generalized covariance matrix $\\mathbf {A} = [{\\begin{matrix} 2 & 6 \\\\ 6 & 8 \\end{matrix}}]$ with a negative determinant, and thus not positive semi-definite." ], [ "Experimental Results", "In this section, we carry out an image reconstruction and denoising experiment using the EEF kernel based PCAs, $\\ell _2$ -PCA and the recursive $\\ell _1$ -PCA to illustrate the robustness of the EEF kernel introduced in Section .", "Image reconstruction example is the same as the experiment in [22].", "The source code of [22] is available in [29], so we only set the tolerance parameter of the recursive $\\ell _1$ -PCA method as $1\\times 10^{-8}$ as suggested by the author P. Markopoulos.", "For convenience, we name our method based on Eq.", "(REF ) as \"MF-$\\ell _1$ \"-PCA, method based on Eq.", "(REF ) as \"min-$\\ell _1$ -PCA-1\" and method based on Eq.", "(REF ) as \"min-$\\ell _1$ -PCA-2\", respectively, in Table REF and Table REF .", "In the first row of Fig.", "REF , we have three $128\\times 128=16384$ \"clean\" gray-scaled images ($\\mathbf {I}\\in \\lbrace 0, \\frac{1}{255}, ..., \\frac{255}{255}\\rbrace ^{128\\times 128}$ ).", "We assume that the image $\\mathbf {I}$ is not available but we have $N=10$ occluded versions $\\mathbf {I}_1, \\mathbf {I}_2, ..., \\mathbf {I}_{10}$ , are available as shown in the second row of Fig.", "REF and Fig.", "REF .", "The occluded images are created by partitioning the original image $\\mathbf {I}$ into sixteen tiles of size $32\\times 32$ and replacing three arbitrarily selected tiles by $32\\times 32$ gray-scale-noise patches.", "The noise patches are in the uniformly random distribution in the interval $(0, 1)$ .", "In the second experiment, we add salt and pepper noise to images and restore the original images using various PCA methods.", "We assume that the image $\\mathbf {I}$ is not available but we have $N=10$ corrupted versions $\\mathbf {I}_1, \\mathbf {I}_2, ..., \\mathbf {I}_{10}$ , are available as shown in the third column (Fig.", "REF ) and the forth column (Fig.", "REF ) of Fig.", "REF , respectively.", "The corrupted images are created by adding salt and pepper noise to the original image $\\mathbf {I}$ with noise density 0.1.", "In other words, this affects 10% pixels by making them either 0 or 1 assuming that the image pixel values are in the range of $[0, 1]$ .", "Figure: Samples of image reconstruction results.", "Images in each columns are ordered as the original image (1 row), the noise patches occluded image (2 row, 1 and 2 columns) or salt-and-pepper noise corrupted image (2 row, 3 and 4 columns), results of ℓ 2 \\ell _2-PCA (3 row), recursive ℓ 1 \\ell _1-PCA (4 row), MF-ℓ 1 \\ell _1-PCA (5 row), min-ℓ 1 \\ell _1-PCA-1 (6 row) and min-ℓ 1 \\ell _1-PCA-2 (7 row), respectively.Table: PSNR (dB) of Image Reconstruction Results of Noise PatchesTable: PSNR (dB) of Image Reconstruction Results of Salt and Pepper NoiseWe perform PCA on the set of $\\mathbf {V} = [\\mathbf {v}_1\\ \\mathbf {v}_2\\ ...\\ \\mathbf {v}_{10}]$ , where $\\mathbf {v}_i = \\text{vec}(\\mathbf {I}_i), i=1, 2, ..., 10$ , is the vector form of $\\mathbf {I}_i$ .", "In this way, we obtain the eigenvector matrix $\\mathbf {W}\\in \\mathbb {R}^{16384\\times 2}$ of the covariance or the MF-covariance matrices of $(\\mathbf {V}-\\mathbf {\\bar{v}})$ .", "Then, we recover the image $\\mathbf {I}$ as $\\mathbf {\\hat{v}}_i=\\mathbf {WW}^T(\\mathbf {v}_i-\\mathbf {\\bar{v}})+\\mathbf {\\bar{v}}$ $\\mathbf {\\hat{I}} = \\text{mat}\\mathbf {\\hat{(v})_i}$ where $\\mathbf {\\bar{v}}\\in \\mathbb {R}^{16384\\times 1}$ is the mean value of $[\\mathbf {v}_1\\ \\mathbf {v}_2\\ ...\\ \\mathbf {v}_{10}]$ , $\\mathbf {0.5}$ or $\\mathbf {0}$ , $\\mathbf {I}_i$ is an arbitrary occluded image, and $\\text{mat}(\\cdot )$ is the inverse transform of $\\text{vec}(\\cdot )$ that reshapes a vector back to the matrix form.", "We calculate $\\mathbf {\\hat{v}}_i$ in the method that returns the largest peak signal-to-noise-ratio (PSNR).", "PSNR between the reconstructed image $\\mathbf {\\hat{I}}$ and the original image $\\mathbf {I}$ as the following equations is used for evaluation in Table REF and Table REF : $\\text{MSE} = \\text{mean}((\\mathbf {\\hat{I}}-\\mathbf {I})^2),$ $\\text{PSNR} = 10\\text{log}_{10}(\\frac{\\text{peakval}^2}{\\text{MSE}}),$ where $(\\cdot )^2$ is the element-wise square and “peakval\" is the peak signal value.", "The higher the value of PSNR is, the better the reconstruction result is.", "Our experiment is summarized in Algorithm .", "Results of these PCA methods are shown in Fig.", "REF for four test images and their statistics are provided in Table REF and Table REF .", "Although which method works the best depends on the images, our three methods return larger PSNR than $\\ell _2$ -PCA and the recursive $\\ell _1$ -PCA in both experiments, and the two min-$\\ell _1$ -PCAs are better than the MF-$\\ell _1$ -PCA, globally.", "For example, the min-$\\ell _1$ -PCA produces about $1.4dB$ better than the recursive $\\ell _1$ -PCA in the salt-and-pepper noise removal experiment.", "[htbp] Image Reconstruction Experiment [1] N corrupted images $\\mathbf {I}_1, \\mathbf {I}_2, ..., \\mathbf {I}_N\\in \\mathbb {R}^{D\\times D}$ .", "Reconstructed image $\\mathbf {\\hat{I}}$ .", "$i=1, 2, ..., N$ $\\mathbf {v_i} = \\text{vec}(\\mathbf {I_i})\\in \\mathbb {R}^{D^2\\times 1}$ ; $\\mathbf {V} = [\\mathbf {v}_1\\ \\mathbf {v}_2\\ ...\\ \\mathbf {v}_N]\\in \\mathbb {R}^{D^2\\times N}$ ; $\\mathbf {\\bar{v}} = \\mathbf {0}, \\mathbf {0.5}$ or $\\text{mean}(\\textbf {V})\\in \\mathbb {R}^{D^2\\times 1}$ ; Run PCA on $(\\mathbf {V}-\\mathbf {\\bar{v}})$ to obtain $K$ -dominant eigenvector matrix $\\mathbf {W} = [\\mathbf {w}_1\\ \\mathbf {w}_2 \\ ... \\ \\mathbf {w}_K]\\in \\mathbb {R}^{D^2\\times K}$ ; $\\mathbf {\\hat{v}}_i=\\mathbf {WW}^T(\\mathbf {v}_i-\\mathbf {\\bar{v}})+\\mathbf {\\bar{v}}\\in \\mathbb {R}^{D^2\\times 1}$ ; $\\mathbf {\\hat{I}} = \\text{mat}(\\mathbf {\\hat{v}}_i) \\in \\mathbb {R}^{D\\times D}$ ; $\\mathbf {\\hat{I}}$ .", "Comment: In this experiment, $N=10, D=128$ and $K=2$ .", "Function $\\text{mean}(\\cdot )$ is the mean of each row, so it returns a column vector.", "Function $\\text{vec}(\\cdot )$ reshapes a matrix into the column vector form, and function $\\text{mat}(\\cdot )$ is its inverse transform that reshapes a column vector back to the matrix form.", "$(\\mathbf {V}-\\mathbf {\\bar{v}})$ is defined as $[\\mathbf {v}_1-\\mathbf {\\bar{v}}\\ \\mathbf {v}_2-\\mathbf {\\bar{v}}\\ ...\\ \\mathbf {v}_N-\\mathbf {\\bar{v}}]$ .", "We also compared the computational cost of the PCA algorithms to reconstruct an image in MATLAB.", "As it is shown in Table REF , $\\ell _2$ -PCA is the fastest algorithm, while our proposed kernel methods are slightly slower than $\\ell _2$ -PCA but significantly faster than the recursive $\\ell _1$ -PCA.", "The recursive $\\ell _1$ -PCA is the slowest because it obtains the result by recursion, while $\\ell _2$ -PCA and our three methods return the result straight-forwardly.", "The reason why our kernel PCAs run a little slower than $\\ell _2$ -PCA is that, the time to construct an MF-covariance matrix is slightly slower compared to the sample covariance matrix, which is optimized in MATLAB.", "The computational cost of eigenvalue-eigenvector computations are the same in both $\\ell _2$ -PCA and the proposed kernel-PCAs.", "Table: Computational cost in seconds" ], [ "Conclusion", "In this paper, we proposed three new robust PCA methods.", "We have reached the following conclusions: (i) Proposed novel kernel methods are more energy-efficient than $\\ell _2$ -PCA because their Gram matrices are computed without any multiplication operations.", "(ii) They do not suffer from outliers in the data as in $\\ell _2$ -PCA because they are based on the $\\ell _1$ -norm.", "(iii) They do no require any hyper-parameter optimization as in the recursive $\\ell _1$ -PCA [22] because their Gram matrices are straightforward to compute as described in Eq.", "(REF ).", "We compared the new kernel-based methods with the $\\ell _2$ -PCA and the recursive $\\ell _1$ -PCA on an image reconstruction and salt-and-pepper noise removal tasks and found out that our min-$\\ell _1$ -PCAs returns the largest PSNR among these methods in most scenarios.", "Let $\\mathbf {x}, \\mathbf {y} \\in \\mathbb {R}^N$ .", "We define the min-operator $\\oplus : \\mathbb {R}^N \\times \\mathbb {R}^N \\mapsto \\mathbb {R}$ as following $\\mathbf {x} \\oplus \\mathbf {y} := \\sum _{i=1}^{N} \\text{sgn}(x_iy_i) \\min (|x_i|, |y_i|)$ In the following we will show that the operator $\\oplus $ defines a valid kernel $K(\\mathbf {x},\\mathbf {y})$ .", "A symmetric function $K: \\mathbb {R}^N \\times \\mathbb {R}^N \\mapsto \\mathbb {R}$ is a kernel iff $\\sum _{i=1}^{N} \\sum _{j=1}^{N} a_i a_j K(\\mathbf {x_i}, \\mathbf {x_j) \\ge 0}$ for any reals $a_i, a_j$ and for any vectors $\\mathbf {x_i}, \\mathbf {x_j} \\in \\mathbb {R}^N$ .", "In our case, we are interested in proving that $K(\\mathbf {x_i}, \\mathbf {x_j}) = \\mathbf {x}_i \\oplus \\mathbf {x}_j$ satisfies Eq.", "REF .", "Define a matrix $\\mathbf {K} \\in \\mathbb {R}^{N\\times N}$ such that $\\mathbf {K}_{ij} = \\text{sgn}(x_i x_j)\\text{min}(|x_i|,|x_j|)$ .", "Proving that $K(.,.", ")$ is a valid kernel is equivalent to proving that the matrix $\\mathbf {K}$ is positive semi-definite.", "We will use the following facts to construct our proof that $\\oplus $ is a kernel: Theorem 2 (Schur product theorem) [30] Let $\\mathbf {A}, \\mathbf {B} \\in \\mathbb {R}^{N \\times N}$ be two positive semi-definite matrices, then their Hadamard product $({\\mathbf {A}\\odot \\mathbf {B}})_{ij}:=\\mathbf {A}_{ij}\\mathbf {B}_{ij}$ is also positive semi-definite.", "Lemma 1 [28] R. Nader, A. Bretto, B. Mourad and H. Abbas.", "“On the Positive Semi-definite Property of Similarity Matrices.\"", "Theoretical Computer Science Let $\\mathbf {x} \\in \\mathbf {R}^N$ be a strictly positive vector.", "Then the matrix $\\mathbf {A}_{ij}:=\\min (x_i,x_j)$ is positive semi-definite.", "Our claim is the following: Corollary 2.1 Let $\\mathbf {x} \\in \\mathbf {R}^N$ .", "Then the matrix $\\mathbf {K}_{ij}:= \\text{sgn}(x_i x_j)\\min (|x_i|, |x_j|)$ is positive semi-definite.", "The matrix $\\mathbf {K}_{ij}$ can be written as hadamard product between matrix $\\mathbf {B}_{ij}=\\text{sgn}(x_i)\\text{sgn}(x_j)$ and $\\mathbf {A}_{ij}=\\min (|x_i|,|x_j|)$ , the matrix $\\mathbf {B}$ is a (rank-one) positive semi-definite matrix since it can be written as $\\text{sgn}(\\mathbf {x})\\text{sgn}(\\mathbf {x})^T$ .", "The matrix $\\mathbf {A}$ is positive semi-defnite according to Lemma 1.", "The Hadamard product $\\mathbf {K}=\\mathbf {A}\\odot \\mathbf {B}$ is positive semi-definite according to Theorem 1.", "Thus the $\\oplus $ operator defines a valid kernel." ] ]
2105.11634
[ [ "Clifford Algebras, Spinors and $Cl(8,8)$ Unification" ], [ "Abstract It is shown how the vector space $V_{8,8}$ arises from the Clifford algebra $Cl(1,3)$ of spacetime.", "The latter algebra describes fundamental objects such as strings and branes in terms of their $r$-volume degrees of freedom, $x^{\\mu_1 \\mu_2 ...\\mu_r}$ $\\equiv x^M$, $r=0,1,2,3$, that generalizethe concept of center of mass.", "Taking into account that there are sixteen $x^M$, $M=1,2,3,...,16$, and in general $16 \\times 15/2 = 120$ rotations of the form $x'^M = {R^M}_N x^N$, we can consider $x^M$ as components of a vector $X=x^M q_M$, where $q_M$ are generators of the Clifford algebra $Cl(8,8)$.", "The vector space $V_{8,8}$ has enough room for the unification of the fundamental particles and forces of the standard model.", "The rotations in $V_{8,8}\\otimes \\mathbb{C}$ contain the grand unification group $SO(10)$ as a subgroup, and also the Lorentz group $SO(1,3)$.", "It is shown how the Coleman-Mandula no go theorem can be avoided.", "Spinors in $V_{8,8}\\otimes \\mathbb{C}$ are constructed in terms of the wedge products of the basis vectors rewritten in the Witt basis.", "They satisfy the massless Dirac equation in $M_{8,8}$ with the internal part of the Dirac operator giving the non vanishing masses in four dimensions." ], [ "Introduction", "Unification of fundamental interactions is still an unfinished project in theoretical physics.", "Among many attempts string theory has for long time been a very promising avenue.", "Strings are extended objects that upon quantization “miraculously” give rise to gauge fields.", "Namely, the quantum excitations of a string contain Yang-Mills fields or gravity, depending on whether the string is open or closed.", "In other words, the interaction fields are associated with a string's (quantum) configurations.", "Another possible approach to the unification employs the idea that the arena for physics is not spacetime, but the configuration space of matter[1], [2], [3], [4].", "A matter configuration can be modeled as a multiparticle system represented as a point in configuration space ${\\cal C}$ , whose dimension equals the number of particles multiplied by the dimension of spacetime $M_{1,3}$ .", "In such scenario, spacetime is a subspace of configuration space, associated with a chosen single particle.", "One can then proceed à la Kaluza-Klein and obtain 4D gravity and gauge fields from the higher dimensional gravity in ${\\cal C}$[2].", "This, among others, also brings an insight why a string configuration contains gauge interaction fields.", "In string theory gauge fields occur at the quantum level, but according to the postulated general relativity[1], [2], [3], [4] in configuration space, they occur already at the classical level for any matter configuration and therefore also for a string.", "A string as an extended object has infinite dimensions and can be represented as a point in an infinite dimensional configuration space ${\\cal C}$ .", "Choosing a point $P$ on the string, e.g., its center of mass, or one of its ends, it should then in principle be possible to perform a Kaluza-Klein reduction from ${\\cal C}$ to $M_{1,3}$ , association with $P$ , and so obtain 4D gravity and gauge fields.", "In the latter description we assumed that the string is embedded in 4-dimensional spacetime.", "This is possible at the classical level, while in the quantized theory one encounters inconsistencies which disappear in 26 dimensions.", "A peculiar feature of a string is that though extended, it is still singular, because it is infinitely thin.", "This indicates that a string is an idealized description of an actual physical object.", "Exact strings, as well as exact point particles, do not exist in nature.", "Physical objects are not exactly point-like nor exactly string-like.", "They have thickness.", "In Refs.", "[5], [6], [1], [7], [8], [11], [9], [10], [12], [13] it was explained how a physical object can be described not only in terms of its center of mass coordinates, i.e., the coordinates of a point, but can also be sampled in more detail by means of oriented lengths, areas, volumes and 4-volumes, in general, $r$ -volumes, $r=0,1,2,3,4$ , also called $r$ -areas.", "Convenient mathematical objects that describe $r$ -volumes are Clifford numbers, the elements of a Clifford algebra, in our case the Clifford algebra $Cl(1,3)$ of spacetime.", "An extended objects can thus be described by $2^4=16$ degrees of freedom.", "By starting with spacetime and considering the Clifford algebra over it we thus gain a lot of new room for description of particles and the interactions between them.", "Namely, a $Cl(1,3)$ can be considered as a tangent space at a given point of a 16-dimensional manifold, called Clifford space $C$ which, in general, can be curved.", "Employing the Kaluza-Klein recipe, we thus obtain the unification of interactions in Clifford space[9], [10].", "Once we have a Clifford space $C$ , we can consider it as a manifold of “its own”[14] and forget that we have arrived at it via Clifford algebra $Cl(1,3)$ .", "Thus we just have a 16-dimensional manifold $M_{8,8}$ , such that the tangent space at any of its points is a vector space $V_{8,8}$ .", "By considering the rotations in $V_{8,8}$ we thus have the orthogonal group $SO(8,8)$ whose subgroups are $SO(6,4)$ and $SO(2,4)$ .", "The group Spin(6,4), the cover group of $SO(6,4)$ , is isomorphic to the Pati-Salam[15], [16] grand unification group $SU(4)\\times SU(4)$ .", "If we consider complex valued spinors generated by basis vectors of the vector space $V_{6,4}$ , then we can equivalently consider complex valued spinors generated by, e.g., the vectors of $V_{0,10}$ that we denote $V_{10}$ .", "The group $SO(10)$ of rotations in $V_{10}$ is considered, besides $SU(4)\\times SU(4)$ and $SU(5)$ , in grand unification of particles and forces (see the instructive review by Baez[16]).", "Concerning the group $SO(2,4)$ , it is related to the Stueckelberg theory[17], [18], [19], [20] with evolution parameter and to the two times physics considered by Bars[21].", "It contains the Lorentz group $SO(1,3)$ as a subgroup.", "In this paper we consider the spinors of only one minimal left ideal of $Cl(8,8)$ and show that such a setup incorporates grand unified theories.", "Consideration of the full $Cl(8,8)$ or its subgroup $Cl(8)$ brings a lot of additional possibilities concerning the unification[22], [23], [24], [25], [26], [27], [28], [29]." ], [ "From the distances in spacetime to Clifford space: How a 16D space is “embedded”\nin a 4D space", "Spacetime consists of point events.", "The squared distance between two infinitesimally close events is $\\mbox{\\rm d}s^2 = \\eta _{\\mu \\nu } \\mbox{\\rm d}x^\\mu \\mbox{\\rm d}x^\\nu ~, ~~~~~\\mu , \\nu = 0,1,2,3 ,$ There are two possible ways of taking the square root of the quadratic form (REF ): $(i) ~~~\\sqrt{\\mbox{\\rm d}s^2} = \\mbox{\\rm d}s = \\sqrt{\\eta _{\\mu \\nu } \\mbox{\\rm d}x^\\mu \\mbox{\\rm d}x^\\nu } ,$ where $\\mbox{\\rm d}s$ is the infinitesimal scalar distance; $(ii) ~~~\\sqrt{\\mbox{\\rm d}s^2} = \\mbox{\\rm d}x = \\mbox{\\rm d}x^\\mu \\gamma _\\mu , \\hspace{28.45274pt}$ Here $\\mbox{\\rm d}x$ is the infinitesimal vector that joints the point with the coordinates $x^\\mu $ and $x^\\mu + \\mbox{\\rm d}x^\\mu $ , while $\\gamma _\\mu $ are basis vectors satisfying $\\gamma _\\mu \\cdot \\gamma _\\mu \\equiv \\frac{1}{2} (\\gamma _\\mu \\gamma _\\nu + \\gamma _\\nu \\gamma _\\mu ) = \\eta _{\\mu \\nu }.$ The latter relation defines the generators $\\gamma _\\mu $ of the Clifford algebra Cl(1,3) of spacetime $M_{1,3}$ .", "The vector (REF ) is an oriented line element.", "By wedge products of vectors we can form oriented areas, volumes and 4-volumes: $\\mbox{\\rm d}x \\wedge \\mbox{\\rm d}x^{\\prime } = \\frac{1}{2} \\mbox{\\rm d}x^{\\mu \\nu } \\gamma _\\mu \\wedge \\gamma _\\nu ,$ $\\mbox{\\rm d}x \\wedge \\mbox{\\rm d}x^{\\prime } \\wedge \\mbox{\\rm d}x^{\\prime \\prime }= \\frac{1}{3!}", "\\mbox{\\rm d}x^{\\mu \\nu \\rho } \\gamma _\\mu \\wedge \\gamma _\\nu \\wedge \\gamma _\\rho ,$ $\\mbox{\\rm d}x \\wedge \\mbox{\\rm d}x^{\\prime } \\wedge \\mbox{\\rm d}x^{\\prime \\prime } \\wedge \\mbox{\\rm d}x^{\\prime \\prime }= \\frac{1}{4!}", "\\mbox{\\rm d}x^{\\mu \\nu \\rho \\sigma } \\gamma _\\mu \\wedge \\gamma _\\nu \\wedge \\gamma _\\rho \\wedge \\gamma _\\sigma ,$ where $\\mbox{\\rm d}x^{\\mu \\nu }$ , $\\mbox{\\rm d}x^{\\mu \\nu \\rho }$ and $\\mbox{\\rm d}x^{\\mu \\nu \\rho \\sigma }$ are the antisymmetrized products of the vector components $\\mbox{\\rm d}x^\\mu $ , $\\mbox{\\rm d}x^{\\prime \\mu }$ , $\\mbox{\\rm d}x^{\\prime \\prime \\mu }$ , $\\mbox{\\rm d}x^{\\prime \\prime \\prime \\mu }$ , namely, $\\mbox{\\rm d}x^{\\mu \\nu } = \\mbox{\\rm d}x^\\mu \\mbox{\\rm d}x^{\\prime \\nu }- \\mbox{\\rm d}x^{\\prime \\mu }\\mbox{\\rm d}x^\\nu ,$ $\\mbox{\\rm d}x^{\\mu \\nu \\rho } = \\mbox{\\rm d}x^\\mu \\mbox{\\rm d}x^{\\prime \\nu }\\mbox{\\rm d}x^{\\prime \\prime \\rho }+ \\mbox{\\rm d}x^{\\prime \\mu }\\mbox{\\rm d}x^{\\prime \\prime \\nu }\\mbox{\\rm d}x^\\rho + \\mbox{\\rm d}x^{\\prime \\prime \\mu }\\mbox{\\rm d}x^\\nu \\mbox{\\rm d}x^{\\prime \\rho }- \\mbox{\\rm d}x^\\mu \\mbox{\\rm d}x^{\\prime \\prime \\nu }\\mbox{\\rm d}x^{\\prime \\prime \\rho }-\\mbox{\\rm d}x^{\\prime \\prime \\mu }\\mbox{\\rm d}x^{\\prime \\nu }\\mbox{\\rm d}x^\\rho -\\mbox{\\rm d}x^{\\prime \\mu }\\mbox{\\rm d}x^\\nu \\mbox{\\rm d}x^{\\prime \\prime \\rho },$ and the analogous expression for $\\mbox{\\rm d}x^{\\mu \\nu \\rho \\sigma }$ .", "The wedge product denotes the antisymmetrized product of vectors.", "For the basis vectors we have: $\\gamma _\\mu \\wedge \\gamma _\\nu \\equiv \\frac{1}{2} \\left(\\gamma _\\mu \\gamma _\\nu - \\gamma _\\nu \\gamma _\\mu \\right),$ $\\gamma _\\mu \\wedge \\gamma _\\nu \\wedge \\gamma _\\rho \\equiv \\frac{1}{3!}", "\\left(\\gamma _\\mu \\gamma _\\nu \\gamma _\\rho +\\gamma _\\nu \\gamma _\\rho \\gamma _\\mu + \\gamma _\\rho \\gamma _\\mu \\gamma _\\nu - \\gamma _\\mu \\gamma _\\rho \\gamma _\\nu - \\gamma _\\rho \\gamma _\\nu \\gamma _\\mu - \\gamma _\\nu \\gamma _\\mu \\gamma _\\rho \\right),$ and analogous for $\\gamma _\\mu \\wedge \\gamma _\\nu \\wedge \\gamma _\\rho \\wedge \\gamma _\\sigma $ .", "In a manifold there are infinitely many ways of connecting its point into submanifolds, for instance, into surfaces of a lesser dimensionality.", "In particular, the submanifolds can be associated with physical objects in spacetime.", "For instance, a set of points can lie on a closed string.", "Another possibility is the set of points lying on an open membrane whose boundary is a closed loop.", "An infinitesimal oriented area of that membrane (Fig.", "1) is the wedge product of two tangent vectors $ \\mbox{\\rm d}\\Sigma = \\mbox{\\rm d}\\xi _1 \\wedge \\mbox{\\rm d}\\xi _2$ .", "Expanding $\\mbox{\\rm d}\\xi _1$ and $\\mbox{\\rm d}\\xi _2$ in terms of the tangent basis vectors $e_a$ , $a=1,2$ , on the membrane, we have $\\mbox{\\rm d}\\Sigma = \\mbox{\\rm d}\\xi _1 \\wedge \\mbox{\\rm d}\\xi _2 = \\mbox{\\rm d}\\xi _1^a \\mbox{\\rm d}\\xi _2^b e_a \\wedge e_b = \\frac{1}{2} \\mbox{\\rm d}\\xi ^{ab} e_a \\wedge e_b ,$ where $\\mbox{\\rm d}\\xi ^{ab} = \\mbox{\\rm d}\\xi _1^a \\mbox{\\rm d}\\xi _2^b - \\mbox{\\rm d}\\xi _2^a \\mbox{\\rm d}\\xi _1^b$ .", "Recalling now that using the membrane's embedding functions $X^\\mu (\\xi ^a)$ , we have $\\mbox{\\rm d}x^\\mu = \\partial _a X^\\mu \\mbox{\\rm d}\\xi ^a,$ where $\\partial _a \\equiv \\partial /\\partial \\xi ^a$ .", "Multiplying the right and the left side of Eq.", "(REF ) with $\\gamma _\\mu $ , we obtain $\\mbox{\\rm d}x = \\mbox{\\rm d}x^\\mu \\gamma _\\mu = \\partial _a X^\\mu \\mbox{\\rm d}\\xi ^a \\gamma _\\mu = \\mbox{\\rm d}\\xi ^a e_a = \\mbox{\\rm d}\\xi ,$ from which we have the the following relation between the spacetime basis vectors $\\gamma _\\mu $ and the brane's basis vectors $e_a$ : $e_a = \\partial _a X^\\mu \\gamma _\\mu .$ Inserting (REF ) into Eq.", "(REF ) and integrating the infinitesimal oriented surface elements $\\mbox{\\rm d}\\Sigma $ over the membrane, we obtain[8] $\\int _\\Sigma \\mbox{\\rm d}\\Sigma = \\frac{1}{4} \\int _\\Sigma \\mbox{\\rm d}\\xi ^{ab}\\left(\\partial _a X^\\mu \\partial _b X^\\nu - \\partial _a X^\\nu \\partial _b X^\\mu \\right) \\gamma _\\mu \\wedge \\gamma _\\nu = \\frac{1}{2} x^{\\mu \\nu } \\gamma _\\mu \\wedge \\gamma _\\nu ,$ where $x^{\\mu \\nu } = \\frac{1}{2} \\int _\\Sigma \\mbox{\\rm d}\\xi ^{ab}\\left(\\partial _a X^\\mu \\partial _b X^\\nu - \\partial _a X^\\nu \\partial _b X^\\mu \\right) .$ By the Stokes theorem, Eq.", "(REF ) becomes $x^{\\mu \\nu } = \\frac{1}{2} \\int _B \\mbox{\\rm d}s\\left(X^\\mu \\frac{\\partial X^\\nu }{\\partial s} - X^\\nu \\frac{\\partial X^\\mu }{\\partial s} \\right) ,$ where $X^\\mu (s)$ are the embedding functions of the boundary loop $B$ , $s$ being a parameter along the loop.", "The quantity $x^{\\mu \\nu }$ denotes the effective oriented area of the surface $\\Sigma $ bounded by the loop $B$ .", "From the relation (REF ) we see that $X^{\\mu \\nu }$ depends only on the boundary $B$ and is independent on the shape of $\\Sigma $ .", "Eqs.", "(REF ) and (REF ) tell us that we have a mapping $X^\\mu (\\xi ^a) \\longrightarrow x^{\\mu \\nu },$ or equivalently, $X^\\mu (s) \\longrightarrow x^{\\mu \\nu },$ from an infinite dimensional object, namely a surface $\\Sigma $ , described by $X^\\mu (\\xi ^a)$ , or its boundary $B$ , described by $X^\\mu (s)$ , into the finite dimensional object $x^{\\mu \\nu }$ .", "Figure: An aread element dΣ\\mbox{\\rm d}\\Sigma on a surface with the boundary BB.Such arrangement can describe two physically distinct objects[30]: (i) A loop $B$ can be a closed instantonic string.", "Then $x^{\\mu \\nu }$ are the bivector coordinates associated with the closed instantonic string.", "(ii) A surface $\\Sigma $ can correspond to an open instantonic 2-brane whose boundary is $B$ .", "Then $x^{\\mu \\nu }$ are bivector coordinates associated with the open instantonic 2-brane.", "Analogous holds for objects of arbitrary dimension.", "The corresponding multivector ($r$ -vector) coordinates are then $x^{\\mu _1\\mu _2 ...\\mu _r} = \\frac{1}{r!}", "\\int \\mbox{\\rm d}\\xi ^{a_1 a_2 ...a_r}\\partial _{a_1} X^{[\\mu _1} \\partial _{a_1} X^{\\mu _2} ...\\partial _{a_r} X^{\\mu _r ]} \\hspace{56.9055pt}$ $\\hspace{56.9055pt} = \\frac{1}{r!}", "\\int \\mbox{\\rm d}s^{\\underline{a}_1 \\underline{a}_2 ...\\underline{a}_{r-1}}X^{[\\mu _1} \\partial _{\\underline{a}_1} X^{\\mu _2} ...\\partial _{\\underline{a}_{r-1}} X^{\\mu _r ]}$ Here $\\mbox{\\rm d}\\xi ^{a_1 a_2 ...a_r} \\equiv \\mbox{\\rm d}\\xi ^{[a_1} \\mbox{\\rm d}\\xi ^{a_2}...\\mbox{\\rm d}\\xi ^{a_r ]}$ and $\\mbox{\\rm d}s^{\\underline{a}_1 \\underline{a}_2 ...\\underline{a}_r}\\equiv \\mbox{\\rm d}s^{[\\underline{a}_1} \\mbox{\\rm d}s^{\\underline{a}_2}...\\mbox{\\rm d}s^{\\underline{a}_r]}$ are, respectively, the infinitesimal elements of an $r$ -dimensional surface and of its $(r-1)$ -dimensional boundary.", "The parameters (coordinates) denoting the points in those manifolds are $\\xi ^a$ , $a=1,2,...,r$ , and $s^{\\underline{a}}$ , $\\underline{a} =1,2,...,r-1$ , while $\\partial _a \\equiv \\partial /\\partial \\xi ^a$ and $\\partial _{\\underline{a}} \\equiv \\partial /\\partial s^{\\underline{a}}$ are the derivatives with respect to those parameters.", "The bracket `$[\\ ]$ ' denotes antisymmetrization of the expression.", "The functions $X^\\mu (\\xi ^a)$ , $a=1,2,...,r$ , can describe an $r$ -dimensional surface, shortly $r$ -surface, bounded by an $(r-1)$ -dimensional surface.", "Equation (REF ) determines the mapping from an infinite dimensional object, namely an $r$ -surface, $X^\\mu (\\xi ^a)$ , $a=1,2,...,r$ , or its boundary, $X^\\mu (s^{\\underline{a}})$ , $\\underline{a} = 1,2,...,r-1$ , into the finite dimensional object $x^{\\mu _1,\\mu _2,...,\\mu _r}$ .", "The quantity $x^{\\mu _1,\\mu _2,...,\\mu _r}$ can thus describe[31], [30] two distinct types of physical objects in spacetimeThe objects considered here are point “particles” (events), strings, membranes, in general, branes, in spacetime.", "Usually those names refer to the objects in space (that is, in a three dimensional subspace of spacetime) that in spacetime sweep a worlsline, a worldsheet, a worlvolume, etc.. An $r$ -brane considered here does not sweep an $(r+1)$ -dimensional worldsheet, therefore we call it instantonic $r$ -brane.", "Once this nomenclature is clear, we can omit “instantonic”.", "a) an open instantonic $r$ -brane; b) a closed instantonic $(r-1)$ -brane.", "The mapping is many–to–one, so that $x^{\\mu _1,\\mu _2,...,\\mu _r}$ is associated with a class of those physical objects, whose representative is an oppen instantonic $r$ -brane or, alternatively, a closed instantonic $(r-1)$ -brane.", "Distinction between those two types of objects can be formally made by means of a scalar parameter $\\sigma $ which in the case of an open string is proportional to its length, in the case of an open membrane (2-brane) to its scalar area, and, in general, the scalar $r$ -volume ($r$ -area) of an open $r$ -brane.", "For a closed $(r-1)$ -brane the scalar parameter $\\sigma $ vanishes.", "For instance, in the case of a closed string there is no “material” embraced by the string, therefore the scalar area $\\sigma $ associated with such string is zeroRecall that the corresponding 2-vector (oriented) area of a closed string is different from zero and given by the bivector $X^{\\mu \\nu } \\gamma \\wedge \\gamma _\\nu $ ..", "This is not so so for an open membrane and, in general, for an open $r$ -brane.", "Then $\\sigma = \\frac{1}{A_r} \\int \\mbox{\\rm d}\\xi ^1 \\mbox{\\rm d}\\xi ^2 ... \\mbox{\\rm d}\\xi ^r \\left( {\\rm det}\\frac{\\partial X^\\mu }{\\partial \\xi ^a} \\frac{\\partial X^\\nu }{\\partial \\xi ^b} \\eta _{\\mu \\nu } \\right)^{1/2},~~~a,b,1,2,...,r,$ is different from zero.", "The quantity $A_r$ , $r=1,2,3,4$ , is defined as[30] $A_r = \\sqrt{X^\\ddagger *X}~,~~~~~ X= x^{\\mu _1 \\mu _2...\\mu _r} \\gamma _{\\mu _1} \\wedge \\gamma _{\\mu _2}...\\wedge \\gamma _{\\mu _r} ,$ where $\\sqrt{~~~}$ is the scalar square root of the expression.", "Here we have generalized the case considered in Eqs.", "(REF ) and (REF ).", "The symbol $\\ddagger $ denotes reverions, i.e., the operation that reverses the order of vectors in an expression, e.g., $(\\gamma _\\mu \\gamma _\\nu )^\\ddagger = \\gamma _\\nu \\gamma _\\mu $ ; the star $*$ denotes the scalar product of two multivectors, defined as $A*B = \\langle A B \\rangle _0$ , where $\\langle ~~~ \\rangle _0$ denotes the scalar part of the expression.", "For an open string the multivector grade is $r=1$ and the formula (REF ) gives $X^\\mu = X_2^\\mu - X_1^\\mu ,$ where $X_1^\\mu $ and $X_2^\\mu $ are the coordinates of the string's ends.", "Equation (REF ) is the result of the integration of the oriented line elements (REF ), i.e, the infinitesimal vectors, along the string.The result of such integration is a finite vector $X^\\mu \\gamma _\\mu $ , whose components are given in Eq.", "(REF ).", "The scalar associated with an open string is determined by Eq.", "(REF ) for $r=1$ : $\\sigma = \\frac{1}{A_1} \\int \\mbox{\\rm d}\\xi \\left(\\frac{\\partial X^\\mu }{\\partial \\xi }\\frac{\\partial X^\\nu }{\\partial \\xi }\\eta _{\\mu \\nu } \\right)^{1/2},$ where $A_1 = \\sqrt{(X^\\mu \\gamma _\\mu )^2} = \\sqrt{X^\\mu X^\\nu \\eta _{\\mu \\nu }} \\equiv \\sqrt{( X_2^\\mu - X_1^\\mu )( X_2^\\nu - X_1^\\nu ) \\eta _{\\mu \\nu }} ,$ which, in general, is differrent from the string length $\\int \\mbox{\\rm d}\\xi \\left(\\frac{\\partial X^\\mu }{\\partial \\xi }\\frac{\\partial X^\\nu }{\\partial \\xi }\\eta _{\\mu \\nu } \\right)^{1/2}$ .", "This scalar $\\sigma $ is obtained if we integrate the infinitesimal scalar distances (line elements) along the string.", "Let us now take into account that a string is an idealization and that the actual object is not a string but a thick string.", "Then also the 2-vector, 3-vector and 4-vector coordinates $X^{\\mu \\nu }$ , $X^{\\mu \\nu \\rho }$ , can be different from zero.", "The multivector coordinates $X^{\\mu _1,\\mu _2,...,\\mu _r}$ , $r=0,1,2,3,4$ , describing an extended object in spacetime, are components of a Clifford number $X = \\sigma \\underline{1} + x^\\mu \\gamma _\\mu + \\frac{1}{2} x^{\\mu \\nu } \\gamma _\\mu \\wedge \\gamma _\\nu + \\frac{1}{3!}", "x^{\\mu \\nu \\rho } \\gamma _\\mu \\wedge \\gamma _\\nu \\wedge \\gamma _\\rho + \\frac{1}{4!}", "x^{\\mu \\nu \\rho \\sigma } \\gamma _\\mu \\wedge \\gamma _\\nu \\wedge \\gamma _\\rho \\wedge \\gamma _\\sigma $ $= \\sum _{r=0}^4 x^{\\mu _1 \\mu _2 ... \\mu _r} \\gamma _{\\mu _1 \\mu _2 ... \\mu _r}\\equiv \\sum x^M \\gamma _M \\equiv x^M \\gamma _M ~~(\\text{Einstein's summation convention}).$ Here $\\gamma _{\\mu _1 \\mu _2 ... \\mu _r} \\equiv \\gamma _{\\mu _1} \\wedge \\gamma _{\\mu _2}\\wedge ... \\wedge \\gamma _{\\mu _r}|_{\\mu _1 < \\mu _2 < ... < \\mu _r}$ , therefore the factor $1/r!$ does not take place in the expansion of $X$ in terms of $\\gamma _{\\mu _1 \\mu _2 ... \\mu _r}$ .", "In the last step we have condensed the notation even more by introducing $\\gamma _M\\equiv \\gamma _{\\mu _1 \\mu _2 ... \\mu _r}$ and $x^M\\equiv x^{\\mu _1 \\mu _2 ... \\mu _r}|_{\\mu _1 < \\mu _2 < ... < \\mu _r}$ .", "The Clifford number $X$ , the so called polyvector, is the sum of multivectors of the grades from $r=0$ to $r=4$ .", "It denotes position in the 16-dimensional Clifford space $C$ , a manifold that has at any of its points the Clifford algebra $Cl(4)$ , more precisely, $Cl(1,3)$ , as the tangent space.", "The objects described by $x^{\\mu _1 \\mu _2 ... \\mu _r}$ are “instantonic” in the sense that they occupy a finite region of spacetime, so that they are localized not only in 3-space but also in time.", "They are not infinitely extended along a time-like direction as in the case of a particle's worldline or a string's worldsheet.", "When we talk about a “particle” we usually have in mind an object that looks like a point in 3-space, and as a line in 4D spacetime $M_4$ .", "Similarly, by “string” we usually mean anobject whose spatial form is a string, while it is a 2-dimensional sheet in $M_4$ .", "In this work we consider the concept of instantonic string, instantonic membrane, and, in general, an instantonic $r$ -brane, $r=0,1,2,3,4$ which generalize the concept of event in spacetime.", "The configuration spaces of those objects, described by the embedding functions $X^\\mu (\\xi ^a)$ , $a=1,2,3,4$ , are infinite dimensional.", "The description of $r$ -branes in terms of the multivector coordinates $X^{\\mu _1,\\mu _2,...,\\mu _r}$ , $r=0,1,2,3,4$ , is achieved by the mapping $X^\\mu (\\xi ^a) \\longrightarrow x^{\\mu _1 \\mu _2 ...\\mu _r}~,~~~~~~\\mu =0,1,2,3;~~a=1,2,...,n; ~~~~n=1,2,3,4,$ according to Eqs.", "(REF ), (REF ).", "The cases $n=1,2,3,4$ denote, respectively, an instantonic string, instantonic membrane, instantonic 3-brane and instantonic 4-brane.", "The latter object fils a 4-dimensional region of spacetime; it is like a spacetime filling brane that not fills all but only a portion of spacetime.", "The description of an instantonic object by a polyvector $X$ , as defined in Eq.", "(REF ), comprises the usual $p$ -branes, $p=1,2,3$ , as limiting cases in which their time-like extension goes to infinity.", "A polyvector $X=x^M \\gamma _M$ , $M = 1,2,...,16$ , thus encodes a configuration of an extended object, not in all its infinite detail, but in terms of the oriented $r$ -areas associated with the object" ], [ "Clifford space as the arena for physics", "The dynamics of the objects described by polyvectors has been investigated in Refs.", "[13], [31], where the concept of relativity in Clifford space ($C$ -space) was developed[5], [1], [11], [10], [13] and pointed out how it leads to the unification of particles and forces[9], [10], [32].", "This is possible because the arena for physics is taken to be the 16-dimensional Clifford space $C$ .", "The points of $C$ correspond to the instantonic extended objects in 4D spacetime modeled by the polyvector coordinates $x^{\\mu _1 \\mu _2 ... \\mu _r} \\equiv x^M$ .", "Taking the differential of the polyvector (REF ), $\\mbox{\\rm d}X = \\mbox{\\rm d}x^M \\gamma _M ,$ the quadratic form in Clifford space is, $\\mbox{\\rm d}S^2 = \\mbox{\\rm d}X^\\ddagger * \\mbox{\\rm d}X = \\mbox{\\rm d}x^M \\mbox{\\rm d}x^N \\eta _{MN},$ where the metric is $\\eta _{MN} = \\gamma _M^\\ddagger * \\gamma _N = \\langle \\gamma _M^\\ddagger \\gamma _N \\rangle _0 .$ With such a definition of the metric, the signature is $(8,8)$ , so that the explicit form of the expression (REF ) is $&&\\mbox{\\rm d}S^2 = \\mbox{\\rm d}\\sigma ^2 + (\\mbox{\\rm d}x^0)^2 - (\\mbox{\\rm d}x^1)^2 - (\\mbox{\\rm d}x^2)^2 - (\\mbox{\\rm d}x^3)^2 \\nonumber \\\\&& \\hspace{28.45274pt} -(\\mbox{\\rm d}x^{01})^2 - (\\mbox{\\rm d}x^{02})^2 - (\\mbox{\\rm d}x^{03})^2 + (\\mbox{\\rm d}x^{12})^2 + (\\mbox{\\rm d}x^{13})^2 +(\\mbox{\\rm d}x^{23})^2 \\nonumber \\\\&&\\hspace{28.45274pt}- (\\mbox{\\rm d}{\\tilde{x}}^0)^2 + (\\mbox{\\rm d}{\\tilde{x}}^1)^2 + (\\mbox{\\rm d}{\\tilde{x}}^2)^2+(\\mbox{\\rm d}{\\tilde{x}}^3)^2 - \\mbox{\\rm d}{\\tilde{\\sigma }}^2 ,$ which comes after rewriting the polyvector (REF ) according to $X = \\sigma {\\underline{1}} + x^\\mu \\gamma _\\mu + \\frac{1}{2} x^{\\mu \\nu } + {\\tilde{x}}^\\mu \\gamma _5 \\gamma _\\mu + {\\tilde{\\sigma }} \\gamma _5 ,$ where ${\\tilde{x}}^\\mu = \\frac{1}{3!}", "{\\epsilon ^\\mu }_{\\nu \\rho \\sigma } x^{\\nu \\rho \\sigma }~,~~~~~~~{\\tilde{\\sigma }} = \\frac{1}{4!}", "\\epsilon _{\\mu \\nu \\rho \\sigma } x^{\\mu \\nu \\rho \\sigma }~,$ $\\gamma _\\mu \\wedge \\gamma _\\nu \\wedge \\gamma _\\rho = {\\epsilon _{\\mu \\nu \\rho }}^\\sigma \\gamma _5 \\gamma _\\sigma ~,~~~~~~~~ \\gamma _\\mu \\wedge \\gamma _\\nu \\wedge \\gamma _\\rho \\wedge \\gamma _\\sigma = \\epsilon _{\\mu \\nu \\rho \\sigma } \\gamma _5 .$ A worldline in $C$ , described by the equation $x^M = X^M (\\tau )$ , represents the evolution of a `thick’ particle in spacetime (Fig. 2).", "Thick particle can be an aggregate of $p$ -branes for various $p=0,1,2,…$ .", "But such interpretation is not obligatory.", "Thick particle may be a conglomerate of whatever extended objects that can be sampled by polyvector coordinates $X^M \\equiv X\\,^{\\mu _1 \\mu _2 ...\\mu _r }$ , $r=0,1,2,3$ .", "The action for such a system is $I = \\kappa \\int _{}^{} {\\mbox{\\rm d}\\tau \\,(\\eta _{MN} \\dot{X}^M \\dot{X}^N } )^{1/2},$ where $\\kappa $ is a constant having the role of mass in $C$ .", "This is just like the point particle action, only that the “point particle” is now in 16-dimensional Clifford space $C$ .", "The equation of motion $\\ddot{X}^M \\, \\equiv \\,\\,\\frac{{\\,{\\rm {d}}^{\\rm {2}} X^M }}{{{\\rm {d}}\\tau ^2 }}\\,\\, = \\,\\,0$ describes a flat worldline in $C$ , which corresponds to a tensionless brane in $M_4$ .", "For the branes with tension one has to introduce curved Clifford space in which instead of the flat space metric $\\eta _{MN}$ we take a curved metric $g_{MN}$ .", "Then Eq.", "(REF ) generalizes to the equation of a geodesic in $C$ .", "For a particular choice of metrc[31] one obtains the description of the Dirac-Nambu-Goto brane sampled the coordinates $X^M \\equiv X\\,^{\\mu _1 \\mu _2 ...\\mu _r }$ .", "Figure: A worldline in Clifford space CC corresponds to a thick worldline inspacetime M 4 M_4 (up).", "A wordlsheet in CC corresponds to a thick worldsheet in M 4 M_4 (down).A world sheet in $C$ , described by the equation $x^M = X^M (\\tau ,\\sigma )$ , represents the evolution of a `thick’ string in spacetime $M_4$ (Fig. 2).", "Thick string can be an aggregate $p$ -branes for various $p=0,1,2,…$ .", "But such interpretation is not obligatory.", "Thick string may be a conglomerate of whatever extended objects that can be sampled by polyvector coordinates $X^M \\equiv X\\,^{\\mu _1 \\mu _2 ...\\mu _r }$ .", "The action for a $C$ -space string in conformal gauge is $I = \\frac{\\kappa }{2}\\int \\mbox{\\rm d}\\tau \\,\\mbox{\\rm d}\\sigma \\,({\\dot{X}}^M {\\dot{X}}^N - X^{\\prime M} X^{\\prime N} )\\,\\eta _{MN}$ The space in which such string lives is Clifford space.", "Its dimension is 16, and signature $(8,8)$ .", "In Ref.", "[33] it was shown that upon quantization there are no central terms in the Virasoro algebra, if the space in which the string lives has signature $(8,8)$ , provided that the definition of vacuum as considered in Refs.", "[36], [34], [37], [35], [38] is used.", "According to Refs.", "[38], [39], such vacuum definition gives the correct quantization of the harmonic oscillator in a pseudo-Euclidean space, because it gives the correct classical limit." ], [ "Promoting the Clifford algebra $Cl(1,3)$ to a vector space {{formula:7784f041-8505-4387-953e-03bdf50e448a}}", "Clifford algebra is a vector space.", "In particular, $Cl(1,3)$ is a 16D vector space with signature $(8,8)$ .", "Let us denote it $V_{8,8}$ .", "We will now adopt the view of Ref.", "[14] and consider $V_{8,8}$ as a vector space spanned by the basis vectors $q_M$ that satisfy $q_M \\cdot \\,q_N \\, = \\frac{1}{2}\\left( {q_M q_N \\, + q_N q_M } \\right)\\, = \\,\\eta _{MN} ~,~~~~~~~M,N=1,2,3,...,16,$ where, as before, $\\eta _{MN} = {\\rm diag} (1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1)$ , i.e., the metric with signature $(r,s) =(8,8)$ .", "Therefore, if, instead of the basis $\\gamma _M \\equiv \\gamma _{\\mu _1 \\mu _2 ... \\mu _r}$ of the Clifford algebra $Cl(1,3)$ , we take the generators $q_M$ of $Cl(8,8)$ , satisfying Eq.", "(REF ), we obtain the same quadratic form (REF ).", "The $V_{8,8}$ is a tangent space to the 16-dimensional manifold $M_{8,8}$ .", "In general one has to distinguish between the coordinate frame field and the orthonormal frame field.", "Let us denote with $q_M$ the coordinate basis vectors and with $q_A$ , $A =1,2,3,...,16$ , the orthonormal basis vectors.", "Here for simplicity we use the same symbol $q$ and distinguish those two different sorts of objects by the indices $M$ and $A$ .", "A possible local decomposition of $M_{8,8}$ is $M_{8,8} = M_{1,1} \\,\\dot{+}\\, M_{1,3}\\, \\dot{+} \\,M_{6,4}$ The subspace $ M_{1,3}$ is the Minkowski spacetime.", "The subspace $M_{1,1}$ together with $M_{1.3}$ forms the six-dimensional space $M_{2,4}$ .", "Good features of $M_{2,4}$ : a) It is the arena for 2T physics (Bars[21]).", "b) It enables the Stueckelberg theory[17], [18], [19], [20], [1].", "c) It is the arena for the conformal group.", "The subspace $M_{6,4}$ has the role of the internal space that brings into the game the additional interactions, besides the gravity in 4D spacetime.", "A good feature of $M_{6,4}$ is that it serves as the arena for the SO(10) grand unification.", "In this scheme the overall arena for physics is $M_{8,8}$ .", "A point in $M_{8,8}$ has coordinates $X^M$ , $M = 1,2,...,16$ .", "The group SO(8,8) acting within a tangent space $V_{8,8}$ of $M_{8,8}$ contains: $SO(8,8) \\supset \\,SO(2,4) \\times SO(6,4),$ where $SO(2,4) \\supset \\,SO(1,3) \\times \\,SO(1,1)$ Here $SO(1,3)$ is the Lorentz group which together with $SO(1,1)$ is a subset of the conformal group $SO(2,4)$ .", "For the second factor in Eq.", "(REF ) we have $SO(6,4) \\supset SO(6) \\times SO(4) \\leftarrow SU(4) \\times SU(2) \\times SU(2),$ which gives the Pati-Salam unified model[15], [16].", "We have thus arrived at the framework which enables the unification of the Lorentz group with the Pati-Salam group, a descent of which is the Standard model gauge group $SU(3) \\times SU(2) \\times U(1)$ .", "What about the Coleman-Mandula theorem[40]?", "Its starting assumption is the unitarity of the $S$ -matrix in $M_{1,3}$ , which holds for a wave function that depends on position in spacetime.", "As a consequence, no mixing of spacetime and internal symmetries is possible within such a setup.", "This is not true for a theory whose starting point is a higher dimensional space, such as, e.g., the Clifford space or the space of the Kaluza-Klein theory, or the space $M_{8,8}$ considered here, in which the wave function is a function of all $N>4$ coordinates, and thus unitarity holds in the higher dimensional space.", "Then, fundamentally, there is no distinction between external and internal symmetries, and therefore they can mix among themselves.", "Only after a symmetry breaking, eg., in Kaluza-Klein theories due to isometries along the internal dimensions or a compactification of small extra dimensions, a wave function is effectively a function of spacetime coordinates only.", "Then, of course, for such effective theory in four dimensions one gets that the spacetime and internal symmetries do not mix.", "However, for an underlying more fundemental theory in a higher dimensional space, the Coleman-Mandula theorem is not applicable.", "To sum up, the spacetime and the internal symmetries can only not mix after a symmetry breaking which sets apart the 4D spacetime and a higher dimensional \"internal\" space.", "That there is a loophole in the Coleman-Mandula theorem was shown in Ref.", "[23] by a different argumentation, namely, that before symmetry breaking, there is no metric and thus no S-matrix.", "Vector fields in $M_{8,8}$ Upon the (first) quantization of the classical system described by the action (REF ) we obtain the Klein-Gordon equation in $M_{8,8}$ : $(\\partial ^M \\partial _M - \\kappa ^2 )\\Phi (X^M ) = 0~,~~~~M=1,2,3,...,16,$ where $X^M$ denotes coordinates of position in $M_{8,8}$ , and $\\partial _M \\equiv \\partial /\\partial x^M$ .", "The latter equation comes from the classical momentum constraint $P_M P^M - \\kappa ^2=0$ which becomes the operator equation acting on a state $\\Phi $ .", "For a state we take a vector field in $M_{8,8}$ , $\\Phi = \\phi ^A q_A~,~~~~~q_A \\cdot q_B = \\eta _{AB}~,~~~~~A,B=1,2,3,...,16.$ If the components $\\phi ^A$ are complex valued, then we can choose a new basis vectors $q^{\\prime A}$ such that the same vector field $\\Phi $ can be expanded in terms of those new basis vectors according to $\\Phi = \\phi ^{\\prime A} q^{\\prime }_A~,~~~~~q^{\\prime }_A \\cdot q^{\\prime }_B = \\eta ^{\\prime }_{AB}~,~~~~~A,B=1,2,3,...,16,$ where the signature of the new metric $\\eta ^{\\prime }_{AB}$ differs from the signature of the old metric $\\eta _{AB}$ .", "The new signature, $(r,s)$ , can be arbitrary.", "In particular it can be $r=2$ , $s=14$ , which corresponds to $V_{2,14} \\, = \\,V_{2,4} {\\dot{+}} V_{0,10} .$ Instead of the basis of the original vector space $V_{8,8}$ we can thus take the basis of the space $V_{2,14}$ that can be split into the space $V_{2,4}$ and $V_{0,10}$ .", "Over the latter vector space we can construct the spinors of the $SO(10)$ grand unification.", "Spinor fields in $M_{8,8}$ We have considered complex vector fields in the manifold $M_{8,8}$ .", "Let us now consider complex spinor fields in $M_{8,8}$ .", "At any of its points the tangent space is $V_{8,8}$ .", "Its basis vector $q_A$ , $A=1,2,3,...,16$ , can be split into the time-like and space-like part according to $\\begin{array}{l}q_A = (q_a ,\\tilde{q}_a )\\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,a = 1,2,3,...,8\\, \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,q_a^\\dag = \\,q_a \\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,q_a \\cdot q_b \\, = \\delta _{ab} \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\tilde{q}_a^\\dag = \\,\\, - \\tilde{q}_a \\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\tilde{q}_a \\cdot \\tilde{q}_b^{} \\, = \\, - \\delta _{ab} \\\\\\end{array}$ An alternative basis is the Witt basis: $\\begin{array}{l}\\chi _a \\, = \\frac{1}{2}\\,\\left( {q_a + \\tilde{q}_a } \\right) \\\\\\chi _a^\\dag \\, = \\frac{1}{2}\\,\\left( {q_a - \\tilde{q}_a } \\right) \\\\\\end{array}$ Writing $\\tilde{q}_a = i \\bar{q}_a$ we have $\\begin{array}{l}q_A = (q_a ,i\\,\\bar{q}_a )\\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,a = 1,2,3,...,8\\, \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,q_a^\\dag = \\,q_a \\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,q_a \\cdot q_b \\, = \\delta _{ab} \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\bar{q}_a^\\dag = \\,\\bar{q}_a \\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\bar{q}_a \\cdot \\bar{q}_b^{} \\, = \\,\\delta _{ab}\\end{array}$ The Witt basis then reads $\\begin{array}{l}\\chi _a \\, = \\frac{1}{2}\\,\\left( {q_a + \\,i\\,\\bar{q}_a } \\right) \\\\\\chi _a^\\dag \\, = \\frac{1}{2}\\,\\left( {q_a - \\,i\\,\\bar{q}_a } \\right)\\end{array}$ It satisfies the fermionic anticommutation relations $\\lbrace \\chi _a ,\\chi _b^\\dagger \\rbrace \\, = \\,\\delta _{ab} ~,~~~~\\lbrace \\chi _a ,\\chi _b \\rbrace \\, =\\lbrace \\chi _a^\\dagger ,\\chi _b^\\dag \\rbrace \\, =0.$ Spinors are given in terms of the creation operators $\\chi _a^\\dagger $ acting on the vacuum[41], [42], [43], [44], [45], [46], [47], [48] $\\Omega = \\prod \\limits _{a = 1}^8 {\\chi _a }$ A generic spinor field in $M_{8,8}$ is: $\\Psi = \\left( {\\psi ^0 1 + \\psi ^{a_1 } \\chi _{a_1 }^\\dag + \\psi ^{a_1 a_2 } \\chi _{a_1 }^\\dag \\chi _{a_2 }^\\dag + ...\\psi ^{a_1 a_2 ...a_8 } \\chi _{a_1 }^\\dag \\chi _{a_2 }^\\dag ...\\chi _{a_8 }^\\dag } \\right)\\Omega \\equiv \\psi ^{\\tilde{A}} \\xi _{\\tilde{A}},$ which is a superposition of spinor components $\\psi ^{\\tilde{A}}$ and basis spinors $\\xi _{\\tilde{A}}$ .", "We assume that it depends on position in $M_{8,8}$ and satisfies the generalized Dirac equation [9], [10], [49]: $\\left( i\\gamma ^M \\partial _M + \\kappa \\right)\\Psi = 0,$ where $\\kappa $ is the mass in sixteen dimensions.", "This is analogous to the usual procedure where spinors are assumed to satisfy the Dirac equation in $M_{1,3}$ .", "In general, the components $\\psi ^{a_1 a_2 ...a_r }$ , $r = 0,1,2,...,8$ , are complex.", "Therefore the same spinor can be generated in terms of the basis vectors $q_A$ of any signature $p,q$ .", "For instance, instead of constructing spinors over $V_{8,8} \\otimes \\mathbb {C}$ we can construct them over $(V_{2,4} \\, \\dot{+} \\,V_{0,10} ) \\otimes \\mathbb {C}.$ In the subspace $V_{2,4}$ live the spinors of Minkowski space, whilst in $V_{0,10}$ live the spinors of $SO(10)$ grand unification.", "Spinors in $M_{8,8}$ are functions of position $X^M$ in $M_{8,8}$ .", "They have values (as members of minimal ideals) in the complexified $Cl(8,8)$ which can be written as $\\begin{array}{l}Cl(8,8) \\otimes C = Cl(2,14) \\otimes C \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, = \\left( {Cl(2,4) \\otimes Cl(10)} \\right) \\otimes C\\,\\,\\, = Cl(16) \\otimes C \\\\\\end{array}$ Crucial here are the two subalgebras: (i) In $Cl(2,4)$ are contained the spinors of Minkowski space, as a part of a larger theory which can be, as mentioned before, the 2T physics by Bars, the Stueckelberg theory, or conformal theory.", "(ii) In $Cl(10)$ have values the spinors of $SO(10)$ grand unification.", "In the following, instead of considering the full theory in $M_{8,8}$ , we will confine us here to the theory in the subspace $M_{7,7}= M_{1,3} \\dot{+} M_{6,4}$ and so neglect the piece $M_{1,1}$ which together with spacetime forms $M_{2,4}$ .", "A spinor field in $M_{7,7}$ can be writen as $\\Psi = \\psi ^{\\tilde{A}} \\xi _{\\tilde{A}} \\equiv \\psi ^{\\alpha i} \\xi _{\\alpha i}\\,\\,,\\,\\,\\,\\,\\,\\,\\,\\alpha = 1,2,3,4,~~i =1,2,...,32,$ where $\\xi _{\\tilde{A}}$ , $\\tilde{A} =1,2,3,...,128$ are the basis spinorsWe now use for the basis spinors and the spinor components in $M_{7,7}$ the same symbols as we did in Eq.", "(REF ) for those in $M_{8,8}$ .", "in $M_{7,7}$ and $\\psi ^{\\tilde{A}}$ the spinor components.", "If the manifold $M_{7,7}$ is flat, then $\\xi _{\\tilde{A}}$ can be written as the product $\\xi _{\\tilde{A}} = \\xi _\\alpha \\eta _i$ of the basis spinors $\\xi _\\alpha $ in $M_{1,3}$ and the basis spinors $\\eta _i$ in the internal space $M_{6,4}$ .", "The spinor $\\Psi $ depends on position in $M_{8,8}$ and hence also on position in the subspace $M_{7,7}$ .", "From now on, $x^M$ will denote coordinates in $M_{7,7}$ , and the index ${M}$ will take values $1,2,3,...,14$ .", "By splitting the coordinates into those of spacetime $M_{1,3}$ , and those of the internal space, we have $\\Psi = \\Psi (X^M )\\,,\\,\\,\\,\\,\\,\\,\\,X^M = (x^\\mu ,x^{\\bar{M}} )\\,,\\,\\,\\,\\,\\,\\mu = 0,1,2,3;\\,\\,\\,\\,\\bar{M} = 5,6,...,14$ Considering the Dirac equation we find $(i \\gamma ^M \\partial _M + \\kappa )\\Psi = i(\\gamma ^\\mu \\partial _\\mu + \\gamma ^{\\bar{M}} \\partial _{\\bar{M}} + \\kappa )\\Psi = 0$ If the higher dimensional mass $\\kappa $ is zero, then the Dirac equation becomes $\\gamma ^M \\partial _M \\Psi = \\gamma ^\\mu \\partial _\\mu \\Psi + \\gamma ^{\\bar{M}} \\partial _{\\bar{M}} \\Psi = 0$ For the higher dimensional spinor field $\\Psi $ we can take the ansatz $\\Psi (x^\\mu ,x^{\\bar{M}} ) = \\Psi ^{(4)} (x^\\mu )\\exp \\,[ - ip_{\\bar{M}} x^{\\bar{M}} ]\\,{\\rm {U}}^{(10)} ,$ which is the product of a spinor field in $M_{1,3}$ and a spinor field in $M_{0,10}$ .", "This is a simplified model in which the spinor field in the ten dimensional internal space is just a plane wave spinor.", "With such ansatz the Dirac equation (REF ) becomes $\\gamma ^\\mu \\partial _\\mu \\Psi ^{(4)} U^{(10)} + \\Psi ^{(4)} ( - i)\\gamma ^{\\bar{M}} p_{\\bar{M}} {\\rm {U}}^{(10)} = 0$ Using the relation $\\gamma ^{\\bar{M}} p_{\\bar{M}} {\\rm {U}}^{(10)} = m\\,{\\rm {U}}^{(10)}$ we obtain $i\\gamma ^\\mu \\partial _\\mu \\Psi ^{(4)} {\\rm {U}}^{(10)} \\, + \\,m\\,\\Psi ^{(4)} {\\rm {U}}^{(10)} = 0$ which is the massive Dirac equation in four dimensions.", "A more general ansatz is $\\Psi (x^\\mu ,x^{\\bar{M}} ) = \\Psi ^{(4)} (x^\\mu )\\Psi ^{(10)} (x^{\\bar{M}} )$ which is not constrained to plane waves.", "Then from Eqs.", "(REF ) and (REF ) we obtain $i\\gamma ^{\\bar{M}} \\partial _{\\bar{M}} \\Psi ^{(10)} = m\\,\\Psi ^{(10)},$ and $i\\gamma ^\\mu \\partial _\\mu \\Psi ^{(4)} \\Psi ^{(10)} \\, + \\,m\\,\\Psi ^{(4)} \\Psi ^{(10)} = 0 .$ According to Eq.", "(REF ), the internal states are eigenstates of the operator $ i\\gamma ^{\\bar{M}} \\partial _{\\bar{M}}$ with $m$ being an eigenvalue.", "The mass term in the 4D Dirac equation (REF ) thus comes from the internal space.", "This is a well know feature of Kaluza-Klein theories which work in curved higher dimensional manifolds.", "Therefore, in the following we will assume that $M_{7,7}$ is curved and such that $M_{7,7}= M_{1,3} \\dot{+} M_{6,4}$ still holds locally.", "The operator $i\\gamma ^{\\bar{M}} \\partial _{\\bar{M}}$ is responsible for generations.", "When acting on the spinor field $\\Psi ^{(10)} = \\psi ^i \\eta _i$ it consists of two parts: $\\gamma ^{\\bar{M}} \\partial _{\\bar{M}} \\Psi ^{(10)} = \\left( {\\gamma ^{\\bar{M}} \\partial _{\\bar{M}} \\psi ^i + \\gamma ^{\\bar{M}} \\Gamma _{\\bar{M}\\,j}^{\\,\\,i} \\psi ^j } \\right)\\eta _i \\equiv \\mbox{\\rm D}_{\\bar{M}} \\psi ^i \\eta _i.$ The first part is the “orbital” contribution, while the second part comes from the action of the derivativeIn ref.", "[10] it was explained that the same symbol $\\partial _M$ can be used for the derivative operator acting on different kinds of objects.", "For instance, if acting on a scalar field, then it behaves as partial derivative, if acting on a basis vector it gives the connection, and if acting on a basis spinor it give the spin connection.", "Usage of the same symbol for the derivative in all such cases much simplifies the calculations.", "$\\partial _{\\bar{M}}$ on the basis spinors $\\eta _i$ of the internal space $\\partial _{\\bar{M}} \\eta _i = \\Gamma _{\\bar{M}\\,i}^{\\,\\,j} \\eta _j .$ We see that the second part in Eq.", "(REF ) is the contribution due to Yukawa coupling, where the spin connection in the “internal” space, $\\Gamma _{\\bar{M}\\,i}^{\\,\\,j}$ , has the role of Higgs fields.", "The eigenvalue equation (REF ) in the internal space yields the possible values of the particle mass $m$ .", "According to Eq.", "(REF ), mass does not arise from the Higgs field only, but also from the orbital momentum $\\gamma ^{\\bar{M}} \\partial _{\\bar{M}} \\psi ^i$ .", "Namely, if we multiply Eq.", "(REF ) from the left by $-i \\gamma ^{{\\bar{N}}} \\partial _{\\bar{N}}$ and then also by ${\\eta ^{j}}^\\dagger $ , we obtain, after taking the scalar part and renaming the indices, the following equation[4]: $g^{\\bar{M} \\bar{N}} \\mbox{\\rm D}_{\\bar{M}}\\mbox{\\rm D}_{\\bar{N}} \\psi ^i +{(\\sigma _{\\bar{M} \\bar{N}})^i}_j {R_{\\bar{M} \\bar{N}}^j}_k \\psi ^k= m^2 \\psi ^i ,$ where $g^{\\bar{M} \\bar{N}} \\mbox{\\rm D}_{\\bar{M}}\\mbox{\\rm D}_{\\bar{N}} \\psi ^i =g^{\\bar{M} \\bar{N}}\\left(\\partial _{\\bar{M}} \\mbox{\\rm D}_{\\bar{N}} \\psi ^i- \\Gamma _{\\bar{M} \\bar{N}}^{\\bar{K}} \\mbox{\\rm D}_{\\bar{K}} \\psi ^i + \\Gamma _{\\bar{M} j}^i \\partial _{\\bar{N}} \\psi ^j \\right).$ and $g^{\\bar{M} \\bar{N}} \\partial _{\\bar{M}} \\mbox{\\rm D}_{\\bar{N}} \\psi ^i= g^{\\bar{M} \\bar{N}} \\partial _{\\bar{M}}\\left( \\partial _{\\bar{N}} \\psi ^i + {\\Gamma _{\\bar{N}}^i}_j \\psi ^j \\right).$ Here ${(\\sigma _{\\bar{M} \\bar{N}})^i}_j = \\frac{1}{2}{([\\gamma _{\\bar{M}},\\gamma _{\\bar{N}}]^i}_j$ is the spin tensor and ${R_{\\bar{M} \\bar{N}}^i}_k$ the curvature tensor expressed in term of the spin connection.", "The term $g^{\\bar{M} \\bar{N}} \\partial _{\\bar{M}}\\partial _{\\bar{N}} \\psi ^i$ , written in terms of the spherical coordinates in the internal space, becomes an expression that contains the “orbital” momentum operator acting on the internal state $\\psi ^i$ .", "In this setup the number of generations is given by the number of eigenstates and eigenvalues determined by the equation (REF ).", "Moreover, the fact that masses are not determined by Higgs fields only, but also by the orbital momentum in the internal space should be taken into account and recalculate, e.g., the proton life time.", "In general, in Eq.", "(REF ), the derivative $\\partial _M$ , acting on the basis spinors $\\xi _{\\tilde{A}}$ , gives the connection $\\Gamma _{M \\tilde{B}}^{~\\tilde{A}}$ that includes the ordinary spin connection, a Yang-Mills field and a multiplet of Higgs fields.", "All those fields are contained in the covariant derivative $\\partial _M \\Psi = (\\partial _M \\psi ^{\\tilde{A}} +\\Gamma _{M \\tilde{B}}^{~\\tilde{A}} \\psi ^{\\tilde{B}})\\xi _{\\tilde{A}}$ .", "A first step in this direction was proposed in Ref.", "[10], where the internal piece of the connection was not yet recognized as a Higgs multiplet.", "That the covariant Dirac derivative contains all those fields, including the Higgs multiplet, was observed in Ref.", "[23], but without using the concept of geometric spinors defined according to Eq.", "(REF )." ], [ "Conclusion", "Fundamental objects such as strings and branes can be described in terms of their spacetime volume degrees of freedom represented by Clifford numbers belonging to $Cl(1,3)$ .", "The latter algebra is considered as a tangent space of a 16-dimensional manifold, called Clifford space.", "Clifford algebra as a vector space can be considered as being spanned over 16 basis vectors that are generatore of $Cl(8,8)$ or its complexified version.", "We thus have a 16-dimensional vector space $V_{8,8}$ that have enough room for the unification of fundamental particles and interactions including gravity.", "Spinors are members of left ideals of $Cl(8,8)$ .", "They satisfy the Dirac equation in sixteen dimensions.", "The extra dimensions give rise not only to the presence of the first generation particles of the $SO(10)$ grand unification, but also to different generations.", "Namely, the effective mass in four dimensions is due to the presence of extra dimensions, a known feature of Kaluza-Klein theories.", "But in those theories there is also a problem of how to reconcile the chiral properties of spinors in higher dimensions with the observed parity non conservation in four dimensions.", "Within the framework considered here, such problem does not exist.", "Namely, considering the concept of Clifford space whose tangent space at any of its points is a Clifford algebra, with its members being spinors, automatically leads[45], [49], [12] to mirror particles[50], [51], [52], [53], [54], [55], [56] that are coupled to mirror gauge fields and are unobservable by means of ordinary gauge fields.", "Therefore, as extensively investigated in Refs.", "[56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [49], [12], they are candidates for dark matter." ] ]
2105.11808
[ [ "Dynamic Semantic Graph Construction and Reasoning for Explainable\n Multi-hop Science Question Answering" ], [ "Abstract Knowledge retrieval and reasoning are two key stages in multi-hop question answering (QA) at web scale.", "Existing approaches suffer from low confidence when retrieving evidence facts to fill the knowledge gap and lack transparent reasoning process.", "In this paper, we propose a new framework to exploit more valid facts while obtaining explainability for multi-hop QA by dynamically constructing a semantic graph and reasoning over it.", "We employ Abstract Meaning Representation (AMR) as semantic graph representation.", "Our framework contains three new ideas: (a) {\\tt AMR-SG}, an AMR-based Semantic Graph, constructed by candidate fact AMRs to uncover any hop relations among question, answer and multiple facts.", "(b) A novel path-based fact analytics approach exploiting {\\tt AMR-SG} to extract active facts from a large fact pool to answer questions.", "(c) A fact-level relation modeling leveraging graph convolution network (GCN) to guide the reasoning process.", "Results on two scientific multi-hop QA datasets show that we can surpass recent approaches including those using additional knowledge graphs while maintaining high explainability on OpenBookQA and achieve a new state-of-the-art result on ARC-Challenge in a computationally practicable setting." ], [ "Introduction", "Multi-hop QA is one of the most challenging tasks that benefits from explainability as it mimics the human question answering setting, where multi-hop QA requires both the collection of information from large external knowledge resources and the aggregation of retrieved facts to answer complex natural language questions [53].", "Figure: The AMR of the hypothesis (black), Fact 1 and Fact 2.", "A hypothesis is a statement derived from a question and a choice.", "The hypothesis AMR can be inferred by relevant fact AMRs.Currently, external knowledge is mostly stored in two forms – textual and graph structure (e.g.", "Knowledge Graph (KG)).", "Textual corpora contain rich and diverse evidence facts, which are ideal knowledge resources for multi-hop QA.", "Especially with the success of pretrained models [12], [31], [23], we can get powerful representations for such textual facts.", "However, retrieving relevant and useful facts to fill the knowledge gap for inferring the answer is still a challenging problem.", "In addition, the reasoning process over the facts is hidden by the unexplainable neural network, which hinders the deployment of real-life applications.", "On the other hand, KG is able to provide structural clues about relevant entities for explainable predictions [15], [41], [49].", "But it is known to suffer from sparsity, where complex question clues are unlikely to be covered by the closed-form relations in KG [56], [55].", "Another issue is that KG requires large human labor and is easy to become outdated if not maintained timely.", "To take advantages of both rich textual corpora and explicit graph structure and make it compatible to all textual knowledge, we explore the usefulness of Abstract Meaning Representation (AMR) as a graph annotation to a textual fact.", "AMR [3] is a semantic formalism that represents the meaning of a sentence into a rooted, directed graph.", "Figure REF shows some examples of AMR graphs, where nodes represent concepts and edges represent the relations.", "Unlike other semantic role labeling that only considers the relations between predicates and their arguments [42], the aim of AMR is to capture every meaningful content in high-level abstraction while removing away inflections and function words in a sentence.", "As a result, AMR allows us to explore textual facts and simultaneously attributes them with explicit graph structure for explainable fact quality assessment and reasoning.", "In this paper, we propose a novel framework that incorporates AMR to make explainable knowledge retrieval and reasoning for multi-hop QA.", "Our framework works on textual knowledge, which is easy to obtain and allows us to get informative facts.", "The introduced AMR serves as a bridge that enables an explicit reasoning process over a graph structure among questions, answers and relevant facts.", "As exemplified in Figure REF , a hypothesis is first derived from a question and an answer choice.", "We then parse the hypothesis and a large number of facts to corresponding AMRs.", "After that, we dynamically construct AMR-SG for each question-choice pair by merging the AMRs of its hypothesis and relevant facts.", "Unlike previous works on multi-hop QA that rely on existing KGs to find relations among entities [47], [15], our proposed AMR-SG is dynamically constructed, which reveals intrinsic relations of facts and can naturally form any-hop connections.", "After construction, we analyze all connected paths starting from the question to the answer on AMR-SG.", "We focus the consideration of facts on those paths because they together connect the question with the answer, indicating their active roles in filling the knowledge gap.", "The connections of facts on AMR-SG can be further used as the supervision for downstream reasoning.", "Therefore, we adopt GCN [21] to model the fact-level information passing.", "Experimental results demonstrate that our approach outperforms previous approaches that use additional KGs.", "It obtains 81.6 accuracy on OpenBookQA [32], and pushes the state-of-the-art result on ARC-Challenge [8] to 68.94 in a computationally practicable setting." ], [ "Multi-hop QA with External Resource.", "Despite the success of pretrained model in most Natural Language Processing (NLP) tasks, it performs poorly in multi-hop QA, where some information is missing to answer questions [60].", "Figure: Overall architecture of our proposed model.", "The black dash lines in AMR-SG indicate that we cut the connection between question nodes and choice nodes.", "The pink arrows indicate two paths that can be spotted in AMR-SG.", "Facts with red background are active facts detected.", "The dashed node Active Fact-level Connection Graph indicates fact4 is not considered as a valid node as it is not an active fact.Textual corpora contain rich and diverse knowledge, which is likely to cover the clues to answer complex questions.", "[5] demonstrate some carefully designed queries can effectively retrieve relevant facts.", "[50], [11] extract groups of evidence facts considering the relevance, overlap and coverage, but such method requires exponential computation in the retrieval step.", "[14], [51] construct a fact chain by iteratively reformulating the query to focus on the missing information.", "However, the fact chain often grows obliquely as a result of the failure of first fact retrieval, making the QA model brittle.", "As some recent QA datasets [53], [32], [20] annotate a gold evidence fact for each question, it enables training supervised classifier to identify the correct fact driven by a query [36], [37], [45], [4].", "[34] take a further step to jointly predict the answer span and select evidence facts in a unified model.", "Though these supervised retrievers have achieved impressive improvement, they heavily rely on the annotated gold facts, which are not always available in real-world applications.", "In addition, previous works also explore the effectiveness of structured knowledge by either encoding the nodes [52], [48], triples [33], [47], paths [28], [25] or tabular [59] to capture the missing information.", "Other works avoid the sparsity of KGs by constructing KGs directly from textual knowledge.", "OpenIE [40] is widely used in knowledge base question answering to extract entity-relation triples [6], [56], [10].", "However, OpenIE favors precision over recall, which is not necessarily effective to form connections among diverse evidence facts for multi-hop QA.", "Wikipedia contains internal hyperlinks, which are effective to build graph connections from unstructured articles [2], [29].", "However, such hyperlinks are not available in most textual corpora." ], [ "AMR.", "Recent success in AMR research makes it possible to benefit downstream tasks, such as summarization [44], [13], [27], event detection [26] and machine translation [42].", "In the domain of QA, AMR has been used to form logic queries and conduct symbolic reasoning [35], [18].", "Comparing to name entity [57] or other cross-sentence annotations [24], [54], we use AMR to build our semantic graph because it is align-free and can be easily adapted to powerful pretrained models." ], [ "Framework Description", "In this paper, we consider the multi-hop QA in the form of multi-choice, where a question $Q_i$ is provided with $J$ answer choices $C_{ij}, j \\in \\lbrace 1,2,...,J\\rbrace $ .", "As shown in Figure REF , our framework consists of three components: (1) a Fact Retrieval component to retrieve evidence facts $\\hat{F}=\\lbrace \\hat{F}^1,...,\\hat{F}^m\\rbrace $We omit the subscript $ij$ for simplicity.", "for each question-choice pair from a large textual corpus; (2) a Semantic Graph Construction & Analytics component that dynamically constructs a semantic graph, named AMR-SG, to select active facts $F=\\lbrace F^1,...,F^n\\rbrace $ from $\\hat{F}$ and capture their relations $A$ ; and (3) a Hypothesis Assessment component that classifies whether the question-choice is correct, given the active facts and their relations in (2)." ], [ "Hypothesis Generation.", "As shown in Figure REF , we first generate a hypothesis $H_{ij}$ for the $i^{th}$ question and the $j^{th}$ choice.", "A hypothesis is a completed statement derived from each question-choice pair.", "Comparing to simply concatenating the question and the choice, a hypothesis contains less meaningless words and maintain a good grammatical structure, which can avoid retrieving noisy facts and allow AMR parser to generate high-quality AMR graphs.", "We generate hypotheses by the rule-based model of [9].", "For some unsolvable cases, we directly concatenate the question and the choice.", "We apply this process for all training, develop and test sets." ], [ "Fact Extraction.", "We retrieve a pool of evidence facts $\\hat{F}$ for each hypothesis separately using Elasticsearch [16].", "We set a large size $m$ of the fact pool to cover as many valid facts as possible." ], [ "Semantic Graph Construction & Analytics", "Active facts $F$ are facts that really fill the knowledge gap between question and choice.", "The activeness of a fact cannot be simply determined by comparing it with the hypothesis, as multi-hop QA requires multiple facts to complete the reasoning chain.", "Therefore, we need to filter out facts that are just partially related and focus on the consideration of active facts and their roles in the reasoning chain.", "In this component, we first construct AMR-SG.", "Then, we propose a path-based analytics approach to extract active facts and construct an Active Fact-level Connection Graph to capture their relations with the question and the answer choice." ], [ "AMR-SG Construction", "As the nodes of AMR are high-level abstraction of concepts conveyed in the corresponding textual fact, two AMRs sharing the same node indicate that they concern about the same concept, which shows their correlation.", "This motivates us to construct AMR-SG, shown in Figure REF , to represent the relations of the corresponding hypothesis and evidence facts for each question-choice pair.", "We leverage the state-of-the-art AMR parser [7] to generate AMR $G=\\lbrace G^H, G^1,...,G^m\\rbrace $ for a hypothesis and all facts in the corresponding fact pool, where $G^H$ , $G^i$ are the AMR of the hypothesis and the $i^{th}$ fact respectively.", "AMR is also a directed and edge-labeled graph, which implies information specified in the edge is propagated in one pre-defined direction.", "However, such inner-AMR (edge labels and directions) information does not contribute to inter-AMR relations.", "Therefore, we only care about if there exists an edge between two nodes but ignore the edge labels and directions.", "During construction, we regard $G^H$ as the start point of AMR-SG.", "Then, we incrementally find one fact AMR in the fact pool sharing some nodes with it and add this fact AMR onto it by merging the shared nodes.", "The merging operation stops when no AMR can be added onto AMR-SG or the fact pool is empty.", "In fact, as shown in Figure REF , we do not change the architecture of each individual AMR, but reuse some shared nodes as the nodes in AMR-SG.", "Note that, some nodes are over-general, which are not appropriate to connect two AMRs (e.g.", "(p/planet :name(n/name :op1\"Earth\")), the node n/name is an over-general concept).", "Fortunately, such over-general nodes always have non-node attributes (e.g.", "Earth of n/name) that shows the specific referent.", "Therefore, we replace the nodes with their non-node attributes if any to address this issue." ], [ "Path-based Analytics", "Current multi-hop QA models are hindered by the quality of retrieved facts [5].", "We address this issue by a path-based analytics approach to guarantee the selected facts having a positive effect to answer the question.", "As shown in Figure REF , AMR-SG reveals any-hop relations of the hypothesis and all facts.", "Completed paths can be spotted out of $G^H$ to connect the question nodes with the choice nodes by passing through multiple facts.", "These facts, which together provide the missing knowledge to maintain complete reasoning chains, are active facts that we want to extract.", "Specifically, we split the nodes of $G^H$ into question nodes $Q^H$ and choice nodes $C^H$ .", "Question nodes represent the concepts extracted in the question text.", "As one question is provided with $J$ choices, where we can generate $J$ hypothesis AMRs.", "We take the shared nodes of these AMRs as $Q^H$ , while the remaining as $C^H$ : $Q^H_{ij} = \\cap _{j=1}^J\\lbrace v| v \\in G^H_{ij}\\rbrace $ $C^H_{ij} = \\lbrace v| v \\in G^H_{ij}, v \\notin Q^H_{ij}\\rbrace , j=1,...,J$ We cut the edges between $Q^H$ and $C^H$ to guarantee the paths are spotted outside $G^H$ .", "Then we apply depth-first search on AMR-SG to find all paths that connect at least one question node and one choice node, including the path that does not have a minimum length (e.g.", "the path passing through fact3 in Figure REF ).", "All facts that the paths pass through (one node in and another node out) are considered as active facts.", "This is because we try to cover more facts as long as they do not deviate from the correct reasoning direction to provide enough information for QA model.", "In addition, the any-hop relations of the hypothesis and active facts in AMR-SG can be used for a hypothesis to precisely aggregate knowledge from relevant facts to reduce ambiguity during the reasoning process.", "Therefore, we construct an Active Fact-level Connection Graph from AMR-SG to capture such relations among the hypothesis and all active facts.", "As shown in Figure REF , each node in Active Fact-level Connection Graph is either the hypothesis or an active fact.", "We draw an edge between two facts (include hypothesis) if they share one concept node in AMR-SG." ], [ "Hypothesis Assessment with Fact-level Reasoning", "As shown in Figure REF , we concatenate the hypothesis with all active facts, where [SEP] token is inserted between the two texts and [CLS] is put at the beginning of the sequence.", "We feed the whole sequence into a pretrained model based on RoBERTa [31] architecture to get the hidden representation of each token.", "Then, Active Fact-level Connection Graph is used as an additional supervision in fact-level modeling to guide the reasoning process.", "Formally, let $s^{H}_{1:{l_H}} \\in \\mathbb {R}^{l_H \\times d}$ , $s^{i}_{1:{l_i}} \\in \\mathbb {R}^{l_i \\times d}$ be the hidden representations of the hypothesis and the $i^{th}$ active fact respectively, where $l_H$ , $l_i$ denote the length and $d$ is the dimension of the representation.", "A max pooling layer is applied over these hidden representations to get the node representations respectively: $\\begin{array}{l}x_H = \\textbf {MaxPool}(s^{H}_{1:{l_H}}) \\in \\mathbb {R}^{1 \\times d}\\\\x_i = \\textbf {MaxPool}(s^{i}_{1:{l_i}}) \\in \\mathbb {R}^{1 \\times d}, i=1,...,n\\end{array}$ The connections of hypothesis (0th) and active facts in Active Fact-level Connection Graph can be viewed as an adjacency matrix $A \\in \\mathbb {R}^{(n+1) \\times (n+1)}$ , where $A_{ij}=\\left\\lbrace \\begin{array}{ll}1& \\text{if } F^i \\text{ is connected with } F^j\\\\0& \\text{otherwise}\\end{array}\\right.$ As there is no edge information in the graph, a simple GCN is enough to model the knowledge fusion among the hypothesis and multiple active facts in the reasoning process.", "We also introduce multi-head mechanism [46] to stabilize the learning of different knowledge: $X^{(k)}=[head_1^{(k)}:...:head_h^{(k)}]$ where $[:]$ denotes concatenation operation, $X^{(k)}$ is the node states at the $k^{th}$ layer, $X^{(0)}=[x_H;x_1;...;x_n]$ , $[;]$ denotes the sequential concatenation operation, $head_i$ is the $i^{th}$ head.", "Specifically, we compute the nodes states by aggregating knowledge from their neighboring nodes in each layer: $head_i^{(k)}=\\mathrm {ReLU}(\\Lambda X^{(k-1)}W_i^{(k)})$ where $W_i^{(k)}\\in \\mathbb {R}^{d \\times (d/h)}$ is the projection matrix of $head_i$ at the $k^{th}$ layer, $h$ is the head number.", "$\\Lambda $ is the normalization constant to avoid scale changing: $\\begin{array}{l}\\Lambda =D^{-1/2}AD^{-1/2}\\\\D_{ii}=\\sum _jA_{ij}\\end{array}$ After that, a $\\sigma $ gate is applied to calculate how much knowledge can be propagated to score the question-choice pair: $\\lambda = \\sigma ( W^{\\lambda }[x_{cls}:x^{(K)}_{H}]+b^{\\lambda })$ $s(q,a) = W^o(\\lambda x^{(K)}_{H} +(1-\\lambda )x_{cls}) + b^o$ where $W^{\\lambda } \\in \\mathbb {R}^{1 \\times 2d}$ , $W^o \\in \\mathbb {R}^{d \\times d}$ , $b^{\\lambda }$ , $b^o$ are the parameters.", "We get the final probability by normalize all question-choice pairs with softmax." ], [ "Datasets", "We evaluate our approach on two multi-choice multi-hop QA datasets: ARC-Challenge [8] and OpenBookQA [32].", "The textual corpus we use for both datsets is ARC Corpus [8], which contains about 14M science facts.", "OpenBookQA and ARC-Challenge have their leaderboards with train, develop and test sets publicly available.", "we follow [1] to combine the training set of OpenBookQA (4957), ARC-Easy (2251), ARC-Challenge (1119) and RegLivEnv (665) as the final training set of ARC-Challenge task.", "The data splits is shown in Table REF .", "Table: Number of instances in each dataset.For ARC-Challenge, we retrieve 100 facts to form the fact pool.", "Based on this, we select up to 20 active facts using our approach as the context for each question-choice pair.We can only reproduce the results similar to [1] using 20 facts as the context.", "OpenBookQA provides an accompanying open-book of 1326 science facts, which are highly related to the questions in this dataset.", "Therefore, for OpenBookQA, we retrieve 10 facts from the open-book and another 90 facts from ARC Corpus, forming the 100 facts in the fact pool.", "We then select up to 15 active facts using our approach as the context." ], [ "Implementation", "We implement our approach on two pretrained models: RoBERTa [31] and AristoRoBERTa [1].", "AristoRoBERTa employs the RoBERTa architecture but uses RACE [22] to first fine-tune the RoBERTa model.", "We prepare active facts as the context to further fine-tune the model with the target dataset.", "For OpenBookQA, we continue to fine-tune the QA model following the same procedure as [1], where the initial learning rate is 2e-5, the batch size is 12 and the max sequence length is 256.", "For ARC-Challenge, the initial learning rate, the batch size and the max sequence length are 1e-5, 6, and 416 respectively.", "We use grid search to find optimal hyper-parameters, where the learning rate is chosen from {5e-6, 1e-5, 2e-5}, the batch size is chosen from {4, 6, 8, 12, 16}.", "The number of GCN layer $K$ is chosen from {1,2,3,4}, while the head number $h$ is the RoBERTa-Large default value.Our code is available at: https://github.com/wwxu21/AMR-SG We introduce 6M parameters of the fact-level reasoning module in addition to 355M of RoBERTa-Large.", "We run all experiments on one TITAN RTX card, which takes about 1 hour and 3 hours to complete the training of OpenBookQA and ARC-Challenge respectively." ], [ "Comparison Methods", "We compare with recent existing methods that make use of similar power of pretrained models in order to conduct a fair comparison.", "These include the baseline AristoRoBERTaV7 [1] finetuned on top of AristoRoBERTa, KF-SIR [4] that exploits the knowledge fusion among facts, FreeLB [58] that tackles the robustness issue and another three methods leveraging an additional knowledge graph [43] in addition to the textual knowledge: PG [47], MHGRN [15], AlBERT + KB.", "PG(albert + gpt2, roberta + gpt2) are two implementations with different pretrained model architectures [31], [23], [38], where the latter is more fair to compare with us." ], [ "OpenBookQA.", "The test set accuracy is shown in Table REF .", "AMR-SG-Full is our full model based on AristoRoBERTa.", "Results show that AMR-SG-Full can surpass models leveraging additional KG.", "It demonstrates that the fundamental improvement of AMR-SG-Full comes from the knowledge mining of the textual corpus.", "However, such knowledge resource has not been fully investigated by existing methods and contains richer and more diverse evidence facts than KGs.", "We do not compare with UnifiedQA [19] and T5 3B [39] as they rely on extremely large pretrained models (at least 3B parameters), which are not fair for comparison." ], [ "ARC-Challenge.", "We also implement AMR-SG-Full on another difficult multi-hop QA dataset: ARC-Challenge.", "It consists of the questions only answered incorrectly by both a retrieval-based algorithm and a word co-occurrence algorithm [8], which theoretically is not friendly to our approach.", "As shown in Table REF , we can still obtain 2.47 accuracy improvement comparing to AristoRoBERTaV7 and achieve a new state-of-the-art performance in a computationally practicable setting.", "Table: Test accuracy on ARC-Challenge.", "All models use RoBERTa architecture for the pretrained model and do not leverage additional KG." ], [ "Ablation Study", "We conduct ablation study by incrementally adding each component of AMR-SG-Full to investigate its effectiveness on two pretrained models in Table REF .", "We include the analysis on RoBERTa because it is a more general and widely used pretrained model.", "We start from the vanilla pretrained models, where no textual facts are provided (denoted as No Fact).", "We retrieve 15 facts as the context to create the first variant (denoted as + Fact Context).", "The purpose is to test the contribution of the facts retrieved by the simple information retrieval (IR) system (Elasticsearch).", "We continue to add the path-based fact analytics component (denoted as + Fact Analytics).", "In fact, this variant merely use the facts selected from AMR-SG to fine-tune the pretrained models.", "On top of both two pretrained models, we observe a great performance improvement, where the improvement brought by + Fact Analytics is higher than + Fact Context on top of RoBERTa, which demonstrates this component can effectively select useful facts to fill the knowledge gap that have not been covered by the IR system.", "We finally equip our model with the fact-level reasoning component (denoted as + Fact-level Reasoning).", "From the results, we can observe that this component performs well on top of RoBERTa, but has very little effect on top of AristoRoBERTa.", "This is because this component tries to infuse some fact-level connections to ease the reasoning process of the model.", "Such information can be learned automatically by the model itself if exposed to enough in-domain data (AristoRoBERTa).", "Nevertheless, the fact-level reasoning is a more general method when such data is unavailable." ], [ "Impact of Evidence Facts.", "As discussed above, the major improvement of our approach comes from more useful facts selected for each question-choice pair.", "In this section, we take a deep look at the quality and the composition of those facts on OpenBookQA.", "We derive five variants by varying the composition of core (facts retrieved from open-book) or common (facts from ARC Corpus) facts.", "For core facts, as open-book annotates one gold core fact for each question, the retrieval accuracy of the gold fact is a natural way to evaluate the quality.", "For common facts, we conduct human analysis to evaluate the quality from three aspects: (1) Relatedness: Does the retrieved fact related to the question or the answer?", "(2) Informativeness: Does the retrieved fact provided useful information to answer the question?", "(3) Completeness: Do all retrieved facts together fill the knowledge gap to completely answer the question?", "We randomly sample 50 questions and evaluate the evidence facts corresponding to the correct answer choice, where one fact would contribute 1 score if it meets the requirement of Relatedness or Informativeness respectively and all 15 facts contribute 1 score if they together meet the requirement of Completeness.", "Evaluation results are presented in Table REF .", "When varying the fact composition of IR variants, we find the gold core fact retrieval accuracy has a positive impact on the final accuracy on top of RoBERTa.", "At this stage, some questions can be inferred sufficiently with the gold core facts.", "Higher retrieval accuracy accounts for more questions of this kind to be correctly answered.", "However, this advantage is not as obvious for AristoRoBERTa.", "Our human evaluation reveals that such facts are unlikely to form a complete reasoning chain, making it hard for real multi-hop reasoning.", "On the other hand, our approach directly models the intrinsic fact relations, where the path-based analytics ensures that the facts selected are in the reasoning chain from the question to the answer.", "Results show that our approach makes an overall improvement with regard to Relatedness, Informativeness and Completeness and is less harmful to core fact retrieval.", "We also find that AMR-SG (10/100) can make a further improvement compared to AMR-SG (10/30) by including more facts to construct AMR-SG.", "It demonstrates that AMR-SG has the capability of detecting useful facts from a large and noisy fact pool, thus making up for the deficiency of the IR system." ], [ "Impact of AMR Consistency.", "We investigate the quality consistency of AMR graphs to see how it affects the construction of AMR-SG and thus affects the QA model.", "We prepare AMR in three consistency levels, where Fully-Automatic is generated by automatic AMR parser; Mixed is that we manually annotate the error-free AMRs for the core facts in open-book (1326 in total) and use the error-free core fact AMRs and other automatically generated AMRs to construct AMR-SG; Error-Free-Adapted is that we use the error-free AMRs annotated to fine-tune the AMR parser and use the tuned parser to generate AMR for all the remaining facts (including hypotheses and common facts, about 900k in total).", "The test set accuracy are 81.6, 80.2, 80.4 for Fully-Automatic, Mixed and Error-Free-Adapted respectively.", "It is interesting to note that using Fully-Automatic AMRs results in higher QA accuracy than Mixed and Error-Free-Adapted, where the latter two contain a mix of AMRs with different levels of quality.", "This phenomenon has also been observed in other AMR applications [30], [17], where automatic parses perform well than manual parses.", "We conjecture that this can be attributed to the discrepancy between the error-free AMRs and the automatically parsed AMRs in the choices of AMR concepts with similar meaning.", "This small difference in concept choices may omit potential connections, results in some important facts failing to be detected.", "In contrast, automatically parsed AMRs contain errors, but they are consistent in their concept choices, which is more likely for AMRs to form connections.", "The 0.2 accuracy improvement between Mixed and Error-Free-Adapted also demonstrates our assumption, since the parser is finetuned on the error-free AMRs, where its parsed AMRs should be more consistent with the error-free AMRs.", "Table: A case study showing how our framework selects useful facts to completely fill the knowledge gap.Figure: Analysis of fact-level reasoning on OpenBookQA.", "(a) presents the distribution of prediction confidence with or without fact-level reasoning module.", "(b) shows the QA performance with different GCN layer K. Size 0 denotes the original pretrained model." ], [ "Case Study", "Table REF shows one case study of evidence facts selected by our framework.", "Since the important term earthquake is missing from the search query, the IR system assigns low retrieval scores for the two facts, causing a low ranking.", "However, the two facts can form a complete reasoning chain with the question and the answer via several concept nodes, where our approach can successfully extract the two facts despite the low retrieval scores.", "More cases can be found in Appendix REF ." ], [ "Why Fact-level reasoning.", "Figure REF (a) shows that fact-level reasoning improves the performance by making a more confident prediction for the correct answer.", "This is because the fact-level connections of AMR-SG inform the model how these active facts are intrinsically related, which allows the model to precisely receive knowledge from related facts." ], [ "Impact of Number of Hops (K).", "We vary the hyper-parameter K to consider the impact of K-hop neighbors on OpenBookQA.", "As show in Figure REF (b), the performance reaches the top at $K=2$ .", "It indicates that most of the questions can be well answered using two evidence facts, which is consistent with the construction of this dataset.", "However, the performance drops when $K>2$ .", "It might be attributed to exponential noise found in longer reasoning chains." ], [ "Conclusion", "We propose to dynamically construct AMR-SG that can reflect the intrinsic relations of relevant facts leveraging AMR, a graph annotation.", "AMR-SG combines the advantages of rich textual corpus and graph structure, where we can select useful facts that completely form the reasoning chain and make fact-level modeling.", "Experimental results show that AMR-SG can maintain high explainability, and successfully couple with strong pretrained models to achieve significant improvement on OpenBookQA and ARC-Challenge over approaches leveraging additional KGs." ], [ "Case Study", "More case studies can be found in Table REF .", "Table: More case studies in addition to Table" ] ]
2105.11776
[ [ "A generalized configuration model with triadic closure" ], [ "Abstract In this paper we present a generalized configuration model with random triadic closure (GCTC).", "This model possesses five fundamental properties: large clustering coefficient, power law degree distribution, short path length, non-zero Pearson degree correlation, and existence of community structures.", "We analytically derive the Pearson degree correlation coefficient and the clustering coefficient of the proposed model.", "We select a few datasets of real-world networks.", "By simulation, we show that the GCTC model matches very well with the datasets in terms of Pearson degree correlations and clustering coefficients.", "We also test three well-known community detection algorithms on our model, the datasets and other three prevalent benchmark models.", "We show that the GCTC model performs equally well as the other three benchmark models.", "Finally, we perform influence diffusion on the GCTC model using the independent cascade model and the linear threshold model.", "We show that the influence spreads of the GCTC model are much closer to those of the datasets than the other benchmark models.", "This suggests that the GCTC model is a suitable tool to study network science problems where degree correlation or clustering plays an important role." ], [ "Introduction", "Network science emerges as a multi-disciplinary study of problems related to graphs in physics, social science, computer science and biology.", "To study these problems, especially in a mathematical way, researchers usually need to select a random network model.", "The selected random network model must resemble real-world networks as closely as possible in order for the research results to be convincing.", "Newman [1] classified real-world networks into four categories.", "They are social networks, engineering networks, information networks and biological networks.", "Although real-world networks arise in seemingly very different disciplines, they share five fundamental properties [1], [2].", "The five common properties are large transitivity; small-world property or short expected path lengths; power law degree distributions; assortative or disassortative degree correlations, and existence of community structures.", "Social networks tend to be assortative, and networks in the other three categories tend to be disassortative.", "The goal of this paper is to propose a random network model that possesses all the five properties listed above.", "There has been a tremendously large number of random network models proposed in the network science literature.", "Many important and influential random models were proposed to explain how the fundamental properties listed above emerge.", "Although these models successfully explained how certain properties emerge, they may not possess other properties in the list.", "For instance, the seminal paper by Watts and Strogatz [3] proposed the well-known small world model, which successfully captures large transitivity and short path lengths.", "However, it does not possess other properties.", "Another example is the preferential attachment model (also called the BA model) proposed by Barabási and Albert [4].", "This model elegantly explains how power law degree distributions arise.", "However, this model lacks the rest four properties.", "Another example is the configuration model proposed original by Bender et al.", "[5].", "One can construct a configuration model for a specific degree sequence.", "Thus, one can sample a degree sequence from a power law distribution and create a configuration model from that degree sequence.", "In other words, configuration models can possess power law degree distributions.", "In addition, configuration models also possess short path lengths due to the nature of their construction methods.", "One advantage of the configuration models is that they are mathematically simple.", "So, they are used to study many networking problems including giant component sizes, percolation, epidemic networks and etc.", "We refer the reader to Newman [1] for their applications.", "Lee et al.", "[6] proposed a generalized configuration model.", "Instead of randomly connecting stubs like how a configuration model does, a generalized configuration model first sorts stubs according to their degrees.", "Then, the stubs are divided into blocks.", "Depending on whether positive or negative degree correlations are desirable, one chooses properly a permutation function that associates each block with another block.", "In each block, a fraction $q$ of all stubs are randomly assigned as type 1 stubs.", "The rest of fraction $1-q$ stubs are called type 2 stubs.", "Type 1 stubs in a block are randomly connected to type 1 stubs in its associated block determined by the permutation function.", "Type 2 stubs are randomly connected to any unconnected type 2 stubs in the network.", "Lee et al.", "[6] showed that this network can be assortative or disassortative depending how the permutation function is selected.", "The generalized configuration model inherits many properties from the configuration model.", "In short, the generalized configuration model can have power law degree distributions, small world property, and non-zero Pearson degree correlations.", "In this paper we propose a new model by adding two features to the generalized configuration model.", "First, we use two layers of blocks in the new model.", "Specifically, stubs are divided into macroscopic blocks, which model communities.", "In each community, stubs are divided into microscopic blocks, which are used to create non-zero Pearson degree correlations.", "The new model with the first feature possesses community structures.", "The second feature added to the new model is called triadic closure [7], [8].", "Triadic closure means that there is an increased likelihood that two people, who have a friend in common, will become friends [7].", "Triadic closure has been observed in many real-world networks.", "For instance, Kossinets and Watts found clear evidence of triadic closure by taking multiple snapshots on the communication network using an email dataset [9].", "Moreover, Leskovec et al.", "[10] analyzed the properties of triadic closure in online social networks of LinkedIn, Flickr, Del.icio.us, and Yahoo!", "Answers.", "In addition, many random models also adopt triaic closure operations.", "For example, in several extensions of the BA model, arriving nodes not only add links to their preferentially selected nodes, but also to the neighbours of the selected nodes [11].", "With these two new features, the new model possesses large transitivity and community structures.", "The new model is named as configuration model with triadic closure (abbreviated CTC).", "The detail construction algorithm of the CTC model will be presented in Section .", "One contribution of this paper is that we derive closed form expressions for the Pearson degree correlation and the clustering coefficient of the CTC model.", "We mention that there are other proposals of random networks that possess the five fundamental properties listed above.", "For example, Toivonen et al.", "[2] proposed a growth model that possesses all the five properties.", "Toivonen et al.", "analyzed the degree distribution and the clustering coefficient of this model.", "However, it is not clear if the Pearson degree correlation coefficient of this model can be analyzed.", "One important application of the CTC model is that it can serve as a benchmark network to evaluate community detection algorithms.", "To assess the performance of a community detection algorithm, researchers need to test the algorithm on a network that has known community structures to serve as ground truth.", "Very few real-life networks possess ground-truth community structures.", "The real-life networks that do have ground-truth communities tend to be small.", "The well-known Karate-club network is one such example.", "In order to evaluate a community detection algorithm on a large network, researchers usually resort to random networks that have artificial community structures.", "These artificial communities serve as ground truth.", "Since the CTC model has communities, one application of our model is that it can be used as a benchmark for community detection algorithms.", "Other models that were often used to benchmark community detection algorithms include the Girvan-Newman models [12], LFR models [13], [14] and ABCD models [15].", "We remark that it is not clear if the models above possess all the five fundamental properties.", "The LFR model and the ABCD model are quite complicated.", "Mathematical analysis of their Pearson degree correlation and clustering coefficient seems difficult.", "In Section , we shall test several well known community detection algorithms using our model, the LFR and ABCD models.", "This paper is organized as follows.", "In Section , we present the construction algorithm of the CTC model.", "In Section we analyze the Pearson degree correlation coefficient of the CTC model.", "In Section we analyze the clustering coefficient of the CTC model.", "In Section we present numerical results.", "Finally, we present the conclusions of this paper in Section .", "Table: List of notations" ], [ "CTC network and its Construction Algorithm", "In this section we present a construction algorithm for the configuration model with triadic closure (CTC).", "Recall that in Newman's construction algorithm of a standard configuration model, two ends of an edge are called “stubs\".", "To construct a standard configuration model, randomly connect stubs.", "In other words, an unconnected stub is connected to another randomly selected stub among all unconnected stubs.", "This construction algorithm creates a random network that has asymptotically vanishing Pearson degree correlation function as the network becomes large.", "To introduce non-zero Pearson degree correlation, Lee et al.", "[6] partition stubs into blocks according to their degrees.", "Properly select a permutation function that associates a block with another block.", "To introduce positive (resp.", "negative) correlation, the selected permutation function associates blocks of large (resp.", "small) degrees with another block of large degrees.", "Stubs in a block are designated into type 1 stubs and type 2 stubs.", "An unconnected type 1 stub in a block is connected to a randomly selected unconnected type 1 stub in the associated block.", "An unconnected type 2 stub is connected to an unconnected type 2 stub randomly selected in all blocks.", "Connection of type 1 stubs introduces non-zero Pearson degree correlations.", "Connection of type 2 stubs resembles the construction algorithm of the standard configuration model.", "The construction algorithm of a CTC model is similar to that of a generalized configuration model, except that it has two additional features.", "First, a CTC model has two layers of blocks.", "Stubs are divided into macroscopic blocks, which model communities.", "In each macroscopic block, stubs are further divided into microscopic blocks, which are used to create non-zero Pearson degree correlations.", "To achieve this, each stub is designated to one of three types.", "Type 1 and type 2 stubs provide intra-community connections and non-zero degree correlations.", "Type 3 stubs provide intra-community as well as inter-community connections.", "We describe the construction algorithm of CTC in details.", "There are $c$ communities, where $c\\ge 1$ .", "Community $i$ , where $i=1, 2, \\ldots , c$ , has $n_i$ vertices.", "Let $n$ be the total number of vertices in the network.", "It follows that $n=\\sum _{i=1}^c n_i.$ We note that the community sizes can be distinct.", "That is, it is possible that $n_i\\ne n_j$ for some $i\\ne j$ .", "Suppose that a probability mass function (pmf) $\\lbrace p_k, k=0, 1, \\ldots \\rbrace $ is given.", "We repeatedly use this pmf sequence to sample $c$ degree sequences.", "We obtain a double sequence of degrees $\\lbrace k_{ij},i=1, 2, \\ldots , c, j=1, 2, \\ldots , n_i\\rbrace $ , where $k_{ij}$ denotes the degree of the $j$ -th vertex in community $i$ .", "Define $m_i$ such that $ 2 m_i=\\sum _{j=1}^{n_i} k_{ij}.$ The quantity $2m_i$ in (REF ) is the total number of stubs that vertices in community $i$ have.", "The total number of edges in the network is $m$ , where $m=\\sum _{i=1}^c m_i.$ A degree $k$ vertex has $k$ stubs.", "We arrange the stubs associated with the vertices in each community in an ascending order according to their degrees.", "We then partition the stubs in each community into $b$ blocks evenly.", "That is, each block has the same number of stubs.", "To create correlations, we associate each block with another block by a permutation function $h$ .", "Specifically, block $i$ is associated with block $j$ , if $h(i)=j$ .", "In addition, $h$ is selected such that $h(h(i))=i$ for all $i=1, 2, \\ldots , b$ .", "We then classify the stubs in each block into three types proportionally.", "Denote the ratio of type 1 and type 2 stubs to the total stubs in each block as $r\\in [0, 1]$ and the ratio of type 1 stubs to the type 1 and type 2 stubs in each block as $q\\in [0, 1]$ .", "Suppose $r$ and $q$ are given.", "For block $j$ of community $i$ , randomly designate $\\lceil 2m_i qr/b\\rceil $ stubs as type 1 stubs, and randomly designate $\\lceil 2m_i(1-q)r/b\\rceil $ stubs as type 2 stubs.", "Designate the rest stubs in block $j$ as type 3 stubs.", "To make a connection, one randomly picks an unconnected stub, say stub $s$ .", "If $s$ is a type 1 stub in block $j$ of community $i$ , connect it with a randomly selected unconnected type 1 stub in block $h(j)$ in community $i$ and connect it to $s$ .", "If $s$ is a type 2 stub, randomly select an unconnected type 2 stub in community $i$ and connect it to $s$ .", "If $s$ is a type 3 stub, randomly select an unconnected type 3 stub in the network and connect the stub to $s$ .", "These edges are referred to as regular edges.", "Next, we apply triadic closure operations to increase the number of triangles in the network.", "The edges added into the network by the triadic closure operations are called transitive edges.", "We examine all pairs of unconnected vertices in the network.", "For each pair of unconnected vertices, say vertices $A$ and $B$ , if there exists at least one common neighbor, we connect vertices $A$ and $B$ with probability $a$ .", "The construction algorithm for the CTC model is shown in Algorithm .", "Finally, we note that the generalized configuration model is a special case of the CTC model with $c=1, r=1$ and $a=0$ .", "Construction Algorithm Inputs: Double degree sequence $\\lbrace k_{ij}: i=1, 2, \\ldots , c,j=1, 2, \\ldots , n_i\\rbrace $ and parameters $b, q, r, h, a$ .", "Outputs: graph $(G,V,E)$ [1] $i=1, 2, \\ldots , c$ For community $i$ , create $2m_i$ stubs from degree sequence $\\lbrace k_{ij}: j=1, 2, \\ldots , n_i\\rbrace $ and arrange the stubs in an ascending order according to the degrees; Divide $2m_i$ stubs into $b$ blocks evenly; $j=1,2,\\ldots , b$ For block $j$ of community $i$ , randomly designate $\\lceil 2m_i qr/b\\rceil $ stubs as type 1 stubs, and randomly designate $\\lceil 2m_i(1-q)r/b\\rceil $ stubs as type 2 stubs; Designate rest stubs in block $j$ as type 3 stubs; there are unconnected stubs Randomly select a stub.", "Assume that the stub is in block $j$ of community $i$ ; type 1 stub connect this stub with a randomly selected type 1 unconnected stub in block $h(j)$ in community $i$ ; type 2 stub connect this stub with a randomly selected type 2 unconnected stub in community $i$ ; connect this stub with a randomly selected type 3 stub among all type 3 stubs that are unconnected in the network; each unconnected pair of vertices these two vertices have at least one common neighbor due to regular edges with probability $a$ connect these two vertices with a transitive edge, and with probability $1-a$ leave these two vertices unconnected;" ], [ "Review of the Generalized Configuration Model", "In this section we review some basic results of the generalized configuration model in [6].", "These results will be used to derive Pearson degree correlation function and the clustering coefficient of the CTC model in Section and Section , respectively.", "The following assumption is crucial to the analysis.", "Assumption 1 The degree distribution $\\lbrace p_k\\rbrace $ is said to satisfy this assumption if one can find mutually disjoint sets $H_1, H_2, \\ldots , H_{b}$ , such that $\\bigcup _{i=1}^b H_i=\\lbrace 0, 1, 2, \\ldots \\rbrace $ and $ \\sum _{k\\in H_{i}} k p_k ={\\bf \\sf E}[Z]/b$ for all $i=1, 2, \\ldots , b$ .", "In addition, we assume that the degree sequence $k_1, k_2, \\ldots , k_n$ sampled from the distribution $\\lbrace p_k\\rbrace $ can be evenly placed in $b$ blocks.", "Specifically, there exist mutually disjoint sets $H_1, H_2, \\ldots , H_{b}$ that satisfy $\\bigcup _{i=1}^b H_i=\\lbrace 1, 2, \\ldots , n\\rbrace $ , $k_i\\ne k_j$ for any $i\\in H_{\\ell _1}$ , $j\\in H_{\\ell _2}$ , $\\ell _1\\ne \\ell _2$ , and $\\sum _{j\\in H_i} k_j=2m/b$ for all $i=1, 2, \\ldots , b$ .", "Let $Z$ be the degree of a randomly selected vertex in the generalized configuration network.", "Then, the pmf of $Z$ is $\\lbrace p_k\\rbrace $ and ${\\bf \\sf E}[Z]=\\sum _{k=0}^\\infty k p_k.$ We randomly select a stub in the range $[1, 2m]$ .", "Denote this stub by $t$ .", "Let $v$ be the vertex, with which stub $t$ is associated.", "Let $Y$ be the degree of vertex $v$ .", "Since the stub is randomly selected and vertices with degree $y$ have $n y p_y$ stubs.", "We have $\\Pr (Y=y)=\\frac{n y p_y}{2m}= \\frac{y p_y}{{\\bf \\sf E}[Z]}.$ Now connect stub $t$ to a randomly selected stub according to the construction algorithm with $c=1, r=1, a=0$ in Section .", "Let this stub be denoted by $s$ .", "Let $u$ be the vertex, with which $s$ is associated, and let $X$ be the degree of vertex $u$ .", "Next we study ${\\bf \\sf P}(X=x | Y=y)$ .", "We assume that Assumption REF holds.", "Suppose $x$ is a degree in set $H_i$ .", "The total number of stubs which are associated with vertices with degree $x$ is $n x p_x$ .", "By Assumption REF , all $nx p_x$ stubs are in block $i$ .", "There are two cases, in which stub $t$ connects to stub $s$ .", "In the first case, stub $t$ is of type 1.", "This occurs with probability $q$ .", "In this case, stub $s$ must be a type 1 stub and belong to a vertex with a degree in block $h(i)$ .", "With probability $ \\frac{qnxp_x}{2mq/b-\\delta _{i, h(i)}},$ the construction algorithm in Section connects $t$ to stub $s$ .", "In (REF ) $\\delta _{i,j}$ is the Kronecker delta, is equal to one if $i=j$ , and is equal to zero otherwise.", "In the second case, stub $t$ is of type 2.", "This occurs with probability $1-q$ .", "In this case, stub $s$ can be associated with a degree in any block.", "With probability $ \\frac{(1-q)nx p_x}{2m(1-q)-1}$ the construction algorithm connects stub $t$ to stub $s$ .", "Combining the two cases in (REF ) and (REF ), we have $ \\Pr (X=x | Y=y)=\\frac{q^2 nxp_x}{2mq/b-\\delta _{i, h(i)}}+\\frac{(1-q)^2 nx p_x}{2m(1-q)-1}$ for $y\\in H_{h(i)}$ .", "If $y\\in H_j$ for $j\\ne h(i)$ , $ \\Pr (X=x | Y=y)= \\frac{(1-q)^2 nxp_x}{2m(1-q)-1}.$ Now assume that the network is large.", "That is, we consider a sequence of constructed graphs, in which $n\\rightarrow \\infty $ , $m\\rightarrow \\infty $ , while keeping $2m/n={\\bf \\sf E}[Z]$ .", "Under this asymptotic, Eqs.", "(REF ) and (REF ) converge to $\\Pr (X=x | Y=y)\\rightarrow \\left\\lbrace \\begin{array}{ll}\\frac{qb+(1-q)}{{\\bf \\sf E}[Z]}xp_x, & \\quad y\\in H_{h(i)} \\\\\\frac{1-q}{{\\bf \\sf E}[Z]}xp_x, & \\quad y\\in H_j, j\\ne h(i).\\end{array}\\right.$ From (REF ) and (REF ) we obtain $&{\\bf \\sf P}(X=x, Y=y)={\\bf \\sf P}(X=x | Y=y){\\bf \\sf P}(Y=y)\\nonumber \\\\&=\\left\\lbrace \\begin{array}{ll}\\frac{qb+(1-q)}{({\\bf \\sf E}[Z])^2}xy p_xp_y,& \\quad x\\in H_i, y\\in H_{h(i)}\\\\\\frac{1-q}{({\\bf \\sf E}[Z])^2}xy p_xp_y, &\\quad x\\in H_i, y\\in H_j, j\\ne h(i)\\end{array}\\right.", "$ We next analyze the expected value of $Y$ and the product $XY$ , respectively.", "From (REF ), we obtain ${\\bf \\sf E}[Y]=\\sum _{y}y\\Pr (Y=y)=\\sum _{j=1}^{b}\\sum _{y\\in H_j}\\frac{y^2 p_y}{{\\bf \\sf E}[Z]}=\\frac{1}{{\\bf \\sf E}[Z]}\\sum _{j=1}^{b}u_j.$ where $u_j = \\sum _{y\\in H_j} y^2 p_y.$ From (REF ), we have ${\\bf \\sf E}[XY]&=\\sum _{x}\\sum _{y}xy{\\bf \\sf P}(X=x, Y=y)\\nonumber \\\\&=\\frac{1-q}{({\\bf \\sf E}[Z])^2}\\sum _{i=1}^{b}\\sum _{j=1}^{b}u_{i}u_{j}+\\frac{qb}{({\\bf \\sf E}[Z])^2}\\sum _{i=1}^{b}u_{i}u_{h(i)}.$ We now consider ${\\bf \\sf E}[Y | X]$ .", "Denote the conditional expectation ${\\bf \\sf E}[Y | X=x]$ by $g(x)$ .", "Assume that $x\\in H_i$ for some $i$ .", "From (REF ), we have $g(x)&= {\\bf \\sf E}[Y | X=x] \\nonumber \\\\&=\\sum _{i=1}^b\\frac{(1-q) u_{i}}{{\\bf \\sf E}[Z]}+\\frac{qb u_{h(i)}}{{\\bf \\sf E}[Z]}.", "$ In addition, the analysis of the clustering coefficient needs the probability that two specific vertices are connected by a regular edge.", "Randomly select two vertices, say vertices $A$ and $B$ .", "Denote the degrees of $A$ and $B$ by $X_A$ and $X_B$ .", "Let the blocks of $A$ and $B$ be $Q_A$ , $Q_B$ , respectively.", "We now consider the conditional connection probability $&p_c(A, B)=\\nonumber \\\\&{\\bf \\sf P}(\\mbox{vertices $A$ and $B$ are connected}\\, |\\,X_A=k_A, X_B=k_B,\\nonumber \\\\&\\quad Q_A=i, Q_B=j).$ If $h(i)\\ne j$ , vertices $A$ and $B$ can only be connected through a pair of type 2 stubs.", "A type 2 stub of vertex $A$ connects to a type 2 stub of vertex $B$ with probability $(1-q)k_B/(2m(1-q))$ , since $B$ has on average $(1-q) k_B$ type 2 stubs, there are totally $2m(1-q)$ type 2 stubs in the network, and the connection is randomly selected.", "Since vertex $A$ has $(1-q)k_A$ type 2 stubs on average, it follows that $p_c(A, B)=\\dfrac{(1-q)^2 k_A k_B}{2m(1-q)}=\\dfrac{(1-q) k_A k_B}{2m}.$ If $h(i)=j$ , vertices $A$ and $B$ can be connected through a pair of type 1 or type 2 stubs.", "In this case, $p_c(A, B)= \\dfrac{(1-q)^2 k_A k_B}{2m(1-q) }+\\dfrac{q^2 k_A k_B}{2mq/b}=\\dfrac{(1-q+qb) k_A k_B}{2m}.$" ], [ "The Pearson Degree Correlation", "In the section we shall analyze the Pearson degree correlation of the CTC model.", "Recall that the CTC model presented in Section has multiple communities.", "Stubs corresponding to vertices are first divided into communities.", "In each community, stubs are divided into blocks to create a non-zero Pearson degree correlation.", "This two layer structure of stubs does not change the mathematical nature on how Pearson degree correlation being derived.", "However, it does increase the number of cases and the complexity of notations significantly.", "For this reason, we assume that there is one community in this section.", "In this special case, $c=1$ and $r=1$ .", "Pearson degree correlation is defined as the Pearson correlation coefficient of degrees at the two ends of a randomly selected edge.", "The CTC model has two types of edges, regular edges and transitive edges.", "In this paper we analyze the correlation coefficient of degrees at the two ends of a randomly selected regular edge.", "Randomly select an edge among regular edges in the network.", "Let $X$ and $Y$ be the number of regular edges that the two ends of the edge have.", "Let $X^{\\prime }$ and $Y^{\\prime }$ be the number of transitive edges that the two ends of the edge have.", "We shall analyze $\\rho (X+X^{\\prime },Y+Y^{\\prime })\\stackrel{\\scriptstyle \\rm def}{=}\\dfrac{{\\bf \\sf Cov}(X+X^{\\prime },Y+Y^{\\prime })}{\\sigma _{X+X^{\\prime }}\\sigma _{Y+Y^{\\prime }}}$ where ${\\bf \\sf Cov}(X+X^{\\prime },Y+Y^{\\prime })$ is the co-variance of random variables $X+X^{\\prime }$ and $Y+Y^{\\prime }$ and $\\sigma _{X+X^{\\prime }}$ is the standard deviation of $X+X^{\\prime }$ .", "It is well known that $&{\\bf \\sf Cov}(X+X^{\\prime },Y+Y^{\\prime })\\nonumber \\\\& ={\\bf \\sf E}\\left[(X+X^{\\prime })(Y+Y^{\\prime }) \\right]-{\\bf \\sf E}[X+X^{\\prime }]{\\bf \\sf E}[Y+Y^{\\prime }]\\nonumber \\\\& =\\left({\\bf \\sf E}[XY]-{\\bf \\sf E}[X]{\\bf \\sf E}[Y]\\right)+\\left({\\bf \\sf E}[X^{\\prime }Y]-{\\bf \\sf E}[X^{\\prime }]{\\bf \\sf E}[Y]\\right)\\nonumber \\\\& \\quad +\\left({\\bf \\sf E}[XY^{\\prime }]-{\\bf \\sf E}[X]{\\bf \\sf E}[Y^{\\prime }]\\right)+\\left({\\bf \\sf E}[X^{\\prime }Y^{\\prime }]-{\\bf \\sf E}[X^{\\prime }]{\\bf \\sf E}[Y^{\\prime }]\\right).$ Due to symmetry, the product of standard deviations in the denominator of (REF ) is equal to $\\sigma _{X+X^{\\prime }}\\sigma _{Y+Y^{\\prime }} &=\\sigma _{X+X^{\\prime }}^2 \\nonumber \\\\&={\\bf \\sf E}[(X+X^{\\prime })^2]-({\\bf \\sf E}[X+X^{\\prime }])^2\\nonumber \\\\&={\\bf \\sf E}[X^2]-({\\bf \\sf E}[X])^2+2({\\bf \\sf E}[XX^{\\prime }]-{\\bf \\sf E}[X]{\\bf \\sf E}[X^{\\prime }])\\nonumber \\\\&\\quad +{\\bf \\sf E}[(X^{\\prime })^2]-({\\bf \\sf E}[X^{\\prime }])^2.$ In the following subsections, we shall derive the terms on the right-hand side of (REF ) and (REF ).", "In Subsection REF , we show that all these quantities on the right-hand of (REF ) and (REF ) are in the form of ${\\bf \\sf E}[X^i Y^j ({\\bf \\sf E}[Y | X])^k({\\bf \\sf E}[X | Y])^l]$ for some integers $i$ , $j$ , $k$ and $l$ .", "Besides, we analyze the expected value ${\\bf \\sf E}[X^i Y^j ({\\bf \\sf E}[Y | X])^k({\\bf \\sf E}[X | Y])^l]$ for integers $i$ , $j$ , $k$ and $l$ in Subsection REF ." ], [ "Analysis of ${\\bf \\sf E}[X^{\\prime }]$ , {{formula:92301632-eba6-453f-9118-7cdf7afe6117}} , and {{formula:a3a87646-743c-4fe2-9d99-d79e4ffb9b4f}}", "In this section we analyze the terms needed to compute the covariance in (REF ).", "We have analyzed the ${\\bf \\sf E}[Y]$ and ${\\bf \\sf E}[XY]$ in Section .", "Next, we analyze ${\\bf \\sf E}[X^{\\prime }]$ , ${\\bf \\sf E}[X^{\\prime }Y]$ and ${\\bf \\sf E}[X^{\\prime }Y^{\\prime }]$ .", "We first analyze the expected value of $X^{\\prime }$ .", "Recall that $X$ and $Y$ are the number of regular edges that two vertices at the two ends of a randomly selected regular edge.", "Let $A$ and $B$ denote the two vertices.", "$X^{\\prime }$ and $Y^{\\prime }$ are the number of transitive edges that $A$ and $B$ have, respectively.", "Number the $X$ regular edges such that the first edge connects to $B$ .", "Along the $i$ -th regular edge of $A$ to reach the other side, where $i=2, 3, \\ldots , X$ , one finds $Y_i$ regular edges.", "A graphical illustration is shown in Figure REF .", "Figure: Edge ABAB is a randomly selected regular edge.", "Vertex AA has XX regularedges.", "The first edge connects toBB, which has YY regular edges.", "The ii-th edge connects to vertex B i B_i, whichhas Y i Y_i regular edges.Along the first edge, $A$ has $Y-1$ second neighbors and along the $i$ -th edge, $A$ has $Y_i-1$ second neighbors.", "Totally, vertex $A$ has $Y-1+\\sum _{i=2}^{X} (Y_i-1)$ second neighbors.", "The total number of second neighbors in (REF ) can be overestimated, as some second neighbors can be counted more than once.", "However, as the network size is large, the error is asymptotically small.", "We also remark that random variables $Y_i$ , $i\\le 2$ , are identically distributed for large networks.", "Their common distribution is the same as that of $Y$ .", "The number of transitive edges that vertex $A$ has, given $X$ and $Y$ , is $X^{\\prime }=\\sum _{j=1}^{Y-1}B_{1,j}+\\sum _{i=2}^X \\sum _{j=1}^{Y_i-1} B_{ij},$ where $\\lbrace B_{ij}: i\\ge 1, j\\ge 1\\rbrace $ is a double sequence of independent and identically distributed Bernoulli random variables with success probability $a$ .", "The conditional expectation of $X^{\\prime }$ , given $X$ and $Y$ , is ${\\bf \\sf E}[X^{\\prime }|X,Y]&={\\bf \\sf E}\\left[\\sum _{j=1}^{Y-1}B_{1,j}+\\sum _{i=2}^X\\sum _{j=1}^{Y_i-1} B_{ij}\\Biggl |X,Y\\right]\\nonumber \\\\&=a\\cdot (Y-1)+\\sum _{i=2}^X a\\cdot {\\bf \\sf E}[Y_i-1 | X].$ Since $Y_i$ and $Y$ are identically distributed for all $i$ , the preceding equation can be rewritten as ${\\bf \\sf E}[X^{\\prime }|X,Y]= a\\left(Y-1+(X-1){\\bf \\sf E}[Y|X]-(X-1)\\right).$ Taking expectation with respect to $X$ and $Y$ , we have ${\\bf \\sf E}[X^{\\prime }]&=a({\\bf \\sf E}[Y]-1+{\\bf \\sf E}[XY]-{\\bf \\sf E}[Y]-{\\bf \\sf E}[X]+1)\\nonumber \\\\&=a({\\bf \\sf E}[XY]-{\\bf \\sf E}[X]).$ Next, we analyze ${\\bf \\sf E}[X^{\\prime }Y]$ .", "From (REF ) and similar to (REF ), we have ${\\bf \\sf E}[X^{\\prime }Y|X,Y]&={\\bf \\sf E}\\left[\\left(\\sum _{j=1}^{Y-1}B_{1,j}+\\sum _{i=2}^X\\sum _{j=1}^{Y_i-1} B_{ij}\\right)Y\\Biggl |X,Y\\right]\\nonumber \\\\&=a((Y-1)Y+ (X-1) Y\\cdot {\\bf \\sf E}[Y-1 | X])\\nonumber \\\\&=a(Y^2-XY+XY{\\bf \\sf E}[Y | X] -Y{\\bf \\sf E}[Y | X]).$ It follows that ${\\bf \\sf E}[X^{\\prime }Y] = a({\\bf \\sf E}[Y^2]-{\\bf \\sf E}[XY]+{\\bf \\sf E}[XY{\\bf \\sf E}[Y | X] ]-{\\bf \\sf E}[Y{\\bf \\sf E}[Y | X] ]).$ We see that the terms on the right of the preceding expression are in the form of ${\\bf \\sf E}[X^i Y^j ({\\bf \\sf E}[Y | X])^k({\\bf \\sf E}[X | Y])^l]$ for some integers $i$ , $j$ , $k$ and $l$ .", "We shall analyze these terms in the next subsection.", "Finally, we analyze ${\\bf \\sf E}[X^{\\prime }Y^{\\prime }]$ .", "From (REF ), we have ${\\bf \\sf E}[X^{\\prime }Y^{\\prime }|X,Y]&={\\bf \\sf E}\\left[\\left(\\sum _{j=1}^{Y-1}B_{1,j}+\\sum _{i=2}^X\\sum _{j=1}^{Y_i-1} B_{ij}\\right)\\right.\\nonumber \\\\&\\quad \\left.\\cdot \\left(\\sum _{j=1}^{X-1}C_{1,j}+\\sum _{i=2}^Y\\sum _{j=1}^{X_i-1} C_{ij}\\right)\\Biggl |X,Y\\right],$ where $\\lbrace C_{ij}: i\\ge 1, j\\ge 1\\rbrace $ is a double sequence of independent and identically distributed Bernoulli random variables with success probability $a$ .", "Double sequences $\\lbrace B_{ij}\\rbrace $ and $\\lbrace C_{ij}\\rbrace $ are independent.", "Thus, we have ${\\bf \\sf E}[X^{\\prime }Y^{\\prime }|X,Y] &= a^2(Y-1+(X-1){\\bf \\sf E}[Y|X]-(X-1)) \\\\&\\quad \\cdot (X-1+(Y-1){\\bf \\sf E}[X|Y]-(Y-1)).$ Then, we obtain ${\\bf \\sf E}[X^{\\prime }Y^{\\prime }] &= a^2(-2{\\bf \\sf E}[X^2]+2{\\bf \\sf E}[X^2Y]\\nonumber \\\\&\\quad -2{\\bf \\sf E}[XY{\\bf \\sf E}[Y|X]]+2{\\bf \\sf E}[Y{\\bf \\sf E}[Y|X]]\\nonumber \\\\&\\quad +{\\bf \\sf E}[{\\bf \\sf E}[Y|X]{\\bf \\sf E}[X|Y]]\\nonumber \\\\&\\quad -2{\\bf \\sf E}[Y{\\bf \\sf E}[Y|X]{\\bf \\sf E}[X|Y]]\\nonumber \\\\&\\quad +{\\bf \\sf E}[XY{\\bf \\sf E}[Y|X]{\\bf \\sf E}[X|Y]]).$ Again, we see that the terms on the right of (REF ) are in the form of ${\\bf \\sf E}[X^i Y^j ({\\bf \\sf E}[Y | X])^k({\\bf \\sf E}[X | Y])^l]$ for some integers $i$ , $j$ , $k$ and $l$ .", "We will analyze these terms in the next subsection." ], [ "Analysis of ${\\bf \\sf E}[X^i Y^j ({\\bf \\sf E}[Y | X])^k({\\bf \\sf E}[X | Y])^l]$", "We have observed that all quantities of expected values in (REF ) and (REF ) are in the form of ${\\bf \\sf E}[X^i Y^j ({\\bf \\sf E}[Y | X])^k({\\bf \\sf E}[X | Y])^l]$ for some integers $i$ , $j$ , $k$ and $l$ .", "Instead of tediously presenting the derivations of all expectation terms needed to compute the covariance and the variance, we choose a more complex term, that is ${\\bf \\sf E}[X^2({\\bf \\sf E}[Y|X])^2]$ , and derive it in this subsection.", "The derivation of other terms is similar, and is omitted.", "We simply present the result in Appendix A.", "With (REF ), we have ${\\bf \\sf E}[X^2({\\bf \\sf E}[Y|X])^2]&= \\sum _x(x g(x))^2 {\\bf \\sf P}(X=x) \\\\&=\\sum _{i=1}^b \\sum _{x\\in H_i} \\left(\\sum _{i=1}^b\\frac{(1-q) u_{i}}{{\\bf \\sf E}[Z]}+\\frac{qb u_{h(i)}}{{\\bf \\sf E}[Z]}\\right)^2 \\\\&\\quad \\cdot x^2\\cdot \\frac{x p_x}{{\\bf \\sf E}[Z]} \\\\&= (1-q)^2\\frac{({\\bf \\sf E}[Z^2])^2{\\bf \\sf E}[Z^3]}{({\\bf \\sf E}[Z])^3}\\\\&\\quad +2(1-q)qb\\frac{{\\bf \\sf E}[Z^2]}{({\\bf \\sf E}[Z])^3}\\sum _{i=1}^bu_{h(i)}t_i\\\\&\\quad +q^2b^2\\frac{1}{({\\bf \\sf E}[Z])^3}\\sum _{i=1}^b(u_{h(i)})^2t_i,$ where $t_{i} &=\\sum _{x\\in H_{i}}x^{3}p_{x}.$" ], [ "Analysis of ${\\bf \\sf Cov}(X+X^{\\prime },Y+Y^{\\prime })$", "Finally, substituting (REF ), (REF ), (), (REF ), and (REF ) into (REF ), we obtain ${\\bf \\sf Cov}(X+X^{\\prime },Y+Y^{\\prime }) =\\alpha _{0} + \\sum _{i=1}^5 \\beta _i W_i,$ where $& \\alpha _{0} =2a\\frac{a({\\bf \\sf E}[Z^{2}]-{\\bf \\sf E}[Z])+{\\bf \\sf E}[Z]}{({\\bf \\sf E}[Z])^{3}}\\left({\\bf \\sf E}[Z]{\\bf \\sf E}[Z^3]-({\\bf \\sf E}[Z^{2}])^{2}\\right)\\nonumber \\\\& \\beta _{1}= \\frac{q}{({\\bf \\sf E}[Z])^2}+2a \\frac{q}{({\\bf \\sf E}[Z])^{2}}\\left(\\frac{(1-q){\\bf \\sf E}[Z^{2}]}{{\\bf \\sf E}[Z]}-1\\right)\\nonumber \\\\&\\ \\ +a^{2}\\frac{q\\left(((1-q){\\bf \\sf E}[Z^{2}]+q{\\bf \\sf E}[Z])^{2}-2(2-q^{2}){\\bf \\sf E}[Z^{2}]{\\bf \\sf E}[Z]\\right)}{({\\bf \\sf E}[Z])^{4}}\\nonumber \\\\& \\beta _{2} =2a^{2}\\frac{q}{({\\bf \\sf E}[Z])^{2}}\\nonumber \\\\& \\beta _{3} =-2a\\frac{q^{2}}{({\\bf \\sf E}[Z])^{2}}\\left((1-a)+a\\frac{(1-q){\\bf \\sf E}[Z^2]}{{\\bf \\sf E}[Z]}\\right)\\nonumber \\\\& \\beta _{4} =2a\\frac{q^{2}b}{({\\bf \\sf E}[Z])^{3}}\\left((1-a)-aq+a\\frac{(1-q){\\bf \\sf E}[Z^2]}{{\\bf \\sf E}[Z]}\\right) \\nonumber \\\\& \\beta _{5} = a^{2}\\frac{q^{3}b^{2}}{({\\bf \\sf E}[Z])^{4}} \\nonumber \\\\&W_1=b\\sum _{i=1}^b u_i u_{h(i)}-\\sum _{i=1}^bu_i\\sum _{j=1}^b u_j \\\\&W_2=b\\sum _{i=1}^b t_i u_{h(i)}-\\sum _{i=1}^bu_i \\sum _{j=1}^b t_j\\\\&W_3= b\\sum _{i=1}^{b}u_{i}u_{i}-\\sum _{i=1}^{b}u_{i}\\sum _{j=1}^{b}u_{j}\\\\&W_4= b\\sum _{i=1}^{b}u_iu_{h(i)}u_{i}-\\sum _{i=1}^{b}u_{i}u_{h(i)}\\sum _{j=1}^{b}u_{j}\\\\&W_5=b\\sum _{i=1}^{b}u_{i}u_{h(i)}u_{i}u_{h(i)}-\\sum _{i=1}^{b}u_{i}u_{h(i)}\\sum _{j=1}^{b}u_{j}u_{h(j)}.$ In (REF ) and (), sequences $\\lbrace u_i\\rbrace $ and $\\lbrace t_i\\rbrace $ are defined in (REF ) and (REF ), respectively.", "One of the main results of this paper is the following theorem.", "Its proof is presented in Appendix B at the end of this paper.", "Theorem 2 If $h(i)=i$ , then ${\\bf \\sf Cov}(X+X^{\\prime }, Y+Y^{\\prime })\\ge {\\bf \\sf Cov}(X, Y) \\ge 0.$" ], [ "Analysis of $\\sigma _{X+X^{\\prime }}$", "In this subsection we analyze $\\sigma _{X+X^{\\prime }}$ in the denominator in Eq.", "(REF ).", "Among the expectation terms needed in the $\\sigma _{X+X^{\\prime }}$ , ${\\bf \\sf E}[X]$ and ${\\bf \\sf E}[X^{\\prime }]$ were already analyzed.", "We next analyze ${\\bf \\sf E}[X X^{\\prime }]$ and ${\\bf \\sf E}[(X^{\\prime })^2]$ .", "Since $Y_i$ and $Y$ are identically distributed and ${\\bf \\sf E}[Y_i | X]={\\bf \\sf E}[Y|X]$ .", "From (REF ) we have $&{\\bf \\sf E}[XX^{\\prime }] \\nonumber \\\\&= {\\bf \\sf E}[{\\bf \\sf E}[XX^{\\prime } | X, Y, Y_i,\\forall i]] \\nonumber \\\\&= {\\bf \\sf E}\\left[{\\bf \\sf E}\\left[X \\left(\\sum _{j=1}^{Y-1}B_{1,j}+\\sum _{i=2}^X \\sum _{j=1}^{Y_i-1} B_{ij}\\right)\\Biggl | X, Y, Y_i,\\forall i\\right]\\right]\\nonumber \\\\&= {\\bf \\sf E}\\left[{\\bf \\sf E}\\left[X \\left(a(Y-1)+a\\sum _{i=2}^X (Y_i-1)\\right)\\Biggl | X, Y, Y_i,\\forall i\\right]\\right] \\nonumber \\\\&= {\\bf \\sf E}\\Big [a X({\\bf \\sf E}[Y|X]-1+(X-1)({\\bf \\sf E}[Y|X]-1))\\Big ]\\nonumber \\\\&= a\\left({\\bf \\sf E}[X^2 Y]-{\\bf \\sf E}[X^2]\\right).$ Next, we analyze ${\\bf \\sf E}[(X^{\\prime })^2]$ .", "From (REF ), we have $&{\\bf \\sf E}[(X^{\\prime })^2]\\nonumber \\\\&={\\bf \\sf E}\\left[{\\bf \\sf E}\\left[\\left(\\sum _{j=1}^{Y-1}B_{1,j}+\\sum _{i=2}^X \\sum _{j=1}^{Y_i-1} B_{ij}\\right)^2\\Biggl |X, Y, Y_i, \\forall i\\right]\\right]\\nonumber \\\\&= {\\bf \\sf E}\\left[{\\bf \\sf E}\\left[\\left(\\sum _{j=1}^{Y-1}B_{1,j}\\right)^2\\Biggl |X, Y, Y_i, \\forall i\\right]\\right] \\nonumber \\\\&\\ +{\\bf \\sf E}\\left[{\\bf \\sf E}\\left[2\\left(\\sum _{j=1}^{Y-1}B_{1,j}\\right)\\left(\\sum _{i=2}^X \\sum _{j=1}^{Y_i-1} B_{ij}\\right)\\Biggl |X, Y, Y_i, \\forall i\\right]\\right]\\nonumber \\\\&\\ + {\\bf \\sf E}\\left[{\\bf \\sf E}\\left[\\left(\\sum _{i=2}^X \\sum _{j=1}^{Y_i-1} B_{ij}\\right)^2\\Biggl |X, Y, Y_i, \\forall i\\right]\\right].$ The first term on the right side of Eq.", "(REF ) is equal to $& {\\bf \\sf E}\\left[{\\bf \\sf E}\\left[\\sum _{i=1}^{Y-1}\\sum _{j=1}^{Y-1}B_{1,i}B_{1,j}\\Biggl |X, Y, Y_i, \\forall i\\right]\\right] \\\\&= {\\bf \\sf E}\\left[{\\bf \\sf E}\\left[\\sum _{i=1}^{Y-1}B_{1,i}^2+\\sum _{i=1}^{Y-1}\\sum _{{j=1}\\atop {j\\ne i}}^{Y-1}B_{1,i}B_{1,j}\\Biggl |X, Y, Y_i, \\forall i\\right]\\right] \\\\&=a({\\bf \\sf E}[Y]-1)+a^2 {\\bf \\sf E}[({\\bf \\sf E}[Y | X]-1)({\\bf \\sf E}[Y | X]-2)]\\\\&=a^2{\\bf \\sf E}[({\\bf \\sf E}[Y | X])^2]+(a-3a^2){\\bf \\sf E}[Y]+2a^2-a.$ The second term on the right side of Eq.", "(REF ) is equal to $&{\\bf \\sf E}\\left[{\\bf \\sf E}\\left[2\\left(\\sum _{j=1}^{Y-1}B_{1,j}\\right)\\left(\\sum _{i=2}^X \\sum _{j=1}^{Y_i-1} B_{ij}\\right)\\Biggl |X, Y, Y_i, \\forall i\\right]\\right] \\\\&={\\bf \\sf E}\\left[{\\bf \\sf E}\\left[2(Y-1)\\cdot a^2\\cdot \\left(\\sum _{i=2}^X Y_i - (X-1)\\right)\\Biggl |X, Y, Y_i, \\forall i\\right]\\right].$ Taking average of the preceding with respect to $Y_i$ for all $i$ , while conditioning on $X$ and $Y$ , we have $&{\\bf \\sf E}\\left[{\\bf \\sf E}\\left[2\\left(\\sum _{j=1}^{Y-1}B_{1,j}\\right)\\left(\\sum _{i=2}^X \\sum _{j=1}^{Y_i-1} B_{ij}\\right)\\Biggl |X, Y, Y_i, \\forall i\\right]\\right] \\\\&=2a^2{\\bf \\sf E}\\left[(Y-1)(X-1)({\\bf \\sf E}[Y | X]-1)\\right]\\\\&=2a^2(-2{\\bf \\sf E}[XY]+3{\\bf \\sf E}[X]+{\\bf \\sf E}[XY{\\bf \\sf E}[Y | X]]\\\\&\\quad -{\\bf \\sf E}[Y{\\bf \\sf E}[Y | X]]-1).$ Now consider the third term on the right of (REF ).", "It can be written as $&\\left(\\sum _{i=2}^X \\sum _{j=1}^{Y_i-1} B_{ij}\\right)^2\\nonumber \\\\&= \\sum _{i=2}^X \\sum _{j=1}^{Y_i -1} B_{i,j}^2 +\\sum _{i=2}^X \\sum _{j=1}^{Y_i-1} \\sum _{{k=2}\\atop {k\\ne i}}^X\\sum _{\\ell =1}^{Y_k-1}B_{i,j}B_{k,\\ell } \\nonumber \\\\&\\quad +\\sum _{i=2}^X\\sum _{j=1}^{Y_i-1}\\sum _{{\\ell =1}\\atop {\\ell \\ne j}}^{Y_i-1} B_{i,j}B_{i,\\ell }.$ The conditional expectation of (REF ), given $X$ , $Y$ and $Y_i$ for all $i$ , is $&a\\sum _{i=2}^X (Y_i-1)+a^2\\sum _{i=2}^X\\sum _{{k=2}\\atop {k\\ne i}}^X(Y_i-1)(Y_k-1) \\nonumber \\\\&\\ \\ +a^2\\sum _{i=2}^X(Y_i-1)(Y_i-2).$ Taking average on (REF ) with respect to $Y_i$ for all $i$ , we have $&a(X-1)({\\bf \\sf E}[Y | X]-1)\\nonumber \\\\&+a^2(X-1)(X-2){\\bf \\sf E}[(Y_2-1)(Y_3-1)] | X)) \\nonumber \\\\&+a^2 (X-1){\\bf \\sf E}[(Y-1)(Y-2) | X].$ Since $Y_2$ and $Y_3$ are conditional independent, given $X$ , the preceding quantity is equal to $&a(X-1)({\\bf \\sf E}[Y | X]-1)\\nonumber \\\\&+a^2(X-1)(X-2)({\\bf \\sf E}[Y| X-1 ])^2) \\nonumber \\\\&+a^2 (X-1){\\bf \\sf E}[(Y-1)(Y-2) | X].$ Finally, taking average on (REF ) with respect to $X$ , we obtain $&a({\\bf \\sf E}[XY]-2{\\bf \\sf E}[X]+1)+\\nonumber \\\\&a^2(-{\\bf \\sf E}[X^2Y]+3{\\bf \\sf E}[XY] -2{\\bf \\sf E}[Y]\\nonumber \\\\&\\quad +{\\bf \\sf E}[X^2{\\bf \\sf E}[Y | X]{\\bf \\sf E}[Y | X]]\\nonumber \\\\&\\quad -3{\\bf \\sf E}[X{\\bf \\sf E}[Y | X]{\\bf \\sf E}[Y | X]]\\nonumber \\\\&\\quad +2{\\bf \\sf E}[{\\bf \\sf E}[Y | X]{\\bf \\sf E}[Y | X]]).$ This is the third term on the right side of (REF )." ], [ "Clustering Coefficient", "In this section we analyze the local clustering coefficient of the CTC model.", "We remark that in this section we study the clustering coefficient of a special case in which there is only one community, i.e.", "$c=1$ and $r=1$ .", "We also remark that it is quite easy to extend the analysis to CTC models with more than one community.", "We choose to present the result of the special case in order to keep notational simplicity.", "Let $A$ be a randomly selected vertex in a random network.", "Let $k$ be the degree of $A$ .", "The local clustering coefficient of vertex $A$ is defined as $&C_A(k)=\\frac{\\mbox{number of connected pairs of neighbors of $A$}}{\\mbox{number of pairs of neighbors of $A$}}\\nonumber \\\\&=\\frac{\\mbox{number of connected pairs of neighbors of $A$}}{k(k-1)/2}.$ If $k\\le 1$ , $C_A(k)$ is defined to be zero.", "We distinguish between regular edges and transitive edges.", "Assume that vertex $A$ has $k$ regular edges and $k^\\prime $ transitive edges.", "We denote the local clustering coefficient of $A$ by $C_A(k, k^{\\prime })$ .", "Let the $k$ vertices connected with $A$ by regular edges be denoted by $U_1, U_2, \\ldots , U_k$ .", "Let the $k^\\prime $ vertices connected with $A$ by transitive edges be denoted by $V_1, V_2, \\ldots , V_{k^\\prime }$ .", "We analyze the numerator of (REF ).", "Obviously, if $k=0$ , $C_A(0, k^\\prime )=0$ .", "For general $k\\ge 1$ and $k^\\prime \\ge 0$ , we claim that $C_A(k,k^{\\prime })=\\frac{\\displaystyle \\left(\\begin{array}{c}k\\\\2\\end{array}\\right)a+k^\\prime +\\left(\\begin{array}{c}k^\\prime \\\\2\\end{array}\\right)\\times \\frac{a}{k}}{\\left(\\begin{array}{c}k+k^\\prime \\\\2\\end{array}\\right)},$ with the convention that $\\left(\\begin{array}{c}i\\\\j\\end{array}\\right)=0\\ \\mbox{if $i<j$}.$ To analyze (REF ) we consider six types of triangles as shown in Figure REF .", "We consider type 1 triangles shown in panel (a) of Figure REF .", "The expected number of type 1 triangles is $\\sum _{k_1, k_2} \\left(\\begin{array}{c} k \\\\ 2 \\end{array}\\right) p_c(U_1, U_2) p_{k_1} p_{k_2},$ where $k_1$ and $k_2$ are the degrees of vertices $U_1$ and $U_2$ , respectively.", "Since $2m=n {\\bf \\sf E}[Z]$ , it follows that the expected number of type 1 triangles in the last expression approaches to zero as the network size $n$ is large.", "Note that type 5 triangles in panel (e) of Figure REF also require vertices $U_1$ and $U_2$ be connected by regular edges.", "By the same argument, it is easy to see that the expected number of type 5 triangles also goes to zero as the network gets large.", "Now we consider the second type of triangles shown in panel (b) of Figure REF .", "Vertices $U_1$ and $U_2$ are connected by a transitive edge.", "This transitive edge is formed because vertice $U_2$ is an unconnected second neighbor of $U_1$ through vertex $A$ .", "Thus, the expected number of type 2 triangles is $\\sum _{k_1, k_2} \\left(\\begin{array}{c} k \\\\ 2 \\end{array}\\right)\\left(1- p_c(U_1, U_2) p_{k_1} p_{k_2}\\right)\\cdot a=\\left(\\begin{array}{c} k \\\\ 2 \\end{array}\\right)\\cdot a.$ This is the first term in the numerator of (REF ).", "We next analyze type 3 triangles shown in panel (c) of Figure REF .", "Note that transitive edge $AV_1$ can be formed in two types of event.", "The first type of event is the successful event of random triadic closure of the connected triples of $A$ , $V_1$ and $U_1$ .", "The second type of event is the successful event of random triadic closure of the connected triples of $A$ , $V_1$ and one of the first neighbors of A in the set $\\lbrace U_1, U_2, \\ldots , U_k\\rbrace $ except $U_1$ .", "If the transitive edge $AV_1$ in type 3 triangle is formed from the first type of event, the number of type 3 is $k^{\\prime }$ .", "If the transitive edge $AV_1$ is formed from the second type of event, to form a type 3 triangle, $U_1$ and $V_1$ need to be connected by a regular edge.", "From the analysis of type 1 triangles, we know that the connected probability of $U_1$ and $V_1$ approaches to zero as the network size $n$ is large.", "To sum, the expected number of type 3 triangles is $k^{\\prime }$ , which is the second term in the numerator of (REF ).", "Now we consider type 4 triangles shown in panel (d) of Figure REF .", "In order to form a transitive edge between $U_1$ and $V_1$ , these two vertices must have at least one common neighbor by regular edges.", "Besides vertices $A$ , $U_1$ and $V_1$ , there are $n-3$ vertices in the network.", "Let $E_{n-3}$ be the event that there is at least one vertex in $n-3$ vertices that connects to both $U_1$ and $V_1$ .", "Then, the expected number of type 4 triangles is $k k^{\\prime } a {\\bf \\sf P}(E_{n-3})=k k^{\\prime } a (1-{\\bf \\sf P}(E_{n-3}^c)),$ where $E_{n-3}^c$ is the complement of event $E_{n-3}$ .", "Denote the $n-3$ vertices by vertices $1, 2, \\ldots , n-3$ .", "${\\bf \\sf P}(E_{n-3}^c)=&\\sum _{u, v, k_1, k_2, \\ldots , k_{n-3}}\\prod _{j=1}^{n-3} (1-p_c(j,U_1))(1-p_c(j, V_1))\\nonumber \\\\&\\times p_u p_v p_{k_1} p_{k_2}\\cdots p_{k_{n-3}},$ where $u$ and $v$ are the degrees of $U_1$ and $V_1$ , and $k_j$ is the degree of vertex $j$ for $j=1, 2, \\ldots , n-3$ .", "To evaluate ${\\bf \\sf P}(E_{n-3})$ , we substitute (REF ) or (REF ) into (REF ).", "For example, denote $b_{U_1}$ , $b_{V_1}$ and $b_{j}$ as the block index of $U_1$ , $V_1$ and $j$ , respectively.", "Suppose $b_{U_1} \\ne h(b_{j})$ and $b_{V_1} \\ne h(b_{j})$ , we substitute (REF ) into (REF ) and have $&{\\bf \\sf P}(E_{n-3})=1-\\nonumber \\\\&\\sum _{u, v, k_1, k_2, \\ldots , k_{n-3}}\\prod _{j=1}^{n-3} \\prod _{j=1}^{n-3} \\left(1-\\frac{(1-q)u k_j}{2m}\\cdot \\frac{(1-q)v(k_j-1)}{2m}\\right) \\nonumber \\\\&\\times p_u p_v p_{k_1}p_{k_2}\\cdots p_{k_{n-3}}\\nonumber \\\\&=1-\\sum _{u,v}\\left(1-\\frac{(1-q)^2 u v ({\\bf \\sf E}[Z^2]-{\\bf \\sf E}[Z])}{(n {\\bf \\sf E}[Z])^2}\\right)^{n-3}p_u p_v \\nonumber \\\\&\\le 1-\\left(1-\\frac{(1-q)^2 ({\\bf \\sf E}[Z])^2({\\bf \\sf E}[Z^2]-{\\bf \\sf E}[Z])}{(n{\\bf \\sf E}[Z])^2}\\right)^{n-3} \\\\&\\rightarrow 0\\ \\mbox{as $n\\rightarrow \\infty $,}\\nonumber $ where inequality (REF ) is due to Jensen's inequality [16].", "It follows that the expected number of type 4 triangles is zero in large networks.", "Note that substitution of (REF ) into (REF ) leads to the same result, i.e.", "the expected number of type 4 triangles is zero in large networks.", "Finally, we consider the expected number of type 6 triangles shown in panel (f) of Figure REF .", "Note that to form transitive edges $AV_1$ and $AV_2$ , vertices $V_1$ and $V_2$ must be unconnected second neighbors of $A$ through some first neighbors of $A$ in the set $\\lbrace U_1, U_2, \\ldots , U_k\\rbrace $ .", "There are two cases.", "In the first case shown in panel (a) of Figure REF , $V_1$ and $V_2$ have distinct common neighbors with $A$ .", "Vertices $V_1$ and $V_2$ randomly and independently select first neighbors from the set $\\lbrace U_1, U_2, \\ldots , U_k\\rbrace $ .", "The probability that their selections are distinct is $(k-1)/k.$ Thus, the expected number of type 6 triangles in the first case is $\\left(\\begin{array}{c}k^{\\prime } \\\\ 2 \\end{array}\\right)\\cdot \\dfrac{k-1}{k}\\cdot a\\cdot {\\bf \\sf P}(E_{n-5}).$ By the same argument in (REF ), it is easy to show that the quantity above goes to zero as $n$ goes to infinity.", "Now we consider the second case shown panel (b) of Figure REF .", "Vertices $V_1$ and $V_2$ share a common first neighbor $U_i$ with $A$ .", "The probability that the random selections of $V_1$ and $V_2$ are the same is $1/k$ .", "Thus, the expected number of type 6 triangles is $\\left(\\begin{array}{c}k^{\\prime } \\\\ 2 \\end{array}\\right)\\cdot \\dfrac{a}{k}.$ This is the third term in the numerator on the right side of (REF ).", "Figure: Six types of triangles.", "Solid lines denote regular edgesof the CTC model.", "Dashed lines denotetransitive edges due to triadic closure operations.Figure: Type 6 triangles.", "Vertices V 1 V_1 and V 2 V_2 must be unconnected secondneighbors of AA through some first neighbors U i U_i and U j U_j.", "In panel(a), the two first neighbors of AA are distinct.", "In panel (b), verticesV 1 V_1 and V 2 V_2 have a common first neighbor of AA." ], [ "Numerical and Simulation Results", "In this section, we first verify our derivation of the Pearson degree correlation and the clustering coefficient by comparing numerical calculation with simulation.", "We then test the community structures of the CTC model using three well known community detection algorithms.", "We now verify the correctness of our derivation.", "We first present numerical and simulation results on the Pearson degree correlation of the special case of the CTC model where $c=1$ and $r=1$ .", "In our experiment, we assume that there are 10000 vertices.", "We choose power law degree distributions.", "In addition, we set the block size $b=2$ .", "To obtain a data point we repeat 50 simulations and take an average.", "We discuss numerical and simulation results on the $\\mbox{cov}(X+X^{\\prime }, Y+Y^{\\prime })$ .", "Let $h(i)=i$ .", "Figure REF and Figure REF show plots of $\\mbox{cov}(X+X^{\\prime }, Y+Y^{\\prime })$ with respect to $a$ and $q$ , respectively.", "These two figures show that the difference between simulation result and analytical result is very small.", "We note that the values of $\\mbox{cov}(X+X^{\\prime }, Y+Y^{\\prime })$ in Figure REF and Figure REF are all positive, i.e., the simulated graphs are all assortative mixing when $h(i)=i$ .", "From Figure REF and Figure REF , we observe that $\\mbox{cov}(X+X^{\\prime }, Y+Y^{\\prime })$ is increasing with $a$ and $q$ , respectively.", "Moreover, the plots in Figure REF and Figure REF confirm Theorem REF .", "Next, let $h(i)=b+1-i$ .", "Figure REF and Figure REF show that the difference between simulation result and analytical result is also very small.", "From Figure REF , we find that some part of the red line as well as the purple line is below the zero line, indicating that the CTC model with small values of $a$ can also be disassortative.", "Besides, Figure REF shows that the relationship between the covariance of the CTC model without triadic closure operations and $q$ (the blue line) is linear while the covariance of the CTC model with triadic closure operations is nonlinear with $q$ (the green line, the purple line and the red line).", "Moreover, Figure REF shows simulation result and analytical result on the Pearson degree correlation.", "We also observe that the analytical result is very close to the simulation result.", "Figure: Plot of cov(X+X ' ,Y+Y ' )\\mbox{cov}(X+X^{\\prime }, Y+Y^{\\prime }) versus aa with h(i)=ih(i)=i.Figure: Plot of cov(X+X ' ,Y+Y ' )\\mbox{cov}(X+X^{\\prime }, Y+Y^{\\prime }) versus qq with h(i)=ih(i)=i.", "In the upperpanel, a=0a=0 and 0.10.1.", "In the lower panel, a=0.5a=0.5 and 0.90.9.Figure: Plot of cov(X+X ' ,Y+Y ' )\\mbox{cov}(X+X^{\\prime }, Y+Y^{\\prime }) versus aa with h(i)=b+1-ih(i)=b+1-i.Figure: Plot of cov(X+X ' ,Y+Y ' )\\mbox{cov}(X+X^{\\prime }, Y+Y^{\\prime }) versus qq with h(i)=b+1-ih(i)=b+1-i.In the upper panel, a=0a=0 and 0.10.1.", "In the lower panel, a=0.5a=0.5 and 0.90.9.Figure: Plot of Pearson degree correlation versus qq.Next, we numerically compute and simulate the local clustering coefficient of the CTC model.", "The results are shown in Figure REF .", "We find that the simulation result and the analytical result obtained by Eq.", "(REF ) is very similar.", "Figure: Plot of local clustering coefficient versus aa.Finally, we test the community structure of the CTC model using three well known community detection algorithms.", "The three community detection algorithms are the walktrap algorithm [17], the leading eigenvector algorithm [18] and the fast unfolding algorithm [19].", "We also compare the CTC model with other benchmark models that are the ABCD networks [15] and the LFR networks [13].", "Source code to generate ABCD networks and LFR networks has been provided in the github.", "Note that the Girvan-Newman model [12] has also been used to benchmark community detection algorithms.", "Girvan-Newman models are actually interconnection of Erdős and Rényi networks, and are a special case of stochastic block models.", "Since ER models have Poisson degree distributions, it is hard to make a fair comparison with the other three models.", "Thus, we have not used the GN model as a benchmark.", "To make a fair comparison, the three network models must have the same degree sequences or distributions.", "Since the random triadic closure operations modify the original degrees in the CTC networks, we need to modify the degrees in the ABCD networks and the LFR networks in order to achieve a fair comparison.", "We take the following steps.", "We generate an ABCD network with 10000 vertices using the github source codehttps://github.com/bkamins/ABCDGraphGenerator.jl.", "The github source code also requires a power exponent for a power law degree distribution and a power exponent for a power law community size distribution.", "In our experiment, we let both exponents be 2.", "The source code also needs a mixing parameter $u$ , which is the proportion of external edges for a given community.", "Let $c$ be the number of communities of the ABCD network generated in the last step.", "We obtain the degree sequence and the community assignment of nodes from the ABCD network, and use them as the inputs the double degree sequence $\\lbrace k_{ij}: i=1, 2, \\ldots , c, j=1, 2, \\ldots , n_i\\rbrace $ to generate a CTC network.", "Other input parameters of the CTC network are set according to $q=0.1$ , $a=0.1$ , $b=2$ , $h(i)=i$ , and $r$ ranges between $0.1$ and $0.9$ with a step of $0.1$ .", "For each generated CTC graph, we measure the degree distribution, including regular edges and transitive edges, and fit a power law distribution.", "We also calculate the new mixing parameter $u$ of the generated CTC graph.", "We then generate an ABCD network again using the fitted power law exponent, the calculated mixing parameter $u$ , and the same power law exponent of 2 for community size distribution.", "We also run the github codehttps://github.com/eXascaleInfolab/LFR-BenchmarkUndirWeightOvp to construct an LFR network using the fitted power law exponent, the calculated mixing parameter $u$ , and the same power law exponent of 2 for community size distribution.", "We apply the walktrap algorithm, the leading eigenvector algorithm and the fast unfolding algorithm on the three random network models.", "We have two partitions of the same network.", "The first partition corresponds to the community structure of the network and serves as the ground truth.", "The second partition corresponds to the communities predicted by a community detection algorithm.", "A commonly used measure of community detection algorithms is the normalized mutual information (NMI) [20].", "We repeat the graph generation of each model for 30 times and take an average of the NMI values of each community detection algorithm on each benchmark model.", "From Figure REF and Figure REF , we observe that the performance of these three community detection algorithms on the CTC network is worst.", "It implies that the community structure of the CTC model is more difficult to detect than those of the other two benchmark models.", "Figure: The performance of walktrap algorithm on three benchmark models.Figure: The performance of leading eigenvector algorithm on three benchmark models.Figure: The performance of fast unfolding algorithm on three benchmark models." ], [ "Conclusions", "In this study, we present a configuration model with triadic closure, which is an extension of the generalized configuration model.", "The CTC model possesses five most important properties of graphs that arise in network science.", "We derive closed forms for the Pearson degree correlation and clustering coefficient of the CTC model.", "We verify the correctness of our analytical results with simulations.", "Moreover, we compare the CTC model with other two benchmark models using three well-known community detection algorithms.", "Appendix A In this appendix we list all expectations needed to compute the covariance in Eq.", "(REF ) and the variance in Eq.", "(REF ).", "Note that ${\\bf \\sf E}[X]$ and ${\\bf \\sf E}[XY]$ have been derived in [6].", "We repeat them here for easy reference.", "${\\bf \\sf E}[X] &= \\frac{{\\bf \\sf E}[Z^2]}{{\\bf \\sf E}[Z]} \\\\{\\bf \\sf E}[X^2] &= \\frac{{\\bf \\sf E}[Z^3]}{{\\bf \\sf E}[Z]} \\\\{\\bf \\sf E}[XY] &=\\frac{1-q}{({\\bf \\sf E}[Z])^2}\\sum _{i=1}^{b}\\sum _{j=1}^{b}u_{i}u_{j}\\nonumber \\\\&+\\frac{qb}{({\\bf \\sf E}[Z])^2}\\sum _{i=1}^{b}u_{i}u_{h(i)} \\\\{\\bf \\sf E}[XY^2] &= {\\bf \\sf E}[X^2 Y]\\nonumber \\\\&= {\\bf \\sf E}[X^2{\\bf \\sf E}[Y|X]]\\nonumber \\\\&=\\frac{(1-q){\\bf \\sf E}[Z^2]{\\bf \\sf E}[Z^3]}{({\\bf \\sf E}[Z])^2}\\nonumber \\\\&+\\frac{qb}{({\\bf \\sf E}[Z])^2}\\sum _{i=1}^{b}u_{i}t_{h(i)} \\\\{\\bf \\sf E}[X{\\bf \\sf E}[Y|X]] &= {\\bf \\sf E}[Y{\\bf \\sf E}[X | Y]]={\\bf \\sf E}[XY] \\\\{\\bf \\sf E}[Y{\\bf \\sf E}[Y|X]] &= {\\bf \\sf E}[X{\\bf \\sf E}[X | Y]] \\nonumber \\\\&={\\bf \\sf E}[({\\bf \\sf E}[Y|X])^2]\\nonumber \\\\&=\\frac{(1-q^2)({\\bf \\sf E}[Z^2])^2}{({\\bf \\sf E}[Z])^2}\\nonumber \\\\&+\\frac{q^2b}{({\\bf \\sf E}[Z])^2}\\sum _{i=1}^{b}u_{i}u_{i} \\\\{\\bf \\sf E}[X({\\bf \\sf E}[Y|X])^2] &=\\frac{(1-q)^2({\\bf \\sf E}[Z^2])^3}{({\\bf \\sf E}[Z])^3}\\nonumber \\\\&+\\frac{2(1-q)qb{\\bf \\sf E}[Z^2]}{({\\bf \\sf E}[Z])^3}\\sum _{i=1}^{b}u_{i}u_{h(i)}\\nonumber \\\\&+\\frac{q^2b^2}{({\\bf \\sf E}[Z])^3}\\sum _{i=1}^{b}u_{i}u_{h(i)}u_{h(i)}\\\\{\\bf \\sf E}[X^2({\\bf \\sf E}[Y|X])^2]&= (1-q)^2\\frac{({\\bf \\sf E}[Z^2])^2{\\bf \\sf E}[Z^3]}{({\\bf \\sf E}[Z])^3}\\nonumber \\\\&\\quad +2(1-q)qb\\frac{{\\bf \\sf E}[Z^2]}{({\\bf \\sf E}[Z])^3}\\sum _{i=1}^bu_it_{h(i)}\\nonumber \\\\&\\quad +q^2b^2\\frac{1}{({\\bf \\sf E}[Z])^3}\\sum _{i=1}^b(u_i)^2t_{h(i)}\\\\{\\bf \\sf E}[XY{\\bf \\sf E}[Y|X]] &= {\\bf \\sf E}[XY{\\bf \\sf E}[X|Y]]\\nonumber \\\\&=\\frac{(1-q)^2({\\bf \\sf E}[Z^2])^3}{({\\bf \\sf E}[Z])^3}\\nonumber \\\\&+\\frac{2(1-q)qb{\\bf \\sf E}[Z^2]}{({\\bf \\sf E}[Z])^3}\\sum _{i=1}^{b}u_{i}u_{h(i)}\\nonumber \\\\&+\\frac{q^2b^2}{({\\bf \\sf E}[Z])^3}\\sum _{i=1}^{b}u_{i}u_{h(i)}u_{h(i)}\\\\{\\bf \\sf E}[{\\bf \\sf E}[Y|X]{\\bf \\sf E}[X|Y]] &= \\frac{(1-q)(1+q+q^2)({\\bf \\sf E}[Z^2])^2}{({\\bf \\sf E}[Z])^2}\\nonumber \\\\&+\\frac{q^3b}{({\\bf \\sf E}[Z])^2}\\sum _{i=1}^{b}u_{i}u_{h(i)}\\\\{\\bf \\sf E}[Y{\\bf \\sf E}[Y|X]{\\bf \\sf E}[X|Y]] &= \\frac{(1+q)(1-q)^2({\\bf \\sf E}[Z^2])^3}{({\\bf \\sf E}[Z])^3}\\nonumber \\\\&+\\frac{(1-q^2)qb{\\bf \\sf E}[Z^2]}{({\\bf \\sf E}[Z])^3}\\sum _{i=1}^{b}u_{i}u_{h(i)}\\nonumber \\\\&+\\frac{(1-q)q^2b{\\bf \\sf E}[Z^2]}{({\\bf \\sf E}[Z])^3}\\sum _{i=1}^{b}u_{i}u_{i}\\nonumber \\\\&+\\frac{q^3b^2}{({\\bf \\sf E}[Z])^3}\\sum _{i=1}^{b}u_{i}u_{i}u_{h(i)}\\\\{\\bf \\sf E}[XY{\\bf \\sf E}[Y|X]{\\bf \\sf E}[X|Y]] &= \\frac{(1-q)^3({\\bf \\sf E}[Z^2])^4}{({\\bf \\sf E}[Z])^4}\\nonumber \\\\&+\\frac{3qb(1-q)^2({\\bf \\sf E}[Z^2])^2}{({\\bf \\sf E}[Z])^4}\\sum _{i=1}^{b}u_{i}u_{h(i)}\\nonumber \\\\&+\\frac{q^2b^2(1-q)}{({\\bf \\sf E}[Z])^4}(\\sum _{i=1}^{b}u_{i}u_{h(i)})^2\\nonumber \\\\&+\\frac{2q^2b^2(1-q){\\bf \\sf E}[Z^2]}{({\\bf \\sf E}[Z])^4}\\sum _{i=1}^{b}u_{i}u_{i}u_{h(i)}\\nonumber \\\\&+\\frac{q^3b^3}{({\\bf \\sf E}[Z])^4}\\sum _{i=1}^{b}u_{i}u_{i}u_{h(i)}u_{h(i)}$ Appendix B In Appendix B, we present the proof of Theorem REF .", "Recall that stubs corresponding to the degrees are partitioned evenly into $b$ blocks.", "Recall also that our construction algorithm arranges degrees in ascending order (descending order will also work).", "That is, $x\\le y\\ \\mbox{for all}\\ x\\in H_i\\ \\mbox{and}\\ y\\in H_j,$ where $i\\le j$ .", "Due to Assumption REF , (REF ) holds.", "We break down the proof of Theorem REF into several lemmas listed as follows.", "Lemma 3 Suppose that $h(i)=i$ .", "If sequences $\\lbrace x_i, i=1, 2, \\ldots , b\\rbrace $ and $\\lbrace y_i, i=1, 2, \\ldots , b\\rbrace $ are both non-decreasing or both non-increasing, then $b\\sum _{i=1}^{b}x_{i}y_{h(i)}-\\sum _{i=1}^{b}x_{i}\\sum _{j=1}^{b}y_{j}\\ge 0.$ Proof of Lemma REF .", "If sequences $\\lbrace x_i\\rbrace $ and $\\lbrace y_i\\rbrace $ are both non-increasing, then $x_{[i]}&= x_i \\\\y_{[i]}&= y_i,$ where $x_{[i]}$ denotes the $i$ -th largest element in sequence $\\lbrace x_i\\rbrace $ .", "On the other hand, if sequences $\\lbrace x_i\\rbrace $ and $\\lbrace y_i\\rbrace $ are both non-decreasing, then $x_{[i]}&= x_{b-i+1} \\\\y_{[i]}&= y_{b-i+1}.$ In either cases, we have $ \\sum _{i=1}^b x_i y_{h(i)}=\\sum _{i=1}^b x_i y_i = \\sum _{i=1}^b x_{[i]} y_{[i]}.$ Now consider circular shift permutation $\\sigma _j(\\cdot )$ with $\\sigma _j(i)=(i+j-1\\ \\mbox{mod}\\ b)+1$ for $j=1, 2, \\ldots , b$ .", "From symmetry, we have $\\sigma _j(i)=\\sigma _i(j)$ .", "Thus, $ \\sum _{i=1}^b \\sum _{j=1}^b x_i y_j=\\sum _{i=1}^b \\sum _{j=1}^b x_i y_{\\sigma _i(j)}=\\sum _{j=1}^b \\sum _{i=1}^b x_i y_{\\sigma _j(i)}.$ Let $v_i=y_{\\sigma _j(i)}.$ Thus, $ \\sum _{i=1}^b x_i y_{\\sigma _j(i)} = \\sum _{i=1}^b x_i v_i\\le \\sum _{i=1}^b x_{[i]} v_{[i]}.$ The last inequality in (REF ) is due to the well known Hardy, Littlewood and Pólya rearrangement inequality (see e.g., the book [21], pp.", "141).", "Clearly, sequence $\\lbrace v_i\\rbrace $ is a shifted version of $\\lbrace y_i\\rbrace $ .", "Thus, $v_{[i]}=y_{[i]}.$ Substituting the preceding equation and (REF ) into (REF ), we obtain $\\sum _{i=1}^b \\sum _{j=1}^b x_i y_j \\le \\sum _{j=1}^b \\sum _{i=1}^b x_{[i]} y_{[i]}=b \\sum _{i=1}^ b x_i y_i,$ where the last equality in the preceding is due to (REF ).", "Lemma 4 If stubs corresponding to degrees are arranged in ascending order evenly into blocks, then $\\lbrace u_i: 1\\le i\\le b\\rbrace $ $\\lbrace t_i: 1\\le i\\le b\\rbrace $ $\\lbrace t_i-u_i: 1\\le i\\le b\\rbrace $ $\\lbrace u_i^2: 1\\le i\\le b\\rbrace $ $\\lbrace u_i(u_i-c): 1\\le i\\le b\\rbrace $ for any constant $c$ are all non-deceasing sequences.", "Proof of Lemma REF .", "Let $i$ and $j$ be two blocks, where $i<j$ .", "From Assumption REF , $ \\sum _{x\\in H_i} x p_x = \\sum _{x\\in H_j} x p_x.$ Let $x_i^{\\mbox{\\small max}}$ denote the maximum degree in block $i$ .", "Then, $u_i &= \\sum _{x\\in H_i} x^2 p_x \\\\&\\le x_i^{\\mbox{\\small max}} \\sum _{x\\in H_i} x p_x \\\\&=x_i^{\\mbox{\\small max}} \\sum _{x\\in H_j} x p_x \\\\&\\le \\sum _{x\\in H_j} x^2 p_x \\\\&=u_j.$ Other sequences can be proved similarly.", "Lemma 5 If $h(i)=i$ , then $W_i \\ge 0$ for $i=1, 2, 3, 4, 5$ .", "Proof of Lemma REF .", "Lee et al.", "[6] proved that $W_1\\ge 0$ if $h(i)=i$ .", "We now prove that $W_2 \\ge 0$ .", "From part 2 of Lemma REF , it follows that sequence $\\lbrace t_i: 1\\le i\\le b\\rbrace $ is non-decreasing.", "It follows from Lemma REF that $W_2\\ge 0$ .", "Proof for $W_i\\ge 0$ , $3\\le i\\le 5$ is similar.", "Lemma 6 If $h(i)=i$ , then $W_2 \\ge W_1.$ Proof of Lemma REF .", "One can express $W_2 - W_1 &= b\\sum _{i=1}^b u_i(t_{h(i)}-u_{h(i)})-\\sum _{i=1}^b\\sum _{j=1}^b u_i(t_j-u_j)\\\\&=b\\sum _{i=1}^b u_i(t_{i}-u_{i})-\\sum _{i=1}^b\\sum _{j=1}^b u_i(t_j-u_j).$ From part 3 of Lemma REF , it follows that sequence $\\lbrace t_i-u_i: 1\\le i\\le b\\rbrace $ is non-decreasing.", "The claim of the lemma follows from Lemma REF .", "Lemma 7 If $h(i)=i$ , then $&{\\bf \\sf E}[Z]{\\bf \\sf E}[Z^3]-({\\bf \\sf E}[Z^{2}])^{2} \\ge W_{1} \\\\&W_4-\\frac{{\\bf \\sf E}[Z]}{b} W_3 \\ge 0 $ Proof of Lemma REF .", "We first prove (REF ).", "Note that $&{\\bf \\sf E}[Z^3]{\\bf \\sf E}[Z]-({\\bf \\sf E}[Z^2])^2-W_1 \\\\&= bz\\sum _{i=1}^b t_i -(\\sum _{i=1}^b u_i)^2-(b\\sum _{i=1}^b u_i^2-\\sum _{i=1}^b u_i \\sum _{j=1}^b u_j) \\\\&= b\\sum _{i=1}^b (t_i z-u_i^2),$ We claim that $t_i z \\ge u_i^2$ for all $i$ .", "Inequality (REF ) follows directly from this claim.", "We now prove the claim.", "From the definition of $u_i$ and $t_i$ in (REF ) and (REF ) respectively, we have $&t_i z-u_i^2 \\\\&=\\sum _{x\\in H_i}\\sum _{y\\in H_i} x^3 y p_x p_y - \\sum _{x\\in H_i}\\sum _{y\\in H_i} x^2 y^2 p_x p_y \\\\&=\\sum _{x\\in H_i}\\sum _{y\\in H_i} x^2 y (x-y)p_x p_y \\\\&=\\sum _{x, y\\in H_i, x> y} x^2 y (x-y)p_x p_y + \\sum _{x, y\\in H_i, x < y} x^2 y (x-y)p_x p_y \\\\&=\\sum _{x, y\\in H_i, x> y} x^2 y (x-y)p_x p_y - \\sum _{x, y\\in H_i, x < y} x^2 y (y-x)p_x p_y\\\\&=\\sum _{x, y\\in H_i, x> y} x^2 y (x-y)p_x p_y - \\sum _{x, y\\in H_i, y < x} y^2 x (x-y)p_y p_x,$ where the last equality follows by exchanging symbols $x$ and $y$ in the second term of the last equation.", "The preceding difference equals $\\sum _{x, y\\in H_i, x> y} x y (x-y)^2 p_x p_y,$ which is non-negative.", "Next we prove ().", "Note that $\\frac{{\\bf \\sf E}[Z]}{b}=z \\le u_{i}$ for $i=1,2,...,b$ .", "Substituting () and () into (), we obtain $ &W_{4}-\\frac{{\\bf \\sf E}[Z]}{b}W_{3}\\nonumber \\\\&=\\left(b\\sum _{i=1}^{b}u_{i}u_{i}u_{i}-\\sum _{i=1}^{b}u_{i}u_{i}\\sum _{j=1}^{b}u_{j}\\right)\\nonumber \\\\&\\quad -\\left(b\\sum _{i=1}^{b}u_{i}u_{i}z-\\sum _{i=1}^{b}u_{i}z\\sum _{j=1}^{b}u_{j}\\right)\\nonumber \\\\& =b\\sum _{i=1}^{b}u_{i}u_{i}(u_{i}-z)-\\sum _{i=1}^{b}u_{i}\\sum _{j=1}^{b}u_{j}(u_{j}-z).\\nonumber $ From part 3 of Lemma REF , it follows that sequence $\\lbrace u_i(u_i-z): 1\\le i\\le b\\rbrace $ is non-decreasing.", "Inequality () follows from Lemma REF .", "Now we prove Theorem REF .", "Proof of Theorem REF .", "Through extensive algebraic manipulation, we express the sum of the first three terms and the sum of the last three terms in (REF ) in a different manner, i.e.", "$&\\alpha _{0} + \\beta _{1}W_{1}+ \\beta _{2}W_{2}= D_{1}+D_{2}+D_{3}+D_{4}+D_{5} \\\\&\\beta _{3}W_{3}+ \\beta _{4}W_{4}+ \\beta _{5}W_{5} = D_{6} + D_{7},$ where $&D_{1} = \\frac{q}{({\\bf \\sf E}[Z])^2}W_{1}={\\bf \\sf Cov}(X,Y) \\\\&D_{2}=2a^{2}\\frac{{\\bf \\sf E}[Z^2]}{({\\bf \\sf E}[Z])^{3}}\\left({\\bf \\sf E}[Z]{\\bf \\sf E}[Z^3]-({\\bf \\sf E}[Z^{2}])^{2}\\right)\\nonumber \\\\&\\quad - 2a^{2}q\\frac{{\\bf \\sf E}[Z^2]}{({\\bf \\sf E}[Z])^3} W_{1}\\nonumber \\\\&\\quad +2a\\frac{q(1-q){\\bf \\sf E}[Z^2]}{({\\bf \\sf E}[Z])^3} W_{1}\\nonumber \\\\&\\quad -2a^{2}\\frac{q(1-q){\\bf \\sf E}[Z^{2}]}{({\\bf \\sf E}[Z])^{3}}W_{1}\\\\&D_{3}=2a(1-a)\\frac{1}{({\\bf \\sf E}[Z])^{2}}\\left({\\bf \\sf E}[Z]{\\bf \\sf E}[Z^3]-({\\bf \\sf E}[Z^{2}])^{2}\\right) \\nonumber \\\\&\\quad +2a^{2}\\frac{q}{({\\bf \\sf E}[Z])^{2}}W_{2} - 2a\\frac{q}{({\\bf \\sf E}[Z])^2} W_{1} \\\\&D_{4}=a^{2}\\frac{q(1-q)^{2}({\\bf \\sf E}[Z^{2}])^{2}}{({\\bf \\sf E}[Z])^{4}}W_{1} \\\\&D_{5}=a^{2}\\frac{q^{3}}{({\\bf \\sf E}[Z])^{2}}W_{1} \\\\&D_6=2a \\frac{q^{2}b}{({\\bf \\sf E}[Z])^{3}}\\left(1-a+\\frac{(1-q)({\\bf \\sf E}[Z^{2}])^{2}}{{\\bf \\sf E}[Z]}\\right) \\nonumber \\\\&\\quad \\times \\left(W_4-\\frac{{\\bf \\sf E}[Z]}{b}W_3\\right) \\\\& D_{7} = a^{2}\\frac{q^{3}b^{2}}{({\\bf \\sf E}[Z])^{4}}\\left(W_{5} -2\\frac{{\\bf \\sf E}[Z]}{b}W_{4}\\right).$ We claim that $D_i\\ge 0, \\quad i=1, 2, 3, \\ldots , 7.$ Since $D_1={\\bf \\sf Cov}(X,Y)$ , the claim implies (REF ).", "In the rest of the proof, we focus on the proof of the claim.", "From Lemma REF , $W_1\\ge 0$ .", "It follows from () and () that $D_4\\ge 0$ and $D_5\\ge 0$ .", "For $D_2$ , we place ${\\bf \\sf E}[Z]{\\bf \\sf E}[Z^3]-({\\bf \\sf E}[Z^{2}])^{2}$ with $W_1$ and obtain a lower bound for $D_2$ , i.e.", "$D_2 \\ge 2a(1-q)(a+(1-a)q)\\frac{{\\bf \\sf E}[Z^2]}{({\\bf \\sf E}[Z])^3} W_{1},$ which is greater than or equal to zero, because $W_1\\ge 0$ .", "Now we consider $D_3$ .", "We replace ${\\bf \\sf E}[Z]{\\bf \\sf E}[Z^3]-({\\bf \\sf E}[Z^{2}])^{2}$ and $W_2$ with $W_1$ and obtain a lower bound for $D_3$ , i.e.", "$D_{3} \\ge 2a(1-a)(1-q)\\frac{1}{({\\bf \\sf E}[Z])^2} W_{1},$ which is greater than or equal to zero, again because $W_1\\ge 0$ .", "Now we analyze $D_6$ .", "From () it is clear that $D_6\\ge 0$ , if and only if $W_4-\\frac{{\\bf \\sf E}[Z]}{b} W_3 \\ge 0.$ From () in Lemma REF , $W_4-\\frac{{\\bf \\sf E}[Z]}{b} W_3 \\ge 0$ holds.", "Then, we have $D_6\\ge 0$ .", "Finally we analyze $D_{7}$ .", "From (), $D_7\\ge 0$ , if and only if $W_{5}-2\\frac{{\\bf \\sf E}[Z]}{b}W_{4}\\ge 0.$ Replacing $W_4$ and $W_5$ with their definitions in () and (), we have $&W_{5}-2\\frac{{\\bf \\sf E}[Z]}{b}W_{4}\\nonumber \\\\&=\\left(b\\sum _{i=1}^{b}u_{i}u_{i}u_{i}u_{i}-\\sum _{i=1}^{b}u_{i}u_{i}\\sum _{j=1}^{b}u_{j}u_{j}\\right)\\nonumber \\\\&\\qquad -2\\left(b\\sum _{i=1}^{b}u_{i}u_{i}u_{i}z-\\sum _{i=1}^{b}u_{i}u_{i}\\sum _{j=1}^{b}u_{j}z\\right)\\nonumber \\\\&=b\\sum _{i=1}^{b}u_{i}u_{i}u_{i}(u_{i}-2z)-\\sum _{i=1}^{b}u_{i}u_{i}\\sum _{j=1}^{b}u_{j}(u_{j}-2z).$ From case (4) and case (5) of Lemma REF and Lemma REF , it follows that the right side of (REF ) is non-negative.", "The proof of Theorem REF is completed." ] ]
2105.11688
[ [ "Exploring Autoencoder-based Error-bounded Compression for Scientific\n Data" ], [ "Abstract Error-bounded lossy compression is becoming an indispensable technique for the success of today's scientific projects with vast volumes of data produced during the simulations or instrument data acquisitions.", "Not only can it significantly reduce data size, but it also can control the compression errors based on user-specified error bounds.", "Autoencoder (AE) models have been widely used in image compression, but few AE-based compression approaches support error-bounding features, which are highly required by scientific applications.", "To address this issue, we explore using convolutional autoencoders to improve error-bounded lossy compression for scientific data, with the following three key contributions.", "(1) We provide an in-depth investigation of the characteristics of various autoencoder models and develop an error-bounded autoencoder-based framework in terms of the SZ model.", "(2) We optimize the compression quality for main stages in our designed AE-based error-bounded compression framework, fine-tuning the block sizes and latent sizes and also optimizing the compression efficiency of latent vectors.", "(3) We evaluate our proposed solution using five real-world scientific datasets and comparing them with six other related works.", "Experiments show that our solution exhibits a very competitive compression quality from among all the compressors in our tests.", "In absolute terms, it can obtain a much better compression quality (100% ~ 800% improvement in compression ratio with the same data distortion) compared with SZ2.1 and ZFP in cases with a high compression ratio." ], [ "Introduction", "Today's scientific applications are producing extremely large amounts of data during simulation or instrument data acquisition.", "Advanced instruments such as the Linac Coherent Light Source (LCLS) [1] and Advanced Photon Source [2], for example, may produce vast amounts of data with a very high data acquisition rate (250 GB/s [3]).", "Consequently, reducing the data volumes with user-tolerable data distortion is critical to the efficient data storage and transfer.", "Error-bounded lossy compression is arguably the most efficient way to significantly reduce the data volumes for scientific applications with big data issues.", "Unlike lossless compressors [4], [5], [6], [7] that suffer from very low compression ratios (generally $\\sim $ 2:1) on floating-point datasets, error-bounded lossy compressors can obtain fairly high compression ratios (10+ or even several hundreds [8], [9], [3]).", "Moreover, error-bounded lossy compressors are able to keep a high fidelity of the reconstructed data for the user's post hoc analysis based on the user's required bounds on data distortion, as verified by many recent studies [10], [11], [12].", "Error-bounded lossy compressors can be split into two models: prediction-based and transform-based models.", "Prediction-based compressors (such as SZ [8], [9]) may suffer from low reconstructed data quality at high compression ratios because they have to predict each data point using the reconstructed data values nearby instead of original data, in order to guarantee the bounded errors during the decompression.", "To obtain a high compression ratio, therefore, one has to set the error bound relatively large; and as a result, the data prediction accuracy can be degraded significantly because of large errors in the reconstructed data, leading to limited compression ratios in turn.", "For transform-based compressors (such as ZFP [13]), large compression ratio means that a very limited number of coefficients or bit-planes can be preserved for reconstructing the data and thus will considerably lower the data reconstruction quality.", "As a classic type of deep learning model, the Autoencoder (AE) has been gaining more and more attention.", "Such a deep neural network (DNN) architecture is composed of both an encoder (encoding the input data) and a decoder (decoding the encoded data) and is trained to minimize the error between the reconstructed data and the initial data.", "In general, because the trained encoder and decoder can be used separately, AE can be used to learn efficient data representation (or coding), typically for dimensionality reduction.", "The corresponding DNN will be trained to reconstruct the main patterns in the dataset effectively based on the reduced information generated from the original data.", "Recently, several variations of the AE have been developed with different model frameworks and training paradigms for improving the effectiveness of data reconstruction and for handling more tasks such as data generation.", "Nevertheless, although AE has been widely used in the image compression domain, few studies explored the possibility of leveraging it for error-bounded compression models for scientific datasets.", "In this paper we explore the possibility of leveraging the AE model to improve the error-bounded lossy compression significantly.", "Such a study faces several challenges.", "First, many types of autoencoders exist, each with different architectures or training methods, so that determining the most effective AE model is challenging.", "Second, applying AE in the error-bounded model with a proper configuration setting is nontrivial.", "Third, latent vectors from AE need to be stored in the compressed data, so minimizing the latent vector overhead while maintaining a high reconstruction quality is challenging and critical to getting a good rate distortion in the high-compression cases.", "In this work we propose a novel error-bounded lossy compressor, AE-SZ, which combines the classic prediction-based error-bounded compression framework SZ [8], [9] and the Sliced-Wasserstein Autoencoder (SWAE) model with convolutional neural network implementations.", "The key contributions of the paper are summarized as follows: Our autoencoder-based error-bounded compression framework is designed on the basis of a blockwise model, which can adapt to diverse data changes in a dataset well.", "To the best of our knowledge, AE-SZ is the first AE-based error-bounded lossy compressor that exhibits a better rate distortion than the three state-of-the-art models SZauto [14], SZ [15], and ZFP [13].", "We investigate various autoencoder models and identify the most effective one for the error-bounded lossy compression model and also carefully optimize the related configurations, such as block sizes and strategies of compressing latent vectors.", "We evaluate the proposed AE-SZ by using the scientific datasets generated by five different real-world high-performance computing (HPC) applications across different domains.", "We identify the effectiveness of AE-SZ by comparing it with two other AE-based lossy compression methods and four other state-of-the-art error-bounded lossy compressors.", "Our experiments show that AE-SZ is the best compression method in the category of AE-based compressors.", "AE-SZ also exhibits competitive rate distortion compared with existing state-of-the-art error-bounded lossy compressors.", "Specifically, when the compression ratio is greater than 100, AE-SZ can get 100%$\\sim $ 800% higher compression ratios than can SZ2.1 and ZFP, with the same peak signal-to-noise ratio (PSNR).", "The rest of this paper is organized as follows.", "In Section  we discuss related work.", "In Section  we formulate the research problem.", "In Section  we present the overall design of AE-SZ as well as the detailed optimization strategies.", "In Section  we evaluate our solution using multiple real-world scientific simulation datasets.", "In Section we conclude the paper with a vision of the future work." ], [ "Related Work", "Error-bounded lossy compression techniques have been studied for years, since lossless compression suffer from very low compression ratios (generally 2:1 [16]).", "Many error-bounded lossy compressors have been developed for compressing scientific datasets [13], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30].", "These can be categorized into prediction-based models (e.g., SZ [8], [9] and FPZIP [17]) and transform-based models (e.g., ZFP [13]).", "Among these compressors, SZ2.1 [8], [9], [15] and ZFP [13] are the two main state-of-the-art works with wide public usage.", "Several works have also been developed based on SZ2.1.", "For example, SZauto [14] merges second-order regression/Lorenzo and automatic parameter tuning in the SZ framework; and SZinterp [31] applies dynamic spline interpolation into data prediction and achieves significant improvement in the prediction accuracy.", "For assessing the lossy compressors, [32] provides an effective framework.", "With the fast growth of deep learning, a recent research trend is leveraging deep learning models such as autoencoders on data compression tasks.", "Successful works for AE-based image compression include [33], [34], [35], [36], [37], [38], [39], which design different convolutional networks for image feature extracting and reconstruction and combine them with quantization and encoding algorithms.", "Unlike scientific lossy compressors, however, those autoencoder-based image compression models are not designed to compress floating-point data and do not provide a strict error-controlling scheme based on scientific user’s requirements on post hoc analysis.", "Recently, a few works have used an autoencoder to compress scientific data.", "Glaws et al.", "[40] presented a convolutional autoencoder for lossy compression of turbulence flow simulation data (with a fixed compression ratio of 64).", "The authors proposed an AE model including 12 residual blocks (i.e., skip connections [41]) to extract features and 3 compression layers to reduce features in both the encoder and decoder.", "However, different from our work that can provide a strict control of local error (e.g., relative/absolute error) and can be adapted to any scientific datasets, this AE-based scientific compressor is not error-bounded and is designed only for turbulence data.", "The fixed compression ratio is also its limitation.", "Choi et al.", "[42] proposed another specific variational autoencoder approach for physics plasma simulation data compression.", "The proposed AE model focuses on minimizing loss of information under physics constraints (e.g., mass, energy, moment) by adopting physics-informed optimization functions and refinement layers.", "However, unlike this work using quantized latent vectors (integer based), our solution applies lossy compression to floating-point latent vectors, which provides a highly flexible tradeoff between compression and accuracy.", "In [42] the authors presented a brief version that lacks some important details for replication of their experiments, so we did not compare their performances in our paper.", "Liu et al.", "[43] developed an autoencoder method for scientific data compression.", "Their proposed AE model includes three fully connected layers for both encoder and decoder (i.e., total of seven layers including the latent vector), and the size of each layer is reduced/increased by $8\\times $ compared with its previous layer, thus leading to an overall compression ratio of $512\\times $ .", "The limitations of this work are that the autoencoder in their model processes only 1-D data, and the experiments are also based mainly on small-scale 1D scientific data.", "Our AE-SZ framework overcomes the limitations by designing and training convolutional autoencoders that are aware of dimensional information and are well adapted to large-scale data with relatively high prediction speed and accuracy." ], [ "Background and Problem Formulation", "In this section we describe the research background and formulate the research problem." ], [ "Research Background – Autoencoder", "We describe autoencoder briefly as follows.", "A stereotype autoencoder model is composed of an encoder network and decoder network.", "The former encodes the input data to a latent vector in reduced size, and the latter decodes the latent vector to an approximate reconstruction of data.", "The latent vector stands as a compressed representation of the input data, and different autoencoders have different technical details for computation of the latent vector.", "The nature of the autoencoders grants them the potential for being leveraged for data reduction, in that the reconstructed data based on the latent vector can approximate the original data to a certain extent.", "Figure REF shows visualization of the reconstructed data versus the original data with the autoencoder [40] (reduction ratio = 64$\\times $ ) on a turbulence dataset.", "Figure: Reconstructed data of AE (64×\\times ) on a turbulence dataset(original value range: [--3.06 , 2.64], max pointwise absolute error = 1.2)As a historical and well-researched neural network model, multiple variations have been proposed for the AE model.", "In what follows, we mainly present SWAE, which is to be used as the fundamental AE model in our designed AE-SZ compressor.", "SWAE [44] is a derivation of Wasserstein Autoencoder (WAE) [45], and regularizes the autoencoder loss with the sliced-Wasserstein distance between the distribution of the encoded training samples and a predefined samplable distribution.", "From [44], marking $\\phi $ as the encoder and $\\psi $ as the decoder, given a latent dimension $d$ , a regularization coefficient $\\lambda $ , a number of random projections $L$ , and a predefined latent distribution $q_Z$ , SWAE optimizes the following loss function: $\\begin{array}{l}\\hspace{-8.53581pt}\\mathcal {L}(\\phi , \\psi )=\\frac{1}{M} \\sum _{m=1}^{M} c\\left(x_{m}, \\psi \\left(\\phi \\left(x_{m}\\right)\\right)\\right)\\\\\\hspace{31.29802pt}+\\frac{\\lambda }{L M} \\sum _{l=1}^{L} \\sum _{m=1}^{M} c\\left(\\theta _{l} \\cdot \\tilde{z}_{i[m]}, \\theta _{l} \\cdot \\phi \\left(x_{j[m]}\\right)\\right) \\hspace{-8.53581pt},\\end{array}$ in which $\\left\\lbrace x_{1}, \\ldots , x_{M}\\right\\rbrace $ is sampled from training set (i.e.", "$p_X$ ), $\\left\\lbrace \\tilde{z}_{1}, \\ldots , \\tilde{z}_{M}\\right\\rbrace $ is sampled from $q_Z$ , $\\left\\lbrace \\theta _{1}, \\ldots , \\theta _{L}\\right\\rbrace $ is sampled from $\\mathrm {S}^{d-1}$ (K-dimensional unit sphere), $i[m]$ and $j[m]$ are the indices of sorted $\\theta _{l}\\cdot \\tilde{z}_{m}\\mathrm {~s}$ and $\\theta _{l}\\cdot \\phi \\left(x_{m}\\right)$ , respectively, and $c(x,y)=||x-y||_2^2 .$ Kolouri et al.", "[44] proved that optimizing this loss function is equal to optimizing $\\operatorname{argmin}_{\\phi , \\psi } W_{c}\\left(p_{X}, p_{Y}\\right)+\\lambda S W_{c}\\left(p_{Z}, q_{Z}\\right) ,$ in which $W_{c}\\left(p_{X}, p_{Y}\\right)$ is the Wasserstein distance from $p_{X}$ (distribution of input data $X$ ) to $p_{Y}$ (distribution of decoded data $Y$ ) and $S W_{c}\\left(p_{Z}, q_{Z}\\right)$ is the sliced-Wasserstein distance from $p_{Z}$ (distribution of encoded latent $Z$ ) to $q_{Z}$ .", "Kolouri et al.", "[44] also show the efficiency of computing Eq.", "REF .", "The autoencoder itself cannot bound the compression errors, which is a significant gap to scientific user's demand for error controls.", "As shown in Figure REF , the maximum point-wise compression error is up to 1.2, which is about 20% of the original data value range ($-$ 3.06, 2.64].", "In comparison, scientists often need to control the point-wise errors to a much smaller bound such as 1% of the original value range [9], [15].", "In this work we aim to develop a deep learning based error bounded lossy compressor.", "Specifically, for some scientific applications, we train neural networks based on a certain amount of training data, and then apply the trained networks to compress the testing data generated by the same applications.", "We separate the training data and testing data because we expect that the pre-trained networks can be used to compress new data for the same applications, such that the training time and model size can be excluded from the compression time and size.", "In our experiments, the training and test data are from different time steps or the simulation running with different configuration settings in the same application." ], [ "Math Formulations for Error-bounded Lossy Data Compression", "The compression ratio (denoted by $\\rho $ ) is defined as $\\frac{|D|}{|D^{\\prime }|}$ , where $|D|$ and $|D^{\\prime }|$ denote the original data size and compressed data size (both in bytes), respectively.", "Error-bounded lossy compression has one important constraint, namely, that the reconstructed data respect a user-specified error bound (denoted by $e$ ) strictly.", "Under this constraint, the rate distortion often serves as a criterion to assess the compression quality, which involves two critical terms: bit rate and data distortion.", "The bit rate is defined as the average number of bits used to represent one data point after the compression; hence, the lower the bit rate, the higher the compression ratio.", "In the lossy compression community rate distortion is often evaluated by the PSNR, defined as shown below: $P\\hspace{-0.85358pt}S\\hspace{-0.85358pt}N\\hspace{-0.85358pt}R = 20\\log _{10}{(vrange(D)} \\hspace{-0.85358pt}-\\hspace{-0.85358pt} 10\\hspace{-0.85358pt}\\log _{10}{(mse(D,\\hspace{-0.85358pt}D^{\\prime }))} \\hspace{-2.27621pt}\\vspace{-2.84526pt},$ where $D^{\\prime }$ is the reconstructed dataset after decompression (i.e., decompressed dataset), vrange($D$ ) represents the value range of the original dataset $D$ (i.e., the difference between its highest value and lowest value), and mse refers to mean squared error.", "The higher the PSNR value is, the smaller the mean squared error, which means higher precision of the decompressed data.", "Our objective is to obtain higher compression ratios than other related works obtain (including other deep-learning-based compressor and traditional error-bounded lossy compressors) with the same PSNR value, while also strictly respecting the user's error bound, especially aiming at optimizing the use cases with high compression ratio.", "We can write the research problem formulations as follows: $\\begin{array}{l}maximize\\hspace{5.69054pt}\\rho \\\\s.t.", "\\hspace{11.38109pt}PSNR(D,D^{\\prime }) = \\lambda \\\\\\hspace{22.76219pt}\\hspace{2.84526pt}|d_i - d_i^{\\prime }| \\le e\\end{array},$ where $\\lambda $ is a particular PSNR value representing a specific data distortion level and $d_i$ and $d_i^{\\prime }$ refer to any data point in the original dataset $D$ and decompressed dataset $D^{\\prime }$ , respectively." ], [ "AE-SZ: Autoencoder-based Error-bounded Lossy Compression Framework", "In this section we present the design overview of AE-SZ and describe the detailed optimization strategies for AE-SZ." ], [ "Design Overview of AE-SZ", "We present the overall framework of our designed autoencoder-based error-bounded lossy compression framework AE-SZ as shown in Figure REF .", "The overall compression involves two stages: offline training and online compression.", "During the offline training, we split the training data snapshots into multiple small fixed-size blocks (such as 32$\\times $ 32 for a 2D data field or 8$\\times $ 8$\\times $ 8 for a 3D data field) and train the network with numerous small blocks.", "The advantage of such a design is twofold: (1) the AE model works more efficiently on the divided data blocks, which can catch fine-grained data features; (2) such a data-splitting design creates numerous training samples (i.e., data blocks), so that the AE model is tractable.", "During the online compression, AE-SZ executes four steps as shown in Figure REF : (1) splitting the input data to be compressed into many small blocks (with the same block size as during the training stage), (2) prediction, (3) linear-scale quantization, and (4) entropy/dictionary encoding.", "Specifically, in each block, the data are predicted by a predictor (either autoencoder or Lorenzo), and the prediction errors will be quantized based on the user's error bound, followed by Huffman encoding and Zstd [5].", "The Lorenzo predictor is similar to the one used in SZ2.1.", "Specifically, under the Lorenzo predictor, the $i$ th data point $d_i$ is predicted by three nearby data values in 2D data ($d_{i,j}$$\\leftarrow $$d_{i,j-1}$ + $d_{i-1,j}$ $-$ $d_{i-1,j-1}$ ) or by 7 nearby values in 3D data ($d_{i,j,k}$$\\leftarrow $$d_{i-1,j,k}$ + $d_{i,j-1,k}$ + $d_{i,j,k-1}$ $-$ $d_{i-1,j-1,k}$ $-$ $d_{i-1,j,k-1}$ $-$ $d_{i,j-1,k-1}$ + $d_{i-1,j-1,k-1}$ ).", "We refer the readers to read our prior work [9] for more details.", "We note that the only difference between the Lorenzo predictor in AE-SZ and [9] is that AE-SZ makes the selection between classic Lorenzo and mean-Lorenzo separately on each block instead of a global switching mechanism as in [9].", "That is, if a data block can be better predicted by its mean value than by classic Lorenzo, AE-SZ will use the mean value for prediction, and all the involved mean values will be saved losslessly.", "We find that this mean-Lorenzo predictor can make up for the deficiencies of classic Lorenzo and AE under extremely high error-bounds (such as $\\sim $ 1E-1).", "The compressed data generated by AE-SZ consists of three parts: a header containing metadata (with trivial space cost), lossy compressed latent vectors from autoencoders, and quantization bins (losslessly encoded).", "The main difference between AE-SZ and SZ2.1[15] is that SZ2.1 includes two data predictors for compression: linear regression [15] and Lorenzo [46], whereas AE-SZ replaces the linear regression predictor by a pre-trained autoencoder.", "For scientific datasets in which the data changes could be diverse, autoencoders can overcome the limitation of linear regression, which can only approximate the data using flat hyperplanes.", "Figure: Design overview of AE-SZ (highlighted parts include our specifically optimized design compared with SZ compressor)The pseudo code of the AE-SZ compression procedure is presented in Algorithm REF .", "As mentioned before, AE-SZ compresses the input data block by block, and the compression of each block follows the same routine.", "Thus, in the following, we describe the compression procedure mainly on a single data block (i.e., line 2$\\sim $ 16), without loss of generality.", "For any block, AE-SZ first generates predicted data based on two predictors (Lorenzo and autoencoder) for this block respectively (line 3$\\sim $ 8).", "Then, the predictor with lower element-wise $l$ 1-loss is selected out for this block (line 9$\\sim $ 13).", "The reason is that the smaller the prediction errors are, the more uneven the distribution of quantization bins in general, and hence the higher compression ratio of quantization bins.", "Then, AE-SZ uses linear-scale quantization to quantize the prediction errors based on the user-specified error bound $e$ (line 14).", "Similar to SZ2.1 [9], [15], we need to set a maximum number of quantization bins (65,536 by default) for the linear quantization, in order to keep high performance.", "The total quantization range may not cover all predicted values as the prediction errors may be large.", "The corresponding data points, called unpredictable data, will be saved separately (denoted as $U$ in line 14).", "For more details about linear-scale quantization, we refer readers to read our paper [9].", "AE-SZ Compression Algorithm Input: Input data $D$ , block size $S$ , error-bound $e$ , latent error-bound $e_l$ .", "Output: Compressed data $D^{\\prime }$ ={$\\hat{z}$ , $\\hat{Z}$ , $U$ }.", "[1] Split $D$ into blocks of Size S(1D), S$\\times $ S(2D), or S$\\times $ S$\\times $ S(3D).", "(each block $B$ in the data) $z$ $\\leftarrow $ $Eec(B)$ .", "Encode $B$ with the encoder network $Eec$ .", "$z^{\\prime }$ $\\leftarrow $ $f$ ($z$ ,$e_l$ ).", "Get decompressed latent vector $z^{\\prime }$ based on $e_l$ $B^{\\prime }=Dec(z^{\\prime })$ .", "Get Decoded $B^{\\prime }$ using decoder network $Dec$ $loss_1=||B-B^{\\prime }||_1$ .", "Compute $l$ 1 loss of $B^{\\prime }$ vs. $B$ .", "$B^{\\prime \\prime }=Lorenzo(B)$ .", "Predict $B$ with Lorenzo.", "$loss_2=||B-B^{\\prime }||_1$ .", "Compute $l$ 1 loss of Lorenzo predictor.", "$loss_2\\le loss_1$ $B_p=B^{\\prime \\prime }$ .", "Select Lorenzo-predicted values $B_p=B^{\\prime }$ .", "Select Autoencoder-predicted values $Q,U=Quantize(B,B_p,e)$ .", "linear-scale quantization with $e$ , to get quantization codes $Q$ and unpredictable data $U$ .", "Compress all saved coefficients from AE and Lorenzo.", "$H$ $\\leftarrow $ Huffman_Encode($Q$ ).", "Huffman encoding $\\hat{Z}$ $\\leftarrow $ Zstd($H$ ).", "Zstd compression In the following subsections, we present several critical optimization strategies for AE-SZ, which are developed in terms of fundamental takeaways we summarized from our in-depth analysis or comprehensive experimental evaluation." ], [ "Design Detail: AE network structure in AE-SZ", "The structure of our designed autoencoder network used in AE-SZ is illustrated in Figure REF .", "Like most of the autoencoders, it consists of an encoder network to generate the latent vectors as the compressed representation of input original data and a decoder network to reconstruct the data from latent vectors.", "The input of the network are (batchs of) data blocks, which will be linearly normalized to the range of [-1, 1] based on the global maximum and minimum of data before being put in the network, and the output of the network needs to be denormalized to generate the final prediction values.", "The encoder and decoder networks are both formed with several convolutional/deconvolutional blocks and a fully connected layer for resizing latents, and their structures are mirror-symmetric except for an additional final output layer-set in the decoder network.", "Figure: Our Designed Blockwise Convolutional AE network for CompressionAs shown in Figure REF , the convolutional blocks in encoder network are composed of the layer sequence of Convolution(Stride 1)-Convolution(Stride 2)-GDN, and the ones in decoder network are composed of the layer sequence of Deconvolution(Stride 1)-Deconvolution(Stride 2)-iGDN.", "The size of each (de)convolutional kernel in the network is 3$\\times $ 3 (2D case) or 3$\\times $ 3$\\times $ 3 (3D case).", "We take some experiments for the block design from the image compressive autoencoder in [34] and [35].", "The reason we apply stride-1 convolutions before stride-2 convolutions is to increase the number of parameters of the network without fast reducing the size of feature maps.", "As reported in [33] and [34], consecutively stacking stride-2 convolutions will harm the performance of the network.", "In our AE-SZ autoencoder, we do not use traditional activation functions, but use Generalized Divisive Normalization (GDN) [47] as the activation function.", "In fact, according to Balle et al.", "'s work [48], GDN can provide better image reconstruction quality with trivial additional parameters compared with traditional activation and normalization functions such has Relu, LeakyRelu [49] and Batch Normalization [50].", "Several existing lossy image compression autoencoder models [37], [38], [39], [35], [51], [52] have leveraged GDN and proved its advantages.", "Our primary experiments also confirm that GDN outperforms other tested activation functions on scientific data lossy compression task.", "More details about GDN can be found in [47] and [48].", "Following the common configurations, we apply original GDN in convolutional blocks and apply its reverse iGDN in deconvolutional blocks.", "Figure: (a) The Convolutional blocks used in AE-SZ encoder network.", "(b) The Deconvolutional blocks used in AE-SZ decoder network.To adapt to different datasets, the number of Convolutional blocks and the number of channels in each block may vary, but the overall structure remains the same.", "For example, The main difference between autoencoders used for 2D/3D datasets is just the dimension (2D or 3D) of convolutional/deconvolutional operation in the network layers (see Figure REF ).", "The network model in AE-SZ is saved separately against the compressed data because it can be reused by different time steps or other simulations with different parameter settings, which is verified in our experiments (see Section )." ], [ "Design Details: Choosing the Autoencoder Type", "Takeaway 1: Sliced-Wasserstein Autoencoder is particularly suitable for data prediction in scientific data compression compared with other AE models.", "A key point in designing AE-SZ is that we need to select the most appropriate model for scientific data prediction from multiple variations of autoencoder models.", "In AE-SZ, we select sliced-Wasserstein autoencoder (SWAE) for the AE compressor and predictor.", "The advantages of SWAE in data compression are as follows: Compared with the other tested autoencoders, SWAE shows less reconstruction loss on scientific data.", "Different from traditional variational autoencoders (VAEs), the encoding and decoding computation in SWAE are both determinant.", "VAEs such as [53], [54], [55], [56], [57] actually compute means and variances with input data and sample latent vectors with the means and variances from the prior distribution.", "Therefore, in multiple runs with the same input, the latent vector as the output of encoder in a VAE will differ, which makes the VAE being unstable for data compression tasks.", "Compared with Wasserstein autoencoders (WAE), the computation of training loss in SWAE is more numerically efficient.", "Similar to SWAE, WAE computes Wasserstein distances for training losses, and its computation cost is higher than the computation of sliced-Wasserstein distances.", "With both $n$ samples from the training set and prior distribution, the computational cost of the Wasserstein distance is $O(n^2)$ whereas the computational cost of the sliced-Wasserstein distance is $O(n\\log {n})$ .", "Table REF presents the reconstruction quality (PSNR) on different types of autoencoders that we explored.", "We trained 8 types of autoencoders on a split of snapshots of the CESM-CLDHGH data field: a vanilla autoencoder, vanilla variational autoencoder [53], $\\beta $ -VAE [54], DIP-VAE [55], Info-VAE [56], LogCosh-VAE [57], WAE [45], and SWAE [44].", "After training, the AEs were tested by using another split of data snapshots.", "From this table, we observe that SWAE has the best prediction accuracy with highest PSNR, which motivates us to use it as our final predictor in AE-SZ.", "Table: Average prediction PSNR of different types of autoencoders on CESM-CLDHGH data field" ], [ "Design Detail: Optimizing AE Configurations", "Takeaway 2: The performance of AE may differ a lot with different configurations under the same model structure.", "Optimizing the AE configurations, especially the input block size and the latent vector size, is critical to the final performance of AE-SZ.", "As presented in Section REF , the AE network in AE-SZ has a flexible structure, which can accept various configurations such as different input block sizes, latent vector sizes, block numbers, and channel numbers.", "From those, the input block size and latent vector size are two critical hyperparameters for the performance, corresponding to the scale of learned data patterns and the representation compactness of latent vectors.", "We need to optimize the input block size to have the best scope of data, and we need to optimize the latent vector size to balance prediction accuracy and latent overhead.", "Table REF shows the average prediction PSNR and AE-SZ compression ratio (error bound = 1E-2) of different input block sizes under the same latent ratio (input block size divides latent vector size, 64 for CESM-CLDHGH and 32 for NYX-baryon_density).", "We conclude that optimizing the input block size is of great importance because autoencoders can achieve apparently different performances under the same latent overhead but various input block sizes.", "In our work we optimize the input block size of the autoencoder in AE-SZ separately for each field, and we find that 32$\\times $ 32 input block fits most of the 2D data fields tested and that 8$\\times $ 8$\\times $ 8 input block fits most of the 3D data fields tested.", "Table: Average prediction PSNR and AE-SZ compression ratio (1e-2 error bound) of different input block sizesTable REF presents the final compression ratio of AE-SZ under the error bound of 1E-2 with AEs of different latent sizes on the Hurricane-U data field.", "The input block size is 8$\\times $ 8$\\times $ 8, and the rest part of the network remains the same for different latent sizes.", "We can see that different latent sizes bring a 40%+ difference in final compression ratios, which motivates us to choose an appropriate latent size in our design.", "In what follows, we discuss how we reduce the latent vector size while maintaining high prediction accuracy of AEs.", "Table: Compression ratio of AE-SZ under the error bound of 0.01 with AEs of different latent sizes on the Hurricane-U data field" ], [ "Design Detail: Lossy compression of AE latent vectors", "Takeaway 3: Predicting the data with error-bounded lossy decompressed latent vectors can maintain a very small loss of prediction accuracy, while greatly reducing the latent vector size (i.e., latent overhead).", "One main disadvantage of autoencoders is the overhead of storing latent vectors, which can be reduced but cannot be eliminated.", "To maximize the compression ratio with autoencoders, instead of using the original encoder output latent vectors for compression, AE-SZ compresses the latent vectors with a built-in customized compressor and then uses the decompressed latent vectors for decoding.", "In this approach, the compressed latents are to be stored.", "The computation of compressed latents and autoencoder predictions in AE-SZ is shown in Figure REF .", "For the original latent vector $z$ as the encoder network, a lossy compressor generates the compressed latent $z_c$ in reduced size and the decompressed $z_d$ (which can also be directly computed from $z_c$ ); then the decoder network computes the prediction with $z_d$ as its input.", "Figure: The autoencoder with customized latent compressorWe customize an efficient method for compressing AE-SZ latent vectors (called customized or custo.", "for short) with two steps: (1) quantize the original value using error bound of 0.1$e$ , where $e$ is the user-specified error bound (the error bounds are value-range based) for the dataset; and (2) use Huffman + Zstd to compress the quantization codes.", "The advantage of such a design is twofold.", "First, this can get better compression ratios than SZ2.1, as shown in Table REF .", "The key reason is that the latent vector data are not quite smooth across adjacent elements, based on our observation, while SZ2.1 strongly relies on the spatial smoothness.", "Second, the custo.", "design is consistent with an important constraint required in the AE-SZ: the compression of each data block must be independent of other data blocks, which is explained as follows.", "Note that we select the better prediction method between AE and Lorenzo based on their prediction accuracy for each block.", "After this step, all the blocks (either AE-predicted blocks or Lorenzo-predicted blocks) have corresponding predicted data, which can be applied with the quantization directly.", "Obviously, in order to minimize the latent overhead, we should not store the AE latents for the Lorenzo-predicted blocks, but that requires that the compression of latents be independent across data blocks.", "SZ2.1 has data dependency across blocks, which makes it unsuitable for the latent vector compression here.", "Table: Compression ratios of our customized compressor vs. SZ2.1 on latent vectors under different error bounds ϵ\\epsilon Through masses of experiments using different datasets, we note that choosing a reasonable error bound can achieve a relatively high compression ratio of latents with small loss of prediction accuracy.", "Figure REF presents two rate-distortion plots of the AE prediction values with different compression ratios of latent vectors.", "The prediction accuracy (w.r.t.", "PSNR) does not degrade at all when the latent vectors are compressed with a ratio such as 4 (corresponding to bit-rate 0.25 in the figure as the original latent size is $\\frac{1}{32}$ of the input size).", "That is, compressing latent vectors with a relatively high compression ratio (under a certain error bound) does not affect the compression of quantization bins much.", "Figure: Rate distortion of SWAE (without quantization)" ], [ "Design Detail: Combination of AE and Lorenzo", "Takeaway 4: The autoencoder model has a high ability to represent the data roughly with a high reduction ratio, but it is not as effective as Lorenzo in high-precision use cases.", "Therefore, a combination of AE and Lorenzo can effectively mitigate their own particular limitations in data prediction.", "Although AE has a great ability in learning the distribution of data, it still has two critical drawbacks that prevent it from being directly used as a data predictor especially for high-precision error-bounded compression use-cases.", "The first drawback is that, similar to linear regression, the latent vectors generated by AE for decompression sometimes bring cost due to redundancies of space.", "Specifically, we observe that quite a few data blocks may have constant or approximately constant values in scientific data.", "For these blocks, applying a simple and low-cost predictor is accurate enough, while being able to reduce the storage size as much as possible.", "Second, to maintain the learning effectiveness and efficiencies, the reconstructed data blocks from the autoencoder always suffer from certain noises, making it inadequate for extremely high-precision compression.", "By comparison, we note that the Lorenzo predictor outperforms the autoencoder especially when a relatively small error bound is used.", "Figure: ErrBound=1E-4We use Figure REF to illustrate the pros and cons of the autoencoder and Lorenzo predictor under different error bounds.", "This figure demonstrates the prediction error distributions of Lorenzo predictor, linear regression predictor, and our trained autoencoder under an error bound of 1E-2 and 1E-4, respectively (the input data is a snapshot of CESM-FREQSH data field).", "One can clearly observe that under the large error bound 1E-2, the autoencoder has a better (sharper) prediction error distribution.", "In contrast, the prediction accuracy of the Lorenzo predictor grows rapidly as the error bound decreases to a small value 1E-4.", "During the online compression, AE-SZ selects a predictor between autoencoder and Lorenzo for each data block.", "The selection criterion is checking which predictor has lower prediction errors (i.e., loss) for the given block.", "The details can be found in Algorithm REF (see line 6$\\sim $ 13)." ], [ "Performance Evaluation", "In this section we present the experimental setup and then discuss the results." ], [ "Experiment Environment", "We perform the experiments on the gpu_v100_smx2 nodes of the Argonne National Laboratory Joint Laboratory for System Evaluation computation cluster.", "Each node is driven by two Intel Xeon GOLD 6152 processors with 188 GB of DRAM and NVIDIA TESLA V100 GPUs." ], [ "Data Used in Experiments", "We perform the evaluation using five real-world application datasets in different domains that are commonly used in testing lossy compressors.", "Most of the datasets such as CESM, NYX, Hurricane can be downloaded from SDRBench [58].", "CESM [59]: A well-known climate simulation package.", "We use its atmosphere model [58] in our experiments.", "These datasets are 2D, although some fields exhibit three dimensions in their metadata.", "For the CLOUD field (26$\\times $ 1800$\\times $ 3600), for instance, SZ2.1 has a better compression ratio (31.1 vs. 22.6) if we compress it with the range-based error bound 1E-3 in 2D mode instead of 3D mode.", "RTM: Reverse time migration (RTM) code for seismic imaging in areas with complex geological structures [60].", "NYX [61]: An adaptive mesh, hydrodynamics code designed to model astrophysical reacting flows on HPC systems.", "Two separate simulations are performed for generation of training and test data.", "Hurricane [62]: A simulation of a hurricane from the National Center for Atmospheric Research in the United States.", "EXAFEL [63]: An Exascale Computing Project for analyzing molecular structure X-ray diffraction data generated by the LCLS [1].", "The data contains groups of 32 2D arrays of size 185$\\times $ 388.", "We discard the groups with nearly uniform data points; and following [64], we concatenate the 2D arrays in each group to form a single 5920$\\times $ 388 2D array for each group.", "More detailed information of the datasets (all in single precision ) is shown in Table REF .", "The fields of NYX are transformed to their logarithmic value before compression for better visualization, as suggested by domain scientists.", "Table: Basic information about application datasets" ], [ "Comparison Lossy Compressors in Our Evaluation", "In our experiment, we compare AE-SZ with six other lossy compressors.", "The first four are classic error-bounded compressors: SZ2.1 [8], [9], [15] and ZFP0.5.5 [13], which have been widely used in the community, and two recent works based on the SZ framework and developed from SZ2.1: SZauto [14] and SZinterp [31].", "The fifth one is a recent work of an autoencoder-based scientific data compressor [43], called AE-A in our evaluation.", "The sixth one is a pure convolutional autoencoder model [40], called AE-B, proposed for compressing turbulence data, which is not error bounded." ], [ "Experimental Configurations", "For SZ2.1, ZFP0.5.5, SZauto, and SZinterp we adopt value-range-based error bounds and use default configurations for other parameters.", "For the training phase of the autoencoders in AE-SZ, we train different autoencoders for different data fields on selected parts of the data, then test and compare all compressors on the remaining parts.", "Table REF shows the input block size, length of latent vectors, number of the convolutional blocks in encoder network, and number of channels in convolutional blocks of encoder network.", "The number of deconvolutional blocks is the same as the encoder's, and the channel numbers of the decoder network are symmetric with those in the encoder network.", "Table REF shows the training-test split for all datasets.", "All autoencoders in AE-SZ are trained for 100 epochs.", "Table: Autoencoder Configurations for Each data FieldTable: Train-test split for each datasetFor AE-A [43], we download their from https://github.com/tobivcu/autoencoder, which supports only double-precision floating data originally.", "We improve the code by enabling it to compress single-precision floating data, in that most of datasets in our test are stored in single-precision.", "We trained its model using the same training data split for 100 epochs.", "The .dvalue files generated by the model are compressed by SZ2.1 following the instruction of [43].", "After fine-tuning, we applied the same value-range-based relative error bound to compress the .dvalue file.", "For AE-B, since Glaws et al.", "[40] does not provide enough details for training from scratch, following the paper's recommendation, we fine-tuned a pretrained autoencoder (from https://github.com/NREL/AEflow indicated by [40]) on different data fields for 5 epochs each." ], [ "Evaluation Metrics", "We evaluate the seven lossy compressors based on the critical metrics described below.", "Rate distortion: Rate distortion is the most commonly used metric by the lossy compression community to assess compression quality.", "Rate distortion involves and plots with two critical metrics: peak signal-to-noise ratio (PSNR) and bit rate.", "The definition of PSNR is introduced in section , and bit rate is defined as the average number of bits used per data point in the compressed data.", "Generally speaking, Bit rate equals $Sizeof(datatype)/cr$ , in which $Sizeof(datatype)$ is the byte size of input data (32 for single-precision data for example), and $cr$ is the compression ratio.", "Therefore, smaller bit rate means better compression ratio, and vice versa.", "Visualization with the same compression ratio (CR): Compare the visual quality of the reconstructed data based on the same CR.", "Compression speed and decompression speed: $\\frac{original \\hspace{1.42262pt} size}{compression\\hspace{1.42262pt} time}$ (MB/s) and $\\frac{reconstructed\\hspace{1.42262pt} size}{decompression\\hspace{1.42262pt} time}$ (MB/s).", "In the following experimental results, when it comes to error bound values, without loss of generality, we adopt value-range-based error bounds (denoted as $\\epsilon $ ), which takes the same effect with absolute error bound (denoted $e$ ) because $e=\\epsilon \\cdot (\\max (D)-\\min (D))$ ." ], [ "Rate distortions of different lossy compressors", "We present the rate distortion results of all seven lossy compressors on all tested data fields, illustrating the PSNR of final decompression results with bit rates.", "Figure REF shows the rate distortion plots for each lossy compressor on eight data fields.", "Only four compressors are shown in Figure REF (a), (b), and (c) because the other three compressors (SZauto, SZinterp, and AE-B) support only 3D data, while CESM and EXAFEL are both 2D datasets.", "We observe that AE-SZ is significantly better than the other two AE-based lossy compressors (AE-A and AE-B) in term of rate distortion.", "That is, our developed AE-SZ compression method is arguably the best AE-based lossy compressor to date.", "We also compare the most competitive error-bounded lossy compressors (to the best of our knowledge): SZinterp [31], SZauto [14], SZ2.1 [15], and ZFP [13].", "Generally speaking, AE-SZ obtains much better rate distortions than SZauto, SZ2.1 and ZFP do under low bit rates (i.e., in high-compression-ratio cases) and have a comparable quality with SZ2.1 for high bit rates.", "We observe that AE-SZ generally has 100%$\\sim $ 800% higher compression ratios than SZ2.1 has in the high-compression-ratio cases on both 2D and 3D datsets.", "In the 2D datasets, for example, AE-SZ exhibits the best rate distortion (240% higher compression ratio than the second best at the same PSNR around 44) for the CESM-FREQSH data field.", "On the EXAFEL dataset, AE-SZ has a 200% higher compression ratio than the second best (SZ2.1) in the high-compression cases.", "On the 3D datasets, AE-SZ also exhibits very competitive rate distortions from among all the seven compressors.", "Its compression quality is close to that of SZinterp in the low-bit-rate range (e.g., [0,1]) and may also exhibit the best rate distortion in a few cases (e.g., Figure REF (e)).", "Figure: RTM (3D)" ], [ "Decompression data visualizations of different lossy compressors", "We present data visualizations in Figure REF on the NYX-baryon_density field to verify the effectiveness of the reconstructed data of AE-SZ at high compression ratio use cases.", "We clearly observe that the reconstructed data at the PSNR of 46.8 under AE-SZ has a very good visual quality.", "Other prior works [15], [65] show that PSNR in the range of [30,60] is good enough to have a high visual quality for different scientific applications.", "Moreover, Figure REF demonstrates that with the same compression ratio of 180, AE-SZ has a much better visual quality compared with that of the three state-of-the-art lossy compressors, SZauto, SZ2.1, and ZFP0.5, and is also better than SZinterp.", "Figure: ZFP (PSNR:30.2,CR:161)" ], [ "Performances of AE-SZ predictors under different error bounds", "To better understand how the autoencoder and Lorenzo predictor in AE-SZ cooperatively contribute to the compression ratios, we record the percentage of data blocks predicted by AE-SZ autoencoders on three different data fields, as shown in Figure REF .", "For better vision of the plots, the x-axis is logged error bounds.", "The plots show that autoencoders in AE-SZ achieve advantages over Lorenzo under a range of medium error bounds (about 5E-3 to 2E-2), under which most of the data blocked can be better predicted by autoencoders.", "As the error bound decreases, the Lorenzo predictor becomes better than autoencoders on more data blocks.", "When the error bound becomes very high, the latents need to be compressed with a high error bound, so the prediction error of autoencoders may drop rapidly, and Lorenzo may also turn better.", "To understand the effectiveness of our adaptive prediction design, we present the rate distortion in three situations: predicting data with only AE, predicting data with only Lorenzo, and combining both, as shown in Figure REF .", "The figure shows that AE+Lorenzo achieves the best quality at all bit rates since it can take advantage of both predictors adaptively.", "Figure: Percentage of blocks predicted by AE under different error boundsFigure: Hurricane (U)" ], [ "Compression speeds and autoencoder training speeds", "The average compression speed of each error-bounded lossy compressor on all the tested datasets under the error bound of 1E-3 are shown in Table REF in units of Mb/s (SZauto, SZinterp, and AE-B have speeds only on 3D data because they currently do not support 2D data).", "Because of the relatively high computation cost of neural networks, AE-SZ cannot achieve comparable compression throughput with traditional lossy compressors (its speed is about 10%-40% as fast as that of SZ2.1 and SZinterp).", "In fact, the current version of AE-SZ code is in the experimental stage, so it is not as optimized as the off-the-shelf compressors such as SZ2.1 and ZFP.", "We believe that with further optimization AE-SZ can be much accelerated.", "In fact, the throughput of AE-SZ has been significantly better than the other autoencoder-based compressors such as AE-A [43] by 30$\\times $ to 200$\\times $ (mainly due to the complicated data preprocessing and postprocessing procedures in AE-A) and AE-B [40] by up to 4$\\times $ speedup in compression and 9$\\times $ speedup in decompression (note that AE-B is not error bounded so its running speeds include only the AE prediction process).", "Table: Compression/decompression speeds (MB/s): error bound=1E-3Second, table REF shows the training time of autoencoders in AE-SZ and AE-A [43] using the same training data and the same number of training epochs.", "We can conclude that the autoencoders in AE-SZ outperforms AE-A [43] with similar or shorter training time.", "For AE-B, the tested networks are only fine-tuned so we are unable to present its training time.", "Table: Autoencoder Training Time (in hours)" ], [ "Conclusion and Future Work", "In this paper we explored leveraging convolutional autoencoders to improve error-bounded lossy compression.", "To this end, we developed an efficient method called AE-SZ, by integrating autoencoders in the SZ compression model with a series of optimizations.", "We comprehensively evaluated AE-SZ by comparing it with six related works on five real-world simulation datasets, with the following key findings.", "AE-SZ is competitive in the low-bit-rate range (i.e., high-compression-ratio cases).", "Specifically, it exhibits the best rate distortion results in 2D datasets.", "On 3D datasets, it obtains a much better rate distortion than SZ2.1 and ZFP do (about 100%$\\sim $ 800% improvement with the same PSNR).", "AE-SZ also exhibits very close rate distortions with those of SZinterp in high compression cases, demonstrating its great potential in error-bounded lossy compression.", "AE-SZ has a higher visual quality at the same compression ratio compared with SZauto, SZ2.1, and ZFP.", "AE-SZ is slower than SZ2.1 and ZFP, but is 30$\\times $$\\sim $ 200$\\times $ faster than other autoencoder-based error-bounded lossy compressors.", "In the future, we plan to improve AE-SZ in several ways, including (1) optimizing the network structure and the hyperparameters of autoencoders in AE-SZ and (2) speeding up the compression and decompression speeds for AE-SZ." ], [ "Acknowledgments", "This research was supported by the Exascale Computing Project (ECP), Project Number: 17-SC-20-SC, a collaborative effort of two DOE organizations – the Office of Science and the National Nuclear Security Administration, responsible for the planning and preparation of a capable exascale ecosystem, including software, applications, hardware, advanced system engineering and early testbed platforms, to support the nation’s exascale computing imperative.", "The material was supported by the U.S. Department of Energy, Office of Science, under contract DE-AC02-06CH11357, and supported by the National Science Foundation under Grant OAC-2003709 and OAC-2003624/2042084.", "We acknowledge the computing resources provided on Bebop (operated by Laboratory Computing Resource Center at Argonne) and on Theta and JLSE (operated by Argonne Leadership Computing Facility)." ] ]
2105.11730
[ [ "Towards open-ended evolutionary simulator for developing novel tumour\n drug delivery systems" ], [ "Abstract Tumours behave as moving targets that can evade chemotherapeutic treatments by rapidly acquiring resistance via various mechanisms.", "In Balaz et al.", "(2021, Biosystems; 199:104290) we initiated the development of the agent-based open-ended evolutionary simulator of novel drug delivery systems (DDS).", "It is an agent-based simulator where evolvable agents can change their perception of the environment and thus adapt to tumour mutations.", "Here we mapped the parameters of evolvable agent properties to the realistic biochemical boundaries and test their efficacy by simulating their behaviour at the cell scale using the stochastic simulator, STEPS.", "We show that the shape of the parameter space evolved in our simulator is comparable to those obtained by the rational design." ], [ "Introduction", "Precision oncology is a novel approach in tumour treatments that uses patient-specific information about the tumour to define the best possible targeted treatment [1].", "Even when the right drug is identified for the right tumour profile, the problem remains of how to develop an appropriate drug delivery system that will pass all physiological barriers and reach the tumour in a high enough dose to be efficacious [9].", "The sources of difficulty are numerous [8], [4].", "Here, we will focus only on the problem of optimal parameter identification.", "Our question is how to identify an optimal set of parameters, such as binding affinity, uptake, and diffusion rate in order to reach desired treatment efficacy, having in mind that tumour can rapidly develop drug resistance.", "With the recent increase in clinical research on developing targeted therapies with multiple drugs to deal with the adaptable tumour, two modelling problems came into focus.", "The first one is how to design optimal treatment that includes several drugs [10], [11].", "The second, more general problem, is how to design treatment for the adaptable tumour even when future phenotypes are unknown.", "Recently we have developed a simple agent-based, open-ended evolutionary engine to automatically design combinatorial oncological treatments [2].", "There, we demonstrated that open-ended evolution coupled with multidimensional optimization can lead to the emergence of successful virtual therapies.", "Here we introduce several improvements described below." ], [ "Model description and simulation results after mapping parameters to realistic values", "Details of the original model and the range of all parameters are given in [2] and here we only give a brief overview.", "In the model we previously introduced the simulated world is represented as a 2D grid where evolvable nano-agents (NA) learn to eliminate tumour cells (CC) while not harming healthy cells (HC).", "NA can move, have internal memory, and can learn about their environment.", "They can only “see” the contents of the grid position they occupy.", "When the NA “steps-on” a cell, it will check whether it already has it in its memory (by looking at the visible properties of the cells), and memorize it if not.", "If the memory is full - the new cell information will take place of the oldest memorized cell.", "At the beginning of the simulation all agent memories are empty.", "Interaction of NA agents with cell-agents is modelled as a Michaelis-Menten reaction network: $\\mathit {NA}_f=R \\overset{p_a, p_d}{\\longleftrightarrow } C \\overset{p_i}{\\longrightarrow } \\mathit {NA}_i + R, $ where $\\mathit {NA}_f$ is a free NA, $R$ is a receptor at the cell surface, $C$ is a NA-cell complex and $\\mathit {NA}_i$ is an internalized NA.", "Probabilities of association ($p_a$ ), disassociation ($p_d$ ) and internalization ($p_i$ ) govern the dynamics of the NA-cell interaction.", "If NA encounters a “familiar cell”, the probability of association with that agent is $p_a$ .", "Otherwise, the probability of association is reduced by multiplying $p_a$ with the “curiosity” factor.", "The local fitness value is assigned to each NA agent as a sum of killed CC minus the sum of killed HC.", "After each 10 time steps selection/mutation routine takes place.", "Mutable parameters are movement speed, $p_a$ , $p_d$ , $p_i$ and probability of killing ($p_k$ ).", "CCs can evade NA by mutating their visible properties by which NA recognize them.", "To mimic tumour resistance, 10$\\%$ of the total number of tumour cells were chosen randomly and assigned randomly-chosen resistance modifiers (one per cell).", "Resistance modifiers can change $p_a$ , $p_d$ , $p_i$ , and $p_k$ of NA that interact with them.", "Resistance strength is randomly chosen in the interval of $30-80\\%$ and is inherited after cell division.", "The novel feature we introduce here is the split of the model into two modes: the learning mode and the simulation mode.", "In the learning mode, the goal is to find the best collective strategy to fight the tumour, regardless of the duration of the treatment.", "After interacting with a cell, NA agents do not perish but continue with movement.", "We ran the learning mode for $10,000$ time steps and the efficacy of top performing NAs is evaluated in the “simulation mode” and in the STEPS simulator.", "In the simulation mode we include the realistic temporal dimension based on the following assumptions: (i) one grid cell equals one cell agent, (ii) average tumour cell diameter is 10 µm, (iii) the diffusion coefficient range for nanoparticles in a fluid is in the range of $10^{-10} cm^{2}\\slash s$ [6], [7].", "Given that we are in 2D space, NA then needs $t = (10^{-3})^{2} \\div 2 \\times 10^{-10} = 0.5 \\times 10^{4}$ seconds to move from one empty grid cell to a neighbouring cell.", "As we fixed NA speed in an empty cell to 1, one time-step in the simulation mode equals 5000 seconds.", "To calculate injection dynamics of NAs in simulation mode, we took pharmacokinetics values for a typical anticancer drug Doxorubicin [3].", "There, a drug reaches maximum concentration in the tumour in 20 hours ($\\sim $ 14 simulated time steps) after the injection.", "For the following 80 hours ($\\sim $ 72 time steps) a drug concentration slowly declines until it reaches zero.", "Therefore, during the first 14 simulation steps, at each step we inject 1/14 of the predefined total dose.", "As expected, most efficacious option is when the maximum tolerated dose of NA is injected.", "This eliminated $7\\%$ of tumour cells after the single dose treatment (data not shown).", "To test how the spatial distribution of cells affects nanoparticle transport, we used the Stochastic Engine for Pathway Simulation (STEPS) [12] where we incorporates spatiality through the discretisation of the domain into tetrahedral well-mixed subregions.", "To normalize evolved NA parameters to realistic values we take the range of association rate constant ($ka$ ) to be between $10^4$ and $10^6$ [1/Ms] [5].", "Since the time step is $5,000$ seconds, $ka$ is $5x10^7$ to $5x10^9$ [1/M time step].", "If we assume that one NA represents $10^5$ particles ($1.66x10^-7$ [M]), then the $ka$ range is $8.3$ to $8.3x10^2$ [particles / time step].", "The normalization factor is then $1.66x10^-7$ [M] / $5,000$ [$s$ ] = $1.2x10^3$ .", "Therefore, for example $p_a = 0.3$ translates to ka=$3.61x10^2$ [1/Ms].", "Using the same approach we calculated the normalization factor for disassociation constant $kd$ and internalization constant ($ki$ ) to be $2x10^{-4}$ .", "In the simulations, maximum penetration depth is 6 cells Fig.REF and NA were able to kill all cells into which they were internalized." ], [ "Conclusions", "At this stage it is hard to directly compare our results to clinically relevant ones due to the differences in tumour size scaling, drug-action mechanisms and influence of tumour heterogeneity on drug diffusion.", "Nevertheless, it is important to note that the distribution of NA parameters we obtained as a result of in silico evolution is similar to those used to synthesize nanoparticle-based DDS (which is result of highly skilled rational design) as well as to parameters obtained via deterministic modelling [5].", "Therefore we believe that our evolutionary approach can generate useful leads in designing novel drug-delivery systems, especially since our engine is designed to deal with tumour changeability which is one of the main problems in long term efficacy of oncology treatments.", "Therefore, our ongoing work is focused on further closing the gap between our evolvable simulator and pre-clinical investigations, mostly by developing closer mapping between the parameter space of the simulator and the corresponding biochemical counterparts." ], [ "Acknowledgements", "This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No 800983." ] ]
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[ [ "Empirical Investigation of Factors that Influence Human Presence and\n Agency in Telepresence Robot" ], [ "Abstract Nowadays, a community starts to find the need for human presence in an alternative way, there has been tremendous research and development in advancing telepresence robots.", "People tend to feel closer and more comfortable with telepresence robots as many senses a human presence in robots.", "In general, many people feel the sense of agency from the face of a robot, but some telepresence robots without arm and body motions tend to give a sense of human presence.", "It is important to identify and configure how the telepresence robots affect a sense of presence and agency to people by including human face and slight face and arm motions.", "Therefore, we carried out extensive research via web-based experiment to determine the prototype that can result in soothing human interaction with the robot.", "The experiments featured videos of a telepresence robot n = 128, 2 x 2 between-participant study robot face factor: video-conference, robot-like face; arm motion factor: moving vs. static) to investigate the factors significantly affecting human presence and agency with the robot.", "We used two telepresence robots: an affordable robot platform and a modified version for human interaction enhancements.", "The findings suggest that participants feel agency that is closer to human-likeness when the robot's face was replaced with a human's face and without a motion.", "The robot's motion invokes a feeling of human presence whether the face is human or robot-like." ], [ "INTRODUCTION", "There is a wide variety of means of digital communication.", "For example, the telephone, e-mail, SNS, and video conferencing are representative and the most common.", "In recent years, communication via telepresence robots has been attracting attention.", "Telepresence robots called “mobile robotic presence systems,” that is physical robotic platforms with a video-conferencing system mounted on a robotic mobile platform .Since remote operators can control this physical embodiment which is telepresence robot in a remote location, these systems chance to increase for video-conference and audio-communication and imbue communicators with a strong sense of presence .", "Other researchers have mentioned that telepresence robots provide a new communication platform in various situations.", "For example, business, education, and Medical related .", "Furthermore, recently, these robots have become a new platform for people with disabilities to use so that they can join in society and perform labor .", "In the field of telepresence robots, it is said that the visibility of a face is a factor that makes people feel the presence of a person .", "Previous research on movement with telepresence robots has shown that robots that can express themselves socially through movement are more immersive and desirable than those that do not move .", "In some cases, a robot that does not have a human face can make people feel the presence of a person through its movements .", "For telepresence robots, the effect the face has depends on whether the face is human-like or robot-like, and people may feel that one type gives a greater sense of agency or presence.", "Also, which makes people feel more agency or presence in a robot, the case with or without arm motion which is moving or static?", "In this paper, we conducted a web-based experiment featuring video of a telepresence robot ($n$ = 128, $2\\times 2$ between-participant study, robot face factor: video-conference, robot-like face; arm motion: moving, static) to investigate which factors significantly affect human presence and agency.", "One of the most important factors in human-robot interaction (HRI) is the ability to show facial cues.", "In literature, McGinn conducted various studies on service robots by changing their heads and facial cues to determine the effect on social interface between human and robots.", "To facilitate better communication between human and robot, it is common for robots to possess head-like features that is capable of providing social feedback.", "Although it is not a human’s face and expression, research has shown that relation between human and the robot can be enhanced when the robot is equipped with human-like “robotic” face that can express and show motion like humans .", "In addition, interaction with robot gets even better when the robot exhibits social behaviors with anthropomorphic characteristics .", "Effect of human’s facial cue on robots were discussed in Mollahosseini et al.", ".", "The study indicated that eye gaze and certain facial expressions from the actual human can further improve the relation between human and robots by physically displaying human’s face on the 2D screen using telecommunication either telepresence or a virtual agents .", "For example, Beam is one of the known telepresence robotic system that utilizes this method by replacing robot’s head with the LED screen to display human’s face via video-conference.", "Several research and surveys indicated that people can really feel the presence of human in the robot ." ], [ "Non-verbal Cues in HRI", "Previous research indicated that, non-verbal cues are becoming important factors in HRI technology as it plays a significant role in human-human interaction.", "Telepresence robot that has an ability to show social expression by the motion movement can make user to feel engagement and likeable .", "OriHime is one of the telepresence and avatar robot with human-like behavior that was designed by Ory Laboratory Inc.", "The purpose of this technology is to aid people to engage in life-like social behavior over distance, facilitate human-to-human social interaction, and serve as an avatar to the user .", "Although OriHime does not have an expressive facial cue, but its limb movement and the voice from the user relaxes the social-relation between human and the robot and still feel the presence of human in them .", "In addition, having social expression embodiment like Orhime-D can join society and social events for disabled people and fulfill their mentality .", "Spatial configuration and body orientation of a telepresence robot affected people enable to arrange themselves, robot tend to copy human-like action and theyt detected by the surrounded motion .", "This greatly increases interaction quality between human and robots.", "In the research using teleoperated robot with the Wizard of OZ method, participants was not affected enjoyment by the knowledge of whether the robot was being controlled by a program or a human .", "Yamada et al.", "proposed motion-based ASE(Artificial Subtle Expressions) in which a robot slowly hesitates by turning to a human before giving advice with low confidence.", "As they conclude, the long or short-wait expressions might be applicable to expressing a robot's confidence, fast or slow-motion as a motion-ASE is more suitable for such expressions .", "The other research suggest that synchronized on-screen and in-space gestures significantly improved viewers(participants)’ interpretation of the action compared to on-screen or in-space gestures alone and addition of proxy motion also improved measures of perceived collaboration .", "Furthermore, in-space gestures positively influenced perceptions of both local and remote participants ." ], [ "Platform", "We used a humanoid robot, Rapiro , which is widely used for different applications, such as for education and hobbies.", "The Arduino and Raspberry Pi boards in the robot enable users (developers) to communicate with the robot by only sending command signals from a PC, and they also allow for the system to be extended easily.", "Therefore, we used this robot as a telepresence robot for our experiment.", "For the experiment, we fixed Rapiro's eye color to blue due to color bias.", "We modified Rapiro's head as a prototype for experimental study and to show the remote user's face." ], [ "Hardware Spec", "Rapiro http://www.rapiro.com/ has 12 degrees of freedom (DoF), a USB camera, a microphone in its forehead, and a speaker inside of its head.", "Fig.", "REF shows an overview of Rapiro.", "Modifications to Rapiro, we modified the head of another Rapiro to show the face of a remote user.", "We used a 5-inch portable monitor, and the head was made of PLA using a 3D printer.", "This Rapiro also had 12 DoF, a USB camera, microphone, and speaker inside of its head.", "Fig.", "REF shows an overview of the modified Rapiro." ], [ "User Interface and Robot Motion", "To control Rapiro, we made a keyboard input interface.", "When the operator (from a geographically separate location) presses the number “2\" on the keyboard, a local PC receives the signal from the operator's location via Wi-Fi, and the robot makes that specific motion.", "We generated robot motions in accordance with the following principle.", "For both of the robots, we used preset motions and the original motion which we developed for the experiments, such as “hands up\" and “wave the both hands\"and etc.", "In total, we used six motions in the video task and motions list showed in Table REF .", "The preset motions included motion like going forward or backward, but we did not use these at this time.", "Table: List of Motions" ], [ "Experimental Design", "We conducted the experiment using a $2\\times 2$ (robot face: video-conference vs. robot-like face; arm motion: moving vs. static) between-participant design.", "To explore how people interact widely with our design, we used G*Power sample size calculation (n =128), and we ran our experiment using online questionnaire surveys along with showing the video.", "Participants were recruited from the Yahoo!", "Crowdsourcing service and after finished experiments we used Gooogle form for survey.", "For most of the methods for the online experiment, we referred to Sirkin et al.", ".", "We wanted the remote operator's speech and gestures to have precise timings and the same interaction content.", "While online responses may differ from in-person experiments, from Powers et al.", ", it was found that remote robots could be used in experiments and be more sociable and engaging than co-located robots  .", "Furthermore, studies comparing live and video-based HRI trials were both broadly equivalent in most cases .", "Therefore, we chose to run the experiment online.", "Our dependent values were presence and agency.", "We designed the online experiment so that we could compare which condition affect from our dependent values and perceptions of the remote operator who communicated with them via the telepresence robot across four robot conditions: Human face and moving: The face of the robot used a video-conference style screen from which the remote operator was shown, and the robot's arm made motions.", "An overview of this robot is shown in Fig.", "REF .", "Human face and Static: The face of the robot used a video-conference style screen from which the remote operator was shown, and the robot's arm was static.", "An overview is shown in Fig.", "REF .", "Robot face and moving: The face of the robot was robot-like, and the robot's arm made motions.", "An overview is shown in Fig.", "REF .", "Robot face and Static: The face of the robot was robot-like, and the robot's arm was static.", "An overview is shown in Fig.", "REF .", "Figure: Robot face and Static." ], [ "Hypotheses", "We formulated four hypotheses for our experiment.", "As mentioned above, we conducted the experiment using a between-participant design (robot face: video-conference vs. robot-like face; arm motion: moving vs. static) to investigate which factors significantly affect presence and agency.", "H1 Face affects agency.", "H2 Motion affects agency.", "H3 Face affects presence.", "H4 Motion affects presence." ], [ "Participants", "A total of 216 participants took part in the experiment online (male: 147, female: 69).", "Their ages ranged from 18 to 63 (M = 44.2, SD = 10.6).", "We recruited the participants from Yahoo!", "Crowdsourcing, which is a service provided by Yahoo!", "Japan." ], [ "Task", "The participants watched one video from among the four different conditions as shown in Fig.1,2,3,4.", "We created videos in which a remote operator communicated via a telepresence robot and discussed moon survival and item ranking.", "We created this Moon Survival scenario from the Desert Survival Problem and also a NASA exercise since the Desert Survival Problem is used by many social scientists and robotics researchers .", "The video was about an astronaut who had crash-landed on the moon and was discussing how to select 5 items that he needed from the 15 items left to return to his distant home planet.", "Due to the video length, we only discussed ranking up to five items because if we had gone up to 15 items, the video length would have been too long.", "The ranking of the five items is shown in the Table REF .", "Table: Rank of Items" ], [ "Procedure", "The procedure is shown in Fig.", "REF .", "The participants viewed the instructions and watched one video from the four conditions.", "In the instructions, we stated that, “In the experiment, you will watch a video of a human talking to a robot controlled by a human via remote control\" and “Watch as if you were talking to the robot.\"", "Afterward, we also told them that the task of the video was to have a discussion on moon survival.", "When participants finished watching the video, they were asked to rate their agreement on a seven-point Likert scale, 1 = strongly disagree, 7 = strongly agree, in two questionnaire surveys (in total, 30 statements).", "When they finished the experiment, there was an additional comment or question space.", "We paid 100 yen (about $1.00US), and the average time to complete the study was about 15 to 30 minutes.", "Figure: Flowchart of experiment" ], [ "Questionnaire Survey", "We used two different questionnaires, the Godspeed series and one for social presence.", "Godspeed is a standardized measurement tool for HRI .", "There are five key concepts for the measurement used: anthropomorphism, animacy, likeability, perceived intelligence, and perceived safety.", "For the second questionnaire, we used Networked Minds Measure of Social Presence, which is measure of presence .", "We modified this questionnaire due to some of the statements not fitting into our experiment.", "A few of the questions from the first and second questionnaires are listed in Tables 1 and 2.", "Table: GODSPEED's QuestionnaireTable: Networked Minds Measure of Social Presence's Questionnaire" ], [ "result", "To test our hypotheses, we used a two way analysis of variance (ANOVA).", "For the G*Power calculation , the sampling size was 128.", "For each condition, we used 32 participants for analysis.", "Before participants watched a video, they were asked to rate their agreement on a seven-point Likert scale, 1 = not very familiar, 7 = very familiar, for two statements (“Are you familiar with video conferences?” and “Are you familiar with robots?”).", "Most participants were not familiar with robots (mean = 2.46, SD = 1.57 ), but they were familiar with video conferences (mean = 3.39, SD = 1.84).", "We used anthropomorphism from the GODSPEED questionnaire series to measure agency .", "To measure presence, we used Networked Minds Measure of Social Presence's questionnaire ." ], [ "Meausrement", "The results of the ANOVA showed in Table REF and Table REF , also result of simple mian effect for agency showed in Table REF .", "As well as the means and standard deviations(S.D.)", "for all of the dependent variables can be seen in Figs.", "REF and REF as well as in Table REF and Table REF .", "Also, conditions explain again in Table REF .", "For agency, we found that the interaction was significant(p <0.01).", "In the Static group, the simple main effect of face was significant(p <0.0001).", "It was higher in the group with face.", "For presence, there was a significant interaction between the two factors.", "We found that the main effect was significant only for the motion factor(p <0.01).", "Furthermore, motion with moving was the highest.", "Figure: Averages for score for motion perceived for each condition in experiment.", "Anthropomorphism as dependent value.Figure: Averages for score for motion perceived for each condition inexperiment.", "Presence as dependent value.Table: Conditions in the experimentsTable: Agency of S.D.", "and MeanTable: Presence of S.D.", "and MeanTable: CONCLUSIONS" ] ]
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